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 Volume 36, Issue 4, 1988
Geophysical Prospecting  Volume 36, Issue 4, 1988
Volume 36, Issue 4, 1988


ENERGY‐INTERACTION: THE LONG‐RANGE INTERACTION OF SEISMIC SOURCES^{1}
More LessABSTRACTThe quantity of energy that is radiated as sound by marine seismic sources is examined. When more than one source is used, the total radiated acoustic energy depends on the separation of the sources and is not constant, even when the separation is sufficiently large that the individual signatures are substantially unaffected. We believe that this long‐range interaction has not been described before in the literature and have called it energy‐interaction.
The existence of energy‐interaction is demonstrated experimentally for airguns. In. the experiment presented, the acoustic output is more than doubled because of energy‐interaction.
Two methods are described for computing the energy of the wavefield of an array of sources. One method is simple and direct; the energy that is radiated into the far field is calculated by integrating the directivity pattern over all directions. The other method finds the energy that is radiated into the near field. Because the medium is assumed lossless, these two energies are the same. Although the near‐field method is conceptually more difficult, it is faster, more accurate and provides a more detailed description of the energy budget.



SEISMIC AND ELECTRICAL PROPERTIES OF UNCONSOLIDATED PERMAFROST^{1}
Authors M. S. KING, R. W. ZIMMERMAN and R. F. CORWINABSTRACTA model has been developed to relate the velocities of acoustic waves Vp and Vs in unconsolidated permafrost to the porosity and extent of freezing of the interstitial water. The permafrost is idealized as an assemblage of spherical quartz grains embedded in a matrix composed of spherical inclusions of water in ice. The wave‐scattering theory of Kuster and Toksoz is used to determine the effective elastic moduli, and hence the acoustic velocities. The model predicts Vp and Vs to be decreasing functions of both the porosity and the water‐to‐ice ratio. The theory has been applied to laboratory measurements of Vp and Vs in 31 permafrost samples from the North American Arctic. Although no direct measurements were made of the extent of freezing in these samples, the data are consistent with the predictions of the model. Electrical resistivity measurements on the permafrost samples have demonstrated their essentially resistive behaviour. The ratio of resistivity of permafrost in its frozen state to that in its unfrozen state has been related to the extent of freezing in the samples.
Electromagnetic and seismic reflection surveys can be used together in areas of permafrost: firstly an EM survey to determine the extent of freezing and then the acoustic velocity model to predict the velocities in the permafrost. The necessary transit time corrections can thus be made on seismic reflection records to compensate for the presence of permafrost.



THE MAXIMUM ENTROPY APPROACH TO THE INVERSION OF ONE‐DIMENSIONAL SEISMOGRAMS^{1}
By E. RIETSCHABSTRACTDetermination of impedance or velocity from a stacked seismic trace generally suffers from noise and the fact that seismic data are bandlimited. These deficiencies can frequently be alleviated by ancillary information which is often expressed more naturally in terms of probabilities than in the form of equations or inequalities.
In such a situation information theory can be used to include ‘soft’information in the inversion process. The vehicle used for this purpose is the Maximum Entropy (ME) principle. The basic idea is that a prior probability distribution (pd) of the unknown parameter(s) or function(s) is converted into a posterior pd which has a larger entropy than any other pd which also accounts for the information. Since providing new information generally lowers the entropy, this means that the ME pd is as non‐committal as possible with regard to information which is not (yet) available. If the information used is correct, then the ME pd cannot be contradicted by new, also correct, data and thus represents a conservative solution to the inverse problem.
In the actual implementation, the final result is, generally, not the pd itself (which may be quite broad) but rather the expectation values of the desired parameter(s) or function(s).
A general problem of the ME approach is the need for a prior pd for the parameter(s) to be estimated. The approach used here for the velocity is based on an invariance criterion, which ensures that the result is the same whether velocity or slowness is estimated. Unfortunately, this criterion does not provide a unique prior pd but rather a class of functions from which a suitable one must be selected with the help of other considerations.



THE NON‐LINEAR INVERSION OF SEISMIC WAVEFORMS CAN BE PERFORMED EITHER BY TIME EXTRAPOLATION OR BY DEPTH EXTRAPOLATION^{1}
Authors A. TARANTOLA, G. JOBERT, D. TREZEGUET and E. DENELLEABSTRACTThe classical aim of non‐linear inversion of seismograms is to obtain the earth model which, for null initial conditions and given sources, best predicts the observed seismograms. This problem is currently solved by an iterative method: each iteration involves the resolution of the wave equation with the actual sources in the current medium, the resolution of the wave equation, backwards in time, with the current residuals as sources; and the correlation, at each point of space, of the two wavefields thus obtained.
Our view of inversion is more general: we want to obtain a whole set of earth model, initial conditions, source functions, and predicted seismograms, which are the closest to some a priori values, and which are related through the wave equation. It allows us to justify the previous method, but it also allows us to set the same inverse problem in a different way: what is now searched for is the best fit between calculated and a priori initial conditions, for given sources and observed surface displacements. This leads to a completely different iterative method, in which each iteration involves the downward extrapolation of given surface displacements and tractions, down to a given depth (the‘bottom’), the upward extrapolation of null displacements and tractions at the bottom, using as sources the initial time conditions of the previous field, and a correlation, at each point of the space, of the two wavefields thus obtained. Besides the theoretical interest of the result, it opens the way to alternative numerical methods of resolution of the inverse problem. If the non‐linear inversion using forward‐backward time propagations now works, this non‐linear inversion using downward‐upward extrapolations will give the same results but more economically, because of some tricks which may be used in depth extrapolation (calculation frequency by frequency, inversion of the top layers before the bottom layers, etc.).



BOREHOLE COUPLING OF SEISMIC WAVES IN A PERMEABLE SOLID^{1}
Authors J. E. WHITE and E. WELSHABSTRACTExpressions are derived for the acoustic pressure in a fluid‐filled borehole due to the passage of plane compressional or shear waves propagating in a porous and permeable medium. The derivation is based on a quasistatic description, yielding results applicable at low frequencies. Computations for a plausible sandstone (with porosity of 0.21 and permeability of 300 millidarcies) show accoustic pressure to be a surprisingly distortion‐free version of the stress waveform in a plane compressional wave. For a plane shear wave, the pressure waveform is visibly distorted.



A NON‐GEOMETRICAL SH‐ARRIVAL^{1}
Authors P. F. DALEY and F. HRONABSTRACTThe existence of‘*‐waves’has, in recent years, prompted a renewed interest in these non‐geometrical arrivals, which are generated by point sources located adajcent to plane interfaces. It has led to the re‐evaluation of seismic data aquisition techniques and to the question of how to use this real phenomena in enhancing existing seismic interpretation methods.
This paper considers a non‐geometrical SH‐arrival which is generated by a point torque source unrealistically buried within a half‐space. The method of solution is essentially the same as presented in an earlier paper, with the modification that the limitation placed on the distance of the source from the interface has been removed in the saddle point method used to obtain a high‐frequency approximate solution. In the earlier paper, a preliminary assumption forced the saddle point, which corresponded to the *‐wave arrival, to be real when it is generally complex. However, for offsets removed from the distinct ray, the imaginary part of this complex quantity is negligible.
A problem which arose when comparing exact synthetic traces with those obtained using zero‐order saddle point methods, was the inability to match either the amplitude or phase of the geometrical arrival in the range of offsets when the *‐wave and this corresponding geometrical ray were well separated. For this range of offsets the geometrical arrival was approaching grazing incidence and another term in the saddle point expansion of the integral was necessary to rectify this error. This method is also being used to validate the results for higher order terms obtained using asymptotic ray theory.
Analytical formulae are given for both the *‐wave and the higher order expansion of the geometrical event, together with a comparison of synthetic seismograms using the method developed here and a numerical integration algorithm.



APPROXIMATE INVERSION OF AIRBORNE EM DATA FROM A MULTILAYERED GROUND^{1}
More LessABSTRACTA complex transfer function c (or generalized skin depth) can be derived from data for the secondary magnetic field measured by a dipole system with small coil spacing at height h above the ground. This function has a useful property: For a uniform or layered ground, the real part of c yields the‘ centroid depth’z* of the in‐phase current system as a function of frequency.
This parameter can be combined with the apparent resistivity ρa derived by conventional methods. The function ρa(z*), if known over a broad frequency range, yields a smoothed approximation of the true distribution ρ(z) without an initial model. The relations between ρa(z*) and ρ(z) are studied for a number of multilayer models. An example of the application of the ρa*) algorithm to data from a groundwater survey is given.

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Volume 72 (2023  2024)

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