Spherical Divergence Research Articles

Turbulence-induced edge image waviness: theory and experiment

A theoretical model for the edge image waviness effect is developed for the ground-to-ground imaging scheme and validated by use of IR imagery data collected at the White Sands Missile Range. It is shown that angle-of-arrival (AA) angular anisoplanatism causes the phenomenon of edge image waviness and that the AA correlation scale, not the isoplanatic angle, characterizes the edge image waviness scale. The latter scale is determined by the angular size of the imager and a normalized atmospheric outer scale, and it does not depend on the strength of turbulence along the path. Spherical divergence of the light waves increases the edge waviness scale. A procedure for estimating the atmospheric and camera-noise components of the edge image motion is developed and implemented. A technique for mitigation of the edge image waviness that relies on averaging the effects of AA anisoplanatism on the image is presented and validated. The edge waviness variance is reduced by a factor of 2-3. The time history and temporal power spectrum of the edge image motion are obtained. These data confirm that the observed edge image motion is caused by turbulence.

AVO investigations of shallow marine sediments

Amplitude‐variation‐with‐offset (AVO) analysis is based on the Zoeppritz equations, which enable the computation of reflection and transmission coefficients as a function of offset or angle of incidence. High‐frequency (up to 700 Hz) AVO studies, presented here, have been used to determine the physical properties of sediments in a shallow marine environment (20 m water depth). The properties that can be constrained are P‐ and S‐wave velocities, bulk density and acoustic attenuation. The use of higher frequencies requires special analysis including careful geometry and source and receiver directivity corrections. In the past, marine sediments have been modelled as elastic materials. However, viscoelastic models which include absorption are more realistic. At angles of incidence greater than 40°, AVO functions derived from viscoelastic models differ from those with purely elastic properties in the absence of a critical angle of incidence. The influence of S‐wave velocity on the reflection coefficient is small (especially for low S‐wave velocities encountered at the sea‐floor). Thus, it is difficult to extract the S‐wave parameter from AVO trends. On the other hand, P‐wave velocity and density show a considerably stronger effect. Attenuation (described by the quality factor Q) influences the reflection coefficient but could not be determined uniquely from the AVO functions. In order to measure the reflection coefficient in a seismogram, the amplitudes of the direct wave and the sea‐floor reflection in a common‐midpoint (CMP) gather are determined and corrected for spherical divergence as well as source and streamer directivity. At CMP locations showing the different AVO characteristics of a mud and a boulder clay, the sediment physical properties are determined by using a sequential‐quadratic‐programming (SQP) inversion technique. The inverted sediment physical properties for the mud are: P‐wave velocity α=1450±25 m/s, S‐wave velocity β=90±35 m/s, density ρ=1220±45 kg/m3, quality factor for P‐wave QP=15±200, quality factor for S‐wave QS=10±30. The inverted sediment physical properties for the boulder clay are: α=1620±45 m/s,β=360±200 m/s,ρ=1380±85 kg/m3,QP=790±660,QS=25±10.

Fracture Detection Using P-wave AVO Measurements

Physical modelling of seismic wave propagation can provide insights into complex structural, stratigraphic and reservoir characterisation problems. The detection and delineation of fracture zones is very important in development drilling. Determination of fracture orientation within a reservoir may play an important role in EOR programs. The fundamental principle used in AVO studies is the extraction of shear wave information present in the reflection amplitude variation with source-receiver distance. The objective of this study was to determine if the shear information in 3-D AVO surveys can be utilised for fracture detection and delineation.A 3-D physical modelling experiment was conducted over a simulated fracture system at the Allied Geophysical Laboratories at the University of Houston. The ‘fracture system’ consisted of three isotropic homogeneous layers with a transversely isotropic Phenolite disc embedded in the central portion of the middle layer. The three layers are: an upper layer of black resin 2741LV; a middle layer of blue Stycast resin 2850MT (containing the anisotropic disc); and a lower layer of Evercoat casting resin. Two 3-D seismic surveys were acquired over the model, one with acquisition lines oriented parallel to, and a second survey with the acquisition lines perpendicular to, the fracture direction. Multi-offset and multi-azimuth seismic experiments were also conducted over the simulated fracture system. After applying corrections for spherical divergence and source-receiver directivity the reflection amplitudes from the black/Stycast resin interface were analysed.The results show that the ‘fractured’ disc is characterised by low AVO gradients in both surveys. In addition, the 3-D AVO gradient estimates from the survey perpendicular to the fracture orientations are 30% to 40% lower than the AVO gradient estimates from the survey with line orientation parallel to the fracture direction. Also, AVO effects along different line orientations suggest that at near offsets the amplitude behavior is similar for all azimuths but at far offsets an anomalous amplitude is seen along an azimuth perpendicular to the fracture zone.

High‐resolution velocity determination: Statistical phase correlation and image processing

We address three areas of the problem of the stacking velocity determination: (1) the development of a new high‐resolution velocity determination technique, (2) the choice of an optimal velocity trial scenario, and (3) a unified approach to the comparison of time‐velocity spectra produced by various methods. We present a class of high‐resolution coherency measures providing five‐eight times better velocity resolution than conventional measures. The measure is based on the rigorous theory of statistical hypothesis testing and on the statistics of directional data. In its original form, our method analyzes only the phase distributions of the data, thus making unnecessary careful spherical divergence corrections and other normalization procedures. Besides the statistical one, we develop an “instantaneous” version of the conventional coherency measure. This measure is based on the concept of the trace envelope, thus eliminating the need for an averaging procedure. Finally, we design a hybrid high‐resolution coherency measure, incorporating the latter and the statistical one. Carrying out a systematic comparison of various measures of coherency, we present a simple estimate of an attainable velocity resolution. Based on this estimate, we define an optimal velocity grid, providing uniform coverage of all details of the time‐velocity spectrum. To facilitate quantitative comparisons of different coherency functions, we develop a unified normalization approach, based on techniques known in image processing. Described methods are tested on synthetic and field data. In both cases, we obtained a remarkable improvement in the time‐velocity resolution. The methods are general, very simple in implementation, and robust and reliable in application.

Quantifying crustal fracture heterogeneity by seismic scattering

SUMMARY Seismic waves travelling through the crust generate trains of scattered waves called the seismic coda. Coda motion u(t) excited by a source pulse σ(t) and recorded at times t > ?2t0 by a sensor located at traveltime t0 from the source can be written as the convolution of σ(t) with the gradient c′(r) =∂c(r) of spatial velocity fluctuations c(r) encountered by the wave front at r?tc0/2: u(t)? 4πu0(t0/t)(c′(tc0/2)> *σ(t), where t0/t accounts for source wavelet spherical divergence, c0 is the mean seismic velocity of the medium, and 〈〉 denotes the scattering amplitude c′(r) averaged over the 4π solid angle of the wave front. It follows that the coda frequency spectrum u(f) is proportional to vc(v)σ(f), where v is the spatial frequency v= 2f / c0, vc(v) the spectrum of c′(r) and σ(f) the source spectrum. Sonic-velocity fluctuations cBH(r) logged at 15 cm intervals over a 1.5 km length of deep borehole in crustal rock show that the major cause of seismic velocity fluctuations c(r) in the brittle crust are fracture distributions with a power-law spectral dependence on fracture spacing, c(v)?v−0.4. The resulting velocity gradient power-law spectrum is vc(v)?v0.6. In accordance with the above scattering expressions, the power-law scattering amplitude vc(v) causes the high-frequency power-law enrichment of the coda spectrum relative to the source spectrum u(f)/σ(f)?f0.6 as observed in coda waves recorded in a borehole at a depth of 2.5 km in the crust. The coda wave phenomenology observed in deep crustal borehole data does not emerge from treatments of seismic scattering based on correlations between randomly distributed elastic heterogeneities. This is because scattering is proportional to the derivative of fluctuations in material properties; for power-law-distributed velocity fluctuations in crustal rock, differentiation enhances scattering while correlation smoothing of fluctuations reduces scattering.

Inversion of Reflection Seismic Amplitude Data For Interface Geometry

SUMMARY Reflection seismic tomography using only traveltime data may be unable to resolve the ambiguity caused by trade-off between reflector position and velocity anomaly. The inclusion of amplitude data in the inversion may help to resolve this problem because the amplitudes and traveltimes are sensitive to different features of the model, therefore providing us with more accurate information about underground structures and velocity distribution. The amplitude of a reflected seismic wave is determined partly by the reflection coefficients and partly by the curvature of the reflector. The latter causes the spherical divergence of the seismic rays to be modified at the reflection point (focused or defocused) and can be represented using a simplified analytical expression. We show, using geologically relevant synthetic models, that the information contained in amplitude versus offset data (here excluding traveltime data) suffices to constrain accurately the geometry of an arbitrary 2-D reflector separating constant velocity layers. The most effective inversion method is a subspace gradient algorithm using a model parametrization in which the interface is described as a discrete Fourier series with fixed upper and lower bounds on the wavenumber. Model parameters are allocated to separate subspaces first on the basis of different physical dimensionality. We also found that declaring separate subspaces for those parameters defining short, intermediate and long wavelength components of the interface geometry, based on the magnitude of singular values of the Frechet derivative matrix, is very effective in accelerating convergence and obtaining a more accurate solution. The inversion is robust with respect to data errors and poor initial estimates.

Radiation transfer in gamma-ray bursts

We have calculated gamma-ray radiative transport in regions of high energy density, such as gamma-ray burst source regions, using a discrete ordinate, discrete energy group method. The calculations include two-photon pair production and annihilation, as well as three-photon pair annihilation. The radiation field itself acts as an absorbing medium, and the optical depth depends on its intensity, so the problem is intrinsically nonlinear. Spherical divergence produces effective collimation of the flux. At high optical depth the high energy ($E > 1$ MeV) portion of the emergent spectrum assumes a nearly universal form. An approximate limit is derived for the high energy flux from a gamma-ray burst source region of given size, and the implications of this limit for the distance to the March 5, 1979 event are briefly discussed. We discuss more generally the problem of very luminous bursts, and implications of Galactic halo distances for flare models.

Array responses for plane and spherical incidence

The design of array filters for attenuation of source‐generated seismic noise is founded, almost without exception, on the assumption of plane‐wave incidence. We examine the theoretical limitations of this assumption and assess their practical significance. A time‐domain array response is developed, incorporating spherical wavefront geometry and divergence. Departures from conventional plane‐wave response are related to systematic differences between the spectral composition of respective waveforms within the array’s aperture. The study also develops a hybrid‐array response based on a modified plane‐wave assumption, retaining plane wavefront geometry but incorporating a model‐consistent approximation to spherical divergence. Deviations between the spherical‐wave response and those arising under plane‐wave assumptions are analyzed in terms of effective implementation errors required to compensate. Analysis reveals that the magnitude of these effective deviations can significantly exceed those expected for actual implementation errors. Findings establish that errors stemming from a plane‐wave approximation are controlled by the ratio of reflector depth to aperture width and diminish as the distance between the source and array midpoint becomes large compared with the former parameters.

Amplitude‐versus‐offset measurement errors in a finely layered medium

Recently Spratt (1987) showed how amplitude‐versus‐offset analysis (AVO) can be sensitive to small residual velocity errors. However, even when the velocity is determined perfectly, serious AVO distortions remain due to normal‐moveout stretch, differential tuning as a function of offset, spherical divergence, and source and receiver directivity patterns. I have found that all of these errors can be expanded in a Taylor series about the zero‐offset event time, assuming it is much larger than the wavelet width. The first term of this series represents the residual velocity error term found by Spratt, while the second term encompasses the remaining effects mentioned. In practice, either term can be larger than the underlying amplitude variations being estimated. For example, Ricker wavelet stretch leads to a peak AVO error which is 61 percent of the peak zero‐offset reflectivity, even though the velocity field is uniform and correct. This result is independent of the wavelet frequency, and the range of incidence angles used in the analysis. Positive gradients in moveout velocity amplify this error, while narrowband filtering of the data prior to AVO analysis greatly widens its temporal extent. Aligning a particular event with static shifts instead of normal‐moveout correction can eliminate stretch, but not differential tuning error, in a finely layered target zone whose wavelets overlap.

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.

VSP processing with full downgoing - wavefield deconvolution applied to the total wavefield

In this articIe a new processing sequence for vertical seismic profile (VSP) data is described, in which full downgoing - wavefield deconvolution is applied to the total wavefield before spherical divergence correction and wavefield separation. Two field examples are used to illustrate its effectiveness over a more conventional VSP processing sequence.

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.

Three-component amplitude versus offset analysis

Three-component (3-C) amplitude versus offset (AVO) inversion is the AVO analysis of the three major energies in the seismic data, P-waves, S-waves and converted waves. For each type of energy the reflection coefficients at the boundary are a function of the contrast across the boundary in velocity, density and Poisson's ratio, and of the angle of incidence of the incoming wave. 3-C AVO analysis exploits these relationships to analyse the AVO changes in the P, S, and converted waves. 3-C AVO analysis is generally done on P, S, and converted wave data collected from a single source on 3-C geophones. Since most seismic sources generate both P and S-waves, it follows that most 3-C seismic data may be used in 3-C AVO inversion. Processing of the P-wave, S-wave and converted wave gathers is nearly the same as for single-component P-wave gathers. In split-spread shooting, the P-wave and S-wave energy on the radial component is one polarity on the forward shot and the opposite polarity on the back shot. Therefore to use both sides of the shot, the back shot must be rotated 180 degrees before it can be stacked with the forward shot. The amplitude of the returning energy is a function of all three components, not just the vertical or radial, so all three components must be stacked for P-waves, then for S-waves, and finally for converted waves. After the gathers are processed, reflectors are picked and the amplitudes are corrected for free-surface effects, spherical divergence and the shot and geophone array geometries. Next the P and S-wave interval velocities are calculated from the P and S-wave moveouts. Then the amplitude response of the P and S-wave reflections are analysed to give Poisson's ratio. The two solutions are then compared and adjusted until they match each other and the data. Three-component AVO inversion not only yields information about the lithologies and pore-fluids at a specific location; it also provides the interpreter with good correlations between the P-waves and the S-waves, and between the P and converted waves, thus greatly expanding the value of 3-C seismic data.

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.

Effects of small, intense laser spots on thin films

Two-dimensional hydrodynamic simulations show that small diameter laser spots burn through flat foils considerably faster than larger spots. The physical mechanisms are (1) more nearly spherical divergence from the smaller spots which allows the debris to rapidly convect radially away from the small volume of laser heated plasma and (2) a variation on thermal self-focusing in which light is refracted into this volume, increasing the local heating rate.

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.

Spall observations and mechanisms in alluvium

Spall, as identified by minus 1 g dwells on vertical accelerometers followed by impulsive rejoin signals, has been observed in ground motion data recorded in alluvium. These data resulted from explosions ranging from fully contained to surface burst configurations. The variability of the loading environments and the observational data support the conclusion that the phenomena result from multiple mechanisms. The data show that relatively low accelerations (<20 g) accompany the process, but owing to the long dwell times involved, resulting displacements may be large. Five spall models are discussed and shown to operate in some aspect of the data base. These mechanisms include (1) simple tensile reflection from the free surface, (2) Rayleigh/shear waves from surface sources, (3) spherical and cylindrical wave field divergence, (4) positive and negative phase airblast effects on porous materials, and (5) material properties of porous particulate matter.

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.

IN‐SITU INVESTIGATION OF SEISMIC BODY WAVE ATTENUATION IN HETEROGENEOUS MEDIA*

AbstractField experiments have been carried out to study the nature and magnitude of seismic wave attenuation for a variety of lithologies. In each survey two three‐component sets of geophones with wall clamping mechanisms were lowered down boreholes and signals originating from surface compressional and shear‐wave sources were recorded. The data collected were corrected for spherical divergence and analyzed to determine intrinsic attenuation. For situations of anomalous wavefront expansion and in cases where multiple reflection losses may be significant, corrections were supplied by a synthetic seismogram programme to improve the estimates of intrinsic attenuation.Values of attenuation were obtained for pure sandstone, sandstone‐marl sequences and fissured‐unfissured chalk sequences. These formations were all near surface and relatively porous. Significant differences in the relative values of compressional and shear‐wave attenuation in the various lithologies were noted. In particular compressional absorption in the fissured zones of chalk was twice the absorption associated with the seismic velocities and the absorption of shear waves fluctuated much less. A large contrast in P‐wave attenuation was also observed between pure Bunter Sandstone and the sandstone‐marl sequence (absorption was over three times as small in the former). A smaller contrast was noted for shear attenuation.The results obtained suggest that, for the relatively homogeneous formations such as pure sandstone and “unfissured” chalk, “shear” absorption was dominant over “bulk” absorption. In contrast bulk absorption was larger than shear absorption for other formations, e.g. the “fissured” chalk sequences and a “partially saturated” chalk section. Attenuation was found to be approximately proportional to frequency in all experiments.

Effect of propagation distance on rotary wing aircraft sound duration

Military aircraft noise contouring programs employ an empirical factor describing the change in duration of a flyby with distance. In the past the SEL or EPNL was calculated from the maximum A-weighted level by the use of a duration correction given by 10 times the logarithm of the distance ratio, applicable to spherical divergence without absorption. Recent measurements have indicated that a better factor for a variety of military aircraft is six times the logarithm of the distance ratio (J. D. Speakman, “Effect of Propagation Distance on Aircraft Flyover Sound Duration,” AFAMRLTR-81-28, May 1981). This paper describes the results of an experimental and theoretical study of the duration factors for rotary wing aircraft. The theoretical study investigates the effect of surface conditions, coherence, and atmospheric absorption on the duration correction.