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Inequality constraint in least-squares inversion of geophysical data


This paper presents a simple, generalized parameter constraint using a priori information to obtain a stable inverse of geophysical data. In the constraint the a priori information can be expressed by two limits: lower and upper bounds. This is a kind of inequality constraint, which is usually employed in linear programming. In this paper, we have derived this parameter constraint as a generalized version of positiveness constraint of parameter, which is routinely used in the inversion of electrical and EM data. However, the two bounds are not restricted to positive values. The width of two bounds reflects the reliability of ground information, which is obtained through well logging and surface geology survey. The effectiveness and convenience of this inequality constraint is demonstrated through the smoothness-constrained inversion of synthetic magnetotelluric data.


  • Constable, S. C., R. L. Parker, and C. G. Constable, Occam’s inversion: A practical algorithm for generating smooth models from electromagnetic sounding data, Geophysics, 52, 289–300, 1987.

    Article  Google Scholar 

  • deGroot-Hedlin, C. and S. Constable, Occam’s inversion to generate smooth, two-dimensional models from magnetotelluric data, Geophysics, 55, 1613–1624, 1990.

    Article  Google Scholar 

  • Rijo, L., W. H. Pelton, E. C. Feitosa, and S. H. Ward, Interpretation of apparent resistivity data from Apodi Valley, Rio Grande Do Norte, Brazil, Geophysics, 42, 811–822, 1977.

    Article  Google Scholar 

  • Sasaki, Y., Two-dimensional joint inversion of magnetotelluric and dipole-dipole resistivity data, Geophysics, 54, 254–262, 1989.

    Article  Google Scholar 

  • Tarantola, A., Inverse Problem Theory: Methods for Data Fitting and Model Parameter Estimation, 613 pp., Elsevier, Amsterdam, 1987.

    Google Scholar 

  • Tikhonov, A. N. and V. Y. Arsenin, Solutions of Ill-posed Problems, 258 pp., Winston & Sons, Washington, D.C., 1977.

    Google Scholar 

  • Uchida, T., Smooth 2-D inversion for magnetotelluric data based on statistical criterion ABIC, J. Geomag. Geoelectr., 45, 841–858, 1993.

    Article  Google Scholar 

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Correspondence to Hee Joon Kim.

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Kim, H.J., Song, Y. & Lee, K.H. Inequality constraint in least-squares inversion of geophysical data. Earth Planet Sp 51, 255–259 (1999).

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  • Equality Constraint
  • Inequality Constraint
  • Geophysical Data
  • Inversion Process
  • Positiveness Constraint