# Identification of subsurface structures using electromagnetic data and shape priors

## Abstract

We consider the inverse problem of identifying large-scale subsurface structures using the controlled source electromagnetic method. To identify structures in the subsurface where the contrast in electric conductivity can be small, regularization is needed to bias the solution towards preserving structural information. We propose to combine two approaches for regularization of the inverse problem. In the first approach we utilize a model-based, reduced, composite representation of the electric conductivity that is highly flexible, even for a moderate number of degrees of freedom. With a low number of parameters, the inverse problem is efficiently solved using a standard, second-order gradient-based optimization algorithm. Further regularization is obtained using structural prior information, available, e.g., from interpreted seismic data. The reduced conductivity representation is suitable for incorporation of structural prior information. Such prior information cannot, however, be accurately modeled with a gaussian distribution. To alleviate this, we incorporate the structural information using shape priors. The shape prior technique requires the choice of kernel function, which is application dependent. We argue for using the conditionally positive definite kernel which is shown to have computational advantages over the commonly applied gaussian kernel for our problem. Numerical experiments on various test cases show that the methodology ismore »

- Authors:

- Uni CIPR, Uni Research, Bergen 5020 (Norway)
- (Norway)
- Department of Mathematics, Faculty of Science, Assiut University, Assiut 71516 (Egypt)

- Publication Date:

- OSTI Identifier:
- 22465604

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Journal of Computational Physics; Journal Volume: 284; Other Information: Copyright (c) 2015 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; 97 MATHEMATICAL METHODS AND COMPUTING; ALGORITHMS; DEGREES OF FREEDOM; DISTRIBUTION; ELECTRIC CONDUCTIVITY; GAUSS FUNCTION; INFORMATION; KERNELS; MATHEMATICAL SOLUTIONS; OPTIMIZATION; SUBSURFACE STRUCTURES

### Citation Formats

```
Tveit, Svenn, E-mail: svenn.tveit@uni.no, Department of Mathematics, University of Bergen, Bergen 5020, Bakr, Shaaban A., E-mail: shaaban.bakr1@gmail.com, Uni CIPR, Uni Research, Bergen 5020, Lien, Martha, E-mail: martha.lien@octio.com, Octio AS, Bøhmergaten 44, Bergen 5057, Mannseth, Trond, E-mail: trond.mannseth@uni.no, and Department of Mathematics, University of Bergen, Bergen 5020.
```*Identification of subsurface structures using electromagnetic data and shape priors*. United States: N. p., 2015.
Web. doi:10.1016/J.JCP.2014.12.041.

```
Tveit, Svenn, E-mail: svenn.tveit@uni.no, Department of Mathematics, University of Bergen, Bergen 5020, Bakr, Shaaban A., E-mail: shaaban.bakr1@gmail.com, Uni CIPR, Uni Research, Bergen 5020, Lien, Martha, E-mail: martha.lien@octio.com, Octio AS, Bøhmergaten 44, Bergen 5057, Mannseth, Trond, E-mail: trond.mannseth@uni.no, & Department of Mathematics, University of Bergen, Bergen 5020.
```*Identification of subsurface structures using electromagnetic data and shape priors*. United States. doi:10.1016/J.JCP.2014.12.041.

```
Tveit, Svenn, E-mail: svenn.tveit@uni.no, Department of Mathematics, University of Bergen, Bergen 5020, Bakr, Shaaban A., E-mail: shaaban.bakr1@gmail.com, Uni CIPR, Uni Research, Bergen 5020, Lien, Martha, E-mail: martha.lien@octio.com, Octio AS, Bøhmergaten 44, Bergen 5057, Mannseth, Trond, E-mail: trond.mannseth@uni.no, and Department of Mathematics, University of Bergen, Bergen 5020. Sun .
"Identification of subsurface structures using electromagnetic data and shape priors". United States.
doi:10.1016/J.JCP.2014.12.041.
```

```
@article{osti_22465604,
```

title = {Identification of subsurface structures using electromagnetic data and shape priors},

author = {Tveit, Svenn, E-mail: svenn.tveit@uni.no and Department of Mathematics, University of Bergen, Bergen 5020 and Bakr, Shaaban A., E-mail: shaaban.bakr1@gmail.com and Uni CIPR, Uni Research, Bergen 5020 and Lien, Martha, E-mail: martha.lien@octio.com and Octio AS, Bøhmergaten 44, Bergen 5057 and Mannseth, Trond, E-mail: trond.mannseth@uni.no and Department of Mathematics, University of Bergen, Bergen 5020},

abstractNote = {We consider the inverse problem of identifying large-scale subsurface structures using the controlled source electromagnetic method. To identify structures in the subsurface where the contrast in electric conductivity can be small, regularization is needed to bias the solution towards preserving structural information. We propose to combine two approaches for regularization of the inverse problem. In the first approach we utilize a model-based, reduced, composite representation of the electric conductivity that is highly flexible, even for a moderate number of degrees of freedom. With a low number of parameters, the inverse problem is efficiently solved using a standard, second-order gradient-based optimization algorithm. Further regularization is obtained using structural prior information, available, e.g., from interpreted seismic data. The reduced conductivity representation is suitable for incorporation of structural prior information. Such prior information cannot, however, be accurately modeled with a gaussian distribution. To alleviate this, we incorporate the structural information using shape priors. The shape prior technique requires the choice of kernel function, which is application dependent. We argue for using the conditionally positive definite kernel which is shown to have computational advantages over the commonly applied gaussian kernel for our problem. Numerical experiments on various test cases show that the methodology is able to identify fairly complex subsurface electric conductivity distributions while preserving structural prior information during the inversion.},

doi = {10.1016/J.JCP.2014.12.041},

journal = {Journal of Computational Physics},

number = ,

volume = 284,

place = {United States},

year = {Sun Mar 01 00:00:00 EST 2015},

month = {Sun Mar 01 00:00:00 EST 2015}

}