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.

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.

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.

Summary. In this paper wavelet transforms and a logarithmic barrier method are applied to the inversion of large-scale magnetic data to recover a 3-D distribut

We develop three methods to invert induced polarization (IP) data. The foundation for our algorithms is an assumption that the ultimate effect of chargeability is to alter the effective conductivity when current is applied. This assumption, which was first put forth by Siegel and has been routinely adopted in the literature, permits the IP responses to be numerically modeled by carrying out two forward modelings using a DC resistivity algorithm. The intimate connection between DC and IP data means that inversion of IP data is a two-step process. First, the DC potentials are inverted to recover a background conductivity. The distribution of chargeability can then be found by using any one of the three following techniques: (1) linearizing the IP data equation and solving a linear inverse problem, (2) manipulating the conductivities obtained after performing two DC resistivity inversions, and (3) solving a nonlinear inverse problem. Our procedure for performing the inversion is to divide the earth into rectangular prisms and to assume that the conductivity sigma and chargeability eta are constant in each cell. To emulate complicated earth structure we allow many cells, usually far more than there are data. The inverse problem, which has many solutions, is then solved as a problem in optimization theory. A model objective function is designed, and a 'model' (either the distribution of sigma or eta ) is sought that minimizes the objective function subject to adequately fitting the data. Generalized subspace methodologies are used to solve both inverse problems, and positivity constraints are included. The IP inversion procedures we design are generic and can be applied to 1-D, 2-D, or 3-D earth models and with any configuration of current and potential electrodes. We illustrate our methods by inverting synthetic DC/IP data taken over a 2-D earth structure and by inverting dipole-dipole data taken in Quebec.

The inversion of magnetic data is inherently non- unique with respect to the distance between the source and observation locations. This manifests itself as an ambiguity in the source depth when surface data are inverted and as an ambiguity in the distance between the source and boreholes if borehole data are inverted. Joint inversion of surface and borehole data can help to reduce this nonuniqueness. To achieve this, we develop an algorithm for inverting data sets that have arbitrary observation locations in boreholes and above the surface. The algorithm depends upon weighting functions that counteract the geometric decay of magnetic kernels with distance from the observer. We apply these weighting functions to the inversion of three-component magnetic data collected in boreholes and then to the joint inversion of surface and borehole data. Both synthetic and field data sets are used to illustrate the new inversion algorithm. When borehole data are inverted directly, three-component data are far more useful in constructing good susceptibility models than are single-component data. However, either can be used effectively in a joint inversion with surface data to produce models that are superior to those obtained by inversion of surface data alone.