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
32nd SCARCE ANNUAL CONFERENCE: Integrated modelling and monitoring at different river basin scales
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
42nd SCARCE ANNUAL CONFERENCE: Integrated modelling and monitoring at different river basin scales
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
52nd SCARCE ANNUAL CONFERENCE: Integrated modelling and monitoring at different river basin scales
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:
62nd SCARCE ANNUAL CONFERENCE: Integrated modelling and monitoring at different river basin scales
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
72nd SCARCE ANNUAL CONFERENCE: Integrated modelling and monitoring at different river basin scales
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
2nd SCARCE ANNUAL CONFERENCE: Integrated modelling and monitoring at different river basin scales
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[ ] [ ] [ ]⎪ ⎪⎩ ⎭
2nd SCARCE ANNUAL CONFERENCE: Integrated modelling and monitoring at different river basin scales
( 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
102nd SCARCE ANNUAL CONFERENCE: Integrated modelling and monitoring at different river basin scales
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= − −
112nd SCARCE ANNUAL CONFERENCE: Integrated modelling and monitoring at different river basin scales
[ ] [ ] [ ]( ){ } [ ] [ ]( ){ }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:
122nd SCARCE ANNUAL CONFERENCE: Integrated modelling and monitoring at different river basin scales
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
132nd SCARCE ANNUAL CONFERENCE: Integrated modelling and monitoring at different river basin scales
Basic hydrological modelBasic hydrological model
TETIS modelTETIS model
142nd SCARCE ANNUAL CONFERENCE: Integrated modelling and monitoring at different river basin scales
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
Ppt MeanSimulatedObserved
40
50
60
s]
0
1
2
3
Ppt MeanSimulatedObserved
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
Hydrograph - Validation Event 2 - Station Q006 Correction Factors calibrated in station Q001
25 0
1Ppt MeanSi l t d
Hydrograph - Validation Event 3 - Station Q007 Correction Factors calibrated in station Q001
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
192nd SCARCE ANNUAL CONFERENCE: Integrated modelling and monitoring at different river basin scales
Results: spatial validationResults: spatial validation
202nd SCARCE ANNUAL CONFERENCE: Integrated modelling and monitoring at different river basin scales
Results: temporal validationResults: temporal validation
212nd SCARCE ANNUAL CONFERENCE: Integrated modelling and monitoring at different river basin scales
Results: spatial-temporal validationResults: spatial-temporal validation
222nd SCARCE ANNUAL CONFERENCE: Integrated modelling and monitoring at different river basin scales
Results: performance vs sub-basin areaResults: performance vs. sub-basin area
232nd SCARCE ANNUAL CONFERENCE: Integrated modelling and monitoring at different river basin scales
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.
242nd SCARCE ANNUAL CONFERENCE: Integrated modelling and monitoring at different river basin scales
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
252nd SCARCE ANNUAL CONFERENCE: Integrated modelling and monitoring at different river basin scales