A structural approach to joint inversion of hydrogeophysical data

Abstract

We have developed a flexible methodology to jointly invert different types of geophysical and hydrogeological data, where the resulting models honour the spatial correlation structure derived from geophysical borehole logs. Additionally, the individual models are internally consistent. The key components in this work are: (1) an efficient method to estimate stochastic regularization operators based on geostatistical models; (2) to minimize the cross product of the gradients of two models, which is used as a measure of structural dissimilarity. Electrical resistances and Ground Penetrating Radar (GPR) traveltime data were collected between boreholes in unsaturated Sherwood Sandstone close to Eggborough, UK. These data were inverted for three-dimensional models, where geophysical logs were used to estimate the geostatistical model that we used to define appropriate regularization operators. The forward computations and the calculations of the Jacobians were performed with a finite element code for the electrical resistances and a finite difference travel time algorithm for the traveltime data. The inverse problem was solved using LSQR. The resulting models are more horizontally layered when the regularization is based on the stochastic regularization operator instead of the commonly used smoothness constraints; a pronounced layering is evident in both EM conductivity and gamma logs at the Eggborough site. The interfaces between lithological units are better defined in the models that were derived from joint inversions compared to individual inversions. An advantage of this methodology is that petrophysical relationships can be evaluated a posteriori without a priori assumptions. The methodology allows us to test what features are resolved by the data and to evaluate possible models that honour the data. This can be achieved by defining a set of regularization operators that are based on different possible geostatistical models and by giving different weights to the cross-gradient function. The methodology could readily be adapted to joint inversion of tracer test and crosshole geophysical data.