mapping groundwater quality in the netherlands · mapping groundw ater quality in the netherlands...

MAPPINGGROUNDWATERQUALITY IN THE NETHERLANDS

CIP-GEGEVENS,KONINKLIJKE BIBLIOTHEEK,DEN HAAG

Pebesma,EdzerJan

MappingGroundwaterQuality in theNetherlands/ EdzerJanPebesma.- Utrecht:FaculteitRuimtelijkeWetenschappenUniversiteitUtrecht.- Ill., tab.ProefschriftUniversiteitUtrecht.- Met lit. opg.- Metsamenvattingin hetNederlands.ISBN 90-6266-127-0Trefw.:grondwaterkwaliteit ; kartering; Nederland

ISBN 90-6266-127-0(Thesis)ISBN 90-6809-216-2(NGS)

Copyright FaculteitRuimtelijkeWetenschappenUniversiteitUtrecht 1996

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Omslagontwerp:Mark Oonk

MAPPING GROUNDWATER QUALITY IN THE NETHERLANDS

HET KARTEREN VAN DE GRONDWATERKWALITEIT IN NEDERLAND(meteensamenvattingin hetNederlands)

PROEFSCHRIFT

TER VERKRIJGING VAN DE GRAAD VAN DOCTOR AAN DE

UNIVERSITEIT UTRECHT OP GEZAG VAN DE RECTOR MAGNIFICUS,PROF.DR. J.A. VAN GINKEL, INGEVOLGE HET BESLUIT VAN HET

COLLEGE VAN DECANEN IN HET OPENBAAR TE VERDEDIGEN OP

1MAART 1996DESMIDDAGSTE 14:30UUR

DOOR

EDZER JAN PEBESMA

GEBORENOP 3 JULI 1967,TE BEETSTERZWAAG

promotor: Prof.dr. P.A. Burrough FaculteitRuimtelijkeWetenschappenUniversiteitUtrecht

co-promotor: Dr.ir. J.W. deKwaadsteniet RijksinstituutvoorVolksgezondheidenMilieu

Dit proefschrift werd mogelijk gemaaktmet financiële steun van het Rijksinstituut voorVolksgezondheidenMilieu (RIVM)

foar AnneenAndrys

Chapter 1 General intr oduction 1

1.1 Problemdefinition 1

1.2 Groundwaterquality—whatis it? 2

1.3 Monitoringgroundwaterquality 2

1.4 Modellinggroundwaterquality 4

1.5 Groundwaterqualitymonitoringnetwork optimization 4

1.6 Objectivesandapproach 5

1.7 Outlineof thisthesis 5

Chapter 2 Statistical Mapping 7

2.1 Introduction 7

2.2 Uncertainty, accuracy andstatistics 9

2.3 Linearmodels 11

2.4 Changeof supportanderrorstructure 15

2.5 Geostatistics 16

2.6 Confidenceinterval maps 19

2.7 Mappinggroundwaterquality 20

Chapter 3 Mapsof thegroundwater quality in theNetherlandsat 5-17metredepthin 1991 22

3.1 Introduction 22

3.2 An aggregatedmapcomprisingsoil typeandlanduse 23

3.3 Dataselection 23

3.4 Spatialinterpolation 25

3.5 Thegroundwaterqualitymaps 29

3.6 Theeffectsof stratification 32

3.7 Theeffectsof monitoringnetwork density 32

3.8 Discussion 41

Chapter 4 Maps of temporal changes in the groundwater quality in theNetherlandsat 5-17metredepth 44

4.1 Introduction 44

4.2 An aggregatedmapcomprisingsoil typeandlanduse 45

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4.3 Dataselection 45

4.4 Changesin groundwaterqualityvariablesat thewell screens 46

4.5 Spatialinterpolation 48

4.6 Results 51

4.7 Discussion 54

Chapter 5 Impr oving estimateswith ancillary information 57

5.1 Introduction 57

5.2 Ancillary information 58

5.3 Spatialinterpolationwith ancillaryinformation 59

5.4 ExampleA. Nitrateleachingfrom agriculturalsoils 61

5.5 ExampleB. Atmosphericdepositionof zinc 66

5.6 ExampleC.Measurementsof sulphatein shallow groundwater 71

5.7 Theeffectsof ancillaryinformation 74

5.8 Discussion 74

Chapter 6 Discussionand conclusions 80

6.1 Introduction 80

6.2 Optimizingthegroundwaterqualitymonitoringnetworks 80

6.3 Generalaspectsof thediscussions 81

6.4 Conclusions 83

6.5 Futuredirections 83

Summary 87

Samenvatting in het Nederlands 91

Appendix A Tables 96

References 99

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2.1 Nitrate-N concentration measured in the groundwatermonitoring networksat5-17 m depthin 1991.Bulletsare centred at measurementlocations.Bullet sizeisproportionalto themeasuredconcentration 8

3.1 Soil typeand land usemap.Dominantsoil-land usecategoriesfor 2 km × 2 kmcells 26

3.2 Monitoringnetworklocations.Sitesof thenationalgroundwaterqualitymonitoringnetwork(•) and theprovincial groundwaterquality monitoringnetworks(+) thatcontributedto thegroundwaterqualitymaps 27

3.3 Variogramsof log(Al)e , stratified by soil typeand land use. A numberreflectsthenumberof observationpairs usedfor an estimate(+) of a variogrampoint,codesusedfor a variogrammodel(—)areexplainedin TableA3,page97 28

3.4 Map of aluminiumin thegroundwater. 95%Confidenceintervalsfor 4 km× 4 kmblock medianvaluesrelatedto four concentrationlevels 30

3.5 Map of aluminium in the groundwater (alternative display of Fig. 3.4). 95%Confidenceintervals for 4 km × 4 km block median values related to fourconcentrationlevels 31

3.6 Theeffectsof nostratificationfor aluminium.95%Confidenceintervalsfor 4 km×4 kmblock medianvalueswhensoil andland useinformationis ignored,relatedtofour concentrationlevels.Onlyresultsthat differ fromFig.3.4areshown 33

3.7 Variogramof log(Al)e ,whenstratificationis omitted.Codesusedfor thevariogrammodel(—)areexplainedin TableA3,page97 34

3.8 Variogramsof log(Al)e ,stratifiedby land useonly.A numberreflectsthenumberofobservationpairs usedfor an estimate(+) of a variogrampoint,codesusedfor avariogrammodel(—)areexplainedin TableA3,page97 34

3.9 The effects of stratification by land use only for aluminium.95% Confidenceintervalsfor 4 km × 4 km block medianvalueswhensoil informationis ignored,related to four concentration levels.Only results that differ from Fig. 3.4 areshown 35

3.10 Variogramsof log(Al)e ,stratifiedbysoil typeandland use,usingonly informationfromthenationalgroundwaterquality monitoringnetwork.A numberreflectsthenumberof observationpairsusedfor anestimate(+) of a variogrampoint,codesusedfor a variogrammodel(—)areexplainedin TableA3,page97 37

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3.11 Theeffectsof networkdensityfor aluminium(total). 95% Confidenceintervalsfor 4 km × 4 km block medianvaluesbasedon measurementsfrom thenationalgroundwaterquality monitoring network only (provincial monitoring networkinformationis left out),relatedto four concentrationlevels.OnlyresultsthatdifferfromFig.3.4areshown 38

3.12 Theeffectsof networkdensityfor aluminium(secondorder information).95%Confidenceintervalsfor 4 km× 4 kmblock medianvaluesbasedon(i) first orderinformation from both monitoring networks,and (ii) secondorder informationfrom the national groundwaterquality monitoringnetworkonly, relatedto fourconcentrationlevels.Onlyresultsthat differ fromFig.3.4areshown 39

3.13 Theeffectsof networkdensity(first order information).95%Confidenceintervalsfor 4 km × 4 km block medianvaluesbasedon (i) first order informationfromthenationalgroundwaterquality monitoringnetworkonly, and (ii) secondorderinformationfrombothnetworks,relatedto four concentration levels.Only resultsthat differ fromFig.3.4areshown 40

4.1 Monitoringnetworklocations.Sitesof thenationalgroundwaterqualitymonitoringnetworkthat contributedto themapsof changesin groundwaterquality 47

4.2 Schematicrepresentationof ‘new observations’andtheir accuracy(aserror bars)at a singlemonitoringwell screen(noactualmeasurementsdrawn) 49

4.3 Variogramsof log(K)e , stratifiedby soil typeand land use, obtainedfromthe1991measurements(nationalandprovincial monitoringnetworks).Anumberreflectsthenumberof observationpairs usedfor an estimate(+) of a variogrampoint,codesusedfor a variogrammodel(—)areexplainedin TableA3,page97 50

4.4 Map of potassiumin thegroundwateraround1980,where it differs fromFig. 4.5(‘around2000’).95%Confidenceinterval for 4 km × 4 km block medianvalues,relatedto four concentrationlevels 52

4.5 Map of potassiumin thegroundwateraround2000,where it differs fromFig. 4.4(‘around 1980’).95%Confidenceinterval for 4 km × 4 km block medianvalues,relatedto four concentrationlevels 53

5.1 Mapof nitrogenleachingratefromagricultural soilstoshallowgroundwater.Valuesareaveragedfromtheoriginal 500m× 500mmodelcalculationsto yield 2 km× 2kmmeanvalues.Onlycellsin thecategorygrasslandonsandareshown 62

5.2 NO3-N concentration in groundwaterand nitrogen leaching rate. Plot of NO3-Nconcentrationmeasuredin thegroundwaterversusnitrogenleachingratefromsoilsto shallowgroundwater, for thecategorygrasslandonsand 62

5.3 Variogramsof (a) log(NO3-N)e

,and(b)residualsfor thecategorygrasslandonsand.Calculationof thesamplevariogramsis explainedin TablesA2(page 96) and A6(page97).Numbersreflectthenumberof (a)observationpairsor (b) residualpairsusedfor an estimate(+) of a variogrampoint.Codesusedfor a variogrammodel(—)areexplainedin TableA3,(page97) 63

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5.4 Mapof NO3-N in thegroundwater(categorygrasslandon sand).95%Confidenceintervalsfor 4 km× 4 kmblock medianvalues,relatedto four concentrationlevels.Estimatesare obtainedby ordinary kriging (on log-scale),usinginformationfromthemonitoringnetworksonly 64

5.5 The effects of using information on nitrogen leaching rates.95% Confidenceintervalsfor 4 km× 4 kmblock medianNO3-N concentrations,relatedto four levels.Estimatesare obtainedby universal kriging (on log-scale),usingbothinformationfrom the monitoringnetworksand ancillary information(nitrate leaching rates).Onlyresultsthat differ fromFig.5.4areshown 65

5.6 Map of atmosphericdepositionof zinc.Valuesare interpolatedfromtheoriginal10 km× 10 kmmodelcalculationsto 2 km× 2 kmcells.Only cells in thecategorysemi-natural vegetationonsandareshown 67

5.7 Zincconcentrationin groundwaterandatmosphericdepositionof zinc.Plot of zincconcentrationmeasuredin thegroundwaterversusatmosphericdepositionof zinc,for thecategorysemi-natural vegetationonsand.A bullet (•) marksa measurementat a seepagelocation 67

5.8 Variograms of (a) log(Zn)e , and (b) residuals for the category semi-naturalvegetation on sand,non-seepage locations.Numbers reflect the number of (a)observationpairs or (b) residualpairs,usedfor an estimate(+) of a variogrampoint.Codesusedfor a variogrammodel(—) are explainedin TableA3, page 97.Calculationof thesamplevariogramsis explainedin TablesA2(page 96) and A6(page97) 68

5.9 Map of zinc in the groundwater (category semi-natural vegetation on sand).95% Confidenceintervals for 4 km × 4 km block medianvalues,relatedto fourconcentration levels.Estimatesare obtainedby ordinary kriging (on log-scale),usinginformationfromthemonitoringnetworksonly (resultsapplyto non-seepagelocationsonly) 69

5.10 Theeffectsof usinginformationonatmosphericdepositionof zinc.95%Confidenceintervalsfor 4 km× 4 kmblock medianzincconcentrations,relatedto four levels.Estimatesareobtainedbyuniversalkriging (onlog-scale),usingbothinformationfrom the monitoring networks and ancillary information (zinc atmosphericdepositionrates).Only resultsthat differ fromFig.5.9areshown(resultsapplytonon-seepagelocationsonly) 70

5.11 Map of sulphatemeasurementsites.Shallowgroundwatermeasurementsites(+)andgroundwaterqualitymonitoringnetworksites(•) in thecategorysemi-naturalvegetationonsand 72

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5.12 Variograms of sulphate. Variogram for (a) log(SO4)e

measurementsfrom thegroundwaterquality monitoringnetworks(category semi-natural vegetationon

sand),for (b) log(SO4)e

fromtheshallowgroundwatermeasurements,and(c)crossvariogramfor (a)and(b).Anumberreflectsthenumberof observationpairsusedfor anestimate(+) of avariogrampoint,codesusedfor avariogrammodel(—)areexplainedin TableA3,page97.Calculationof thesamplevariogramsis explainedin TablesA2(page96) andA7 (page98) 73

5.13 Map of sulphatein thegroundwater(category semi-natural vegetationon sand).95%Confidenceintervalsfor 4 km × 4 km block medianvalues,relatedto fourconcentration levels.Estimatesare obtainedby ordinary kriging (on log-scale),usingmeasurementsfromthegroundwaterquality monitoringnetworksonly andthevariogramof Fig.5.12a 75

5.14 The effects of using shallow groundwater measurement information. 95%Confidenceintervals for 4 km × 4 km block mediansulphateconcentrations,relatedto four levels.Estimatesare obtainedby cokriging (on log-scale),usingmeasurementsfrom shallow groundwater and from the groundwater qualitymonitoringnetworks,and thevariogramsof Fig. 5.12a-c.Only resultsthat differfromFig.5.13areshown 76

5.15 Local scatterplotsof measurementsversusancillary variablefor NO3-N (a) andzinc (b). For five arbitrarily chosen(but distinct) local kriging neighbourhoodsR, scatterplots showconcentration in groundwater(a: NO3-N in 3g/m , b: zinc

in 3mg/m) versustheancillary variable (a: nitrate leaching rate in kg/(ha.yr),b:atmosphericdepositionof zincin mol/(ha.yr)).Arrowslink thelocationfor whichkriging neighbourhoodinformationis shownto thecorrespondingscatterplot 78

6.1 Monitoringnetworkoptimization:total costasa functionof measurementintensity.Thearrowpointsto an‘optimal’ measurementintensity 81

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3.1 Basicinformation:(1) soil-landusecategory, (2) areacovered by thecategory, (3)numberof selectedmeasurementsfromthenationalgroundwaterqualitymonitoringnetwork for each category and (4) numberof selectedmeasurementsfrom theprovincial groundwaterqualitymonitoringnetworkfor each category.Figs.3.1and3.2showthespatialinformationon(2),(3) and(4) 24

3.2 Theeffectsof stratificationfor aluminium.Numberof cellsfor which thevalueofa 95%confidenceinterval for a 4 km × 4 kmblock medianconcentration relatedtoa referencelevel(possiblevalues:higher,lowerandnotdistinguishable)changedasa resultof omittingstratificationby(a) land use,(b) soil type,and(c)both(totalnumberof cells:8197).Changed cells for columns(b) and (c) are shownin Figs.3.9and3.6 respectively 36

3.3 The effectsof monitoring network densityfor aluminium.Numberof cells forwhich the valueof a 95%confidenceinterval for the 4 km × 4 km block medianconcentration relatedto a referencelevel (possiblevalues:higher, lower and notdistinguishable)changedasa resultof omittingtheinformationfromtheprovincialgroundwaterquality monitoringnetworks(PGMs)(total numberof cells: 8197).Changesare a resultof (a) lossof informationaboutvariograms(secondorderinformation),and/or (b) loss of information about the meanlevel and locationspecificinformation (first order information).Changes from each one of thesecomponentsare alsoshown.Changed cellsare shownin Figs.3.11(total),3.12 (a)and3.13(b) 36

5.1 Theeffectsof ancillary information.For each variableand level thenumberandpercentage of cells for which the addition of ancillary informationchanged theoutcome(possiblevalues:lower,higherandnot distinguishable) 77

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1.1 Problemdefinition

Pollution from agricultural and industrial origin threatensthe groundwater quality in theNetherlands(VanDuijvenbooden,1989).Locally, thispollution is measuredin thegroundwaterattensof metresdepth(CCRX,1994,1995).Sincegroundwateristhemainsourceof freshwater,pollutioncausesadecreasein thelong-termresourcesof watersuitablefor humanconsumption,andcurrentlywatercompanieshave to increasetheir efforts for makinggroundwatersuitablefor drinking (Versteegh et al., 1995).Furthermore,becausemostgroundwaterfinally flows tothesurface,adeteriorationof groundwaterqualityaffectstheecohydrologicalconditionsof thereceiving seepageareasaswell asthereceiving seas.

In orderto get insight in thecurrentsituationof groundwaterquality andthesystematicchangesof groundwaterqualityovertime,thenationalgroundwaterqualitymonitoringnetwork(NGM) wasestablishedfrom 1978to 1984(Van Duijvenboodenet al., 1985).This networkconsistsof 370permanentwells spreadfairly evenly over thecountry(Fig.3.2, page27), withscreensat 8-10and23-25metrebelow thesoil surface.Thenetwork is sampledyearly, at firstfor 19variables,andmorerecentlyfor 25variables.In recentyearsall provincescommencedtheinstallationof similar monitoringnetwork sites,theprovincial groundwaterqualitymonitoringnetworks(PGMs).Currently, they provideadoublingof thenumberof wellsin theNGM.

Becausethe monitoringnetwork is a major financial investment,questionslike ‘do wehavesufficientmeasurements?’or ‘canareducedmonitoringnetwork besufficient?’arise.Suchquestionscanonly beansweredwhenwe have answersto thequestions‘sufficient for what?’and‘what exactly do we needto know aboutgroundwaterquality?’Clearly, this calls for thequantificationof what canbe inferredfrom availableinformationaboutthe quality of all thegroundwaterof interest.Thereforewe need(a) to choosea suitable,effective and estimablemeasureof groundwater quality (what characteristicof groundwater quality do we needtoknow?),and(b) tohandletheestimationuncertaintyin ameaningfulway(towhatextentcanweinfer thischaracteristicfromavailableinformation?).Theuncertaintyaboutthetruegroundwaterqualitythatresultsfrom limitedestimationaccuracy is thekey parameterin optimizationstudies,wherea trade-off betweenphysicalcostandlossesdueto limited knowledgeshouldbemade:if we needto know everythingaboutgroundwater thenwe will have to measureeverything,whichwill costtoomuch;if wedonotwantto spendmoney, wewill haveto livewith completeuncertaintyaboutthegroundwaterquality.Somewherebetweenthesetwoextremaliesanoptimalgroundwaterqualitymonitoringnetwork.

1

2 GENERALINTRODUCTION

1.2 Groundwater quality—what is it?

Groundwatermaybedefinedasthesubsurfacewaterin soilsandrocksthatarefully saturated(WardandRobinson,1989).Duetoaprecipitationsurplus,partof theinfiltratedrainwaterflowsthroughthesoil andreachesthesaturatedzone,whereit becomesgroundwater. In general,freshgroundwaterflowsin thedirectionof thelowestgroundwaterheads,usuallythelowerelementsin the landscape(e.g.,a sea,lake, river, depressionor local drainagepoint), whereit finallyexfiltratesassurfacewater. Theflow speeddependson thegradient(slope)in thegroundwatertableandthe permeability(or resistance)of the soil. Conceptually, groundwaterflow canbedivided into verticalflow andhorizontalflow. In the areawith sandysoils (Fig. 3.1, page26),the vertical, downward flow speedof groundwater near the groundwater table is estimatedto be approximately1 metreper year (Meinardi, 1994) in recharge (infiltration) areas.Thehorizontalflow mayrangefrom afew metresto hundredsof metresperyear, dependingonlocalcircumstances.

Wheninfiltrating waterreachesthegroundwatertableit hasa chemicalcompositionthatdependson the compositionof the rainwater, on the condensationdueto evapotranspiration,andon theabsorption,desorption,chemicalandbiochemicalreactionsthatoccuredduringtheflow through the possiblycontaminatedsoil. The chemicalcompositionof groundwater isvariablein spaceandtimeasaresultof variabilityin theamountandin thechemicalcompositionof rain, of variability in the chemicalcompositionof infiltrating water (possibly resultingfrom agriculturalor industrialpollution), variability in evapotranspiration,asa result of soilheterogeneity, of interactionof infiltrating waterwith thesolid phaseandof theheterogeneityof thegeochemicalandgeohydrologicalenvironment.It is oftenestablishedthat thevariationin chemicalcompositionis very largein themostshallow groundwater(i.e.theuppermetreofgroundwater).Duringtheflow of groundwater, mixing occurs,andit is expectedthatasa resultof thismixing thelarge(horizontalandvertical)short-distancevariationdecreaseswith depth.

Groundwater quality is the suitability of groundwater for a certainpurpose.Here,wewill define groundwater quality as the suitability of groundwater for humanconsumption.Thissuitabilitydependsmostlyon thechemicalcompositionof groundwater(notethatin somecasesmicrobiologicalcomponentsmaybeimportantaswell). As far asgroundwaterquality isdeterminedby chemicalcomposition,it canbemappedby showing thechemicalcompositionin relationto critical levelsfor humanconsumption,e.g.(multiplesof) targetconcentrationsormaximumtoleratedconcentrations.

1.3 Monitoring groundwater quality

The national groundwater quality monitoring network (NGM) was establishedfrom 1978to 1984.The objectivesof this network are (Van Duijvenboodenet al., 1985) to investigate(i) the quality of the groundwater in the upperaquifer, (ii) the extent of humaninfluenceongroundwater quality and (iii) the changesof groundwater quality over time; and to collectsufficientinformation(iv) to allow a goodmanagementof groundwaterresources,and(v) forthedevelopmentof groundwaterqualitymanagementmodels.

Human influence on groundwater quality would be detectableat first in the most

MONITORINGGROUNDWATERQUALITY 3

shallow groundwater, andthereforeit seemssensibleto monitoronly theupperfew metresofgroundwater. However, due to the large short-distancevariation in shallow groundwater thiswouldcall for somany monitoringsitestobeeffectivethatthecostsof suchanetwork wouldbetoohigh.For thisreason,theupperscreensof theNGM wereinstalledsomewhatdeeper,at8-10m depth.In rechargeareasthegroundwatermeasuredat thisdepthhasinfiltratedapproximately10 yearsbeforethetime of measurement.Screensof 2 m lengthresultin anintegratedsampleof a 2 m layer of groundwater, thusrepresentinggroundwater that infiltrated in a periodofapproximately2 years.Thescreenlengthchoicehasbeena compromise:shorterscreensmighthave revealedvariationsat shortdepthintervalsthatresultfrom (irrelevant)seasonalvariation,whereaslongerscreensresultin averagingovera longertime(depth)intervalwhichwouldhavehinderedthedetectionof systematicchangesof groundwaterqualityover time.

Themonitoringwell locationsof theNGM wereselectedwith thefollowingcriteriain mind(VanDuijvenboodenetal.,1985):

(i) thelocationsshouldbespreadratherevenlyover thecountry(e.g.seeFig.3.2)

(ii) awell shouldbesitedon theprevalentsoil typeandlanduseof thelocation

(iii) a well siteshould,asfar aspossible,beon theoff-streamsideof a contiguousareawithconstantsoil typeandlanduse,thusenablinga non-ambiguousdeterminationof soil typeandlanduseof theinfluenceareafor well screens

(iv) practicalconsiderations,e.g.it mustbepossibleto reachthelocationby car.

Monitoringwellsaresampledyearlybecauseonly thestructuralchangein groundwaterqualityover severalyearsis of direct interest.In orderto avoid possibleseasonalfluctuationsto showup in thetime seriesfor a well, eachwell is visitedevery yearin thesamemonth.Thesamplethat is analysedin the laboratoryis taken after the withdrawal of approximately100 litres ofgroundwater. Therefore,at a sedimentporosityof 0.3,this samplerepresentsthegroundwaterpresentin a sedimentvolumeof approximatelyonethird of a cubicmetre.Thesetof variablesmeasuredhas grown over the years.Most of the variablesmeasuredin the first yearsofmeasuringarelistedin TableA5 (page97), thevariablesmeasuredin 1991arelistedin TableA1.Obviously, thechoiceof variablesthataremeasuredis guidedby therelevanceof thevariablesfor groundwaterqualityassessment.

Theareaof influencefor awell screenis theareawheregroundwatersampledatthatscreenhasinfiltrated.Horizontalgroundwaterflow causesa horizontaldisplacementof the influencearearelativeto ameasurementlocation.

It is expectedthat factorslike land use,soil type andgeohydrologicalsituationhave aninfluenceon groundwaterquality. For themappingof groundwaterquality—theestimationofcurrent situationand systematictemporalchangesof groundwater quality variables—thesefactorsshould,asfaraspossible,betakeninto account.


1.4 Modelling groundwater quality

Whenwemodelgroundwaterquality, wetry to summarizethespatialandtemporalvariationofgroundwaterqualityvariableswith alimitedsetof conceptsor mathematicalequations.Becauseof thecomplexity of variationin groundwaterqualityandourlimitedknowledgeof it, any modelmustbeasimplificationof reality.

Not all modelsaresuitablefor mappingpurposes.For instance,a model that needsthechemicalcompositionof thegroundwaterasinput for assigningacertaincharacteristicto it (e.g.Frapportiet al., 1993)canonly beappliedat measurementsites,i.e.wherethis compositionisknown.Suchamodelis oftenadescriptivemodel:it summarizes(describes)themeasurements.For mappingpurposesit is necessarythatamodelis predictive:it mustbecapableof estimatingthegroundwaterquality at locationswhereit wasnot measured.In additionto measurements,predictivemodelstypically useoneor moreindependentvariableswhosevalueis known at allestimation(mapping)locations.

Tobecomplete,aphysicallyandchemicallybaseddeterministicgroundwaterqualitymodelwouldcontainat leastthree-dimensionalnon-stationaryconvectiveanddispersivetransportandaccountfor geochemicalreactions.For the transportmodelling(e.g.DomenicoandSchwartz,1990)it would thenbenecessaryto know for every locationthe initial groundwatertable,thegroundwaterrechargeasafunctionof time,andfor everylocationanddepth(or moreprecisely,for every spatialunit thatdiscretizesthethree-dimensionalmodelspace)thepermeabilityanddispersioncoefficientsof thesediment.For thegeochemicalmodelling(e.g.AppeloandPostma,1993)it would thenbenecessaryto know theinitial chemicalcompositionof groundwater, thechemicalcharacteristicsof the sediment,the groundwater temperatureandmany equilibriumconstants,again for all model units. Collecting the necessaryinformation—if possible—isa majoreffort. For practicalreasons,a modelon a nationalscalewould have units thesizeofhundredsof metresto kilometres,and even if we had acceptableestimatesof the necessaryvariablesandparameters,it remainsquestionablewhethertheconstantsandequationsusedinsuchamodelarestill relevantwhenappliedto spatialunitsthissize(e.g.,Beven,1985).

As will be shown later, mappinggroundwaterquality is possibleby usingmuchsimplermodelsthat lump most of the above mentionedfactorsinto a spatiallydependentstochasticterm.

1.5 Groundwater quality monitoring network optimization

The groundwaterquality monitoringnetworksaccountfor a substantialpart of the total costof environmentalquality monitoringnetworks,andquestionsariseasto whetherthe way wecurrentlymonitorgroundwaterqualityisworththecost.Monitoringnetworkoptimizationwouldatleastinvolvetheevaluationof thetotalcostof anetworkwith respecttomeasurementintensityandparameterchoice,wheretotal costincludescostof measuringandthe lossdueto limitedknowledgeof thegroundwaterquality. Thespecificationof sucha lossfunction is non-trivial,andis beyondthescopeof thisstudy.

Beforewecanmakeajudgmentaboutthevalueof thecurrentmonitoringnetwork,wehavetoestablishtowhatextentwecaninfer thecurrentsituationandchangesin timeof groundwater

MONITORINGNETWORK OPTIMIZATION 5

quality for the whole of the Netherlands.For this, we must map(estimate)the groundwaterquality, usingasmuchaspossiblerelevant information:measurementsfrom the nationalandprovincialgroundwaterqualitymonitoringnetworks,mapinformationonsoil typeandlanduse,hydrologicalparametersanddeterministicprocessmodels.Theestimationaccuracy limits thepossibilityto infer groundwaterquality from estimates.

1.6 Objectivesand approach

Theobjectivesof thisstudyareto

• map groundwater quality in the Netherlands,using available measurementsfrom thenationalandprovincial groundwaterqualitymonitoringnetworksandmapinformationonsoil typeandlanduse

• show theeffectsof monitoringnetwork densityandtheeffectsof usingsoil typeandlanduseinformationon theresultinggroundwaterqualitymaps

• map the systematic,temporalchangesin groundwater quality in a way similar to themappingof groundwaterquality

• show how groundwater quality maps can be improved by using relevant ancillaryinformation in the estimationprocedure,whereancillary information is obtainedfromdeterministicprocessmodelsor from othermeasuredvariables.

Theprimaryaim is,with theseobjectives,to try to answerthebasicquestionsof describingthecurrentsituationandsystematictemporalchangesof groundwaterquality for thewholeof theNetherlands.

In this studyfairly simplemodelsareusedin which many of the factors(or processes)that could be usedin a complex deterministicphysical-chemicalapproacharelumpedinto astochastictermthatmaybespatiallydependent.Themodelsusedallow anexplicit quantificationof theaccuracy of resultingestimates.Thisaccuracy is takeninto accountin theresultingmaps,anticipatingthequestionaboutthevalueof thecurrentmonitoringnetworksfor inferringcurrentsituationandsystematicchangesin timeof groundwaterquality.

This thesisexplainsthemethodsused,illustratedby resultson a few groundwaterqualityvariables.Completeresultsfor all groundwaterquality variablesmeasuredin the monitoringnetworksfor mapsof 1991(Chapter3) andfor mapsof changesin groundwaterquality (1980vs.2000,Chapter4) arefoundin separatereports(PebesmaandDeKwaadsteniet,1994,1995).

1.7 Outline of this thesis

The statisticalmappingof quantitative spatial variableslike groundwater quality variablesis introducedin Chapter2. It explainsthe basicsof spatialvariation,errors,uncertaintyandstatistics,why we usegeostatisticalmodelsfor thespatialestimationof variables,andwhy wepresentresultswith limited but known accuracy asconfidenceinterval maps,andit concludes


with anoutlineof themethodsusedfor mappingthegroundwaterquality.Chapter3 describeshow mapsweremadefor all groundwaterquality variablesthatwere

measuredat 5-17m depthin 1991.Basedon a stratificationby soil typeandlanduse,selectedmeasurements,andaninterpolationmethodthattakesspatialdependencebetweenmeasurementsinto account,estimatesof 4 km × 4 km local mean(block median)concentrationshave beenpresentedon mapsin termsof theapproximate95%confidenceintervals.The resultingmapsandtheeffectsof stratificationandmonitoringnetworkdensityonthesemapsareillustratedwithexamplesonaluminiumconcentration.

In Chapter4 short-termpredictionsfor two extrapolationmoments(1980 and 2000),obtainedat themonitoringsitesfrom measurementseriesareusedto obtainmapsof statisticalindicationsof time-trends.Thisis illustratedby resultsonpotassiumconcentration.

Chapter5 shows how mapsof groundwaterquality variablescanbe improved by usingancillary information obtainedfrom deterministicprocessmodelsor from other measuredvariables.

Chapter6 discussessomegeneralaspectsof Chapters3-5: it addressesmonitoringnetworkoptimization,it shortly lists the main conclusions,and it givessomefuture directions.Morespecificaspectsof monitoringnetwork optimization,i.e.thevalueof resultsfrom thisstudyforgroundwaterqualitymonitoringnetwork optimizationareaddressedin Chapters3 (monitoringdensity),4 (monitoringfrequency) and5 (improving estimateswith ancillaryinformation).Theaspectof uncertainty, resultingfrom limited samplesizeis addressedthroughoutthisstudy.

2.1 Intr oduction

Thischapterdescribesstatisticalmapping:why weneedit, whatstatisticalmapsare,how wecancreatethem,why we usegeostatistics,why we presentstatisticalmapsasconfidenceintervalsandhow we make statisticalmapsof groundwaterquality variables.Without claiming to becomprehensive, this chapterintroducesthe most obvious statisticalapproachesto mappinggroundwaterquality variables,i.e. variablesthat have a valueat every locationbut which aresampledat a limited numberof locationsonly. Althoughgeostatisticswasnot developedfromclassicalstatistics(Cressie,1990),hereit is introducedastheobviousgeneralizationof classicalstatistical‘prediction’ for spatialapplications.This chapterdoesnot give a full introductioninstatisticsor geostatistics,but merelydescribesthemethodsusedin thefollowingchapters.Morecompletereferencesaree.g.Cressie(1991),DraperandSmith (1981),JournelandHuijbregts(1978),andChristensen(1987,1991).

A mapis a drawing of someattributeof an areaasit would appearif it wasseenfromabove: it is a specialtype of graphthat shows observationsin geographicalspacemappedintwo dimensionsby makingascaled(andthereforeusuallysimplified,generalized)image.Mapstell uswhere somethinghappens.Mapsoftenshow directlyobserved, i.e.visually observedormeasuredphenomenathatcanfor instancebediscreteeventssuchasthelocationof a city or aroad,numberssuchasnumberof inhabitantsat thelocationof thecity, or continuousvariablessuchasthemeasuredvalueof agroundwaterqualityvariableat themeasurementlocations(Fig.2.1). For technicalreasons,elementson a mapcanonly bedisplayedwith limited accuracy. Inorderto show not morethanwhatis known, thisdisplayaccuracy shouldnot exceedtheextentto which the elementsareknown, and generallythis is solved by choosinga properscalinganddisplayresolution.In theenvironmentalsciencesit is very commonthat the‘observations’shown onamapdonotdirectlyportrayobservedphenomenabut quantitiesthatareonly knownapproximately, andin thiscasetheneedto limit thedisplayaccuracy becomesmoreimportant.

Errorsin maps,thediscrepanciesbetweenwhatthemapshowsandthepartof realityaimedat, canbe ascribedto locationalerrorsandattributeerrors.Errorsthat accruefrom locationaluncertaintywill notbeaddressedhere.Attributeerroristhediscrepancybetweenthevalueshownatacertainlocationonamapandthereal,truevaluethatthemapaimedtoshow.Attributeerrorsusuallystemfrom incompleteknowledgeof theattributein themaparea,andtheseerrorsoccureasilywhenwehaveto estimatetheattributefrom measurements:themeasuredsampleis oftendifficult andexpensivetoobtain,andreflectsonlyasmallfractionof thepopulation.Furthermorethespatialvariationin themeasuredvaluescanbelarge,asin Fig.2.1.

7

8 STATISTICAL MAPPING

< 0.11 g/m3

2.3 g/m3�5.6 g/m3

11.3 g/m3

22.6 g/m3�45.2 g/m3�

Figure2.1 Nitrate-Nconcentrationmeasuredin thegroundwatermonitoringnetworksat 5-17mdepthin 1991.Bulletsarecentredat measurementlocations.Bullet sizeis proportionalto themeasuredconcentration

Theaccuracy of amapistheclosenesstorealityof themap,anddependsontheobservationson which themapis basedandon whatexactly is beingmapped.Fig.2.1is a mapshowing theNitrate-Nconcentrationasmeasured in thegroundwaterin theNetherlands(at 5-17m depth,in 1991),andeachmeasurementappliesto a volumegroundwaterof at mosta cubic metre.Consequently, thetotal volumeof themeasurementsis a tiny fractionof thetotal volumefromwhich thesamplesweretaken.SinceFig. 2.1doesnot tell usanything aboutthegroundwaterqualityat unsampledlocations,thevalueof it asa mapof Nitrate-Nin thegroundwaterfor thewholeof theNetherlandsisratherlimited.AlthoughFig.2.1showssomepattern,noclaimismadeaboutthe concentrationat unsampledlocations.For mappingthe groundwaterquality for thewholeof theNetherlands,simplyshowingthemeasurementsonamapisnotenoughandwehaveto estimatethevalueof groundwaterqualityvariablesatall unsampledlocationsin themappingareausingmeasurementsand,aswewill see,otherrelevantinformationwheneverpossible.

UNCERTAINTY,ACCURACY AND STATISTICS 9

2.2 Uncertainty, accuracyand statistics

If wewantto know thegroundwaterqualityat somelocationwith maximumcertainty—withinthe measurementprecision—thenwe have to measureit. Often,we cannotmeasureit at alldesiredlocationsfor practicalreasons,andwehave to besatisfiedwith anestimateof it, whichwill beanapproximate,uncertainanswer.Weestimatethevalueatanunsampledlocationusuallybyaveragingmeasurementsfromsimilar locations,i.e.locationswheretheconditionsaresimilarto thoseof theestimationlocation.How far this estimatemaydepartfrom thetruevalue—theestimationaccuracy—canbederivedfrom thenumberof andthevariationin themeasurementsatsimilar locations.

If we areinterestedin the meanvalueof a groundwaterquality variableover the wholecountry, simply takingmoremeasurements(e.g.at randomlychosenlocations)would increasetheaccuracy of theestimateof themean.However, it wouldberathernaivetousethisgrandmeanasanestimatefor thevalueataspecificlocation:thisestimatewouldbe(i) notspecific(it wouldbethesameeverywhere)and(ii) highly inaccurate,becauseweknow thatlocalconditionscausegroundwaterqualityvariablesto deviatelocally from thisgrandmean.

In ordertoobtainestimatesthataremorespecificthanthis,for anestimateof agroundwaterquality variableat a specificlocationwe shouldselectmeasurementsthat areobtainedfromlocationswith conditionsthataresimilar to thoseat theestimationlocation.Only on thebasisof external,independentinformationaboutlocal conditionscanwe decidewhich observationsmaybeconsideredfor thisselection.For mappingpurposes,theexternal,independentvariablesthatdefinetheseconditionsneedto beknown for thewholeareaof interest,i.e.atmeasurementlocationsaswell asatestimationlocations.

Ideally, ‘similar conditions’ for an unsampledlocation would be specifiedin such away that thesetof measurementsat connectedlocations—having thesameconditions—is(i)homogeneous(ii) largeenoughto yield goodestimates(iii) dissimilarto measurementsetsatotherlocations.This would result in estimatesthat arespecificandaccurateenoughto reveallocaldifferencesin themappingvariable.

External,independentvariablesExternal,independentvariablesthatdefinesimilarity of conditionscanbecategoricalvariables,continuousvariables,or combinationsof them.An exampleof acategoricalvariablethatwouldbe useful for the mappingof many groundwaterquality variableswould for instancebe theareawith marineinfluence:if weknew thisvariable(assumingwecoulddefineit), noneof theestimatesfor fresh groundwater would be ‘contaminated’with measurementsfrom the areawith marineinfluence.An exampleof a continuousvariabledefiningsimilarity of conditionscouldbetheatmosphericdepositionof a variable.If atmosphericdepositionis themainsourcefor avariablemeasuredin thegroundwater, themeasurementsat locationswith depositionratessimilar to thedepositionrateat theestimationlocationareexpectedto yield thebestestimateofthevariableat theestimationlocation.

In general,independent,externalvariablesthat aresuitablefor the purposeof mappinggroundwaterquality arevariablesthat cause(or explain) a major part of the spatialvariationin groundwaterquality, suchasgeochemicalvariables(e.g.demarcatingthe areawith marineinfluence,or oxygen-or calcium-richgroundwater),or variablesthatdefinefluxesfrom human


supply(e.g.from manuring,pesticideapplication,atmosphericdepositionfrom industrialorigin,etc.).Theconstraintof beingknown at all locationsin themappingarealimits thepossibilities(Sections3.1, 5.2).

ModelstructureThemathematicalformulationof thesetof relationsconnectingindependent,conditiondefiningvariablesto themeasurementsis themodel.Which independentvariablesareused,andhow themeasurementvariabledependson themis themodelstructure.

The choice of the model structureshould be guided both by availability of data,i.e.measurementsandindependent,conditiondefiningvariables,andby theory, theapriori ideaswehaveabouttheprocesswearemodelling.Theoryanddatashouldbebalancedwhenchoosingthemodelstructure.If amodelistoocomplex it isover-specified,andthisis inefficientbecausemoreparametersthannecessaryhavetobeestimated.In anextremecasesomeof theparameterscannotbeidentifiedbythedataandit islikelythatsuchamodelwill produceinadequateestimateswhichmorereflectthecreator’s fantasythantherealityaimedat.On theotherhand,if a modelis toosimpleit is under-specified.This is inefficientbecauseit will resultin unnecessarilyinaccurateestimates.Sincethe complexity of the modelstructuredependson the dataavailability, thereis not onecorrectmodel,andmoredatawill usuallysuggestandallow a morecomplex modelstructure.The compromisein model structurecomplexity we usuallychooseis the simplestadequatemodelthatis parsimoniousin its parameters.

EstimationOnly afterchoosinga modelstructurethat is suitablefor theproblemat hand,it is possibletoestimatethe quantityof interest.This demandsan explicit statementof the objective: whichcharacteristicof a measuredgroundwaterquality variabledo we want to estimate?We canforinstancebeinterestedin thevaluethatwould bemeasuredat a specificlocation,thevaluethatwould bemeasuredat a locationrandomlychosenfrom a specifiedarea,themeanor medianvalueof all measurementsin a specificarea,or thefractionof anareathathasvaluesabove acritical level.Not all modelsareequallysuitablefor estimatinga specificcharacteristic.Givena setof relationsthatdefinesthemodelstructure,we have to choosea statisticalproceduretoestimatethecharacteristicof interest.

AccuracyWhenwe estimatean unknown quantity, we never know the estimationerror. However, frommeasurementswecanretrieveinformationaboutthesizeof theerrorwecanexpect,asaveragedoversimilarestimationproblems.In someinstancesobservationsaresoregularor sonumerousthattheaccuracy of thevariablewewish to mapis of noconcern:theaccuracy is goodenough.For groundwaterqualityvariablestheaccuraciesof estimatesareusuallynotnegligible,andwecannotignorethiswhenpresentingestimatesin maps.Why andhow estimationaccuraciesareaccountedfor whenmappingestimateswill beexplainedin Section2.6, but wewill first explainsomebasicconceptsof statisticalandgeostatisticalmodels.

StatisticsStatisticsisthedisciplinethatprovidestheoryandtoolsfor decidingin thedata-theorycomplexity

UNCERTAINTY,ACCURACY AND STATISTICS 11

trade-off, for inferring (estimating)populationcharacteristicsfrom limited sampleinformationand for assigningaccuracy measuresto the inference.Here,statisticalmappingis definedasmappingresultsfrom an (inferential)statisticalanalysis,i.e. combiningestimatedvalue andaccuracy measurein onemap.Themostcommonlyusedmodelsin statisticsaresomeform ofa linearmodel.

2.3 Linear models

For mappinggroundwaterquality—estimatinggroundwaterqualityvariablesat locationswherethey arenotmeasured—weneedamodel.Fromall possiblemodelswechoosestatisticalmodels,and the family of linear modelsprovides a comprehensive framework for most commonlyusedstatisticalmodels.A limitation of linear modelsis that they only allow additive effects.For groundwaterqualityvariables(beingnon-negativeandhighly skewed)additivity of effectsis often a reasonableassumptionafter log-transformingthe measurements.Wider classesofproblemscanbeformed,e.g.by non-lineartransformationof variables(asin generalizedlinearmodels,McCullaghandNelder, 1989),or by defininglinear relationslocally (approximatingmorecomplex relationswith local linearmodels,asin generalizedadditivemodels,HastieandTibshirani,1990).

Linearmodelsarea meansof expressingreal-life problemsin a mathematicalform, andare thereforehelpful in the abstractionof problemsand in communication.Comprehensivetreatmentsof linear modelsarefound in Rao(1973),Searle(1971),Christensen(1987),andin DraperandSmith(1981).Cressie(1991)andChristensen(1991)extendthesetreatmentstospatial(i.e.mapping)problems.A noteonterminology:in classicalstatisticstheword‘estimation’is reservedfor assigningvaluesto modelparameters,whereas‘prediction’is usedto denotetheevaluationof randomvariables,but hereI will use‘estimation’for both,following mostof thetraditionalgeostatisticalliterature(e.g.Matheron,1971,1989(page4-5),JournelandHuijbregts,1978).

ThreesimpleproblemsLinear modelsprovide a flexible way of expressinga wide rangeof problemsin a compactnotation.Considerthethreefollowing simpleproblems:

(i) n groundwaterqualitymeasurementsarecollectedrandomlyfrom a homogeneousspatialunit, andwe want to estimatethemeanvalueof themeasuredvariablein thisunit anditsestimationvariance

(ii) ngroundwaterqualitymeasurementsarecollectedin twospatialunits,q in thefirst,n � q inthesecond,andwewantto estimatethemean(andits estimationvariance)of eachgroup

(iii) n nitrate concentrationmeasurementsare collected in a spatial unit, and for everymeasurementlocationthe manuringrateis known. We assumethat nitrateconcentrationincreaseslinearlywith manuringrate,andwewanttoknow therateof increase(theincreasein nitrateconcentrationataunit increasein manuringrate)

For thesesimpleproblemstheobservationsz(xi) at locationsxi, i �� 1…n canbewrittenas


(i) z(xi) �� m+ d(xi),�

i : i �� 1…n

(ii) z(xi) �� m1 + d(xi),�

i : i �� 1…q

z(xi) �� m2 + d(xi),�

i : i �� q + 1…n

(iii) z(xi) �� 1 + r(xi)� 2 + d(xi),�

i : i �� 1…n

Here,conceptuallythemeasurementsaretakenasthesumof a structural,systematicpartandaresidual,unsystematicpart.Thestructuralpartconsistsof (i) m, themeanvalueof thespatialunit;(ii) m1 andm2, themeanvaluesof units1and2; (iii) r(xi), themanuringrateat locationxi, � 2 thechangein nitrateconcentrationataunit changein manuringrate,and� 1 thenitrateconcentrationatnomanuringrate.Theunsystematicpartin all threeproblemsis d(xi): thedeviationof thei-thmeasurementfrom thestructuralpart.

In a linearmodeltheobservationz(xi) is representedby a randomvariableZ(xi), andZ(xi)is modelledasthesumof its expectedvalueE(Z(xi)) �� m(xi) andarandomdeviationfrom m(xi),e(xi)

Z(xi) �� m(xi) + e(xi), E(e(xi)) �� 0,

(heree refersto ‘error,’ not in thesenseof a mistake,but natural,residualvariationthat is notaccountedfor by m(x)). Theexpectedvaluem(x) is modelledasa linear functionof p known,independentvariablesthathaveacausalinfluenceuponZ(x), andp unknown coefficients� j thatrelatetheseindependentvariablesto theobservations

m(xi) ��p

j �� 1f j(xi)� j ,

whichgives,usingvectornotation

Z(xi) �� f (xi)� + e(xi)

with f (xi) �� (f 1(xi), f 2(xi),… ,f p(xi)) therow vectorwith thevaluesof theindependentvariablesatlocationxi, and� �� (� 1,… , � p)′, thecolumnvectorwith theunknown coefficients.Representingall observations,themodelcanbewrittenas

Z(x) �� Fx� + e(x)

with Z(x) �� (Z(x1),… ,Z(xn))′,Fx�� [ f j(xi)]n� p �� (f 1(x),… , f p(x))with f j(x) �� (f j(x1),… , f j(xn))′,and

e(x) �� (e(x1),… ,e(xn))′.WhenwedefineFx and� accordingly, thismodelcoversthethreesimpleproblemsdiscussedabove:

(i) p �� 1, f (xi) �� 1, i �� 1…n, and� 1 �� m

(ii) p �� 2, if i q thenf (xi) �� (1,0) elsef (xi) �� (0,1), and� �� (m1,m2)′

LINEAR MODELS 13

(iii) p �� 2, f (xi) �� (1,r(xi)), i �� 1…n, and� �� (� 1, � 2)′.

Thuswe canstructurethe problemby choosingFx (andthusdefiningthe sizeof � ). The j-thcolumnin Fx, f j(x) definesthestructureof therelationof themeasurementvariablez(x) to thej-thparameter� j. If � j is theoverallmeanasm in problem(i) or aninterceptas� 1 in problem(iii),thenf j(x) isacolumnof ones.If themodelcontainscategoriesasproblem(ii) and� j is themeanof the j-th category, thenf j(x) is thebinaryvariablethat for every observationdenoteswhetherit belongsto thej-th categorywith aone,or not,with azero.In caseof aregression-typerelationlike in problem(iii), f j(x) containsthe valueof the regressor, the independentvariableat theobservationlocations.

Beforewe canestimate� or thevalueof observationsat unsampledlocations,we have tospecifythestructureof theerrors.

Linear modelswith independent,identicallydistributederrorsIn thesimplestcasewe assumethat theerrorse(x) areindependentandidenticallydistributed(IID), resultingin themodel

Z(x) �� Fx� + e(x), E(e(x)) �� 0, Cov 2(e(x)) �� I , (2.1)

which leadsto theordinaryleastsquares(OLS)estimate(providedthatFx hasfull rank)

^� �� (Fx′Fx� 1) Fx′z(x), (2.2a)

andestimationvariancesandcovariancesfor (� � ^� )

Cov(� � ^� ) �� (Fx′Fx� 1) 2 , (2.2b)

andwhere 2 is estimatedby

2s �� z(x)′(I � Fx(Fx′Fx� 1) Fx′)z(x)/ (n � R), (2.3)

with R therank(thenumberof columns)of Fx.

At anunsampledlocationx0, giventhisestimate^� , thevalueof z(x0) (andthemeanvalue

of r independentreplicationsof z(x0)) is estimatedby

^z(x0) �� f (x0)^� , (2.4)

with f (x0) thevalueof theindependentvariablesat locationx0. Theestimationvarianceof theestimatorof z(x0) (r �� 1) or themeanof r independentreplicationsof z(x0) is givenby

2r (x0) �� (1

9r+ f (x0)(Fx′Fx

� 1) f (x0)′)2 . (2.5)

If theassumptionof IID errorsisappropriate,theequations(2.2a)-(2.3) leadto theanswerof the


threesimplequestionsposedat thebeginningof thissection.

Linear modelswith dependenterrorsA wider classof problemsthantheonewith IID errorsis obtainedwhentheerrorsareallowedto bedependent

Z(x) �� Fx� + e(x), E(e(x)) �� 0, Cov(e(x)) �� V (2.6)

with V �� [Cov(e(xi),e(xj))] n� n. Thisleadsto weightedleastsquares(WLS)estimatesof �^� � �� (Fx′

� 1V Fx� 1) Fx′

� 1V z(x), (2.7a)

with estimationcovariances

Cov(� � ^� � ) �� (Fx� 1′V Fx

� 1) . (2.7b)

Underthismodel,given^� � by (2.7a), theestimateof z(x0) is

^zwls(x0) �� f (x0)^� � + v0′

� 1V (z(x) � Fx

^� � ) (2.8)

wherev0′ �� (Cov(e(x1),e(x0)),… ,Cov(e(xn),e(x0))), andhasestimationvariance:

2wls(x0) �� C(0) � v0

� 1′V v0 + (f (x0) � v0� 1′V Fx)(Fx

� 1′V Fx� 1) (f (x0) � v0

� 1′V Fx)′ (2.9)

with C(0) �� Var(e(x0)). In statisticalterms,thevalueestimatedin (2.8) isexpressedasthesumof

thebestlinearunbiasedestimate(BLUE) of m(x0),^m(x0) �� f (x0)

^� � andthebestlinearunbiased

predictor(BLUP)of the(correlated)error, ^e(x0) �� v0′� 1V (z(x) � Fx

^� � ).MultivariableestimationWhensvariablesZk(x),k �� 1…seachfollow alinearmodelZk(x) �� Fk,x� k +ek(x),andtheek(x) arecorrelated,thenit makessensetoextendtheweightedleastsquaresmodeltoallow multivariableestimation.Without lossof generality, assumes �� 2. Whenz(x) �� (z1(x),z2(x))′ andB �� (� 1, � 2)′aresubstitutedfor z(x) and� , andwhen

f (x0) �� [ f 1(x0)

00

f 2(x0) ], Fx�� [ F1,x

00

F2,x ], V �� [ V11

V21

V12

V22 ], v0 �� [ v11v21

v12v22 ],

with V21 �� [Cov(e2(xi),e1(xj))],v21 �� (Cov(e2(x1),e1(x0)),… ,Cov(e2(xn),e1(x0)))′,and0aconformingzeromatrixor vector, aresubstitutedfor f (x0), Fx, V andv0, thentheleft-handsidesof both(2.8)and(2.9) yield themultivariableestimates:theleft-handsideof (2.8) thenbecomestheestimatevector ^z(x0) �� (^z1(x0),

^z2(x0))′, andthe left-handsideof (2.9) becomesthe (2 × 2) matrix withestimationcovariances.

LINEAR MODELS 15

ConfidenceintervalsWhentheestimationerrorz(x0) � ^z(x0) isnormallydistributedwith zeromeanandvariance 2 (x0),confidenceintervalscanbeconstructedfor z(x0) (from,dependingon themodelused,(2.4) and(2.5), or (2.8) and(2.9)), andtheinterval

[^z(x0) � 2 (x0) , ^z(x0) + 2 (x0)] (2.10)

is a95%confidenceinterval for z(x0).

2.4 Changeof support and error structur e

Usuallyameasurementis notacharacteristicof apoint in space,but anaverageoversomeareaor volume,thesupportof themeasurement(JournelandHuijbrechts,1978).For groundwaterquality, thevolumemeasuredappliesto somethingbetweenthevolumepumpedfrom thewellbeforea sampleis taken (i.e. approximatelyhundredlitres) and the volume that is actuallymeasured(i.e. a few millilitres). We hopethat enoughmixing occurredduring samplingandthat the measurementon thesmall volumeis representative for the larger volume.In practicethemeasurementsupportis determinedby practicallimitations:if it wereeasywe would havemeasuredtherelevantpopulationcharacteristic(e.g.meangroundwaterqualityin acertainarea)directly. The supportof measurementsalsodeterminesthe variation in the valuesobserved:measuredmeansof largerspatialunitsshow lessvariation,becausevariationwithin theunitsisaveraged(JournelandHuijbregts,1978).

Meaningfully mapping groundwater quality variables for units of the size of themeasurements—estimatingvaluesthatwould actuallybemeasuredat unsampledlocations—isnotpossiblebecausethevariationin themeasurementsis frequentlytoolargeandconsequentlywe cannotestimategroundwaterquality with a reasonableaccuracy at that scale.At a lowerspatial resolution(e.g. for meanvaluesof adjacentor overlappingsquarecells on a grid)it is possibleto estimategroundwater quality variablesbecausethe patternof local averagegroundwaterqualityissmootherthanthepatternof themeasurements.For theestimationof thissmoothpatternof localaveragesweneedamodelthatallowsestimationof spatiallocalaveragevalues.

It would be convenientto usethe linear modelwith independent(IID) errorspresentedin the previoussectionbecauseit is simpleandit is supportedby a rich body of researchinclassicalstatistics.However, if we wantto obtainindependencefrom a design-basedargument(e.g.Hansenetal.,1983,DeGruijterandTerBraak,1990),thenthismodelis only adequateforestimatingthepatternof localmeanvalueswhen(i) theobservationsarecollectedrandomlyfromtheareaswith constantvaluesfor thef (xi) (i.e.from thecategoriesdistinguishedor from anareawith aspecificvaluefor theregressors)and(ii) theareasfor whichwewantestimatesof aspatialaveragecoincidewith theseareasof constantf (xi). Theseconditionslimit thesuitabilityof thismodelfor estimationlocal averagesseverely.

Therefore,for the mappingof groundwater quality variables,we will continue withmodelsthat allow errorsto be spatially dependent.Moreover, althoughpart of the residualerror can usuallybe attributed to measurementerror, it is very likely that in a linear model


intendedfor spatial estimationa large proportion of the residual,unexplained variation iscausedby (unknown) spatiallysmoothfactors,resultingin a spatiallydependenterror, andformappingpurposeswecanusethisspatialdependenceto capturethespatialstructurepresentinthe measurementsbeyond the part explainedby the independentvariables.Averagevaluesofarbitrarilyshapedelementscanbeestimatedefficientlyif wearewilling toassumeamodelwithaspatiallydependenterrorstructure.Estimatinglocalaveragesusingamodelthatallowsspatialdependenterrorsis a typicalproblemin geostatistics.

2.5 Geostatistics

Geostatisticsis thedisciplinethatprovidesasetof modelsandtoolsfor theestimationof blockaveragesor localaveragesfrom sampleobservations,takingbothlargescalevariation(thetrend)andsmallscalevariation(spatialcorrelation)into account.Typically, in a geostatisticalmodeltheobservationsz(x) areconsideredasobtainedfrom a realizationof therandomfield Z(x). Forthisrandomfield weassumethattheuniversalkriging model

Z(x) �� m(x) + e(x) (2.11)

holds,with expectation

E(Z(x)) �� m(x) ��p

j �� 1f j(x)� j

�� Fx� (2.11a)

anderror

E(e(x)) �� 0, Cov(e(x)) �� V, (2.11b)

with m(x) thetrend,thelargescalefluctuationof Z(x), ande(x) the(possibly)spatiallycorrelatederror. Thus,the universalkriging model is a linear model with dependenterrors,like (2.6).Ordinarykriging is the simplestform of universalkriging with m(x) �� m, a constantmean.In the pastuniversalkriging hasbeenheld synonymouslywith non-stationarykriging usingpolynomialsof thespatialcoordinatesasthef j(x), but herewewill only considervariablesthatcarryaphysicalsignificancewith respecttoZ(x) asclientsfor theindependentvariablesf j(x),andwe will accountfor regionaldifferences(non-stationarity)by assumingthat (2.11a) holdsonlylocally (JournelandRossi,1989,Journel,1992).

The challengein geostatisticalmodellingis to find a meaningfulstructurefor the trendand suitablevaluesfor the spatialcorrelation.Since it is impossibleto obtain estimatesofCov(e(xi),e(xj)) experimentallyfrom a singlerealizationof Z(x), additionalassumptionshave tobeimposedontheerrore(x).Commonlyusedmodelsfor theerrorarethecloselyrelatedsecondorderstationaryandintrinsicallystationaryprocesses.Secondorderstationarityimpliesthattheerrorcovariancebetweentwo locationsxi andxj isa functionof theseparationvectorh �� xi � xj

only, writtenas

Cov(e(xi),e(xj)) �� Ce(xi,xj) �� Ce(h),

GEOSTATISTICS 17

whereCe(h) is thecovariogramof e(x). Intrinsicstationarityimpliesthatthesemivarianceof theerrorprocess� e(xi,xj) �� 1⁄2E(e(xi) � e(xj

2)) is a functionof theseparationvectoronly

� e(xi,xj) �� e(xi � xj) �� e(h),

where � e(h) is the variogramof e(x). A further assumptionthat facilitatesthe estimationofcovariancesis isotropy. Isotropicprocesseshaveacovariogramor variogramthatdependsontheseparationdistanceonly,disregardingdirection:Ce(h) �� Ce( � h � )and� e(h) �� e( � h � ).An advantageof usingthevariogramover thecovariogramis that if theexpectationof Z(x) is a constant,thevariogramcanbedirectlyexpressedasa functionof theobservationvariableZ(x)

� e(xi,xj) �� Z(xi,xj) �� 1⁄2E(Z(xi) � Z(xj2)) ,

andhenceit canbe estimateddirectly from the measurements(TableA2, page96), whereascovarianceestimationadditionallyinvolvestheestimationof themeanvalue.

For estimation,variogramvaluesfor any distanceh maybeneeded.Thereforewe have tochooseafunctionthatguaranteespositiveestimationvariances(e.g.TableA3,page97)andfit thisfunctionto thesamplevariogram^� e(hj), calculatedfrom sampledataatregulardistanceintervalshj (e.g.TableA2, page96andTableA6, page97).

If thetrendcontainsanintercept(whichwill usuallybetrue,e.g.anunknown meanvaluefor eachvariableor an unknown meanfor eachcategory),andif a generalizedcovarianceisdefinedasKe(xi, xj) �� e(xi, xj) + d for anarbitraryconstantd, thensubstitutingKe(xi, xj) forCov(e(xi),e(xj)) in (2.6) is sufficientfor (2.8) to yield thekriging estimateof z(x0) andfor (2.9) to

yield thekriging variance 2k(x0) �� 2

wls(x0) (Christensen,1991,VI.3; Cressie,1991,5.4).AveragedvaluesZ(B0) for rectangularor otherwiseshapedblocksB0 are estimatedby

replacingthepoint-to-pointsemivariance� e(xi,x0)with thepoint-to-blocksemivariance� e(xi,B0),themeanof all point semivariancesbetweenxi andthepointsthatdefineB0

� e(xi,B0) ��B0� 1��

B0� e(xi,u)du,

with �B0 � theareaor volumeof B0; by replacing� e(x0,x0) with theblock-to-blocksemivariance� e(B0,B0), themeanof all point semivariancesbetweenthepairsof pointsdefiningtheblockB0

� e(B0,B0) �� B0� 2� �

B0�

B0� e(u,v)dudv,

andby replacingf (x0) with f (B0) �� (f 1(B0),… , f p(B0)), with f j(B0) �� B0� 1� �

B0f j(u)du, theaverage

valueof f j( � ) overblockB0.In Chapter4wehave‘new observations’thatareshort-termpredictionsfor anextrapolation

moment,obtained from a linear regressionmodel for each monitoring site. These ‘newobservations’aresubjectto a location-specifictime-predictionerror, andcanthereforenot beadequatelyrepresentedby astationarymodel.However,whenthese‘new observations’estimatean underlyingvariablez(x) that can be modelledby a stationarymodel,then we can model


such‘new observations’y(x) asthe sum of a stationarycomponentZ(x) (asof (2.11)) and anon-stationary‘measurementerror’ � (x):

Y(x) �� Z(x) + � (x). (2.12)

Whenwe assumethat(i) Var( � (xi)) �� 2� (xi) is known andCov(� (xi), � (xj)) �� 0,�

i, j : i �� j, and(ii) Cov( � (xi),e(x)) �� 0,

�i,x,and(iii) thevariogram� e(h) isknown,then,from ‘new observations’

y(xi), z(x0) is estimatedwith (2.8) and 2k(x0) with (2.9) when Ke(0) + 2� (xi) is substituted

for Cov(e(xi),e(xi)) and generalizedcovariances(settingd �� 0) are substitutedfor all othercovariances,asdescribedabove(Delhomme,1978).

Whenmorethanonemeasuredvariableisusedfor theestimationof thevariableof interestand the variablesarespatially inter-correlated,the multivariableextensionof (2.8) and (2.9)canbe usedfor multivariableestimation(Ver Hoef andCressie,1993).Inter-variablespatialdependenceisusuallymodelledwith thecrossvariogram(Section5.3; TableA7, page98) or thecrosscovariogram.Multivariableestimationof pointor blockaveragesis calledcokriging.

Ordinarykriging within categoriesthat subdivide theareawith variogramsmodelledpercategory is calledstratifiedkriging (Chapter3).Stratifiedkriging canbeseenasa specialcaseof multivariablekriging(with thevariablesin thedifferentcategoriestakenasdifferentvariablesandnocrossvariograms),andwith alittle moreeffort asaspecialcaseof universalkriging(withthetrendmodelledasa seriesof binaryvariables—asteptrend—reflectingthecategories,anda category-specificvariogramthat is only defined(i.e.not a constant)for pairsof observationsfrom thesamecategory).

An importantadaptionof theuniversalkrigingmodelthatleadstoawiderandmoreflexibleclassof modelsis thelocal applicationof (2.11): theobservationsareassumedto follow (2.11a)and(2.11b)onlywithin alocalneighbourhood,e.g.definedbythenumberof nearestobservationsor thesizeof aneighbourhoodareaconsidered.Thisallowsthelocalexpectationtodeviatefroma globally fixed expectation,essentiallyweakeningthestationarityhypothesis(2.11a) to localstationarity(i.e.globalnon-stationarity).Furthermorelocalestimationonly callsfor variogramvaluesat distancesthatallow a meaningfulestimationof thevariogram(i.e.valuesat distanceslessthanapproximatelyhalf thedimensionsof theareausedfor variogramcalculation,JournelandHuijbregts,1978,III.B.7; Matheron,1989)andthatarelessaffectedby biasedestimationofthetrendparameters� (CressieandZimmerman,1992).

In practiceV andv0 arenot known but estimatedwith^V and ^v0 from sampledata,causing

^ 2k(x0) �� f (

^V, ^v0) to underestimatethe true estimationerror 2

k (Harville, 1985,Christensen,1991,VI.5). Thus,when ^ k(x0) is substitutedfor (x0) in (2.10), theprobabilitythattheresultingintervalswill covertherealvaluez(x0) is lessthan95%.In thisstudy,however, intervalsobtainedin thiswaywill beusedasfirst approximationsto 95%confidenceintervals.

Estimationusing the multivariableextensionof model (2.11) and variogrammodellingwereimplementedin theGSTAT computerprogram(Pebesma,1995),developedduringthisstudy.This programcanbeusedasanextensionto theGISPCRASTER (WesselingandVanDeursen,1995,VanDeursen,1995),andit allowsawiderangeof possibilitiesfor theintegrationof pointinformation(measurements)andmapinformationfor themappingof themeasuredvariable.Ithasbeenusedfor all geostatisticalcomputationsthroughoutthisstudy, andPCRASTER hasbeen

GEOSTATISTICS 19

usedfor theprocessingandpresentationof all rastermaps.

2.6 Confidenceinterval maps

If theestimationaccuracy (or moreprecisely, theestimationprecision)of groundwaterqualityvariablesislow,wehavetoshow thisin mapsbecauseshowingonlytheestimatedvalue—silentlypretendingit is a good estimateof reality—is misleading.The estimationprecisioncan beexpressedastheestimationvariance((2.5), (2.9)), but if wepresenttheestimationvariancein aseparatemap,wesplit theinformationaboutestimateandprecision,suggestingthatthey aretwoseparateentitiesandinviting thereaderto ignoretheprecisioninformation.

Theobvioussolutionto this problemis to combineestimateandestimationvariancein aconfidenceinterval(e.g.,(2.10))andtopresentthisonamap.Thiscanonlybedoneif assumptionsontheerrordistributionaremade(whichisnotnecessarilybad:thequestionariseswhatthevalueof anerrorvarianceiswhentheform of theerrordistributionisunknown).Confidenceintervalsshow wherethe true valueprobablylies (mostly betweenthe upperandlower boundsof theinterval),andwhattheprecisionof theestimateis (thewidthof theinterval).Whentheprecisionis high,theconfidenceinterval mapreducesto theclassicalmapthatshowstheestimatedvalueonly, becausebothsidesof theconfidenceinterval contractto thisvalue.

Theconfidencelevel of a confidenceinterval controlsthe fractionof intervalsthat coverthe true valueif all modelassumptionsaremet (i.e., in a mapwith many independent,valid95%confidenceintervals,approximately95%of the intervalswill cover the true value).Thislevel shouldbechosenwith theapplicationof themapin mind.At first sight,choosinga lowconfidencelevel (e.g.50% or 70%) seemsattractive becauseit leadsto mapswith narrowerintervals,but the ‘precision’achieved in this way is false:the fraction of matchesis simplysmaller. Moreover, choosinga low confidencelevel seriouslyunderminestheintuitivemeaningof theword ‘confidence,’ especiallybecausetheconfidencelevel is valid only whenall modelassumptionsaremet,which will in practiceonly be true approximately. The choicefor 95%confidenceintervalsfor thegroundwaterquality mapsin this studyis a mixtureof a matteroftasteandtradition(e.g.,Fisher, 1935).

Thedisplayof confidenceintervalsin mapsraisestheproblemof showing twovalues(bothsidesof theinterval)atonelocation.Splittingtheinterval andshowing thetwo sidesin separatemaps(asin Pebesma,1992)isnotrecommended,sincetheuppersideof aconfidenceintervalmayseemalarminglyhighwhile thelowersideisverylow,onlymeaningthatwehavenoinformationaboutthetruevalue.Thenotionof precision,thewidth of theinterval is lostwhenonly onesideof theinterval is considered.

Consideringboth sidesof the interval simultaneouslyleadsto the desiredaim of a map:to show wherewecandistinguishestimatesfrom specificvalues.For asinglespecificvalue,thiscanbeeasilyachievedin a map:any interval is entirelylower thanthespecificvalue,is entirelyhigherthanthevalue,or straddlesthevalue(in whichcasetheestimateandthespecificvaluearenot distinguishable, basedontheinformationavailable).For example,Fig.3.4(page30) givesamapwherethepositionof theconfidenceintervalsrelative to four referencelevelsis shown infour separatesub-maps,onefor eachreferencelevel.Thedisadvantageof thismapis thateachseparatesub-mapdoesnot show theprecisionof theestimates:consideringonly onesub-map,


notdistinguishablecanbecloseto (precise)aswell ascompletelyunknown, andthisinformationis only revealedby consideringall sub-mapsatonce.

An attemptto achievea combinedpresentationof bothpositionandwidth of theintervalsin onesinglemapis shown in Fig.3.5(page31). Thismaphassolidcolouredsquarecells—thusreducingto the‘classical,’ deterministicmap—wherea legendclasscoversaconfidenceintervalcompletely, and strongly contrastingupper and lower trianglesin a squaredcell when theconfidenceinterval iswide(noinformationavailable).Again,thismapmightbemisleadingsincereduppertrianglesin green-redcellsmayunjustlybetakenasalarming.

2.7 Mapping groundwater quality

Thissectiondrawstheline from theory, assetout in theprevioussections,to theapplicationofmappinggroundwaterquality which is the main objective of this study, describedin detail inChapters3-5.

All groundwaterquality variablesconsideredin this thesisareconcentrationsof solutesandarethereforephysically limited betweenzeroandsaturationlevel. In practice,relative tosaturationlevel, only low concentrationsoccur, andtheupperlimit canbe ignored.In generalthedistributionof themeasurementsishighlyskewed:many measurementsarelow, few areverylarge.Becauseof thephysicallower boundandtheshapeof thedistributions,thenaturalscaleto modelthesevariablesis thelog-scale.This impliesthaton thelog-scaleeffectsareassumedto beadditiveandallowstheuseof linearmodelswith stationaryerrordistributions(e.g.Myers,1989).

In thesimplestgeostatisticalapproachweassumethattheobservationsin theareaof interestcomefrom a realizationof an intrinsically stationaryprocess.This meansthata modelwith alocally constantmeanvalueandhomogeneousvariationsufficientlycharacterizesthevariablefrom which we have observationsin thearea.If thismodelis toosimpleto berealistic,we canfor instancesubdividetheareain severalsub-areasfor whichthismodelisappropriate,or wecantransformtheobservationsnon-linearly, or wecanassumea morecomplex modelfor themeanvalue.

The groundwater quality variablesmeasuredin the monitoring networks cannot beconsideredas having a constantmean or having homogeneousvariation throughout theNetherlands.Locally however, undersimilarconditionsthismaybeareasonableassumption.InordertodividetheNetherlandscompletelyin sub-areasthatmaybethoughtof ashavingalocallyconstantmeanandbeinghomogeneous,weneeda suitableexternal,independent,criterionthatis known atall locationsin themappingregion(Section2.3). In thesimpleapproachof Chapter3wechosethiscriterionto bethesoil-landusecategoryof eachlocation(Fig.3.1, page26).

Within soil-landusecategories,fromavailablemeasurements(in 1991,at5-17metredepth)weselecteda suitablesubsetof measurementsthatcouldbemarkedas‘similar’ measurements,allowingtheuseof astationarymodel.Fromthesemeasurements,within soil-landusecategories,estimatesof 4 km × 4 km block meanvalues,asobtainedby ordinarykriging on thelog-scale,werepresentedasapproximate95%confidenceintervalsfor blockmedianconcentrationontheoriginalscale(Chapter3,PebesmaandDeKwaadsteniet,1994).Theeffectsof stratificationwerestudiedbysystematicallyleavingouteithersoil typeor landuse,orboth;theeffectsof monitoring

MAPPINGGROUNDWATERQUALITY 21

network densitywerestudiedby leaving outasubsetof themeasurements.For studyingtime-trendsin groundwater quality (Chapter4), attentionwas focusedon

therecognitionof steady(i.e.monotonous)long-termcomponentsin changesin groundwaterquality. Within soil-landusecategories,usinga regressionmodel,short-termpredictionsfor1980and2000wereobtainedat250monitoringsitesfrom long-term(1987-1993)measurementseriesof thenationalgroundwaterqualitymonitoringnetwork,andusedin aspatialinterpolation,takingtime-predictioninaccuraciesintoaccount.For thespatialinterpolationlocallevel,thesizeof spatialvariation,spatialdependencein spatialvariationandlocationspecifictime-predictionvarianceswereexplicitly takenintoaccountfor eachmappingvariablebyusingaslightlyadaptedversionof ordinarykriging,modifiedasdescribedin Section2.5(2.12),andvariogramsobtainedfrom the1991-situation(Chapter3).

Mapsof groundwaterquality variablescanbe improved by incorporatingotherrelevant,ancillary information in the mappingprocedure.This ancillary information may consistofeithermapinformation(e.g.obtainedfrom deterministicprocessmodels),or point information(e.g.othermeasuredvariables).Chapter5 explorestheuseof model(2.11) andits multivariableextension for this purpose,and gives several examples.Using the first type of ancillaryinformation,mapinformationfrom deterministicprocessmodelsled to usinguniversalkrigingwith thedeterministicprocessmodelastheindependentvariable.Thesecondtypeof ancillaryinformationled to theuseof cokriging.

(from:PebesmaandDeKwaadsteniet,1994)

3.1 Intr oduction

In thefirst half of the1980sthenationalgroundwaterqualitymonitoringnetwork (NGM) wasinstalledfor themonitoringof groundwaterquality andchangesin groundwaterquality in theNetherlands(Van Duijvenboodenet al., 1985).For this purpose370 monitoringsites,spreadfairly evenlyover thecountry, aresampledyearlyat two depths(from well screensat about10and25metrebelow surface).At first sampleswereanalysedfor 19,andlater25,groundwaterqualityvariables.

Morerecently(duringthefirsthalf of the1990s)the12provincesof theNetherlandsstartedto install similar monitoringsites(theprovincial groundwaterqualitymonitoringnetworks,thePGMs),in ordertostudygroundwaterqualityatadenserspatialresolution.Currently, thePGMshaveasmany monitoringsitesastheoriginalNGM.

For all variablesmeasured(TableA1, page96) mapsaremade,usingmeasurementsfrombothmonitoringnetworks.Uncertaintiesin predictedvaluesarequantifiedandpresentedonthemaps.Full resultsarepresentedin aseparatereport(PebesmaandDeKwaadsteniet,1994).In thischapterweillustratethemethodsusedwith resultsononegroundwaterqualityvariable,namelyaluminiumconcentration.

Fromgeohydrologyweknow thatmany factorsinfluencegroundwaterquality,likesoil type,landuse,andgeohydrologicalsituation(e.g.depthof groundwatertable,infiltration or seepage,marineinfluence,precipitation).Soil type (Steuret al., 1985)andlanduse(Thunnissenet al.,1992)canbe distinguishedfrom the other factorsbecausethe dataareavailableasdigitizedmaps.Hence,wecanusethemto helpmakegroundwaterqualitymaps,andwestratifiedby soiltypeandlanduseprior to theextrapolation.Theeffectsof partialor no stratificationwerealsoinvestigated.

For eachsoil-landusecategory andfor eachgroundwaterquality variable,attentionwaspaidto thelevel of, thevariationin andthespatialdependencein thevaluesmeasured(atafixedmoment)by usingkriging interpolation(JournelandHuijbregts,1978)within categories.

Becausea groundwater quality monitoring network that has a national or provincialextentby nomeanscanprovideaccurateestimatesof groundwaterqualityvariablesfor specificunmeasuredlocations,acoarserspatialresolutionfor thegroundwaterqualitymapswasadopted,

22

INTRODUCTION 23

andblock meanvaluesof 4 km × 4 km squareelementswereestimatedfor a 2 km × 2 km gridcoveringthewholecountryusingblock kriging (JournelandHuijbregts,1978).Thesizeof thesquareelementswaschosenwith thespatialdifferentiationof soil typeandlandusein mind.Theresultingmapsof block meanvaluesof groundwaterquality (or, moreprecisely, block medianvalues)can be regardedasa translationof point information obtainedfrom the monitoringnetworks into information on spatial units, the size of units usedin regional groundwatermodels.

Theeffectof monitoringnetwork densitywasquantifiedby comparingmapsbasedon thefull dataset(NGM + PGMs)with mapsbasedon theoriginally availableinformationfrom theNGM.

Sections3.2, 3.3and3.4describethestratification,dataselectionandspatialinterpolation.Section3.5 explains the groundwater quality maps.Effects of stratificationand monitoringdensityarediscussedin sections3.6and3.7. Section3.8concludeswith adiscussion.

3.2 An aggregatedmap comprising soil type and land use

Theavailablemapsregardingsoil type(Steuretal.,1985)andlanduse(Thunnissenetal.,1992)aresodetailedthat hundredsof combinationsof classescanbemade.For thepurposeof thisstudysuchdetaildoesnot make sense:many categorieswould representonly a small fractionof thetotalareaandwouldconsequentlycontaintoo few (or no)measurements.Therefore,andanalogouswith considerationson the monitoringnetwork design(Van Duijvenboodenet al.,1985),weaggregatedclassestothe13categories—apartfrombuilt-upareas—shownin Table3.1.Here,semi-naturalvegetationcomprisesforests,heathsandnaturereserves;thecategory‘dunes’is strictly spoken not a soil-landusecategory, but wasobtainedfrom a mapwith ecologicaldistricts(Klijn, 1988).Foreachelementona2km× 2kmgridthedominantsoil-landusecategorywasdeterminedfrom anoverlayof bothaggregatedmaps.Theresultis shown in Fig.3.1(page26). Thesoil typeandlanduseinformationfor measurementswasderivedfrom thismap.

3.3 Data selection

For thegenerationof groundwaterqualitymapsanumberof measurementswereleft outbecausethey seemedunsuitablefor thispurpose.For instance,monitoringsitesfromthecategory‘built-uparea’wereleft outbecausethesoilmap(Steuretal.,1985)doesnotprovideinformationaboutsoiltypein built-upareas.Extrapolationfrommeasurementinformation,takingsoil typeintoaccountis thereforeimpossiblein built-up areaswith theinformationavailable,andnoestimatesfor thisclassweremade.Monitoring sitesthat could not be matchedto a soil-landusecategory (Fig.3.1) becausethey wereinfluenceddirectlyby waterfrom riversor theseawereleft out too.Fig.3.2showsthelocationsof the425monitoringnetwork sitesthatcontributedto thegroundwaterqualitymaps.

The groundwater quality mapswere basedon measurementsof the 25 groundwaterqualityvariablesavailablefrom 1991(TableA1, page96) in theselectiondescribedabove,withmonitoringscreenscompletelybetween5 and17 metrebelow groundsurface.Detectionlimit

24 MAPSOFGROUNDWATERQUALITY IN THE NETHERLANDS

Table 3.1 Basicinformation:(1) soil-land usecategory, (2) areacovered by thecategory, (3)numberof selectedmeasurementsfromthenationalgroundwaterquality monitoringnetworkfor each category and (4) numberof selectedmeasurementsfromtheprovincial groundwaterqualitymonitoringnetworkfor each category.Figs.3.1and3.2showthespatialinformationon(2),(3) and(4)

category area number

(%) NGM PGM total

semi-naturalvegetationonsand 10 36 26 62

grasslandonsand 25 79 45 124

arablelandonsand 7.1 29 15 44

dunes 2.8 11 4 15

semi-naturalvegetationonfluvial clay 0.2 0 2 2

grasslandonfluvial clay 8 25 26 51

arablelandonfluvial clay 0.5 1 0 1

semi-naturalvegetationonpeat 0.3 1 4 5

grasslandonpeat 8.6 24 21 45

arablelandonpeat 0.2 2 1 3

semi-naturalvegetationonmarineclay 1 2 2 4

grasslandonmarineclay 9 16 10 26

arablelandonmarineclay 15.5 26 17 43

built-up area 11.8

total: 100 252 173 425

valueswereassignedto measurementsbelow thedetectionlimit (usingthehighestvaluewhendifferentdetectionlimits occurred).

Somedoubtfulmeasurementswereleft out altogether. In thequality control,we usedthemeasurementsfrom theNGM asa referenceset.Themeasurementsfrom theNGM themselveswerechecked only marginally: a measurementfrom the NGM-set was left out for mappingonly if (i) themeasurementdifferedmorethana factor10 from all othermeasurementsof thatvariablein thesamemonitoringscreenin theperiod1985-1992and(ii) thefactor10differencedid notariseby achangein thedetectionlimit. FromthePGMwellsoneor moremeasurementsof a variablewereleft out whenthey couldbemarkedasanoutlier (asa resultof samplingoranalysiserrors,or localpollution).Thepresenceof outlierswassuspectedwhenadditionof thePGM-setto theNGM-setresultedin asubstantialincreaseof thewithin-categoryvariation(thelong-distancevalueof thevariogramconcerned,Section3.4). A suspectedvaluewasclassifiedasanoutlier whenit did not belongto theinterval [ �z � 3 s , �z+ 3 s], where�z andsarethemeanvalueandthestandarddeviationof thelog-transformedmeasurementsin theNGM-setthatwere

DATA SELECTION 25

notequalto thedetectionlimit. Lessthan0.2%of themeasurementsweremarkedasoutliers.Very few monitoringnetwork siteshadtwo screensat the depthinterval considered.For

thesesites,themeanvalueof the log-transformedvalueswasusedasoneobservation for thespatialinterpolation.Thevaluesfrom separatescreenswereusedindividually for thevariogram,thus contributing to the short distancevariation, the start of the variogram(assumingthatvariationof groundwaterquality variablesat a verticaldistanceof tenmetresis similar to thespatialvariationatahorizontaldistanceof a few kilometres).

3.4 Spatial interpolation

Maps of groundwater quality variableswere obtainedby meansof ordinary block-kriging(JournelandHuijbregts,1978)of log-transformedmeasurementswithin eachof the soil-landusecategoriesof Table3.1, usingthemeasurementsavailable.For categorieswith only veryfewmeasurementsamodifiedprocedurewasused,asdescribedat theendof thissection.

For the spatialinterpolationof the measurementsa model conceptwasadoptedwhichconsideredlog-transformedmeasurementsof a groundwaterquality variablez(x) asa samplefrom arealizationof therandomfield Z(x) with thefollowing properties:

Z(x) �� m+ e(x) (3.1)

with

mconstant, (3.1a)

E(e(x)) �� 0,�

x � R, (3.1b)

and

1⁄2E[(Z(x1) � Z(x22)) ] �� ( � x1 � x2 � ), � x1,x2 � R. (3.1c)

Thatis,within anareaof sizeR insideasoil-landusecategoryweusedanadditivemodel(onlog-scale)with (i) aconstantexpectedvalueand(ii) aspatialdependencebetweenmeasurementsthat is a function of the distanceh betweenthe measurementlocationsonly. Additionally weassumedthat in a soil-landusecategory for a groundwaterquality variableonly onefunction� (h) is needed(independentof thelocationsof theareaof sizeR).

Dependingon the samplevariogram(Table A2, page96), the form of the theoreticalvariogram� (h) waschosenfrom a nuggetmodel,a sphericalmodel,or a mixtureof thesetwomodels(TableA3, page97). An ad hocmethodfor fitting thevariogrammodelto the samplevariogramwas used:the short distancevariance(the nugget)and the maximumcorrelationdistance(therangeof a sphericalmodel)werechosen,whereasthelongdistancevariance(thesill) wasfitted (usingweightedleastsquaresfit with weightsequalto thenumberof pairsusedfor eachsamplevariogramestimate).Fig.3.3showsthevariogramsfor aluminium.TheareasizeR usedwasdefined(in orderof preference)as(i) theclosest20observationswhenmorethan20observationswereavailablewithin adistanceof 100km,(ii) 100km when10to20observations


semi-natural vegetation, sandgrassland, sandarable land, sanddunessemi-natural veg., fluvial claygrassland, fluvial clayarable land, fluvial claysemi-natural veg., peatgrassland, peatarable land, peatsemi-natural veg., marine claygrassland, marine clayarable land, marine claybuilt-up area

Figure 3.1 Soil typeand land usemap.Dominantsoil-land usecategoriesfor 2 km × 2 kmcells

SPATIAL INTERPOLATION 27

Figure3.2 Monitoringnetworklocations.Sitesof thenationalgroundwaterqualitymonitoringnetwork(•) andtheprovincial groundwaterqualitymonitoringnetworks(+) thatcontributedtothegroundwaterqualitymaps


0� 25000� 50000� 75000� 100000

Distance (m)

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0� 25000� 50000� 75000� 100000

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�18

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0� 25000� 50000� 75000� 100000

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0� 25000� 50000� 75000� 100000

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0� 25000� 50000� 75000� 100000

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0

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201$ 191

165

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205$ 234$ 256$

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271$ 281$

276$ 292$

253$ 251$

Grassland on sand = 1.3 NUG(0) + 1.07 SPH(30000)

0� 25000� 50000� 75000� 100000

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0� 25000� 50000� 75000� 100000

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Figure 3.3 Variogramsof log(Al)e , stratified by soil typeand land use. A numberreflectsthenumberof observationpairs usedfor an estimate(+) of a variogrampoint,codesusedfor avariogrammodel(—)areexplainedin TableA3,page97


wereavailablewithin thisdistance,(iii) the10nearestobservations.Theestimatesgivenin themapsconcernmeanvalueson log-scaleof 4 km × 4 km blocks.

Estimatesarepresentedasapproximate95%confidenceintervalscalculatedby

[^z(x0) � 2 k(x0) , ^z(x0) + 2 k(x0)] (3.2)

with ^z(x0) thepoint estimate(kriging estimate)on log-scale,and k(x0) theestimationstandarddeviation (kriging standarddeviation) of the estimate(Journeland Huijbregts, 1978).Backtransformationof this interval to theoriginalscale(takingtheexponentof bothsides)yieldsanapproximate95%confidenceinterval for theblockmedianvalue.

All 4 km × 4 km blocksconsideredarecentredat thecellsof the2 km × 2 km grid of Fig.3.1(assumingthatsoil-landusedominanceof 2 km × 2 km cellsstill holdsfor the4 km × 4 kmblocks).In thisway two adjacent4 km × 4 km blockshavea 50%overlap,resultingin thefinerspatialresolutionof 2 km × 2 km cells,while the limited estimationvarianceof 4 km × 4 kmblock meanvaluesremainsrelevant.Theresultshavebeenpresentedin maps(e.g.seeFigs.3.4and3.5). Only the2 km × 2 km centresof blocksfor which estimatesweremadeareshown onthemaps.

For the small soil-land usecategorieswith lessthan 10 observationson a variabletheestimationprocedurewasmodified,becausetoolittle informationisavailablein thesecategoriesfor modellingthevariogramandfor estimatingblock meanvaluesin thewaydescribedbefore.For spatialinterpolationin sucha small category, for a specificvariable(i) the variogramofthe largestcategory with thesamesoil typewasused(implying thatsoil type is consideredtobethemajorstructuredeterminingfactor)(ii) theobservationsof thecategory from which thevariogramwasusedwereaddedto thesetof observationsof thesmallcategory, but only whenthe80-percentilerangeof theobservationsin themaincategorycoveredtheobservationsof thesmallcategory, implying that thevaluesof bothcategoriesshouldbesimilar:in any othercasenoestimatesweremadefor thesmallcategory. Thesmallcategoriescover2.2%of themaps.

3.5 The groundwater quality maps

In thegroundwaterqualitymapspresentedthe95%confidenceintervalsfor 4 km × 4 km blockmedianvaluesarerelatedto four referencelevelsobtainedfrom multiplesof target levelsorcritical levelsfor humanconsumption.Thesetargetlevelsor critical levelshowever, aredefinedin relationto individualsamples.If ablockmedianvalueexceedsacritical level,morethanhalfof theindividualsamplesin theblockexceedthislevel.If ablockmedianvaluedoesnotexceedacritical level,still many individualsampleswithin theblockmayexceedit.

The mapsare shown in two forms. In the basemaps(the black-and-whitemaps,e.g.aluminium concentrationin Fig. 3.4) the estimatesare relatedto the four referencelevelsconsideredin fourseparatesub-maps.Eachof thesub-mapsshowsthepositionof theconfidenceintervalsrelative to onereferencelevel.Theinterval canbecompletelybelow thelevel (lower),completelyabove the level (higher) or it canstraddlethe level (in which caseestimatedvalueandreferencelevel arenot distinguishable, basedon the informationavailable).In the colourmaps(e.g.aluminiumconcentrationin Fig.3.5) four black-and-whitesub-mapsaremergedinto


level: 50 mg/m3

lowernot distinguishablehigher

level: 25 mg/m3


level: 100 mg/m3


level: 200 mg/m3


Figure 3.4 Map of aluminiumin thegroundwater. 95%Confidenceintervalsfor 4 km× 4 kmblock medianvaluesrelatedto four concentrationlevels

THE GROUNDWATERQUALITY MAPS 31

concentration: mg/m3

upper boundary of confidence interval

lower boundary of confidence interval

0 - 25 25 - 50 50 - 100 100 - 200 > 200

Figure 3.5 Map of aluminium in the groundwater(alternativedisplay of Fig. 3.4). 95%Confidenceintervalsfor 4 km× 4 kmblock medianvaluesrelatedto four concentrationlevels


asinglemap(holdingexactlythesameinformation).In general,theblack-and-whitemapsareeasierto read,but thecolourmapsimmediately

show the accuracy of the estimates:cells with confidenceintervals falling completelyin onelegendclasshave a solid colour, whereasgreen-redcellsarecellsfor which no informationisavailableat theresolutionconsidered.Thedoublingof thereferencelevelsin themapsreflectsthechoicefor equidistantlevelson log-scale.

3.6 The effectsof stratification

Groundwaterqualitymapscanalsobeobtainedwhile completelyignoringthegeohydrologicalknowledge that resultedin stratificationby soil type and land use.For nine groundwaterquality variables(TableA4, page97) the effectsof stratificationwereinvestigatedby makinggroundwaterquality mapswith partialstratificationandwithout stratification.Thesemapsarebasedonthesamedataasthemainmaps,usingthesamemodel(3.1) but now (a)for onecategoryonly (containingall 13 categoriesof Fig. 3.1), (b) for land usecategoriesonly, or (c) for soiltypecategoriesonly. For aluminiumtheeffectsof no stratificationareshown in Figs.3.7and3.6, andtheeffectsof stratificationby landuseonly—ignoringsoil type—areshown in Figs.3.8and3.9. Figs.3.6and3.9only show thedifferencesfrom thebasemap(Fig.3.4). Theresultsaresummarizedin Table3.2.In generalit canbesaidthat

(i) Theeffectsof omittingor partiallyapplyingstratificationin thisstudyarelarge,evenin thecurrentsituationwhereusuallyverywideconfidenceintervalsarerelatedto four referencelevels.Foralmostall groundwaterqualityvariablesconsideredpartialstratificationresultsin10to 25%changein at leastoneof thefour maps(PebesmaandDeKwaadsteniet,1994).

(ii) Thesimplertheapproach,thelessthespatialdifferentiation.

(iii) Thelargedifferencesbetweenresultingmapsdemonstratethatit is essentialto incorporategeohydrologicalknowledgeinto amappingprocedurelikethis(DeKwaadsteniet,1990).

(iv) Thefull stratificationusedfor thebasemapsrepresentsthemostsafeapproachcomparedtothesimplerapproaches,becauseall simplermodelsareaspecialcaseof thefully stratifiedmodel.Onlywhenonehasgoodreasontobelievethatfor somegroundwaterqualityvariablestratificationshouldhave beenomitted(partially),onemaypreferthecorrespondingmapresultingfrom no(or partial)stratification.

3.7 The effectsof monitoring network density

The effectsof the densityof the monitoringnetwork on the groundwaterquality mapswerestudiedby comparinggroundwater quality mapsbasedon the limited information from theobservationsof theNGM with themapsbasedon thefull setof observations(NGM + PGMs).This was done for nine groundwater quality variables(Table A4, page97). The resultsfor

THE EFFECTSOFMONITORINGNETWORK DENSITY 33

level: 50 mg/m3


level: 25 mg/m3

not distinguishablehigher

level: 100 mg/m3


level: 200 mg/m3


Figure 3.6 Theeffectsof no stratificationfor aluminium.95%Confidenceintervalsfor 4 km× 4 km block medianvalueswhensoil and land useinformation is ignored,relatedto fourconcentrationlevels.Onlyresultsthat differ fromFig.3.4areshown


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Figure3.7 Variogramof log(Al)e ,whenstratificationis omitted.Codesusedfor thevariogrammodel(—)areexplainedin TableA3,page97

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Figure 3.8 Variogramsof log(Al)e , stratified by land useonly. A numberreflectsthenumberof observationpairsusedfor anestimate(+) of a variogrampoint,codesusedfor a variogrammodel(—)areexplainedin TableA3,page97

aluminiumarepresentedin Figs.3.10and3.11.Changescanbeassignedto(i) lossof informationaboutthemeanlevelpersoil-landusecategory,and/orlocationspecificinformation(lossof firstorder information)or (ii) lossof spatialdependence(variogram)information(lossof secondorder information).Changesthat result from eitheroneof thesetwo specificcomponentsareshown in Figs.3.13and3.12respectively.Theresultsaresummarizedin Table3.3.

It is evident that the extensionof the NGM with the PGMs hasresultedin importantimprovementsof thegroundwaterqualitymaps(Fig.3.11, ‘total’ columnin Table3.3).Foralarge


level: 50 mg/m3


level: 25 mg/m3


level: 100 mg/m3


level: 200 mg/m3


Figure 3.9 The effects of stratification by land use only for aluminium.95% Confidenceintervalsfor 4 km× 4 kmblock medianvalueswhensoil informationis ignored,relatedto fourconcentrationlevels.Onlyresultsthat differ fromFig.3.4areshown


Table 3.2 Theeffectsof stratificationfor aluminium.Numberof cells for which thevalueofa 95%confidenceinterval for a 4 km× 4 kmblock medianconcentrationrelatedto a referencelevel (possiblevalues:higher, lower and not distinguishable)changed asa resultof omittingstratificationby (a) land use, (b) soil type,and (c) both(total numberof cells:8197).Changedcellsfor columns(b) and(c) areshownin Figs.3.9and3.6 respectively

level numberof changedcellsasa resultof omitting

3(mg/m) landuse soil type both

25 1276 1200 2227

50 975 1030 1668

100 959 1992 2314

200 825 1444 1898

Table3.3 Theeffectsof monitoringnetworkdensityfor aluminium.Numberof cellsfor whichthevalueof a 95%confidenceinterval for the4 km× 4 kmblock medianconcentrationrelatedtoareferencelevel(possiblevalues:higher,lowerandnotdistinguishable)changedasaresultofomittingtheinformationfromtheprovincial groundwaterqualitymonitoringnetworks(PGMs)(total numberof cells:8197).Changesarea resultof (a) lossof informationaboutvariograms(secondorder information),and/or (b) lossof informationaboutthemeanlevel and locationspecificinformation(first order information).Changesfromeach oneof thesecomponentsarealsoshown.Changedcellsareshownin Figs.3.11(total),3.12(a) and3.13(b)

level numberof changedcells

3(mg/m ) total asa resultof (a) asa resultof (b)

25 2765 2275 598

50 1649 1981 599

100 2643 2558 665

200 1828 1681 496

parttheseimprovementsresultfromtheincreaseof spatialdependenceinformation(Fig.3.12andcolumn(a)in Table3.3),especiallyfrom informationabouttheshortdistancevariation(compareFig. 3.3with Fig. 3.10). The effectsof increasedfirst order informationalsoled to importantimprovementsof thegroundwaterqualitymaps(Fig.3.13andlastcolumnof Table3.3).


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Figure3.10 Variogramsof log(Al)e ,stratifiedbysoil typeandland use,usingonly informationfrom the national groundwaterquality monitoringnetwork.A numberreflectsthe numberofobservationpairs usedfor an estimate(+) of a variogrampoint,codesusedfor a variogrammodel(—)areexplainedin TableA3,page97


level: 50 mg/m3


level: 25 mg/m3


level: 100 mg/m3


level: 200 mg/m3


Figure3.11 Theeffectsof networkdensityfor aluminium(total).95%Confidenceintervalsfor4 km× 4kmblock medianvaluesbasedonmeasurementsfromthenationalgroundwaterqualitymonitoringnetworkonly(provincial monitoringnetworkinformationis left out),relatedto fourconcentrationlevels.Onlyresultsthat differ fromFig.3.4areshown


level: 50 mg/m3


level: 25 mg/m3


level: 100 mg/m3


level: 200 mg/m3


Figure 3.12 Theeffectsof networkdensityfor aluminium(secondorder information).95%Confidenceintervalsfor 4 km× 4 kmblock medianvaluesbasedon (i) first order informationfrombothmonitoringnetworks,and(ii) secondorderinformationfromthenationalgroundwaterqualitymonitoringnetworkonly,relatedtofour concentrationlevels.Onlyresultsthatdiffer fromFig.3.4areshown


level: 50 mg/m3


level: 25 mg/m3


level: 100 mg/m3


level: 200 mg/m3


Figure3.13 Theeffectsof networkdensity(first order information).95%Confidenceintervalsfor 4 km × 4 km block medianvaluesbasedon (i) first order informationfrom the nationalgroundwaterquality monitoring networkonly, and (ii) secondorder information from bothnetworks,relatedto four concentrationlevels.Onlyresultsthat differ fromFig.3.4areshown


3.8 Discussion

In this study, estimatesof 4 km × 4 km block medianvalueswerepresentedasapproximate95% confidenceintervals:a compactway of combiningthe presentationof expectedvalues(pointestimates)anduncertainty(estimationvariances)thatpreventsmisinterpretationof pointestimates.If thereisanareawhereconfidenceintervalsarewideandwhereblockmedianvaluescannotbedistinguishedfrom animportantcritical level,onecoulddecideto placeanadditionalmonitoringsitein or closeto thatarea.Ontheotherhand,wheresuchadistinctioncaneasilybemade,onemaydecidethat fewer measurementswould besufficient.For considerationsof thiskind,thecritical levelsshouldapplytoblockmedianvaluesandnot to individualmeasurements.In thisway,monitoringnetwork optimizationcanbedirectedefficientlyandlocationspecificallyto relevant aspectsof groundwater quality maps.In generalthe mapsgive little reasonforreducingthemonitoringnetwork density(they havewideconfidenceintervals).

From a policy point of view the maps can be used directly either for quantifyingdiffusegroundwatercontamination(becausethe measurementsdo not representstrictly localcontaminations),or for obtaininglocation-specificbackgroundconcentrationse.g.for assistinglocal contaminationassessment.More indirectly themapscanbeusedasan input for andforvalidationof policy supportingregionalor nationalgroundwaterqualitymodels.Themapscanbeconsideredasa translationof theinformationat themeasurementsizeinto spatialaveragesthesizeof spatialunits,asusedin regionalgroundwatermodels.

The spatial resolutionof the groundwater quality maps(half overlapping4 km × 4km blocks)canbe seenasa maximum,given the availablemonitoringnetwork densityandthe observed variation in groundwater quality. For purposeswhere the propertiesof largerspatialunitsaresufficient,usingthecurrentapproachbut with simply largerblockswould beunsatisfactory:becauseof thespatialvariationof soil typeandlanduse,themeaningof blockdominantsoil-landuseasa basisfor stratificationwould loseits relevance.Whenestimatesofgroundwaterqualityvariablesareneededatalowerspatialresolution,it maybebettertocombinethe availablemeasurementswithin a soil-landusecategory, treatthemasa (random)sample,and useclassicalstatistics(design-basedinference).This may alsopermit the estimationofhigherquantiles(e.g.estimationof the fractionof individual samplesin a region exceedingacritical level in thetail of thesampledistribution).Providedthatthislatterapproachincludesbothspatialandtemporalaspectsof groundwaterquality, it canbeconsideredasasecond,alternativeapproachto monitoringnetwork evaluation(andoptimization).

Stratificationby land useand soil type precedingspatialinterpolationturnedout to beessential:thenecessityof usingprocessknowledgebecauseof thenon-uniquenessof a strictlygeostatisticalapproach(De Kwaadsteniet,1990)is not just a theoreticalpoint,but it resultsinmapsthat differ substantiallyfrom mapsobtainedwhenno or partial stratificationis applied.Differencescanbeviewedasthetranslationof thewayin which,with respectto thegroundwaterquality variables,stratadiffer in level, in order of magnitudeof variation, and in spatialdependencebetweenmeasurements.For thelasttwo points,seealsoFig.3.3.

Moregenerallyit canbesaidthatthecurrentapproachcloselyfollowsaclassicalstatistical


approach:westartwith astratificationbysoil typeandlanduse.It improvesonapurelyclassicalstatisticalapproachby explicitly taking accountof location specificinformationand spatialdependence,by using stratified kriging. The effects of monitoring network density clearlydemonstratethe addedvalue of using location specificinformation and spatialdependence:especiallytheeffectof shortdistancevariationbut alsotheeffectof local informationresultsinimportantmodificationsof thegroundwaterqualitymaps.

Thestratificationusedfor thisstudyis not ideal.Soil typeinformationwasobtainedfroma nationalsoil survey (Steuret al., 1985)that describesthe top metreof thesoil, whereasthesoil characteristicsfrom soil surfaceto the depthof the monitoringscreenswould have beenmorerelevant.Landuseinformation(Thunnissenetal.,1992)describesthe(estimated)situationin 1986-87,whereaslanduset yearsago(wheret dependson local groundwaterflow) wouldyield more adequateinformation.The sub-optimalinformation availability resultsin largerwithin-categoryvariation(widerconfidenceintervals)thanthevariationthatwouldbeobtainedwith the idealsoil andlanduseinformation.Also errorsoccurringin bothmaps,clusteringofcategoriesandusingdominantsoil-landuseresultin an increaseof within-category variation.Finally an important part of the within-category variation may stem from the absenceofrelevant geohydrologicalattributes,of which no mapsareavailable.Whenmapsof relevantgeohydrologicalattributes(e.g.on infiltration versusseepage)becomeavailable,it might beworthwhileto includethemin thestratification,whichwould resultin fewermeasurementspercategory, but with smallerwithin-categoryvariationmoreaccurateestimatesmaybeobtained.

Mathematicalaspectswere kept simple,primarily becausewe aim for (a) an optimalaccessibilityof the variousmethodologicalitemsmentioned,and (b) the avoidanceof morecomplex assumptionsthannecessary, basedon theobjective of this studyandthe informationavailable.Thesimplicityreferstothemodelused(3.1), thechoiceandfit of variogrammodels,thechoiceof R, thewayapproximateconfidenceintervalsfor blockmedianvalueswerecalculated,andtheoutlierprocedure.Somegeneralremarksaboutthemathematicalaspectsare:

(i) Kriging asappliedin thisstudy(usingmodel(3.1) for eachvariablefor eachsoil-landusecategory) canbe consideredasa first stepbeyond andan improvementof the classicalstatisticalapproach(for eachvariablefor eachcategory) in its four formsbasedon spatialdependenceor independence,andonemeanpercategoryor localmeansof regionswithin acategory, respectively.Thekrigingapproachisanimprovementontheseclassicalstatisticalapproachesbecauselocal informationis usedmoreeffectively.

(ii) The (approximate)95% confidenceintervals presentedin this studyareconsistentwithinterpolationson a log-scale:anobviouschoicegiven thecharacterof themeasurementsat hand.For a first quantificationof uncertainties(thecalculationof confidenceintervalsusing(3.2)) theestimationerrorof blockmeanvaluesonlog-scaleisassumedtobenormallydistributed,and uncertaintyof the (kriging) standarddeviation is ignored.This impliesthatonly aroughindicationof confidenceis obtained,andthatthewidth of theconfidenceintervalsshouldbeconsideredasthe lower boundto morecarefullyobtainedconfidenceintervals.

(iii) In thecurrentcontext (blockkriging ona log-scale)thecentralmeasurethatis estimatedis

DISCUSSION 43

theblockmedian,not theblockmean.Whentheblockmeanis needed(e.g.for comparingthemapswith groundwaterquality modelresults)first indicationsof block meanscanbeobtainedfrom themapsof blockmediansby usinganapproximaterelationbetweenblockmeanm andblockmedianM , givenby

m �� M exp(s2/ 2), (3.3)

wheres2 is the short distancevariance,the short distancevalue of the variogramfortherelevantvariableandcategory. Relation(3.3) is derived from thewell known relationbetweenmeanandmedianof a lognormaldistribution(AitchisonandBrown,1957).Firstindicationsof 95%confidenceintervalsfor 4 km × 4 km block meanvaluesareobtainedwhenthelegendclassboundariesof amaparemultipliedwith thefactorof M from (3.3).For a groundwaterqualityvariable,thiswill resultin differentlegendclassboundariesfordifferentsoil-landusecategories.

(from:PebesmaandDeKwaadsteniet,1995)

4.1 Intr oduction

Theprimarygoalof thenationalgroundwaterqualitymonitoringnetwork is to provide insightin thecurrentsituationandsystematicchangesin timeof groundwaterquality.For thispurpose,about370monitoringsites,spreadfairlyevenlyoverthecountry,aresampledyearlyattwodepths(from well screensat about10 and25metrebelow surface).At first sampleswereanalysedfor19,andlater25,groundwaterqualityvariables.

Thecurrentstudyaimsatmappinggroundwaterqualityvariables,usingpointmeasurementsobtainedfrom the monitoring networks, in such a way that uncertaintiesin estimatesarequantifiedstatistically. In particular, the attentionis focusedon the descriptionof changesingroundwaterquality in time.Mappinggroundwaterqualityvariablesatameasurementmoment(in oneyear, 1991)wasdescribedin Chapter3.

Becausebothstartinglevel andtime gradientaresignificantto theassessmentof changesin groundwaterquality, we choseto examinethis subjectin termsof short-termpredictions.Using a simple regressionmodel,extrapolationfrom observed time seriesyields short-termpredictions(‘expectedobservations’)for themonitoringscreens,whichcanbeusedin thespatialinterpolation.Time-predictionuncertainties,expressedas estimationvariancesof ‘expectedobservations’canbeincorporatedin thespatialinterpolation.

Basically, theprocedurefor mapping‘expectedobservations’issimilarto theprocedureformappinggroundwaterqualityatameasurementmoment(Chapter3).Asbefore,mapsof soil type(Steuretal.,1985)andlanduse(Thunnissenetal.,1992)wereaggregatedtoasetof soil-landusecategories.For eachcategory, for everygroundwaterqualityvariable,andfor eachextrapolationtimeattentionwaspaidto thetime-predictionuncertaintyof, thelevelof, thevariationin andthespatialdependencein ‘expectedobservations’byusingavariantof ordinarykriging(JournelandHuijbrechts,1978)thatwasmodifiedto accountfor non-constanttime predictionuncertainties(Delhomme,1978).For eachextrapolationtime,estimatesof 4 km × 4 km blockmedianvalueswerepresentedonmapsasapproximate95%confidenceintervals.

It wasnot possibleto createmapsof changesin groundwaterquality for all variablesor to

44

INTRODUCTION 45

useall measurementsthatwereusedfor themapsof 1991(PebesmaandDeKwaadsteniet,1994,TableA1,Fig.3.2) because(i) many measurementsiteshavebeeninstalledonly recently,and(ii)for theremainingsitesthemonitoringof severalgroundwaterqualityattributeswasstoppedorstartedonly recently. Therefore,thisstudyconcernsonly 12 groundwaterquality variablesandisbasedonamuchsmallersetof observations:themeasurementsof theprovincialgroundwaterqualitymonitoringnetworkshadtobeleft outof thisstudy.Full resultsarepresentedin aseparatereport(PebesmaandDeKwaadsteniet,1995).In thischapterweillustratethemethodsusedwithresultsononegroundwaterqualityvariable,namelypotassiumconcentration.

The mapsof changesin groundwater quality can be usedfor (i) assessmentof diffusegroundwater contaminations(becausethe measurementsdo not representstrictly localcontaminations)(ii) thevalidationof policy-supportingregionalor nationalgroundwaterqualitymodels.

Sections4.2, 4.3, 4.4 and4.5describethestratification,thedataselection,thechangesintimeatthewell screensandthespatialinterpolation.Theresultsarediscussedin Section4.6andthischapterconcludeswith adiscussionin Section4.7.

4.2 An aggregatedmap comprising soil type and land use

Thesoil-landusemapthatwasusedfor thestratificationandclassificationof themeasurementsis identicalto themapdescribedin Section3.2, shown in Fig.3.1(page26). Table3.1presentsbasicdataon thecategories.

4.3 Data selection

For thegenerationof mapsof changesin groundwaterqualityseveralobservationswereleft outbecausethey seemedunsuitablefor thispurpose.As in Section3.3, monitoringsitesin ‘built-upareas’wereleft out,aswell asmonitoringsitesthatwereinfluenceddirectlyby waterfrom riversor thesea.

Becausethecurrentstudyconcentratesontherecognitionof meaningful,steady, long-termcomponentsof the changesin time, the selectiononly comprisestwelve groundwaterqualityvariablesfor which long-termserieswere available (TableA5, page97). As in the previouschapter, only monitoring screensbetween5 and 17 metre below the ground surface wereconsidered.

Visualexaminationof thetimeseriesplotsfor eachvariableandfor eachmonitoringscreenrevealedthatfor many variablesthefirstyearsof monitoring(’85and’86)didnotfit in thegeneralpatternof the seriesfor a substantialfraction of screens,leadingto doubtsaboutthe qualityof thesemeasurements(thequalityof sampling,sampletreatmentor analysis).We chosefor auniform treatment,andfor all variablesandmonitoringscreensonly measurementsfrom 1987andlaterwereused.Usually, seriesupto 1992or 1993wereavailable.Examinationof thetimeseriesplotsfurthermorerevealedthatfor somesamplesaconsiderablepart(morethan3) of thevariablesdid notfit in theseries,whichwasinterpretedasanindicationof erroneoussamplingorsamplingtreatment.Thesesampleswereconsideredasoutliers,andthey wereleft outaltogether.

46 TEMPORALCHANGESIN GROUNDWATERQUALITY

(Threesamplesweremarkedasoutlier.)Detectionlimit valueswereassignedto measurementsbelow thedetectionlimit, for each

variableusingthehighestdetectionlimit whena changein detectionlimit hadoccuredduringthe measurementperiod considered.In exceptionalcases,when only a tiny fraction of themeasurementsweredeterminedwith aconsiderablyhigherdetectionlimit, measurementsbelowsuchahighdetectionlimit weretreatedasmissingvalues.

The measurementswerefinally subjectedto a global quality control, in which, for eachvariableand monitoringscreen,measurementsthat differed more than a factor 10 from theothermeasurementsof the time serieswereconsideredas‘doubtful’ andwereleft out of theselection.Finally,afterapplyingall previouslydescribedselections,only themeasurementsfrommonitoringscreenswith atleast5measurementsin theperiodconsideredwereusedfor mappingchangesin groundwaterquality.

The locationsof the 250 monitoringscreensusedfor mappingchangesin groundwaterquality areshown in Fig. 4.1. The numberof measurementsin eachsoil-landusecategory isshown in thethird columnof Table3.1(exceptfor thecategory‘grasslandonpeat,’ whichhas22measurementsdueto theadditionalselectioncriteriadescribedabove).

4.4 Changesin groundwater quality variablesat the well screens

Becausestartinglevel andtime gradientareboth significantto the assessmentof changesingroundwater quality, we choseto examinethe subjectof changesin groundwater quality intermsof short-termpredictions.In particular,weareinterestedin theassessmentof meaningful,steady(i.e.monotonous),long-termcomponentsin thechangein time of groundwaterqualityvariables.

For eachgroundwaterquality variableandfor eachseparatemonitoringwell screen,thevariationof aconcentrationvariableonthelog-scalein timey(t) is representedbyalinearmodelwith time,having independent,identicallydistributederrors

Y(t) ��' 0 + ' 1t + e(t), E(e(t)) �� 0, Cov(e(t)) �� 2 I , (4.1)

which is a specialcaseof thelinearmodel(2.1), writtenasY(t) �� Ft� + e(t) whenf (ti) �� (1, ti), i�� 1…n, is thei-th row of Ft,and� �� (' 0, ' 1)′.Thus,estimateandestimationvariancefor themean

valueof a short-termpredictionat extrapolationtime tnew, given^� (2.2a), using(2.4) and(2.5),

are

^y(tnew) �� f (tnew)^� �� ^' 0 + ^' 1tnew (4.2)

and

2 (tnew) �� f (tnew)(Ft′Ft� 1) f (tnew)′ 2 , (4.3a)

and 2 (tnew) is estimatedby 2s (tnew) when,using(2.3),

CHANGESAT THE WELL SCREENS 47

Figure4.1 Monitoringnetworklocations.Sitesof thenationalgroundwaterqualitymonitoringnetworkthat contributedto themapsof changesin groundwaterquality


2s �� y(t)′(I � Ft(Ft′Ft� 1) Ft′)y(t)/ (n � 2) (4.3b)

issubstitutedfor 2 in (4.3a).Usingmodel(4.1) impliesthatonalog-scaleweassumeaconstantincreaseor decreaseof concentrationin time:afirst approachfor a trendin whichthedeviationfrom a straight line is consideredas ‘white noise’. Using the log-transformationnot onlyguaranteespredictionsthat arestrictly positive on the original scale,it alsoyieldspredictionsat themonitoringnetwork siteson themostobviousscalefor spatialinterpolation(Section2.7,Chapter3).

For creatingmapsof changesin groundwaterquality theshort-termpredictionsfrom (4.2)weretakenas‘new observations’atthemonitoringnetwork sites.These‘new observations’wereinterpolated,andlocationspecificobservationaccuracy wasderivedfrom(4.3a) and(4.3b).Morespecifically, for twomoments(i.e.timeperiods)atequaltimesfrom thecentreof theobservationtimeseries1987-1993(at t �� 1980.5andt �� 2000.5), asetof ‘new observations’wascomputed.For two‘new observations’ataspecificmonitoringsite,thisequitemporalchoiceof extrapolationtimesresults—onthe log-scale,and by absenceof a trendon the original scaleaswell—inequalobservationaccuraciesfor bothextrapolationmoments(Fig.4.2). Theaimof choosinganextrapolationspanof 1980-2000is to emphasizetrendsthatarevisibledespitetheobservationinaccuracies.

Quantifyingtheobservationaccuracy with (4.3a) impliesthat the‘new observations’onlyrelateto the situationaround 1980 and around 2000:for ‘new observations’ the short-termcomponentsof temporalvariation,theyear-to-yeardeviationfromtheregressionline,arefilteredout,within thecontext of thecurrentmodelstructure.

For a monitoring well with two screensin the depth interval considered,each ‘newobservation’ used for the spatial interpolation of a groundwater quality variable for anextrapolationmomentwascalculatedasasimpleaverageof both‘new observations’^y1(tnew) and^y2(tnew)

^y(tnew) �� (^y1(tnew) + ^y2(tnew))/2.

Theestimatedobservationaccuracy for suchamonitoringwell wascalculatedas

2s (tnew) �� (s21(tnew) + s2

2(tnew))/4,

wheres21(tnew) ands2

2(tnew) werederivedfrom (4.3a) and(4.3b).

4.5 Spatial interpolation

Mapsof short-termpredictionsfor groundwaterquality variableswereobtainedby meansofordinaryblock-krigingof log-transformeddata(JournelandHuijbregts,1978)within eachofthecategoriesin Table3.1,usingtheshort-termpredictionsobtainedfor everywell in acategory:the‘new observations’obtainedfrom (4.2) at t �� 1980.5andt �� 2000.5.

For the spatial interpolationa model conceptwas adoptedwhich basicallyconsideredobservationsof amappingvariablewithin anareaR insideasoil-landusecategoryasasample


1980( 1985( 1990) 1995) 2000

time

log

-co

nce

ntr

atio

n

*

domain of measurements+regression line

‘new observation’ for 1980

‘new observation’ for 2000

Figure4.2 Schematicrepresentationof ‘new observations’andtheir accuracy(aserror bars)at a singlemonitoringwell screen(noactualmeasurementsdrawn)

from arealizationof therandomfield Z(x) with thefollowing properties:

Z(x) �� m+ e(x) (4.4)

with

mconstant, (4.4a)

E(e(x)) �� 0,�

x � R, (4.4b)

and

1⁄2E[(Z(x1) � Z(x22)) ] �� ( � x1 � x2 � ), � x1,x2 � R. (4.4c)

Thatis,within anareaof sizeR insideasoil-landusecategoryweusedanadditivemodel(onlog-scale)with (i) a constantexpectedvalueand(ii) spatialdependencebetweenmeasurementsthat is a function of the distanceh betweenthe measurementlocationsonly. Additionally weassumedthat in a soil-landusecategory for a groundwaterquality variableonly onefunction� (h) is needed(independentof R), andthat this function (the ‘variogram’)is identicalto thevariogramthat wasobtainedfrom the measurementsof 1991(Section3.4, compareFig. 3.3),for thecorrespondingsoil-landusecategory. Fig.4.3shows thesevariogramsfor potassium,asobtainedfrom the1991measurements(PebesmaandDeKwaadsteniet,1994).

The areasizeR wasdefined(in order of preference)as(i) the closest20 observations


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Figure4.3 Variogramsof log(K)e , stratifiedby soil typeand land use,obtainedfromthe1991measurements(nationalandprovincial monitoringnetworks).A numberreflectsthenumberofobservationpairs usedfor an estimate(+) of a variogrampoint,codesusedfor a variogrammodel(—)areexplainedin TableA3,page97


whenmorethan20 observationswereavailablewithin a distanceof 100km, (ii) 100km when10 to 20 observationswereavailablewithin thisdistance,(iii) the10 nearestobservations.Theinterpolationprocedurethat wasusedherediffersfrom standardblock-krigingbasedon (4.4)(JournelandHuijbregts,1978)in that locationspecificuncertaintyfor each‘new observation’obtainedwith (4.3a) and(4.3b) wastaken into account,asdescribedin Section2.5(2.12), andDelhomme(1978).

The estimatesconcernmeanvalueson log-scaleof 4 km × 4 km blocks.Estimatesarepresentedasapproximate95%confidenceintervalscalculatedby

[^z(x0) � 2 k(x0) , ^z(x0) + 2 k(x0)] (4.5)

with ^z(x0) the point estimate(kriging estimate)on the log-scale,and k(x0) the correspondingestimationstandarddeviation (kriging standarddeviation), using the modeldescribedabove(Delhomme,1978,JournelandHuijbregts,1978).Back transformationof this interval to theoriginalscale(takingtheexponentof bothsides)yieldsanapproximate95%confidenceintervalfor theblockmedianvalue.

Thedisplayof valuesfor 4 km × 4 km blockson the2 km × 2 km block centres,andthetreatmentof smallsoil-landusecategorieswith lessthan10 observationsis identicalto thatofSection3.4.

4.6 Results

In the mapsof changesin groundwater quality, the 95% confidenceintervals for 4 km × 4km block medianvaluesare relatedto four referencelevels, as obtainedfrom multiplesoftarget levelsor critical levels for humanconsumption(thesetarget levelsor critical levelsaredefinedin relationto individual samples).For eachreferencelevel a sub-mapshowswheretheconfidenceintervalsarecompletelyabove or below the level, or wherethey straddlethe level.In thelattercase,blockmedianvaluesandreferencelevel cannotbedistinguished,basedontheinformationavailable.Thedoublingof the referencelevels in themapsreflectsanequidistant(linear)choiceof levelson thelog-scale.For eachvariablechangesin groundwaterqualityarerevealedby showing mapsfor 1980and2000sideby side,andby only showing their mutualdifferences—wherethemapsareidentical,novaluesareshown.Figs.4.4and4.5show themapsof changesin potassiumconcentration.

Themapsreflectthedevelopmentof groundwaterquality over a 20 yearspan,within thecontext of thecurrentmodelstructure.Thedifferencesshown arisedirectly from theslopeofall or partof theregressionlines,asfitted to themeasuredtimeseries:if theslopesarezerothemapsareidentical.

Theindicatedtrenddirectionin a cell canbederivedfrom thedirectionof shift in legendclassfrom oneextrapolationtime to the other. In general,trendmagnitudecannotbe derivedfrom thisshift, for instance,in caseof a singlelegendclassshift. It canbestatedthatall trendindicationsshown areof vital importancein thesensethat they concerna shift in estimatesof4 km × 4 km block medianconcentrationthat resultsin a differentmapfor at leastoneof thereferencelevels.


level: 10 g/m3


level: 5 g/m3


level: 20 g/m3


level: 40 g/m3


Figure 4.4 Map of potassiumin thegroundwateraround1980,where it differs fromFig. 4.5(‘around2000’).95%Confidenceinterval for 4 km× 4 kmblock medianvalues,relatedto fourconcentrationlevels

RESULTS 53

level: 10 g/m3


level: 5 g/m3


level: 20 g/m3


level: 40 g/m3


Figure 4.5 Map of potassiumin thegroundwateraround2000,where it differs fromFig. 4.4(‘around1980’).95%Confidenceinterval for 4 km× 4 kmblock medianvalues,relatedto fourconcentrationlevels


The maps(Figs. 4.4 and 4.5; Pebesmaand De Kwaadsteniet,1995) show a wealth oftrend indications.Primarily we can concludethat a trend approachwith sucha high spatialdifferentiationis attainableandadvisable.Advisablebecause(i) cellswith a pronouncedtrendcanbedistinguishedfrom a neighbourhoodwith lesspronouncedtrends,(ii) trendsin adjacentor nearbycells(alsowithin thesamesoil-landusecategory)canhaveoppositedirections.

4.7 Discussion

The changesin predictedgroundwater quality as presentedon the mapsarise from (i) themeasurementseriesfrom 250monitoringsitesand(ii) themodelstructureused.Somefurthercommentsbelonghere.

For thedescriptionof temporalchangein a groundwaterquality variablewe useda firstorderlinear regressionmodelfor measurementson the log-scalewith time,assumingthat thedeviationfrom aregressionline couldbeconsideredas‘white noise’.

Usingthelog-scaleastheworkingscaleyields(i) temporalpredictionsonthemostobviousscalefor spatialinterpolation(Chapter3), (ii) a uniformly quantifiedlocationspecificmeasureof accuracy for thepredictionat themonitoringsites,directlyon thespatialinterpolationscaleand(iii) strictly positivepredictionson theoriginalscale.

Initially, the intuitive and, except for large factorial trends,most obvious and simpleapproachfor modellingsteadytrendswould beto usea linearmodelwith time on thenatural(i.e. measurement)scale.However, when we consider10-year-extrapolations,this approachdifferssignificantlyfrom modellinglineartrendson thelog-scaleonly whenfactorialtrendsofat leasta factor2 occurandwhenpredictioninaccuraciesareignored.For largefactorialtrendsthelog-scaleis preferredon intuitivegrounds,for smallfactorialtrendsthedifferencesbetweenbothapproachesfadewhenpredictioninaccuraciesaretakenintoconsideration.Thequantitativeincorporationof locationspecificpredictionaccuraciesisanessentialaspectof thecurrentstudy.As theworkingscale,thelog-scalepermitsthis.

Themapsarebasedona conceptwherethevariablewewantto map—thespatialvariablez(x) at an extrapolationtime at an observation location xi—is not observed,but whereonlyapproximateobservations^y(xi) areavailable,whicharesubjectto anunknown observationerror.Wecanmodelsuchapproximateobservationsasthesumof astationaryvariableZ(xi) (4.4) thatrepresentsthevariableof interestz(xi), andameasurementerror � (xi)

Y(xi) �� Z(xi) + � (xi), (4.6)

andfor thisobservationerror � (xi) weassumethatit hasnosystematiccomponent

E(� (xi)) �� 0,�

i, (4.7)

andthatit hasa locationspecificvariance 2� (xi)

Var( � (xi)) �� 2� (xi),�

i. (4.8)

DISCUSSION 55

In thisstudy,weassumethat 2� (xi) isknown,asit isderivedfrom (4.3a) and(4.3b).Furthermore,weassumethattheobservationerrorsat two differentlocationsarenotcorrelated

Cov( � (xi), � (xj)) �� 0,�

i �� j, (4.9)

andthattheobservationerrorsarenotcorrelatedwith thespatialvariableZ(x)

Cov( � (xi),Z(x)) �� 0,�

i,�

x. (4.10)

Thisimpliesthatbeyondthestandardkrigingassumptionsthemapsarebasedontheassumptionthat at a monitoringsite we do not know the exact valueof z(xi), but that (for convenience’ssake,in caseof a normallydistributedobservationerror)independentlyof everythingelse95%confidenceintervalsfor z(xi) aregivenby

[^y(xi) � 2 � (xi) , ^y(xi) + 2 � (xi)],�

i.

The adaptionof the standardkriging procedureto handleobservationsthat aresubjectto anobservationerror that satisfies(4.6)-(4.10) wasgiven in Section2.5andin Delhomme(1978).Comparedto the situationwhere the observation error would have beenignored,applyingthisadaptionrepresentsa morecarefulapproachthatresultsin wider confidenceintervals,andanapproachin which lessweight is assignedto observationswith relatively largeobservationerrors.

Thestatisticalindicationof steadytimetrendsin blockmedianvalues(steadytrendsin firstorderstatistics)is theprimarygoalof thisstudy. Fromamethodologicalpointof view, in a firsttrendstudy(like thisone)it is a correctfirst approachto work with time-invariantsecondorderstatistics(spatialvariation,spatialdependencein variation,thevariogram)within soil-landusecategories.Thedecisionfor thatisbasedonthefollowingarguments.(i) Statisticaldistributionalpropertiesof variogramsarecomplex, anda statisticalcomparisonof variogramsis a subject,thecomplexity of which is beyondthecontext of thisstudy. (ii) Usingadifferentvariogramforeachextrapolationtimewouldcompriseamodellingdetailthatwasnotstatisticallyshown to benecessary(andthatprobablycannotbeshown tobenecessary,giventhenumberof observationspersoil-landusecategory, Table3.1), andwouldrepresentamodellingapproachin whichmapsnomoreexclusivelyreflecttheprimarygoal:differencesin mapswouldin thatcasereflectsteadytimetrendsaswell asthenewly introducedtime-differencesin secondorderstatistics.

For the variogramsof Z(x) we used the variogramsof the 1991 measurementsthatwereobtainedfor the mapsof the groundwaterquality in 1991(Chapter3, PebesmaandDeKwaadsteniet,1994).The primary reasonfor this is that for studyingspatialvariation of agroundwaterquality variablewithin a soil-landusecategory, in the previous studywe hadamuchlargerobservationset.Apartfromthemeasurementsfromthenationalgroundwaterqualitymonitoringnetworkwecoulduseabout170measurementsfromprovincialmonitoringnetworks.Distinguishingthe1991-situationandthe‘around1991-situation’with respectto thevariogramspecificationwas—withall othersimplificationsin mind—consideredto beoverdone.

In this study, estimatedsecondorderstatistics(variogramsand‘observationaccuracies’)are taken asdeterministicquantities.This aspect,togetherwith the contentsand foundation


of themodelstructureimpliesthat themapsonly yield a first stepin thedirectionof spatiallydifferentiatedstatisticaltrends.Therefore,for theinterpretationof themapscautionis required:only first statistical indicationsof meaningfulsteadytrendsin block medianvalueswereobtained.

Taking the estimated‘observation accuracy’ asa deterministicquantitywhen,asin thisstudy, a ‘new observation’ (a short-termprediction)in a monitoringwell screenis basedon atimeseriesof 5to7measurements,impliesthatin thisfirst approachfor thedeterminationof the‘observationaccuracy’ theestimatedvariance 2s (tnew) (3),basedon3degreesof freedomwhen5observationsareavailable,wasnotdistinguishedfrom thetruevarianceof (4.3a). Thisresultsinaseriousunderestimationof the‘observationaccuracy,’ andunderlinesthecautionthatshouldbetakenwith respectto theinterpretationof themaps.Thereasonwhy weused‘new observations’basedon time seriesof 5 measurements,despitetheshortcomingsof themethodsused,is theobservationfrom practicethattheinfluenceof suchseriescannotbeignored.

More generally, it canbestatedthat in trendstudiestheaccuraciesof trendobservationsdeserve explicit, quantifiedattention.In orderto be ableto presentwell-foundedestimatesofchangesin groundwaterquality over a reasonabletime scale,the—now yearly—measurementfrequency shouldnotbereduced.

Tobecomplete,it shouldbenoticedthatthemapsdonotpretendtoshow theactualchangesthatoccuroverthe20yeartimespanconsidered.Thechangesshownonlyconcernextrapolationsfrom observedchangesin themeasurementseries1987-1993,within thecontext of thecurrentmodelstructure.

Seriousmodificationsof themapsin this studymaybeexpectedwhen,aftera few years,measurementseriesof theprovincialgroundwaterqualitymonitoringnetworksbecomeavailableandwhentheseseriescanbeincorporatedin thestudy.

Finally, the currentstudy shouldbe consideredas a supplementto the previous study(Chapter3,PebesmaandDeKwaadsteniet,1994).Themajorityof theitemsdiscussedin Section3.8applyto thisstudyaswell.

5.1 Intr oduction

Thegoalof bothmonitoringandmodellinggroundwaterquality is to gainunderstandingof thespatialandtemporalvariationin groundwaterquality. Thisvariationis understoodbetterwhenmoremeasurementsareavailableor with bettermodels,i.e. modelsthat explain the variationin theavailablemeasurementsbetter. Consequently, theoptimizationof a groundwaterqualitymonitoringnetwork canbeaimedat thenumberandfrequency of measurementsaswell asata betterexplanationof thevariationin measurementswith models.Thischaptershowshow wecanusedifferenttypesof relevant,ancillaryinformationin models,to increasetheaccuracy ofmapsof groundwaterquality.

Chapter3describesasimpleapproachfor mappinggroundwaterqualityin theNetherlands.Maps of groundwater quality variableswere obtained by within soil-land use categoryinterpolationof measurementsobtainedfrom the groundwater quality monitoring networks(Van Duijvenboodenet al., 1985), taking the spatial dependencebetweenmeasurementswithin categoriesinto account(stratifiedordinaryblock kriging,JournelandHuijbregts,1978).Estimatesof 4 km × 4 km block meanvalues(on log-scale,yieldingestimatesof block medianvalueson theoriginal scale)werepresentedasapproximate95%confidenceintervalson mapshaving a2 km × 2 km grid.

In generalthe resultingmapsturnedout to be rather inaccuratewith wide confidenceintervals.If available,moregroundwaterquality measurementsor betterstratificationcriteriamight beusedto increasetheaccuracy of maps,by usingthemodelof thesimplestratificationapproach.Otherpotentiallyrelevant informationtypically consistingof eithermeasurementsonother(related)variablesor resultsfrom deterministicprocessmodelsareavailable,but cannotbehandledby this model.This chapterwill show how this other, ancillary informationcanbeincorporatedin themappingprocedureto improvetheaccuracy of resultinggroundwaterqualitymaps.For threegroundwaterqualityvariables(nitrate,zincandsulphateconcentration)thesizeof theimprovementswill beshown by comparingthemapsthatuseancillaryinformationwiththemapsfrom thesimplestratificationapproach.

Thenatureof theancillary informationusedin theexamplesis describedin Section5.2.The simpleapproachof Chapter3 (stratification–dataselection–interpolation)is the point ofdeparturefor the currentstudy, andSection5.3 describeshow the interpolationprocedureisadaptedto usetheancillaryinformationwithin a soil-landusecategory. Two examplesof usingmap information obtainedfrom modelsare presented:one exampleusesnitrogen leaching

57

58 IMPROVING ESTIMATESWITH ANCILLARY INFORMATION

rate from soils in the agriculturalarea(seeFig 3.1, page26) to aid the estimationof nitrateconcentrationin deepergroundwater(Section5.4), theotherusesatmosphericdepositionof zincfor estimatingzincconcentrationin groundwaterin areaswith semi-naturalvegetation(Section5.5).Theexampleof usingothermeasuredvariablesis theapplicationof sulphateconcentrationmeasuredin shallow groundwater in areaswith semi-naturalvegetationfor the estimationofsulphateconcentrationin deepergroundwaterin thatarea,andthisexampleis givenin Section5.6. Section5.7 summarizesthe sizeof the effectsof ancillary information,and section5.8concludeswith adiscussion.

5.2 Ancillary information

In groundwater quality studiesour major concernis groundwater pollution, and in thischapterwe limit thestudyareato theareaof sandysoilsin theNetherlands(Fig.3.1, page26)that arenot influencedby strictly local pollution. In theseareasgroundwaterpollution occursbecausepollutantsappliedto the soil surfacecandissolve and infiltrate to a depthof about10m within a few decades(Meinardi,1994).Furthermore,theseareasallow thebestconnectionbetweengroundwaterqualitymeasurementsat10m depthandsoil surfacecharacteristicsat themeasuredlocation,becausethehorizontaldisplacementis generallysmall (which is necessarybecauseinformation on the actualgroundwater flow patternsis lacking). The main factorsinfluencinggroundwaterqualitythatareavailableasmaps(andthereforeusefulfor themappingof groundwaterquality)aresoil type(Steuret al.,1985)andlanduse(Thunnissenet al.,1992):they wereusedfor thestratificationin thesimpleapproachdescribedin Chapter3.

Variationsbetweenmeasurementsof groundwaterqualityasobservedwithin soil-landusecategoriescanbeconsideredasstemmingfrom (i) measurementerrorin thegroundwaterqualityvariable,(ii) classificationerrors(measurementerror in thesoil-landuseinformationresultingin wrong assignmentof measurementsto categories),or (iii) true variation in groundwaterquality. Estimatesof groundwaterqualitycanbeimprovedby reducingthefirst two sourcesofvariation,butherewefocusonthethird.Truevariationof groundwaterqualitycannotbereduced(althougha changein the methodof measurementmayreducethe variationobserved),but itcanbemodelled.If informationonaprocessthatcausesthevariationof thegroundwaterqualityvariablewithin a soil-landusecategory is available(e.g.from deterministicprocessmodels),this informationmayprovidea betterexplanationof thevariationin themeasurements,andthesubsequentreductionof theresidual,unexplainedvariationmayyield moreaccurateestimation.Alsoif measurementsareavailablefor other,associatedvariables,thevariationof whichiscausedby thesameprocessesthatcausethevariationof thegivengroundwaterquality variable,theymayhelpto improveits estimation.

Deterministicprocessmodelsdo not necessarilywork with thesamesizespatialunitsasusedin Chapter3. In examplesA andB thespatialunitsare500m and10 km, andthemodelcalculationswerenotobtainedspecificallyfor explainingvariationin groundwaterquality.Whenalargepartof thevariationof groundwaterqualitymeasurementsoccurswithin areasthesizeofthespatialunitsof a deterministicprocessmodel,thenthisvariationcannever beexplainedbyusingthismodelinformation.It mayhoweverbepossibletoapplythesamedeterministicprocessmodelfor smallerspatialunits,andtuningthespatialunit sizeof theseprocessmodelsto thesize

ANCILLARY INFORMATION 59

of groundwaterqualitymeasurements,i.e.downscalingtheprocessmodelsat themeasurementsitesto themeasurementscalemight furtherimprove thevalueof theseprocessmodelsfor theestimationof groundwaterqualityvariables,which is discussedin section5.8.

5.3 Spatial interpolation with ancillary information

In thesimpleapproach(Chapter3)weobtainedmapsof groundwaterqualitybyordinarykriging(JournelandHuijbregts,1978)of thegroundwaterqualityvariablesstratifiedbysoiltypeandlanduse.For this,we adopteda modelwherelog-transformedmeasurementsz(x) within a soil-landusecategorywereconsideredasa samplefrom a realizationof a locally intrinsic randomfieldZ(x) with thefollowing properties:

Z(x) �� m+ e(x) (5.1)

with

mconstant, (5.1a)

E(e(x)) �� 0,�

x � R, (5.1b)

and

1⁄2E[(Z(x) � 2Z(x + h)) ] �� Z( � h � ), � x,x + h � R. (5.1c)

The expectedvalueof the variableZ(x) within a local neighbourhoodR is m, the deviatione(x) from thismeanvaluemight bespatiallycorrelated,andthespatialcorrelationbetweentwomeasurementsisafunctionof theseparationdistanceh only.Additionallyweassumedthatonlyonefunction � Z(h) is neededpersoil-landusecategory. TheneighbourhoodsizeR waschosenasthenearest10to 20measurementswithin adistanceof 100km.

Whentheancillaryinformationf (x) is obtainedfrom adeterministicprocessmodel(andisknown atall locationsx) wecanextend(5.1) to themoregeneralform:

Z(x) �� g(f (x)) + e(x).

Theexpectationof Z, g(f (x)) is now a functionof f (x), theform of which mustbespecified.Iff (x) is a continuousvariableandwe assumea linearrelation,theobviouschoicefor g( � ) is thelinearmodel,extending(5.1) with f (x) to:

Z(x) �� 1 + f (x)� 2 + e(x) (5.2)

with

� 1, � 2 constant, (5.2a)

E(e(x)) �� 0,�

x � R, (5.2b)

and


1⁄2E[(e(x) � 2e(x + h)) ] �� e( � h � ), � x,x + h � R. (5.2c)

In (5.2), � 1 is theconstantparameter(likem in (5.1)), andf (x)� 2 representsa linearcontributionfrom the deterministicprocessmodel to the explanationof the measurements.Model (5.2)expressesthatwithin a local neighbourhoodR theexpectationof Z(x) hastheshapeof f (x). Intheestimationprocedurethis shapewill befitted (locally) to themeasurementsby scalingf (x)by a factor � 2 andshifting this with a value � 1. Whenthescatterplot of theobservationsandf (x) revealsnonlinearities,a non-lineartransformationof the observationsor of f (x) possiblymakes(5.2) moreacceptable(e.g.,in ourexamplesthemodel(5.2) is usedfor measurementsonlog-scale),or, whenjustifiedby themeasurements,amorecomplex modelcanbeadopted.

The variogram � e( � h � ) can be modelled by calculating the sample variogram fromordinaryleastsquaresresiduals^e(x) (TableA6, page97), andfitting a suitablevariogrammodelto this sampleresidualvariogramusing weightedleastsquaresfitting (Cressie,1985).Thisvariogramwill be biased,but somerecentpapers(CressieandZimmerman,1992,Kitanidis,1993,Christensen,1993)arguethat this biaswill not seriouslydisturbestimatesbasedon thisvariogram,andweaccepttheirarguments.

Estimationof block meanvaluesof Z using(5.2) is doneby universalkriging (Section2.3and2.5; Matheron,1971).Prerequisitesfor thisare(i) f (x) isknown atall measurementlocationsand(ii) blockaveragevaluesof f (x) areknown for theestimationlocations.

If ancillaryinformationconsistsof measurementsz2(x) onavariablewhichis relatedtoourprimaryvariableof interestz1(x), we cannotuseit asa fixed,explanatoryvariable(like f (x) in(5.2)) becauseit isknownonlyatthemeasurementlocations.If, apartfrom themeasurements,nootherancillaryinformationisavailablethantheknowledgethatbothvariablesbelongtothesamesoil-landusecategory,wecanadoptamodelsimilarto(5.1) for bothvariablesin thecategoryandallow for inter-variablespatialdependence.Thisdependencemayactuallyimprovetheestimationof z1(x). Thus,weextendmodel(5.1) with asecondaryvariable:

Z1(x) �� m1 + e1(x), Z2(x) �� m2 + e2(x) (5.3)

with

m1,m2 constant, (5.3a)

E(e1(x)) �� 0, E(e2(x)) �� 0,�

x � R, (5.3b)

1⁄2E[(Z1(x) � Z12(x + h)) ] �� 11( � h � ), � x,x + h � R, (5.3c)

1⁄2E[(Z2(x) � Z22(x + h)) ] �� 22( � h � ),

�x,x + h � R, (5.3d)

and

1⁄2E[((Z1(x) � m1) � (Z2(x + h) � m22)) ] �� 12( � h � ), � x,x + h � R. (5.3e)

Thecrossvariogramof (5.3e) isusedbecauseit doesnot requiremeasurementsat identical

SPATIAL INTERPOLATION WITH ANCILLARY INFORMATION 61

locations.Ordinarycokriging is usedfor estimationof the variableof interestZ1(x) (Section2.5, JournelandHuijbregts,1987,Myers,1982).As originally proposedby Clark et al. (1989),Myers(1991)andVer Hoef andCressie(1993)discusstheuseof thecrossvariogram(5.3e) inmultivariableestimation.

Topreventmisinterpretationof pointestimates,estimatedvaluesonthegroundwaterqualitymapsarepresentedasapproximate95%confidenceintervals,calculated(on log-scale)by

[^z(x0) � 2 k(x0) , ^z(x0) + 2 k(x0)] (5.4)

with ^z(x0) the estimatedvalue (block kriging estimate),and k(x0) the estimationstandarddeviation (kriging standarddeviation).Back transformationto theoriginal scaleby takingtheexponentof both sidesyields approximate95%confidenceintervals for 4 km × 4 km blockmedianconcentrations(themedianof all point valuesin a 4 km × 4 km block).Becauseof thedifficulty of showing two values(bothsidesof theconfidenceinterval)atonelocationin amap,every groundwaterquality mapis split into four sub-maps,eachof themshowing thepositionof the intervals relative to one of four referencelevels.This relative position can be lower(the interval is completelybelow thereferencelevel), higher (the interval is completelyabovethe referencelevel), or not distinguishable(the interval straddlesthe referencelevel, and theestimatedvaluecannotbe distinguishedfrom that level given the informationavailable).Thereferencelevelsarechosenas(multiplesof) critical levelsor target levels.It shouldbe notedthat thesecritical levelsor target levelsusuallyapplyto individual samples.If a block medianconcentrationexceedsa critical level, more thanhalf of the individual samplesin the blockexceedsthislevel.If ablockmedianconcentrationis below acritical level,still many individualsamplesin theblockmayexceedthislevel.Thedoublingof thelevelsreflectsequidistant(linear)levelson thelog-scale.

5.4 ExampleA. Nitrate leachingfr om agricultural soils

Regional nitrateleachingratesin agriculturalareaswereestimatedby the NLOAD model(VanDrecht,1993a,1993b),an empiricalmodelthat predictssteadystateleachingof nitratefromthesoil to theshallow groundwaterasafunctionof manuringrate,soil type,typeof agriculturallanduseanddepthof thegroundwatertable.Thefinestspatialresolutionof NLOAD calculationscurrentlyavailableis500m × 500m.Only thesoil-landusecategoryof grasslandonsandwill beconsideredin thisexample.Fig.5.1showsNLOAD modelcalculationsfor 1989averagedto2km× 2km cells,derivedfromtheoriginalmapin Maas(1992).Fig.5.2showsascatterplotof thenitrateconcentrationsmeasuredat the sitesof the groundwaterquality monitoringnetworksandtheNLOAD calculationsobtainedfromthe500m × 500mcellsatthecorrespondinglocations.Fig.5.2showsthatnitrateconcentrationin groundwatertendstobegreaterwhennitrateleachingratesarehigher.Thenitratemeasurementsusedfor thisstudywereobtainedfrom thecategory‘grasslandonsand,’ andwereselectedfrom themonitoringnetwork dataaccordingto theselectioncriteriaof Section3.3.

Fig. 5.3a shows the variogram of the (log) nitrate concentrationmeasurements.Fig.5.4 shows the mapwith estimatesof nitrateconcentrationin thegroundwater, basedon these


kg/(ha.yr) N,

0 - 35 35 - 70 70 - 105 105 - 140 140 - 175 175 - 210

Figure5.1 Mapof nitrogenleachingratefromagricultural soilstoshallowgroundwater.Valuesare averaged fromtheoriginal 500m × 500m modelcalculationsto yield 2 km× 2 kmmeanvalues.Onlycellsin thecategorygrasslandonsandareshown

0� 50 100� 150- 200�NO3-N leaching from soil (kg/(ha.yr)).

0.1

1

10

100

NO

3-N

in g

rou

nd

wat

er (

g/m

3)

Figure 5.2 NO3-N concentration in groundwaterand nitrogen leaching rate. Plot of NO3-Nconcentrationmeasuredin thegroundwaterversusnitrogenleachingratefromsoilsto shallowgroundwater, for thecategorygrasslandonsand

EXAMPLE A. NITRATE 63

a

0� 25000� 50000� 75000� 100000

Distance (m)

0

2

4

6S

emiv

aria

nce

�

log(NO3-N) measurements2.8 Nug(0) + 2.67 Sph(2e+04)

40%

133

158

188208$ 197

173

204$

215$ 212$ 236$ 259$

306� 287$

274$

284$

284$

294$ 257$

257$

b

0� 25000� 50000� 75000� 100000

Distance (m)

0

2

4

6

Sem

ivar

ian

ce

�

log(NO3-N) residuals1.97 Nug(0) + 2.09 Sph(1.5e+04)

40%

133158

187205$

194

165

199

211$ 210$

235

257$ 304�

286$

271$ 281

278$

276

252$

248

Figure5.3 Variogramsof (a) log(NO3-N)e

,and(b)residualsfor thecategorygrasslandonsand.Calculationof thesamplevariogramsis explainedin TablesA2 (page 96) and A6 (page 97).Numbers reflectthenumberof (a) observationpairs or (b) residualpairsusedfor an estimate(+) of a variogram point.Codesusedfor a variogram model(—) are explainedin TableA3,(page97)


level: 5.6 g/m3

lowernot distinguishable

level: 2.8 g/m3


level: 11.3 g/m3


level: 22.6 g/m3


Figure5.4 Mapof NO3-N in thegroundwater(categorygrasslandon sand).95%Confidenceintervalsfor 4 km× 4 kmblock medianvalues,relatedto four concentration levels.Estimatesareobtainedbyordinarykriging(onlog-scale),usinginformationfromthemonitoringnetworksonly

EXAMPLE A. NITRATE 65

level: 5.6 g/m3


level: 2.8 g/m3


level: 11.3 g/m3


level: 22.6 g/m3


Figure5.5 Theeffectsof usinginformationonnitrogenleachingrates.95%Confidenceintervalsfor 4km× 4kmblockmedianNO3-N concentrations,relatedtofour levels.Estimatesareobtainedby universal kriging (on log-scale),usingboth informationfromthemonitoringnetworksandancillary information(nitrateleachingrates).Onlyresultsthat differ fromFig.5.4areshown


measurementsandthisvariogram,computedusingordinarykrigingasin Section3.4,asobtainedfrom PebesmaandDeKwaadsteniet(1994).

Fig. 5.3b shows thevariogramof theestimatedresiduals,obtainedfrom simple(ordinaryleastsquares)linear regressionof these(log) measurementsZ(x) andnitrateleachingratesatcorrespondinglocations,L(x)

Z(x) ��/� 1 + L(x)� 2 + e(x), E(e(x)) �� 0,

asdescribedin TableA6 (page97).Fig.5.5showsthemapof nitrateconcentrationin thegroundwaterwhenbothmeasurements

and NLOAD calculations(Fig. 5.1) are usedwith the variogramof Fig. 5.3b (using universalkriging).Toshow themagnitudeof theeffectsof theNLOAD information,Fig.5.5showsonlythosecellsthatdiffer from Fig.5.4.

5.5 ExampleB. Atmospheric depositionof zinc

Total atmosphericdepositionof zinc in 1985wasmodelledfor 10 km × 10 km blockswiththeTREND model(VanJaarsveld,in prep.)by combininginformationaboutsources(emissions,mainly from areassouthof the Netherlandsandareaswith heavy industry)andair transport.Becausegroundwaterquality wasmodelledusingsoil typeandlanduseinformationon 2 km× 2 km cells,the10 km × 10 km modelpredictionshave beeninterpolatedto 2 km cell values(by simply considering10 km × 10 km block estimatesasthe valueat the block centre,andusinginversedistanceinterpolationwith thenearest4 block-centrevalues).Sincetheoriginal10 km × 10 km mapwasvery smooth,the interpolationto 2 km × 2 km cellshasnot affectedtheoriginalpattern,asshown in Fig.5.6. Wewill focusontheareawith semi-naturalvegetationonsand,whereair depositionisexpectedtobethemainsourcefor zincin groundwater.Thezincmeasurementsusedfor thisstudywereobtainedfrom thecategory ‘semi-naturalvegetationonsand,’ andwereselectedfrom themonitoringnetwork dataaccordingto thecriteriaof Section3.3.

The scatterplot of zinc concentrationmeasuredin the groundwater and the modelledatmosphericdepositionof zinc (Fig. 5.7) revealsthat in generalzinc concentrationtendsto belarger at locationswith higherdepositionrates.In Fig. 5.7 the measurementwith the highestdepositionratedisturbsthispattern,andthismeasurementturnedout to belocatedin a seepagesituation,which is a plausiblereasonfor deviating from thegeneralpattern(watersampledatthatsiteis likely tohaveadifferentorigin).Thereforethemeasurementsatseepagesites(markedwith abullet in Fig.5.7) wereleft outfor themappingof zincconcentration,bothfor thesituationwith andwithout thedepositioninformation.

Fig.5.8ashowsthevariogramfor the(log)zincconcentrationmeasurementsselectedfromthemonitoringnetwork data.Fig.5.9shows themapof zincconcentrationestimates,basedonthesemeasurementsandthisvariogram,asobtainedby ordinarykriging at thelog-scale.

Fig 5.8b shows thevariogramof estimatedresidualsobtainedfrom simple(ordinaryleastsquares)regressionof these(log)measurementsZ(x) andatmosphericdepositionrates,D(x)

EXAMPLE B. ZINC 67

mol/(ha.yr) Zn0 0 - 0.4 0.4 - 0.8 0.8 - 1.2 1.2 - 1.6 1.6 - 2 2 - 2.4

Figure 5.6 Map of atmosphericdepositionof zinc.Valuesare interpolatedfromtheoriginal10 km× 10 kmmodelcalculationsto 2 km× 2 kmcells.Onlycellsin thecategorysemi-naturalvegetationonsandareshown

0.5 1.0 1.5 2.0

Atmospheric deposition of zinc (mol/(ha.yr)).

10

100

1000

Zin

c in

gro

un

dw

ater

(m

g/m

3)

1

Figure 5.7 Zinc concentration in groundwaterand atmosphericdepositionof zinc.Plot ofzinc concentration measured in the groundwaterversusatmosphericdepositionof zinc,forthecategory semi-natural vegetationon sand.A bullet (•) marksa measurementat a seepagelocation


a

0� 20000� 40000� 60000� 80000�Distance (m)

0.0

0.5

1.0

1.5S

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log(Zn) measurements0.274 Nug(0) + 1.5 Exp(3.5e+04)

25$

50� 77

!83"

90#

96#

81"

107

81"

88"

80"

b

0� 20000� 40000� 60000� 80000�Distance (m)

0.0

0.5

1.0

1.5

Sem

ivar

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ce

�

log(Zn) residuals0.0974 Nug(0) + 1.23 Exp(1.8e+04)

25

50� 77

83"

90# 96#

81"

107

81"

88"

80"

Figure5.8 Variogramsof (a) log(Zn)e ,and(b)residualsfor thecategorysemi-natural vegetationon sand,non-seepage locations.Numbers reflect the numberof (a) observationpairs or (b)residualpairs,usedfor anestimate(+) of avariogrampoint.Codesusedfor avariogrammodel(—)are explainedin TableA3, page 97. Calculationof thesamplevariogramsis explainedinTablesA2(page96) andA6 (page97)

EXAMPLE B. ZINC 69

level: 30 mg/m3


level: 15 mg/m3


level: 60 mg/m3


level: 120 mg/m3


Figure 5.9 Map of zinc in thegroundwater(category semi-natural vegetationon sand).95%Confidenceintervalsfor 4 km× 4 kmblock medianvalues,relatedto four concentration levels.Estimatesareobtainedbyordinarykriging(onlog-scale),usinginformationfromthemonitoringnetworksonly(resultsapplyto non-seepagelocationsonly)


level: 30 mg/m3


level: 15 mg/m3


level: 60 mg/m3


level: 120 mg/m3


Figure5.10 Theeffectsof usinginformationonatmosphericdepositionof zinc.95%Confidenceintervalsfor 4 km × 4 km block medianzinc concentrations,relatedto four levels.Estimatesare obtainedby universal kriging (on log-scale),usingboth informationfrom the monitoringnetworksandancillary information(zincatmosphericdepositionrates).Onlyresultsthat differfromFig.5.9areshown(resultsapplyto non-seepagelocationsonly)

EXAMPLE B. ZINC 71

Z(x) �� 1 + D(x)� 2 + e(x), E(e(x)) �� 0,

asdescribedin TableA6 (page97).Fig. 5.10 shows the modificationsto the zinc concentrationmap(obtainedby universal

kriging) thatresultedfrom additionof themodelinformationonatmosphericdepositionof zinc.Estimatesbasedon explicitly selectednon-seepagesitesonly apply to non-seepagelocations,andsinceall seepagelocationsarecurrentlynotknown,Figs.5.9and5.10shouldbeinterpretedwith care.

5.6 ExampleC. Measurementsof sulphatein shallow groundwater

Sulphateconcentrationin shallow groundwaterwasmeasuredin forestareasduringa one-timeinvestigation of soil and groundwater quality in semi-naturallyvegetatedareasin 1989-90(Boumansand Beltman,1991) as part of the national acidification researchprogram.Themeasurementscomprise1560 individual samples,collectedin 156 separate500 m × 500 msquares(in eachsquare10 samplesweretakenon a 50 m inter-spacedtransect).As before,themeasurementsarerestrictedto areaswith semi-naturalvegetationon sand.Thelocationsof the156squaresareshown in Fig.5.11, alongwith thesitesfor thegroundwaterqualitymonitoringnetworks.Themeanvaluesfor individual squaresareusedfor thevariogrammodelling,sincethey will be usedfor the interpolation. Crossvariogramand direct (sample)variogramsforsulphatemeasurementsareshown in Fig. 5.12. To assurevalid estimates,the linear modelofcoregionalization(LMC) is usedfor modellingthevariograms(JournelandHuijbregts,1978).TheLMC requiresthatfor mvariablesall directandcrossvariogrammodelsarederivedfrom alinearcombinationof n directvariogramfunctions� i(h):

� kk′(h) ��n

i �� 1bi

kk′� i(h), with bikk′ �� bi

k′k ,�

k,k′ : k,k′ �� 1…m,�

i : i �� 1…n, (5.5)

and that for every i, the coefficientmatrix [bikk′] is positive semidefinite.For two variablesit

sufficesto show that

� bi12 � 2 9bi

11 � bi22 ,

�i : i �� 1…n. (5.6)

Since(i) any valid crossvariogramplus a constant3� kk′ �� kk′ + C gives identicalestimatesirrespectiveof thevalueof C (Myers,1991),and(ii) theestimationof block meanvaluesfromtwo variableswith noidenticallocationsnever requirestheactualvalueof � kk′(0), it is irrelevantthatcondition(5.6) doesnot hold for the ‘nugget’variance.Theapparent‘nugget’varianceofFig.5.12c canbetakenasconsistingof a truenuggetvariancethatconformswith (5.6) andanunknown constantthatneedsno explicit specification(e.g.resultingfrom theestimationerrorfor themeanvaluesm1 andm2). Calculationof variogramsandcrossvariogramis explainedinTablesA2 (page96) andA6 (page97).

Using the LMC limits the possiblemodelsfor the crossvariogramto symmetriccrossvariogramsonly (i.e. � kk′(h) �� kk′(� h)). In our examplethis limitation is not serious,since


Figure 5.11 Mapof sulphatemeasurementsites.Shallowgroundwatermeasurementsites(+)andgroundwaterqualitymonitoringnetworksites(•) in thecategorysemi-natural vegetationonsand

EXAMPLE C.SULPHATE 73

a

04

250004

500004

750004

1000004

Distance (m)50.0

0.5

1.0

Sem

ivar

ian

ce

6

log(SO4), deep groundwater0.1 Nug(0) + 0.621 Sph(3e+04)4

18

347 397 548 659

639

73:

77:

699

689 78:

85;

588 75:

679

99< 74

94<

83;

90<

b

04

250004

500004

750004

1000004

Distance (m)50.0

0.2

0.4

Sem

ivar

ian

ce

6

log(SO4), shallow groundwater0.125 Nug(0) + 0.273 Sph(3e+04)4

204=

206= 259= 3507 3997

448> 425>

3907

3847

3337

3177

288=

283=

286=

3097

245=

286=

3097 3987

3767

c

04

250004

500004

750004

1000004

Distance (m)50.0

0.5

1.0

Cro

ss S

emiv

aria

nce

?

log(SO4), shallow x deep [email protected] Nug(0) + 0.4 Sph(3e+04)4

87; 132

152

157219=

232=

236

254

271= 281=

270

273=

290=

3537

3747 416>

418>

407>

5298

485>

Figure 5.12 Variograms of sulphate. Variogram for (a) log(SO4)e

measurementsfrom thegroundwaterquality monitoringnetworks(category semi-natural vegetationon sand),for (b)

log(SO4)e

fromtheshallowgroundwatermeasurements,and(c)crossvariogramfor (a)and(b).Anumberreflectsthenumberof observationpairsusedfor anestimate(+) of avariogrampoint,codesusedfor a variogrammodel(—)are explainedin TableA3, page 97. Calculationof thesamplevariogramsis explainedin TablesA2(page96) andA7 (page98)


directional(asymmetric)samplecrossvariogramsdid notsuggestaseriousasymmetry.Fig. 5.13 shows the groundwater quality map of sulphateconcentrationbased on

informationfrom thegroundwaterqualitymonitoringnetworksonly (usingordinarykrigingandthevariogramof Fig. 5.12a).Fig. 5.14 shows themodificationsto this mapresultingfrom theincorporationof theshallow groundwatermeasurements,usingcokrigingandthevariogramsofFigs.5.12a-c.

5.7 The effectsof ancillary information

Theeffectsof addingancillaryinformationtotheinterpolationprocesscanbeseenbycomparingthe resultswith the mapsresultingfrom the simpleapproach;i.e. compareFig. 5.5 with Fig.5.4 (additionof NLOAD nitrateleachinginformation),Fig. 5.10 with Fig. 5.9(additionof TREND

zinc depositioninformation),andFig. 5.14 with Fig. 5.13 (additionof sulphatemeasurementsfrom shallow groundwater).Table5.1shows the numberandpercentageof changedcells atevery referencelevel for eachexample. In general,the modificationsto the nitrogenmapareconsiderable,whereasthemodificationsto thezincandsulphatemapsarerathermodest(thoughnotnegligible).

At first sight the differencein magnitudeof the modificationsbetweennitrateandzincmapsmaybesurprising,sincefor both thestrengthof the relation(Figs.5.2 and5.7) andthedifferencebetweenmeasurementandresidualvariograms(in Figs.5.3and5.8) is similar. Thespatialvariationof therespectivemapsof nitrateleachingestimatesandzincdepositionestimates(Figs.5.1 and 5.6) however is very different,and the variation in the local relationbetweenmeasurementvariableandancillaryvariablediffersfor nitrate(Fig.5.15a)andzinc(Fig.5.15b).For zinc therelationbetweenmeasurementsandancillaryvariableis negligible in thenortherntwo neighbourhoodsof Fig.5.15b,andlocalestimationof themean(asin (5.1)) will accountforanimportantpartof thegradualchangein meanzincconcentration(suggestedby Figs.5.6and5.7). On thecontrary, the largeshortdistancevariationof the NLOAD map(Figs.5.1and5.15a)contributeslocally moreto theestimationof nitrateconcentration.

The modifications to the sulphate map resulting from the shallow groundwatermeasurements(Fig.5.14) obviously occurmainly at locationswith many shallow groundwatermeasurementsandfew monitoringnetwork locationsnearby.

5.8 Discussion

Chapter 3 explained the procedureused to make maps of groundwater quality using astratificationby soil type and land use,relevant measurementsfrom available groundwaterqualitymonitoringnetworks,andaninterpolationmethodthattakesspatialdependencebetweenmeasurementswithin strataintoaccount.Thisstudyshowshow ancillaryinformation,consistingeitherof resultsfrom deterministicmodelsof processesthat causevariation in groundwaterqualityor of measurementsonavariablethevariationof whichiscausedby thesameprocessesasthegroundwaterquality variable,canbeincorporatedin themappingprocedureto improvethe accuracy of the groundwater quality maps.Threeexamples(on nitrate,zinc and sulfate

DISCUSSION 75

level: 30 g/m3


level: 15 g/m3


level: 60 g/m3


level: 120 g/m3


Figure 5.13 Map of sulphatein thegroundwater(category semi-natural vegetationon sand).95%Confidenceintervalsfor 4 km× 4 kmblock medianvalues,relatedto four concentrationlevels.Estimatesareobtainedby ordinary kriging (on log-scale),usingmeasurementsfromthegroundwaterqualitymonitoringnetworksonlyandthevariogramof Fig.5.12a


level: 30 g/m3


level: 15 g/m3


level: 60 g/m3


level: 120 g/m3


Figure 5.14 The effects of using shallow groundwater measurement information. 95%Confidenceintervals for 4 km × 4 km block mediansulphateconcentrations,relatedto fourlevels.Estimatesare obtainedby cokriging (on log-scale),usingmeasurementsfrom shallowgroundwaterand from the groundwaterquality monitoringnetworks,and the variogramsofFig.5.12a-c.Onlyresultsthat differ fromFig.5.13areshown

DISCUSSION 77

Table 5.1 Theeffectsof ancillary information.For each variable and level the numberandpercentage of cells for which the addition of ancillary information changed the outcome(possiblevalues:lower,higherandnot distinguishable)

variable level %cellschanged numberof cellschanged

NO3-N 2.8 3g/m 22 505

5.6 20 47711.3 20 47122.6 10 240

Zn 15 3mg/m 7 64

30 7 6060 6 57

120 5 51

SO4 15 3g/m 9 87

30 2 2060 5 46

120 5 48

concentration)haveshown thattheadditionof ancillaryinformationdoesimprovetheresultingmapsin termsof the ability to distinguishestimated4 km × 4 km block medianvaluesfromcritical or target levelsandtheir multiples(compareFig. 5.5 with 5.4, 5.10 with 5.9 and5.14with 5.13). Themodelspresentedin thischapterareextensionsto thesimplemodelof Chapter3 thatareeasyto apply, providedthat therelevantancillaryinformationis available.Sincetheimprovementis considerable,it hasbeenwell worth theeffort—only extra computinganddatahandlingcostswereinvolved.

ExamplesA, B andC haveshown thatwegainedunderstandingof thespatialvariationofgroundwaterqualityvariablesby includingnew, relevantinformationin themodel,resultinginabetterexplanationof thevariationof themeasurementsandin improvedmaps(moreaccurateestimates)of groundwater quality variables.From the perspective of a monitoring networkoptimizationit is important to know how many addedmeasurementswould have given anequivalentimprovement(e.g.Burroughetal.,1995),but thisis a topicfor futurestudy.

Themethodspresentedherearegenerallyapplicablein environmentalstudies,andprovidea convenientway for incorporatingresultsfrom (oftencomplex) deterministicprocessmodelsand measurementsof other variablesin the estimation(spatialinterpolation)of the variableof interestat non-measuredlocations.Both models(5.2) and (5.3) include model (5.1) as aspecialcaseandwill reduceto (5.1) whentheaddedinformationhasno relationto theprimarymeasurementvariable.Thereforethey arerobusttooverspecificationof themodel(only for (5.2)will overspecificationcostsomeefficiency,sinceparametersthatarezerowill beestimated).Mostapplicationsof universalkriging use(polynomialsof) thecoordinatesasexplanatoryvariables(replacingf (x) in (5.2),e.g.Steinetal.,1991).ExamplesA andB insteadshow how variablesthatcarryaphysicalsignificanceto themeasurementvariable(basedon theory, i.e.thehydrological


a

0A

50A

100A

150B 200A0

1

10

100

0A

50A

100A

150B 200A0

1

10

100

0A

50A

100A

150B 200A0

1

10

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0A

50A

100A

150B 200A0

1

10

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0A

50A

100A

150B 200A0

1

10

100

b

0.5C 1.0C 1.5C10

100

1000

0.5 1.0 1.5

10

100

1000

0.5 1.0 1.5

10

100

1000

0.5 1.0 1.5

10

100

1000

0.5 1.0 1.5

10

100

1000

Figure 5.15 Local scatter plots of measurementsversusancillary variable for NO3-N (a)and zinc(b).For fivearbitrarily chosen(but distinct)local kriging neighbourhoodsR, scatterplots show concentration in groundwater(a: NO3-N in 3g/m , b: zinc in 3mg/m) versus theancillary variable (a: nitrate leaching rate in kg/(ha.yr),b: atmosphericdepositionof zinc inmol/(ha.yr)).Arrowslink thelocationfor which kriging neighbourhoodinformationis showntothecorrespondingscatterplot

DISCUSSION 79

cycle andchemicalbalances)canbeutilized.We felt no needto further ‘prove’ thevalidity ofthemodels(e.g.with externalcriteria)sinceboththeoryandobservations(in particularFigs.5.2and5.3, 5.7and5.8, and5.12) supportthemodelschosen.

The examplespresentedin this studycanbe improved further. In particularin examplesA andB theprocessmodelsconcernaveragesof spatial‘model units’ thesizeof 500m × 500m (nitrateleaching)or of 10km × 10km (atmosphericzincdeposition),andtheseaveragesarecomparedwith groundwatermeasurementson a volumelessthana cubicmetre.Thesevalueson ‘modelunits’canneverexplain thevariationof thegroundwaterqualityvariablethatoccurswithin theseunits.Whenthevariationof themeasurementvariablewithin these‘model units’is relatively largeandwhenthis variationis mostly truevariation(smallmeasurementerrors),thenit maybe worthwhile to modify (or simply apply) the processmodelin sucha way thatit yieldsvaluesspecificallyfor ‘model units’ that correspondto theareaof influence(Section1.3) of themonitoringscreens.Suchmodelvaluesarespecificallyintendedto explainvariationin themeasurements,andit is expectedthatusingthemwill leadto a smallerresidualvariationand result in improved (more accurate)estimatesof groundwater quality variables.For theNLOAD modelsucha modificationmay be feasiblebecausethe short-distancevariationof themeasurementsis relatively large(Fig.5.3), andall input variablesareavailableat a finer spatialresolution(soil maps1:50,000(De Vries and Denneboom,1992),land useinformationat a25 m cell resolution(Thunnissenet al., 1992),andmanuringrateinformationat municipalitylevel).Obtainingzinc depositionestimatesfor smallerelementsmayrequirea differentmodelstructure,takingaccountof localeffects(suchaslocal landscaperoughnessandthepositionofthegroundwaterqualitymeasurementsiterelativeto forestedgesexposedto theprevailingwinddirection,Draaijers,1993).Whenthesizeof thespatialunitsof theprocessmodelis adjustedtothesizeof theareaof influence,theneedtoknow thelocationsof theinfluencearea(usinglocalgroundwaterflow patterns)becomesmoreprofound.

Transformationof theestimatedfluxesin ExamplesA andB into estimatedconcentrationswould simplify the interpretationof the regressionslopein (5.2). Here,the regressionslopeis a purely conventionalparameterof no interestby itself, and becausethe rainfall excesswithin a soil-landusecategory (ascurrentlyknown) is nearlya constant(Meinardi,1994),thistransformationwouldhardlyaffect theconcentrationestimates.

In this study mathematicalaspectswere kept simple becausethe primary goal is toillustratethe effectsof addingancillary informationto the estimationof groundwaterquality.The simplicity refersto the modelsused(5.1-5.3), the estimationof the variograms(5.1c),(5.2c) and(5.3c-e), the choiceandfit of variogrammodels,the choiceof the linear modelofcoregionalizationfor modelling (5.3c-e), the choiceof the local neighbourhoodR, the wayapproximate95%confidenceintervalsfor blockmedianvaluesareestimated(5.4), andthedataselectionprocedure.Theapproximationof 95%confidenceintervalswith (5.4) is known to betoooptimistic(biased,cf. Rao,1973),asexplainedin Sections2.5and3.8.

6.1 Intr oduction

In the context of a full monitoringnetwork optimizationthis studycanbe consideredasthefirst basicstep:theinferenceof groundwaterquality from availableinformation.Therelevanceof thisstudyfor groundwaterqualityassessment,andsomeof thequestionsthatneedanswersin a network optimizationarediscussedin Section6.2. A numberof enhancementsthat areworthwhileconsideringfrom thenetwork optimizationpointof view or from amathematicalorstatisticalpoint of view arelistedin Section6.5. A numberof methodologicalchoicesthatarecommonthroughoutthisstudyarediscussedin Section6.3,andSection6.4summarizesthemainconclusions.

6.2 Optimizing the groundwater quality monitoring networks

The groundwaterquality monitoringnetworksaccountfor a substantialpart of the total costof theenvironmentalquality monitoringnetwork in theNetherlands.During thefirst yearsofoperationthenetworksprovidedaninitial survey of thespatialvariationin groundwaterquality,andsomeyearslater the first analysisof changesin time appeared.Now, after several yearsof measuring,the questionariseswhetherthe way we currentlymonitor groundwaterqualityis worth thecost.Aspectsof agroundwaterqualitymonitoringnetwork thatcontrolthecostarelabourandcapitalcostsof boringandsettingup thewells,measurementintensity(spatialwelldensity, numberof screenssampledper well, andmonitoringfrequency), choiceandcostofanalysisof thechemicalvariablesmeasured,andmodellingeffort (thedevelopmentof modelsthatbetterexplainvariationin themeasurements).

Optimizing the monitoring with respect to measurementintensity can be done byminimizing the total costasa functionof themeasurementintensity. The total costat a givenmeasurementintensitycouldbecalculatedby summingthecostof measurementandthe lossdueto limited knowledge(e.g.Fig. 6.1). This would be fairly easyif the lossdueto limitedknowledge(thecostof ‘not monitoring’)wereknownasafunctionof themeasurementintensity.Sucha lossfunctionwould call for anexplicit statementof thevalueof knowing groundwaterquality to a certainextent.From this valuewe could determinethe function of expectedlossdueto limited knowledge,which obviously increaseswhenmeasurementintensitydecreases.Themeasurementintensityfor whichthetotalcostisat itsminimumvaluewill yield theoptimal(cheapest)measurementintensity. Parameterchoicecouldbeevaluatedin asimilarway.

Specifyinga loss function for groundwater quality monitoringnetworks is not a trivialtask,becauseit callsfor theassignmentof a valueto a particular‘degreeof knowledge’.Such

80

OPTIMIZING THE GROUNDWATERQUALITY MONITORINGNETWORKS 81

measurement intensityD

costE cost of measurement F

loss due to limited knowledgetotal costG

Figure6.1 Monitoringnetworkoptimization:total costasa functionof measurementintensity.Thearrowpointsto an‘optimal’ measurementintensity

a function is only somethingwe canagreeon, and it may changein time and with specificapplication.Clearlythisfunctiondependsontheuncertaintyattachedtoestimatesof groundwaterquality, given a certainmonitoringnetwork. Becausein this study ‘degreeof knowledge’ isexpressedastheability to distinguishestimatesfrom critical concentrationlevels(target levelsor maximumtoleratedlevelsfor humanconsumption),it candirectlybeinterpretedasthedegreeof knowledgeof groundwaterquality, anda meaningfullossfunctioncanbederived from it.Othermeasuresof ‘degreeof knowledge’thathavebeencommonlyusedin optimizationstudies(e.g.theestimationvarianceasin Rodríguez-IturbeandMejía (1974)andin VanGeer(1987),or therelativeestimationvarianceonthelog-scaleasin HoekstraandHeuberger(1995))donotlink estimatesto qualityassessment,but focusexclusivelyon estimationuncertainty. Therefore,with respecttothedegreeof knowledgeof groundwaterquality,andthusfor groundwaterqualitymonitoringnetwork optimization,theseothermeasuresareinappropriate.Cressie(1991,4.6.2)discussesawiderclassof optimizationcriteria.

Thisroughsketchof ameasurementintensityoptimizationassumesthatthecharacteristicofgroundwaterqualitywefocusonandthemethodfor obtainingthischaracteristicfrom availableinformationhavebeenfixedin advance.Chapter5showedthatbyincorporatingrelevantancillaryinformationin the interpolationprocedurewe canimprove estimatesof groundwaterquality,thusextendingthe‘degreeof knowledge’of groundwaterqualitywithoutchangingmeasurementintensity, andmoreimprovementcanbeexpected(Sections3.8, 5.8, 6.5). Althoughmodelscanneverreplacethecrucialroleof measurements(e.g.becausemeasurementsareneededfor modeldevelopment,validationandcalibration),from amonitoringnetwork optimizationpointof viewit couldbeworthwhileto studyto whatextentanincreasein explained(or described)variationobtainedby improved modelling could compensatefor a certaindecreasein measurementintensity, whenacertain‘degreeof knowledge’of groundwaterqualityshouldberetained.

6.3 Generalaspectsof the discussions

This sectioncoverssomeaspectsof thediscussionsthatarerelevant to thewholestudy. Morespecificaspectsof thediscussionson thetopicsof Chapters3-5 have beenplacedat theendofthesechapters.

82 DISCUSSIONAND CONCLUSIONS

The estimatesof groundwater quality variablesin this study concernblock medianconcentrationsof 4 km × 4 km blocks:a balancedchoicein which themeasurementvariation,thespatialmonitoringnetwork densityandthecharacterof soil typeandlanduseinformationavailablewereconsidered.If, for somereasonestimatesfor mediansof largeror smallerareasareneeded,thesamemethodscanbeused(but seealsoMeasurementerror reductionin Section6.5).Forestimatesof othercharacteristics(e.g.meanvaluesof blocksorquantilesof largerareas)somemodificationswould be necessary(Sections3.8, 6.5), or an approachbasedon classicalstatistics(design-basedinference)maybemoreappropriate(Section3.8).

In thisstudy, noinformationongroundwaterflow or transportof solublesasobtainedfromdeterministicprocessmodelshasbeenused.If this informationis available it canreadily beusedto improve estimatesof measuredvariables,usingthe methodspresentedin Chapter2,and demonstratedin Chapters3 and 5. For the development(validation,calibrationor inputspecification)of deterministicgroundwaterqualitymodels,themapsobtainedin thisstudycanbeconsideredasa translationof measurement(point) informationinto informationon spatialunitsthesizeof unitsusedin regionalor nationalgroundwatermodels(Sections1.4, 3.8).

Themodelsusedin thisstudymergetheadvantagesof thetraditional,classicalstatisticalapproach(independent,non-measurementinformation is used to explain variation in themeasuredvariable)andthegeostatisticalapproach(local informationis usedby modellingthespatialdependencestructurethat is often presentin the residual,unexplainedvariation), toestimate(map)themeasurementvariables.

In chapter2 geostatisticalmethodswerederivedfrom a classicalstatisticalpoint of view.This is not theway it evolvedhistorically(Cressie,1990),but it emphasizedthatgeostatisticalmodelscanprovide the obviousgeneralizationof modelswith independenterrors,whentheyareusedfor spatialestimation(mapping)purposes.In thecontext of mapping,whennoappealis madeto independenceof errorsfrom a design-basedargument(Section2.4), theautomaticassumptionof independenceof errors(e.g.,Boumans,1994,AlkemadeandVanEsbroek,1994,RIVM, 1995)is unjustifiedandcanbehighly inefficient.

Throughoutthis studytherehasbeenno validationof the modelstructurethroughsomequantitativetestcriterion.However, themodelsusedare(i) simple,(ii) basedonboththeoryandobservationsand(iii) availablerelevant informationis usedasmuchaspossible.If themodelsusedarein someinstancesover-specified,thenthey containthe ‘true’ modelasa specialcase,andthiswill resultin possiblyinefficient,but notinvalidestimates(e.g.CressieandZimmerman,1992,Sections3.6, 5.8). But whenweconsiderthereducedmodelsevaluatedin section3.6, thevariogramsof thefull model(Fig.3.3) revealalreadythata variogrampooledover severalsoiltype or land usecategoriesyieldsunnecessarilyinadequate(i.e. non-specific)error estimates.Under-specificationof modelsis inevitablewhenit isaconsequenceof limited dataavailability.If availabledatasuggestthatmorecomplex modelsresultin betterestimates,thenestimatesfromthesimpler, under-specifiedmodelarenot necessarilywrong,but areinefficientandshouldbeconsideredasa ‘lost opportunity’(Switzer,1992).Consideringthecasestudiesin Chapter5,thesimplemodelisnotnecessarilywrong,but themorecomplex modelisbetter—itresultsin moreaccurateestimates.

By definition, the exponentof the meanon the log-scaleis the geometricmeanon theoriginal scale.However, when on the log-scalethe meanand the mediancoincide,then the

GENERALASPECTSOFTHE DISCUSSION 83

geometricmeanandmediancoincideon the original scale,andthe exponentof a confidenceinterval for the meanon the log-scaleis a confidenceinterval for the medianon the originalscale.This last conditionis met whenon the log-scaleindividual sampleswithin a block aresymmetrically(e.g.normally,AitchisonandBrown,1957)distributed.In general,thissymmetryconditionon thelog-scaleseemsreasonablymild, giventheinformationathand.

6.4 Conclusions

Themainconclusionsare:

• Despitethelargeestimationinaccuraciesresultingfrom naturalvariationandlimited dataavailability, groundwaterqualityvariablesthataremeasuredin thenationalandprovincialgroundwaterqualitymonitoringnetworkscanbemappedat thespatialresolutionof 4 km× 4 km blockswhen(i) soil typeandlandusemapinformationis taken into account,(ii)suitablemeasurementsfrom availablemonitoringnetworksareselected,and(iii) attentionis paidto localvariationin level,sizeof spatialvariation,andspatialdependencein spatialvariationof measurements

• Theeffectsof stratificationbysoil typeandlanduseareprofound,partialornostratificationresultsin essentiallydifferentmaps

• Theeffectsof monitoringnetwork density, asshown by comparingmapsresultingfrom areducedmonitoringnetwork with themapsfrom thefull network,areconsiderable

• Spatiallydifferentiatedstatisticalindicationsof long-term,steadychangesin groundwaterqualityvariablescanbeobtainedwhenshort-termpredictions,obtainedfor eachmonitoringwell screenby a linearregressionmodel,areusedin theinterpolationprocedure,andwhenlocation-specifictime-predictioninaccuraciesaretakeninto account

• Maps of groundwater quality can be improved by using ancillary information in theestimationprocedure,wherethe ancillary informationmay consistof informationfromdeterministicprocessmodels,or of measurementsof relatedvariables

• It is expectedthat mapsof groundwaterquality andof changesof groundwaterqualitycanbeimprovedfurtherwhentherelevantinformationongroundwaterflow is usedin themappingprocedure

6.5 Futur edir ections

Thissectionlistsanumberof subjectsthathavenotbeenstudiedin depth,but whichareworthyof futureresearch.Partly they aredirectly relatedto mappinggroundwaterquality, partly theyaremathematicalor statisticalitems,relevantfor themethodsused.


Incorporatinggeohydrological informationThe most obvious extensionof the presentstudy is to useinformationon groundwaterflowfor mapping groundwater quality variables.This would reveal a better relation betweenmeasurementsandmapswith soil typeandlanduseinformation,sincegroundwaterflow causesthe influence(infiltration) area(Section1.3) to besomewhere‘upstream’from themonitoringwell. With detailedflow information available for monitoring well locationsas well as formappinglocations(Section2.3), connectionsbetweendeterministicmodelsandmeasurementscanbemadeat a finer spatialresolution(Section5.8), andthiswould allow a betterprediction(more accurateestimation)of the groundwater quality at unsampledlocations,blocks orareas.Other relevant geohydrologicalinformationcould be informationaboutthe subsurfacesedimentaryenvironment(in additiontosoilmapsdescribingthetopsoil)andinformationaboutchemicalbalancesfoundthere(e.g.redoxpotential,calciumor oxygenavailability, pH).

DetectionlimitsGroundwaterquality variableshave a thresholdbelow which measurementsareunreliable.Ameasurementbelow this thresholdis a ‘below detectionlimit measurement’(BDL). Somegroundwaterquality variablesmeasuredin thegroundwaterquality monitoringnetworkshavea high fraction (e.g.50-75%)of BDLs. Several methodshave beendesignedfor choosingavaluebetweenzeroandthedetectionlimit so that estimationof thesamplemeanor varianceis unbiased,usuallyassumingthatthedistributionof themeasurementsis known (e.g.HaasandScheff, 1990).Theproblemof applyingany of thesemethodsin a geostatisticalcontext is thatthesamplevariogramwill not revealthespatialdependenceof thetrue,underlyingvariable.

In this study the valueof the detectionlimits is assignedto all BDLs becauseit is thehighestvaluepossible.The disadvantageof this choiceis that it leadsto underestimationofvariogramvaluesandconsequentlyto too optimisticestimationerrors,andit is not consideredasa satisfactorysolution.Whenvariablesaremodelledon a log-scale,theeffect of thevaluesassignedtoBDLsisprofound:assigningasmallfractionof thedetectionlimit toBDLswill yieldavariogramthatis dominatedby thespatialdistributionof BDLs versusnon-BDLs.

Measurementerror reductionMeasurementerrorin agroundwaterqualityvariablecanbeconsideredasasourceof variationthat causespart of the nuggetvariance,the variogramvalue at nearly-zerodistance.It onlyneedsexplicit quantificationwhenestimatesof singlemeasurementsareneededatmeasurementlocations(Christensen,1991,VI.4), andthereforeit is not explicitly consideredin this study.Furthermore,it is unlikely thatmeasurementerrorof thiskind canbereduced.

Theindependentvariables(e.g.soil typeandlandusein Chapter3,nitrateleachingrateinsection5.4) arealsoproneto measurementerror.A largermeasurementerrorin theindependentvariableswill alwaysresult in a weaker relationbetweengroundwater quality variablesandindependentvariables.If theindependentvariablesarecontinuousandtheirmeasurementerroris (i) independent,normallydistributedwith zeromean,(ii) independentof thetrue(underlying)values,and(iii) independentof thegroundwaterqualitymeasurements,thenthismeasurementerrorwill notinvalidatetheestimatesof thegroundwaterqualityvariables(Fuller,1987).Similarconditionsmay be expressedfor categorical independentvariables.Reducinga measurementerrorof thiskind will leadto modelswith smallerresidualvariations,andincreasetheaccuracy

FUTUREDIRECTIONS 85

of estimates(Section5.8).Usingdominantsoil-landusefor 4km × 4km blocksinsteadof actualsoil-landuseinduces

a measurementerror: when matchingthis new variablewith the measurements,part of themeasurementsaremisclassified(i.e. at measurementlocationswheredominantsoil-landusediffersfrom actualsoil-landuse),resultingin largerwithin-classvariancesthanthewithin-classvariancesthat would have beenfound whenactualsoil-landusewereused(Section3.8). Anenhancementof thisprocedurethatwould eliminatemostof thesemisclassificationswould betousetheactualsoil-landuseatthemonitoringwell locations(preferablyof theinfluenceareaofthemonitoringwell,atthetimeof infiltration)asobtainedfromtheoriginalsoil typeandlandusemaps,toseparatelyestimatethegroundwaterqualityvariablesfor everysoil-landusecategoryina4 km × 4 km block,andtocalculatethefinal estimatesby weighingseparateestimateswith theareafractionsof eachsoil-landusecategory in a block.Thisapproachwill beespeciallyusefulin caseswheredominantsoil-landuselosesits relevance(i.e. whenestimatesareneededforaveragesof areasthatarelargeror moreheterogeneousthanthe4 km × 4 km blocksconsideredin Chapters3-5), but mayyield moreaccurateestimatesfor 4 km × 4 km blockmedianvaluesaswell.

ApproximateconfidenceintervalsWithin the context of a valid model structure,confidenceintervals obtainedby substituting

k(x0) for (x0) in (2.10) arevalid 95%confidenceintervalsonly when(i) theestimationerroris normallydistributedand(ii) H , � 0 and� e(B0,B0) areknown.In practice,(i) cannotbecheckedexperimentallybecauseblock averageson thelog-scalecannotbemeasuredand(ii) is not truewhenthesequantitiesareestimatedfrom sampledata.This leadsto theconclusionthat in thiscaseintervalscalculatedwith (2.10) resultin approximate95%confidenceintervalsthathaveatrueconfidencelevel lessthan95%(i.e.biasedconfidenceintervals,cf. Rao,1973).

In this study, we usetheseapproximate95%intervalsbecausecurrentlyno efficient andgenerallyacceptedalternatives for obtaining‘better’ 95% confidenceintervals are available.Attemptsin the directionof solving (ii) in a Bayesianframework are for instancegiven byKitanidis(1986)andHandcock(1994),whereasBrus(1993)presentsa design-basedapproachfor this problem.Obviously modelling variogramsfrom smallersamplesdecreasesthe trueconfidencelevel for intervalsobtainedwith (2.10).

Changeof supportandnon-lineartransformationsBlock averagescanbe estimatedwith linear universalkriging but estimatingblock averagesfrom a non-linearmodel(e.g.a linearmodelfor non-linearlytransformedmeasurements)is nottrivial (Cressie,1991(5.2),Cressie,1992,Myers,1994).In section3.8anapproximatesolutionfor lognormalvariablesisproposed,but tobeeffectiveit maybenecessarytouseeitherCressie’s(1992)adjustedvariance-unbiasedestimatorto obtainbetterestimatesof block averages,oranapproachbasedon averagingconditionallysimulatedpoint valuesin a block (e.g.Journel,1992).

Space-timegeostatisticsanduncertainobservationsIn most geostatisticalapplicationsthe startingpoint is the availability of measurementsthatare possiblyproneto a measurementerror having constantvariance,allowing the useof a


stationarymodel.With time-predictedobservations,we cannotuse this as a startingpointbecausetime-predictionerrorvariancesaretypically very diverse(non-stationary).In Chapter4 simple methodsfor handlinguncertaintime-predictionsin block kriging were proposed,essentiallyassumingthatthetime-predictionerrorhasknown varianceandis independentfromtheunderlyingspatialvariable.This shouldbeconsideredasa startingpoint for morerelevantapproachesin space-timegeostatistics.

1. General introductionGroundwater quality is the suitability of groundwater for a certainpurpose(e.g.for humanconsumption),andismostlydeterminedby itschemicalcomposition.Pollutionfromagriculturaland industrial origin threatensthe groundwater quality in the Netherlands.Locally, thispollution is measuredat tensof metresdepth.Sincegroundwateris themainsourcefor freshwater, this pollution causesa decreasein the long-termresourcesof watersuitablefor humanconsumption.

In order to get insight into the currentsituationof groundwater quality and systematicchangesof groundwaterqualityovertime,thenationalgroundwaterqualitymonitoringnetworkwasestablishedfrom 1978to 1984.Thisnetwork consistsof 370permanentwells,spreadfairlyevenlyover thecountry(Fig.3.2, page27), with screensat 8-10and23-25metrebelow thesoilsurface.Thewell screensaresampledyearly.Morerecently, theprovinceshaveinstalledsimilarmonitoringnetworksthatdoubledthemeasurementdensity.Becausethemonitoringnetworksareamajorfinancialinvestment,thequestionariseswhethertheinformationongroundwaterquality,asobtainedfromthecurrentmonitoringnetworks,issufficient.Thiscallsfor thequantificationofwhatcanbeinferredfrom thisinformationaboutthequalityof all thegroundwaterof interest.

For the modelling of the spatialand temporalvariation in groundwater quality, usinga physically and chemicallybaseddeterministicmodelwould call for informationon manyvariables(e.g.initial andboundaryconditions,modelparameters),thatareatpresentnotavailableon a nationalscale.However, mappinggroundwaterquality is possibleby usingmuchsimplermodels,thatlumpmuchof theunknown factorsinto aspatiallydependentstochasticterm.Theobjectivesof thisstudyareto

• map groundwater quality in the Netherlands,using available measurementsfrom thenationalandprovincial groundwaterqualitymonitoringnetworksandmapinformationonsoil typeandlanduse

• show theeffectsof monitoringnetwork densityandtheeffectsof usingsoil typeandlanduseinformationon theresultinggroundwaterqualitymaps

• map the systematic,temporalchangesin groundwater quality in a way similar to themappingof groundwaterquality

• show how groundwater quality maps can be improved by using relevant ancillaryinformation in the estimationprocedure,whereancillary information is obtainedfromdeterministicprocessmodelsor from othermeasuredvariables.

The primary aim is, with theseobjectives, to answerthe basicquestionsof describingthe

87

88 SUMMARY

currentsituationandsystematictemporalchangesof groundwaterquality for thewholeof theNetherlands.In this studywe usefairly simplemodelsthat allow anexplicit quantificationoftheaccuracy of resultingestimates.This accuracy is taken into accountin the resultingmaps,anticipatingthequestionaboutthevalueof thecurrentmonitoringnetworksfor inferringcurrentsituationandsystematicchangesin timeof groundwaterquality.

2. StatisticalmappingTraditionally, mapsshow directly observed phenomena.When they do so, most often theobservationaccuracy doesnot exceedthe displayaccuracy becauseof the scalingapplied.Intheenvironmentalscienceshowever, it isverycommonthatthe‘observations’shown onthemapdonot directlyportrayobservedphenomenabut quantitiesthatareonly known approximately,e.g.estimatesof somequantitybasedon limited sampleinformation.In ordernot to suggestmore‘knowledge’on a mapthanwe have, in the lattercaseit is necessaryto limit thedisplayaccuracy.

Whenobservationsareavailable,statisticalmodelscanbeusedfor theestimationof thevalueof avariableatanunsampledlocationor themeanvaluein aprespecifiedarea,andfor theassessmentof estimationaccuracy. Geostatisticalmodels—aparticularclassof linear modelswith dependentobservations—arespecificallysuitedfor the mappingproblem,i.e., they aresuitedfor estimatingthevalueat a specificlocation,or themeanvalueof a specificarea.Thesemodelscanbeviewedasa generalizationof themostcommonlyusedstatisticalmodels,whicharebasedon independentobservations.

In ordernot to invite thereaderof a mapto ignoretheaccuracy information,estimateandestimationaccuracy shouldbe presentedcombinedon the mapasa confidenceinterval. Thiswasdoneby showing thepostitionof theconfidenceintervalsrelative to eachof four referencelevelsin four separatesub-maps(e.g.Fig.3.4, page30). An alternativedisplayform is to showbothsidesof theinterval in asinglemapthatreducesto theclassical,‘deterministic’mapwhereconfidenceintervalsarecompletelycoveredby a legendclassinterval (e.g.Fig.3.5, page31).

Thegeneralapproachto mappinggroundwaterqualityvariablesis thatwithin soil-landusecategories,afteracarefuldataselection,estimatesfor 4 km × 4 km blocksareobtainedby localordinarykriging on thelog-scale.After back-transformationto theoriginal scale,approximate95% confidenceintervals for block medianconcentrationlevels are shown in maps.Thisprocedureisslightlymodifiedtoaccountfor locationspecific(i.e.non-stationary)timepredictionvariancesin Chapter4,andto improveestimatesby usingancillaryinformationin Chapter5.

3.Mapsof thegroundwaterquality in theNetherlandsat 5-17metredepthin 1991Mapsof 25groundwaterqualityvariableswereobtainedbyestimating4km× 4kmblockmedianconcentrations(PebesmaandDeKwaadsteniet,1994).Estimateswerepresentedasapproximate95%confidenceintervalsrelatedto 4 concentrationlevelsmostlyobtainedfrom critical levelsfor humanconsumption.All mapswerebasedon425measurementsfromnationalandprovincialgroundwaterqualitymonitoringnetworks.Theestimationprocedurewasbasedonastratificationby soil typeandlanduse.Within eachsoil-landusecategory measurementswereinterpolated.Regionaldifferencesin meanlevelandin spatialdependencebetweenmeasurementsweretakeninto account.Stratificationturnedout to be essential:no or partial stratification(usingeithersoil type or land use)resultsin essentiallydifferentmaps.The effect of monitoringnetwork

89

densitywasstudiedby leavingoutthe173measurementsof theprovincialmonitoringnetworks.Important changesin resultingmapsare assignedto loss of information on short distancevariation,aswell aslossof locationspecificinformation.

Froma policy point of view theresultingmapscanbeusedeitherfor quantifyingdiffusegroundwatercontaminationandlocationspecificbackgroundconcentrations(in orderto assistlocal contaminationassessment),or for input and validation of policy supportingregionalor nationalgroundwaterquality models.Themapscanbeconsideredasa translationof pointinformation obtainedfrom the monitoring networks into information on spatial units, thesizeof unitsusedin regionalgroundwatermodels.Themapsenablelocationspecificnetworkoptimization.In general,the mapsgive little reasonfor reducingmonitoringnetwork density(wideconfidenceintervals).In Chapter3themethodsusedareillustratedbyresultsonaluminiumconcentrationin groundwater.

4. Mapsof temporal changesin groundwaterquality in theNetherlandsat 5-17metredepthFor 12groundwaterqualityvariablesstatisticalmapsof temporalchangesweremade,focusingon the long-termcomponentsof temporalchanges(Pebesmaand De Kwaadsteniet,1995).For eachmonitoringwell screen,short-termpredictionswerecalculatedfrom time seriesofyearlymeasurementsover 5-7years,by usinga simpleregressionmodel.Within eachsoil-landusecategory thesepredictionswere interpolatedspatially, taking locationspecificpredictionaccuracies,local level, and the size and structureof spatial variation into account.Spatialestimatesconcern4 km × 4 km blockmedianconcentrations.For two extrapolationtimes(1980and2000)estimatesof block medianvaluesareshown in mapsas95%confidenceintervals,relatedto four levelsthatwerederivedfrom critical levelsfor humanconsumption.

The resultingmaps(e.g.Fig. 4.4 and 4.5 on pages52 and 53) show a wealth of trendindications.Primarily, we can conclude that a trend approachwith such a high spatialdifferentiationis attainableand advisable.Advisablebecausecells with a pronouncedtrendcanbe distinguishedfrom a neighbourhoodwith lesspronouncedtrends,andbecausetrendsin adjacentor nearbycells (also within the samesoil-land usecategory) can have oppositedirections.Becauseof thecontentsandfoundationof severalaspects,somecautionis requiredfor theinterpretationof themaps,andthemapsreflectonlyafirststepin thedirectionof spatiallydifferentiatedstatisticaltime-trends.Reductionof the—now yearly—measurementfrequency isdiscouraged.Seriousmodificationsof themapsin thisstudymaybeexpectedwhen,aftera fewyears,measurementseriesfromtheprovincialgroundwaterqualitymonitoringnetworksbecomeavailableandcanbeincorporatedin thestudy. In Chapter4 themethodsusedareillustratedbyresultsonpotassiumconcentrationin groundwater.

5. Improvingestimateswith ancillary informationMaps with estimatesof groundwater quality variables,as previously obtainedby ordinarykriging within soil-land usecategoriescan be improved by incorporatingrelevant ancillaryinformationin theinterpolationprocedure.Typically, thisancillaryinformationisobtainedfromdeterministicmodelsof theprocesscausingthevariationin thegroundwaterqualityvariable,orfrom measurementsonarelatedvariable,thevariationof whichwascausedby thesameprocess.Incorporationof thefirst typeof ancillaryinformationleadsto theuseof theuniversalkrigingmodelwith basefunctionsthathave a causalinfluenceuponthemeasuredvariable,thesecond

90 SUMMARY

typeleadsto theuseof cokriging.In threeexamplesit is shown how estimatesof zinc, nitrateandsulphateconcentration

in groundwaterimproved by addinginformationfrom a modelof atmosphericdepositionforzinc, informationfrom a modelof nitrateleachingfrom the soil, andsulphatemeasurementsfrom shallow groundwater. Especiallynitrateestimatesimproved considerably. Sincethe sizeof theelementsmodelledwith deterministicprocessmodelsis muchlargerthanthesizeof theareaof influencefor monitoringscreens,further improvementmay be expectedwhen,at themeasurementsites,theprocessmodelsareadjustedto thelattersize.

6. DiscussionandconclusionsOptimizing the monitoring network with respectto measurementintensity would requireknowledgeof the loss(or cost)dueto ‘not monitoring’asa function of monitoringintensity.Assessingsucha function is not trivial: several partieswould have to agreeon it. Becauseinthisstudy‘degreeof knowledge’is expressedastheability to distinguishestimatesfrom criticalconcentrationlevels(targetlevelsor maximumtoleratedlevelsfor humanconsumption),it candirectly be interpretedasthe degreeof knowledgeof groundwaterquality, anda meaningfullossfunctioncanbederivedfrom it. Chapter5 showedthat improving estimateswith ancillaryinformationmay play a role that is similar to increasingthe measurementintensitywhenthemethodfor interpretationof themeasurementsis fixed.

Flow information or other information from deterministicgroundwater quality modelscouldreadilybeusedin theestimationprocedureto improve estimates.Alternatively themapsthatresultedfrom thisstudycouldbeusedfor thedevelopmentof suchmodels.Severalapectsof themethodsasusedin this studycanbeimproved.Whenestimatesof groundwaterqualityvariablesarerequiredat anotherspatialresolutionthanthecurrent4 km × 4 km blocks,slightmodificationsof theestimationprocedureappliedheremayberequired.Themainconclusionsarelistedanddirectionsfor futureresearcharegiven.

HET KARTEREN VAN DE GRONDWATERKWALITEIT IN NEDERLAND

1. AlgemeneinleidingGrondwaterkwaliteit isdegeschiktheidvangrondwatervooreenbepaalddoel(bijvoorbeeldvoormenselijkeconsumptie),enwordt hoofdzakelijk bepaalddoordechemischesamenstellingvangrondwater.Degrondwaterkwaliteit in Nederlandwordtbedreigddoorvervuilingvanagrarischeenindustriëleoorsprong.Lokaalwordtdezevervuilinggemetenopmeerdantienmeterdiepte.Omdatgrondwaterdebelangrijkstebron is voor schoonwater(drinkwater),veroorzaaktdezevervuiling op de langetermijn eenafnamevan de voorraadgrondwaterdat geschiktis voormenselijkeconsumptie.

Ominzicht tekrijgenin dehuidigetoestandvandegrondwaterkwaliteit endetrendmatigeveranderingendaarin,werd tussen1978 en 1984 het landelijk meetnetgrondwaterkwaliteitingericht. Dit meetnetbestaatuit ongeveer 370 regelmatig over Nederlandverspreiddemeetputten(figuur 3.2, pag.27), die jaarlijks wordenbemonsterdop ongeveer10 en25 meterbenedenhetmaaiveld.In hetbeginvandejaren’90 zijn deprovinciesbegonnenmetdeinrichtingvanvergelijkbaremeetnetten,teneindeeenverhoogde(thansverdubbelde)ruimtelijkedichtheidtebewerkstelligen.Omdatdemeetnetteneenaanzienlijkefinanciëleinvesteringbetekenen,ishetbelangrijkteevaluerenof deinformatieovergrondwaterkwaliteitdiemetdehuidigemeetnettenwordtverzameld,voldoendeis.Dit vraagtomdekwantificeringvandematewaarindekwaliteitvanal hetgrondwaterwaarinwegeïnteresseerdzijn kanwordenafgeleiduit dezeinformatie.

Voor het modellerenvan ruimtelijke en temporelevariatie in grondwaterkwaliteit metbehulpvanfysischeenchemischedeterministischemodellen,ishetnoodzakelijk velevariabelen(bijvoorbeeldinitiële voorwaardenengrensvoorwaarden,systeemparameters)tekennen.Opditmomentzijn dezevariabelenniet beschikbaarop eennationaleschaal.Het is echtermogelijkgrondwaterkwaliteit te karterendoorgebruikte makenvaneenvoudigeremodellen,waarbijdeonbekendefactorenwordensamengevoegd tot eenruimtelijk afhankelijke stochastischeterm.Hetdoelvandezestudieis om

• de grondwaterkwaliteit in Nederlandte karteren,gebruik makendevan metingenvannationaleen provinciale meetnettengrondwaterkwaliteit en kaartinformatieaangaandelandgebruikenbodemtype

• de effectenvan meetnetdichtheiden de effectenvan het gebruikvan informatieomtrentlandgebruikenbodemtypeopderesulterendekaartbeeldente latenzien

• de monotone,langjarigesystematischeveranderingenin grondwaterkwaliteit in beeldtebrengenopeenwijzedienauwaansluitbij dewijzewaaropdegrondwaterkwlaiteitin beeld

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92 SAMENVATTING IN HET NEDERLANDS

werdgebracht

• te laten zien hoe grondwaterkwaliteitskaartenkunnenwordenverbeterddoor relevanteaanvullendeinformatie in de schattingsprocedurete betrekken, waarbij de aanvullendeinformatie afkomstig is van deterministischeprocesmodellenof van anderegemetenvariabelen

Het primairedoel is om doordezedoelendehoofdvragenomtrentdehuidigetoestandvandeNederlandsegrondwaterkwaliteit en de trendmatigeveranderingendaarin te beantwoorden.In dezestudiewordenbetrekkelijk simpelemodellengebruikt die het mogelijk maken omde nauwkeurigheidvan schattingenexpliciet te kwantificeren.Deze nauwkeurigheid is inde resulterendekaartbeeldenbetrokken, vooruitlopendop de vraagin hoeverre de kwaliteitvan al het grondwater waarin we geïnteresseerdzijn kan worden afgeleid uit de huidigemeetnetinformatie.

2. Statistisch karterenNormaalgesprokentonenkaartenverschijnselendie direct waargenomenzijn. Wanneerdit zois, danovertreftdeafbeeldingsnauwkeurigheiddoordetoegepasteschalinggewoonlijk niet dewaarnemingsnauwkeurigheid.In milieukundigetoepassingenishetechternormaaldatdein eenkaartafgebeelde‘waarnemingen’nietbetrekkinghebbenopdirectwaargenomenverschijnselenmaaropverschijnselendieslechtsbij benaderingbekendzijn,zoalsschattingenvaneengrootheidgebaseerdop beperktesteekproefinformatie.Om in dit geval niet meer‘kennis’te suggererendanvoorhandenis, is hetnoodzakelijk omdeafbeeldingsnauwkeurigheidtebeperken.

Wanneerwaarnemingenbeschikbaarzijn, dan kunnen statistischemodellen wordengebruikt om een schatting te maken van de waarde van de waarnemingsvariabele opeen niet-bemetenlocatie of van het gemiddeldeover een bepaaldgebied, en om eenschattingsnauwkeurigheidhieraantoe te kennen.Geostatistischemodellen— eenspecifiekeklassevanlineairemodellenmetgecorreleerdewaarnemingen— zijn speciaalgeschiktvoorhetkarterenvaneenvariabele:hetschattenvandewaardevaneenvariabeleopeenspecifiekelocatieof van het gemiddeldeover eenspecifiekgebied.Geostatistischemodellenkunnenwordengezienalseengeneralisatievandestatistischemodellendie zijn gebaseerdop onafhankelijkewaarnemingen.

Om de lezervan eenkaartniet uit te nodigende nauwkeurigheidsinformatiete negerenis het noodzakelijk schattingen schattingsnauwkeurigheidin eenkaart gecombineerdweerte geven als een betrouwbaarheidsinterval. Dit is gedaandoor de relatieve ligging van hetbetrouwbaarheidsinterval ten opzichtevan elk van vier referentieniveausaf te beeldenin éénvanvier deelkaarten(zie het voorbeeldvan aluminium,figuur 3.4, pag.30). Eenalternatiefisombeidezijdenvanhetinterval in éénenkelekaartaf tebeelden,diereduceerttot deklassieke,‘deterministische’kaartwaareenbetrouwbaarheidsintervalgeheelbinneneenlegendaklassevalt(figuur3.5, pag.31).

In grote lijnenis devolgendeprocedure voor het in kaartbrengenvandegrondwaterkwa-liteit gehanteerd.Binnencategorieënbodemtypeenlandgebruik werdenvanuitzorgvuldig gese-lecteerdewaarnemingenschattingenvoor4 km × 4 km-ruitgemiddeldenverkregendoorgebruikte maken van ordinary kriging op de log-schaal.Na terugtransformatie naarde oorspronkelij-

93

keschaalwerdenbenaderende95%betrouwbaarheidsintervallenvoorruitmedianeconcentratiesverkregen,welkezijn afgebeeldin kaarten.Dezeprocedureisenigszinsaangepastomrekeningtehoudenmetlocatiespecifieke(niet-stationaire)tijd-voorspellingsvariantiesin hoofdstuk4,enomschattingenteverbeterendoorgebruik temakenvanaanvullendeinformatie in hoofdstuk5.

3. Kaartenvandegrondwaterkwaliteitin Nederlandop5-17meterdieptein 1991Van25grondwaterkwaliteitsvariabelenwerdenkaartengemaaktdoorruitmedianeconcentratiesvoor 4 km × 4 km-ruitenteschatten(PebesmaenDeKwaadsteniet,1994).Schattingenwerdengepresenteerdin de vorm van benaderende95%betrouwbaarheidsintervallen,gerelateerdaaneenviertalveelalaannormstellingontleendereferentieniveaus.Dekaartenzijn gebaseerdop425metingenuit landelijk enprovincialemeetnettengrondwaterkwaliteit.De schattingsprocedureis gebaseerdop eenstratificatienaarbodemtypeenlandgebruik.Binnenelke categoriewerdende waarnemingengeïnterpoleerd,rekening houdendmet regionaleverschillenin niveauenin ruimtelijke afhankelijkheid tussenwaarnemingen.De gebruiktestratificatieis essentieelgebleken:hetnietof slechtspartieeluitvoerenvandebetreffendestratificatieleidt tot wezenlijkafwijkendekaartbeelden.Deeffectenvanmeetnetdichtheidzijn onderzochtdoorkaartbeeldentegenererenopbasisvandemeetpuntenverzamelingvanhetlandelijkmeetnetgrondwaterkwaliteitalleen.Wezenlijke aanpassingenvoortkomenduit toevoeging van de provinciale meetnettenkunnenwordentoegeschrevenaandetoegenomeninformatieaangaandekorte-afstand-variatieendetoegenomenhoeveelheidlocatiespecifiekeinformatie.

Beleidsmatigkunnende kaarten(a) rechtstreekswordenbenutenerzijdsin relatietot demeerdiffusevormenvan grondwaterverontreiniging,anderzijdsvoor het verkrijgenvan eenruimtelijk gedifferentieerdbeeld van achtergrondwaardenwaartegen strikt locatiespecifiekeverontreinigingssituatieskunnen worden beoordeeld;(b) meer indirect worden benut alsinvoer- en toetsingsmateriaalvoor de de beleidsonderbouwingten dienstestaanderegionaleen landelijke modellenvoor grondwaterkwaliteitsontwikkeling. De kaartenkunnenwordengezienals eenvertalingvan de uit de meetnettenafkomstigepuntinformatienaarinformatieop eenruimtelijk schaalniveauwaaropbijvoorbeeldgangbareregionalemodelleringaansluit.Meetnetoptimalisatiekan efficiënt en locatiespecifiekworden gericht op aspectenvan hetgegenereerdelandelijke beeld.In het algemeenkan worden gesteld,dat de gepresenteerdekaarten weinig aanleidinggeven tot meetnetreductie(brede betrouwbaarheidsintervallen).In hoofdstuk 3 wordt de gehanteerdemethodiek geïllustreerdmet resultatenvoor dealuminiumconcentratiein hetgrondwater(figuur3.4en3.5, pag.30en31).

4. Kaartenvanveranderingenin degrondwaterkwaliteitin Nederlandop5-17meterdiepteVaneen12-talgrondwaterkwaliteitsparameterswerdenlandsdekkendestatistischekaartbeeldenvanveranderingenin detijd gepresenteerd(PebesmaenDeKwaadsteniet,1995).In hetbijzondergingdebelangstellingdaarbijuit naarhetonderkennenvanmonotonelangjarigecomponenteninhetveranderingsbeeld.Aandebasiswerden,aansluitendbij langjarigewaarnemingstijdreeksenin een250-talmeetpuntenvanhet landelijk meetnetgrondwaterkwaliteit, via regressieanalysekorte-termijn-voorspellingenin de betreffendemeetpuntenbepaald,die vervolgensruimtelijkwerdengeïnterpoleerd.Schattingsonnauwkeurigheidvan de korte-termijn-voorspellingenindemeetpuntenwerdmedein beschouwinggenomen.Bij ruimtelijkeinterpolatiewerdexplicietrekeninggehoudenmethet (locatiespecifieke)niveau,degroottevanruimtelijkevariatieende

94 SAMENVATTING IN HET NEDERLANDS

ruimtelijke samenhangin deruimtelijke variatievaneente interpolerenvariabele.Ruimtelijkeinterpolatievond plaatsper categorie landgebruik/bodemtype.De ruimtelijke resolutievandegepresenteerdekaartbeeldenis beperkttot 4 km × 4 km-ruiten.In dekaartenworden,voortwee extrapolatietijdstippen(1980 en 2000), schattingenvan ruitmedianewaardengegevenin de vorm van benaderende95% betrouwbaarheidsintervallen, gerelateerdaan een 4-talparameterafhankelijke,veelalaannormstellingontleendeniveaus.

De resulterendekaarten(kaliumconcentratiein figuren 4.4 en 4.5, pag.52 en 53) lateneen rijkdom aan trend-indicatieszien. Primair kan worden geconcludeerddat een sterkruimtelijk gedifferentieerdetrend-benaderinghaalbaaren zinvol is; zinvol omdat (i) ruitenmet eensterk geprononceerdetrend daarbijkunnenwordenonderscheidenvan eenmindergeprononceerdeomgeving en (ii) trendsin onmiddellijk aangrenzendeof nabijgelegenruiten(ook binneneenzelfdecategorie landgebruik/bodemtype)van tegengestelderichting kunnenzijn. Voorzichtigheidbij de interpretatievan de kaartbeeldenis echter, geziende inhoud enonderbouwingvan eenaantalaspecten,geboden;de kaartbeeldenweerspiegelenslechtseeneerstestap in de richting van een ruimtelijk gedifferentieerdestatistischetrend-benadering.In hoofdstuk 4 wordt de gehanteerdemethodiek geïllustreerdmet resultatenvoor dekaliumconcentratiein hetgrondwater.

Het terugbrengenvan de nu gangbarejaarlijkse bemonsteringsinspanningnaar eenmeer-jaarlijkse dient te worden ontraden.Wezenlijke aanpassingenvan het gepresenteerdelandsdekkendebeeldmogenwordenverwacht,wanneermetdetijd langjarigemeetreeksenin demeetpuntenvandeprovincialeMeetnettenGrondwaterkwaliteit terbeschikkingkomenendezemedein debeschouwingenwordenbetrokken.

5. Het verbeterenvanschattingenmetbehulpvanaanvullendeinformatieKaarten met schattingenvoor grondwaterkwaliteitsvariabelenzoals deze eerder werdenverkregendoormiddelvanordinarykrigingbinnenbodemtype/landgebruikcategorieënkunnenwordenverbeterddoorrelevanteaanvullendeinformatiein deinterpolatieproceduretebetrekken.Dezeaanvullendeinformatieis ontleendaandeterministischemodellenvan het procesdat devariatiein degrondwaterkwaliteitsvariabeleveroorzaakte,of aanmetingenaaneengerelateerdevariabele,van welke de variatiewerd veroorzaaktdoor hetzelfdeproces.Het betrekken vanheteerstetypeaanvullendeinformatiein deinterpolatieprocedureleidt tot hethetgebruikvanuniversal kriging metbasis-functiesdieeencausaleverbandhebbenmetdegemetenvariabele,hetbetrekkenvanhettweedetypeleidt tot hetgebruikvancokriging.

In drievoorbeeldenwordtgetoondhoeschattingenvanzink-,nitraat-ensulfaatconcentratiein het grondwater verbeterdendoor respectievelijk modelinformatieomtrent atmosferischedepositie van zink, modelinformatie omtrent nitraatuitspoelinguit de bodem, en doormeetinformatieomtrentsulfaatin ondiepgrondwatertebetrekkenin deschattingsprocedure.Metnamedenitraatschattingenverbeterdenaanzienlijk.Omdatdeomvangvandeelementenwaaropde deterministischeprocesmodellenbetrekkinghebbenvele malengroter is dan de omvangvan de invloedsgebiedenvoor waarnemingsfilters,is eenverdereverbeteringte verwachtenwanneerop dewaarnemingslocatieshet procesmodelwordt aangepastaandeomvangvan deinvloedsgebieden.

95

6. DiscussieenconclusiesHet optimaliserenvanhetmeetnetmetbetrekkingtot demeetintensiteit(ruimtelijkedichtheid,meetfrequentie)vergt debepalingvandeverlieskostenalsgevolgvan‘niet meten’alseenfunctievan de meetintensiteit.Het vaststellenvan eendergelijke functie is niet triviaal:verschillendepartijenzoudenhieroverconsensusmoetenbereiken.Omdatin dezestudiede‘matevankennis’wordt uitgedruktin termenvandemogelijkheidom schattingenteonderscheidenvankritischeconcentratieniveaus(streefwaarden,maximaaltoelaatbarewaarden),kandit directalsdematevan kennisomtrent grondwaterkwaliteit worden geïnterpreteerd,en kan een betekenisvolleverlieskostenfunctiehiervanwordenafgeleid.In hoofdstuk5 werdgetoonddat het verbeterenvanschattingenmetbehulpvanaanvullendeinformatieeenrol kanvervullendie vergelijkbaaris methetverhogenvandemeetintensiteitondereenvastgelegdewijze vaninterpretatievandemetingen.

Informatie omtrent grondwaterstromingof andereinformatie vanuit deterministischeprocesmodellenzouden enerzijds eenvoudig in de schattingsprocedurekunnen wordenbetrokkenom schattingente verbeteren,anderzijdskunnendekaartenuit dezestudiewordengebruiktvoordeontwikkelingvandergelijkemodellen.Verschillendeaspectenvandegebruiktemethodenkunnenwordenverbeterd.Voor schattingenvan grondwaterkwaliteitsvariabelenopeenandereruimtelijke resolutiedande4 km × 4 km-ruitenkunnenkleineaanpassingenin deschattingsprocedurenoodzakelijk zijn.Debelangrijksteconclusieswerdengetoondenrichtingenvoor toekomstigonderzoekwerdengenoemd.

TableA1. 25Groundwaterqualityvariables

Al, As, Ba, Ca, Cd, Cl, Cr, Cu, DOC, EC, Fe, pH, HCO3, K, Mg, Mn, Na, NH4-N, Ni,NO3-N, Pb, P-tot, SO4, Sr, Zn

TableA2. Thesamplevariogram

Thesamplevariogram,^� Z(hi) is calculatedfrom observationsz(x) by:

^� Z(hi) �� 192Nh(i)

j,k [z(xj) � z(xk)2] (A1)

where

z(xj) theobservationat locationxj

and,for every i (i �� 1…q):

j,k sumfor all observation pairsz(xj),z(xk) in thecategory for whichhjk�� xj � xk �I� [(i � 1) J x , i J x)

hi themeanvalueof thecorrespondinghjk’s

Nh(i) numberof observationpairsin thei-th summation

q, J x dependingon thenumberof observations(in Chapter3,q �� 20and J x �� 5000)

96

97

TableA3. Threevariogrammodels

name equation code

� (h) �� 0, h �� 0� (h) �� a, a constant, h K 0

nugget a Nug(0)

� (h) �� b( 3h/2c � 3h / 32c ), b,c constant, 0 h c

� (h) �� b, h K c

spherical b Sph(c)

exponential � (h) �� d(1 � exp( � h/e)), h L 0 d Exp(e)

TableA4. 9 Groundwaterqualityvariables

Al, Ca, Cd, Cu, K, NO3-N, Pb, P-tot, Zn

TableA5. 12Groundwaterqualityvariables

Ca, Cl, EC, pH, HCO3, K, Mg, Na, NH4-N, NO3-N, P-tot, SO4

TableA6. Theresidualsamplevariogram

Whenz(x), then measurementsz(xj) canbe representedby a linear modelin p basefunctionsf p(x)

Z(xj) ��p

i �� 1f i(xj)� i + e(xj),

�j �� 1…n,

ordinaryleastsquares(OLS)residualsarecalculatedby

^e(xj) �� z(xj) � f (xj)^� ,

with^� �� (Fx′Fx

� 1) Fx′z(x) where � �� (� 1, … , � p)′, Fx�� (f 1(x),… , f p(x)) and f i(x) ��

(f i(x1),… , f i(xn))′. For model(5.2) andtheexamplesA andB, p �� 2, f 1(x) �� 1,�

x, andf 2(x) isthe f (x) in (5.2). Thesampleresidualvariogram^� e is calculatedasin TableA2 by substitutingtheOLSresiduals^e(xj) for observationsz(xj).

98 TABLES

TableA7. Thesamplecrossvariogram

Thesamplecrossvariogram^� 12(hi) for two observationvariablesz1(x) andz2(x) is calculatedby

^� 12(hi) �� 192Nh(i)

j,k [(z1(xj) � ^m1) � (z2(xk) � ^m22)]

with ^m1 and ^m2 thesamplemeansof z1(x) andz2(x), therestof thesymbolsasin TableA2.

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104 REFERENCES

DM

ANKWOORD

Dit proefschriftwas er niet geweestzonderde directeof indirectehulp en steunvan veleanderen,waarvanik hiereenaantalwil noemen.Mijn samenwerkingmetHansdeKwaadstenietgedurendedit projecthebik in veelopzichtenalserg inspirerendervaren.PeterBurroughwil ikbedankenvoor de interesse,ondersteuningenkritiek tijdenshetonderzoek.GerardHeuvelinkwil ik bedankenvoorzijn kritischebeoordelingvanvroegeenlateversiesvanhetmanuscript,enGerardHeuvelink enMarcBierkensvoordevelediscussies— waarof wanneerook.

Verdergaatmijn dankuit naarCeesWesselingvoordehulpbij hetprogrammerenennaarPetervanHorssenvoordevele‘plakavonden’.Ookgaatmijn dankuit naarArthur vanBeurden,Rob van de VeldeandJaapWillems voor het zitting nemenin mijn begeleidingscommissie,naarCeesMeinardi en Wil van Duijvenboodenvoor de kritischeopmerkingenbij hoofdstuk1, naarRobAlkemade,Leo Boumans,Gerardvan Drecht,HermanPrins,andHansReijndersvoor develediscussiesover meetnettenenstatistiek,ennaarJanKunst,RolandLiesteenHansVerlouwvoordehulpbij demoeilijkedelenvanhetGIS-werk.Tot slotbedankik Ellenvoordevoortdurendesteun,metnamegedurendedezomervan’95.

105

CURRICULUMN

VITAE

Ik bengeborenop3 juli 1967teBeetsterzwaag,alwaarik van1971tot 1979hetbasisonderwijsgenoot.Van 1979 tot 1985 volgde ik Atheneum-Baan ‘RSG het DrachtsterLyceum’ teDrachten.In 1985begon ik in Utrechtaande universitaireopleidingFysischeGeografieaande toenmaligeInterfakulteit Aardrijkskundeen Prehistorieaande Rijksuniversiteit Utrecht,de huidige Faculteit der Ruimtelijke Wetenschappenaande Universiteit Utrecht.In januari1991heb ik dezestudieafgerondmet als afstudeerrichtingproceskunde.Mijn eindveldwerkbetrof het onderwerp‘ruimtelijke en temporelevariatie van bodemfysischeparametersinde Ardèche,’ mijn afstudeerscriptiebetrof de steekproefonzekerheidvan variogramschatters.Op 1 februari 1991ben ik bij de vakgroepFysischeGeografievan de Universiteit UtrechtbegonnenaaneenAIO-onderzoek,gefinancierddoorenuitgevoerdbij hetLaboratoriumvoorBodem-enGrondwateronderzoekvanhet toenmaligeRijksinstituutvoor VolksgezondheidenMilieuhygiëne, thanshet Rijksinstituutvoor Volksgezondheiden Milieu (RIVM). In october1992 werd mijn voltijdse dienstverbandomgezetin een 8/10 dienstverband,waardoorditdoorlieptot 1september1995.Vanaf1september1995tot 1februari1996werdmijn aanstellingverlengddoor het RIVM, om eenbegin te maken met de optimalisatievan de provincialemeetnettengrondwaterkwaliteit. Vanaf 1 februari1996werk ik als post-docbij de vakgroepfysischegeografieen bodemkundevan de Universiteit van Amsterdamaan het tweejarigeEG-project ‘Assessmentof cumulative uncertainty in spatial decision support systems:applicationsto examinethecontaminationof groundwaterfrom diffusesources.’

mapping groundwater quality in the netherlands · mapping groundw ater quality in the netherlands...

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