A multi-resolution approach for imaging hydraulic conductivity

Heterogeneities of hydraulic conductivity across multiple spatial scales can significantly affect the flow of groundwater, yet the sparsity of subsurface observations does not allow accurate reconstruction at all such scales. Instead, a common approach is to estimate an effective parameter which predicts large scale variations in flow. We build a new framework upon maximum a posteriori inversions to directly address the problems of scale. This framework consists of (1) a fractal prior model for conductivity which is an autoregressive (AR) process evolving from coarse to fine scale, and (2) a measurement model in which each observation is well-approximated by a linear functional of the AR process at some scale. This framework efficiently incorporates multiple measurement sources, which consist of point measurements of hydraulic conductivity and a piezometric head. The multi-resolution framework produces a conductivity estimate with spatially varying resolution tailored to the measurement sampling geometry