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Arnim B. Haase

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Proceedings Papers

Publisher: Society of Exploration Geophysicists

Paper presented at the 2007 SEG Annual Meeting, September 23–28, 2007

Paper Number: SEG-2007-3100

Abstract

ABSTRACT Q-estimation based on the spectral ratio method works perfectly only for a noise free homogeneous medium under far-field conditions. The Sommerfeld integral for layered earth situations is utilized in this contribution to compute the down going wave field of synthetic anelastic VSP's by numerical integration. Q is then estimated from the generated model traces and compared to the original model input parameter Q. Q-estimation errors are found to be considerable in the vicinity of single interfaces investigated and in the near-field.

Proceedings Papers

Publisher: Society of Exploration Geophysicists

Paper presented at the 2007 SEG Annual Meeting, September 23–28, 2007

Paper Number: SEG-2007-0269

Abstract

Summary Spherical-wave reflection coefficients are normally calculated using a zero-phase wavelet. The purpose of this study is to clarify whether phase affects reflectivity, and to explain the observed results. Our numerical experiments show that zero-phase, rotatedphase, and linear-phase wavelets give identical reflectivities, but that minimum-phase wavelets give slightly different AVO results in a region beyond the critical point, even though they share the same amplitude spectrum. Expressing the reflectivity calculation as a weighted integral of plane-wave coefficients provides insight into these results. The weighting functions for zeroand minimum-phase wavelets differ from each other. In particular, although the central part of the weighting function does not differ appreciably, the edges differ significantly in ways that mimic the differences between reflection coefficient curves. Introduction Previous studies of spherical-wave AVO behavior (Haase, 2004; Ursenbach, Haase and Downton, 2005) have shown the need for spherical-wave modeling near critical points. These studies have employed three different types of wavelet: Ormsby, Ricker, and Rayleigh. All wavelets in these studies have been zero-phase, which is relevant to most cases of AVO modeling and inversion. Does phase have an influence on the reflectivity? Plane-wave reflectivity is of course unaffected by phase. However, in the spherical-wave method the reflection coefficient is a quotient of two integrals, each including the wavelet. Thus phase contributions in the numerator and denominator could cancel, but strictly speaking this would only occur if they can be taken outside of the integrals. If the phase is frequency dependent this would not be the case, so it is reasonable to investigate how large of an effect the phase can have on practical AVO. To approach the question we have chosen an appropriate set of test wavelets, and have used these to produce reflection coefficient curves. We have also analyzed certain weighting functions which are part of the spherical-wave reflection coefficient calculation, and which give further insight into the results. Theory As described elsewhere, a code developed by Haase (2004) calculates spherical-wave reflection coefficients by performing a numerical p -integration to obtain the ray-parallel displacement spectrum at several frequency points (cf. Aki and Richards, 1980). Given the displacement spectrum, a time trace, u , can be obtained by multiplying by the wavelet spectrum and integrating over frequency, and from this trace the maximum of the envelope yields a reflectivity estimate. This is converted to a reflection coefficient by dividing by an estimate employing R PP = 1. The current program based on equation 1 was modified to accept complex-valued wavelet spectra. AVO with non-zero-phase spherical waves For Rayleigh wavelets, W is an analytic function (Ursenbach, Haase and Downton, 2005), but, in principle, the spherical-wave reflection coefficient for any wavelet can be written in the form of equation 2. The only difference is that in general W is not known analytically. It can be represented numerically though, and as this will be useful in our later analysis, we give the expression for this in equation 3 (derived in the appendix)

Proceedings Papers

Publisher: Society of Exploration Geophysicists

Paper presented at the 2006 SEG Annual Meeting, October 1–6, 2006

Paper Number: SEG-2006-0304

Abstract

Introduction 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. A plane-wave reflection coefficient (PWRC) is a solution to the Zoeppritz equations and is usually described as depending on six earth parameters and a direction parameter (Aki & Richards, 1980). A common choice consists of the following: a1 : P-wave velocity of upper layer a2 : P-wave velocity of lower layer ß1 : S-wave velocity of upper layer ß2 : S-wave velocity of lower layer ?1 : density of upper layer ?2 : density of lower layer ?i: angle of incidence ( = sin -1 (a 1 p), where p is the ray parameter) It can be shown however that the six earth parameters can be replaced by three velocity ratios and a density ratio. For amplitude-versus-offset (AVO) studies, a convenient set are the following ? a/ a = (a2 – a1) / [ (a1 + a2) / 2 ] ? ß/ ß = (ß2 – ß1) / [ (ß1 + ß2) / 2 ] ? ?/ ? = (?2 – ?1) / [ (?1 + ?2) / 2 ] ß/ a = (ß1 + ß2) / (a1 + a2) In moving to a normalized spherical-wave reflection coefficient (SWRC), in addition to the four ratios above one also requires z (the depth), a1 (the overburden velocity, defined above), and any wavelet parameters. An Ormsby wavelet for instance would add four frequency parameters, given as f 1 / f 2 – f 3 / f 4 . It is often assumed that spherical-wave effects are only important in the near surface region. However it has been demonstrated that near critical angles an SWRC can differ significantly from a PWRC even at considerable depth (Haase, 2004). (Here, as in other figures in this abstract, all spherical divergence effects have been normalized out.) AVO inversion techniques have recently been extended to the supercritical regime (Downton and Ursenbach, 2005). This was carried out with plane-wave reflection Coefficients. AVO inversion techniques have recently been extended to the supercritical regime (Downton and Ursenbach, 2005). This was carried out with plane-wave reflection Coefficients. It was previously shown that employing a wavelet of a particular exponential form can simplify calculation of the SWRC by permitting an analytic integration over frequency (Ursenbach, Haase and Downton,2005). Conclusions Identifying fundamental parameters simplifies considerably the study of spherical wave reflection coefficients by reducing the parameter space to more manageable proportions. It is expected that the results obtained here for Ricker and Ormsby wavelets could be readily duplicated for other wavelets of interest as well.

Proceedings Papers

Publisher: Society of Exploration Geophysicists

Paper presented at the 2006 SEG Annual Meeting, October 1–6, 2006

Paper Number: SEG-2006-3472

Abstract

ABSTRACT This paper investigates transmission loss contributions to seismic attenuation in VSP data from the Ross Lake heavy- oil field, Saskatchewan. We compensate for transmission losses when estimating seismic attenuation (Q) by the analytic signal method. Major Q-estimate changes are not observed when compared to results obtained from the analytical signal method without this compensation. 1D scattering is not a major contributor to apparent attenuation. 2D scattering shows offset dependent stratigraphic “attenuation/amplification”. Stratigraphic “amplification” is one possible explanation for the peaking of shallow depth Q-estimates from actual VSP data.

Proceedings Papers

Publisher: Society of Exploration Geophysicists

Paper presented at the 2005 SEG Annual Meeting, November 6–11, 2005

Paper Number: SEG-2005-0202

Abstract

Summary 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. Introduction Here A is a scaling factor, w is the frequency, t is time, x is the vertical P-wave slowness in the upper layer, J 0 is the zero-order Bessel function, r is the horizontal receiver coordinate (with the horizontal source position equal to zero), h is the source elevation, and z is the vertical receiver coordinate. The displacement is obtained by applying a gradient in the receiver position to the above potential. Weighting this by the wavelet and applying an inverse Fourier transform yields the displacement time trace observed at the receiver from which AVO information can be extracted. The above method has been implemented in numerical calculations by Haase (2004). Our objective in this study is to develop a semi-analytic approximation that is both highly efficient and quantitatively accurate. Spherical-wave results in this paper have been calculated using two different methods, the semi-analytical weighting function method of Eq. (4), and, for purposes of comparison, the fully numerical approach of Haase (2004). Exponential wavelets have been employed with both methods, but the Ormsby and Ricker wavelets can only be handled by Haase’s method. Figure 3 shows spherical-wave results for two wavelets along with plane-wave results. There are significant amplitude deviations near the critical point, as noted by Haase (2004). The f 5(w) wavelet appears to give a stronger spherical effect than the Ormsby wavelet. The Ormsby wavelet displays some oscillatory character just past the critical point. This is likely related to the slope discontinuities in the Ormsby wavelet. Both wavelets yield a qualitatively similar deviation from the plane wave result, so that spherical effects are only mildly dependent on the precise shape of wavelet. Figure 4 shows a comparison of the results for the Ricker wavelet and three exponential wavelets. While all the plots agree well for most q i , there is essentially quantitative agreement is between the f 5(w) and Ricker wavelets for this model. Figure 5 displays the difference between the f 5(w)and Ricker wavelets. The maximum difference in this case is<0.01, which is negligible in practice. Two other small quantities are also plotted. One is the difference in | R PP| when the f 5(w) wavelet is calculated by Haase’s full numerical method and by Eq. (4). Essentially this constitutes a measure of the error introduced by setting t = R /á1 rather than calculating the entire trace and finding t max. The other line is obtained by calculating the displacement perpendicular to the ray for the f 4(w)wavelet (using a modification of Haase’s method).

Proceedings Papers

Publisher: Society of Exploration Geophysicists

Paper presented at the 2005 SEG Annual Meeting, November 6–11, 2005

Paper Number: SEG-2005-2617

Abstract

ABSTRACT 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.

Proceedings Papers

Publisher: Society of Exploration Geophysicists

Paper presented at the 2005 SEG Annual Meeting, November 6–11, 2005

Paper Number: SEG-2005-0316

Abstract

Summary The AVO response of two-layer VTI models for AVO Class I is investigated. Graebner/Rueger reflection coefficients and the “Weyl-integral for anisotropic media” are utilized for the computation. Spherical wave results are compared with the plane-wave reflectivity. Depth dependence of spherical wave AVO is found to be strongest near critical angles, as was observed in the isotropic situation. Particle motion perpendicular to the ray angle is strongest just beyond critical angles and increasing with anisotropy. Anelasticity also modifies VTI AVO responses. When reflection amplitude losses due to attenuation are compensated for by unit reflectivity scaling, AVO-characteristics similar to the elastic situation are found. Q-factor dependence of spherical wave AVO is found to be strongest near critical angles. This Qdependence, to some degree, mimics depth dependence of elastic comparisons. Particle motion perpendicular to the ray angle is also Q-factor dependent. Introduction Previous spherical wave AVO investigations by the authors are restricted to isotropic media (Haase, 2004; Haase and Ursenbach, 2004a). However it is well known that in many situations anisotropy is present either in the form of apparent anisotropy caused by layering or intrinsic anisotropy caused by, for example, shale layers. This type of anisotropy is usually called VTI (transversely isotropic with a vertical axis of symmetry). Rock fractures can cause HTI (horizontal symmetry axis TI) or also orthotropic anisotropy and these are not considered in this study. Early work on spherical wave AVO by Hron et al. (1986) investigates anisotropy using asymptotic ray theory. They note that “anisotropic media produce noticeable differences in both amplitude- and time-distance curves as a function of the degree of anisotropy”; they also show amplitudedistance plots. Previous work by the authors involved plane wave particle motion reflection coefficients given by Zoeppritz’s equations and the Weyl/Sommerfeld integral for computing isotropic spherical-wave potentials. Plane-wave particle motion reflection coefficients for VTI media have been presented by Graebner (1992) and in refined form by Rueger (1996). The Weyl-integral for anisotropic media is given by Tsvankin (2001). Their “exact” equations are utilized in this study. Approximations are introduced by performing numerical integrations Q-factor dependence has also been observed in previous investigations of isotropic spherical-wave AVO (Haase, 2004; Haase and Ursenbach, 2004b). This modeling study seeks to quantify the sensitivity of spherical-wave AVO responses with respect to finite Q-factors of VTI media. Theory The displacement from a point force located at the origin is given by the following summation over plane waves..

Proceedings Papers

Publisher: Society of Exploration Geophysicists

Paper presented at the 2004 SEG Annual Meeting, October 10–15, 2004

Paper Number: SEG-2004-0263

Abstract

ABSTRACT 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, .

Proceedings Papers

Publisher: Society of Exploration Geophysicists

Paper presented at the 2004 SEG Annual Meeting, October 10–15, 2004

Paper Number: SEG-2004-2497

Abstract

ABSTRACT 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 (Q-factors). The VSP surveys used both vertical and horizontal vibrators as sources and a downhole five-level, three-component 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 200–1200m, 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.

Proceedings Papers

Publisher: Society of Exploration Geophysicists

Paper presented at the 2002 SEG Annual Meeting, October 6–11, 2002

Paper Number: SEG-2002-0304

Abstract

Summary 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 30º of angle, in most cases, the spherical VTIresponse 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.

Proceedings Papers

Publisher: Society of Exploration Geophysicists

Paper presented at the 2001 SEG Annual Meeting, September 9–14, 2001

Paper Number: SEG-2001-2025

Abstract

Introduction Although the energy recorded on vertical geophones is predominantly due to P-wave arrivals, it is not uncommon to also observe converted wave energy on conventional sections (Ferber, 1990; Claerbout, 1992; Ogilvie and Purnell, 1996; Holmes and Edwards, 2000; Guevara and Cary, 2000). This fact is sometimes described as leakage or crosstalk. There are even reports of hydrophones detecting doubly converted waves (Jain, 1988; Bevc et al., 2000). Generally, this converted wave energy on conventional P-wave data is considered noise because it is relatively weak, occurs at larger offsets and is successfully dealt with by conventional multiple attenuation. However sometimes converted waves leak through to the stack and could potentially lead to misinterpretation; in these cases identification of converted wave events is crucial (Ogilvie and Purnell, 1996; Holmes and Edwards, 2000; Bevc et al., 2000).

Proceedings Papers

Publisher: Society of Exploration Geophysicists

Paper presented at the 2000 SEG Annual Meeting, August 6–11, 2000

Paper Number: SEG-2000-2177

Abstract

Summary All rocks encountered in nature are anelastic to some degree, implying that velocities are frequency dependent. One of the consequences of this is anelastic wavelet stretch, which has been demonstrated by experiments and model studies. For low Qfactors, anelastic stretch before deconvolution can be of comparable size to NMO stretch. Removing slow-traveling low frequency components from wavelets reduces anelastic stretch. Trace by trace spiking deconvolution appears to overcompensate for anelastic stretch at large offsets. Introduction Wavelet stretch caused by NMO correction of reflection data, also called NMO stretch, is a well known phenomenon. Signal distortion resulting from application of NMO correction is described and discussed in early papers by Buchholtz(1972) as well as by Dunkin and Levin(1973).

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