MODFLOW-2005 Ground-Water Model - User Guide to the Adjoint State Based Sensitivity Process (ADJ)

This report describes the adjoint state based sensitivity process for MODFLOW-2005 that calculates the sensitivity of observations to parameters. The process is composed of three basic components, described here for one observation and one parameter; first is the solution of the ground-water flow problem, next this solution is used to calculate the adjoint state for the observation, and finally the sensitivity of the of the observation to the parameter is determined by summing the product of the adjoint state with the derivative of the ground-water flow equations with respect to the parameter for each time step of the flow simulation. The theoretical development presents the mathematical basis for the second two steps in the process. Sensitivity information is useful as part of a parameter estimation process, for reliability analysis, in uncertainty analysis, and to describe error propagation. The program described herein only determines the sensitivities. To implement any of the above analyses the program must be used in conjunction with other software. Sensitivity information can be determined using other approaches such as the direct sensitivity calculations of MODFLOW-2000 (Hill and Others, 2000) and the parameter perturbation method implemented in UCODE_2005 (Poeter and others, 2005) and PEST (Doherty, 2004). This report describes when the adjoint sensitivity process can be expected to be more efficient than these other methods. As a general rule, but not always, adjoint sensitivities require the equivalent computational effort of a head solution simulation for each observation. The other approaches require the computational effort of a head solution for each parameter. If a full matrix of sensitivities is needed, the adjoint state based sensitivities is expected to be more efficient that the other approaches if there are more parameters than observations. There are variations of the basic process that can significantly influence the efficiency of the calculations, depending on the use of the sensitivities and on the structure of the flow simulation. The highest efficiency is reached if the gradient of a weighted sum-of-square error is needed. Only a single adjoint state calculation is performed in this case. The gradient is used by truncated Newton, variable-metric, conjugate gradient, and quasi-Newton optimization procedures. The MODFLOWP (Hill, 1992) implemented parameter estimation using adjoint state based gradients with a conjugate gradient routine for an earlier version of MODFLOW (Harbaugh and McDonald, 1988). A different form of computational efficiency can be gained If the flow simulation has constant time steps. Significant increases in efficiency in determining the