Charles P. Ursenbach

Linearized amplitude variation with offset (AVO) inversion with supercritical angles

Contrary to popular belief, a linearized approximation of the Zoeppritz equations may be used to estimate the reflection coefficient for angles of incidence up to and beyond the critical angle. These supercritical reflection coefficients are complex, implying a phase variation with offset in addition to amplitude variation with offset (AVO). This linearized approximation is then used as the basis for an AVO waveform inversion. By incorporating this new approximation, wider offset and angle data may be incorporated in the AVO inversion, helping to stabilize the problem and leading to more accurate estimates of reflectivity, including density reflectivity.

Optimal Zoeppritz approximations

New approximations are developed for RPP and RPS that are optimal in the sense that they preserve as much accuracy as possible with as much simplicity as possible. In particular, they are expressed in a form that is pseudo-linear after the form of the AkiRichards approximation. Thus they would be very simple to apply in an essentially exact non-linear inversion. Accurate analogues of the two-term Shuey equation are also given for both RPP and RPS.

Two-term AVO inversion : Equivalences and new methods

Most amplitude-variation-with-offset (AVO) studies use two-parameter inversion methods that are approximations of a more general three-parameter method based on the Aki-Richards approximation. Two-parameter methods are popular because the three-parameter inversion is often plagued by numerical instability. Reducing the dimensionality of the parameter space stabilizes the inversion. A variety of constraints can accomplish this, and these lead to the multiplicity of current two-parameter methods. It would be useful to understand relationships between various two-parameter methods. To this end, we derive formal expressions for inversion errors of each method. Using these expressions, conversion formulas are obtained that allow the flexibility to convert results of any two-parameter method to those of any other two-parameter method. The only requirement for the equivalence of methods is that the maximum angle of incidence be at least a few degrees less than the critical angle. In addition, error expressions result in a new formulation for a two-parameter AVO tool that combines strengths of two commonly used methods. The expressions also suggest a simple way to incorporate information from well-log calibration into legacy AVO inversions. These results should be helpful in resource exploration.

Efficient spherical-wave AVO modeling

Point-source effects are becoming recognized as important in some long-offset AVO studies. In this article, we introduce the ideas of spherical-wave AVO, caused by point sources, and then present a new and simple way of calculating spherical-wave reflection coefficients.

Spherical-wave computational AVO modelling in elastic and anelastic isotropic two-layer media

Summary Compressional-wave AVO responses and converted-wave AVO responses in elastic and anelastic two-layer isotropic Class 1 models are investigated. These responses are computed by utilizing Zoeppritz reflection coefficients and the Weyl/Sommerfeld-integral. Spherical-wave depth dependence for PP and PSv Class 1 models is found to be strongest near the critical angle. The constant-Q approximation is used to introduce anelastic effects. AVO responses of two-layer isotropic models are sensitive to anelasticity. This Q-factor dependence is strongest near critical

Playing with fire: Noise alignment in trim and residual statics

It is well known the cross-correlation procedure in trim statics can produce spurious reproduction of signal. This phenomenon is quantified and is shown, in the limits of large fold, small correlation window, and large maximum allowable shift, to behave as a simple function of these variables for physically reasonable wavelet lengths. These results allow one to predict what choices of crosscorrelation parameters are likely to result in spurious alignment of noise. It is also shown that this function can help to indicate whether apparent signal is a result of the desired signal alignment, or simply constructed from random background noise. The effect of residual statics can also function as an indicator in this context.

Linearized AVO inversion with supercritical angles

Contrary to popular belief, a linearized approximation of the Zoeppritz equations may be used to accurately estimate the reflection coefficient for angles of incidence up to and beyond the critical angle. These supercritical reflection coefficients are complex implying a phase variation with offset in addition to the amplitude variation with offset. This linearized approximation is used as the basis for a new AVO waveform inversion capable of truly incorporating wide angle information.

Simulation of elastic moduli of porous media

Summary A simulation method is employed to simulate elastic moduli of porous materials, including error estimation. The ratio of porous elastic moduli to their non-porous counterparts is obtained for isotropic materials with a Poisson’s ratio of ¼. Two cases are modeled: 1) The pores are non-overlapping but otherwise randomly distributed spheres. 2) The pores are randomly distributed (overlapping) spheres. In both cases the pores are empty and uniform in size. The results show that Norris’ differential effective medium theory describes overlapping spheres well, particularly compressional properties, and the Kuster-Toksoz model is moderately accurate for nonoverlapping spheres.

Fundamental parameters of spherical-wave reflection coefficients

Summary Spherical-wave reflection coefficients for a two-layer system depend on more parameters than do their corresponding plane-wave analogues. The additional parameters to be specified are depth, overburden velocity, and any parameters required to define the wavelet. For a Rayleigh wavelet it has been shown analytically that the additional parameters can be reduced to a set of two. For other wavelets numerical investigations can be used to explore the possibility of similar simplification. Ricker wavelet reflection coefficients are shown to depend on only one additional parameter, and the Ormsby wavelet on two.

Prestack considerations for the migration of oblique reflectors

A zero-offset stacked section that contains an oblique reflector can be migrated successfully by raising the migration velocity. What prestack processing can be performed to improve the diffracted energy on the stacked section? A special moveout correction can be applied to enhance the oblique reflection on the stacked section and modified prestack migration operators may also be used to enhance the oblique reflection. The benefit of these special processes will depend on the linear extent and angle of the oblique reflector. Satisfactory results can also be obtained when stacking with the RMS velocity and when stacking with an RMS velocity that is increased by the inverse of the cosine of the angle of obliquity. INTRODUCTION A special 2-D poststack migration of oblique reflectors was described by French (1975) in which the migration velocity Vmig is increased over the RMS velocity Vrms as given in equation (1) γ cos rms mig V V = , (1) where γ is the angle of obliquity, as illustrated in Figure 1.