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 Volume 37, Issue 4, 1989
Geophysical Prospecting  Volume 37, Issue 4, 1989
Volume 37, Issue 4, 1989


STOCHASTIC INVERSION BY RAY CONTINUATION: APPLICATION TO SEISMIC TOMOGRAPHY^{1}
Authors A. HAAS and J. R. VIALLIXAbstractThe conventional tomographic inversion consists in minimizing residuals between measured and modelled traveltimes. The process tends to be unstable and some additional constraints are required to stabilize it. The stochastic formulation generalizes the technique and sets it on firmer theoretical bases.
The Stochastic Inversion by Ray Continuation (Sirc) is a probabilistic approach, which takes a priori geological information into account and uses probability distributions to characterize data correlations and errors. It makes it possible to tie uncertainties to the results.
The estimated parameters are interval velocities and B‐spline coefficients used to represent smoothed interfaces. Ray tracing is done by a continuation technique between source and receivers. The ray coordinates are computed from one path to the next by solving a linear system derived from Fermat's principle. The main advantages are fast computations, accurate traveltimes and derivatives.
The seismic traces are gathered in CMPs. For a particular CMP, several reflecting elements are characterized by their time gradient measured on the stacked section, and related to a mean emergence direction.
The program capabilities are tested on a synthetic example as well as on a field example. The strategy consists in inverting the parameters for one layer, then for the next one down. An inversion step is divided in two parts. First the parameters for the layer concerned are inverted, while the parameters for the upper layers remain fixed. Then all the parameters are reinverted.
The velocity‐depth section computed by the program together with the corresponding errors can be used directly for the interpretation, as an initial model for depth migration or for the complete inversion program under development.



VELOCITY‐STACK PROCESSING^{1}
More LessAbstractA conventional velocity‐stack gather consists of constant‐velocity CMP‐stacked traces. It emphasizes the energy associated with the events that follow hyperbolic traveltime trajectories in the CMP gather. Amplitudes along a hyperbola on a CMP gather ideally map onto a point on a velocity‐stack gather. Because a CMP gather only includes a cable‐length portion of a hyperbolic traveltime trajectory, this mapping is not exact. The finite cable length, discrete sampling along the offset axis and the closeness of hyperbolic summation paths at near‐offsets cause smearing of the stacked amplitudes along the velocity axis. Unless this smearing is removed, inverse mapping from velocity space (the plane of stacking velocity versus two‐way zero‐offset time) back to offset space (the plane of offset versus two‐way traveltime) does not reproduce the amplitudes in the original CMP gather. The gather resulting from the inverse mapping can be considered as the model CMP gather that contains only the hyperbolic events from the actual CMP gather. A least‐squares minimization of the energy contained in the difference between the actual CMP gather and the model CMP gather removes smearing of amplitudes on the velocity‐stack gather and increases velocity resolution. A practical application of this procedure is in separation of multiples from primaries.
A method is described to obtain proper velocity‐stack gathers with reduced amplitude smearing. The method involves a t^{2}‐stretching in the offset space. This stretching maps reflection amplitudes along hyperbolic moveout curves to those along parabolic moveout curves. The CMP gather is Fourier transformed along the stretched axis. Each Fourier component is then used in the least‐squares minimization to compute the corresponding Fourier component of the proper velocity‐stack gather. Finally, inverse transforming and undoing the stretching yield the proper velocity‐stack gather, which can then be inverse mapped back to the offset space. During this inverse mapping, multiples, primaries or all of the hyperbolic events can be modelled. An application of velocity‐stack processing to multiple suppression is demonstrated with a field data example.



NUMERICAL SOLUTION OF THE ACOUSTIC AND ELASTIC WAVE EQUATIONS BY A NEW RAPID EXPANSION METHOD^{1}
Authors DAN KOSLOFF, ANIBAL QUEIROZ FILHO, EKKEHART TESSMER and ALFRED BEHLEAbstractWe present a new rapid expansion method (REM) for the time integration of the acoustic wave equation and the equations of dynamic elasticity in two spatial dimensions. The method is applicable to spatial grid methods such as finite differences, finite elements or the Fourier method. It is based on a Chebyshev expansion of the formal solution to the appropriate wave equation written in operator form. The method yields machine accuracy yet it is faster than methods based on temporal differencing. Its disadvantages are that it does not apply to all types of material rheology, and it can also require much storage when many snapshots and time sections are desired. Comparisons between numerical and analytical solutions for simple acoustic and elastic problems demonstrate the high accuracy of the REM.



SENSITIVITY ANALYSIS OF MAGNETOTELLURIC MEASUREMENTS IN RELATION TO STATIC EFFECTS^{1}
More LessAbstractThe relation between the horizontal profiles of the subsurface resistivity and surface magnetotelluric data can be described by the input and output of a moving‐average filter. The impulse response of this spatial filter, which characterizes the averaging process of the magnetotelluric measurements, is given by the sensitivity profile. Thus, the sensitivity analysis can provide insight into the characteristics of the measurements and hence the mechanism of the static effects. The sensitivity analysis presented here consists of constructing the vertical section of the sensitivity distribution using the finite‐element method and then Fourier transforming the selected horizontal profiles. When the dipole is assumed for measuring the electric field, the static effects can be explained by the high‐pass filter characteristics for the near‐surface. When the electrode separation is taken into account, the sensitivity can be obtained by averaging the sensitivities for the dipoles over the horizontal distance equal to the electrode separation. Therefore, the higher‐frequency components at each depth decrease with increasing electrode separation. Thus, although the static effects can be reduced simply by increasing the electrode separation, information on the resistivity variation at depth is also lost. However, such an adverse effect can be reduced by making the EMAP‐type measurements followed by the spatial filtering of the profile data using the tapered weighting function.



3D ELECTROMAGNETIC INVERSION USING INTEGRAL EQUATIONS^{1}
More LessABSTRACTUtilizing electromagnetic data in geophysical exploration work is difficult when measured responses are complicated by the effects of 3D structures. 1D and 2D models may not be capable of accurately simulating the physical processes that contribute to a measured response. 3D conductive‐host modelling is difficult, costly and time‐consuming. Using a 3D inverse procedure it is possible to automate the interpretation of controlled‐source electromagnetic data. This procedure uses an inverse formulation based on frequency‐domain, volume integral equations and a pulse‐basis representation for the internal electrical field and anomalous conductivity. Beginning with an initial model composed of a 3D inhomogeneous region residing in a laterally homogeneous (layered‐earth) geoelectrical section, iterative least‐squares algorithms are used to refine the geometry and the conductivity of the inhomogeneity. This novel approach for 3D electromagnetic interpretation yields a reliable and stable inverse solution provided constraints on how much the variable can change at each iteration are incorporated. Integral‐equation‐based inverse formulations that do not correctly address the non‐linearity of this inverse problem may have poor convergence properties, particularly when dealing with the high conductivity contrasts that are typical of many exploration problems.
While problems associated with contamination of the data by random noise and non‐uniqueness of solutions do not usually influence the inverse solution in an adverse manner, problems associated with model inadequacy and errors in an assumed background conductivity structure can produce undesirable effects.



MAGNETIC WAVETILT MEASUREMENTS FOR GEOLOGICAL FRACTURE MAPPING^{1}
More LessAbstractThe measurement of wavetilt is diagnostic for determining the electrical characteristics of the upper layers of the ground at VLF and LF frequency ranges. Theoretical and field studies have indicated that electric wavetilt using the transverse magnetic (TM) waves detects lateral inhomogeneities virtually instantly as abrupt changes in electrical properties are encountered. Theoretical studies have also indicated that magnetic wavetilt measurements using transverse electric (TE) waves are superior to electric wavetilts for such purposes.
An experimental survey was conducted at two locations near Atikokan, Ontario, to verify the theoretical predictions. The survey area, forming a part of a large granitic pluton, was mapped earlier by various geophysical techniques, including the ground VLF‐EM method, to detect weak conductors formed either by the presence of fractures in the bedrock filled with water and/or clay, or overburden filling bedrock depressions.
A small, multi‐turn, horizontal loop was used during the survey as the transmitter to generate TE waves at eleven frequencies from 10.7 to 58.5 kHz. The magnetic wavetilt measurements detected all previously known conductors at the two locations. In addition, the survey detected several weak conductors that were missed by the VLF survey. Thus, the survey indicated the usefulness of magnetic wavetilt results for detection of weak conductors at shallow depths, which may have application in engineering geophysical surveys. The multi‐frequency wavetilt data also provided some indications of the depth and depth extent of such conductors.

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