Iranian Journal of Geophysics

Iranian Journal of Geophysics

Comparison of standard least squares and structured total least norm approaches for geophysical models inversion

Document Type : Research Article

Authors
Asghar Rastbood 1
Behnam Rostami 1
Abolfazl Ranjbar 2
1 Assistant Professor, Department of Geomatics Engineering, Faculty of Civil Engineering, University of Tabriz, Tabriz, Iran
2 Assistant Professor, Geomatics Engineering, Marand Engineering Faculty, University of Tabriz, Tabriz, Iran
10.30499/ijg.2025.479132.1639
Abstract
The problem of model inversion, which emerges within the realm of geophysical sciences, is under our consideration. The solution to the problem, irrespective of whether it is framed deterministically or stochastically, involves the minimization of a suitable loss function with respect to the unknown parameters. Efficient local minimization plays a vital role in such optimizations, but the intricate nature of the models involved often poses limitations on the usability of derivative-based approaches. Our focus lies in considering the utilization of advanced computer algebra programs to compute the necessary derivatives automatically.
    To demonstrate the broad applicability of the proposed procedure, we present its application to two distinct ground deformation models, both of which are straightforward in nature, i.e., Mogi and Okada models. Furthermore, we employ two different solution techniques in our analysis: the classical nonlinear least squares method, widely recognized as the most commonly employed approach, and the structured total least norm approach. Assuming that the parameters of the volcano and fault reference models are known, vertical displacements for both models are simulated at the Earth surface and inversion is performed with synthetic vertical displacements without error, with Gaussian error, and with several outliers.
    The results show that in the case that the observations are error-free, the source characteristics are correctly retrieved by both methods, and increasing the number of observation grid points does not affect the result and only increases the execution time. For error-free data, the least squares method requires less time and number of iterations to recover the parameters than the structured total least norm. Adding the Gaussian error to the simulated observations increases the number of iterations and the processing time, especially for structured total least norm method, but the accuracy of the estimated source parameters by both methods is the same. The structured total least norm approach using computer algebra when the observations are affected by several outliers leads to better results than the least squares method in the same conditions, but the number of iterations and computation time for the structured total least norm method is more than the least squares method. The performance results of both approaches are almost identical for both Mogi and Okada models. The Okada model is more complicated than the Mogi model. As the model becomes more complicated, the number of iterations and the calculation time increases. When the data is affected by large errors, the parameters detected by the least squares get away from the correct values with the increase in the number and range of errors. On the contrary, with the values used in this research, the parameters identified by the structured total least norm do not seem to be affected by large errors. This shows that, even in the nonlinear case, the use of the L1-norm error cost function with the structured total least norm approach leads to an algorithm that is able to recover parameters more correctly in the presence of measurement errors of arbitrary magnitude.
 
Keywords
Optimization, inverse problem, computer algebra, geodetic data inversion, structured total least norm
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Iranian Journal of Geophysics
Volume 19, Issue 5
November and December 2025
Pages 37-56