Overview
Geophysical inversion is very much an interdisciplinary field, combining aspects of mathematics, computer science, physics and Earth sciences. Geophysical inversion generates models (three-dimensional spatial representations) of the physical properties of the Earth's subsurface that could have given rise to measured data from a geophysical survey: for example, gravity, magnetic, electrical and electromagnetic measurements taken at, above or below ground level. Inversion is a computationally intensive procedure, relying on accurate numerical solution of the differential equations that describe the physical phenomena involved, and development of numerical optimization routines tailored to the specific inverse problem at hand. Geophysical inversion provides a non-invasive method for building models of the subsurface that can be used to inform decision makers before invasive inspection is employed. These techniques have proven helpful for addressing a wide range of practical problems in various fields, including resource exploration, natural hazard mitigation, archaeology, agriculture and many other environmental and civil engineering investigations. However, most existing popular geophysical modelling algorithms apply early, perhaps even obsolete, numerical optimization methods in their solution. Such methods have proven inadequate for more complicated geophysical modelling problems and further research is needed to design, test, apply and tune new numerical methods.
AARMS Geophysical Inverse Problems Discussion Series - Part 1
Each meeting we will discuss a paper that introduces a geophysical inverse problem that involves a more difficult numerical optimization problem to solve. A first stage of discussions will help to identify and better understand some of these types of more complicated inverse problems, from an optimization perspective, before a second stage of discussions on how to solve them. I hope this discussion series helps to educate us all on aspects of geophysical inverse problems that perhaps we have not thought about before, whether you are a student, more senior researcher, code developer or practitioner. Please contact the administrator (plelievre@mta.ca) if there are any papers that you would like the group to discuss.
This paper discussion series will take place every second Friday, online via zoom using the following link:
Time (in various time zones):
- 16:30-17:30 UTC
- 9:30-10:30pm Pacific Canadian time
- 1:30-2:30pm Atlantic Canadian time
- 2:00-3:00pm Newfoundland time
Very tentative schedule (please watch for changes, can also be found
here
):
- May 13, 2022:
Zhang et al., 2022, Geophysical inversions on unstructured meshes using non-gradient based regularization.
The paper above, the main one we will discuss, seeks to improve upon the method in the paper below, which increases some computational demands. We'd like to better understand how well the paper above improves upon the one below.
Lelievre et al., 2013, Gradient and smoothness regularization operators for geophysical inversion on unstructured meshes.
- May 27, 2022:
Farquharson, 2008, Constructing piecewise-constant models in multidimensional minimum-structure inversions.
Also available on the author's webpage:
https://www.esd.mun.ca/~farq/special.html
Considering the discussion at our previous meeting, it seems natural to next look at methods for recovering sharp/blocky/compact features. This one is a relatively short paper, going back to a time when most inversion work was on rectilinear meshes, and it would be sensible to return to these ideas now that unstructured meshes are seeing more use.
- June 10, 2022:
Gündoğdu & Candansayar, 2018, Three-dimensional regularized inversion of DC resistivity data with different stabilizing functionals.
Also available on the author's
ResearchGate page.
This paper compares several regularization functionals that have appeared in geophysical inversion literature. It can provide an overview for us before perhaps looking at some others in more detail, if needed.
- June 24, 2022:
Farquharson & Oldenburg, 2004, A comparison of automatic techniques for estimating the regularization parameter in non-linear inverse problems.
Also available
here.
This paper compares some common approaches for choosing the value of the regularization parameter in underdetermined, minimum-structure-type inverse problems. It can provide an overview for us before looking at more recent ideas.
- July 15, 2022:
Note the THREE week hiatus.
Bijani et al., 2017, Physical-property-, lithology- and surface-geometry-based joint inversion using Pareto Multi-Objective Global Optimization.
If you are unable to access the paper through that link, please email me (plelievre@mta.ca) and I will send you a copy.
This paper introduces the concept of multi-objective optimization and Pareto-optimality. In this work, instead of attempting to select a single value for the regularization parameter, the methods sample the entire Pareto-curve (Tikhonov curve), or for more than two objective functions the Pareto-(hyper)surface. The challenge is then deciding how to visualize and otherwise explore all the possible models provided, extract useful information from them, and make reasonable interpretations.
- July 29, 2022:
Lelièvre et al., 2012, Joint inversion of seismic traveltimes and gravity data on unstructured grids with application to mineral exploration.
If you are unable to access the paper through that link, please email me (plelievre@mta.ca) and I will send you a copy.
This paper looks at various different methods to enforce the similarity of the different models in a joint inversion. This paper will help us discuss previous themes in relation to joint inversion, including the deterioration of numerical behavior for particular objective functions, constructing piecewise-constant models, and determining the value of the regularization parameter(s).
- August 12, 2022:
Vatankhah et al., 2022, Large-scale focusing joint inversion of gravity and magnetic data with Gramian constraint.
If you are unable to access the paper through that link, please email me (plelievre@mta.ca) and I will send you a preprint, or you could contact the first author (saeed.vatankhah@gmail.com).
Following on from our discussion last meeting, this paper introduces another coupling approach for joint inversion, using a Gramian constraint. The paper also discusses some computational efficiencies for uniform grids, and the use of Lp-norm stabilizers.