application of scaling equations to deal with the ...lluvia.dihma.upv.es/es/publi/congres/060...s1...

Application of scaling equations to deal with the spatial aggregation effect on p gg g

watershed hydrological modelling

M. Barrios and F. Francés

Universitat Politècnica de ValènciaInstituto de Ingeniería del Agua y Medio Ambiente

Research Group in Hydrological and Environmental ModellingResearch Group in Hydrological and Environmental Modellinghttp://lluvia.dihma.upv.es

Spatial scales in soil characteristicsSpatial scales in soil characteristics

O i ti i th i l ( )Organization, in the minor scales (Blöschl and Sivapalan, 1995)General pattern can be explained deterministically by a few number of factorsnumber of factorsIt is possible to delineateCartographic Units but

690000 700000 710000 720000 730000

4380

000

4380

000

Cartographic Units, butSubjectiveScale dependent 43

7000

0

4370

000

.

With modal values

4360

000

4360

000

Leyenda

E.g.: Modal values of vertical saturated permeability in Rambla del Poyo, Spain 690000 700000 710000 720000 730000

4350

000

4350

000

Ks (cm/h)0.004

0.008

0.05

0.07

0.081

0.093

0.1

0.11

0.15

0.15

0.152

0.166

0.192

0.219

0.222

0.23

0.244

0.25

0.314

0.319

0.322

0.33

0.35

0.368

0.397

0.4

0.449

0.456

0.498

0.55

0.69

0.706

0.738

0.75

0.806

0.833

1.929

2.103

2.23

3

p y y , p

22nd SCARCE ANNUAL CONFERENCE: Integrated modelling and monitoring at different river basin scales

Spatial scales in soil characteristicsSpatial scales in soil characteristics

Randomness, in the larger scalesDetailed structure is due to a large number of factorsStochastic models with spatial dependence structure

E.g.: Vertical saturated permeability at each cell in one Cartographic Unit of Rambla del Poyo Spainof Rambla del Poyo, Spain


Problems with heterogeneityProblems with heterogeneity

Scale effects when averaging non linear processes:

Spatial aggregation

25

30

35

40

h

fp = 4 mm/h

5

10

15

20

mm

/h

fp = 65 mm/h

fp = 43 mm/h

0

intervalos de 10 min


Effective parametersEffective parameters

Scale effects when averaging non linear processes:

The mean process is not the result with the mean parameter and/or input

Effective parameter: parameter value which reproduces th t th l b tthe mean process at the mesoscale, but:

Different to the spatial mean of the point scale valuesThey can lose their physical meaning at the meso or macroscaleThey can lose their physical meaning at the meso or macroscale

Non-stationary


Non-linear processesNon-linear processes

Static Storage[ ]1 10;2 uX Max X H H= − +

D X X=

Static StorageWater excedence:

Capillary infiltration:

[ ]1 1;Y Min ETP Hλ= ⋅

1 1 2D X X= −Capillary infiltration:

Evapotranspiration:

[ ]X Mi X kΔ

Surface StorageG it ti l i filt ti [ ];3 2 sX Min X t k= Δ ⋅Gravitational infiltration:

Gravitational Storage[ ]34 ; pX Min X t k= Δ ⋅

Gravitational StoragePercolation:


Synthetic heterogeneitySynthetic heterogeneity

Generation of random parameter fields (H k and k )Generation of random parameter fields (Hu , ks and kp )

PDF of Hu [Beta(a,b)]:( )( ) ( ) ( ) 11 1 ba

u ua b

f H Ha b

−−Γ += −Γ Γu [ ( )]

PDF of ks and kp [LN(µ,σ)]:

( ) ( )a bΓ Γ

( )22

ln21

2

sk

f ek

μ

σ

⎡ ⎤−⎢ ⎥−⎢ ⎥⎣ ⎦=s p

Exponential spatial autocorrelation:

2sk σ π

( )3hah eρ

⎛ ⎞−⎜ ⎟⎝ ⎠=

Sampling algorithm (Pinder and Celia, 2006):Latin Hypercube SamplingCholesky Factorization


Synthetic heterogeneitySynthetic heterogeneity

St ti ti f th d fi ldμ (Hu ) μ (ks) μ (kp) CV=σ/μ

Statistics of the random fields

70100

2060

26

0.51

1.52

Spatial scales

2

18 Correlation lengths: a = 2.5,5 10 50 75 100 150 250 500

MicroscaleS1

Mesoscale S2 # of realizations

Size Notation[m2] [m2]

5, 10,… 50, 75, 100, 150, 250, 500,2500 and 5000 m

[m ] [m ]1 x 1 5 x 5 S2a 5001 x 1 15 x 15 S2b 5001 x 1 45 x 45 S2c 25001 1 100 100 S2d 5000

8

1 x 1 100 x 100 S2d 5000


Scale effect: aggregationScale effect: aggregation

A tiAggregation:

Excedence [ ]2 22n

iX S X=∑Excedence

Gravitational infiltrationFlow aggregation

[ ]3 32n

iX S X=∑

[ ]2 21

ii=∑

Mesoscale effective parameters (inverse solution):

Flow aggregation[ ]3 3

1i

i=∑

[ ] [ ] [ ] [ ]1 1 22 2 2 2u tH S X S H S X S= + −

[ ]( )( )

12 3 2

13 3 2

[ 2] [ 2] [ 2]2

[ 2] [ 2] [ 2]s t

X S t X S X Sk S

X S t X S X S

−

−

⎫⎧ ⋅ Δ =⎪ ⎪= ⎨ ⎬⋅ Δ >⎪ ⎪⎩ ⎭ (similar expression for kp)

9

( )3 3 2[ ] [ ] [ ]⎪ ⎪⎩ ⎭


( p p)

Mesoscale effective parametersMesoscale effective parameters

Hu , ks and kp depend strongly on state variables andinput and are sensible to microscale heterogeneity:

Some sensitivity to CVLow sensitivity to the spatial dependence structureLow sensitivity to the spatial dependence structure


Scaling equationsScaling equations

[ ] ( ) ( ) [ ] ( )1 1 1 1 1 1 1 1 0 47ln ln2 1 1 0 93t t t tX H X H

H S X H H Sω ω− − −⎧ ⎫ ⎧ ⎫+ − + −⎡ ⎤ ⎡ ⎤⎪ ⎪ ⎪ ⎪Φ Φ⎨ ⎬ ⎨ ⎬⎢ ⎥ ⎢ ⎥[ ] ( ) ( ) [ ] ( )1 1 1 1 1 1 1 1 0.47

1 1 1 1 22 2

2 1 1 0.93t t t tu t t ut

H S X H H S ω ωω ω−

⎪ ⎪ ⎪ ⎪= + −Φ + Φ −⎨ ⎬ ⎨ ⎬⎢ ⎥ ⎢ ⎥⎪ ⎪ ⎪ ⎪⎣ ⎦ ⎣ ⎦⎩ ⎭ ⎩ ⎭

[ ] [ ] [ ]( ){ } [ ] [ ]( ){ }2 1 1 2 2 2k S k S X S X S X Sε α ε α[ ] [ ] [ ]( ){ } [ ] [ ]( ){ }2 2 22 1 1 2 , 2 2 ,s s t t ttk S k S X S X S X Sε α ε α= − −

[ ] [ ] [ ]( ){ } [ ] [ ]( ){ }3 3 32 1 1 2 , 2 2 ,p p t t ttk S k S X S X S X Sε β ε β= − −


[ ] [ ] [ ]( ){ } [ ] [ ]( ){ }3 3 3p p t t ttβ β

Scaling equationsScaling equations

The relationship between theThe relationship between themicroscale heterogeneity and theparameters of the scaling equationsp g qwas estimated through multilayerperceptron neural networks:


Study siteStudy site

Goodwin Creek experimental basinGoodwin Creek experimental basin

Basin Area: 21.6 km2

Ephemeral riverEphemeral riverHortonian runoff?16 raingauge stations (temporal

l i f i )resolution of 5 minutes)6 flowgauge stations (calibration atthe outlet station)DEM: 30x30 m2

Five flood events were selected inthe period of 1981 to 1983: peakp pflows from 38 to 106 m3/s


Basic hydrological modelBasic hydrological model

TETIS modelTETIS model


TETIS spatial informationTETIS spatial information

Derived from the DEM:Derived from the DEM:Flow directionSlopeSlopeFlow accumulation

2nd SCARCE ANNUAL CONFERENCE: Integrated modelling and monitoring at different river basin scales 15

TETIS spatial informationTETIS spatial information

Hydrological soil parameters:Hydrological soil parameters:Static maximum capacity

(interception + capillary storage)(interception + capillary storage)Upper soil permeabilitySubstrate permeabilitySubstrate permeability

2nd SCARCE ANNUAL CONFERENCE: Integrated modelling and monitoring at different river basin scales 16

Hydrological automatic calibrationHydrological automatic calibration

Calibration of parameter maps correction factors & initialCalibration of parameter maps correction factors & initial conditionsObjective function: RMSE of the outlet dishargesObjective function: RMSE of the outlet disharges

Hydrograph - Calibration Event - Station Q001 Correction Factors calibrated in station Q001

Streamflow:basically overland flow!! 35

40

45

50

s]

0

1

2

3

]

Ppt MeanSimulatedObserved

basically overland flow!!

15

20

25

30

Dis

char

ge[m

³/s

4

5

6

7

Pre

cipi

tatio

n [m

m]

0

5

10

21:1

0:00

21:3

5:00

22:0

0:00

22:2

5:00

22:5

0:00

23:1

5:00

23:4

0:00

0:05

:00

0:30

:00

0:55

:00

1:20

:00

1:45

:00

2:10

:00

2:35

:00

3:00

:00

3:25

:00

3:50

:00

4:15

:00

4:40

:00

5:05

:00

5:30

:00

5:55

:00

6:20

:00

6:45

:00

7:10

:00

7:35

:00

8:00

:00

8:25

:00

8:50

:00

9:15

:00

Time [hours]

8

9

10

Time [hours]

Hydrological validationHydrological validationHydrograph - Calibration Event - Station Q004 Correction Factors calibrated in station Q001

Hydrograph - Validation Event 3 - Station Q001 Correction Factors calibrated in station Q001

8

10

12

s]

0

1

2

3


40

50

60

s]

0

1

2

3


4

6

Dis

char

ge[m

³/s 4

5

6

7

Pre

cipi

tatio

n [m

m]

20

30

Disc

harg

e[m

³/s

4

5

6

7

Prec

ipita

tion

[mm

]

0

2

21:1

0:00

21:3

5:00

22:0

0:00

22:2

5:00

22:5

0:00

23:1

5:00

23:4

0:00

0:05

:00

0:30

:00

0:55

:00

1:20

:00

1:45

:00

2:10

:00

2:35

:00

3:00

:00

3:25

:00

3:50

:00

4:15

:00

4:40

:00

5:05

:00

5:30

:00

5:55

:00

6:20

:00

6:45

:00

7:10

:00

7:35

:00

8:00

:00

8:25

:00

8:50

:00

9:15

:00

Time [hours]

8

9

100

10

22:4

0:00

23:3

0:00

0:20

:00

1:10

:00

2:00

:00

2:50

:00

3:40

:00

4:30

:00

5:20

:00

6:10

:00

7:00

:00

7:50

:00

8:40

:00

9:30

:00

10:2

0:00

11:1

0:00

12:0

0:00

12:5

0:00

13:4

0:00

14:3

0:00

15:2

0:00

16:1

0:00

17:0

0:00

17:5

0:00

18:4

0:00

19:3

0:00

20:2

0:00

21:1

0:00

22:0

0:00

22:5

0:00

23:4

0:00

Time [hours]

8

9

10


25 0

1Ppt MeanSi l t d


18

20 0

1Ppt Mean

15

20

scha

rge[

m³/s

]

2

3

4

5

pita

tion

[mm

]

SimulatedObserved

10

12

14

16

scha

rge[

m³/s

]

2

3

4

5

itatio

n [m

m]

SimulatedObserved

5

10

Di

6

7

8

9

Pre

cip

2

4

6

8Dis

6

7

8

9

Prec

ipi

0

12:5

5:00

14:0

0:00

15:0

5:00

16:1

0:00

17:1

5:00

18:2

0:00

19:2

5:00

20:3

0:00

21:3

5:00

22:4

0:00

23:4

5:00

0:50

:00

1:55

:00

3:00

:00

4:05

:00

5:10

:00

6:15

:00

7:20

:00

8:25

:00

9:30

:00

10:3

5:00

11:4

0:00

12:4

5:00

13:5

0:00

14:5

5:00

16:0

0:00

17:0

5:00

18:1

0:00

19:1

5:00

20:2

0:00

21:2

5:00

22:3

0:00

23:3

5:00

Time [hours]

100

22:4

0:00

23:3

0:00

0:20

:00

1:10

:00

2:00

:00

2:50

:00

3:40

:00

4:30

:00

5:20

:00

6:10

:00

7:00

:00

7:50

:00

8:40

:00

9:30

:00

10:2

0:00

11:1

0:00

12:0

0:00

12:5

0:00

13:4

0:00

14:3

0:00

15:2

0:00

16:1

0:00

17:0

0:00

17:5

0:00

18:4

0:00

19:3

0:00

20:2

0:00

21:1

0:00

22:0

0:00

22:5

0:00

23:4

0:00

Time [hours]

10

Modelling scenariosModelling scenarios

3 information scales and with or w/o scaling equations3 information scales and with or w/o scaling equationsNotation ScenariosS1 Maps with resolution of 30x30 m2, w/o scaling equationsS1+seS2S2+seS3

p , g qMaps with resolution of 30x30 m2, with scaling equationsMaps with resolution of 1740x1740 m2, w/o scaling equationsMaps with resolution of 1740x1740 m2, with scaling equationsM ith th f th h l t h t / li tiS3

S3+seMaps with the average for the whole catchment, w/o scaling equationsMaps with the average for the whole catchment, with scaling equations


Results: spatial validationResults: spatial validation


Results: temporal validationResults: temporal validation


Results: spatial-temporal validationResults: spatial-temporal validation


Results: performance vs sub-basin areaResults: performance vs. sub-basin area


ConclusionsConclusions

Non-linearities + parameter heterogeneity and/or input variability =>Non-linearities + parameter heterogeneity and/or input variability =>non-stationary effective parameters

It is important the s b grid ariabilit representation in h drologicalIt is important the sub-grid variability representation in hydrologicalmodeling.

Particularly, the use of scaling equations implies:For all information scenarios, a significant model performanceimprovement in validations at internal flowgauges and for the smallestg gstorm events.A better performance of S1+se and S2+se in comparison to thereference model S1.


AcknowledgementsAcknowledgements

This work was supported by the Spanish research projects FLOOD-MED (CGL2088-06474-C02-02/BTE) and Consolider-IngenioSCARCE (CSD2009-00065) and by the Programme ALßan, theEuropean Union Programme of High Level Scholarships for Latinp g g pAmerica, scholarship E07D402940DO.

Thanks for your attention


application of scaling equations to deal with the ...lluvia.dihma.upv.es/es/publi/congres/060...s1...

Documents