**Appendix A. Preliminary Calibration Stage**

Consider an agricultural region in which farmers grow *O* crops. The region has access to freshwater, TWW, and brackish water, the regional consumptions of which are limited to the amounts denoted *Q<sup>f</sup>* , *Q<sup>h</sup>* and *Qb*, respectively, and their respective prices are *p<sup>f</sup>* , *ph*, and *pb*, where *p<sup>f</sup>* > *p<sup>h</sup>* > *pb*. Let *q <sup>f</sup> <sup>o</sup>* , *qh*, and *q<sup>b</sup> <sup>o</sup>* denote the per hectare annual water applications of freshwater, TWW, and brackish water to crop *o* (*o* = 1, ... , *O*), respectively, where the sets **q***<sup>f</sup>* = *q f* <sup>1</sup> ,..., *<sup>q</sup> <sup>f</sup> O* , **q***<sup>h</sup>* = *qh* <sup>1</sup>,..., *<sup>q</sup><sup>h</sup> O* , and **q***<sup>b</sup>* = *qb* <sup>1</sup>,..., *<sup>q</sup><sup>b</sup> O* are defined accordingly. The total per hectare annual application to crop *o*, *wo*, is considered constant. The production function is given by *θoeo q f <sup>o</sup>* , *q<sup>b</sup> <sup>o</sup>*, *q<sup>h</sup>* , in which *θ<sup>o</sup>* is a parameter for calibration, and *eo*(•) is the evapotranspiration function of crop *o*, which is taken from Slater et al. [35]. The salinity of brackish water is higher than that of TWW, the salinity of which is higher than that of freshwater; therefore, *<sup>∂</sup>eo ∂q f o* > *<sup>∂</sup>eo ∂q<sup>h</sup> o* > *<sup>∂</sup>eo ∂q<sup>b</sup> o* > 0. We denote by *xo*, the land allocated to crop *o*, which is fixed at the preliminary stage. With the above setting, we first solve the nonlinear optimization problem

$$\max\_{\mathbf{q}^f, \mathbf{q}^b, \mathbf{q}^h} \pi = \sum\_{i=1}^I \mathbf{x}\_o \left[ p\_o \left( \theta\_o \mathbf{c}\_o \left( q\_o^f, q\_o^h, q\_o^b \right) \right) - p^f q\_o^f - p^h q\_o^h - p^b q\_o^b \right] \tag{A1}$$

s.t.

$$q\_o^f + q\_o^h + q\_o^b = w\_o \; \forall \; o = 1, \dots, O \tag{A2}$$

$$\sum\_{o=1}^{O} \mathbf{x}\_o q\_o^f \le Q^f \tag{A3}$$

$$\sum\_{o=1}^{O} \mathbf{x}\_o q\_o^h \le Q^h \tag{A4}$$

$$\sum\_{o=1}^{O} \mathbf{x}\_o q\_o^b \le Q^b \tag{A5}$$

$$\mathbf{q}^f, \mathbf{q}^h, \mathbf{q}^b \ge 0 \tag{A6}$$

where the initial values of *q <sup>f</sup> <sup>o</sup>* , *q<sup>h</sup> <sup>o</sup>* , and *q<sup>b</sup> <sup>o</sup>* are set based on the shares of *Q<sup>f</sup>* , *Qh*, and *Q<sup>b</sup>* in the total regional water *Q<sup>f</sup>* + *Q<sup>h</sup>* + *Qb*, and the parameter *θ<sup>o</sup>* is set so as to equate the computed yield to the observed one *Y*ˆ *o*:

$$
\theta\_o \mathfrak{e}\_o \left( q\_o^f, q\_o^h, q\_o^b \right) = \hat{Y}\_o \; \forall \; o = 1, \ldots, O \tag{A7}
$$

The resultant optimal water allocation sets **q***<sup>f</sup>* <sup>∗</sup>, **q***h*∗, and **q***b*<sup>∗</sup> are then used fto recalibrate *θ<sup>o</sup>* based on Equation (A7).
