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Surveys in Geophysics

, Volume 37, Issue 1, pp 109–147 | Cite as

Applied Mathematics in EM Studies with Special Emphasis on an Uncertainty Quantification and 3-D Integral Equation Modelling

  • Oleg Pankratov
  • Alexey Kuvshinov
Article

Abstract

Despite impressive progress in the development and application of electromagnetic (EM) deterministic inverse schemes to map the 3-D distribution of electrical conductivity within the Earth, there is one question which remains poorly addressed—uncertainty quantification of the recovered conductivity models. Apparently, only an inversion based on a statistical approach provides a systematic framework to quantify such uncertainties. The Metropolis–Hastings (M–H) algorithm is the most popular technique for sampling the posterior probability distribution that describes the solution of the statistical inverse problem. However, all statistical inverse schemes require an enormous amount of forward simulations and thus appear to be extremely demanding computationally, if not prohibitive, if a 3-D set up is invoked. This urges development of fast and scalable 3-D modelling codes which can run large-scale 3-D models of practical interest for fractions of a second on high-performance multi-core platforms. But, even with these codes, the challenge for M–H methods is to construct proposal functions that simultaneously provide a good approximation of the target density function while being inexpensive to be sampled. In this paper we address both of these issues. First we introduce a variant of the M–H method which uses information about the local gradient and Hessian of the penalty function. This, in particular, allows us to exploit adjoint-based machinery that has been instrumental for the fast solution of deterministic inverse problems. We explain why this modification of M–H significantly accelerates sampling of the posterior probability distribution. In addition we show how Hessian handling (inverse, square root) can be made practicable by a low-rank approximation using the Lanczos algorithm. Ultimately we discuss uncertainty analysis based on stochastic inversion results. In addition, we demonstrate how this analysis can be performed within a deterministic approach. In the second part, we summarize modern trends in the development of efficient 3-D EM forward modelling schemes with special emphasis on recent advances in the integral equation approach.

Keywords

Quantification of uncertainties Bayesian approach  Metropolis–Hastings Gradient and Hessian Adjoint sources approach Finite elements Contracting integral equations 

Notes

Acknowledgments

The authors would like to thank Alexander Grayver for many seminal discussions over the course of this work, and Alexey Geraskin and Alexander Grayver for their input to Sect. 11. We wish to thank William Lowrie who helped us to improve the English presentation of this paper. We extend our gratitude to Chester Weiss and an anonymous reviewer for constructive comments on the manuscript. This work has been supported by the European Space Agency through ESTEC contract No. 4000102140/10/NL/JA and in part by the Russian Foundation for Basic Research under Grant No. 13-05-12111. Oleg Pankratov acknowledges the support of ETH during his stay in Zurich as a visiting professor.

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Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.Pushkov Institute of Terrestrial Magnetism, Ionosphere and Radiowave PropagationRussian Academy of SciencesTroitsk, MoscowRussia
  2. 2.Institute of GeophysicsETH ZurichZurichSwitzerland

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