## 2 Physical Scenario  
The scenario studied is shown schematically in Figure 1. In the saturated 2D rectangular aquifer, with impervious top and bottom surfaces, the groundwater penetrates through the left vertical border with constant velocity (v\({}_{\rm o}\)) which is maintained throughout the aquifer due to the imposed piezometric gradient. Thermal conditions of the first class (Diritlech), with different temperature values, are imposed at the left, upper and bottom boundaries. It is assumed that the aquifer is large enough so that the region of development of the vertical temperature profiles, limited by \(\rm l_{T}^{\star}\), is less than the length of the aquifer (L). Beyond this region, the vertical temperature profiles are straight lines. In order for this scenario to reproduce the behavior of real aquifers, no other conditions or assumptions regarding the natural processes of heat advection-diffusion within the aquifer are added. The imposition of horizontal thermal gradients assumed by other authors [11; 12], for example, would radically change the solution of the problem, since the vertical temperature profile would change throughout the entire scenario regardless of its horizontal size. Assuming a homogeneous and isotropic 2-D aquifer, the mathematical model is formed via the heat balance governing equation, which involves diffusion, convection and storage processes, plus the boundary thermal conditions:  
\[\rm k\left(\frac{\partial^{2}T}{\partial x^{2}}+\frac{\partial^{2}T}{\partial y ^{2}}\right)-\rho_{e,w}c_{e,w}\left(v_{o}\frac{\partial T}{\partial x}\right) =\rho_{e}c_{e}\frac{\partial T}{\partial t} \tag{1}\]  
\[\rm T_{(x=0,y,t)} =T_{w} \tag{2}\] \[\rm T_{(x,y=0,t)} =T_{s}\] (3) \[\rm T_{(x,y=H,t)} =T_{b}\] (4) \[\rm T_{(x,y,t=0)} =T_{ini} \tag{5}\]  
against advection to make the influence of velocity, as opposed to horizontal diffusion, predominant and that the time necessary to reach the steady state has elapsed. Finally, parameters \(\alpha\) and H are assumed to be known, as well as temperatures \(T_{\rm s}\) and \(T_{\rm b}\). \(T_{\rm ini}\) is irrelevant since we work at steady-state conditions. In short, the input data of the inverse problem are H, \(\alpha\), \(T_{\rm s}\), \(T_{\rm b}\), \((T-y)_{x_{\rm a},\rm j}\), \((T-y)_{x_{\rm b},\rm j}\) and \(x_{\rm o}\) (the distance \(x_{\rm a}-x_{\rm b}\)). The input temperature of the water, \(T_{\rm w}\), is also an unknown. The protocol for the solution of the inverse problem synthesizes as follows:  
Step 1. A scenario is numerically simulated with the known data. A value of \(T_{\rm w,1}\) is proposed that can be initially adjusted from the shape of the measured temperature profiles, determining whether it is an inlet temperature outside the limits of these profiles (upper or lower) or between such limits [10]. A first random velocity \(v_{\rm o,J}\) (I refers to first iteration) is also imposed, greater than the supposedly expected one. The numerical solution provides the temperature field in the aquifer that is named \(T(x,y)_{\rm sim,J}\). Step 2. The discrete profile \(T-y\) closest to the water entry boundary \((T-y)_{x_{\rm b}}\) is compared with those resulting from the simulation in successive horizontal positions, starting with values close to locations where the profile is nearly linear. The following functional is defined:  
\[\Psi_{\rm b,x_{\rm sim}}=\sum_{i=1}^{i=N}\Bigl{[}(T-y)_{x_{\rm b,i}}-(T-y)_{x_ {\rm sim,i}}\Bigr{]}^{2} \tag{7}\]  
where \((T-y)_{x_{\rm sim,J}}\) is the set of N values of the simulation corresponding to the horizontal position. Once obtained, \(\Psi_{\rm b,x_{\rm sim}}\) and the new \(\Psi_{\rm b,x_{\rm sim}}\) are evaluated from successive locations, \(x_{\rm sim}=x_{\rm sim}-\Delta x_{\rm sim}\), until reaching the location for which \(\Psi_{\rm b,x_{\rm sim}}\) is minimum. \(\Delta x_{\rm sim}\) is a finite value sufficiently small to produce an appreciable change in the functional. Let us name \(x_{\rm b,J}\) and \(\Psi_{\rm b,min,J}\) the final convergent values of this iteration. Step 3.