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Current information for 2023/2024

Information from 2021/2022

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AARMS Collaborative Research Group:
Numerical Solution of Geophysical Inverse Problems
2023/2024

If you are interested in joining this collaboration, please contact the administrator: plelievre@mta.ca

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.

Participants

Please contact the administrator (plelievre@mta.ca) if you would like to be listed as a participant below, or removed from those lists.

Please contact the discussion group coordinator (svatankhah@mta.ca) if you would like to be on the email reminder list, or removed from it.

Academic Administrator

• Dr. Peter Lelièvre, Dept. of Mathematics and Computer Science, Mount Allison University

Discussion Group Coordinator

• Dr. Saeed Vatankhah, Dept. of Mathematics and Computer Science, Mount Allison University

Participants from AARMS Member Universities

• (will be filled as we progress)

Participants from Other Universities

• (will be filled as we progress)

AARMS Geophysical Inverse Problems Discussion Series - Part 1

Are you looking for the list of papers from the first part of our discussion series, from 2022? Click here and scroll down.

AARMS Geophysical Inverse Problems Discussion Series - Part 2

As in Part 1, in these meetings we plan to discuss papers that introduce geophysical inverse problems that involve more difficult numerical optimization problems to solve. Our first stage of discussions in 2022 helped to identify and better understand some of these types of more complicated inverse problems. In this second stage of discussions, we hope to dive a little deeper in to the mathematics of how those optimization problems are solved. We 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 organizers if there are any papers that you would like the group to discuss. We are very open to suggestions.

We recognize that participants are coming with very different backgrounds. We hope that we can all help each other understand and gain insight from the material that we discuss. We encourage participants to ask any questions they want on any level of the mathematics or geophysics involved.

Links to papers will be provided below. We may also refer to one or more optimization textbooks throughout the discussion series. This one aimed at an undergraduate audience and we will likely use this:

• An Introduction to Optimization, 4th Edition (Chong and Żak)

These two are fairly well known to some of us but aimed at graduate level so perhaps a little heavier and less digestible than you may want:

• Practical Optimization (Gill, Murray and Wright)
• Numerical Optimization (Nocedal and Wright)

Note that you can find all three of those in pdf format with a web search. Legality unclear.

This paper discussion series will take place every second Monday, online via zoom using a different link each week. Please scroll to the bottom of the page to find the Zoom link for the next meeting.

Time, in various time zones, with no accounting for whether your country does the silly daylight savings thing,
so please synchronize to Atlantic Canadian time:

• 15:00-16:00 UTC
• 8:00-9:00 Pacific Canadian time (sorry for the early start)
• 12:00-13:00 Atlantic Canadian time
• 12:30-13:30 Newfoundland time
• 16:00-17:00 UK time

Very tentative schedule (please watch for changes):

• Wednesday, October 18, 2023:
  Li and Oldenburg, 1996, 3-D inversion of magnetic data.
  Also available here and here.
  This paper lays out the basics of "minimum structure" geophysical inversion as (historically?) performed by various groups around the world. We hope this is a good place to start, providing enough mathematics to encourage the geophysicists to start thinking harder about optimization, and providing enough geophysical context for the more mathematically minded. We plan to look at these methods in more depth in the meeting after this one. You may also like to read section 6 of Chong and Żak, along with any of their Part 1 that you may need. We will happily discuss any of the mathematics in more detail in the meeting, as required.
  
  Saeed prepared a document that outlines some of his thoughts on the questions raised during our meeting.
  
  
• Wednesday, November 1, 2023:
  Oldenburg and Li, 1994, Subspace linear inverse method.
  Also available here.
  This paper goes into more detail on the methods used in the paper from our previous meeting. It is a little longer but if you already read the previous paper then a lot of this material can be skimmed quickly. If you can read sections 1 through 3 then we'll be in good shape for a discussion on the various ways to pose the optimization problem, and on the subspace minimization approach. But as always, this next discussion can go in whatever direction it naturally goes.
• Wednesday, November 15, 2023:
  Pilkington, 2009, 3D magnetic data-space inversion with sparseness constraints.
  Also available here.
  This paper introduces the concepts of data-space inversion and solution appraisal. It also has a discussion on depth weighting for general model norms. It covers various concepts in not-so-much detail. In later meetings we can dig further into any concepts that interest us.
  
  Here is some information from Tarantola's inverse problem theory book that explains the matrix identity that Pilkington refers to with respect to data-space inversion.
  
  Saeed and Peter prepared a document that summarizes the data space approach and some thoughts on the questions raised during our meeting.
  
  
• Wednesday, November 29, 2023:
  Wei & Sun, 2021, Uncertainty analysis of 3D potential-field deterministic inversion using mixed Lp norms.
  Also available here.
  The group is showing interest in the topic of solution appraisal. This paper covers that topic and uses Lp norms in the process.
• Wednesday, Dec 13, 2023:
  Boulanger & Chouteau, 2001, Constraints in 3D gravity inversion.
  Also available here.
  This paper connects to the previous one through use of different model regularization options. Other concepts covered include sensitivity compression, depth weighting, multigrid solution, and Lagrangian formulation of the inverse problem.
• Monday, Jan 15, 2024 (12:00 noon Atlantic Canadian time)
  Fournier & Oldenburg, 2019, Inversion using spatially variable mixed Lp norms.
  Also available here.
  In this paper on voxel-based inversion, mixed Lp-norm regularization is employed to provide an algorithm for recovering various types of targets (sparse, smooth, or blocky) in each sub-volume of the model domain. This paper presents an interesting discussion on the optimization approach, the scaling of each component of the regularization term, and the selection of a set of suitable p-values for the final inversion.
• Monday, Jan 29, 2024 (12:00 noon Atlantic Canadian time)
  Astic & Oldenburg, 2019, A framework for petrophysically and geologically guided geophysical inversion using a dynamic Gaussian mixture model prior.
  Also available here.
  In this paper, a strategy for incorporating petrophysical and geological information into the traditional voxel-based inversion algorithm is presented. Geological and petrophysical information is integrated into geophysical data inversion through the smallness term. This term replaces the Gaussian prior in the Tikhonov-type approach with a Gaussian mixture model that represents both petrophysical and geological knowledge. The paper provides insights into the methodology and discusses its application through various examples.
• Monday, Feb 12, 2024 (12:00 noon Atlantic Canadian time)
  Giraud et al., 2021, Generalization of level-set inversion to an arbitrary number of geologic units in a regularized least-squares framework.
  A preprint also available here.
  In this paper a generalized level-set inversion approach is developed to reconstruct the geometry of an arbitrary number of geological units. The subsurface is discretized into prisms. A constant physical property for each geological unit is assigned. The model represents each unit through the signed distance to its interface with other units. The interfaces between units are determined using the level-set method. This method facilitates consideration of any number of rock units by formulating the problem and solving it through a locally linearized least-squares approach. Additionally, regularization is employed to integrate prior knowledge and promote particular characteristics in the reconstructed model.
  
  Jeremie Giraud (the first author) was able to attend our meeting and provide us with some additional information related to the paper:
  
  
  • Supplementary test examples for the same methods in the paper.
  • Preprint of recent work by Giraud et al. (see also link at bottom to a related journal article).
  • An article related to transdimensional Bayesian inversion.
  • Another article related to transdimensional Bayesian inversion (this one found by Peter).
  
  
• Monday, Feb 26, 2024 (12:00 noon Atlantic Canadian time)
  Kim & Nakata, 2018, Geophysical inversion versus machine learning in inverse problems.
  Also available here.
  This paper provides a comparison between traditional geophysical inversion methods and machine learning approaches for solving inverse problems. It discusses the similarities and differences of these approaches and considers the pros and cons of each.

  Zoom link: Meeting ID: 896 3533 4247, Passcode: 659904