Walter L. Anderson

Computer program; numerical integration of related Hankel transforms of orders O and 1 by adaptive digital filtering

A linear digital filtering algorithm is presented for rapid and accurate numerical evaluation of Hankel transform integrals of orders 0 and 1 containing related complex kernel functions. The kernel for Hankel transforms is defined as the non‐Bessel function factor of the integrand. Related transforms are defined as transforms, of either order 0 or 1, whose kernel functions are related to one another by simple algebraic relationships. Previously saved kernel evaluations are used in the algorithm to obtain rapidly either order transform following an initial convolution operation. Each order filter is designed with identical abscissas over a large range so that an adaptive convolution procedure can be applied to a large class of kernels. Different order Hankel transforms with related kernels are often found in electromagnetic (EM) applications. Because of the general nature of this algorithm, the need to design new filters should not be necessary for most applications. Accuracy of the filters is comparable t...

Fast Hankel Transforms Using Related and Lagged Convolutions

A heuristic algorithm is presented for fast and accurate evaluation of complex Hankel transforms of orders 0 and 1. Concepts using linear digital convolution are introduced, where Bessel function evaluations are not required. Related convolution is defined over a set of transforms whose integrands are related algebraically. Given any transform argument range, fast lagged convolution is performed m place within a predeslgned digital filter By arranging related and lagged convolutions m matrix form, a whole matrix of complex Hankel transforms is evaluated with a minimum of integrand function calls. For each point in the matrix, an adaptive convolution algorithm (based on function behavior and a given truncation tolerance) further reduce unnecessary evaluations along the decreasing digital-filter tails. The class of mtegrands (excluding the Bessel factor) must be continuous monotonic decreasing complex (or real) functions of a real variable defined in (0, oo). Higher integerorder Hankel transforms may be expressed in terms of orders 0 and 1 by recursion, and can be rapidly evaluated by related convolution. Several elementary and practical examples are given to demonstrate convolution accuracy and efficiency (m terms of total functmn evaluations), where minimum relatwe errors are approximately 10 -s using 32-bit floating-point words. (A double-precision real version is also provided, which yields minimum relative errors approximately 10-12.) Some suggestions are made for future algorithm enhancements. The algorithm is written in portable FORTRAN IV.

A hybrid fast Hankel transform algorithm for electromagnetic modeling

A hybrid fast Hankel transform algorithm has been developed that uses several complementary features of two existing algorithms: Anderson’s digital filtering or fast Hankel transform (FHT) algorithm and Chave’s quadrature and continued fraction algorithm. A hybrid FHT subprogram (called HYBFHT) written in standard Fortran-77 provides a simple user interface to call either subalgorithm. The hybrid approach is an attempt to combine the best features of the two subalgorithms in order to minimize the user’s coding requirements and to provide fast execution and good accuracy for a large class of electromagnetic problems involving various related Hankel transform sets with multiple arguments. Special cases of Hankel transforms of double‐order and double‐argument are discussed, where use of HYBFHT is shown to be advantageous for oscillatory kernel functions.

Effect of transmitter turn-off time on transient soundings

Abstract A general procedure for computing the effect of non-zero turn-off time on the transient electromagnetic response is presented which can be applied to forward and inverse calculation methods for any transmitter-receiver configuration. We consider in detail the case of a large transmitter loop which has a receiver coil located at the center of the loop (central induction or in-loop array). For a linear turn-off ramp of width t 0 , the voltage response is shown to be the voltage due to an ideal step turn-off averaged over windows of width t 0 . Thus the effect is similar to that obtained by using averaging windows in the receiver. In general when time zero is taken to be the end of the ramp, the apparent resistivity increases for a homogeneous half-space over a limited time range. For time zero taken to be the start of the ramp the apparent resistivity is affected in the opposite direction. The effect of the ramp increases with increasing t 0 and first-layer resistivity, is largest during the intermediate stage, and decreases with increasing time. It is shown that for a ramp turn-off, there is no effect in the early and late stages. For two-layered models with a resistive first layer ( ρ 1 > ρ 2 ), the apparent resistivity is increased in the intermediate stage. When the first layer is more conductive than the second layer ( ρ 1 ρ 2 ) and the layer thickness is comparable or greater than the loop radius, similar results are obtained; however, when the layer is thin compared to the loop radius the apparent resistivity is initially decreased and then increases as time increases. Examples are presented which illustrate the strong influence of the geoelectrical section on the turn-off effect. Neglecting the turn-off ramp will affect data interpretation as shown by field examples; the influence is the greatest on near-surface layer parameters.

Shallow subsurface mapping by electromagnetic sounding in the 300 kHz to 30 MHz range: Model studies and prototype system assessment

A new instrument designed for frequency‐domain sounding in the depth range 0–10 m uses short coil spacings of 5 m or less and a frequency range of 300 kHz to 30 MHz. In this frequency range, both conduction currents (controlled by electrical conductivity) and displacement currents (controlled by dielectric permittivity) are important. Several surface electromagnetic survey systems commonly used (generally with frequencies less than 60 kHz) are unsuitable for detailed investigation of the upper 5 m of the earth or, as with ground‐penetrating radar, are most effective in relatively resistive environments. Most computer programs written for interpretation of data acquired with the low‐frequency systems neglect displacement currents, and are thus unsuited for accurate high‐frequency modeling and interpretation. New forward and inverse computer programs are described that include displacement currents in layered‐earth models. The computer programs and this new instrument are used to evaluate the effectiveness ...

Computation of Green’s tensor integrals for three‐dimensional electromagnetic problems using fast Hankel transforms

A new method is presented that rapidly evaluates the many Green’s tensor integrals encountered in three‐dimensional electromagnetic modeling using an integral equation. Application of a fast Hankel transform (FHT) algorithm (Anderson, 1982) is the basis for the new solution, where efficient and accurate computation of Hankel transforms are obtained by related and lagged convolutions (linear digital filtering). The FHT algorithm is briefly reviewed and compared to earlier convolution algorithms written by the author. The homogeneous and layered half‐space cases for the Green’s tensor integrals are presented in a form so that the FHT can be easily applied in practice. Computer timing runs comparing the FHT to conventional direct convolution methods are discussed, where the FHT’s performance was about 6 times faster for a homogeneous half‐space, and about 108 times faster for a five‐layer half‐space. Subsequent interpolation after the FHT is called is required to compute specific values of the tensor integra...

Approximate inversion of high‐frequency electromagnetic soundings using complex image theory

Electromagnetic (EM) soundings in an intermediate range between quasi‐static and radar frequencies (e.g., 30 kHz – 300 MHz) were studied with an approximate inversion method for shallow‐layered earth models using complex image theory. The half‐space image theory formulas for calculating forward soundings for any dipole source are easily extended to accommodate a multilayered earth, and only involve elementary complex functions. Approximate inversion using the modified image formulas is compared with more exact numerical integration inversion for horizontal‐layered models to yield the conductivity, thickness, and dielectric permittivity of each layer. A nonlinear least‐squares algorithm was used to obtain parameter and linear parameter error estimates. The results, using known exact forward models and one field data case, indicate that approximate image inversion is more than an order‐of‐magnitude faster than numerical integration inversion. However, approximate inversion could not always resolve deeper la...

Q‐factor approximation of electromagnetic fields for high‐frequency sounding in the 300 kHz to 30 MHz range over layered media

A new method for rapid approximation of electromagnetic (EM) fields for high‐frequency sounding (HFS) over a layered earth is presented in this paper. The essence of this method uses a Q‐factor correction for extending a closed‐form, half‐space analytic solution to a layered earth model. Use of the Q‐factor in this context was first studied by Wait (1953, 1962). Kraichman (1976) also discusses the problem of when the Q‐factor method can be used to provide a good approximation to an exact layered earth solution.

Effect of conductive host rock on borehole transient electromagnetic responses

Transient electromagnetic (TEM) borehole responses of 3-D vertical and horizontal tabular bodies in a half‐space are calculated to assess the effect of a conductive host. The transmitter is a large loop at the surface of the earth, and the receiver measures the time derivative of the vertical magnetic field. When the host is conductive (100 Ω ⋅ m), the borehole response is due mainly to current channeled through the body. The observed magnetic‐field response can be visualized as due to galvanic currents that pass through the conductor and return in the half‐space. When the host resistivity is increased, the magnetic field of the conductor is influenced more by vortex currents that flow in closed loops inside the conductor. For a moderately resistive host (1000 Ω ⋅ m), the magnetic field of the body is caused by both vortex and galvanic currents. The galvanic response is observed at early times, followed by the vortex response at later times if the body is well coupled to the transmitter. If the host is ve...

Numerical integration of related Hankel transforms by quadrature and continued fraction expansion; discussion and reply

Chave presents an excellent algorithm and computer subprogram that evaluate Hankel transforms by quadrature and continued fraction summation. This subprogram would be a valuable addition to any mathematical computer library. Chave refers to digital filter (convolution) methods as “the standard numerical approach to the computation of Hankel transforms,” and makes numerous references to my published work (Anderson, 1979, 1982). However, some readers may misinterpret some of his statements regarding convolution methods, which I hope to clarify by this discussion.