Arnim B. Haase

An efficient method for AVO modeling of reflected spherical waves

A method is presented for efficiently and accurately calculating the spherical-wave generalization of the Zoeppritz P-wave reflection coefficients. The main assumptions are that the wavelet is an exponential form that allows for analytic integration over frequency, and the direction of propagation and arrival time are as dictated by ray theory. These assumptions result in calculations sufficiently rapid to be carried out interactively on the computer. Results for an AVO Class I model show that this method quantitatively reproduces exact spherical-wave reflection coefficients obtained using a Ricker wavelet.

Spherical Wave AVO Modeling of Converted Waves in Isotropic Media

Summary The AVO-response of two-layer isotropic models for AVO-Classes 1 and 3 is investigated for converted waves. Zoeppritz’s reflection coefficients and the Weyl-integral are utilized for the computations. Spherical wave results for Rps and Rpp are compared with plane wave reflectivity. Depth dependence of spherical wave AVO is found to be strongest near critical angles of Class 1. There is some similarity between Rps and Rpp for Class 1. Normalized Class 3 responses show no depth dependence. There is no similarity between Class 3 Rps and Rpp . Attenuation reduces AVO-response magnitudes. Rps appears to be more sensitive to finite Q-factors than is Rpp.

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.

Testing VSP-based Q-estimation with spherical wave models

Q-estimation with VSP data from Alberta and Saskatchewan led to the conclusion that stratigraphic effects are a significant part of those estimates. As a first step toward a better understanding of stratigraphic attenuation, we investigate the response to spherical waves of a simple density step model. The Sommerfeld integral is utilized to compute synthetic VSP down-going waves by numerical integration. Q-factors are estimated from these down-going wave fields by applying the spectral ratio method, the analytical signal method, a misfit minimization method adapted from Toverud and Ursin (2005) as well as a modification of the spectral ratio method introduced by Taner and Treitel (2003). We find that at larger depths, away from density steps, all methods recover the model Qfactor quite well. A departure of recovered Q from model Q at shallow depths is noticeable for all methods and is thought to be caused by near-field effects. Q-estimation errors for some methods are found to be considerable in the vicinity of single interfaces investigated. Q-estimates obtained by methods based on spectral ratios appear to be least sensitive to step changes in density values, at least in noise-free situations.

Estimating seismic attenuation (Q) by an analytical signal method

The analytical signal method for seismic attenuation (Q) estimation is reviewed and investigated using a 1D surface data model and field VSP data. The error in Q-factor recovery from modelled data using the analytical signal method is better than 7 percent over a range of Q-factors from 25 to 100. The method is applied to the enhanced downgoing P-wave of an offset VSP-survey from Ross Lake, Saskatchewan. The logarithm of the instantaneous-amplitude ratio versus time-increment plot (an internal step in the method) is surprisingly smooth when compared to log spectral ratios of the same data. The depth average of Q is 34 for clastic rocks of the Ross Lake area. This compares to a Q range of 37 to 41 obtained from the drift correction method over a similar depth range in previous work, which is somewhat lower than a Q of 67 obtained from the spectral ratio method. Errors in Q-estimation by the analytical signal method appear to be caused by insufficient moveout compensation.

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

Attenuation estimates from VSP and log data

VSP data and well log information from the Ross Lake oilfield, Saskatchewan (owned by Husky Energy Inc.) are used to estimate P-wave and S-wave attenuation (Qfactors). The VSP surveys used both vertical and horizontal vibrators as sources and a downhole five-level, threecomponent receiver. From the spectral ratio method applied to downgoing waves, results are obtained for QP as well as QS. We estimate an average QP, over an interval of 2001200m, to be 67 from the spectral ratio technique. We also use VSP-sonic drift curves to find a Qp of 40 over the same interval. QS estimates are 23 from the spectral ratio method and about 37 from “guesstimated” S-wave drift curves over the same interval.

Plane waves, spherical waves and angle-dependent P-wave reflectivity in elastic VTI-models

The AVA-response of VTI-models for AVO-Classes 1 to 4 and two special cases is computed utilizing plane-wave reflection coefficients and the Weyl-integral. It is found that below 30° of angle, in most cases, the spherical VTI-response departs more from an isotropic plane-wave comparison than isotropic spherical responses. Depth dependence of isotropic spherical responses is strongest near critical angles, exactly where important information resides. VTI-type anisotropy shifts this point of maximum sensitivity towards larger angles.

Modelling Near-field Effects in VSP-based Q-estimation

Summary The complete spherical wavefield emanating from a P-wave point source surrounded by a homogeneous isotropic medium is computed with the aid of Weyl/Sommerfeld integrals. In a resulting synthetic VSP, we observe a near-field, a far-field and a 90° phase rotation between the two. Depth dependence of magnitude spectra in these two depth regions is distinctly different. Log magnitude spectra show a linear dependence on frequency in the far-field but not in depth regions where the near-field becomes significant. Near-field effects are one possible explanation for large positive and even negative Q-factors in the shallow section that may be estimated from VSP data when applying the spectral ratio method. A near-field compensation method for Q-estimation in homogeneous models reduces errors in Q-values except in the immediate vicinity of the source.

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.