A Reduced-Order Successive Linear Estimator for Geostatistical Inversion and its Application in Hydraulic Tomography

Hydraulic tomography (HT) is a recently developed technology for characterizing high-resolution, site-specific heterogeneity using hydraulic data (n_d) from a series of cross-hole pumping tests. To properly account for the subsurface heterogeneity and to flexibly incorporate additional information, geostatistical inverse models, which permit a large number of spatially correlated unknowns (n_y), are frequently used to interpret the collected data. However, the memory storage requirements for the covariance of the unknowns (n_y × n_y) in these models are prodigious for large-scale 3-D problems. Moreover, the sensitivity evaluation is often computationally intensive using traditional difference method (n_y forward runs). Although employment of the adjoint method can reduce the cost to n_d forward runs, the adjoint model requires intrusive coding effort. In order to resolve these issues, this paper presents a Reduced-Order Successive Linear Estimator (ROSLE) for analyzing HT data. This new estimator approximates the covariance of the unknowns using Karhunen-Loeve Expansion (KLE) truncated to n_kl order, and it calculates the directional sensitivities (in the directions of n_kl eigenvectors) to form the covariance and cross-covariance used in the Successive Linear Estimator (SLE). In addition, the covariance of unknowns is updated every iteration by updating the eigenvalues and eigenfunctions. The computational advantages of the proposed algorithm are demonstrated through numerical experiments and a 3-D transient HT analysis of data from a highly heterogeneous field site.