Parameter Structure Identification Using the Level Set Methods

Abstract

We introduce an inverse approach for efficiently identifying the spatial distribution of binary materials (say, a lower permeable material D embedded in the background material) using the level set method, given head measurements at some locations and times. No assumption has been made on the shape, size, and locations of these embedded zones, as well as the correlation structure and the proportion of two materials. By this method, instead of handling D directly, the lower permeable zones (or their boundaries) are represented by a level set function, which is negative in the interior of D, positive in its exterior, and zero on its boundary. Since a level set function uniquely defines the region of embedded zones, we manipulate D explicitly through evolution of its boundary. Our aim is to find a level set function such that the hydraulic head solved using the spatial distribution of lower permeable zones defined by this level set function matches the observed values. Because D is unknown, so is the level set function. In the method, we generate a sequence of level set functions such that the regions they defined approach D. We start from an initial level set function (i.e., an initial boundary of lower permeable zones) and sequentially propagate the boundary by solving a level set equation. The speed of propagation of boundary, which is an input to the level set equation, is related to the sensitivity of head to permeability and the error between the measured head and modeled head using the current time. A synthetic example shows that this method can locate those embedded zones efficiently.