**Uncertainty Evaluation of Water Budget Model Parameters for Different Environmental Conditions**

Zoubeida Kebaili Bargaoui, Ahmed Houcine and Asma Foughali

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/55236

### **1. Introduction**

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[44] Mahmood, R, & Hubbard, K. G. Simulating sensitivity of soil moisture and evapo‐ transpiration under heterogeneous soils and land uses. Journal of Hydrology (2003). ,

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Modelling time distribution of soil moisture is a key issue for evapotranspiration and bio‐ mass evaluation and is often adopted for deriving drought awareness indices. Water budget models help computing time evolution of soil moisture provided hydroclimatological and soil information. While runoff time series are often used to drive water budget models cali‐ bration, it may conduct to false conclusions about the other model outputs such as percola‐ tion and evapotranspiration fluxes, in the absence of vegetation response observations. Thus a lot of uncertainty is attached to the calibrated model parameters and may constitute a handicap against model application. The aim of this study is to propose a methodology to cope with vegetation information inside the calibration process of a water balance model us‐ ing a qualitative approach. A review of evapotranspiration estimation through water bal‐ ance modelling is reported in Kebaili Bargaoui (2011). In section 2, we present the data used to apply this methodology. In section 3, we present the methodology of uncertainty quantifi‐ cation using kernel distribution of model parameters. In section 4, resuling kernels are pro‐ vided as well as a sensitivity study of results to the choice of soil parameters evaluation method.

### **2. Data**

Two watersheds are studied: the Wadi Sejnane watershed (North Tunisia) and Wadi Chaffar watershed (South Tunisia). They are of comparable moderate sizes (respectively 376 km2 and 250 km2 ). They have distinguishable occupation and climate. Sejnane basin is a forest basin under subhumid climate. Comparatively, for the Chaffar basin, vegetation cover com‐

© 2013 Bargaoui et al.; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2013 Bargaoui et al.; licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Figure 1. Add caption

prises mainly olives under an arid climate. The soil type is principally sandy for Chaffar ba‐ sin while a dominance of clay soils is outlined for Sejnane basin.

1990 is available for Sejnane basin based on a rainfall network of 14 stations. Both basins have water tables. However piezometric data are not included in the study. Fig 1a and Fig.

Uncertainty Evaluation of Water Budget Model Parameters for Different Environmental Conditions

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229

The water budget lumped BBH model presented by Kobayashi et al. (2001) is performed at daily time scale. Table 1 reports model equations. Mean daily rainfall and mean daily poten‐ tial evapotranspiration are model inputs. Soil moisture content W (mm) and actual evapo‐ transpiration ETR (mm) are model results of interest as well as runoff Rs (mm), percolation (Gd >0 mm) and capillary rise (Gd <0 mm). Seven parameters control model input-output transformations: thickness of active soil layer (*D* mm), effective soil porosity *p*; parameter re‐ lated to the field capacity (*a* mm); parameter representing the decay of soil moisture (*b* mm); parameter representing the daily maximal capillary rise (*c* mm); parameter representing the moisture retaining capacity (0< η <1); parameter representing the stomatal resistance of veg‐ etation to evapotranspiration (0< σ <1). The parameter Wmax (mm) which represents the total

According to Kobayachi and al. (2001) a/Wmax is "nearly equal to or somewhat smaller than the field capacity". After Teshima et al., (2006), *b* is a measure of soil moisture recession that depends on hydraulic conductivity and active soil layer depth *D*. In Iwanaga et al. (2005) a sensitivity analysis of BBH model applied to an irrigated area in semi-arid region suggest that soil moisture RMSE is most sensitive to σ, η and c. All parameters are subject to calibration

1b report the time series of ETP, rainfall and runoff during the calibration periods.

water-holding capacity is a key parameter of the model.

using soil, vegetation as well as climatic and hydrologic information.

W(t) : soil moisture content (mm) P: daily precipitation (mm)

Rs: daily surface runoff (mm)

M(t) = Min(1,W(t)/(σ x Wmax))

p: effective soil porosity

Gd(t) = exp ((W(t)-a)/b)-c

ETR: daily actual evapotranspiration (mm)

ETP: daily potential evapotranspiration (mm)

Wmax: total water-holding capacity (mm) D: thickness of active soil layer (mm)

*a*: parameter related to the field capacity (mm)

*b*: parameter representing the decay of soil moisture (mm) *c*: parameter representing the daily maximal capillary rise (mm)

Gd: daily percolation (if Gd >0) or capillary rise (if Gd <0) (mm)

σ : parameter representing the resistance of vegetation to evapotranspiration

Water balance equation ΔW = W(t + 1) - W(t) = P(t) - ETR(t) - Rs(t) - Gd(t) t: time (day)

*W*max=pD

Daily actual evapotranspiration ETR(t) = M(t) ETP(t)

Daily percolation and capillary

rise

**3. Methodology**

**3. Methodology Figure 1.** Time series of observed ETP (mm/day), rainfall (mm/day) and river discharges (m3/s) for (a) Sejnane basin (b) Chaffar basin

The water budget lumped BBH model presented by Kobayashi et al. (2001) is performed at daily time scale. Table 1 reports model equations. Mean daily rainfall and mean daily potential evapotranspiration are model inputs. Soil moisture content W (mm) and actual evapotranspiration ETR (mm) are model results of interest as well as runoff Rs (mm), percolation (Gd >0 mm) and capillary rise (Gd <0 mm). Seven parameters control model input- Potential evapotranspiration series are computed using the Turc formula based on monthly solar radiation and mean air temperature observed series at surrounding meteorological sta‐ tions. A mean daily value is obtained for each month. Runoff (mm) series are estimated us‐ ing observed daily stream discharges at the basin outlets with standard gauging methods. A ten year calibration period from September 1989 to August 1999 is considered for Chaffar basin including daily basin average rainfall evaluating using Thiessen method based on a network of 10 raingauges. A three year calibration period from September 1988 to August 1990 is available for Sejnane basin based on a rainfall network of 14 stations. Both basins have water tables. However piezometric data are not included in the study. Fig 1a and Fig. 1b report the time series of ETP, rainfall and runoff during the calibration periods.

### **3. Methodology**

prises mainly olives under an arid climate. The soil type is principally sandy for Chaffar ba‐

basin)

/s) during the calibration period (Sejnane

ETP Rainfall Runoff

ETP Rainfall Runoff

(a)

Series of input and output data ETP , Raifall and Runoff (mm/day) (Chaffar)

1 60 119 178 237 296 355 414 473 532 591 650 709 768 827 886 945 1004 1063 time (days)

The water budget lumped BBH model presented by Kobayashi et al. (2001) is performed at daily time scale. Table 1 reports model equations. Mean daily rainfall and mean daily potential evapotranspiration are model inputs. Soil moisture content W (mm) and actual evapotranspiration ETR (mm) are model results of interest as well as runoff Rs (mm), percolation (Gd >0 mm) and capillary rise (Gd <0 mm). Seven parameters control model input-

Potential evapotranspiration series are computed using the Turc formula based on monthly solar radiation and mean air temperature observed series at surrounding meteorological sta‐ tions. A mean daily value is obtained for each month. Runoff (mm) series are estimated us‐ ing observed daily stream discharges at the basin outlets with standard gauging methods. A ten year calibration period from September 1989 to August 1999 is considered for Chaffar basin including daily basin average rainfall evaluating using Thiessen method based on a network of 10 raingauges. A three year calibration period from September 1988 to August

**Figure 1.** Time series of observed ETP (mm/day), rainfall (mm/day) and river discharges (m3/s) for (a) Sejnane basin (b)

(b)

1 165 329 493 657 821 985 1149 1313 1477 1641 1805 1969 2133 2297 2461 2625 2789 2953 3117 3281 3445 3609 time (days)

sin while a dominance of clay soils is outlined for Sejnane basin.

Time series of ETP (mm/day), rainfall (mm/day) and runoff (m<sup>3</sup>

Figure 1. Add caption

0

10

20

30

40

ETP, Rainfall, Runoff

50

60

70

ETP, Rainfall, Runoff

228 Evapotranspiration - An Overview

**3. Methodology** 

Chaffar basin

The water budget lumped BBH model presented by Kobayashi et al. (2001) is performed at daily time scale. Table 1 reports model equations. Mean daily rainfall and mean daily poten‐ tial evapotranspiration are model inputs. Soil moisture content W (mm) and actual evapo‐ transpiration ETR (mm) are model results of interest as well as runoff Rs (mm), percolation (Gd >0 mm) and capillary rise (Gd <0 mm). Seven parameters control model input-output transformations: thickness of active soil layer (*D* mm), effective soil porosity *p*; parameter re‐ lated to the field capacity (*a* mm); parameter representing the decay of soil moisture (*b* mm); parameter representing the daily maximal capillary rise (*c* mm); parameter representing the moisture retaining capacity (0< η <1); parameter representing the stomatal resistance of veg‐ etation to evapotranspiration (0< σ <1). The parameter Wmax (mm) which represents the total water-holding capacity is a key parameter of the model.

According to Kobayachi and al. (2001) a/Wmax is "nearly equal to or somewhat smaller than the field capacity". After Teshima et al., (2006), *b* is a measure of soil moisture recession that depends on hydraulic conductivity and active soil layer depth *D*. In Iwanaga et al. (2005) a sensitivity analysis of BBH model applied to an irrigated area in semi-arid region suggest that soil moisture RMSE is most sensitive to σ, η and c. All parameters are subject to calibration using soil, vegetation as well as climatic and hydrologic information.



**Table 1.** Equations and parameters of the BBH model

Moreover, we have introduced pedo transfer functions in the model in order to reduce the number of parameters to be calibrated on the basis of hydrometeorological series (Bargaoui and Houcine, 2010). It is worth noting that Kobayachi et al. (2001) adjusted soil humidity profiles measurements for BBH model calibration. As such observations are not often available; it seems an important task to adapt the original model using pedotransfer submodels espe‐ cially when dealing with ungauged or partially gauged basins. To that purpose, three key soil characteristics are considered: saturated hydraulic conductivity *Ks*, soil water retention curve shape parameter *B* and field capacity SFC. Assuming the percolation function as an exponential decay function, the leakage *L(s)* is identified according to Guswa et al. (2002) model as reported in (Eq. 1) where *s* is the ratio *W*/*Wmax*. Consequently, parameters a, b, c of the original BBH model are obtained by identification (Eq. 2, 3, 4) using the three soil parameters *Ks*, SFC, and *B*.

$$\mathbf{L}\left(\mathbf{s}\right) = \mathbf{K}\_s \frac{\mathbf{e}^{\mathbf{B}\left(s \cdot \mathbf{S}\_{\rm FC}\right)} - 1}{e^{\mathbf{B}\left(1 - \mathbf{S}\_{\rm FC}\right)} - 1} \tag{1}$$

uncertainty related to soil parameters. For the basin of Chaffar, because of lack of detailed information, we assume that *p* as well as *Ks* and SFC parameters are those corresponding to the dominant soil class. For the Sejnane basin, a spatial mean of soil class properties is adopted using the spatial repartition of soil types as well as the area they cover within the basin. On the other hand, for the two cases, *B* =9 is adopted according to Rodriguez-Iturbe and al. (1999).

Uncertainty Evaluation of Water Budget Model Parameters for Different Environmental Conditions

http://dx.doi.org/10.5772/55236

231

Finally, only the set of parameters (*D*, σ, η) remains subject to calibration through fitting observed and predicted runoff time series. The daily time step is adopted to run the model while annual, monthly and decadal time steps are adopted for its fitting. Many trials are firstly performed to adjust *D* choosing simply between three alternatives: *D* =1000 mm; *D* = 500 mm; *D* =300 mm which represent common values adopted in water balance models. Then, once *D* is fixed, the set of parameters (σ, η) is selected according to annual absolute relative runoff error AARE. Based on the idea of equifinality (Beven, 1993), a threshold value AARE s related to AARE is adopted for eliminating poor solutions using a grid of candidate solutions with Δ σ = Δ η= 0.01. Hence, only those pairs for which AARE > AARE s are selected and analyzed in

Eq. (5) reports the objective function. It quantifies the absolute relative runoff bias during the

( ) ( ) 1

=

where yoi is the annual observed runoff (mm) for year *i*; ysi is the annual computed runoff (mm) for year *i*; N is the number of years of the calibration period. Additionally, for the selected solutions Nash coefficient RN,M evaluated on the monthly basis as well as Nash coefficient RN,D evaluated on the decadal basis are reported. The assumption of existence of capillary rise response is tested through the calibration process. It is further believed that if performance criteria (AARE, RN,M, RN,D) are better in presence (or in absence) of capillary rise assumption,

Thus, the adoption of a fixed value for the threshold AAREs will give rise to a number of acceptable solutions (σ, η). Here, the marginal kernel which represents a non parametric estimation of the statistical distribution of a given random variable (here the parameters σ and η) is adopted to represent parameter uncertainty. Similarly, the kernels of resulting outputs are computed in order to analysis the effect on model outputs especially evapotranspiration which is the variable of interest. A Gaussian kernel is adopted to perform the analysis.

*i AARE yyy <sup>N</sup>*

*si oi oi*

= - å (5)

*N*

<sup>1</sup> ,

s h

**3.1. Model calibration**

the following.

calibration period:

then the assumption is retained.

**3.2. Uncertainty quantification**

$$\mathbf{a} = \mathbf{W}\_{\text{max}} \left[ \mathbf{S}\_{\text{FC}} - \frac{1}{B} \text{Ln} \left( \text{Ks} \frac{1}{e^{B\left(1 - S\_{\text{FC}}\right)} - 1} \right) \right] \tag{2}$$

$$\mathbf{b} = \mathbf{W}\_{\text{max}} \frac{1}{\mathbf{B}} \tag{3}$$

$$\mathbf{c} = \left(\frac{1}{e^{B\left(1-S\_{\rm fC}\right)} - 1}\right) \mathbf{K} \mathbf{s} \tag{4}$$

Rawls et al. (1982) model is adopted for estimating *Ks* while SFC is derived according to two different models: Cosby and Saxton model which was recently adopted by Zhan et al., (2008) and Cosby et al. (1984) model. Effectively, this is suggested as a way to take into account uncertainty related to soil parameters. For the basin of Chaffar, because of lack of detailed information, we assume that *p* as well as *Ks* and SFC parameters are those corresponding to the dominant soil class. For the Sejnane basin, a spatial mean of soil class properties is adopted using the spatial repartition of soil types as well as the area they cover within the basin. On the other hand, for the two cases, *B* =9 is adopted according to Rodriguez-Iturbe and al. (1999).

### **3.1. Model calibration**

Water balance equation ΔW = W(t + 1) - W(t) = P(t) - ETR(t) - Rs(t) - Gd(t) t: time (day)

Daily surface runoff Rs(t) = max P(t)−(*WBC* −*W* (*t*))− *ETR*(*t*)−*Gd*(*t*), 0 *WBC* =η*W*max

**Table 1.** Equations and parameters of the BBH model

230 Evapotranspiration - An Overview

W(t) : soil moisture content (mm) P: daily precipitation (mm)

Rs: daily surface runoff (mm)

ETR: daily actual evapotranspiration (mm)

Gd: daily percolation (if Gd >0) or capillary rise (if Gd <0) (mm)

Moreover, we have introduced pedo transfer functions in the model in order to reduce the number of parameters to be calibrated on the basis of hydrometeorological series (Bargaoui and Houcine, 2010). It is worth noting that Kobayachi et al. (2001) adjusted soil humidity profiles measurements for BBH model calibration. As such observations are not often available; it seems an important task to adapt the original model using pedotransfer submodels espe‐ cially when dealing with ungauged or partially gauged basins. To that purpose, three key soil characteristics are considered: saturated hydraulic conductivity *Ks*, soil water retention curve shape parameter *B* and field capacity SFC. Assuming the percolation function as an exponential decay function, the leakage *L(s)* is identified according to Guswa et al. (2002) model as reported in (Eq. 1) where *s* is the ratio *W*/*Wmax*. Consequently, parameters a, b, c of the original BBH model are obtained by identification (Eq. 2, 3, 4) using the three soil parameters *Ks*, SFC, and *B*.

( ) ( )

s 1 e 1 Ls K

B B - - <sup>=</sup>

*e*

max FC (1 ) 1 1 a= W S

*Ln Ks B e* é ù æ ö ê ú - ç ÷ ë û è ø -

> max <sup>1</sup> b=W

> > (1 ) 1

æ ö =ç ÷ è ø -

*e* -

1 *FC B S*

Rawls et al. (1982) model is adopted for estimating *Ks* while SFC is derived according to two different models: Cosby and Saxton model which was recently adopted by Zhan et al., (2008) and Cosby et al. (1984) model. Effectively, this is suggested as a way to take into account

c

( ) FC s-S

1 *FC <sup>S</sup>*


*Ks*

1 *FC B S*

<sup>B</sup> (3)

(1)

(2)

(4)

η : parameter representing the moisture retaining capacity (0< η <1).

Finally, only the set of parameters (*D*, σ, η) remains subject to calibration through fitting observed and predicted runoff time series. The daily time step is adopted to run the model while annual, monthly and decadal time steps are adopted for its fitting. Many trials are firstly performed to adjust *D* choosing simply between three alternatives: *D* =1000 mm; *D* = 500 mm; *D* =300 mm which represent common values adopted in water balance models. Then, once *D* is fixed, the set of parameters (σ, η) is selected according to annual absolute relative runoff error AARE. Based on the idea of equifinality (Beven, 1993), a threshold value AARE s related to AARE is adopted for eliminating poor solutions using a grid of candidate solutions with Δ σ = Δ η= 0.01. Hence, only those pairs for which AARE > AARE s are selected and analyzed in the following.

Eq. (5) reports the objective function. It quantifies the absolute relative runoff bias during the calibration period:

$$AARE\left(\sigma,\eta\right) = \frac{1}{N} \sum\_{i=1}^{N} \left| \left( y\_{si} - y\_{oi} \right) \right| y\_{oi} \Big|\tag{5}$$

where yoi is the annual observed runoff (mm) for year *i*; ysi is the annual computed runoff (mm) for year *i*; N is the number of years of the calibration period. Additionally, for the selected solutions Nash coefficient RN,M evaluated on the monthly basis as well as Nash coefficient RN,D evaluated on the decadal basis are reported. The assumption of existence of capillary rise response is tested through the calibration process. It is further believed that if performance criteria (AARE, RN,M, RN,D) are better in presence (or in absence) of capillary rise assumption, then the assumption is retained.

### **3.2. Uncertainty quantification**

Thus, the adoption of a fixed value for the threshold AAREs will give rise to a number of acceptable solutions (σ, η). Here, the marginal kernel which represents a non parametric estimation of the statistical distribution of a given random variable (here the parameters σ and η) is adopted to represent parameter uncertainty. Similarly, the kernels of resulting outputs are computed in order to analysis the effect on model outputs especially evapotranspiration which is the variable of interest. A Gaussian kernel is adopted to perform the analysis.

#### **3.3. Including vegetation information**

It is now proposed to accurate (σ, η) kernel distribution by introducing the ratio Kv of mean annual actual evapotranspiration to mean annual potential evapotranspiration. In effect, as noticed by Eagleson (1994) after works of Ehleringer (1985), ecologists recog‐ nize three types of vegetation selection and adaptation in response to environmental stress due to water shortage (Type 1: desert annual grasses and humid climate trees; Type 2: semi-arid and sub-humid trees and shrubs; Type 3: perennial desert plants). Considering actual evapotranspiration as surrogate of vegetation productivity, three typi‐ cal curves of Kv versus the inverse of the soil moisture are drawn by Eagleson (1994). Here, we assume the interval 0.45< Kv < 0.55 (mean Kv = 0.5) for type 2 (Sejnane basin) and 0.15< Kv < 0.25 (mean Kv = 0.2) for type 3 (Chaffar basin) which correspond to the values reported into the graph of Eagleson (1994) in case of weak environmental stress. Effectively, such an hypothesis is justified by the fact that the calibration periods repre‐ sent mean water conditions for the two basins.

relatively to the hypothesis of taking account or not for capillary rise (CR) results in not taking it into account for Chaffar basin (CR=0) while taking it into account for Sejnane basin (CR≠0).

Uncertainty Evaluation of Water Budget Model Parameters for Different Environmental Conditions

Kernels of with assumptions on SFC and Kv (Sejnane)

all Kv SFC=0.37 0.45<Kv<0.55 S=0.37 all Kv SFC=0.45 0.45<Kv<0.55 SFC=0.45

http://dx.doi.org/10.5772/55236

233

all Kv; SFC=0.45 0.45< Kv <0.55; SFC=0.45 all Kv SFC=0.37 .45< Kv <0.55 SFC=0.37

More generally, the comparison of model error variances on monthly and decadal time scales suggests that the assumption of SFC = 0.45 is more suitable for this basin. In effect, Fig. 3 which reports variances corresponding to the selected sets with AAREs= 5% under the two assumptions on SFC, shows that smaller variance values are

**Figure 2.** Parameters kernels (*all Kv* signifies that all selected solutions are considered in the Kernel estimation; *0.45 < Kv < 0.55* signifies that only solutions corresponding to this range of Kv are considered in the Kernel estimation)

(b)

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 

(a)

Kernels of with assumptions on SFC and Kv (Sejnane)

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 

Figure 2. Add caption

0 0,5 1 1,5 2 2,5 3 3,5 4 4,5 5

achieved in the case where SFC = 0.45.

So, kernels of parameters and evapotranspiration conditional to the above conditions will also be drawn in order to evaluate the effect of including vegetation information supplementary to runoff observations on model results.

### **4. Results**

Table 2 reports the 5 parameters which are not subject to fitting on basis of hydroclimatological series.


**Table 2.** Soil type model parameters (with 1: Cosby and Saxton model and 2: Cosby model for estimating SFC)

The thresholds AAREs = 5% and AAREs = 20 % have been applied respectively to Sejnane basin and Chaffar basin. Effectively, it was assumed that owing the more important time variability of rainfall and runoff for Chaffar basin series, it was more indicated to enlarge the threshold AARE <sup>s</sup> for this basin. The analysis of simulation results and runoff performance criteria relatively to the hypothesis of taking account or not for capillary rise (CR) results in not taking it into account for Chaffar basin (CR=0) while taking it into account for Sejnane basin (CR≠0).

**3.3. Including vegetation information**

232 Evapotranspiration - An Overview

sent mean water conditions for the two basins.

to runoff observations on model results.

**Thickness of active soil layer D (mm)**

**Sejnane basin** 1000 0.48 213.4

**Chaffar basin** 500 0.34 3634

**4. Results**

series.

It is now proposed to accurate (σ, η) kernel distribution by introducing the ratio Kv of mean annual actual evapotranspiration to mean annual potential evapotranspiration. In effect, as noticed by Eagleson (1994) after works of Ehleringer (1985), ecologists recog‐ nize three types of vegetation selection and adaptation in response to environmental stress due to water shortage (Type 1: desert annual grasses and humid climate trees; Type 2: semi-arid and sub-humid trees and shrubs; Type 3: perennial desert plants). Considering actual evapotranspiration as surrogate of vegetation productivity, three typi‐ cal curves of Kv versus the inverse of the soil moisture are drawn by Eagleson (1994). Here, we assume the interval 0.45< Kv < 0.55 (mean Kv = 0.5) for type 2 (Sejnane basin) and 0.15< Kv < 0.25 (mean Kv = 0.2) for type 3 (Chaffar basin) which correspond to the values reported into the graph of Eagleson (1994) in case of weak environmental stress. Effectively, such an hypothesis is justified by the fact that the calibration periods repre‐

So, kernels of parameters and evapotranspiration conditional to the above conditions will also be drawn in order to evaluate the effect of including vegetation information supplementary

Table 2 reports the 5 parameters which are not subject to fitting on basis of hydroclimatological

**Saturated hydraulic conductivity Ks** (**mm/day)**

**Field capacity**

0.37(after 1) and 0.45 (after2)

0.166 (after 1)

0.108 (after2)

**Soil water retention curve shape parameter B**

9

9

**SFC**

and

**Effective soil porosity p**

**Table 2.** Soil type model parameters (with 1: Cosby and Saxton model and 2: Cosby model for estimating SFC)

The thresholds AAREs = 5% and AAREs = 20 % have been applied respectively to Sejnane basin and Chaffar basin. Effectively, it was assumed that owing the more important time variability of rainfall and runoff for Chaffar basin series, it was more indicated to enlarge the threshold AARE <sup>s</sup> for this basin. The analysis of simulation results and runoff performance criteria

Kernels of with assumptions on SFC and Kv (Sejnane)

More generally, the comparison of model error variances on monthly and decadal time scales suggests that the assumption of SFC = 0.45 is more suitable for this basin. In effect, Fig. 3 which reports variances corresponding to the selected sets with AAREs= 5% under the two assumptions on SFC, shows that smaller variance values are **Figure 2.** Parameters kernels (*all Kv* signifies that all selected solutions are considered in the Kernel estimation; *0.45 < Kv < 0.55* signifies that only solutions corresponding to this range of Kv are considered in the Kernel estimation)

Figure 2. Add caption

achieved in the case where SFC = 0.45.

#### **4.1. Sejnane basin results**

Fig. 2a reports the kernels corresponding to η in case of Sejnane basin. The result is sensitive to the choice of the parameter SFC while kernels do not change with the change of the class of Kv in both assumptions on SFC. Fig. 2b reports the resulting kernels of σ. It is worth noting that in both assumptions on SFC, kernels are of uniform type reflecting the importance of uncertainty about σ. Conversely, the conditioning of results to the appropriate class of Kv (0.45 < Kv < 0.55) in relation with the vegetation and climate conditions of Sejnane watershed, reduces the uncertainty on σ and leads to two different intervals of variability for σ (smaller value of σ under the assumption of smaller value of SFC).

a justification of adopting Eagleson (1994) Kv versus environmental stress condition variable

Uncertainty Evaluation of Water Budget Model Parameters for Different Environmental Conditions

within model fitting. Fig. 5 which reports the kernels of predicted actual evapotranspiration

shows a clear reduction in uncertainty due to the inclusion of the constraint about vegetation

and climate type. As well, it is noticeable that the kernel is less sensitive to the choice of SFC

Mean value of fitting criteria by class of Kv (SFC=0.45, Sejnane)

Kernels of predicted EA in linkage with Kv and SFC (Sejnane)

Mean cond Nash Month Mean cond Nash Decade Mean cond Absol Error year Mean class value of Kv

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235

0.45< Kv <0.55 SFC=0.45 all Kv SFC=0.45 0.45< Kv <0.55 SFC=37 all Kv SFC=0.37

1 7 13 19 25 31 37 43 49 55 61 67 73 79 85 91 97 103 109 115 121 127 133 139 **simulations**

0 500 1000 1500 2000 2500 3000 3500 **predicted evapotranspiration (mm/ three years calibration period)**

**Figure 5.** Kernels of the total calibration period (3 years) evapotranspiration (Sejnane basin)

**Figure 4.** Values of fitting criteria when solutions are sorted according to the range of Kv

when including such a constraint.

0

0 0,0005 0,001 0,0015 0,002 0,0025 0,003 0,0035 0,004

0,5

1

1,5

2

2,5

3

More generally, the comparison of model error variances on monthly and decadal time scales suggests that the assumption of SFC = 0.45 is more suitable for this basin. In effect, Fig. 3 which reports variances corresponding to the selected (σ,η) sets with AAREs= 5% under the two assumptions on SFC, shows that smaller variance values are achieved in the case where SFC = 0.45.

**Figure 3.** Comparison of monthly and decadal error variances

Fig. 4 reports AARE values as well as the values achieved by the other performance criteria (RN,M and RN,D). Results are sorted according to Kv and dispatched by class of Kv. Four classes are considered: 0.35 < Kv < 0.45 ; 0.45 < Kv < 0.55 ; 0.55 < Kv < 0.65 ; Kv > 0.65. It is worth noting that the parameter sets (σ,η) which result in 0.45 < Kv < 0.55 exhibit the best AARE performance criteria while the other criteria are less sensitive to Kv conditions. Such a result might constitute a justification of adopting Eagleson (1994) Kv versus environmental stress condition variable within model fitting. Fig. 5 which reports the kernels of predicted actual evapotranspiration shows a clear reduction in uncertainty due to the inclusion of the constraint about vegetation and climate type. As well, it is noticeable that the kernel is less sensitive to the choice of SFC when including such a constraint.

**4.1. Sejnane basin results**

234 Evapotranspiration - An Overview

the case where SFC = 0.45.

**Figure 3.** Comparison of monthly and decadal error variances

decadal error variance (mm/j)²

under the assumption of smaller value of SFC).

Fig. 2a reports the kernels corresponding to η in case of Sejnane basin. The result is sensitive to the choice of the parameter SFC while kernels do not change with the change of the class of Kv in both assumptions on SFC. Fig. 2b reports the resulting kernels of σ. It is worth noting that in both assumptions on SFC, kernels are of uniform type reflecting the importance of uncertainty about σ. Conversely, the conditioning of results to the appropriate class of Kv (0.45 < Kv < 0.55) in relation with the vegetation and climate conditions of Sejnane watershed, reduces the uncertainty on σ and leads to two different intervals of variability for σ (smaller value of σ

More generally, the comparison of model error variances on monthly and decadal time scales suggests that the assumption of SFC = 0.45 is more suitable for this basin. In effect, Fig. 3 which reports variances corresponding to the selected (σ,η) sets with AAREs= 5% under the two assumptions on SFC, shows that smaller variance values are achieved in

Comparison of monthly and decadal error variance for two models for SFC

Err Var Dec SFC=0.37 Err Var Dec SFC=0.45

60 80 100 120 140 160 180 200 monthly error variance (mm/j)²

Fig. 4 reports AARE values as well as the values achieved by the other performance criteria (RN,M and RN,D). Results are sorted according to Kv and dispatched by class of Kv. Four classes are considered: 0.35 < Kv < 0.45 ; 0.45 < Kv < 0.55 ; 0.55 < Kv < 0.65 ; Kv > 0.65. It is worth noting that the parameter sets (σ,η) which result in 0.45 < Kv < 0.55 exhibit the best AARE performance criteria while the other criteria are less sensitive to Kv conditions. Such a result might constitute

Mean value of fitting criteria by class of Kv (SFC=0.45, Sejnane)

**Figure 4.** Values of fitting criteria when solutions are sorted according to the range of Kv

Kernels of predicted EA in linkage with Kv and SFC (Sejnane)

**Figure 5.** Kernels of the total calibration period (3 years) evapotranspiration (Sejnane basin)

#### **4.2. Chaffar basin results**

Figure 6. Add Caption

Fig. 6a andFig. 6b report respectively RN,M and RN,D values obtained in case where SFC =0.166 and CR≠0. They are reported according to the corresponding Kv. It is noticeable that Kv values with 0< Kv <1 result from such simulations. Negative values of RN,M are often encountered suggesting very poor performances. Also, values of RN,D are sometimes very low. Better results are obtained when assuming CR=0 (Fig. 7a and Fig. 7b) with Kv values lying only in the interval (0.1< Kv < 0.2) which is more coherent with vegetation and climate information (type 3 curve). Finally Fig. 8 reports η kernels in the two cases (CR=0 and CR≠0). It is noticed that the distribution of resulting evapotranspiration is sensitive to the model assumption about CR. The introduction of the constraint about Kv reduces a little the spread of the kernel distribution. The kernels of σ are reported in Fig. 9. It is noticeable that they are of uni‐ form type in the interval (0,1) : U(0,1) in the case where CR=0 and U(0.5, 1) in the case where CR≠0. For the case CR=0, the constraint about Kv reduces the uncertainty and re‐ sults in a uniform distribution U(0, 0.5). Fig. 10 reports the kernel distribution of evapo‐ transpiration in the case CR=0. The constraint about Kv highly reduces the uncertainty

Uncertainty Evaluation of Water Budget Model Parameters for Different Environmental Conditions

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237

(a)

RN,D versus Kv in case of SFC=0.166 and CR=0 (Chaffar)

Kv

RN,M versus Kv in case of CR=0 and SFC=0.106 (Chaffar)

0 0,02 0,04 0,06 0,08 0,1 0,12 0,14 0,16 0,18

(b)

**Figure 7.** Values of the criteria according to Kv (a) criterion RN,M (monthly basis) and (b) criterion RN,D (decade basis)

0 0,02 0,04 0,06 0,08 0,1 0,12 0,14 0,16 0,18 Kv

about this output.

Figure 7. Add Caption

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7




0

0,5

1

Values of RN,M versus corresponding Kv, Chaffat SFC=0.166, with CR

Values of RN,D versus Kv (Chaffar, SFC=0.166, with CR)

**Figure 6.** Values of the criterion RN,D according to the range of Kv (a) with CR=0 assumption (b) with CR ≠ 0 assump‐ tion

Finally Fig. 8 reports η kernels in the two cases (CR=0 and CR≠0). It is noticed that the distribution of resulting evapotranspiration is sensitive to the model assumption about CR. The introduction of the constraint about Kv reduces a little the spread of the kernel distribution. The kernels of σ are reported in Fig. 9. It is noticeable that they are of uni‐ form type in the interval (0,1) : U(0,1) in the case where CR=0 and U(0.5, 1) in the case where CR≠0. For the case CR=0, the constraint about Kv reduces the uncertainty and re‐ sults in a uniform distribution U(0, 0.5). Fig. 10 reports the kernel distribution of evapo‐ transpiration in the case CR=0. The constraint about Kv highly reduces the uncertainty about this output.

**4.2. Chaffar basin results**

236 Evapotranspiration - An Overview

Figure 6. Add Caption

0

tion

0,1

0,2

0,3

0,4

0,5

0,6


RN,M

Fig. 6a andFig. 6b report respectively RN,M and RN,D values obtained in case where SFC =0.166 and CR≠0. They are reported according to the corresponding Kv. It is noticeable that Kv values with 0< Kv <1 result from such simulations. Negative values of RN,M are often encountered suggesting very poor performances. Also, values of RN,D are sometimes very low. Better results are obtained when assuming CR=0 (Fig. 7a and Fig. 7b) with Kv values lying only in the interval (0.1< Kv < 0.2) which is more coherent with vegetation and climate information (type 3 curve).

Values of RN,M versus corresponding Kv, Chaffat SFC=0.166, with CR

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7

(a)

Values of RN,D versus Kv (Chaffar, SFC=0.166, with CR)

Kv

(b)

**Figure 6.** Values of the criterion RN,D according to the range of Kv (a) with CR=0 assumption (b) with CR ≠ 0 assump‐

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 Kv

RN,M versus Kv in case of CR=0 and SFC=0.106 (Chaffar)

**Figure 7.** Values of the criteria according to Kv (a) criterion RN,M (monthly basis) and (b) criterion RN,D (decade basis)

Figure 7. Add Caption

Kernels of h for Chaffar under SFC=0.166 hypothesis

Kernels of evapotranspiration SFC=0.166 CR=0; Chaffar

Uncertainty Evaluation of Water Budget Model Parameters for Different Environmental Conditions

The methodology developed herein aimed to integrate the type of vegetation response within the calibration process of a water budget model at basin scale and daily time step. From developments using two different watershed of moderate size under two different climatic and vegetation conditions, it results in reducing the uncertainty about the parameters σ representing the resistance of vegetation to evapotranspiration and the parameter η repre‐ senting the moisture retaining capacity. Hence, the uncertainty about actual evapotranspira‐ tion predictions has been also reduced due to such an analysis. This methodology is easily transferable to other water balance models as well as vegetation and climate situations.

Université de Tunis El Manar, Ecole Nationale d'ingénieurs de Tunis, ENIT, Laboratoire

[1] Bargaoui, Z, & Houcine, A. (2010). sensitivity to calibration data of simulated soil

SFC=0.166 CR=0 0.15<Kv<0.25 SFC=0.166 CR=0 all Kv

http://dx.doi.org/10.5772/55236

239

0 200 400 600 800 1000 1200 1400 1600 1800 2000 mm/10 years calibration period

**Figure 10.** Kernels of the total calibration period (10 years) evapotranspiration (Chaffar basin)

Zoubeida Kebaili Bargaoui, Ahmed Houcine and Asma Foughali

moisture related drought indices. *Revue Sécheresse,* , 21

0

0,001

**5. Conclusions**

**Author details**

LMHE, Tunis, Tunisia

**References**

0,002

0,003

0,004

0,005

0,006

**Figure 8.** Kernels for the parameter η under various assumptionsKernels for the parameter

Kernels of s under various assumptions and according to Kv

**Figure 9.** Kernels for the parameter ff under various assumptions

Kernels of evapotranspiration SFC=0.166 CR=0; Chaffar

**Figure 10.** Kernels of the total calibration period (10 years) evapotranspiration (Chaffar basin)

### **5. Conclusions**

Kernels of h for Chaffar under SFC=0.166 hypothesis

Kernels of s under various assumptions and according to Kv

SFC=0.166 with CR all Kv SFC=0.166 with CR 0.15<Kv<0.25 SFC=0.166 with CR=0 all Kv SFC=0.166 with CR=0 0.15<Kv<0.25

SFC=0.166 with CR all Kv SFC=0.166 with CR 0.15<Kv<0.25 SFC=0.166 with CR=0 all Kv SFC=0.166 with CR=0 0.15<Kv<0.25

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

**Figure 8.** Kernels for the parameter η under various assumptionsKernels for the parameter

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

**Figure 9.** Kernels for the parameter ff under various assumptions

238 Evapotranspiration - An Overview

0

0,5 1

1,5

2

2,5 3

3,5

The methodology developed herein aimed to integrate the type of vegetation response within the calibration process of a water budget model at basin scale and daily time step. From developments using two different watershed of moderate size under two different climatic and vegetation conditions, it results in reducing the uncertainty about the parameters σ representing the resistance of vegetation to evapotranspiration and the parameter η repre‐ senting the moisture retaining capacity. Hence, the uncertainty about actual evapotranspira‐ tion predictions has been also reduced due to such an analysis. This methodology is easily transferable to other water balance models as well as vegetation and climate situations.

### **Author details**

Zoubeida Kebaili Bargaoui, Ahmed Houcine and Asma Foughali

Université de Tunis El Manar, Ecole Nationale d'ingénieurs de Tunis, ENIT, Laboratoire LMHE, Tunis, Tunisia

### **References**

[1] Bargaoui, Z, & Houcine, A. (2010). sensitivity to calibration data of simulated soil moisture related drought indices. *Revue Sécheresse,* , 21

[2] Beven, K. J. Prophecy, reality and uncertainty in distributed hydrological modeling. *Adv. in Water Resour.* (1993). , 6, 41-51.

**Chapter 12**

**Evapotranspiration of Succulent Plant** *(Sedum*

Fresh water resources available for agriculture are declining quantitatively and qualitative‐ ly. Therefore, the use of less water or lower-quality supplies will inevitably be practiced for irrigation purposes to maintain economically viable agriculture. Globally arid and semiarid areas are facing salinization of soils along with the acute shortage of water resources. The utilization of marginal waters for agriculture is getting considerable importance in such re‐ gions. In hot and dry climate, one of the most successful ground covers is Sedum. It is per‐ ennial plant, which grows by natural moisture even if there is a little soil [1]. As their common name of stonecrop suggests, they do very well in rocky areas, surviving on little soil and storing water in their thick leaves. While some do well in very sunny areas, others thrive in shade and they all tend to like good drainage. Sedums are suitable plants for rock gardens and flower borders. They are very easy to propagate as almost any tiny leaf or piece of stem that touches the ground will root. Some types become rather invasive but are easy to

Sedum is one of the promising plants in dry areas. It has the characteristics of fire preven‐ tion and dry resistance. It has low transpiration value in the daytime compared to other plants. It uses latent heat transmission to control water loss. Generally, succulents, such as Sedum, have been the most studied and used plants for green roofs [2-5]. Greenroofs are in‐ creasingly being used as a source control measure for urban storm water management as they detain and slowly release rainwater. Their implementation is also recognized as having other benefits, including: habitat creation for birds and insects [6] filtering of aerosols; ener‐ gy conservation by providing thermal insulation [7, 8]; improvement of local microclimate through evaporation; reduction of rooftop temperatures [8]. One of the main reasons Se‐

> © 2013 Al-Busaidi et al.; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© 2013 Al-Busaidi et al.; licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

*aizoonvar.floibundum)*

http://dx.doi.org/10.5772/53213

S. Moritani

**1. Introduction**

A. Al-Busaidi, T. Yamamoto, S. Tanak and

Additional information is available at the end of the chapter

control since the roots are never very deep [1].

