296 - Adjoint Modeling of Groundwater Flow Systems: From the Pioneering Work of Dr. William Yeh to the Present

Dr. William Yeh is well known for his original developments and his historic contributions to inverse modeling. One subset of his inverse modeling work was the development of adjoint models for the calculation of parameter sensitivities, which is an important part of sensitivity analysis in the context of inverse modeling. In inverse modeling, forward model parameters are systematically adjusted so that model outputs match a set of observations of system state (e.g., head, concentration, or flux at a small subset of locations and times). Adjoint models are particularly useful for sensitivity analysis because they provide the sensitivity of one system state at one location and time to changes in multiple parameter values at any location and time. Thus, they are efficient if the number of observations is small relative to the number of parameters. On the other hand, the perturbation of forward models (e.g., using a finite difference approach) provides the sensitivities of multiple system states at any location and time to a change in one parameter at one specific location. This approach must be applied once for each parameter, which is relatively less efficient if the number of observations is small relative to the number of parameters. Adjoint modeling involves the solution of the adjoint (i.e., dual) of the forward model, which is similar in form to the forward model if the forward model is linear. The need to develop adjoint forms of various forward models has limited the use of adjoint modeling in practice. In a seminal paper published in 1990, Dr. Yeh presented a straightforward lookup table for the adjoints of various forward operators commonly used in saturated groundwater flow modeling. In our current work, we developed a similar lookup table for adjoint model load terms and post-processing kernels, which allow users to solve for the sensitivity of many different system states to many different system parameters. The combination of Dr. Yeh’s seminal research and our new contributions should make the use of adjoint models more accessible in practice.