horizontal and vertical kriging of soil properties along a transect in southern new mexico
TRANSCRIPT
Horizontal and Vertical Kriging of Soil PropertiesAlong a Transect in Southern New Mexico
M. H. NASH, L. A. DAUGHERTY,* A. GUTJ'AHR, P. J. WIERENGA, AND S. A. NANCE
ABSTRACTThe application of geostatistical analysis to evaluate a soil tran-
sect was explored by measuring soil properties in four equal depths(0-0.3, 0.3-0.6, 0.6-0.9, and 0.9-1.2 m), at 30-m intervals along atransect of 1800-m length in southern New Mexico. Semivariogramswere computed for clay, sand, coarse fragments, and CaCO, percentfor each depth along the transect. The horizontal semivariogramswere combined to form one semivariogram to account for the ani-sotropy in the measurement procedure. A similar procedure was usedfor vertical semivariograms to account for the anisotropy in the mea-surement units. The combination of the equations describing thehorizontal and vertical semivariograms was used in a kriging pro-gram. The kriging program was used to estimate values of soil prop-erties at 2245 grid points from the 240 original measurements. Two-dimensional, kriged contour maps of clay, sand, coarse fragments,and CaCO, percent were constructed. It appears the kriging tech-nique has excellent potential in preparation of two-dimensional ver-tical contour maps of soil properties.
erate a new set of data for unmeasured sites. Transectvariation was examined through vertical contour maps.
MATERIALS AND METHODSStudy Area
The study was conducted on the New Mexico State Univ.College Ranch adjacent to the USDA Jornada Exp. Range,which is about 40 km north of Las Cruces. Data collectionin this study was conducted along a 1800-m transect chosenbecause of the apparent uniformity on the landscape. The1800-m transect is part of a 2700-m transect study fundedby the National Science Foundation, along which data onrain, soil moisture, vegetation, and small mammals are col-lected over an extended time period (Wierenga et al., 1987).The present transect traverses two geomorphic surfaces (Gileand Grossman, 1979). The lower part of the transect is onthe Jornada II geomorphic surface. The upper part of thetransect is on the Organ and Isaack's complex geomorphicsurface.
IN ALMOST ALL SOIL SURVEYS, some attempt is madeto assess the scale on which the soil changes (Web-
ster, 1973). The information required is often ob-tained from transects. By recording soil properties ata constant close spacing on such transects, data canbe used to reveal patterns of variation. Studies by Wang(1982) indicated soil information collected by thetransect method is obtained without bias, therefore,such information provides a better estimate of the realrange of the variation of soil properties than thoserandomly chosen on the landscape. Recent advancesin geostatistics provide ways to quantify and utilizespatial dependence of soil properties and its conse-quences for classification and surveying an area (Web-ster, 1973). Sampling dependence can be used advan-tageously in interpolating between both observationsand mapping. The basis for such a process was intro-duced by Matheron (1971) and is known as kriging.Kriging was applied to soil salinity problems by Hajra-suliha et al. (1980), and to soil survey by Burgess andWebster (1980). Kriging has also been used in hy-drology (Delhomme, 1978) to describe plant cover(Butler, 1981), infiltration rate (Viera et al., 1981) andwater table levels (Kies, 1982).
The kriging technique predicts values for interpo-lation without bias, and with minimum variance. Also,because the variance of the estimates can itself be es-timated, interpolated values can be used with knownconfidence (Olea, 1975). The purpose of the work wasto display the variation of clay, sand, coarse fragment,and CaCO3 percent within map units along a transect.A linear estimation method (kriging) was used to gen-M.H. Nash, L.A. Daugherty, P.J. Wierenga, Dep. of Agronomy andHorticulture, S.A. Nance, Dep. of Experimental Statistics, NewMexico State Univ., Las Cruces, NM 88003-0003; A. Gutjahr,Mathematic Dep., New Mexico Inst. of Mining and Technology,Socorro, NM 87801. Journal Article 1328, Agric. Exp. Stn., NewMexico State Univ. Received 22 June 1987. 'Corresponding author.Published in Soil Sci. Soc. Am. J. 52:1086-1090 (1988).
ClimateClimate at the College Ranch is characterized by an abun-
dance of sunshine, wide range of diurnal temperatures, lowrelative humidity, and precipitation of about 225 mm (Buf-nngton and Herbel, 1965). More than half the annual pre-cipitation occurs from July to October.
TransectThe transect was surveyed by conventional soil survey
methods by observing the steepness, length and shape ofslopes, the general pattern of the drainage ways, macro andmicro topography, the kinds of native plants and the loca-tion of their changes, and many visible soil characteristics(Nash, 1985).
The transect was divided into 60 observation sites spaced30 m apart. At each site, samples were taken to a depth of120 cm divided into four equal intervals (0-0.3, 0.3-0.6,0.6-0.9, and 0.9-1.2 m). There were 240 samples taken asbulk samples with a 7-cm-diam. bulk auger. Samples werecomposited for analysis. Particle-size analysis was deter-mined by the pipette method, and the CaCO3 determinedby the titration method (Soil Survey Staff, 1972).
SoilsThere are four soils separated by conventional soil survey
methods along the transect. Those soils are: Buckelbar, Ber-inp, Onite, and Dona Ana. All soils are Typic Haplargids,mixed, thermic. The family particle-size class for Bucklebarand Berino soils is fine-loamy and for Onite and Dona Anasoils is coarse-loamy.
Bucklebar soils are deep and well drained. They formedin mixed alluvial sediment. They occur on a very gentlysloping broad pediment. The elevation is approximately 1300m and has deep groundwater. Typically, the surface layer isbrown sandy loam about 5-cm thick. The subsoil is brownand reddish brown, heavy sandy loam and sandy clay loamabout 70-cm thick. The substratum is light brown loam andsilty clay loam to a depth of 130 cm or more. Permeabilityof the Bucklebar soil is moderate. The depth of the root zoneis 130 cm or more. The available water capacity is high.Surface runoff is medium and water erosion hazard is mod-
1086
NASH ET AL.: HORIZONTAL AND VERTICAL KRIGING OF SOIL PROPERTIES IN SOUTHERN NEW MEXICO 1087
erate. The Bucklebar soil does not have a calcic horizonwithin 1 m and has 18% or more clay in the control section.
The Berino soils are similar to Bucklebar, except they havea calcic horizon within 1 m.
The Onite soils are similar to Berino soils, except theyhave <18% clay in the control section and have a coarsertexture than Berino soils. The soils are not calcareous orshow no effervescent in the first 5 cm from the soil surface.They have a calcic horizon within 1 m from soil surface.
The Dona Ana soils are similar to Onite soils, except theyare calcereous throughout the soil profile.
GeostatisticsSoil does not always vary isotropically, even in small areas.
Alluvial deposits generally have greater spatial dependencein the direction parallel to the water movement than at rightangle to it. Likewise, where the land surface bevels, a se-quence of alluvial fan deposits or a sequence of depositionand dissection occurs. The calculated semivariogram at thefirst surface depths will be different from the calculated semi-variogram at the other depths. Because samples were takenat four depths along the transect, four semivariograms wereconstructed to reflect more information about the variablebehavior with depth. Figure 1 shows four semivariogramsfor clay percent. The semivariograms behaved differently ateach depth, having different sills and ranges of dependencyin the horizontal direction. To account for this type of an-isotropy, the four semivariograms were added and averagedto form one semivariogram (Journel and Huijbregts, 1978).As an example, if the semivariograms were calculated at lag1 for each depth, the average of the results of the four semi-variograms at this point is the value of the average semi-variogram at this lag (h), i.e.,
[7n(/0 74i(/0]/4 [1]and similarly for the other lags. We will call the result thehorizontal semivariogram.
There is a 100-fold difference in the sampling distancebetween the horizontal (30 m) and vertical (0.30 m) dimen-sion. This type of anisotropy in the measuring scheme wasaccounted for by constructing a semivariogram for each ver-tical measurement, and by constructing a semivariogram foreach site. To obtain the mean equation for the vertical dis-tance, the 60 semivariograms were averaged to form onesemivariogram. The zone of influence surrounding a samplelocation and the sill can now be measured from the hori-zontal and the vertical semivariograms. Thus, the range ofthe semivariogram provides information about the rate ofchange of a variable with respect to distance. Because thehorizontal distance is 100 times more than the vertical dis-tance, however, the influence of horizontal range is muchhigher than the vertical range. To account for this type ofanisotropy, the equations of the horizontal and verticalsemivariograms were combined additionally; i.e., gammavalue is equal to the horizontal equation plus the verticalequation (Journel and Huijbregts, 1978, p. 182).
According to Delfiner and Delhomme (1973), the inter-pretation of the semivariogram equation is focused on thebehavior of the semi-variogram between the origin and therange of dependency. An exponential model was found to
Table 1. The sill and the range of the horizontal and verticalsemivariograms. ___
Horizontal semivariogram Vertical semivariogram
Variable Sill Range, m Sill Range, m
ClaySandCaCO,Coarse fragments
28.242.373.533.4
600660300510
31.332.870.932.6
0.600.300.400.30
best describe the horizontal semivariogramy(h) = A[l - exp(- x/a)] [2]
while a spherical model was the best fit for the vertical semi-variogram
Y(/z) = £[(3/2) (y/a) - (1/2) (y/d?} [3]where K is the sill, a is the range of dependency, and x andy are the horizontal and vertical coordinates.
To calculate gamma (the variogram) in the kriging pro-gram, the two equations should be combinedy(x,y) = horizontal equation + vertical equationy(x,y) = K[\ -exp(- x/a)] + K[(3/2)(y/a)
- (1/2) (y/a)3]. [4]Table 1 shows the range and the sill for the horizontal andvertical semivariogram for the four selected variables.
Equation 4 can now be used in a kriging program to solvethe linear system
[X] = [C]~'[5] [5]
60-
§ 50-
| 40
f ' O
I 2°j10
0-30 cm Depth
1201
10 15 20 25 30 35 40
Log Distance (1=30 m)Fig. 1. Semivariograms for clay percent for four depths along the
transect.
1088 SOIL SCI. SOC. AM. J., VOL. 52, 1988
where [C] is the cpvariance among all experimental pointswithin the estimation neighborhood of a point, and [C]"1 isthe inverse matrix of [C]. The matrix [C] remains the samefor a given set of support data and [C]~' has to be computedone time for all lambdas [Xs]. The matrix [B] is another setof covariances between the point whose value is to be es-timated and each experimental point. The only unknownsare the weights [Xs], which must be solved through the krig-ing program (David, 1977). The best estimator must be un-biased and must have minimum variance. In other words,the following properties should be met:
1. Minimum variance of estimation = min [Z(x) —
60-i
50-
40-
30-
20-
10-
2. Unbiased E [Z(x) - Zk*(x)] = 0.3. Exact interpolation, predicted values at experimental
data points are identical to observed values (Clark,1979) where Z(x) = observed values and Z*(x) =predicted values.
To ensure the expected kriged value Zk* and the actualobserved values Z, for the same position, x, = 0, it is re-quired that
= 1 (Journel and Huijbregts, 1978) [6]
Xs values are a function of the distance from the measuredposition to the kriged position. The general formula is
_ _
<ac/)
0
(xk) = £\j Z(x,) [7]
0 405 10 15 20 25 30 35Lag Distance (1=30 m)
Fig. 2. Average semivariogram of clay percent for four depths forthe horizontal distance (horizontal semivariogram).
where Z* is the estimated value at position Xk, \i are theweights associated with each of the values of Z measured atlocations x,, and N is the number of locations (Clark, 1979).This method is an optimum interpolator because it allowsthe variances of the estimates to be estimated, and is ex-tremely helpful in identifying improved sampling schemes.The kriging interpolation method is becoming more com-mon in soil science studies (Burgess and Webster, 1980). Itshould be pointed out, through its usage for soil observa-tions taken within the soil body (profile) and between ob-
120200 400 600 800 1000
DISTANCE (meters)1200 1400
0.0-3.0%
BUCKLEBAR
3.5-10.0% 10.5-17.0% 17.5-23.0% 23.5-30.0%
Fig. 3. Contour map of coarse fragments percent along the transect.
BERING | ONITE DONA ANA
I600 I800
H 30.5-44.0%
I20200 400 600 800 IOOO
DISTANCE (meters)1200
51.0-54.5% 55.0-61.0% 61.5-68.OOA 68.5-75.0% ^Fig. 4. Contour map of sand percent along the transect.
I400
75.5-82.0%
I600 1800
82.5-86.0%
NASH ET AL.: HORIZONTAL AND VERTICAL KRIGING OF SOIL PROPERTIES IN SOUTHERN NEW MEXICO 1089
Table 2. An example of the vertical kriging of the clay percentfor the soil site.f
Table 3. Details of the conventional soil survey for the transect, t
Verticaldistance
cm30
35
40
45
50
55
60
Horizontal distance
30
3:7.580.008.210.438.730.679.160.769.500.679.800.43
J10.070.00
35
0.438.430.558.820.729.170.789.450.729.660.559.710.43
t Kriged valueKriged variance
40
8.790.698.930.729.150.809.360.849.500.809.560.729.500.67
45
9.450.779.540.789.630.849.700.869.690.849.600.789.440.75
t Measured
50
0.6810.230.72
10.240.80
10.180.84
10.040.809.810.729.530.67
values.
55
10.920.43
10.980.55
10.940.72
10.830.78
10.590.72
10.210.559.810.43
60
11.733:0.00
11.730.43
11.720.67
11.640.75
11.420.67
10.980.43
10.263:0.00
Map unitcomponent
name
BucklebarBerinoOniteDona Ana
Distanceon
transect, m
0-450450-1050
1050-13501350-1800
No. ofsampling
sites
15201015
From-tositesno.
0-1516-3536-4546-60
Familyparticle-size
class
Fine-loamyFine-loamyCoarse-loamyCoarse-loamy
servations, it becomes possible to construct contours of iso-values that are more meaningful than simplistic values ofmeans calculated from two positions. A computer programfrom a SAS (SAS/ETS, 1982) package was used to generatevertical contour maps.
t All soils are Typic Haplargid, mixed, thermic.
RESULTS AND DISCUSSIONThe kriging method used in this study produced
2245 estimated values from 240 original values foreach soil property used in this study. The result of thekriging program of the clay percent for the soil site,as an example, is shown in Table 2. The kriging planwas to krige each 5 m horizontally and each 5 cmvertically. Notice that the variance of the kriged val-ues should be minimum. The variance should be equalto zero whenever the measured values are kriged. Theestimation variance is higher for unmeasured valuesas the distance from the measured values increases(Table 2).
Figures 3 to 6 show the vertical contour maps forcoarse fragments, sand, clay, and CaCO3 percent, re-
120200 400 600 800 1000
DISTANCE (meters)1200 1400 1600 1800
5.0-8.0% 8.5-13.0% 13.5-19.0% 19.5-24.0% 24.5-30.0%
Fig. 5. Contour map of clay percent along the transect.30.5-33.0%
30
a.ua
I20200 400 600 800 IOOO
DISTANCE (meters)I200 I400 I600 I800
0.0-0.4% 4.5-11.0% 11.5-18.0% 18.5-26.0% 26.5-33.0%Fig. 6. Contour map of CaCO, percent along the transect.
33.5-37.0%
1090 SOIL SCI. SOC. AM. J., VOL. 52, 1988
spectively. Map unit component names from the con-ventional soil survey and the family particle-size classare given in Table 3 (Nash, 1985). The family particle-size class were fine-loamy and coarse-loamy. TheBucklebar map delineation has low coarse fragmentpercent in the first depth (0.13-3.5%) (Fig. 3) and highamounts of coarse fragments (23-30%) in the last twodepths (60-120 cm). Examination of Fig. 3, however,shows the coarse fragments change with distance hor-izontally and vertically for the Bucklebar unit. Sandpercent (Fig. 4) for Bucklebar was high at the begin-ning of this map delineation (75-82%), then decreasedwith distance. The clay percent, on the other hand,was low at the beginning of this delineation (7-13%),then increased with increasing distance. For the Buck-lebar map delineation, the highest clay content was atthe lower depths (29-32%) (Fig. 5).
The Berino delineation, located between 450 to 1050m on vertical contour maps, has similar behavior tothe Bucklebar delineation (Fig 5). The Berino mapdelineation, however, has a calcic horizon, i.e., has ahigher amount of CaCO3 (11-37%) than Bucklebar (4-18%) (Fig. 6).
The Onite delineation is located between 1050 to1350 m on the vertical contour maps. This delineationhas lower clay content than the Bucklebar and Berino.Figure 5 shows the Onite unit has 7 to 13% clay.
The Dona Ana delineation, however, should be cal-careous throughout and should have a calcic horizonat a depth of more than 60 cm. This delineation islocated between 1350 to 1800 m. It has low clay (5-13%, Fig. 5), high sand (75-85%, Fig. 4), and a highcontent of coarse fragments (23-33%, Fig. 3). Figure6 shows the vertical contour map of CaCO3. The DonaAna map delineation has the highest amount of CaCO3(11-37%).
The detailed soil survey shows four map units inthe area covered by the contour map (Table 3). Thegeneral distribution pattern of clay, sand, and coarsefragments on the contour maps indicates there are sev-eral areas of increase or decrease for each variablewith distance or with depth. If the soil boundary de-rived by a conventional soil survey is overlaid on thecontour maps, one can observe the distribution of eachvariable inside the soil body for the surveyed area.The vertical contour maps show the variation withinthe map units. These variations are related to the fol-lowing:
1. The landscape is the main source of soil varia-tion because the elevation and the topography ofthe area is not evenly distributed along the tran-sect.
2. The windblown materials are irregularly distrib-uted (Gile et al., 1981).
3. The study area was formed from alluvial depo-sition materials that were likely affected by dif-ferent events of deposition and/or dissection.
CONCLUSIONSThis paper shows results of applying kriging tech-
niques to soil properties that soil scientists use in eval-uating soils in the field during a conventional soil sur-
vey. It shows how strongly point estimates vary fromplace to place and, in particular, the variation of CaCO3percent, which results in a large nugget effect. Thisindicates the effect of sampling schemes on the rangeof dependence. Thus for this kind of soil property, oneshould sample in closer intervals to reduce the largenugget effect. For future research, one can carry out asimilar analysis by sampling genetic horizons insteadof bulk sampling between observations. Thus, one canpredict genetic changes from one site to another onthe landscape. Kriging has promise for use in the anal-ysis of transects that are a necessary component of soilsurveys. The samples must be collected at short enoughintervals so they are spatially dependent to provideaccurate estimates for semivariograms. The samplingdensity usually requires much time and expense.