Joint acoustic full-waveform inversion of crosshole seismic and ground-penetrating radar data in the frequency domain

Abstract

Integrating crosshole ground-penetrating radar (GPR) with seismic methods is an efficient way to reduce the uncertainty and ambiguity of data interpretation in shallow geophysical investigations. We have developed a new approach for joint full-waveform inversion (FWI) of crosshole seismic and GPR data in the frequency domain to improve the inversion results of both FWI methods. In a joint objective function, three geophysical parameters (P-wave velocity, permittivity, and conductivity) are effectively connected by three weighted cross-gradient terms that enforce the structural similarity between parameter models. Simulation of acoustic seismic and scalar electromagnetic problems is implemented using 2D finite-difference frequency-domain methods, and the inverse problems of seismic FWI and GPR FWI are solved using a matrix-free truncated Newton algorithm. The joint inversion procedure is performed in several hierarchical frequencies, and the three parameter models are sequentially inverted at each frequency. The joint FWI approach is illustrated using three numerical examples. The results indicate that the joint FWI approach can effectively enhance the structural similarity among the models, modify the structure of each model, and improve the accuracy of inversion results compared with those of individual FWI approaches. Moreover, joint inversion can reduce the trade-off between permittivity and conductivity in GPR FWI, leading to an improved conductivity model in which artifacts are significantly decreased.

A comparative study of structural similarity and regularization for joint inverse problems governed by PDEs

“…Because this formulation further increases the nonlinearity of the problem, we instead use (5). Next, we derive first and second derivatives of the cross-gradient regularization, interpret these derivatives as PDE operators, and draw analogies with the derivatives of the TV functional R TV or its regularized version (6). For this purpose, we first derive the first and second variation of the TV functional as follows:…”

“…This formulation emerges from the general case above by defining F(m 1 , m 2 ) = [F 1 (m 1 ), F 2 (m 2 )] T and d = [d 1 , d 2 ] T . In the context of subsurface exploration, just a few of the different physical phenomena that can be combined in (2) include electromagnetic and seismic waves [4,5], radar and seismic waves [6], DC resistivity and seismic waves [7], and current resistivity and groundwater flow [8].…”

“…In [7], the authors introduce the cross-gradient term R(m 1 , m 2 ) = 1 2 Ω |∇m 1 × ∇m 2 | 2 dx, which seeks to align gradients of the two parameter fields at each point in the medium, i.e., level sets that have the same shape. This seems to be the most popular choice in geophysics [4,5,6,7,8], and is discussed in section 2.1. Instead of the gradients of the parameter fields, one can use normalized gradients.…”

Data-Driven Joint Inversions for PDE Models

“…The first type of methods introduces additional measurement data to enrich the information content of data {u h }. The additional measurement could come from either the same physics in the original problem [39] or a different but related physical process that is coupled to the original problem [1,17,30] (often called model fusion). The second type of methods uses a priori information we have on the unknown f and g to improve the reconstruction.…”

Joint inversion of seismic and electrical data in saturated porous media

“…Typically, these approaches require the combination of different geophysical models that can be solved by imposing the same structure of the subsurface. This is what in literature is called ‘structural’ joint inversion and helps to reduce the inconsistency of interfaces that could arise in the interpretation of the final models result from individual inversion (Dobróka et al ., 1991; Haber & Oldenburg, 1997; Gallardo & Meju, 2003, 2004; de Nardis et al ., 2005; Hu et al ., 2009; Doetsch et al ., 2010; Moorkamp et al ., 2011; Feng et al ., 2017; Senkaya et al ., 2020). A further step in joint inversion schemes is achieved by using rock physics relationships among the unknown geophysical model parameters to constrain themselves relative to each other in what is called hereafter a ‘physical’ joint inversion.…”