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A comparison of the Gauss-Newton and quasi-Newton methods in resistivity imaging inversion

Loke, MH and Dahlin, Torleif LU (2002) In Journal of Applied Geophysics 49(3). p.149-162
Abstract
The smoothness-constrained least-squares method is widely used for two-dimensional (2D) and three-dimensional (3D) inversion of apparent resistivity data sets. The Gauss-Newton method that recalculates the Jacobian matrix of partial derivatives for all iterations is commonly used to solve the least-squares equation. The quasi-Newton method has also been used to reduce the computer time. In this method. the Jacobian matrix for a homogeneous earth model is used for the first iteration, and the Jacobian matrices for subsequent iterations are estimated by an updating technique. Since the Gauss-Newton method uses the exact partial derivatives, it should require fewer iterations to converge. However, for many data sets, the quasi-Newton method... (More)
The smoothness-constrained least-squares method is widely used for two-dimensional (2D) and three-dimensional (3D) inversion of apparent resistivity data sets. The Gauss-Newton method that recalculates the Jacobian matrix of partial derivatives for all iterations is commonly used to solve the least-squares equation. The quasi-Newton method has also been used to reduce the computer time. In this method. the Jacobian matrix for a homogeneous earth model is used for the first iteration, and the Jacobian matrices for subsequent iterations are estimated by an updating technique. Since the Gauss-Newton method uses the exact partial derivatives, it should require fewer iterations to converge. However, for many data sets, the quasi-Newton method can be significantly faster than the Gauss-Newton method. The effectiveness of a third method that is a combination of the Gauss-Newton and quasi-Newton methods is also examined. In this combined inversion method, the partial derivatives are directly recalculated for the first two or three iterations, and then estimated by a quasi-Newton updating technique for the later iterations. The three different inversion methods are tested with a number of synthetic and field data sets. In areas with moderate (less than 10:1) subsurface resistivity contrasts, the inversion models obtained by the three methods are similar. In areas with large resistivity contrasts, the Gauss-Newton method gives significantly more accurate results than the quasi-Newton method. However, even for large resistivity contrasts, the differences in the models obtained by the Gauss-New-ton method and the combined inversion method are small. As the combined inversion method is faster than the Gauss-Newton method, it represents a satisfactory compromise between speed and accuracy for many data sets. (Less)
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author
organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
imaging, resistivity, optimisation, quasi-Newton, gauss-Newton, 2D
in
Journal of Applied Geophysics
volume
49
issue
3
pages
149 - 162
publisher
Elsevier
external identifiers
  • wos:000175227100003
  • scopus:0036509413
ISSN
0926-9851
DOI
10.1016/S0926-9851(01)00106-9
language
English
LU publication?
yes
id
567dee15-971c-426e-bf25-e70518fc82e7 (old id 339568)
date added to LUP
2007-11-23 12:19:26
date last changed
2017-11-05 04:20:31
@article{567dee15-971c-426e-bf25-e70518fc82e7,
  abstract     = {The smoothness-constrained least-squares method is widely used for two-dimensional (2D) and three-dimensional (3D) inversion of apparent resistivity data sets. The Gauss-Newton method that recalculates the Jacobian matrix of partial derivatives for all iterations is commonly used to solve the least-squares equation. The quasi-Newton method has also been used to reduce the computer time. In this method. the Jacobian matrix for a homogeneous earth model is used for the first iteration, and the Jacobian matrices for subsequent iterations are estimated by an updating technique. Since the Gauss-Newton method uses the exact partial derivatives, it should require fewer iterations to converge. However, for many data sets, the quasi-Newton method can be significantly faster than the Gauss-Newton method. The effectiveness of a third method that is a combination of the Gauss-Newton and quasi-Newton methods is also examined. In this combined inversion method, the partial derivatives are directly recalculated for the first two or three iterations, and then estimated by a quasi-Newton updating technique for the later iterations. The three different inversion methods are tested with a number of synthetic and field data sets. In areas with moderate (less than 10:1) subsurface resistivity contrasts, the inversion models obtained by the three methods are similar. In areas with large resistivity contrasts, the Gauss-Newton method gives significantly more accurate results than the quasi-Newton method. However, even for large resistivity contrasts, the differences in the models obtained by the Gauss-New-ton method and the combined inversion method are small. As the combined inversion method is faster than the Gauss-Newton method, it represents a satisfactory compromise between speed and accuracy for many data sets.},
  author       = {Loke, MH and Dahlin, Torleif},
  issn         = {0926-9851},
  keyword      = {imaging,resistivity,optimisation,quasi-Newton,gauss-Newton,2D},
  language     = {eng},
  number       = {3},
  pages        = {149--162},
  publisher    = {Elsevier},
  series       = {Journal of Applied Geophysics},
  title        = {A comparison of the Gauss-Newton and quasi-Newton methods in resistivity imaging inversion},
  url          = {http://dx.doi.org/10.1016/S0926-9851(01)00106-9},
  volume       = {49},
  year         = {2002},
}