Illustration and Verification of Adjoint Sensitivity Theory for Steady State Groundwater Flow

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Abstract
The application of adjoint sensitivity theory to steady state ground water flow is illustrated with three one‐dimensional flow problems. Adjoint states are analytically derived for four performance measures of these test problems: hydraulic head at a point, spatially average hydraulic head, Darcy velocity at a point, and flux from a prescribed head boundary. Sensitivity coefficients are analytically calculated for average head. The adjoint states are interpreted and their usefulness is discussed. The implementation of a numerical adjoint sensitivity flow code to solve for these adjoint states is described, and the computed adjoint states are used in the code to evaluate the sensitivities of model results to model input parameters. The one‐dimensional flow problems provide a set of verification tests for the numerical code. The numerical code successfully reproduces both the analytically derived adjoint states, including those involving jump conditions, and the sensitivity coefficients for model output values that are nonlinear with respect to model parameters.