Kirchhoff migration using eikonal-based computation of traveltime source-derivatives

Constant-velocity-gradient Model

In a 2D medium of linearly changing velocities, $v(z,x) = v_0 + a x + b z$ where x is the lateral position and z is the depth, the traveltimes and source-derivatives have analytical solutions (Slotnick, 1959). Figure 1 shows the model used in our numerical test and the analytical source-derivative for a source located at $(0,0)$ km. The domain is of size 4km $\times$ 4km with grid spacing $0.01$ km in both directions. We solve for the traveltime tables at five sources of uniform spacing $1$ km along the top domain boundary by FMM and their associated source-derivatives using the method described in Appendix A. Figure 2 compares the errors in computed source-derivative between the proposed approach and a centered second-order finite-difference estimation for the same source shown in Figure 1. The proposed method is sufficiently accurate except for the small region around the source. This is due to the source singularity of the eikonal equation and can be improved by adaptive or high-order upwind finite-difference methods (Qian and Symes, 2002) or by factoring the singularity (Fomel et al., 2009). Since we are aiming at using the interpolated traveltime tables for migration purposes and the reflection energy around the sources is usually low, these errors in current implementation can be neglected. In Figure 3, we interpolate the traveltime table for a source at location $(0,0.25)$ km from the nearby source samples at $(0,0)$ km and $(0,1)$ km by the cubic Hermite, linear and shift interpolations. We use the eikonal-based source-derivative in the cubic Hermite interpolation. The shift interpolation is not applicable for some $\mathbf{q}$ and $\mathbf{x_s}$ if $\mathbf{x = q+x_s}$ is beyond the computational domain. In these regions, we use a linear interpolation to fill the traveltime table. As expected, the cubic Hermite interpolation achieves the best result, while its misfits near the source are related to the errors in source-derivatives. The shift interpolation performs generally better than the linear interpolation, especially in the regions close to the source where the wave-fronts are simple.

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Figure 1. (Left) a constant-velocity-gradient model $v(z,x) = 2 + 0.5 x$ km/s and (right) its analytical traveltime source-derivative for a source at origin $\mathbf {x_s} = (0,0)$ km.

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Figure 2. Comparison of error in computed source-derivative by (left) the proposed method and (right) a centered second-order finite-difference estimation based on traveltime tables. The maximum absolute errors are $0.15$ s/km and $0.56$ s/km, respectively.

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Figure 3. Traveltime interpolation error of three different schemes: (top left) the analytical traveltime of a source at location $(0,0.25)$ km; (top right) error of the cubic Hermite interpolation; (bottom left) error of the linear interpolation; (bottom right) error of the shift interpolation. Using derivatives in interpolation enables a significantly higher accuracy. The $l2$ norm of the error are $1.5$ s, $9.2$ s and $6.0$ s respectively.

The difference between a cubic Hermite interpolation and a linear or shift one is in the usage of source-derivatives. In this regard, one may think of supplying the finite-difference estimated derivatives to the interpolation. Indeed, a refined source sampling and higher-order differentiation may lead to more accurate derivatives. However the additional computation is considerable. For the same model in Figure 1, we carry out both a source sampling refinement experiment and a model grid spacing refinement experiment. The results are shown in Figures 4 and 5. Both figures are plotted for the traveltime at subsurface location $(1.5,-0.5)$ km for the source at location $(0,0)$ km. Although the curves vary for different locations, the source sampling refinement experiment suggests the general need for approximately three times finer source-sampling than that of Figure 2 to achieve the same level of accuracy.

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Figure 4. Source-sampling refinement experiment. The plot shows, at a fixed model grid sampling of $0.01$ km and increasing source sampling, the error in source-derivative estimated by a first-order finite-difference (solid) and a centered second-order finite-difference scheme (dotted) decrease. The horizontal axis is the number of sources and the source sampling is uniform. The vertical axis is the natural logarithm of the absolute error. The flat line (dash) is from the proposed eikonal-based method and is source-sampling independent.

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Figure 5. Gird-spacing refinement experiment. The plot shows, at a fixed source sampling of $1$ km and increasing model grid sampling, the error in source-derivative estimated by the proposed eikonal-based method decreases. Meanwhile, the errors of both first- and second-order finite-difference estimations do not improve noticeably. The horizontal axis is the number of grid points in both directions and the grid sampling is uniform. See Figure 4 for descriptions of the vertical axis and the lines.

Kirchhoff migration can use traveltime source-derivatives in two ways: for traveltime interpolation when the source and receiver of a trace does not lie on the source grid of pre-computed traveltime tables, and for anti-aliasing. Figure 6 shows a synthetic model of constant-velocity-gradient with five dome-shaped reflectors. The model has a $0.01$ km grid spacing in both directions. We solve for traveltimes and source-derivatives by the modified FMM introduced in Appendix A at 21 sparse shots of uniform spacing 0.5 km, and migrate synthetic zero-offset data. The interpolation of source-derivative for the anti-aliasing purpose follows the method described in Appendix B. 48 interpolations are carried out within each sparse source sampling interval. Figures 7 and 8 compare the images obtained by three different interpolations and the effect of anti-aliasing. All images are plotted at the same scale. We do not limit migration aperture for all cases and adopt the anti-aliasing criteria suggested by Abma et al. (1999) to filter the input trace before mapping a sample to the image, where the source-derivative and receiver-derivative (in the zero-offset case they coincide) determine the filter coefficients. As expected, the cubic Hermite interpolation with anti-aliasing leads to the most desirable image. The image could be further improved by considering not only the kinematics predicted by the traveltimes but also the amplitude factors (Vanelle et al., 2006; Dellinger et al., 2000).

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Figure 6. Constant-velocity-gradient background model $v(z,x) = 1.5 + 0.25 z + 0.25 x$ km/s with dome shaped reflectors.

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Figure 7. Zero-offset Kirchhoff migration image with (top) the cubic Hermite interpolation and (bottom) the shift interpolation.

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Figure 8. Zero-offset Kirchhoff migration image with (top) the linear interpolation and (bottom) the cubic Hermite interpolation without anti-aliasing.

Kirchhoff migration using eikonal-based computation of traveltime source-derivatives