Inversion after depth imaging

Transcription

1 Robin P. Fletcher *, Stewart Archer, Dave Nichols, and Weijian Mao, WesternGeco Summary In many areas, depth imaging of seismic data is required to construct an accurate view of the reservoir structure. This is especially true in areas of complex geology and areas with salt tectonics. Having identified the reservoir structure, we of course want to extract further information on lithology, reservoir quality, and fluids. This can be achieved by seismic amplitude variation with offset (AVO) inversion. Traditionally, depth-imaged data are inverted by converting it back into the time domain to enable the seismic data to be represented by convolution with a consistent wavelet, varying only slowly with both time and spatial location. We will describe an alternate approach that allows us to invert the seismic data directly in the depth domain, where it has been correctly located by the depth imaging. This approach also allows us to account for the variability in the image amplitudes that arise due to the complex geology and the spatial variations in the acquisition geometry. This new approach to inversion is demonstrated on synthetic model data, and compared with results obtained from conventional time-domain inversion. Introduction Conventional amplitude inversion assumes that the input migrated image has preserved relative amplitude information and is free from the effects of illumination. Under this assumption, stretching a depth-migrated image back to time and applying inversion based on 1D convolutional modeling can produce reasonable results. However, illumination effects in complex geological settings (such as shadow zones in subsalt imaging) pose a challenge to even the most advanced imaging algorithms such as reverse-time migration (RTM). Traditional approaches to compensate for illumination effects in migrated images are difficult to regularize in areas of very poor illumination. We propose a technique for performing amplitude inversion directly in the depth domain through incorporating a measure of how well the migration algorithm (or possibly a processing workflow) acts as an inverse to the Earth s response that we acquire. After outlining the underlying theory, we compare our new method with conventional time-domain inversion on a synthetic example representative of subsalt inversion using RTM. Method Traditional approaches to migration/inversion regard the recorded data, d, as the result of a linear modeling operator, M, applied to the reflectivity model, r. This can be either a discrete or continuous (integral) operator. The least-squares inverse to this problem is * 1 M M M d, ˆ * r (1) where M *, the adjoint of modeling, is the migration operator. The true model and the migrated image I=M * d are related through I Hr, () where the Hessian operator, H=M * M, can be thought of as demigration followed by migration, and is often thought of as a measure of illumination that reflects the effects of velocity variation and the acquisition footprint. If we relax the requirement that the modeling operator and the migration operator are related to each other, then the operator H is still considered as an operator that blurs the true reflectivity model to give the image. We define a 3D earth model m as the elastic properties (such as acoustic impedance, v p /v s ratio, and density), and represent the (possibly non-linear) plane wave reflectivity calculation as r=r(m). To invert for the best model, a simulated image, HR(m), is compared with the original image, I, and the model is updated to derive the model with an image that best fits the data. We define the objective function to be minimized as J ( m) Rm p p 1 1 Cd HRm 1 1 Cm I m m, where C d is the data covariance operator, m 0 is a prior model, C m is the model covariance operator, and p is the L p -norm parameter. The α parameter is the weighting parameter that determines the relative amount of sparseness that can be brought into the inversion. 0 (3) 01 SEG SEG Las Vegas 01 Annual Meeting Page 1

2 The first term in equation 3 minimizes the L p -norm of the reflectivity. Choosing a p value close to 1.0 introduces sparseness into the reflectivity model as is often used in this type of optimization (Oldenburg et al., 1983; Ma, 00). The second term minimizes the error of fit to the data and the third term minimizes the changes made to the prior model, typically built from well information and the migration velocity model. The model and data covariance specification, together with α, control the relative weights of these three terms in the final model selection. A further requirement is that the final model update varies smoothly spatially with the geologic structure. This is achieved by applying local directional smoothing as a preconditioner to the earth model. To achieve faster convergence at early iterations, and because we were not expecting to formally minimize equation 3, we decided to follow this shaping regularization approach (Fomel, 007), rather than explicitly incorporating a penalty term in the objective function. The dip field used to drive the local directional smoothing is estimated from the migrated image. This dip field is also used in the reflectivity operator, R(), calculation. As equation 3 contains an L p -norm as well as a possibly non-linear reflectivity operator, we choose to minimize equation 3 using a non-linear conjugate gradient algorithm (NLCG) or the limited-memory Broyden-Fletcher- Goldfarb-Shanno (BFGS) algorithm of Nocedal (1980). Examples A poststack inversion was applied to synthetic data from the constant-density SigsbeeA model, and compared to the results of conventional inversion. The SigsbeeA model (Figure 1a) and data are made available by the SMAART consortium. It is described as representative of a geologic setting found on the Sigsbee escarpment in the deep water Gulf of Mexico and having illumination problems due to the complex salt shape with rugose salt top found in this area resulting in sub-salt structure that is difficult to image. Figure displays the prestack RTM image using the migration velocity model displayed in Figure 1. The image clearly shows lateral variability in image amplitude and quality beneath the salt where the velocity model does not indicate any corresponding changes in geology. Note that there are point reflectors added to this model in the subsalt sediments at two depth levels. These also show the lateral variability in the subsalt image. diffractors. The demigration is based on the same modeling kernel used in the RTM algorithm. The response of demigration-remigration of a single point diffractor is known as the point spread function (PSF). PSFs based on raytracing are described by Lecomte (008) and are often used in acquisition survey design. Figure displays our two-way wave-equation modeling-based PSFs computed for a regular grid of point diffractors simultaneously. There is significant lateral variability in the PSFs subsalt. These PSFs are cut out from the image and interpolated spatially on the fly during application of the H operator (i.e., non-stationary convolution with the PSFs). For our current estimate of the model, we compute the reflection coefficient at each depth sample, multiply by the PSF for each location, and then sum them to give the forward modeled image. For locations where we did not compute a PSF, we interpolate one from the surrounding PSFs. Poststack inversion of this image was computed using two conventional inversion processes: simultaneous seismic inversion (Rasmussen et al., 004) and sequential seismic inversion (Poggliagliomi and Allred, 1994) for comparison with the proposed inversion algorithm outlined above. Figure 3 shows the data used for this comparison (Figures 3a and 3b) and compares the results of the three inversion approaches. Figures 3(c) to 3(f) show the acoustic impedance after subtraction of a smooth background model. The new inversion (Figure 3d) shows more continuous reconstruction of the model than either of the conventional inversions. The simultaneous inversion (Figure 3f) method is similar to the new workflow, with the main difference being that it uses a 1D spatially invariant (or perhaps slowly varying) wavelet in the time domain. This clearly prevents this inversion from recovering the true reflectivity from zones where the image amplitudes are distorted. Both of these methods use the same smooth prior model. The sequential inversion (Figure 3e) uses a prior model based on extrapolation of well data to determine the scalar for the contribution of the seismic data to the inversion result. In this case we used the assigned well location shown in Figure 3, along with horizon picks from the structural image to compute the prior model. This allows better recovery of the true reflectivity for this method, although it relies heavily on constructing an accurate prior model. The new inversion shows better recovery of the model through the weak image zone than either of the other two inversion methods. This new approach uses the measured response of the seismic imaging workflow, the PSFs, so it relies less on assumptions on the lateral stability of the data or the accuracy of the prior model. Our estimate of the H operator is obtained by demigration followed by the prestack RTM workflow applied to point 01 SEG SEG Las Vegas 01 Annual Meeting Page

3 Discussion and conclusions We propose an alternative approach to applying conventional inversion based on 1D convolutional modeling, after depth imaging. We invert seismic data directly in the depth domain, accounting for illumination effects in the image by replacing the 1D wavelet in conventional inversion with the point spread function of the depth imaging processing. The synthetic example we presented inverted an RTM image using PSFs generated from finite-difference wave-equation propagation. However, the inversion algorithm could be applied using PSFs (or a complete calculation of the Hessian) and images generated using other cheaper propagators for geologies that do not warrant a full wave solution. Even in simple models, the ability to perform inversion directly in the depth domain, rather than converting vertically to time and having to estimate a consistent wavelet, may be worth the increase in computational effort. Applied on field data, matching filters will be required to calibrate the forward modeled image with measurements from available wells. Whilst we have shown an example of poststack acoustic impedance inversion, the extention of this workflow to perform AVO inversion of angle stacked depth images is under investigation. Acknowledgments The authors thank WesternGeco for permission to publish this work as well as James Rickett and Irina Marin for valuable contributions and discussion. Figure 1: SigsbeeA model stratigraphic velocity, migration velocity. Figure : Prestack reverse-time migration image, Point spread functions. 01 SEG SEG Las Vegas 01 Annual Meeting Page 3

4 (c) (d) (e) (f) Figure 3: Detailed zone of inversion. the image, PSFs, (c) true model acoustic impedance (after removing the background prior model), (d) depth domain inversion, (e) sequential inversion and (f) simultaneous inversion. 01 SEG SEG Las Vegas 01 Annual Meeting Page 4

5 EDITED REFERENCES Note: This reference list is a copy-edited version of the reference list submitted by the author. Reference lists for the 01 SEG Technical Program Expanded Abstracts have been copy edited so t hat 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 Fomel, S., 007, Shaping regularization in geophysical estimation problems: Geophysics, 7, no., R9 R36. Lecomte, I., 008, Resolution and illumination analyses in PSDM: A ray -based approach: The Leading Edge, 7, Ma, X-Q., 00, Simultaneous inversion of prestack seismic data for rock properties using simulated annealing: Geophysics, 67, Nocedal, J., 1980, Updating quasi-newton matrices with limited storage: Mathematics of Computation, 95, Oldenburg, D. W., T. Scheur, and S. Levy, 1983, Recovery of the acoustic impedance from reflection seismograms: Geophysics, 48, Poggliagliomi, E., and R. D Allred, 1994, Detailed reservoir definition by integration of well and 3 -D seismic data using space adaptive wavelet processing: The Leading Edge, 13, Rasmussen, K. B., A. Bruun, and J. M. Pedersen, 004, Simultaneou s seismic inversion: 66th Conference and Exhibition, EAGE, Extended Abstracts, P SEG SEG Las Vegas 01 Annual Meeting Page 5