1D Inversion Of 4D Radar Data To Image Fluid Flow

Abstract

One of the essential steps in geophysics is the conversion of geophysical data to a map of physical properties something which can only truly be achieved through inversion. While full wavefield inverse methods are widely applied in acoustical imaging, their application to radar has been limited for a number of reasons. These include the complexity of antenna – soil coupling, the fact that antenna radiation patterns are unknown and the underdetermined nature of radar data (in that radar data is in general single rather than multi offset) as well as the large number of parameters we have to invert for. Thus, the majority of current radar inversion efforts are actually kinematic inversions. We investigate the feasibility of a 1D full waveform radar inversion effort. We demonstrate that while we can do 1D radar inversion using a mixed simplex/Powell optimization scheme, the results show a fundamental ambiguity between parameters. From theoretical grounds the extension of the inversion to higher spatial dimensions would seem not to be sufficient to resolve this ambiguity. However, an alternative to 1D inversion is 1D timelapse inversion which is more feasible as it adds a whole new level of information. We implement a first simple form of 1D timelapse inversion on a data set collected in Columbia’s subsurface imaging lab which demonstrates both the success and the potential of this method. Introduction – 4D geophysics As discussed in more detail in [1, 2], time lapse 3D or 4D geophysics is one of the main innovations in the world of geophysics in the last 10 years. Time lapse geophysis has found applications both in global seismic ([3],[4] oil industry ([5-7]) and in near surface geophysics ([8], [9]). The attractiveness of 4D geophysics is that it gives us access to processes – something which for many cases is much more of interest than the pure “static” state of the earth. The principle underlying 4D is easy: we collect multiple identical datasets, remove the complexity of the background (which we can often not invert for anyway) and come up with a time-dependent change in physical properties which can be interpreted as processes. However, a core prerequisite for this is that we can invert the geophysical data to yield physical property maps. Inversion of geophysical data for physical property maps One of the fundamental problems in geophysics is the inversion of geophysical data to find 2D or 3D distributions of physical properties. As is known from extensive work in inverse methods by a range of authors (see e.g the excellent papers in [10]) geophysical inversion is in general a mixed under and overdetermined inverse problem in which we have to apply constraints to arrive at a reasonable solution. These constraints can be minor (e.g. we apply smoothness or minimum norm constraints) or can require us to make a priori statements about either the values or distributions of one or more of the parameters that we seek to estimate. In practice, if the constraints we need to apply to get a good result are