Examples

I illustrate the RWE method with several synthetic examples of various degrees of complexity. In all examples, I use extrapolation in 2D orthogonal Riemannian spaces (ray coordinates), and compare the results with extrapolation in Cartesian coordinates. I present images obtained by migration of synthetic datasets represented by events equally spaced in time. In these examples, I use synthetic data from point sources located on the surface. After I migrate these data in depth, I obtain images which are representations of Green's functions from the chosen source point. In all examples, (x,z) are Cartesian coordinates and $(\qt,\qg)$ are ray coordinates for point sources. In all examples, $\qg$ stands for shooting angle and $\qt$ for one-way traveltime from the source.

The first example is designed to illustrate the method in a fairly simple, albeit not completely realistic, model. I use a 2D model with horizontal and vertical gradients $v(\xx,\zz)=250+0.2\;\xx+0.15\;\zz$ m/s which gives waves propagating from a point source a pronounced tendency to overturn (RCsi1.com.ps). The model also contains a diffractor located around $\xx=3800$ m and $\zz=3000$ m.

I use ray tracing to create an orthogonal ray coordinate system corresponding to a point source on the surface at $\xx=6000$ m. RCsi1.com.ps(a) shows the velocity model and the rays in the original Cartesian coordinate system ($\xx,\zz$). RCsi1.com.ps(b) shows the one-to-one mapping of the velocity model from Cartesian coordinates ($\xx,\zz$) into ray coordinate ($\qt,\qg$)using the functions $\xx(\qt,\qg)$ and $\zz(\qt,\qg)$ obtained by ray tracing. The diffractor is mapped to $\qt=2.4$ s and $\qg=-18^\circ$ measured from the vertical. The synthetic data I use is represented by impulses at the source location at every 0.25 s. In ray coordinates, this source is represented by a plane-wave evenly distributed over all shooting angles $\qg$.Ideally, an image obtained by migrating such a dataset is a representation of the acoustic wavefield produced by a source that pulsates periodically.

RCsi1.com.ps(c) shows the image obtained by downward continuation in Cartesian coordinates using the standard $15^\circ$ equation. RCsi1.com.ps(d) shows the image obtained by wavefield extrapolation using the ray-coordinate $15^\circ$ equation. The overlays in panels (c) and (d) are wavefronts at every 0.25 s and rays shot at every $20^\circ$ to facilitate comparisons between the images in ray and Cartesian coordinates.

RCsi1.f15.ps is a direct comparison of the results obtained by extrapolation in the two coordinate systems. The image created by extrapolation in Cartesian coordinates (a) is mapped to ray coordinates (b). The image created by extrapolation in ray coordinates (d) is mapped to Cartesian coordinates (c). Since I use the same velocity for ray tracing and for wavefield extrapolation, I expect the wavefields and the overlain wavefronts to be in agreement. The most obvious mismatch occurs in regions where the $15^\circ$ equation fails to extrapolate correctly at steep dips $\qg=-20^\circ\dots-50^\circ$.This is not surprising since, as its name indicates, this equation is only accurate up to $15^\circ$.However, this limitation is eliminated in ray coordinates, because the coordinate system brings the extrapolator in a reasonable position and at a good angle, although the extrapolator uses an equation of a similar order of accuracy.

Another interesting observation in RCsi1.f15.ps (a) and (c) concerns the diffractor present in the velocity model. When I extrapolate in Cartesian coordinates, the diffraction is only accurate to a small angle relative to the extrapolation direction (vertical). In contrast, the diffraction develops relative to the propagation direction when computed in ray coordinates, thus being more accurate after mapping to Cartesian coordinates.

 

RCsi1.com.ps
Figure 3 Simple linear gradient model: Panels (a) and (c) correspond to Cartesian coordinates, and panels (b) and (d) correspond to ray coordinates. Velocity model with an overlay of the ray coordinate system initiated by a point source at the surface (a); image obtained by downward continuation in Cartesian coordinates with the $15^\circ$ equation (c); velocity model with an overlay of the ray coordinate system (b); image obtained by wavefield extrapolation in ray coordinates with the $15^\circ$ equation (d).

 

RCsi1.f15.ps
Figure 4 Simple linear gradient model: the image obtained by downward continuation in Cartesian coordinates with the $15^\circ$ equation (a); the image in panel (a) interpolated to ray coordinates (b); image obtained by extrapolation in ray coordinates with the $15^\circ$ equation (d); the image in panel (d) interpolated to Cartesian coordinates (c).

We can also observe that the diffractions created by the anomaly in the velocity model are not at all limited in the ray coordinate domain. In a beam-type approach, such diffraction would not develop beyond the extent of the beam in which it arises. Neighboring extrapolation beams are completely insensitive to the velocity anomaly.

The second example is a smooth velocity with a negative Gaussian anomaly that creates a triplication of the ray coordinate system (RCga1.com.ps). Everything other than the velocity model is identical to its counterpart in the preceding example. Similarly to RCsi1.com.ps, panels (a) and (c) correspond to Cartesian coordinates, and panels (b) and (d) correspond to ray coordinates. Using regularization of the ray coordinates parameters, I extrapolate through the triplication. The discrepancy between the wavefields and the corresponding wavefronts highlight the decreasing accuracy in the caustic region caused by the parameter regularization.

 

RCga1.com.ps
Figure 5 Gaussian anomaly model: Panels (a) and (c) correspond to Cartesian coordinates, and panels (b) and (d) correspond to ray coordinates. Velocity model with an overlay of the ray coordinate system initiated by a point source at the surface (a); image obtained by downward continuation in Cartesian coordinates with the $15^\circ$ equation (c); velocity model with an overlay of the ray coordinate system (b); image obtained by wavefield extrapolation in ray coordinates with the $15^\circ$ equation (d).

 

RCga1.f15.ps
Figure 6 Gaussian anomaly model: the image obtained by downward continuation in Cartesian coordinates with the $15^\circ$ equation (a); the image in panel (a) interpolated to ray coordinates (b); image obtained by extrapolation in ray coordinates with the $15^\circ$ equation (d); the image in panel (d) interpolated to Cartesian coordinates (c).

The ``butterfly'' in RCga1.f15.ps (b) is another indication that the ray coordinate system is triplicating and that the Cartesian coordinates are multi-valued function of ray coordinates. None of this happens when I extrapolate in ray coordinates (d) and interpolate to Cartesian coordinates (c) since the mappings $\xx(\qt,\qg)$ and $\zz(\qt,\qg)$ are single-valued.

Comparing panels (a) and (c) of RCga1.f15.ps, we can notice that the triplication tails at, for example, $\xx\approx7000$ m and $\zz\approx4000$ m extend farther with the Cartesian extrapolator (a) than with the Riemannian extrapolator (c). The triplications create internal boundaries in the coordinate system which are better avoided.

The next example uses the more complicated Marmousi model. RCma4.com.ps shows the velocity models mapped into the two different domains, and the wavefields obtained by extrapolation in each one of them. I create the ray coordinate system by ray tracing in a smooth version of the model, and extrapolate in the rough version. The source is located on the surface at $\xx=5000$ m.

In this example, the wavefields triplicate in both domains (RCma4.f15.ps). Since I am using a $15^\circ$ equation, extrapolation in Cartesian coordinates is only accurate for the small incidence angles, as observed in panels (a) and (b). In contrast, extrapolating in ray coordinates (d) does not have the same angular limitation, which can be seen after mapping back to Cartesian coordinates (c).

 

RCma4.com.ps
Figure 7 Marmousi model: Panels (a) and (c) correspond to Cartesian coordinates, and panels (b) and (d) correspond to ray coordinates. Velocity model with an overlay of the ray coordinate system initiated by a point source at the surface (a); image obtained by downward continuation in Cartesian coordinates with the $15^\circ$ equation (c); velocity model with an overlay of the ray coordinate system (b); image obtained by wavefield extrapolation in ray coordinates with the $15^\circ$ equation (d).

 

RCma4.f15.ps
Figure 8 Marmousi model: the image obtained by downward continuation in Cartesian coordinates with the $15^\circ$ equation (a); the image in panel (a) interpolated to ray coordinates (b); image obtained by extrapolation in ray coordinates with the $15^\circ$ equation (d); the image in panel (d) interpolated to Cartesian coordinates (c).

 

RCma4.zom.ps
Figure 9 Marmousi model: Velocity model (a); image obtained by wavefield extrapolation in ray coordinates using the $15^\circ$ equation (b) and the split-step equation (c); image obtained using downward continuation in Cartesian coordinates with the $45^\circ$ equation (d), the $15^\circ$ equation (e) and the split-step equation (f).

 

RCma4.yom.ps
Figure 10 Marmousi model: Velocity model (a); image obtained by wavefield extrapolation in ray coordinates using the $15^\circ$ equation (b) and the split-step equation (c); image obtained using downward continuation in Cartesian coordinates with the $45^\circ$ equation (d), the $15^\circ$ equation (e) and the split-step equation (f).

RCma4.zom.ps is a close-up comparison of the wavefields obtained by extrapolation with different methods in different domains. Panel (a) is a window of the velocity model for reference. Panels (b) and (c) are obtained by extrapolation in ray coordinates using the $15^\circ$ and split-step equations, respectively. Panels (d), (e) and (f) are obtained by downward continuation in Cartesian coordinates using the $45^\circ$, $15^\circ$ and split-step equations, respectively. The ray-coordinate extrapolation results are similar to the Cartesian coordinates results in the regions where the wavefields propagate mostly vertically, but are different in the regions where the wavefields propagate almost horizontally.

RCma4.yom.ps is another close-up comparison of the wavefields obtained by extrapolation with different methods in different domains. The panel structure is similar to the one in RCma4.zom.ps. This window is chosen to capture the portion of the wavefield which is well described kinematically by extrapolation in Cartesian coordinates with the $15^\circ$. We can observe that the amplitude behavior of Riemannian extrapolation coincides with that of Cartesian extrapolation.

RCga2.kin.ps illustrates the difference between wavefield extrapolation using oneway.3d, panel (b) and wavefield extrapolation using oneway.3d.kinematic, panel (c). Kinematically, the two images are equivalent and the main changes are related to amplitudes. Panels (b) and (c) have the same clip to highlight the point that only the amplitudes change but not the kinematics.

 

RCga2.kin.ps
Figure 11 The effect of neglecting the first order terms in Riemannian wavefield extrapolation. From left to right the velocity model with an overlay of the ray coordinate system (a), extrapolation with oneway.3d including the first order terms (b), and extrapolation with the simplified oneway.3d.kinematic (c). Panels (b) and (c) are gained equally, illustrating that the changes caused by neglecting the first-order terms affect the amplitudes and not the kinematics.

RCma2.fdm.ps shows a comparison between time domain acoustic finite-difference modeling (a), and Riemannian wavefield extrapolation (b) for a point source. The Marmousi velocity model is smoothed to avoid backscattered energy in panel (a) in order to facilitate a comparison with the one-way wavefield extrapolator in panel (b).

Despite being computed with a one-way extrapolator, the wavefield in panel (b) captures accurately all the important features of the reference wavefield depicted in panel (a), including triplications and amplitude variations. Some of the diffractions in panel (b) are not as well developed as their counterparts in panel (a) due to the limited angular accuracy of the $15^\circ$ approximation. Regardless of accuracy, the computed Riemannian wavefield could not be achieved with Cartesian-based downward continuation.

 

RCma2.fdm.ps
Figure 12 A comparison of wavefields computed by time-domain acoustic finite-difference modeling (a), and wavefields computed by Riemannian wavefield extrapolation (b).