Analysis of multiple‐source bipole‐quadripole resistivity surveys using the apparent resistivity tensor

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Abstract
Measurements of apparent resistivity made using the bipole‐dipole method depend upon the location and orientation of the current source relative to the body under study. Although it has been recognized that this dependence on orientation can be partially overcome by use of two current bipoles with different orientations, no agreement on the method of analysis of multiple source surveys has been reached. The most general form of presentation of such results is an apparent resistivity tensor. Various rotation invariants derived from the apparent resistivity tensor can be regarded as mean values of apparent resistivity, independent of the direction of the electric field, thus greatly reducing the “false anomalies” typical of single‐source bipole‐dipole survey results. Two of the tensor invariants obey the principle of reciprocity: if the roles of the current and potential electrodes are interchanged, the invariants are unchanged. The properties of the apparent resistivity tensor are demonstrated for selected simple models. For a horizontally layered medium, when the receiver array is far from the current source, the tensor is symmetric and has invariants which depend only on the distance from the current source. The extreme values of apparent resistivity occur when the electric field vector is tangential and radial relative to the current source. These extreme values correspond to the Schlumberger apparent resistivity and the “polar” dipole apparent resistivity, respectively. Lateral discontinuities in resistivity are modeled with both a single vertical discontinuity and a hemispherical model. The source‐dependent variations in the apparent resistivity derived from a single‐current bipole are greatly reduced in plots of the tensor invariants. For a vertical discontinuity, the tensor trace (the sum of the diagonal elements) is close to the resistivity underlying the receiver site, whereas for a hemisphere, the square root of the tensor determinant gives the best representation. Near lateral discontinuities in resistivity, the apparent resistivity tensor indicates strong dependence of apparent resistivity on the direction of the measured electric field. This apparent anisotropy can be used as an indicator of such discontinuities, yielding both position and orientation of the discontinuity.

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