m=[a,b,c,d] T together with the a posteriori covariance

Transcription

1 zimuthal VO analysis: stabilizing the model parameters Chris Davison*, ndrew Ratcliffe, Sergio Grion (CGGVeritas), Rodney Johnston, Carlos Duque, Musa Maharramov (BP). solved using linear least squares to yield the model vector Summary m=[,b,c,d] T together with the a posteriori covariance VO is an important tool in the interpretation of seismic data. zimuthal VO models (VZ) have been used to characterize fracture distributions and directions in HTI (Horizontal Transverse Isotropy) media. The main subject of this paper is a method for the stabilization of VZ parameters. We present an extension of the technique of Whitcombe, Dyce, McKenzie and Hoeber (4) for gradient stabilization in the standard -term VO model given by the Shuey equation to the case of Ruger and Tsvankin s (997) azimuthal VO analysis. We also investigate the estimated errors in the VZ model parameters with varying HTI isotropy plane direction for a selection of offset-azimuth distributions, including azimuthal sectors and Common Offset Vector (COV) classes. The application of the technique to WZ land data from lgeria illustrates the use of the techniques. zimuthal VO: models and uncertainties Ruger and Tsvankin (997) proposed a model for the dependence of the amplitude of a reflection event, R, on the shot-receiver azimuth, φ, and narrow angles of incidence, θ, for an isotropic half space over a HTI anisotropic half space, given by: R ( θ, φ) = + [ + ( )] cos φ φ sin θ. B iso B ani sym matrix C m =σ N [G T G] -, where σ N is the Gaussian noise in the data. The diagonal entries of this matrix give the errors in the linear parameters and the off-diagonal entries the covariances. Downton and Gray (6) and Downton et al. (7) show how the errors in the VO parameters, B iso, B ani and φ iso may be derived from the entries of this matrix, using simplifying assumptions and approximations. Effect of offset-azimuth binning on errors We compare the effect of binning the data into COV classes and azimuth sectors on the VZ error analysis. Dong and Davidson (3) previously carried out such a study for different noise levels and acquisition geometries using a more empirical approach than the theoretical one used here. We consider a synthetic example relating to an event in the reservoir zone of the real data set described in the next section. Here we have VZ parameters of: =., B iso =-.3, B ani =.6. The event has a zero offset time of.36s and the signal to noise ratio is 5. Note that the parameters extracted from the real data were normalised to give an impedance reflectivity of =. as commonly done by authors in similar studies. s errors decrease by the square root of fold, the predictions are normalised with respect to a fold of 5 to allow direct comparison of geometries. The analysis is performed for data restricted to angles of incidence in the range -35 degrees, which again is inspired by the real data. Here, is the P-wave impedance reflectivity, B iso the isotropic gradient, B ani the anisotropic gradient and φ sym the symmetry axis of the HTI media. The angle of the isotropy plane φ iso is related to φ sym by φ iso = φ sym + 9. To solve these non-linear equations, Downton and Gray (6), Downton, Gray and Zuk (7) and earlier authors make the substitutions: B = B iso + B ani, C = B ani cos ( φ ), D = sin( φ ), iso B ani which convert Ruger s equation into the following linear equation: R ( θ, φ) = + B C cos( φ ) D sin( φ ) sin θ. This equation can be written in matrix form d=gm and iso The polar plots illustrated in Figure show how the results vary with the direction of the HTI isotropy plane. For brevity only two cases are shown: case (a) relates to binning in COV classes (3x4m), and case (b) to azimuth sectors sampled every 3 degrees in azimuth and m in offset. For equal fold, the errors predicted for COV classes are smaller than those observed in sectors. These results agree with the current authors analogous study of errors in azimuthal velocity models for HTI media (C. Davison,. Ratcliffe, S. Grion, R. Johnston, C. Duque, J. Neep, M. Maharramov,, zimuthal velocity uncertainty: estimation and application, submitted to SEG 8 st nnual International Meeting). Stabilizing the VZ parameters Whitcombe et al. (4) show a method of stabilizing the VO gradient for the -term Shuey equation. This combines time windowed VO with hodogram technology. Here we extend the methodology to the case of azimuthal SEG SEG San ntonio nnual Meeting 33

2 zimuthal VO analysis: stabilizing the model parameters VO. The general idea of Whitcombe et al. (4) may be briefly described as follows. If we plot the VO gradient G against the VO intercept found using a least squares analysis, the noise correlates along the eigenvector of the largest eigenvalue of the a posteriori covariance matrix. Furthermore, the cross-plot of the gradient against the intercept for all the points in a wavelet associated with a discrete event shows a linear relationship (Ratcliffe and dler, ). Therefore, to stabilize the VO gradient, Whitcombe et al. (4) process the intercept and gradient (,G) over a time window corresponding to a wavelet. The points in the window are first of all rotated to a coordinate frame (,G ) such that G is the eigenvector along which the noise correlates. linear least squares fit is then performed on the points, and the value of G found for the mid-point of the time window. The new point (,G ) is then rotated back to the original (,G) coordinates. In this paper we have extended the method to an azimuthal VO model. Consider the linearization of Ruger s equation in the new variables m=[,b,c,d] T. If we consider how the values of these variables vary over a wavelet, then by a straightforward extension of Ratcliffe and dler s () technique, we find that the linear variables m=[,b,c,d] T lie on a straight line through the origin. We can easily derive these linear parameters for any data given that we have the VO attributes (,B iso,b ani,φ iso ). lso note that from least squares theory and principal component analysis, we find in this case too that the noise correlates along the eigenvector corresponding to the largest eigenvalue of the covariance matrix. In the two dimensional case, the problem is simplified by the fact that any x rotation matrix, performing a rotation in two dimensions from the existing orthogonal coordinate system (,G) to (,G ), can be characterised by a single angle, which is easily derived (Whitcombe et al., 4). For the four-dimensional azimuthal VO case, we can still transform the linear parameters into a frame represented by the eigenvectors of the covariance matrix. However, in this instance we must know the entries of the covariance matrix to calculate the 4x4 orthogonal transformation. Therefore, parameter stabilization is rather cumbersome as a post-processing step, since it requires a very large amount of data to be stored from the least squares analysis. However, experience shows that if we have a good WZ distribution of traces then the correlations in the linear parameters are approximately zero, with the exception of the correlation between the parameters and B. This means that the 4x4 orthogonal transformation matrix takes the approximate form: R T = x I x x. Here, I x is the x identity matrix, x is the x zero matrix and R is a x rotation matrix. The stabilization of the azimuthal VO parameters then proceeds as follows. Over a time window we take the linear parameters (,B,C,D) calculated from the azimuthal VO model and rotate to the new coordinate system (,B,C,D) using the 4x4 matrix T. We can then use linear least squares fitting on the pairs of variables (,B ), (,C) and (,D), as a straight line in four dimensions through the origin can be characterised by its projections onto three two-dimensional subspaces. We take the values of B, C and D at the mid-point of the time window, and rotate the point in 4-dimensions, (,B,C,D), back to the (,B,C,D) coordinate frame. The linear parameters are then transformed back into the non-linear azimuthal VO parameters (,B iso,b ani,φ iso ). To evaluate our stabilization method we carry out an azimuthal VO analysis on gathers of purely random noise, with an offset/azimuth geometry as per the real data example described in the next section. In Figure we show the cross-plots before and after the application of the stabilization method to the linear parameters (,B,C,D). We observe that the method reduces the level of scatter in the data. lso note that parameters and B are strongly correlated, but that the other pairs of parameters do not show any strong correlations. Real data example VO analysis was carried out on WZ land data from the Tiguentourine 3D survey in lgeria. This ~95 km area was covered by surveys carried out in 4 and 7 for the In menas Joint Venture (Sonatrach, BP and Statoil). For this work we concentrated on a 4 km subset of this volume. The data were acquired with a shot line spacing of 5m and a shot spacing of 5m. The receiver line spacing is m, with a receiver spacing of 5m. The nominal bin size is 5x5m, and the nominal fold is -4. The data is binned into COV classes of 3x4m. zimuthal VO analysis is applied after a processing flow that honours azimuth information. Figure 3 shows an inline section of field data for the anisotropic gradient, before and after we apply the stabilization technique. Here we chose the minimal time window of 3 samples and spatial filter of 3x3 points. reduction of the noise in the anisotropic gradient after stabilization is clearly observed in the inline of data. Given that this area is known for having fractured reservoirs, any reduction in the noise of these key seismic indicators is a benefit during interpretation. SEG SEG San ntonio nnual Meeting 33

3 zimuthal VO analysis: stabilizing the model parameters Conclusions We have investigated the effect of the offset-azimuth distribution on errors in the VZ model parameters and discussed a method of stabilizing the parameters in the model. The error analysis with respect to binning shows that the COV classes give smaller errors than azimuth sectors. pplication of the stabilization method to real data delivers substantial noise reduction. cknowledgements This work was the result of a technical collaboration between BP and CGGVeritas. Both companies are thanked equally for their permission to publish this work. lso, the authors wish to thank Sonatrach and BP for their kind permission to publish the real data example. a) b) Figure a) Percentage errors for (3 4m) COV classes, b) Percentage errors for ( x=m, φ=3 ) azimuth sectors. a) b) c) : B vs : C vs : D vs B C D d) e) f) : B vs : C vs : D vs B C D Figure Cross-plots of linear model parameters: (a,b,c) before, and (d,e,f) after stabilization. SEG SEG San ntonio nnual Meeting 33

4 zimuthal VO analysis: stabilizing the model parameters a) b) km km Figure 3 Line of anisotropic gradient from field data: (a) before, and (b) after stabilization (arrows indicating areas of clearly improved resolution of events). SEG SEG San ntonio nnual Meeting 333

5 EDITED REFERENCES Note: This reference list is a copy-edited version of the reference list submitted by the author. Reference lists for the SEG Technical Program Expanded bstracts have been copy edited so that references provided with the online metadata for each paper will achieve a high degree of linking to cited sources that appear on the Web. REFERENCES Dong, W., and M. E. Davidson, 3, Quantifying 3D acquisition adequacy for azimuthal VO analysis: The Leading Edge,, , doi:.9/ Downton, J., and D. Gray, 6, VZ parameter uncertainty estimation: 76th nnual International Meeting, SEG, Expanded bstracts, Downton, J., D. Gray, and T. Zuk, 7, Visualizing VZ parameter estimates with uncertainty due to noise: 69th nnual International Conference and Exhibition, EGE, Extended bstracts, D3. Ratcliffe,., and F. dler,, ccurate velocity analysis for Class II VO events: 7th nnual International Meeting, SEG, Expanded bstracts, Rüger,., and I. Tsvankin, 997, Using VO for fracture detection: nalytic basis and practical solutions: The Leading Edge, 6, , doi:.9/ Whitcombe, D. N., M. Dyce, C. J. S. McKenzie, and H. Hoeber, 4, Stabilizing the VO Gradient: 74th nnual International Meeting, SEG, Expanded bstracts, SEG SEG San ntonio nnual Meeting 334