### Abstract

Geophysical inverse problems try to infer the value of a physical property of the earth from data measured at the boundary of the domain. Model parameterization is a key concept to make the inverse problem less ill conditioned. In this contribution we compare the performance of four different basis functions, Fourier, Pixel, Haar and Daubechies Wavelets, for a 1D linear continuous inverse problem. Adopting the right parameterization also reduces the number of dimensions in which the inverse problem is going to be solved, allowing easy posterior analysis. To compare the different basis expansions we have studied a simple 1D linear inverse problem in gravimetric inversion for a density anomaly with Gaussian shape. For this simple toy-problem we show that the Fourier basis gives a better reconstruction using Filon's quadrature to avoid numerical instabilities caused by highly oscillating functions, both, in the noise-free and noisy cases. Besides, the Fourier base is the one that provides the lowest reconstruction error for the number of basis terms and the highest condition number. The Haar and pixel (piecewise-continuous functions) bases provide similar results, although the pixel base needs more terms to achieve lower reconstruction errors. Also, the pixel basis is the one that has the lowest condition number that is obviously related to the way the energy in the system matrix is distributed. Besides, the Daubechies Db2 basis expansion is the one that has the highest system rank, it is more difficult to apply to project the integral kernel, and provides the worst results, compared to the other basis expansions. Finally we propose how to generalize this methodology to linear inverse problems in several dimensions and to non-linear problems. A separate paper will be devoted to this subject.

Original language | English (US) |
---|---|

Pages (from-to) | 92-102 |

Number of pages | 11 |

Journal | Journal of Applied Geophysics |

Volume | 113 |

DOIs | |

State | Published - Feb 1 2015 |

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### All Science Journal Classification (ASJC) codes

- Geophysics

### Cite this

*Journal of Applied Geophysics*,

*113*, 92-102. https://doi.org/10.1016/j.jappgeo.2014.12.010

}

*Journal of Applied Geophysics*, vol. 113, pp. 92-102. https://doi.org/10.1016/j.jappgeo.2014.12.010

**Comparative analysis of the solution of linear continuous inverse problems using different basis expansions.** / Fernández-Muñiz, Zulima; Fernández-Martínez, Juan L.; Srinivasan, Sanjay; Mukerji, Tapan.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Comparative analysis of the solution of linear continuous inverse problems using different basis expansions

AU - Fernández-Muñiz, Zulima

AU - Fernández-Martínez, Juan L.

AU - Srinivasan, Sanjay

AU - Mukerji, Tapan

PY - 2015/2/1

Y1 - 2015/2/1

N2 - Geophysical inverse problems try to infer the value of a physical property of the earth from data measured at the boundary of the domain. Model parameterization is a key concept to make the inverse problem less ill conditioned. In this contribution we compare the performance of four different basis functions, Fourier, Pixel, Haar and Daubechies Wavelets, for a 1D linear continuous inverse problem. Adopting the right parameterization also reduces the number of dimensions in which the inverse problem is going to be solved, allowing easy posterior analysis. To compare the different basis expansions we have studied a simple 1D linear inverse problem in gravimetric inversion for a density anomaly with Gaussian shape. For this simple toy-problem we show that the Fourier basis gives a better reconstruction using Filon's quadrature to avoid numerical instabilities caused by highly oscillating functions, both, in the noise-free and noisy cases. Besides, the Fourier base is the one that provides the lowest reconstruction error for the number of basis terms and the highest condition number. The Haar and pixel (piecewise-continuous functions) bases provide similar results, although the pixel base needs more terms to achieve lower reconstruction errors. Also, the pixel basis is the one that has the lowest condition number that is obviously related to the way the energy in the system matrix is distributed. Besides, the Daubechies Db2 basis expansion is the one that has the highest system rank, it is more difficult to apply to project the integral kernel, and provides the worst results, compared to the other basis expansions. Finally we propose how to generalize this methodology to linear inverse problems in several dimensions and to non-linear problems. A separate paper will be devoted to this subject.

AB - Geophysical inverse problems try to infer the value of a physical property of the earth from data measured at the boundary of the domain. Model parameterization is a key concept to make the inverse problem less ill conditioned. In this contribution we compare the performance of four different basis functions, Fourier, Pixel, Haar and Daubechies Wavelets, for a 1D linear continuous inverse problem. Adopting the right parameterization also reduces the number of dimensions in which the inverse problem is going to be solved, allowing easy posterior analysis. To compare the different basis expansions we have studied a simple 1D linear inverse problem in gravimetric inversion for a density anomaly with Gaussian shape. For this simple toy-problem we show that the Fourier basis gives a better reconstruction using Filon's quadrature to avoid numerical instabilities caused by highly oscillating functions, both, in the noise-free and noisy cases. Besides, the Fourier base is the one that provides the lowest reconstruction error for the number of basis terms and the highest condition number. The Haar and pixel (piecewise-continuous functions) bases provide similar results, although the pixel base needs more terms to achieve lower reconstruction errors. Also, the pixel basis is the one that has the lowest condition number that is obviously related to the way the energy in the system matrix is distributed. Besides, the Daubechies Db2 basis expansion is the one that has the highest system rank, it is more difficult to apply to project the integral kernel, and provides the worst results, compared to the other basis expansions. Finally we propose how to generalize this methodology to linear inverse problems in several dimensions and to non-linear problems. A separate paper will be devoted to this subject.

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U2 - 10.1016/j.jappgeo.2014.12.010

DO - 10.1016/j.jappgeo.2014.12.010

M3 - Article

AN - SCOPUS:84920913335

VL - 113

SP - 92

EP - 102

JO - Journal of Applied Geophysics

JF - Journal of Applied Geophysics

SN - 0926-9851

ER -