Common-azimuth downward continuation and ray bending

As we previously discussed (), when the propagation velocity is constant common-azimuth downward continuation is exact, within the limits of the stationary-phase approximation. In this case, there is no ray bending and the source and receiver rays propagate straight along the slanted planes shown in Figure  at every depth level, until they meet to image a diffractor at depth. However, when ray bending occurs, it seems that common-azimuth downward continuation introduces an error. This conclusion would be in contradiction with the accurate results obtained by the application of common-azimuth migration shown in the next section, and it requires a closer examination. We first discuss the simpler case of velocity varying with depth, and then the general case of velocity varying laterally.

Inspection of the stationary-phase results [equation (7) and equation (9)] shows that in a horizontally stratified medium the cross-line ray parameters for the source and receiver rays $\left(p_{sy}, p_{ry}\right)$change across boundaries between layers with different velocities. This result contradicts Snell's law that states that the horizontal ray parameters must be constant when the velocity varies only vertically. In common-azimuth downward continuation, by imposing the constraint that the source and receiver ray must lie on the same plane, we force the source and receiver rays to bend across interfaces in a way that may not be consistent with Snell's law. In particular, the ray bending determined by common-azimuth continuation is incorrect when the velocity variations would prescribe the source ray to bend differently than the receiver ray along the cross-line direction. This error in the ray bending can be analyzed by evaluating the difference between the values of $\left(p_{ry}- p_{sy}\right)$ across an interface where the upper layer has velocity V₁ and the lower layer has velocity V₂

$$\begin{eqnarray} \lefteqn{\Delta\left(p_{ry}- p_{sy}\right) = \Delta \left( \fra... ..._{2}}^2} -p_{rx}^2}+ \sqrt{\frac{1}{{V_{2}}^2} -p_{sx}^2}\right)}.\end{eqnarray}$$ (10)

It is straightforward to show from (10) that the error is equal to zero, if and only if, one of the following conditions is fulfilled,

$$\begin{eqnarray} & \left\vert p_{rx}\right\vert = \left\vert p_{sx}\right\vert & \\ & p_{ry}= -p_{sy}& \\ & V_{1}= V_{2}&.\end{eqnarray}$$ (11) (12) (13)

The first condition is fulfilled when either the source ray is parallel to the receiver ray (e.g. zero offset data), or the two rays converge and form opposite angles with the in-line axis x (e.g. flat reflections along the in-line midpoint axis). The second condition is fulfilled in the case of vertical propagation (2-D), while the third condition confirms that when the velocity is constant common-azimuth downward continuation is kinematically exact.

The previous analysis shows that common-azimuth downward continuation introduces an error in the ray bending across velocity interfaces. However, this consideration does not necessarily leads to the conclusion that common-azimuth continuation is inaccurate. On the contrary, we argue that the error in the ray bending causes only second order errors in the continuation results. This claim can be simply verified by recognizing that for downward continuing common-azimuth data we evaluate the phase function $\Phi(\omega,{\bf k}_{m},{\bf k}_{h},{z})$ [equation (3)] at its stationary point $\hat {k}_{hy}^{'}{(z^{})}$ [equation (7)] with respect to the cross-line offset wavenumber k_hy. Since the phase function is stationary at $\hat {k}_{hy}^{'}{(z^{})}$the first order term of its Taylor expansion as a function of k_hy around $\hat {k}_{hy}^{'}{(z^{})}$ is equal to zero. Therefore, an error in k_hy has only a second order effect on the evaluation of the phase function $\Phi_{stat}(\omega,{\bf k}_{m},k_{hx},{z})$ [equation (6)]. In other words, the error introduced by the incorrect ray bending has second order effects on the continuation results, and consequently on the migration results. This conclusion is supported by the accuracy of the migrated images shown in the next section.

When the velocity field varies laterally, the previous analysis becomes more complex because the incorrect ray bending causes errors in the evaluation of the phase function not only through errors in the cross-line offset wavenumber $\hat {k}_{hy}^{'}{(z^{})}$,but also through errors in the horizontal locations where the velocity function is evaluated. The arguments that support the conclusion that the errors in the phase function caused by error in $\hat {k}_{hy}^{'}{(z^{})}$ are of second order are still valid for the general case of lateral velocity variations. However, the magnitude of the error introduced by the mispositioning of sources and receivers at depth when evaluating the velocity function cannot be neglected in principle. These errors are dependent on the spatial variability of the velocity function, and cannot be readily analyzed analytically. In the following section we show accurate migration results obtained over a velocity function varying both laterally and vertically. These results are encouraging and suggest that the range of application of common-azimuth migration to depth migration problems is fairly large.

 

stconv
Figure 2 Schematic showing the ray geometry for common-azimuth downward continuation. For each pair of source ray and receiver ray, both rays are constrained to lie on the same slanted plane. All the propagation planes share the line connecting the source and the receiver locations.

MIGRATION RESULTS We tested common-azimuth migration by imaging two synthetic data sets generated by a modeling program based on the Kirchhoff integral. The Green functions are computed analytically assuming velocity functions with a constant spatial gradient. In both cases the velocity at the origin of the spatial coordinates is equal to $\rm{1.7~km~{s^{-1}}}$.The first data set is generated assuming a vertical gradient of $\rm{1.0~{s^{-1}}}$ while the second data set is generated assuming a gradient with the horizontal component equal to $\rm{0.2~{s^{-1}}}$ and the vertical component equal to $\rm{0.25~{s^{-1}}}$. The horizontal component of the gradient is oriented at an angle of 45 degrees with respect to the offset azimuth (in-line direction) of the acquisition geometry. The reflectivity model is a constant reflectivity function positioned along a half-spherical dome superimposed onto a horizontal planar reflector.

The acquisition geometry has 128 midpoint along both the in-line and cross-line directions; with midpoint spacing of $\rm{25~m}$ in both directions. Each midpoint gather has 64 offsets, spaced every $\rm{40~m}$; the nearest offset traces are actually recorded at zero offset. The offset-azimuth of the data is aligned with the in-line direction. Figure  and show two in-line zero-offset sections extracted from respectively the vertical gradient data set and the oblique gradient data set. The effects of the lateral component of the velocity gradient are evident in both the tilting of the horizontal reflector, and in the asymmetry of the quasi-hyperbolic reflections from the dome.

Figures  and show an in-line section and a depth slice of the results of migrating the vertical gradient data set. The in-line section (Figure ) is taken across the middle of the dome, while the depth slice (Figure ) is taken across the base of the dome $\rm{\left( z = 1.75~km \right)}$.Common-azimuth migration has accurately imaged the data; both the in-line section and the depth slice show a perfect focusing of the reflectors. To visually verify the isotropic response of common-azimuth migration we overlaid a circle onto the depth slice. The migrated dome is perfectly circular, notwithstanding the ray bending caused by the strong vertical velocity gradient.

Figures  and show the result of migrating the oblique gradient data set. The in-line section (Figure ) shows that common-azimuth migration has correctly positioned the reflectors. The planar reflector has been flattened, and the spherical dome has been properly focused. A spatial variability in both the frequency content and the amplitude of the migrated reflector is noticeable in the depth slice. The frequency variability is expected, and it is caused by the widening of the spatial wavelength of the wavefield caused by higher propagation velocities. The wavelet is narrower closer to the origin (upper-left corner) where velocity is lower and it is wider where the velocity is higher (lower-right corner). The amplitude variations cannot be readily explained, and further analysis is needed to determine whether they are artifacts of common-azimuth migration. The combined frequency and amplitude effects creates an ``anisotropic'' appearance to the migrated image. However, by overlaying a circle onto the plot of the seismic data we see that the migrated dome, though not perfectly circular, it is very close to circular. The comparison of these results with the results from the vertical gradient data set confirm the accuracy of common-azimuth migration even in the presence of strong lateral velocity gradients.

 

Mod-vz
Figure 3 In-line zero-offset section extracted from the synthetic data set modeled assuming a constant vertical gradient in velocity equal to $\rm{1.0~{s^{-1}}}$.

 

Mod-grad
Figure 4 In-line zero-offset section extracted from the synthetic data set modeled assuming a constant velocity gradient with the horizontal component equal to $\rm{0.2~{s^{-1}}}$ and the vertical component equal to $\rm{0.25~{s^{-1}}}$.

 

Ph3D-vz-sy
Figure 5 In-line section of the migration results for the vertical gradient data set. This section passes through the middle of the dome.

 

Ph3D-vz-sz-circ
Figure 6 Depth slice of the migration results of the vertical gradient data set. A circle is overlaid onto the seismic data to visually verify the isotropy of the migrated dome.

 

Ph3D-grad-sy
Figure 7 In-line section of the migration results for the oblique gradient data set. This section passes through the middle of the dome.

 

Ph3D-grad-sz-circ
Figure 8 Depth slice of the migration results of the oblique gradient data set. Frequency content and amplitude variations are evident along the circular reflector. However, the imaged reflector is fairly close to be circular.

CONCLUSIONS We have shown that common-azimuth migration () can be successfully applied to imaging 3-D prestack data with laterally varying velocity. This important generalization of common-azimuth migration was made possible by two theoretical insights. The first one led us to recast common-azimuth migration as a recursive application of a new common-azimuth downward continuation operator. The second one is a ray-theoretical interpretation of common-azimuth downward continuation that enables us to analyze the errors in presence of ray bending caused by velocity inhomogeneities.

We implemented common-azimuth depth migration by downward continuation in mixed space-wavenumber domain using a split-step scheme. The application of our depth migration algorithm to a data set with a strong horizontal component of the velocity gradient resulted in an accurate imaging of the reflectors. ACKNOWLEDGEMENTS We thank Arnaud Berlioux for helping to generate the dome model using GOCAD. Biondo Biondi would like to thank Norm Bleistein for a short conversation that motivated him to better understand the errors involved in the stationary-phase derivation of common-azimuth migration.

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A The purpose of this Appendix is to demonstrate the equivalence of the stationary phase derivation of the common-azimuth downward continuation operator [equation (7) in the main text] and the constraint on the propagation directions of the source rays $\left( p_{sx}, p_{sy}, p_{sz}\right)$ and receiver rays $\left( p_{rx}, p_{ry}, p_{rz}\right)$expressed in equation (8).

We start by showing that equation (8) is directly derived by imposing the condition that the source ray and the receiver ray lie on the same plane. For this condition to be fulfilled, the components of the two rays in the direction of the cross-line axis y must be equal. From elementary geometry, these components are

$$\begin{eqnarray} & dy_{s}= v({\bf s},z)p_{sy}dl_{s}= \frac{p_{sy}dz}{p_{sz}} & \... ... \\ & dy_{r}= v({\bf r},z)p_{ry}dl_{r}= \frac{p_{ry}dz}{p_{rz}}, &\end{eqnarray}$$ (14)

where dl_s, dl_r are the differential ray-paths lengths for the source and receiver rays, dy_s, dy_r are their components along the y axis, and dz is the component along the depth axis, constrained to be the same for the source and receiver rays. By equating the two equations in (A-1) we immediately derive equation (8); that is,  

$$\frac{p_{sy}}{p_{sz}} = \frac{p_{ry}}{p_{rz}}.$$ (15)

The second step is to eliminate p_sz and p_rz from equation (A-2) by using the following relationships among the ray parameters

$$\begin{eqnarray} & p_{sx}^2 + p_{sy}^2 + p_{sz}^2 = \frac{1}{v({\bf s},z)^2} & \... ... \\ & p_{rx}^2 + p_{ry}^2 + p_{rz}^2 = \frac{1}{v({\bf r},z)^2} &.\end{eqnarray}$$ (16)

After a few simplifications we get the equivalent of the stationary path relationship [equation (7)] but expressed in terms of ray parameters,  

$$\left(p_{ry}- p_{sy}\right) = \left(p_{ry}+ p_{sy}\right) \f... ...^2} - p_{rx}^2 }+ \sqrt{\frac{1}{v({\bf s},z)^2} - p_{sx}^2 }}.$$ (17)

To derive equation (7) from equation (A-4) it is sufficient to substitute for the ray parameters by applying the relationships

$$\begin{eqnarray} k_{mx}= {k_{rx}+ k_{sx}} = \omega \left(p_{rx}+ p_{sx}\right) \... ...k_{ry}- k_{sy}} = \omega \left(p_{ry}- p_{sy}\right) \nonumber \\ \end{eqnarray}$$