Accuracy and Efficiency of Adjoint-State In MODFLOW-2005

Adjoint state methods provide an efficient means of calculating the Jacobian matrix of local sensitivities when implemented for underdetermined problems for which the number of parameters exceeds the number of observations.  The computational effort for an adjoint state Jacobian matrix is roughly equivalent to one forward model run per observation, whereas for perturbation sensitivities the computational effort required is one (or more) forward model run per parameter.

Implementation of adjoint state calculations typically requires coding within a forward model.  An adjoint-state version of MODFLOW-2005 has been developed that supports many of the common MODFLOW packages.  In addition to the computational efficiency for underdetermined problems described above, adjoint-state sensitivity calculations can be more accurate than perturbation sensitivities as calculated by typical model-independent parameter estimation codes.

We detail systematic testing of the MODFLOW-2005 adjoint sensitivity code through application to synthetic test problems where the sensitivity of head and boundary condition observations to various parameter types is evaluated.  Using the local sensitivity analysis utility JACTEST that is distributed with the PEST software, the assumption of local linearity that underlies local sensitivity calculation is evaluated, together with the effect of the perturbation increment and flow solution convergence on results compared with adjoint-state results.

In all cases evaluated, local sensitivities calculated using the MODFLOW-2005 adjoint process were significantly smoother that those obtained using perturbations sensitivities. On some occasions, noise in calculated perturbation sensitivities resulted in high calculated sensitivities that were solely a result of file reading and writing that is not required by the adjoint process integrated within MODFLOW-2005.  Mitigation of noisy derivatives can be accomplished through techniques such as multiple increments for perturbations and, in some cases, tighter solver convergence criteria.  The adjoint state solution, however, is routinely more accurate even with its greater computational efficiency.