Numerical Stimulation on the Seismic Wave Attenuation in Random Inhomogeneous Media

Numerical stimulation was utilized to study the seismic wave attenuation in random medium. The main reasons of attenuation in elastic media were the spherical divergence and scattering wave caused by in homogeneities with variant scales. The elastic wave attenuation in random media cannot be analyzed by analytic method, so the numerical stimulation was the only way to discuss the problem. Rotated staggered finite difference method was chose to stimulate the situation. 6 random media were constructed with different correlation length. Numerical Stimulation was implemented in them, and particle displacements in certain locations were record. These records were compared with the ones in the corresponding homogeneous media to discuss the attenuation.

True‐amplitude imaging and dip moveout

True‐amplitude seismic imaging produces a three dimensional (3-D) migrated section in which the peak amplitude of each migrated event is proportional to the reflectivity. For a constant‐velocity medium, the standard imaging sequence consisting of spherical‐divergence correction, normal moveout (NMO), dip moveout (DMO), and zero‐offset migration produces a true‐amplitude image if the DMO step is done correctly. There are two equivalent ways to derive the correct amplitude‐preserving DMO. The first is to improve upon Hale’s derivation of F-K DMO by taking the reflection‐point smear properly into account. This yields a new Jacobian that simply replaces the Jacobian in Hale’s method. The second way is to calibrate the filter that appears in integral DMO so as to preserve the amplitude of an arbitrary 3-D dipping reflector. This latter method is based upon the 3-D acoustic wave equation with constant velocity. The resulting filter amounts to a simple modification of existing integral algorithms. The new F-K and integral DMO algorithms resulting from these two approaches turn out to be equivalent, producing identical outputs when implemented in nonaliased fashion. As dip increases, their output become progressively larger than the outputs of either Hale’s F-K method or the integral method generally associated with Deregowski and Rocca. This trend can be observed both on model data and field data. There are two additional results of this analysis, both following from the wave‐equation calibration on an arbitrary 3-D dipping reflector. The first is a proof that the entire imaging sequence (not just the DMO part) is true‐amplitude when the DMO is done correctly. The second result is a handy formula showing exactly how the zero‐phase wavelet on the final migrated image is a stretched version of the zero‐phase deconvolved source wavelet. This result quantitatively expresses the loss of vertical resolution due to dip and offset.

Physical properties of deep crustal reflectors in southern California from multioffset amplitude analysis

Analysis of reflection waveforms before stack can constrain the physical properties of reflectors in the deep crust. To simplify this analysis, recorded amplitudes are assumed to be reflections from weak elastic heterogeneities. With these assumptions, trends in reflection amplitudes with offset may indicate whether the signs of a reflector’s density and rigidity contrast agree with or oppose the sign of its Lamé’s parameter contrast. The slope of the trend indicates the degree of Poisson’s ratio contrast. No attempt is made to invert for the individual modulus or density contrasts. By examining only gross amplitude‐versus‐offset (AVO) trends, deep reflections constrain some crustal properties. Two seismic reflection surveys in the Mojave Desert recorded deep reflections that show amplitude changes with offset. Both the 1985 Calcrust Ward Valley survey in the eastern Mojave and the 87 km COCORP Mojave line 3 in the western Mojave incorporate long offsets of 10 km or more. Prestack traces are equalized using a quantile technique assuming a constant noise level at large time, then corrected for spherical divergence. Gross AVO trends that are summarized for each survey in amplitude trend stacks suggest that the strongest reflectors in the middle and deep crust represent Poisson’s ratio contrasts of at least 10 percent. In the eastern Mojave, a transition to a basal‐crustal zone, at ∼23 km depth, may include an increase in Poisson’s ratio with depth. Poisson’s ratio may also increase at the Moho.

Spherical divergence correction for seismic reflection data using slant stacks

We have developed a method for the spherical divergence correction of seismic reflection data based on normal moveout and stacking of cylindrical slant stacks. The method is illustrated on some Gulf of Mexico data. The results show that our method yields essentially the same traveltime information as does conventional processing. Our amplitudes, however, are more interpretable in terms of reflectivity than are those obtained by using an empirical spherical divergence correction.

Combining attenuation by Q and spherical divergence

A general correlation of cross‐well seismic data and surface seismic data is attempted simply by examining a combination of the two major mechanisms of seismic wave attenuation, anelastic Q, and spherical divergence. High‐frequency cross‐well seismology can be hindered by the assumption that an order of magnitude increase in frequency is accompanied by an order of magnitude decrease in propagation distance such that the anelastic attenuation (described by quality factor Q) remains constant. Such a comparison, however, neglects the effects of geometrical spreading, which is independent of frequency. Through the consideration of both Q and spherical divergence, it is demonstrated that the total attenuation of 2 000 Hz energy at a distance of 613 m is equivalent to the total attenuation of 20 Hz energy at a distance of 7 000 m. Applications of kilohertz cross‐well seismic surveys between wells spaced by over 600 m could be possible with present dynamic‐range capabilities in seismic recording systems. Such applications would allow the use of high‐resolution cross‐well seismic surveys between wells drilled on a 40 acre spacing. One example of cross‐well seismic data is shown to demonstrate the high‐frequency content (kilohertz) which can be obtained. Theoretical calculations indicate that kilohertz energy could be recorded at distances up to 1 km. The assumptions are that present surface seismology measurements define the dynamic‐range capabilities of recording instruments, and that Q with spherical divergence accounts for all attenuation. All calculations neglect the problems of miniaturization required for downhole applications. The goal is that omission of extremely low Q regions near the surface and the noise‐free borehole environment can overcome the miniaturization problems.

Simultaneous pre‐normal moveout and post‐normal moveout deconvolution

Wave theory justifies both pre‐ and post‐NMO deconvolution, but the filters should be estimated simultaneously, not sequentially. A linearized theory enables simultaneous estimation of the two filters. The theory incorporates spherical divergence independently from statistical weighting. Field data test cases show the expected interaction between NMO and deconvolution. The tests were not able to establish that simultaneous estimation is superior to sequential estimation. The difficulty is ascribed to the inadequacy of NMO as a downward continuation process.

Simultaneous correction for divergence and deconvolution without changing industrial practice

In 1978 Mendel and Kormylo (see, also, Kormylo and Mendel, 1980; and Mendel, 1983) proposed a technique for simultaneously correcting for spherical divergence and performing deconvolution. We showed that when the traditional starting point for deconvolution, namely, the convolutional sum model of a seismogram, is cast into state‐variable format, so that deconvolution can be performed by Kalman filtering and optimal smoothing techniques, e.g., minimum‐variance deconvolution (MVD) or maximum‐likelihood deconvolution (MLD), then one should not make the commonly made divergence correction on the data. Instead, that correction, which is a time‐varying one, should be put into the state‐variable model for deconvolution. This is possible because MVD and MLD are applicable to time‐varying and/or nonstationary systems.

A novel approach to seismic signal processing and modeling

This paper demonstrates some results obtained using state‐variable models and techniques on problems for which solutions either cannot be or are not easily obtained via more conventional input‐output techniques. After a brief introduction to state‐variable notions, the following seven problem areas are discussed: modeling seismic source wavelets, simultaneous deconvolution and correction for spherical divergence, simultaneous wavelet estimation and deconvolution, well log processing, design of recursive Wiener filters, Bremmer series decomposition of a seismogram (including suppression of multiples and vertical seismic profiling), and estimating reflection coefficients and traveltimes.