Most bulk elastic media are weakly anisotropic. -The equations governing weak anisotropy are much simpler than those governing strong anisotropy, and they are much easier to grasp intuitively. These equations indicate that a certain anisotropic parameter (denoted 6) controls most anisotropic phenomena of importance in exploration geophysics. some of which are nonnegligible even when the anisotropy is weak. The critical parameter 6 is an awkward combination of elastic parameters, a combination which is totally independent of horizontal velocity and which may be either positive or negative in natural contexts.

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Weak elastic anisotropy

The standard hyperbolic approximation for reflection moveouts in layered media is accurate only for relatively short spreads, even if the layers are isotropic. Velocity anisotropy may significantly enhance deviations from hyperbolic moveout. Nonhyperbolic analysis in anisotropic media is also important because conventional hyperbolic moveout processing on short spreads is insufficient to recover the true vertical velocity (hence the depth).We present analytic and numerical analysis of the combined influence of vertical transverse isotropy and layering on long-spread reflection moveouts. Qualitative description of nonhyperbolic moveout on 'intermediate' spreads (offset-to-depth ratio x/z epsilon . With this expansion, we also show that the weak anisotropy approximation becomes inadequate (to describe nonhyperbolic moveout) for surprisingly small values of the anisotropies delta and epsilon .However, the fourth-order Taylor series rapidly loses numerical accuracy with increasing offset. We suggest a new, more general analytical approximation, and test it against several transversely isotropic models. For P-waves, this moveout equation remains numerically accurate even for substantial anisotropy and large offsets. This approximation provides a fast and effective way to estimate the behavior of long-spread moveouts for layered anisotropic models.

Nonhyperbolic reflection moveout in anisotropic media

Recently Bortfeld (1982) gave a cursory nonmathematical introduction to a procedure for computing the geometrical spreading factor of a primary zero‐offset reflection from the common datum point traveltime measurements of the event. To underline the significance and consequences of this method, a derivation and discussion of geometrical spreading factors is now given for two‐ and three‐dimensional earth models with curved reflecting boundaries. The spreading factors can be used easily to transform primary reflections in a zero‐offset seismic section to true amplitude reflections. These permit an estimation of interface reflection coefficients, either directly or in connection with a true amplitude migration. A seismic section with true amplitude reflections can be described by one physical experiment: the tuned reflector model. Hence the application of the wave equation (in connection with a migration after stack) is justified on such a seismic section. Also the geometrical spreading factors that are deri...

Computing true amplitude reflections in a laterally inhomogeneous earth

The term “simultaneous source” refers to the idea of firing several seismic sources so that their combined energy is recorded into the same set of receivers during a single conventional shotpoint timing cycle. The idea is to collect the equivalent of two or more shots worth of data in the same time as it takes to collect one. The potential advantages include cost or time savings in field acquisition, which is of renewed interest due to the popularity and expense of WATS data. We were motivated to the work presented here by observations made on a 3D dataset acquired over the Petronius field in the Gulf of Mexico with two source vessels. The second source was fired with a random delay compared to the first, so that the energy from secondary source is similar to asynchronous noise. While the random nature of the crosstalk in combination with the two known geometries had been enough to successfully apply relatively standard processing techniques for other studied datasets, we found that this one required an improvement on those techniques. This paper describes a high-resolution (sparse) Radonbased separation technique with that aim. We find that while the technique does not by itself do all the required separation, it sufficiently separates the data to allow subsequent standard noise attenuation techniques to complete the task.

Simultaneous Source Separation By Sparse Radon Transform

We present a technique in which two or more shots are acquired during the time it normally takes to acquire one shot. The two (or more) shots are fired in a near simultaneous manner with small random time delays between the component sources. A variety of processing techniques are applied to produce the same seismic images which would have resulted from firing the simultaneous shots separately. These processing techniques rely on coherency of the wavefield in the common-shot domain and unpredictability in the common-receiver, common-offset, and common-midpoint domains. We present results of its application on synthetic 2D, real 2D, and real 3D data from the Gulf of Mexico. These results demonstrate that, in deep water with modest water-bottom reflectivity, no special processing is required, whereas in shallower water with stronger water-bottom reflectivity, the use of shot-separation techniques is necessary. We conclude that this technique can be used robustly to improve source sampling and, for example, ...

Acquisition using simultaneous sources

It is proved that horizontally stratified media, the various materials of which do not differ in their Poisson number, can be considered as isotropic for reflections of a small dip when taking into account quasi-longitudinal waves which have to be reckoned with in practice. Approximating the surface of the wave by an ellipsoid would, however, result in considerable errors.



Moreover, curves are presented allowing practical calculations for steeper dips to be made.

a Theorem Concerning Anisotropy of Stratified Media and its Significance for Reflection SEISMICS

It is well known that interval velocities can be determined from common-reflection-point moveout times. However, the mathematics becomes complicated in the general case of n homogeneous layers with curved interfaces dipping in three dimensions.



In this paper the problem is solved by mathematical induction using the second power terms only of the Taylor series which represents the moveout time as a function of the coordinate differences between shot and geophone points. Moreover, the zero-offset reflection times of the nth interface in a certain area surrounding the point of interest have to be known. The n—I upper interfaces and interval velocities are known too on account of the mathematical induction method applied. Thus, the zero-offset reflection raypath of the nth interface can be supposed to be known down to the intersection with the (n—1)th interface.



The method applied consists mainly in transforming the second power terms of the moveout time from one interface to the next one. This is accomplished by matrix algebra.



Some special cases are discussed as e.g. uniform strike and small curvatures.

COMPUTATION OF INTERVAL VELOCITIES FROM COMMON REFLECTION POINT MOVEOUT TIMES FOR n LAYERS WITH ARBITRARY DIPS AND CURVATURES IN THREE DIMENSIONS WHEN ASSUMING SMALL SHOT‐GEOPHONE DISTANCES*

Three-dimensional seismic surveys, in general, do not need the same high degree of CDP coverage as 2-D surveys to achieve a certain signal-to-noise ratio after migration.



This can be shown theoretically for Kirchhoff migration and laterally uncorrelated noise. More precisely, there exists a formal relationship between the multiplicity of CDP coverage of a 3-D survey and that of a 2-D survey with the same signal-to-uncorrelated-noise ratio. Frequency and aperture are parameters in the corresponding expression. Heuristically the relationship can be obtained by applying the concept of the Fresnel zone.



Though the mathematics in this paper refer to laterally uncorrelated noise, the underlying concepts can probably also be used for weakly correlated noise, e.g., for multiple reflections and for the low-frequency remnants of surface waves.

Attenuation of random noise by 2‐d and 3‐d cdp stacking and kirchhoff migration*

Nonlinear sweeps have often successfully been employed in the 1960s. However, this area of sweep technology has been neglected since the introduction of digital recording techniques in the Vibroseis system. Now the advent of computerized recording instruments yields a new economical possibility of forming approximately nonlinear sweeps by combining several linear sweeps with or without time gaps to a “Combisweep”. The total duration of a Combisweep may be as long as the maximum available recording time, for example 32 s.



Beside the attenuation of correlation noise, the new method has further merits, such as the weighting of predetermined frequency ranges, in order to effect a certain kind of optimum filtering on the emitter side, or in order to compensate to some degree for frequency dependent absorption.



In all these applications the Combisweep is considered as one signal in the correlation process. But by correlating with the individual sweeps or a partial combination of them and by applying automatic switching at predetermined times within the gaps between the individual sweeps additional possibilities arise, such as obtaining in one run with a twenty-four channel recording unit twenty-four traces with small distances between vibrators and geophones for shallow reflections and another twenty-four traces with larger distances for deeper reflections. Various Combisweeps and their applications are presented.

Combisweep—a contribution to sweep techniques*

The paper presents a discussion of some of the noise features of the VIBROSEIS SYSTEM*** and their bearing on the determination of the optimally weighted stack.

Th. Krey

Papers

a Theorem Concerning Anisotropy of Stratified Media and its Significance for Reflection SEISMICS

COMPUTATION OF INTERVAL VELOCITIES FROM COMMON REFLECTION POINT MOVEOUT TIMES FOR n LAYERS WITH ARBITRARY DIPS AND CURVATURES IN THREE DIMENSIONS WHEN ASSUMING SMALL SHOT‐GEOPHONE DISTANCES*

Attenuation of random noise by 2‐d and 3‐d cdp stacking and kirchhoff migration*

Combisweep—a contribution to sweep techniques*

Remarks on the signal to noise ratio in the vibroseis system