Accounting for imperfect forward modeling in geophysical inverse problems — Exemplified for crosshole tomography

Abstract

Inversion of geophysical data relies on knowledge about how to solve the forward problem, that is, computing data from a given set of model parameters. In many applications of inverse problems, the solution to the forward problem is assumed to be known perfectly, without any error. In reality, solving the forward model (forward-modeling process) will almost always be prone to errors, which we referred to as modeling errors. For a specific forward problem, computation of crosshole tomographic first-arrival traveltimes, we evaluated how the modeling error, given several different approximate forward models, can be more than an order of magnitude larger than the measurement uncertainty. We also found that the modeling error is strongly linked to the spatial variability of the assumed velocity field, i.e., the a priori velocity model. We discovered some general tools by which the modeling error can be quantified and cast into a consistent formulation as an additive Gaussian observation error. We tested a method for generating a sample of the modeling error due to using a simple and approximate forward model, as opposed to a more complex and correct forward model. Then, a probabilistic model of the modeling error was inferred in the form of a correlated Gaussian probability distribution. The key to the method was the ability to generate many realizations from a statistical description of the source of the modeling error, which in this case is the a priori model. The methodology was tested for two synthetic ground-penetrating radar crosshole tomographic inverse problems. Ignoring the modeling error can lead to severe artifacts, which erroneously appear to be well resolved in the solution of the inverse problem. Accounting for the modeling error leads to a solution of the inverse problem consistent with the actual model. Further, using an approximate forward modeling may lead to a dramatic decrease in the computational demands for solving inverse problems.

Estimation and accounting for the modeling error in probabilistic linearized amplitude variation with offset inversion

“…QUANTIFYING THE FORWARD-MODELING ERROR Hansen et al (2014) demonstrate how to quantify the modeling error probabilistically through a probability density, θðdjmÞ. The main idea is to generate a large sample of an assumed (unknown) probability density reflecting the modeling error.…”

“…In some cases, the obtained sample of the modeling error can be described by a Gaussian probability, in which case, a full description of a Gaussian modeling error can be estimated from the sample of the modeling error as a mean and a covariance (Hansen et al, 2014). If the Gaussian model is adopted, then it can account for the Gaussian modeling error in a linear inverse Gaussian problem, such as the one considered by Buland and Omre (2003), by addition of the mean and the covariance of the measurement uncertainty and the modeling error (for details, see Mosegaard and Tarantola, 1995;Tarantola, 2005).…”

“…Assuming that the modeling error is Gaussian, a Gaussian model in the form of a mean and a covariance N ðd Tapp ; C Tapp Þ describing the forward-modeling error θðdjmÞ ∼ N ðd Tapp ; C Tapp Þ can be constructed from a large sample of modeling error realizations, as proposed by Hansen et al (2014) (see Appendix A). The estimated covariance matrix C Tapp of the modeling error using the Buland and Omre forward approximation is depicted in Figure 4 for the two prior models.…”

“…Here, we develop the approach proposed by Hansen et al (2014) to simulate, model, and account for modeling errors in relation to AVO modeling. Specifically, and as a first example, the widely used small-contrast approximation of the Zoeppritz equations will be investigated.…”

Probabilistic electrical resistivity tomography of a CO2 sequestration analog

“…Adequate analysis of these possible sources of error and bias is an active field of research (e.g., Hansen et al, 2014;Kalscheuer and Pedersen, 2007;Ory and Pratt, 1995;Scales and Tenorio, 2001;Trampert and Snieder, 1996). A better understanding of these error sources will improve the resulting inverse models or at least help to better characterize model resolution and uncertainty.…”