DSR artifacts

A straightforward implementation of the DSR equation produces significant artifacts. Several researchers pointed out these artifacts as an inhibiting factor in using the DSR for separate constant-offset sections. A commonly used argument is that the artifacts are due to Fourier domain wraparound along the offset axis. Figure a shows the output of the DSR migration applied to a data cube (t,y,h) containing a single spike in a constant-offset section with an half-offset h₀=190m. The shallower ellipse is an artifact. Compare this with the output of the improved DSR migration in Figure b. Because of the stacking in offset that is implicitly done in the DSR equation, the artifacts are attenuated. However, when each constant-offset section is taken separately, the artifacts are a disturbing presence.

 

Fig1
Figure 1 The output of DSR in offset-midpoint migration. The input cube contains a single spike for the constant-offset section h=190m.
a. Classic DSR migration. The shallower ellipse is an artifact.
b. DSR migration without offset artifacts.

The source of artifacts is equation (4), which limits the interval of existence for the variable k_h. Normally the offset-wavenumber k_h is evenly sampled between the values $(-{\pi \over dh},{\pi \over dh})$ in the DSR offset-midpoint equation, necessary for the FFT along the offset axis. However, the requirement for the square-root expressions to be real limits the available values for k_h. Let's examine equation (4) again:

$${2 \over v}\mid \omega \mid - \mid k_y \mid \; \geq \; \mid k_h \mid .$$

When only a subset of the values of k_h are used from the whole set of discrete values, artifacts are introduced in the phase expression. Consider the form of the DSR equation presented in equation (8):

$$p(t=0,k_y,h,z)= {\int d\omega \; p(\omega,k_y,h) \int dk_h \; e^{ik_z(\omega,k_y,k_h)z-ik_h h}}, \\$$

where the integral in k_h has the integration limits  

$$k_h \in (-{2 \over v}\mid \omega \mid + \mid k_y \mid , {2 \over v}\mid \omega \mid - \mid k_y \mid),$$ (9)

the region of integration can be much smaller than the domain in which k_h is sampled. As a result only several discrete values of k_h are used to evaluate the integral. The missing offset-wavenumbers introduce artifacts, as the interval of existence of the phase is not sufficiently sampled. The correct implementation should subsample the k_h variable between the integration limits and not between some predetermined limits.

Equation (9) is used to migrate a single impulse in a constant-offset section with h=190m, using constant sampling in k_h for all the values $\omega,k_y$. In Figure the only difference between the two migration programs is the offset spacing. Bigger sampling in offset implies smaller sampling in offset-wavenumber. As a result more offset-wavenumber samples are used to compute the phase and reduce the artifacts. The ideal algorithm will use all the available samples, dividing the integration limits in equation (9) by the total number of offsets (assumed to be equal to the number of offset-wavenumbers). This will not preclude the use of FFT to transform the phase from offset-wavenumber domain to offset domain, because for each pair of values $\omega,k_y$, the variable k_h is evenly sampled in the domain of definition.

 

Fig2
Figure 2 Migration of a single constant-offset section using the separable DSR equation. In the two sections I use the same parameters, excepting the k_h spacing. In the right figure dk_h is smaller.
a. Constant-offset phase-shift migration, h=10m.
b. Constant-offset phase-shift migration, h=20m.

In order to use the different results of DSR migration algorithms I create a prestack model consisting of 64 constant-offset sections over three vertical diffractors in a depth variable velocity medium. The velocity model is shown in Figure a while an example of the zero-offset travel-time map is shown in Figure b. Figure a shows a zero-offset common midpoint (CMP) section, while Figure b shows the CMP section corresponding to the longest offset (h=630m).

The prestack model was migrated using several variations of the DSR migration algorithm. Figures a and b compare the results of DSR migration via the classic algorithm implementing equation (1) and the results of DSR migration for separate offsets implementing equation (8) using a FFT to transform the phase. In both cases k_h was constantly sampled. To obtain Figure b all the separate constant-offset sections were stacked in the end. The two results are identical as is expected.

Figure a represents the results of DSR migration by implementing equation (8) directly. The integral in k_h is calculated numerically for each offset. The algorithm is slower than the one evaluating the integral in k_h via FFT, but it does not require the migration of all the separate constant-offset sections at the same time. Figure b represents the results of DSR migration with different sampling in k_h for each $(\omega,k_y)$ pair. Because stacking attenuates the artifacts the difference between the two algorithms is not as visible in the migrated CMP section.

Figure compares the migrated results of the same CMP, for different offsets. The figure was obtained by slicing vertically the migrated data set. The slice passes through the location of the three diffraction events shown in Figure . A correctly migrated image should show the same events for all the offsets. Figure a displays the diagonal artifacts that appear using constant k_h sampling. The diagonal stripes create the ghost ellipse in Figure a. Figure b is obtained using separate sampling of k_h for each $(\omega,k_y)$ pair, and all artifacts are virtually eliminated.

 

Fig3
Figure 3 Depth variable velocity model.
a. Interval velocity used.
b. Zero-offset traveltime map used for modeling.

 

Fig4
Figure 4 Constant-offset sections.
a. First constant-offset section, h=0m.
b. Last constant-offset section, h=630m.

 

Fig5
Figure 5 Migration of the whole prestack cube, using implicit stacking during downward continuation and stacking of the independently migrated constant-offset sections. The two DSR algorithms produce identical results.
a. Classic DSR in $\omega,k_y,k_h$.
b. DSR for individual constant-offset sections, stacked.

 

Fig6
Figure 6 Comparison of constant-offset migration using constant k_h sampling and separate k_h spacings. Stacking greatly attenuates the artifacts in the constant-offset sections.
a. Constant sampling in k_h, stacked offsets.
b. Separate sampling in k_h, stacked offsets.

 

Fig7
Figure 7 Common midpoint section, slicing vertically the three diffraction events. The artifacts in the DSR migration using constant k_h sampling are nonexistent in the DSR migration using separate sampling in k_h for each $\omega,k_y$ pair.
a. Constant sampling in k_h, stacked offsets.
b. Separate sampling in k_h, stacked offsets.