A geostatistical approach to the inverse problem in groundwater modeling (steady state) and one‐dimensional simulations

TLDR

The study revisits estimating hydrogeologic parameters, notably permeability, from input–output data within a geostatistical framework. It models the unknown parameters as a random field and applies a two‑step procedure—iterative structure identification (variogram, trend, MLE, validation) followed by kriging—to obtain minimum‑variance, unbiased estimates, validated through one‑dimensional simulations. The method yields stable, smooth estimates that capture small‑scale variability, improve with higher‑quality measurements, and remain robust to deviations from assumed geostatistical structures.

Abstract

The problem of estimating Hydrogeologic parameters, in particular, permeability, from input‐output measurements is reexamined in a geostatistical framework. The field of the unknown parameters is represented as a ‘random field’ and the estimation procedure consists of two main steps. First, the structure of the parameter field is identified, i.e., mathematical representations of the variogram and the trend are selected and their parameters are established by using all available information, including measurements of hydraulic head and permeability. Second, linear estimation theory is applied to provide minimum variance and unbiased point estimates of hydrogeologic parameters (‘kriging’). Structure identification is achieved iteratively in three substeps: structure selection, maximum likelihood estimation, and model validation and diagnostic checking. The methodology was extensively tested through simulations on a simple one‐dimensional case. The results are remarkably stable and well behaved. The estimated field is smooth, while small‐scale variability is statistically described. As the quality of measurements improves, the procedure reproduces more features of the original field. The results are also shown to be rather insensitive to deviations from assumptions about the geostatistical structure of the field.