Enhancing the resolving power of least‐squares inversion with active constraint balancing

Abstract

Most geophysical inverse problems are solved using least‐squares inversion schemes with damping or smoothness constraints to improve stability and convergence rate. Since the Lagrangian multiplier controls resolution and stability of the inverse problem, we always want to use the optimum multiplier, which is not easy to get and is usually obtained by experience or a time‐consuming optimization process. We present a new regularization approach, in which the Lagrangian multiplier is set as a spatial variable at each parameterized block and automatically determined via the parameter resolution matrix and spread function analysis. For highly resolvable parameters, a small value of the Lagrangian multiplier is assigned, and vice versa. This approach, named “active constraint balancing” (ACB), tries to balance the constraints of the least‐squares inversion according to sensitivity for a given problem so that it enhances the resolution as well as the stability of the inversion process. We demonstrate the performance of the ACB by applying it to a two‐dimensional resistivity tomography problem, which results in a remarkable enhancement of the spatial resolution. Enhancement of the resolution is also verified in the application of resistivity tomography to a field data set acquired at a tunnel construction site.