Technical Note: Analytical Inversion of the Parametric Budyko Equations

The non-parametric Budyko framework provides empirical relationships between a catchment’s long-term mean evapotranspiration (?̅?) and the aridity index, defined as the ratio of mean rainfall depth (?̅?) to mean potential evapotranspiration 10 (E0 ̅̅ ̅). The parametric Budyko equations attempt to generalize this framework by introducing a catchment-specific parameter (n or w), intended to represent differences in catchment climate and landscape features. Many studies have developed complex regression relationships for the catchment-specific parameter in terms of biophysical features, all of which use known values of ?̅?, E0 ̅̅ ̅, and ?̅? to numerically invert the parametric Budyko equations to obtain values of n or w. In this study, we analytically invert both forms of the parametric Budyko equations, producing expressions for n and w only in terms of ?̅?, E0 ̅̅ ̅, and ?̅?. These 15 expressions allow for n and w to be explicitly expressed in terms of biophysical features through the dependence of ?̅?, E0 ̅̅ ̅, and ?̅? on those same features.


Introduction
The non-parametric Budyko framework was developed to explain and describe the distinctive clustering pattern observed for the long-term average evaporative behavior across multiple catchments. This pattern emerges when the 20 evaporative indices, ̅ ̅ (where ̅ is the mean rainfall depth and ̅ is the mean actual evapotranspiration depth), of multiple catchments are plotted against their corresponding aridity indices, 0 ̅̅̅̅ ̅ (where 0 ̅̅̅ is the mean potential evapotranspiration depth). Several empirical relationships of the form, have been proposed to describe this pattern, including (Schreiber, 1904), 25 and (Ol'Dekop, 1911) https://doi.org/10.5194/hess-2020-585 Preprint. Discussion started: 18 November 2020 c Author(s) 2020. CC BY 4.0 License.
In this study, we analytically invert both forms of the parametric Budyko equations. The resulting expressions give 60 and only in terms of ̅ , 0 ̅̅̅ , and ̅ , illustrating that if and depend on any biophysical features, it is due directly to the dependence of ̅ , 0 ̅̅̅ , or ̅ on those same features. Notably, there has not been an analytical derivation illustrating how and relate to biophysical features, though the importance of doing so has been noted many times (Zhang et al., 2004;Yang et al., 2008;Donohue et al., 2012;Xu et al., 2013;Greve et al., 2015;Wang et al., 2016a;Zhang et al., 2018). The expressions we develop here for and satisfy this need, providing a general expression for the dependence of and on any possible 65 biophysical features through the dependence of ̅ , 0 ̅̅̅ , and ̅ on those same features.
where and are constants related to the evaporative and aridity indices, respectively, is an arbitrary complex variable, and is a function of . With the constraint, > > 0, Eq. (12) has a solution of the form (Hochstadt, 2012), p. 81-84, 80 where is an integer index variable, and ( ) is the gamma function. Table 1 illustrates that Eq. (8) and (9) (9)) and for humid climates (Eq. (10) and (11)). From

105
Left to Right: The first column gives the equation and associated climate. The second column gives mathematical constraints that must be true given an arid or humid climate. The third column gives specific mathematical constraints derived from the climatic constraints. The last column gives the condition, > > , for each associated equation and climate, given the climatic and derived constraints.

Version of
Eq. (14) and Climate

Properties of Analytical Expressions for and 110
Here we investigate the mathematical properties of Eq. (14) and (15) to determine that they are valid analytical expressions for and . First, we examine the behavior of Eq. (15) as ̅ → 0 and ̅ → 0 ̅̅̅ or ̅ . Mathematically, the values of are constrained between 0 and ∞ (Yang et al., 2008), and the values of are constrained between 1 and ∞ (Zhang et al., 2004). Therefore, the upper and lower limits of the and versions of Eq. (15) should be equal to these respective constraints.
are equal to the lower and upper constraint for , respectively. Similarly, the lower limit for the version of Eq. (15), 120 and the upper limit, are equal to the lower and upper constraint for , respectively.
Next, we investigate the properties of Eq. (14). This equation contains a convergent infinite series whose value asymptotically approaches or for the and versions, respectively. The asymptotic behavior of the series' terms (e.g., 125 monotonically decreasing or alternating sign and absolute value decreasing) depends on the specific values of ̅ , ̅ , and 0 ̅̅̅ ( Fig. 1-4). To verify that both the and versions of Eq. (14) produce the correct values of and for a given set of ̅ , ̅ , and 0 ̅̅̅ , we numerically invert Eq. (4) and (5) and compare the fitted and values, ̌ and ̌, to successively better approximations of Eq. (14). The numerical inversion of Eq. (4) and (5) consists of numerically solving, for ̌ and ̌, respectively. We compute approximations of Eq. (14) by truncating the infinite series to a finite number of terms, . We successively improve these finite approximations by increasing . As the number of terms in the finite series increase, the approximations asymptotically converge to the ̌ and ̌ values obtained from the numerical inversion of Eq. (4) and (5)  135 for both arid and humid climates ( Fig. 1-4). This convergence is rapid (requiring fewer than ten terms) for typical values of and (i.e., < 4) and provides strong numeric evidence that Eq. (14) yields valid analytical expressions for and .
Notably, the explicit analytical expression for and from Eq. (14) illustrates that the value of the catchment-165 specific parameter is only determined by ̅ , 0 ̅̅̅ , and ̅ . Therefore, if or depend on biophysical features, it is directly due to the dependence of ̅ , 0 ̅̅̅ , or ̅ on those features. In short, this means that Eq. (14)

Author contributions
NGFR conceived the study, performed the analytical inversion and analyses, and drafted the manuscript. All authors contributed in the interpretation of results and manuscript preparation.

Competing interests
The authors declare no conflicts of interest with respect to the results of this manuscript. The outline for portions of these derivations was developed from Hochstadt (2012), p. 81-84.

A.2 The Mellin transform for ( )
The Mellin transform for ( ) of Eq. (12) is, 285 where is a complex number. Whether the improper integral in Eq. (A2) converges or diverges depends on the behavior of ( ) and the value of . Letting = 0 in Eq. (12) gives, Taking the first derivative of Eq. (12) gives, 290 Since > > 0, Eq. (A4) is always negative, meaning that ( ) is a monotonically decreasing function for 0 ≤ < ∞.

A.3 Mellin transformed ( ) in terms of known functions
Next, we evaluate Eq. (A2) explicitly. To do this, we switch the integration from z to y, using Eq. (12). This involves expressing z in terms of , expressing in terms of y and y, 325 and expressing the limits of integration in terms of , We can now rewrite Eq. (A2) in terms of a integration, https://doi.org/10.5194/hess-2020-585 Preprint. Discussion started: 18 November 2020 c Author(s) 2020. CC BY 4.0 License.

A.4 The inverse Mellin transform and solution for ( ) 350
We now take the inverse Mellin transform of Eq. (A26) and solve for ( ) explicitly. The inverse Mellin transform is defined as, where the integral from − ∞ to + ∞ is interpreted as a line integral along a vertical line in the complex plane. For our specific function, the inverse Mellin transform is, 355 The constraint 0 < < 1 is due to Eq. (A14), the constraint on the real part of so that Eq. (A2) would converge. This means that the vertical line in the complex plane over which the line integral is taken must fall between 0 and 1 on the real axis (Fig.   A1). We evaluate the integral in Eq. (A28) to find an explicit form of ( ) using the following methodology: 1) Define an appropriate contour in the complex plane to perform a contour integration of ( ) − . 360 2) Use residue integration to evaluate the value of this contour integral.
3) Show that the only part of this contour that does not vanish is the line integral defined in Eq. (A28), meaning the inverse Mellin transform, and therefore ( ), is equal to the value of the contour integral evaluated in step 2.
First, we choose a semicircle contour in the complex plane, with the straight portion as a vertical line crossing the real axis at = , and the circular portion connecting the ends of this line across the left side of the complex plane (Fig. A1). This contour 365 is consistent with the constraint on the real part of given in Eq. (A14). We can now define the integral of ( ) − over this contour, ( ) can be expressed as the sum of two line integrals, one over the vertical line portion of the contour, and one over the circular arc portion of the contour, 370 where is the radius of the semicircle contour. Allowing → ∞ leads to, We can now evaluate ( ) over the infinitely large contour using residue integration. Residue integration relates the value of a contour integral to the sum of the residues of the function being integrated. Residues occur when the function of interest has 375 singularities within the contour. Inspection of ( ) − (i.e. the integrand in Eq. (A26)) inside the semicircle contour shows https://doi.org/10.5194/hess-2020-585 Preprint. Discussion started: 18 November 2020 c Author(s) 2020. CC BY 4.0 License. which becomes, (1− ) 400 resulting in, Equation (A42) implies that the contribution of the circular arc portion of the contour vanishes, which means, 410 and therefore, which is the solution for the general form, Eq. (12).
https://doi.org/10.5194/hess-2020-585 Preprint. Discussion started: 18 November 2020 c Author(s) 2020. CC BY 4.0 License. Figure A1: Illustration of the semicircular contour in the complex plane, used to evaluate Eq. (A28). The contour is composed of a 415 vertical line crossing the real axis at = and an arc connecting the two ends of the vertical line. The radius of this semicircle is given as . We let → ∞ so the vertical line portion of the contour will encompass the entire imaginary axis.