Analytical sensitivity map of head observations on heterogeneous hydraulic parameters via the sensitivity equation method

Sensitivity maps of drawdown observations on spatial hydraulic parameters are essential information for hydraulic tomographic analysis. It can guide us to design the optimal cross-hole pumping test strategy. In this study, we show how the sensitivity equation method can be employed to derive analytical sensitivity maps of drawdown observations on spatially variable hydraulic parameters during cross-hole pumping test in a two-dimensional, unbounded aquifer. Unlike previously used adjoint state method, the analytical solutions of the sensitivity maps are not in the forms of complex integration and the properties can be evaluated readily. Moreover, we provide discussions on the essess of adjoint method by comparing the derivation of the close-form solution via the sensitivity equation method and the adjoint equation method.