Tutorial : 2-D and 3-D electrical imaging surveys

SUMMARY We present a novel technique for the determination of resistivity structures associated with arbitrary surface topography. The approach represents a triple-grid inversion technique that is based on unstructured tetrahedral meshes and finite-element forward calculation. The three grids are characterized as follows: A relatively coarse parameter grid defines the elements whose resistivities are to be determined. On the secondary field grid the forward calculations in each inversion step are carried out using a secondary potential (SP) approach. The primary fields are provided by a one-time simulation on the highly refined primary field grid at the beginning of the inversion process. We use a Gauss‐Newton method with inexact line search to fit the data within error bounds. A global regularization scheme using special smoothness constraints is applied. The regularization parameter compromising data misfit and model roughness is determined by an L-curve method and finally evaluated by the discrepancy principle. To solve the inverse subproblem efficiently, a least-squares solver is presented. We apply our technique to synthetic data from a burial mound to demonstrate its effectiveness. A resolution-dependent parametrization helps to keep the inverse problem small to cope with memory limitations of today’s standard PCs. Furthermore, the SP calculation reduces the computation time significantly. This is a crucial issue since the forward calculation is generally very time consuming. Thus, the approach can be applied to large-scale 3-D problems as encountered in practice, which is finally proved on field data. As a by-product of the primary potential calculation we obtain a quantification of the topography effect and the corresponding geometric factors. The latter are used for calculation of apparent resistivities to prevent the reconstruction process from topography induced artefacts.

Three‐dimensional modelling and inversion of dc resistivity data incorporating topography – II. Inversion

With the increased availability of faster computers, it is now practical to employ numerical modeling techniques to invert resistivity data for 3-D structure. Full and approximate 3-D inversion methods using the finite-element solution for the forward problem have been developed. Both methods use reciprocity for efficient evaluations of the partial derivatives of apparent resistivity with respect to model resistivities. In the approximate method, the partial derivatives are approximated by those for a homogeneous half-space, and thus the computation time and memory requirement are further reduced. The methods are applied to synthetic data sets from 3-D models to illustrate their effectiveness. They give a good approximation of the actual 3-D structure after several iterations in practical situations where the effects of model inadequacy and topography exist. Comparisons of numerical examples show that the full inversion method gives a better resolution, particularly for the near-surface features, than does the approximate method. Since the full derivatives are more sensitive to local features of resistivity variations than are the approximate derivatives, the resolution of the full method may be further improved when the finite-element solutions are performed more accurately and more efficiently.

3-D resistivity inversion using the finite-element method

Providing a balance between principles and practice, this state-of-the-art overview of geophysical methods takes readers from the basic physical phenomena, through the acquisition and processing of data, to the creation of geological models of the subsurface and data interpretation to find hidden mineral deposits. Detailed descriptions of all the commonly used geophysical methods are given, including gravity, magnetic, radiometric, electrical, electromagnetic and seismic methods. Each technique is described in a consistent way and without complex mathematics. Emphasising extraction of maximum geological information from geophysical data, the book also explains petrophysics, data modelling and common interpretation pitfalls. Packed with full-colour figures, also available online, the text is supported by selected examples from around the world, including all the major deposit types. Designed for advanced undergraduate and graduate courses in minerals geoscience, this is also a valuable reference for professionals in the mining industry wishing to make greater use of geophysical methods. In 2015, Dentith and Mudge won the ASEG Lindsay Ingall Memorial Award for their combined effort in promoting geophysics to the wider community with the publication of this title.

Geophysics for the Mineral Exploration Geoscientist

We simulate the magnetotelluric response to two‐dimensional earth topography using finite elements. Linear interpolation of the secondary field parallel to strike over triangular elements allows accurate modeling of inclined resistivity boundaries, including topographic surfaces. To avoid discontinuities in field derivatives or resistivity, care must be taken that the nodal values of the field parallal to strike used to obtain the auxiliary secondary fields are kept within uniform earth media. The nodal locations may be shifted, but the derivatives still are evaluated at the field points of interest. Correct values may be returned at gentle breaks in slope as well as along straight surfaces. The finite‐element program is verified by comparison with the analytic transverse magnetic response of a hemicylindrical depression and with Rayleigh scattering and transmission surface results for transverse electric and transverse magnetic polarization. Agreement with the other methods generally is excellent, with t...

Two-dimensional topographic responses in magnetotellurics modeled using finite elements

We have made a systematic study of dipole‐dipole apparent resistivity anomalies due to topography and of the effect of irregular terrain on induced‐polarization (IP) anomalies, using a two‐dimensional (2-D), finite‐element computer program. A valley produces a central apparent resistivity low in the resistivity pseudosection, flanked by zones of higher apparent resistivity. A ridge produces just the opposite anomaly pattern—a central high flanked by lows. A slope generates an apparent resistivity low at its base and a high at its top. Topographic effects are important for slope angles of 10 degrees or more and for slope lengths of one dipole‐length or greater. The IP response of a homogeneous earth is not affected by topography. However, irregular terrain does affect the observed IP response of a polarizable body due to variations in the distance between the electrodes and the body. These terrain‐induced anomalies can lead to erroneous interpretations unless topography is included in numerical modeling. A...

Terry J. Killpack

Papers

Topographic effects in resistivity and induced-polarization surveys