Large-scale hydraulic tomography and joint inversion of head and tracer data using the Principal Component Geostatistical Approach (PCGA)

The stochastic geostatistical inversion approach is widely used in subsurface inverse problems to estimate unknown parameter fields and corresponding uncertainty from noisy observations. However, the approach requires a large number of forward model runs to determine the Jacobian or sensitivity matrix, thus the computational and storage costs become prohibitive when the number of unknowns, m, and the number of observations, n increase. To overcome this challenge in large-scale geostatistical inversion, the Principal Component Geostatistical Approach (PCGA) has recently been developed as a matrixfree'' geostatistical inversion strategy that avoids the direct evaluation of the Jacobian matrix through the principal components (low-rank approximation) of the prior covariance and the drift matrix with a finite difference approximation. As a result, the proposed method requires about K runs of the forward problem in each iteration independently of m and n, where K is the number of principal components and can be much less than m and n for large-scale inverse problems. Furthermore, the PCGA is easily adaptable to different forward simulation models and various data types for which the adjoint-state method may not be implemented suitably. In this paper, we apply the PCGA to representative subsurface inverse problems to illustrate its efficiency and scalability. The low-rank approximation of the large-dimensional dense prior covariance matrix is computed through a randomized eigen decomposition. A hydraulic tomography problem in which the number of observations is typically large is investigated first to validate the accuracy of the PCGA compared with the conventional geostatistical approach. Then the method is applied to a large-scale hydraulic tomography with 3 million unknowns and it is shown that underlying subsurface structures are characterized successfully through an inversion that involves an affordable number of forward simulation runs. Lastly, we present a joint inversion of head and tracer test data using MODFLOW and MT3DMS as coupled black-box forward simulation solvers. These applications demonstrate the advantages of the PCGA, i.e., the scalability to high-dimensional inverse problems and the ability to utilize multiple forward models as black boxes.