Comparison of four least‐squares inversion schemes for studying equivalence in one‐dimensional resistivity interpretation

Abstract

The problem of equivalence in dc resistivity inversion is well known. The ability to invert resistivity data successfully depends on the uniqueness of the model as well as the robustness of the inversion algorithm. To study the problems of model uniqueness and resolution, theoretical data are inverted using variations of a nonlinear least‐squares inversion. It is only through model studies such as this one, where the true solutions are known, that realistic and meaningful comparisons of inversion methods can be undertaken. The data are inverted using three schemes of fixed‐layer thickness where only the resistivity varies, and the results are compared to the variable parameter inversion where both the layer resistivities and thicknesses are allowed to vary. The purpose of fixing the layer thicknesses is to reduce the number of parameters solved for during the inversion process. By doing this, nonuniqueness may be reduced. The fixed‐layer thickness schemes are uniform thickness, geometrical progression of thickness, and logarithmic progression of thickness. By applying each inversion scheme to different models, the layer thickness that minimizes the data rms error for various numbers of layers is determined. The curve of data rms error versus model rms error consists of three general regions: unique, nonunique, and no resolution. A good inversion routine simultaneously minimizes the data rms and model rms errors. The variable parameter scheme is best at simultaneously minimizing the data rms and model rms errors for models that can be resolved through the inversion process. The optimum number of layers in the model can be determined by using a modified F‐test.

Assessment of the reliability of 2D inversion of apparent resistivity data

“…In such cases, it is recommended that, as a general guide, the inverted model for the first or second iteration be taken as the optimum model. The results are in agreement with deductions by Simms and Morgan (1991) that in the 1D inversion of resistivity sounding data, the minimum data rms error and the minimum model rms error do not always coincide and that choosing the best model based on the minimum data rms error will not always give the optimum model. Although there could be other suitable statistical measures for comparing the goodness‐of‐fit between the inverted model parameters and the known theoretical model, the model rms error was found to be simple and straightforward.…”

“…This is best carried out with synthetic data since in such cases the true solutions are known. This type of study has been conducted for the 1D inversion of sounding data by Simms and Morgan (1991). However, as far as the authors are aware, no such systematic investigation has been reported for 2D inversion, and for this reason the present work has been carried out.…”

Combined straightforward inversion of resistivity and induced polarization (time-domain) sounding data

“…The applicability of SIS was extended to magnetotellurics sounding data by Gupta et al (1996) and Sri Niwas et al (2007). The efficiency and robustness of SIS are tested extensively and compared with existing popular methods taking a benchmark continuous model (Parker, 1984;Sims and Morgan, 1992),…”

An approach to determine equivalent solutions to the geoelectrical 2D inversion problem

“…Measurement errors further increase the ambiguity of field data. This problem is well investigated in a 1D interpretation of geoelectrical sounding data (Koefoed, 1979;Militzer and Weber, 1985;Telford et al, 1990;Simms and Morgan, 1992). The calculation of ''equivalent models'', i.e., different parameter sets that fit the data equally well within the acceptable error boundaries, is usually implemented with 1D interpretation software (e. g. Resix Plus, v2, Interpex).…”