#### 3.1. Calculation of the Conditional Derivatives

[24] Time series measurements of surface heat fluxes and surface soil moisture and temperature have to be used to calculate the conditional derivatives. The first step in calculating the conditional derivatives is to translate the time series measurements from field experiments into new sequences. The new sequences are obtained by rearranging all the measured time series variables in the ascending (or descending) order of the given parameter *R*. For example, we may sort the original time series in the ascending order of *R*_{n} for calculating the conditional derivatives in equations (18) and (19). Then the resulting sequences are divided into a number of subsequences with equal number of data points. Now these new subsequences are conditioned within a narrow range of a certain given *R*.

[25] The conditional derivatives of a flux variable can be obtained from the regression coefficients of a first-order approximation of its total differential. We use *E* as an example to illustrate the algorithm. According to equation (1), we can write the total differential of *E* as

where superscript 0 stands for a reference point of a subsequence at which the derivatives are defined, *i* is the *i*th point of a certain subsequence, and *m* is the number of data points in the subsequence. The derivatives of other energy fluxes including *G*, *R*^{su}, and *R*^{lu} are calculated in the same manner except that *H* is not an independent variable of these fluxes. A linear regression procedure will be used to obtain the three derivative terms in equation (26) from *m* equations (*m* > 3). Since equation (26) is linear in the unknowns (derivatives), the reference points *T*_{s}^{0}, θ_{s}^{0}, and *H*^{0} need not be specified.

[37] It is widely known that evaporation can be estimated using the water vapor pressure deficit *v*_{d} as an independent variable. Yet the development of our theory assumes *E* is a function of surface variables only. The physical basis of this argument is that the effect of atmospheric conditions on evaporation has been represented by the surface variables due to the strong interaction between the land and the atmosphere. Here we offer a mathematical proof using the parameter restriction methodology combined with the meta-analysis technique [e.g., *Schulze et al.*, 2003]. The statistical analysis will tell us whether including the water vapor deficit *v*_{d} as an additional independent variable of *E*,

leads to significant improvement of the predictability compared to that of equation (1). The null hypothesis under test, H_{0}, is that *v*_{d} does not significantly increase the predictability of evaporation.

[38] The parameter restriction analysis includes the following steps:

[39] 1. Divide the entire multidimensional domain of observed variables, θ_{s}, *T*_{s}, *H*, *v*_{d}, and *R*, into *N* subdomains assuming that *E* can be approximated by piecewise linear functions over the subdomains.

[40] 2. Estimate a linear regression function *E*^{i} of the unrestricted model in equation (27) for all subdomains *i* = 1, 2, …, *N*.

[41] 3. Calculate the unrestricted residual sum of squared errors,

where *E*^{io} is the observed value of *E*.

[42] 4. Estimate a linear regression function *E*^{i} of the restricted model in equation (1) for the same subdomains. This is called restricted since the regression coefficient of *v*_{d} is restricted to be zero.

[43] 5. Calculate the restricted residual sum of squared errors,

[44] 6. Form a test statistic

where *T* is the number of observations used in calculating the regression coefficients, *k* is the number of regressors in the unrestricted model (i.e., five in equation (1)), and *s* is the number of regressors restricted to zero in the restricted model (i.e., one in equation (27)) for each of the *N* subdomains.

[45] It can be shown [e.g., *Pindyck and Rubinfeld*, 1998, p. 134] that ω follows the *F* distribution, *F*_{s,T−k}, under the null hypothesis. Large ω indicates a large decrease in the predictive capability of the restricted model relative to the disadvantage built in by having one fewer regressor. A *p* value associated with the ω can be defined as

which forms a population with a uniform distribution between 0 and 1 under the null hypothesis. Hence the null hypothesis should be rejected at the significance level α when *p* < α for an individual subdomain where the linear regression is performed.

[46] Now we use a standard meta-analysis procedure to create an overall *p* value based on the individual *p* values, *p*_{i}, for each of the *N* subdomains. To do so, we introduce a new statistic Γ,

It turns out that Γ follows the χ_{2N}^{2} distribution under the null hypothesis. The overall *p* value, , can be defined as

so that the null hypothesis should be rejected at the significance level α when < α for the entire domain of the observed independent variables.

[47] Using the data shown in Figure 1 (*v*_{d} not shown), we divide the observed variables into *N* = 50 subdomains. Here we use the daytime data points corresponding to *R*_{n} > 0 when significant evaporation occurs. The overall *p* value is found to be = 0.17, suggesting that the null hypothesis should not be rejected at, say, the 5% significance level. The hypothesis is only rejected at the 17% or higher level, which is unusually high in practice. The result is robust in terms of *N*. In fact, the overall *p* value is much greater for *N* = 10 (fewer fitting parameters), = 0.58, stronger evidence that hypothesis *H*_{0} should be accepted. On the basis of this analysis, we conclude that water vapor deficit does not significantly influence evaporation, and hence equation (1) suffices. This result is consistent with a recent sensitivity study by *Lakshmi and Susskind* [2001], who also found that evaporation is not sensitive to atmospheric humidity.