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Non-linear inverse geothermal problems
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
2017 (English)Licentiate thesis, comprehensive summary (Other academic)
##### Abstract [en]

The inverse geothermal problem consist of estimating the temperature distribution below the earth’s surface using temperature and heat-flux measurements on the earth’s surface. The problem is important since temperature governs a variety of the geological processes including formation of magmas, minerals, fosil fuels and also deformation of rocks. Mathematical this problem is formulated as a Cauchy problem for an non-linear elliptic equation and since the thermal properties of the rocks depend strongly on the temperature, the problem is non-linear. This problem is ill-posed in the sense that it does not satisfy atleast one of Hadamard’s definition of well-posedness.

We formulated the problem as an ill-posed non-linear operator equation which is defined in terms of solving a well-posed boundary problem. We demonstrate existence of a unique solution to this well-posed problem and give stability estimates in appropriate function spaces. We show that the operator equation is well-defined in appropriate function spaces.

Since the problem is ill-posed, regularization is needed to stabilize computations. We demostrate that Tikhonov regularization can be implemented efficiently for solving the operator equation. The algorithm is based on having a code for solving a well- posed problem related to the operator equation. In this study we demostrate that the algorithm works efficiently for 2D calculations but can also be modified to work for 3D calculations.

##### Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press, 2017. , p. 21
##### Series
Linköping Studies in Science and Technology. Thesis, ISSN 0280-7971 ; 1791
##### National Category
Computational Mathematics
##### Identifiers
ISBN: 9789176854044 (print)OAI: oai:DiVA.org:liu-143031DiVA, id: diva2:1157497
##### Presentation
2017-12-04, Ada Lovelace, B-huset, Campus Valla, Linköping, 13:15 (English)
##### Supervisors
Available from: 2017-11-16 Created: 2017-11-16 Last updated: 2017-11-16Bibliographically approved
##### List of papers
1. An efficient regularization method for a large scale ill-posed geothermal problem
Open this publication in new window or tab >>An efficient regularization method for a large scale ill-posed geothermal problem
2017 (English)In: Computers & Geosciences, ISSN 0098-3004, E-ISSN 1873-7803, Vol. 105, p. 1-9Article in journal (Refereed) Published
##### Abstract [en]

The inverse geothermal problem consists of estimating the temperature distribution below the earth's surface using measurements on the surface. The problem is important since temperature governs a variety of geologic processes, including the generation of magmas and the deformation style of rocks. Since the thermal properties of rocks depend strongly on temperature the problem is non-linear.

The problem is formulated as an ill-posed operator equation, where the righthand side is the heat-flux at the surface level. Since the problem is ill-posed regularization is needed. In this study we demonstrate that Tikhonov regularization can be implemented efficiently for solving the operator equation. The algorithm is based on having a code for solving a well-posed problem related to the above mentioned operator. The algorithm is designed in such a way that it can deal with both 2D and 3D calculations.

Numerical results, for 2D domains, show that the algorithm works well and the inverse problem can be solved accurately with a realistic noise level in the surface data.

Elsevier, 2017
##### National Category
Earth and Related Environmental Sciences Mathematics
##### Identifiers
urn:nbn:se:liu:diva-139052 (URN)10.1016/j.cageo.2017.04.010 (DOI)000404697000001 ()
Available from: 2017-06-29 Created: 2017-06-29 Last updated: 2017-11-16Bibliographically approved

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Cite
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