### Abstract

Sparse inversion of gravity data based on L_{1}-norm regularization is discussed. An iteratively reweighted least squares algorithm is used to solve the problem. At each iteration the solution of a linear system of equations and the determination of a suitable regularization parameter are considered. The LSQR iteration is used to project the system of equations onto a smaller subspace that inherits the ill-conditioning of the full space problem. We show that the gravity kernel is only mildly to moderately ill-conditioned. Thus, while the dominant spectrum of the projected problem accurately approximates the dominant spectrum of the full space problem, the entire spectrum of the projected problem inherits the ill-conditioning of the full problem. Consequently, determining the regularization parameter based on the entire spectrum of the projected problem necessarily over compensates for the non-dominant portion of the spectrum and leads to inaccurate approximations for the full-space solution. In contrast, finding the regularization parameter using a truncated singular space of the projected operator is efficient and effective. Simulations for synthetic examples with noise demonstrate the approach using the method of unbiased predictive risk estimation for the truncated projected spectrum. The method is used on gravity data from the Mobrun ore body, northeast of Noranda, Quebec, Canada. The 3-D reconstructed model is in agreement with known drill-hole information.

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

Pages (from-to) | 1872-1887 |

Number of pages | 16 |

Journal | Geophysical Journal International |

Volume | 210 |

Issue number | 3 |

DOIs | |

State | Published - Sep 1 2017 |

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### Keywords

- Asia
- Gravity anomalies and Earth structure
- Inverse theory
- Numerical approximations and analysis

### ASJC Scopus subject areas

- Geophysics
- Geochemistry and Petrology

### Cite this

_{1}inversion of gravity data using truncated unbiased predictive risk estimator for regularization parameter estimation.

*Geophysical Journal International*,

*210*(3), 1872-1887. https://doi.org/10.1093/gji/ggx274

**3-D Projected L _{1} inversion of gravity data using truncated unbiased predictive risk estimator for regularization parameter estimation.** / Vatankhah, Saeed; Renaut, Rosemary; Ardestani, Vahid E.

Research output: Contribution to journal › Article

_{1}inversion of gravity data using truncated unbiased predictive risk estimator for regularization parameter estimation',

*Geophysical Journal International*, vol. 210, no. 3, pp. 1872-1887. https://doi.org/10.1093/gji/ggx274

_{1}inversion of gravity data using truncated unbiased predictive risk estimator for regularization parameter estimation. Geophysical Journal International. 2017 Sep 1;210(3):1872-1887. https://doi.org/10.1093/gji/ggx274

}

TY - JOUR

T1 - 3-D Projected L1 inversion of gravity data using truncated unbiased predictive risk estimator for regularization parameter estimation

AU - Vatankhah, Saeed

AU - Renaut, Rosemary

AU - Ardestani, Vahid E.

PY - 2017/9/1

Y1 - 2017/9/1

N2 - Sparse inversion of gravity data based on L1-norm regularization is discussed. An iteratively reweighted least squares algorithm is used to solve the problem. At each iteration the solution of a linear system of equations and the determination of a suitable regularization parameter are considered. The LSQR iteration is used to project the system of equations onto a smaller subspace that inherits the ill-conditioning of the full space problem. We show that the gravity kernel is only mildly to moderately ill-conditioned. Thus, while the dominant spectrum of the projected problem accurately approximates the dominant spectrum of the full space problem, the entire spectrum of the projected problem inherits the ill-conditioning of the full problem. Consequently, determining the regularization parameter based on the entire spectrum of the projected problem necessarily over compensates for the non-dominant portion of the spectrum and leads to inaccurate approximations for the full-space solution. In contrast, finding the regularization parameter using a truncated singular space of the projected operator is efficient and effective. Simulations for synthetic examples with noise demonstrate the approach using the method of unbiased predictive risk estimation for the truncated projected spectrum. The method is used on gravity data from the Mobrun ore body, northeast of Noranda, Quebec, Canada. The 3-D reconstructed model is in agreement with known drill-hole information.

AB - Sparse inversion of gravity data based on L1-norm regularization is discussed. An iteratively reweighted least squares algorithm is used to solve the problem. At each iteration the solution of a linear system of equations and the determination of a suitable regularization parameter are considered. The LSQR iteration is used to project the system of equations onto a smaller subspace that inherits the ill-conditioning of the full space problem. We show that the gravity kernel is only mildly to moderately ill-conditioned. Thus, while the dominant spectrum of the projected problem accurately approximates the dominant spectrum of the full space problem, the entire spectrum of the projected problem inherits the ill-conditioning of the full problem. Consequently, determining the regularization parameter based on the entire spectrum of the projected problem necessarily over compensates for the non-dominant portion of the spectrum and leads to inaccurate approximations for the full-space solution. In contrast, finding the regularization parameter using a truncated singular space of the projected operator is efficient and effective. Simulations for synthetic examples with noise demonstrate the approach using the method of unbiased predictive risk estimation for the truncated projected spectrum. The method is used on gravity data from the Mobrun ore body, northeast of Noranda, Quebec, Canada. The 3-D reconstructed model is in agreement with known drill-hole information.

KW - Asia

KW - Gravity anomalies and Earth structure

KW - Inverse theory

KW - Numerical approximations and analysis

UR - http://www.scopus.com/inward/record.url?scp=85037100972&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85037100972&partnerID=8YFLogxK

U2 - 10.1093/gji/ggx274

DO - 10.1093/gji/ggx274

M3 - Article

AN - SCOPUS:85037100972

VL - 210

SP - 1872

EP - 1887

JO - Geophysical Journal International

JF - Geophysical Journal International

SN - 0956-540X

IS - 3

ER -