Geophysical Prospecting - Volume 32, Issue 2, 1984

Geophysical inversion involves the estimation of the parameters of a postulated earth model from a set of observations. Since the associated model responses can be nonlinear functions of the model parameters, nonlinear least‐squares techniques prove to be useful for performing the inversion. A common type of inversion applies iterative damped linear least squares through use of the Marquardt‐Levenberg method. Traditionally, this method has been implemented by solving the associated normal equations in conventional ways. However, Singular Value Decomposition (SVD) produces significant improvements in computational precision when applied to the same system of normal equations. Iterative least‐squares modeling finds application in a wide variety of geophysical problems. Two examples illustrate the approach: (1) seismic wavelet deconvolution, and (2) the location of a buried wedge from surface gravity data. More generally, nonlinear least‐squares inversion can be used to estimate earth models for any set of geophysical observations for which an appropriate mathematical description is available.

Bedding on a scale small in comparison to the wavelength results in transverse isotropy. On the other hand, anisotropy has been observed in the field, particularly with horizontally polarized shear waves. In this article velocity data from 23 wells are used to estimate the resulting transverse isotropy and to compare these data with anisotropy observations reported in the literature. Since the well data were compressional velocity (or rather transit time) only, the corresponding shear velocity data were estimated on the basis of a reasonable—though arbitrary—assumption of Poisson's ratio. Actual densities were used wherever they were available; for the remainder of the data, density was assumed to be constant throughout the averaging interval.

The anisotropy data estimated on this basis were generally considerably smaller than the observational data that have been reported in the literature. Although both the above assumptions—functional dependence of Poisson's ratio on transit time and constant density—tend to underestimate anisotropy, and although 23 wells is but a small subset of all available data, it appears unlikely that bedding alone could cause anisotropy of the magnitude reported.

In this study we derive expressions for particle displacement or particle velocity anywhere inside a stratified earth and at its surface due to horizontal torque source located in the top layer. Equivalently, invoking Green's function reciprocity theorem, the solution applies also to the case of a surface or subsurface source when the resulting displacement or velocity is measured within the top layer.

In order to evaluate the closed‐form analytical solution economically and accurately it is advisable to introduce inelastic attenuation. Causal inelastic attenuation also lends the necessary realism to the computed seismic trace. To provide proof that the analytical solution is indeed correct and applicable to the multilayer case, a thick uniform overburden was assumed to consist of many thin layers. The correctness of the computed particle velocity response can be very simply verified by inspection. The computed response can also serve as a check on other less accurate methods of producing synthetic seismograms, such as the techniques of finite differences, finite elements, and various sophisticated ray‐tracing techniques.

It is not difficult to construct horizontal surface torque source. It appears that such source is well suited for seismic exploration in areas with a high‐velocity surface layer. A realistic source function is analyzed in detail and normalized displacement response evaluated at different incidence angles in the near and the far fields.

In an effort to distinguish the features of an SH torque seismogram from a pressure seismogram two models with identical layerings and layer parameters have been set up. As expected the torque seismogram is very different from the compressional seismogram. One desirable feature of a torque seismogram is the fast decay of multiples.

Exact synthetic seismograms have many uses; some of them, such as the study of complex interference phenomena, phase change at wide angle reflection, channeling effects, dispersion (geometrical and material), absolute gain, and inelastic attenuation, can be carried out accurately and effortlessly. They can also be used to improve basic processing techniques such as deconvolution and velocity analysis.

The numerical evaluation of the analytical solution of the wave equation as described in this paper has a long history. Most of the work leading to this paper was carried out by one of us (M. J. K.) in the years 1957 to 1968 at the Geophysical Research Corporation. However, the full testing of the various computer codes was carried out only very recently at the Phillips Petroleum Company.

The use of very long streamers in marine seismic surveys makes the response offset dependent. Line equalization networks are used to balance this effect.

In this paper the response of the transformerless streamer system driving a charge amplifier is analyzed. Examples are given for a seismic cable 2500 m long, showing that the delay and the reduction in amplitude of the streamer impulse response in the absence of leakage are of the order of 1 ms and 0.6 dB, respectively. This means that the line equalization network currently used with the transformerless streamer system is not needed. We have also demonstrated that the presence of leakage resistance across the hydrophone group render the sensitivity of the transformerless streamer system offset dependent.

A hybrid method for wave field computation in two‐dimensional heterogeneous media is proposed. The proposed method is a combination of analytical and numerical techniques. The method is based upon the separation of wave propagation and scattering and upon the description of each process by the most suitable technique. The SH wave scattering problem is used to elucidate the proposed method.

Examples of numerical computations using the hybrid method are considered for a number of simple models. The analysis of the results shows that the hybrid method gives both a detailed and a reasonably accurate description of the total wave field.

In land seismic surveys spectrum equalization can increase the quality of seismic data in a selected frequency band. The power of lower frequencies in the spectrum of input traces is generally greater than that of higher frequencies, particularly in land seismic surveys because of ground roll. In order to improve the quality of seismic data it is necessary to raise the energy of higher frequencies to the same level as that of lower frequencies, without alteration of the phases.

The first step of the method is to compute the amplitude spectrum of each input trace to determine a weighting function which is then applied to the amplitude spectrum in order to balance it. The function is the inverse of the short wavelength variation of the amplitude spectrum. The short wavelength variation can be obtained by interpolation between average values of the modulus of the amplitude spectrum computed in narrow bands within a selected band of frequencies. Another way of obtaining the short wavelength variation is to apply a low‐pass filter to the amplitude spectrum. The calculations are readily performed in the frequency domain by the Fourier transform.

Spectrum equalization is automatically adjusted to each trace and does not modify the average amplitude in the time domain. However, as the frequency band and energy of the ground roll both vary according to the distance from the shot, spectrum equalization tends to make the spectrum of output traces independent of the offset distance.

The use of spectrum equalization before any two‐dimensional filtering improves ground roll elimination. Continuity and resolution of horizons are also increased by spectrum equalization before CDP stack.

Several examples of applications of spectrum equalization to seismic land and marine surveys are shown.

In South and Central Goa iron ore occurs in two parallel belts with the general NW‐SE Dharawar trend. The ore occurrence, however, is not continuous. There are barren zones as well as zones of very high concentration in some of which there are mining activities.

Landsat MSS data have been interpreted over a zone covering both mineral belts in order to delineate the ore occurrences. As a guide line a known ore‐bearing area has been considered along with the unknown zones.

On the basis of two‐dimensional plotting of gray level values it has been found that the MSS bands 4 and 7 are most suitable for the studies over iron‐rich areas in Goa. Two techniques are described here for the processing of the MSS data; the separation of residual from the regional and MSS band‐ratioing. It is observed that (i) the gray level residual maps of MSS bands 4 and 7 are of use in demarcating the iron‐ore‐bearing zones, and (ii) an existing mine, an abandoned mine, and a proved iron ore zone could be delineated by MSS band‐ratioing. On the basis of the latter technique, a few areas with ore occurrence potential have been indicated.

After a brief discussion of MT modeling methods, we expose a particular application of finite elements that may be assimilated to a finite‐difference method. The resulting linear equations obtained are similar to the transmission network equations for two‐dimensional media. The introduction of resistivity discontinuities is more rigorous than in the usual finite‐difference development. The consequence is an improved accuracy.

We also present some conclusions about various problems encountered in modeling, such as the choice of network boundary conditions, linear system resolution and the final derivation of apparent resistivity. Application of successive over‐relaxation is discussed and we detail rules for mesh design that control result accuracy and iterative convergence.

A new mode of operation for the Turam electromagnetic exploration system is proposed in which the transmitter loop is placed across the expected trend of a conductor and the receiver is operated along lines parallel to one side of the transmitter. The concept appears to offer several benefits which include greatly extended traverse length, the use of large coil spacing, rejection of the effects of conductive environments, and consistency in the indication of target dip.

For helicopter‐borne electromagnetic systems, the distance between the transmitting and the receiving coils is small compared with the altitude above ground. For this case, a major simplification can be made for the calculation of model curves. Some two‐layer curves for the interpretation of frequency measurements are presented.

A very simple procedure is demonstrated for the conversion of the relative secondary field into apparent resistivity and apparent distance for the mapping of airborne electromagnetic data. Furthermore, an approximation is described for the determination of the thickness and the resistivity of a layer lying on a perfectly conducting half‐space.