We present a method for inverting surface magnetic data to recover 3-D susceptibility models. To allow the maximum flexibility for the model to represent geologically realistic structures, we discretize the 3-D model region into a set of rectangular cells, each having a constant susceptibility. The number of cells is generally far greater than the number of the data available, and thus we solve an underdetermined problem. Solutions are obtained by minimizing a global objective function composed of the model objective function and data misfit. The algorithm can incorporate a priori information into the model objective function by using one or more appropriate weighting functions. The model for inversion can be either susceptibility or its logarithm. If susceptibility is chosen, a positivity constraint is imposed to reduce the nonuniqueness and to maintain physical realizability. Our algorithm assumes that there is no remanent magnetization and that the magnetic data are produced by induced magnetization only. All minimizations are carried out with a subspace approach where only a small number of search vectors is used at each iteration. This obviates the need to solve a large system of equations directly, and hence earth models with many cells can be solved on a deskside workstation. The algorithm is tested on synthetic examples and on a field data set.

In this paper, the term “depth of investigation” refers generically to the depth below which surface data are insensitive to the value of the physical property of the earth. Estimates of this depth for dc resistivity and induced polarization (IP) surveys are essential when interpreting models obtained from any inversion because structure beneath that depth should not be interpreted geologically. We advocate carrying out a limited exploration of model space to generate a few models that have minimum structure and that differ substantially from the final model used for interpretation. Visual assessment of these models often provides answers about existence of deeper structures. Differences between the models can be quantified into a depth of investigation (DOI) index that can be displayed with the model used for interpretation. An explicit algorithm for evaluating the DOI is presented. The DOI curves are somewhat dependent upon the parameters used to generate the different models, but the results are robust enough to provide the user with a first‐order estimate of a depth region below which the earth structure is no longer constrained by the data. This prevents overinterpretation of the inversion results. The DOI analysis reaffirms the generally accepted conclusions that different electrode array geometries have different depths of penetration. However, the differences between the inverted models for different electrode arrays are far less than differences in the pseudosection images. Field data from the Century deposit are inverted and presented with their DOI index.

A rearrangement of the formula used for the rapid calculation of the gravitational anomaly caused by a two‐dimensional uneven layer of material (Parker, 1972) leads to an iterative procedure for calculating the shape of the perturbing body given the anomaly. The method readily handles large numbers of model points, and it is found empirically that convergence of the iteration can be assured by application of a low‐pass filter. The nonuniqueness of the inversion can be characterized by two free parameters: the assumed density contrast between the two media, and the level at which the inverted topography is calculated. Additional geophysical knowledge is required to reduce this ambiguity. The inversion of a gravity profile perpendicular to a continental margin to find the location of the Moho is offered as a practical example of this method.

We present two methods for inverting surface gravity data to recover a 3-D distribution of density contrast. In the first method, we transform the gravity data into pseudomagnetic data via Poisson’s relation and carry out the inversion using a 3-D magnetic inversion algorithm. In the second, we invert the gravity data directly to recover a minimum structure model. In both approaches, the earth is modeled by using a large number of rectangular cells of constant density, and the final density distribution is obtained by minimizing a model objective function subject to fitting the observed data. The model objective function has the flexibility to incorporate prior information and thus the constructed model not only fits the data but also agrees with additional geophysical and geological constraints. We apply a depth weighting in the objective function to counteract the natural decay of the kernels so that the inversion yields depth information. Applications of the algorithms to synthetic and field data produce density models representative of true structures. Our results have shown that the inversion of gravity data with a properly designed objective function can yield geologically meaningful information.

We investigate the use of general, non-l 2 measures of data misfit and model structure in the solution of the non-linear inverse problem. Of particular interest are robust measures of data misfit, and measures of model structure which enable piecewiseconstant models to be constructed. General measures can be incorporated into traditional linearized, iterative solutions to the non-linear problem through the use of an iteratively reweighted least-squares (IRLS) algorithm. We show how such an algorithm can be used to solve the linear inverse problem when general measures of misfit and structure are considered. The magnetic stripe example of Parker (1994) is used as an illustration. This example also emphasizes the benefits of using a robust measure of misfit when outliers are present in the data. We then show how the IRLS algorithm can be used within a linearized, iterative solution to the non-linear problem. The relevant procedure contains two iterative loops which can be combined in a number of ways. We present two possibilities. The first involves a line search to determine the most appropriate value of the trade-off parameter and the complete solution, via the IRLS algorithm, of the linearized inverse problem for each value of the trade-off parameter. In the second approach, a schedule of prescribed values for the trade-off parameter is used and the iterations required by the IRLS algorithm are combined with those for the linearized, iterative inversion procedure. These two variations are then applied to the 1-D inversion of both synthetic and field time-domain electromagnetic data.