Alan C. Tripp

Two‐dimensional resistivity inversion

Resistivity data on a profile often must be interpreted in terms of a complex two‐dimensional (2-D) model. However, trial‐and‐error modeling for such a case can be very difficult and frustrating. To make interpretation easier and more objective, we have developed a nonlinear inversion technique that estimates the resistivities of cells in a 2-D model of predetermined geometry, based on dipole‐dipole resistivity data. Our numerical solution for the forward problem is based on the transmission‐surface analogy. The partial derivatives of apparent resistivity with respect to model resistivities are equal to a simple function of the currents excited in the transmission surface by transmitters placed at receiver and transmitter sites. Thus, for the dipole‐dipole array the inversion requires only one forward problem per iteration. We use the Box‐Kanemasu method to stabilize the parameter step at each iteration. We have tested our inversion technique on synthetic and field data. In both cases, convergence is rapi...

FDTD simulation of EM wave propagation in 3-D media

A finite‐difference, time‐domain solution to Maxwell’s equations has been developed for simulating electromagnetic wave propagation in 3-D media. The algorithm allows arbitrary electrical conductivity and permittivity variations within a model. The staggered grid technique of Yee is used to sample the fields. A new optimized second‐order difference scheme is designed to approximate the spatial derivatives. Like the conventional fourth‐order difference scheme, the optimized second‐order scheme needs four discrete values to calculate a single derivative. However, the optimized scheme is accurate over a wider wavenumber range. Compared to the fourth‐order scheme, the optimized scheme imposes stricter limitations on the time step sizes but allows coarser grids. The net effect is that the optimized scheme is more efficient in terms of computation time and memory requirement than the fourth‐order scheme. The temporal derivatives are approximated by second‐order central differences throughout. The Liao transmitt...

Inversion of diffusive transient electromagnetic data by a conjugate‐gradient method

Inversion of three-dimensional transient electromagnetic (TEM) data to obtain electrical conductivity and permeability can be done by a time-domain algorithm that extends to diffusive electromagnetic (EM) fields the imaging methods originally developed for seismic wavefields (Claerbout, 1971; Tarantola, 1984). The algorithm uses a conjugate-gradient search for the minimum of an error functional involving EM measurements governed by Maxwells equations without displacement currents. The connection with wavefield imaging comes from showing that the gradient of the error functional can be computed by propagating the errors back into the model in reverse time and correlating the field generated by the backpropagation with the incident field at each point. These two steps (backpropagation and cross correlation) are the same ones used in seismic migration. The backpropagated TEM fields satisfy the adjoint Maxwells equations, which are stable in reverse time. With magnetic field measurements the gradient of the error functional with respect to conductivity is the cross correlation of the backpropagated electric field with the incident electric field, whereas the gradient with respect to permeability is the cross correlation of the backpropagated magnetic field with the time derivative of the incident magnetic field. Tests on two-dimensional models simulating crosswell TEM surveys produce good images of a conductive block scatterer, with both exact and noisy data, and of a dipping conductive layer. Convergence, however, is slow.

Electromagnetic scattering of large structures in layered earths using integral equations

An electromagnetic scattering algorithm for large conductivity structures in stratified media has been developed and is based on the method of system iteration and spatial symmetry reduction using volume electric integral equations. The method of system iteration divides a structure into many substructures and solves the resulting matrix equation using a block iterative method. The block submatrices usually need to be stored on disk in order to save computer core memory. However, this requires a large disk for large structures. If the body is discretized into equal-size cells it is possible to use the spatial symmetry relations of the Greens functions to regenerate the scattering impedance matrix in each iteration, thus avoiding expensive disk storage. Numerical tests show that the system iteration converges much faster than the conventional point-wise Gauss-Seidel iterative method. The numbers of cells do not significantly affect the rate of convergency. Thus the algorithm effectively reduces the solution of the scattering problem to an order of O(N2), instead of O(N3) as with direct solvers.

3-D electromagnetic modeling for near‐surface targets using integral equations

Three‐dimensional electromagnetic (EM) modeling in the frequency range from 100 kHz to about 200 MHz using integral equations is examined. The modeling algorithm is formulated in the frequency domain. Time‐domain responses for ground‐penetrating radar (GPR) and very early time time‐decaying transients are computed via Fourier transforms. Of vital importance to the modeling problem is the computation of the Hankel transforms in the Greens functions. The kernels of those Hankel transforms vary rapidly at high frequencies where displacement currents become important and are even singular for sources in the air, with poles approaching the real axis or branch cuts lying on the real axis. We use high density Hankel filters and a singularity extraction technique to circumvent these problems. Our modeling for GPR applications shows that dielectric targets are very obvious in radargrams, with waves reflected by target boundaries arriving at distinctive times, depending on the path they travel. While GPR signals a...

Optimal survey design using focused resistivity arrays

The first problem which needs to be solved when planning any geoelectrical survey is a choice of a particular electrode configuration that can give the maximal response from a target inhomogeneity. The authors formulate a problem of maximizing the response as an optimization problem for an applied current intensity distribution on the surface. The solution of this problem is the optimal intensity distribution of the current, which maximizes the response from the inclusion. This problem is solved numerically with singular value decomposition of an impedance matrix. The optimal current array is modeled as a current of varying optimal intensity injected at different electrodes. The problem does not need any information about the inclusion but its measured impedance matrix. Thus an optimal current array can be designed for every particular resistivity distribution. The optimal current patterns are found for a number of models of a conductive inclusion, and responses due to the optimal current are compared with responses due to conventional arrays. This method can be applied to any background and inclusion resistivity distribution.

Investigating the resolution of IP arrays using inverse theory

Using a fast 2-D inverse solution, we examined the resolution of different resistivity/IP arrays using noisy synthetic data subject to minimum structure inversion. We compared estimated models from inversions of data from the dipole-dipole, pole-dipole, and pole-pole arrays over (1) a dipping, polarizable conductor, (2) two proximate conductive, polarizable bodies, (3) a polarizable conductor beneath conductive overburden, and (4) a thin, resistive, polarizable dike. The estimated resistivity and polarizability models obtained from inversion of the dipole-dipole data were usually similar to the pole-dipole estimated models. In the cases examined, the estimated models from the pole-pole data were more poorly resolved than the models from the other arrays. If pole-pole resistivity data contain even a fraction of a percent of Gaussian noise, the transformation of such data through superposition to equivalent data of other array types may be considerably distorted, and significant information can be lost using the pole-pole array. Though the gradient array is reputed to be more sensitive to dip than other arrays, it evidently contains little information on dip that does not also appear in dipole-dipole data, for joint inversion of dipole-dipole and gradient array data yields models virtually identical to those obtained from inversion of dipole-dipole data alone.

A block iterative algorithm for 3-D electromagnetic modeling using integral equations with symmetrized substructures

The integral equation method is in many cases a cost effective way of modeling the electromagnetic response of 3-D conductivity structures. Yet the requirements on computer storage and on computation time in forming and inverting the scattering matrix limit its applications for large structures.

Inverse conductivity problem for inaccurate measurements

In order to determine the conductivity of a body or of a region of the Earth using electrical prospecting, currents are injected on the surface, surface voltage responses are measured, and the data are inverted to a conductivity distribution. In the present paper a new approach to the inverse problem is considered for measurements containing noise in which data are optimally chosen using available a priori information at the time of the imaging. For inexact data the eigenvalues of the current-to-voltage boundary mapping show what part of the conductivity function can be confidently restored from the measurements. A special choice of measurements permits simple inversion algorithms reconstructing only this reliable part of the solution, hence reducing the dimension of the inverse problem. The inversion approach is illustrated in application to numerical modelling via a very fast approximate imaging solution.

Electromagnetic and Schlumberger resistivity sounding in the Roosevelt Hot Springs KGRA

One‐ and two‐dimensional modeling of the Schlumberger soundings at the Roosevelt Hot Springs KGRA have indicated a low‐resistivity zone of approximately 5 Ω-m paralleling the dome fault. The low resistivity of this zone is probably due to intensely fractured and altered water‐saturated rock. A zone of resistivity 12 Ω-m extending to the west of the dome fault is probably due to leakage of brine away from the geothermal system through alluvium or moderately altered rock. A resistive basement underlies the conductive zones and is believed to be essentially nonporous and unaltered rock. A major problem in the application of one‐dimensional (1-D) modeling of Schlumberger data in the Roosevelt Hot Springs KGRA is poor resolution of the 1-D parameters. The joint inversion of Schlumberger and electromagnetic sounding data gives a least‐squares 1-D conductivity model in which parameters are much better resolved than are the model parameters estimated by the inversion of Schlumberger data alone. One‐dimensional mo...