Resistivity Inversion Parameter Bounds In the Presence of Equivalence

The interpretation of d.c. resistivity data is greatly affected by equivalence. Because multiple sets of layer parameters (resistivities and thicknesses) can represent nearly the same sounding curve, it is necessary to identify the range of variation in a parameter to determine how well the inverted parameter has been resolved. We investigate three techniques for estimating the range of parameter variation, They are standard error, joint confidence region. and forward modeling. The standard error is calculated using the parameter variance. Error ellipses are constructed based on the joint confidence region. Both the standard error and error ellipse tend to underestimate the range of parameter variation, and are particularly useless when the inverted parameters are associated with a nonunique model. Forward modeling proved to be the most successful technique for estimating the parameter variation. The forward modeling technique is able to provide reasonable error bounds with and without uniqueness, although it is not always successful in identifying the presence of suppressed layers. INTRODUCTION Within the past several years the interest In engineering geophysics has increased dramatically. In particular, surface d.c. resistivity has become very popular. With the increase in use of resistivity surveying, it is important that the interpreter be aware of the problems that may be encountered during the interpretation process. The major problem encountered in resistivity interpretation is equivalence. By equivalence it is meant that multiple sets of layer parameters can represent the same, or nearly the same, sounding curve. The problems associated with equivalence are well known but few attempts at quantifying it have been made. This is mainly due to the acceptance that resistivity inversion will always be plagued with equivalence, and at the present time there is not a quick method to determine the range of variation in parameters which can produce an equivalent solution. Maillet (1947) was one of the first to formulate criteria for equivalence. He found that the resistivity and thickness of a given layer have a one to one correspondence. If the layer resistivity is changed, then the thickness will have a value such that the resistivity (conductivity)-thickness product remains constant, Rocroi (1975) and Koefoed (1976) determined that the equivalence rules given by Maillet (1947) do not strictly apply. They show that the layer resistivity and thickness do not always have a one to one correspondence, and that parameters of different layers can be varied simultaneously to achieve an equivalent solution. The variation in a parameter can be estimated using the standard error and joint confidence region (Flinn, 1965; Inman, 1975; Glen and Ward: 1976; Ward et al., 1976; Herrmann, 1979). These two methods work best when the correlation between parameters is small and the problem is linear. However, the d.c. resistivity problem is nonlinear and parameters are often strongly correlated. Presently-, the interpreter has the option of accepting the resistivity inversion results, which can lead to an incorrect interpretation if no other geologic information is available, or he/she can spend valuable time running different models through the inversion program to identify a range of solutions. Neither option is satisfactory. A method that provides a quick and automatic means of estimating the range of parameter values which result in equivalent models is needed. We compare the parameter ranges obtained using standard error and joint confidence regions, and introduce the use of a posteriori forward modeling to estimate the range of equivalent solutions. The joint confidence region and standard error tend to underestimate the equivalence regions, whereas the forward modeling technique is able to identify models which are nonunique and gives a more accurate estimate of the range of values over which a parameter may vary. METHOD The standard error, joint confidence region, and forward modeling are used to estimate the range of variation of a parameter. The standard error is a measure of precision of the mean and is expressed mathematically as