Sensitivity of soil moisture related drought indices to

calibration data

Zoubeida Bargaoui (Prof., ENIT, Tunisia) Ahmed Houcine (PhD Student, ENIT, Tunisia)

Problem position

• Modelling time distribution of soil moisture is a key issue for natural vegetation and crop survey and is often adopted when deriving drought awareness indices.

• In drought risk analysis, it is very important to analyze uncertainty in model outputs

Climatic context: Average annual P, R nETR estimation

based on dryness index P/Rn or P/ETP

2

1

))]sinh()cosh(1)(tnh( /[ ETRP

Rn

P

Rn

Rn

PPRn +−=

Budyko (1974) approximation of the solution of 1LWBM (P/Rn)

( )( )

K

K

K

ETPP

ETPP1

1 ETP ETR

+=

Hsuen Chun HC (1999) Approximation of Budyko model (P/ETP)

Budyko model is adopted to estimate average annual ETR using a hydrometeorological network composed by 18 stations and to derive average annual regional runoff Rs (Av P = Av ETR + Av Rs)

According to observations from 18 gauged basins estimated Rs match observed Rs for k=1.5

Models of soil moisture dynamics

• (Eagleson (1978), Milly (1993), Milly(1994), Rodriguez-Iturbe et al. (1999) and Rodriguez-Iturbe (2002)).

• Evaluation of soil moisture and evapotranspiration in terms of :– soil and vegetation parameters – climatic variables

• storm depth and duration, • dry season length, • average annual potential evapotranspiration

Water balance representation EAGLESON, P. S. (1978).Climate, Soil, and Vegetatio n. Dynamics of the Annual Water Balance. W. R. R VOL. 14, 5, 749, 764

•A one layer vertically averaged water budget over the root zone taking into account• infiltration, runoff, drainage, leakage, evapotranspiration losses and capillary rise

( ) ( ) ( )

SE- SL- tS,I dt

dSnZr = Rates of Infiltration I; Leakage L

(deep infiltration); evapotranspirationE, relative soil moisture S

n porosity; Zr soil depth,

( )( )

( ) 1

1eKSL 1

S-S

sat

FC

−−= −Β

Β

FCSe

Threshold s* depends on vegetation type s*<< sfc

Loss from leakage and capillary rise

conductivity KsatField capacity SFC :

Water balance representation Rodriguez-Iturbe et al. (1999) Probabilistic modelling of water balance at a point: the role of climate, Soil, and Vegetation. Proc. R. Soc. London A (1999) 455, 3789-3805.

Gd= exp((nS-a)/b) - c

a, b, c (Ksat, SFC, B)capillary rise component (Kobayachi et al., 2001)

Runoff dependence from soil moisture

• Whenever the soil have enough available storage to accommodate the rainfall infiltrating the soil, it results an increment n soil moisture.

• Whenever the rainfall depth exceeds the available storage, the excess water is converted to excess runoff.

For S=WBC

if P(t)>ETR(t)+Gd(t)Rs(t)=P(t)-ETR(t)-Gd(t)

else Rs(t)=0

For S <WBCif P(t)>ETR(t)+Gd(t)+( WBC-S)

Rs(t)= P(t)-ETR(t)-Gd(t)-( WBC-S)else Rs(t)=0

with WBC = η*Wmax

Runoff computation

Capillary rise component of Kobayachi modelLinkage with pedo transfer parameters and

• 0< σ <1 ; 0 < η < 1

( ) 1

1Kc

1sat −= −Β FCSe

( ) ]S) 1

1K(

1[a FC1satmax +

−Β−= −Β FCSe

LnW

1

b max Β= W

Model parameters– (1) soil water retention curve shape parameter B, slope

of log ψ versus log(θ/θs) regression.

– (2) saturated hydraulic conductivity Ks,

– (3) saturated soil water content θs (volume/volume). – (4) σ resistance of vegetation to evapotranspiration– (5) η the moisture retaining capacity– Parameters estimation: – (1) to (3) are evaluated after Cosby et al. (1984)Multiple linear regression analysis of parameters by

textural class ( %Clay, %Sand, %Silt)– (4) and (5) are data driven

Case study• To illustrate the methodology, a case

study of a basin with an area of 250 km² in arid area of South Tunisia is presented.

• Vegetation is mainly olives, with no irrigation.

• The soil type is sandy.P=0.34; %S=93.3; %C=6.7• Climate: arid

Data

• Meteorological data : (at Chaal meteo station 30 km far).

• T, V, H (1989-1990, 2003-2004, 2004-2005• Pan evaporation (3 years), Piche

evapotranspiration (17 years)• P (17 years); (1989-2005)• Profiles of soil temperature (3 years)

• Basin Hydrometeometeorological data • Ten years of P and Rs (1990-1999)

time series

• A time series of meteorological observations allow quantifying the reference crop evapotranspiration ET0.

• Model inputs are daily average basin rainfall and daily ET0 over 10 years (1989-90, 1998-1999).

•• Average long term rainfall is 228 mm/an. • According to Budyko model, average ETR = 213 mm/an.

• Average basin precipitation during the study period varies from 55 mm (1994-95) to 398 mm (1989-90).

•

Epan, Epiche et ET0 (meteo station)

y = 0,8646x

R2 = 0,7202

0

1

2

3

4

5

6

7

8

0 2 4 6 8 10

Evaporation bac (mm)

Eva

potr

ansp

iratio

n de

réf

éren

ce

(mm

)

y = 1,0022x

R2 = 0,6422

0

1

2

3

4

5

6

7

8

0 2 4 6 8Evaporation piche (mm)

Eva

potr

ansp

iratio

n de

réf

éren

ce

(mm

)

0

1

2

3

4

5

6

7

8

1 26 51 76 101 126 151 176 201 226 251 276 301 326 351

Temps (jour)

Evapotranspiration de référence

(mm)

ET0c ET0v

ET0 = 1325mm (2004-2005) ; 1368mm (2003-2004)

Daily Rainfall and runoff (1990-1999)

0

10

20

30

40

50

60

70

80

90

100

1 264 527 790 1053 1316 1579 1842 2105 2368 2631 2894 3157 3420Time (Day)

Rai

nfal

l (m

m)

0

20

40

60

80

100

120

Run

off (

m3/

s)

Rain Runoff

Calibration strategy

• the model is calibrated on the basis of observed runoff at the basin outlet.

• However, it is proposed to also constraint the model outputs on the basis of on average annual evapotranspiration(regional climatic information).

Calibration strategy

• Grid (σ,η) with δσ = ; δ η = 0.01

• Objective function: error Bias and Variance

• Equifinality so,– Consider Bias and Variance of average annual Rs

error– Limit absolute Bias of average annual Rs error to

10% (local data control based on Rs)– Limit absolute Bias of average annual ETR to 10%

(regional data control based on ETR Budyko)

Error Bias and variance

600

700

800

900

1000

1100

1200

1300

1400

1500

5 10 15 20 25 30 35

biais (Rs annuelle)

varia

nce

de l'

erre

ur d

e R

s

cal with Rs

cal with Rs and ETR

Kernel distribution of ETR

80 100 120 140 160 180 200 220 240 2600

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

ETR (mm/an)

Den

sity

est

imat

e fo

r ET

R

cal withRs

cal with Rs ans ETR

Kernel distribution of the relative soil moisture

-0.05 0 0.05 0.1 0.15 0.20

2

4

6

8

10

12

14

16Density estimate for W

W%

Den

sity

est

imat

e fo

r W

cal with Rs

cal with Rs and ETR

Kernel distribution of percolation

-40 -20 0 20 40 60 80 1000

0.02

0.04

0.06

0.08

0.1

0.12

Den

sity

est

imat

e fo

r gra

vity

dra

inag

e

Cal with Rs

cal with Rs and ETR

Kernel distribution of capillary rise

-5 0 5 10 15 20 25 30 35 400

0.01

0.02

0.03

0.04

0.05

0.06

0.07

capillary rise

Den

sity

est

imat

e fo

r cap

illar

y ris

e

Cal with Rs

cal with Rs and ETR

Kernel distribution of σ and η

-0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.40

2

4

6

8

10

12

14

Parameters (sigma and neta)

Den

sity

est

imat

e fo

r par

amet

ers

cal with Rs (neta)

cal with Rs and ETR (neta)cal with Rs (sigma)

cal with Rs and ETR (sigma)

0

0,2

0,4

0,6

0,8

1

1,2

0,2 0,22 0,24 0,26 0,28 0,3 0,32 0,34 0,36 0,38 0,4

neta

Sig

ma

cal with Rs

cal with Rs and ETR

σ versus η through accepted solutions

Sorted η η η η and σσσσ according to the local R s as objective function

Valeur de la fonction objectif Biais annuel en fonc tion de Neta et Sigma

0,1

0,15

0,2

0,25

0,3

0,35

0,4

0 0,2 0,4 0,6 0,8 1 1,2

Sigma

Net

a

classe biais 10-15%

classe biais 15-20%

classe biais 20-25%

classe biais 25-30%

classe biais 30-35%

Sorted η η η η and σσσσ according to the local R s and regionalETR as objective function

Valeur de la fonction objectif Biais annuel en fonc tion de Neta et Sigma

0,1

0,15

0,2

0,25

0,3

0,35

0 0,1 0,2 0,3 0,4

Sigma

Net

a

classe biais 10-15%

classe biais 15-20%

classe biais 20-25%

classe biais 25-30%

classe biais 30-35%

Selection of one optimal solution (median criteria value )

600

700

800

900

1000

1100

1200

1300

1400

1500

5 10 15 20 25 30 35biais (Rs annuelle)

varia

nce

de l'

erre

ur d

e R

s

cal with Rs

cal with Rs and ETR

Optimal solution: cal with Rs

Optimal solution: cal with Rs and ETR

0,00

0,05

0,10

0,15

0,20

0,25

0,30

0,35

1 138 275 412 549 686 823 960 1097 1234 1371 1508 1645 1782 1919 2056 2193 2330 2467 2604 2741 2878 3015 3152 3289 3426 3563

W/Wmax Sfc P/Wmax

gravity drainage periods

capillary rise periods

cal with Rs

0,00

0,05

0,10

0,15

0,20

0,25

0,30

1 138 275 412 549 686 823 960 1097 1234 1371 1508 1645 1782 1919 2056 2193 2330 2467 2604 2741 2878 3015 3152 3289 3426 3563

W/Wmax Sfc P/Wmax

gravity drainage periods

capillary rise periods

cal with Rs and ETR

Effect of regional information on estimated water conten t

-0,20

0,00

0,20

0,40

0,60

0,80

1,00

1 139 277 415 553 691 829 967 1105 1243 1381 1519 1657 1795 1933 2071 2209 2347 2485 2623 2761 2899 3037 3175 3313 3451 3589

W/Wmax Sfc Gd

gravity drainage periods

capillary rise periods

cal with Rs

-0,20

0,00

0,20

0,40

0,60

0,80

1,00

1 139 277 415 553 691 829 967 1105 1243 1381 1519 1657 1795 1933 2071 2209 2347 2485 2623 2761 2899 3037 3175 3313 3451 3589

W/Wmax Sfc Gd

gravity drainage periods

capillary rise periods

cal with Rs and ETR

Effect on Water content, percolation capillary rise

0

0,1

0,2

0,3

0,4

0,5

0,6

1 253 505 757 1009 126115131765 2017226925212773 302532773529

Time (day)

Val

ue

0

20

40

60

80

100

120

cal with Rs and ETR(W/Wmax)cal with Rs (W/Wmax)

Sfc

P

Effect on relative soil moisture

Effect on ETR

0

1

2

3

4

5

6

7

8

9

10

1 267 533 799 106513311597 18632129 239526612927 31933459

Time (day)

ET

R

0

20

40

60

80

100

120

140

P (m

m)

cal with Rs (ETR)

cal with Rs and ETR(ETR)P

What is the performance on thedecadal scale?

Budyko, Milly, Eagleson

Annual scaleDecadal scaleDaily scale

R.I, K.,

Representation of Bias and variance on thedecadal scale

0

5

10

15

20

25

30

-0,06 -0,04 -0,02 0 0,02 0,04 0,06 0,08Biais (Rs décadaire)

Var

ianc

e de

l'er

reur

de

Rs

Cal with RsCal with Rs and ETR

Water budget for the median solution

Cal with RS Cal with Rs and ETRETR (mm/year) 206,1 215,3Wmoy (mm/year) 0,052 0,030Rs (mm/year) 16,72 16,65Rc (mm/year) 22,44 28,00Per (mm/year) 7,0621 3,6301Mmoy 0,245 0,266W/Wbc 0,181 0,117biais decadaire 0,014 0,014variance decadaire 19,88 19,88variance annuelle 1096,90 1096,90Biais annuelle 23,60 23,60

Conclusions

• Due to uncertainty about pedo transfer parameters as well as uncertainty about vegetation response to waterstress, a lot of uncertainty is attached to soil moisturecontent prediction

• Kernel function is a way to quantify this uncertaintyand to transfer it to other soil moisture relatedvariables (median solution)

• Collection of data on vegetation response (LAI,NDVI) (ground and remote sensing) may contribute to decrease this uncertainty