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Sensitivity analysis of the evaporation module of the E-DiGOR model - page 3 / 11

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Et, calculated by the Penman-Monteith equation to changes in a parameter value or input variable, pi, is expressed as follows:

Et= f (p1, p2, p3,…., pN),

where N is the number of parameters and input variables. Then:

Et+∆Et= f (p1+∆ p1, p2+∆ p2, p3+∆ p3,…., pN+∆ pN)

Expanding on the above equation in a Taylor series and ignoring the second-order terms and above leads to the following equation (Beven 1979):

TEt =

2Et 2p1

Tp1 +

2Et 2p2

Tp2 +

2Et 2p3

Tp3 +

p E p t N 2 ......... 2 T + N

where the differentials (Et/pi) define the sensitivity of the estimate to each parameter or variable. A sensitivity index can be calculated for a small change in any variable, while the other parameters are held constant (van Griensven et al. 2006). These sensitivity coefficients are, in themselves, sensitive to the relative magnitudes of Et and pi. According to most of the related literature on SA, the way to do this is by computing derivatives (Saltelli et al. 2004; Cariboni et al. 2007; Huang and Yeh 2007; Masada and Carmel 2008). Then normalised, local, and first-order sensitivity of Et to pi may be determined, and a non- dimensional relative sensitivity is defined as follows (McCuen 1974; Beven 1979; Ginot et al. 2006; Norton 2008):

Si =

2E

t

2p

i

#

p

i

E

t

Si now represents that fraction of the change in pi that is transmitted to change in Et, i.e. an Si value of 0.1 would suggest that a 10% increase in pi may be expected to increase Et by 1%. Negative coefficients would indicate that a reduction in Et will result from an increase in pi. The sensitivity coefficients may vary with differential time steps depending on the current value of all pi and the value of Et. The last equation remains sensitive to the magnitudes of Et and pi, in particular, the relative sensitivity coefficients (Si) may not be a good indicator of the significance of the pi if

M. AYDIN, S. F. KEÇECİOĞLU

either Et or pi tend to zero independently, or if the range of values taken by pi is small in relation to its magnitude (Beven 1979).

In practice, the partial derivatives are calculated as the differences between original (reference) and new parameters and state variables, in incremental ratios (Masada and Carmel 2008). However, sensitivity methods based on local derivatives do not reflect model behaviour over the whole range of input variables, whereas methods based on standardised regression or correlation coefficients cannot detect non-linear and non-monotonic relationships between model input and output (Hamm et al. 2006). In the OAT (One factor-At-a-Time) approach proposed by Morris (1991), local sensitivities get integrated to a global sensitivity measure. On the other hand, Monte Carlo methods have been widely used in sensitivity analyses of environmental models, but may require a large number of simulations and consequently large computational resources (Lim et al. 1989; Sieber and Uhlenbrook 2005; van Griensven et al. 2006). In deterministic models, the outcome of a specific set of parameters is essentially the same for the same initial conditions. In contrast, in stochastic models, the computation of sensitivity involves the comparison of 2 distributions rather than 2 single values (Masada and Carmel 2008).

Materials and methods Derivation of formulas for relative sensitivity

Potential evaporation rates from bare soils were calculated by the Penman-Monteith equation with a surface resistance of zero (Allen et al. 1994; Wallace et al. 1999; Aydın et al. 2005) using standard data of the meteorological stations:

Ep =

d

#(Rn - Gs) +

(

. 8 6 4

  • #

    t# cp #

ra

m#(d+ c)

d

)

(1)

w h e r e E p i s p o t e n t i a l s o i l e v a p o r a t i o n ( k g m 2 d a y 1 m m d a y 1 ) , i s t h e g r a d i e n t o f t h e s a t u r a t e d v a p pressure-temperature curve (kPa °C–1), Rn is the net radiation (MJ m–2 day–1), Gs is the soil heat flux (MJ o u r m 2 d a y 1 ) , ρ i s t h e a i r d e n s i t y ( k g m 3 ) , c p i s t h e s p e c i f i c h e a t o f a i r ( k J k g 1 ° C 1 = 1 . 0 1 3 ) , δ i s vapour pressure deficit of the air (kPa), ra is the t h e

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