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Attenuating tail artifacts

Artifacts in the path-summation images appear due to the absence of weights, which describe probability of the image migrated with a certain velocity and taper out "unlikely" solutions (Landa et al., 2006; Schleicher and Costa, 2009; Landa, 2009). We propose to use analytical Gaussian weighting of images being stacked in path-summation migration, in which velocities in the vicinity of the most likely one are assigned a higher weight. An alternative approach based on the adaptive subtraction method is described in our previous work (Merzlikin and Fomel, 2015).

The following analytical weighting term can be added to the velocity continuation phase-shift:

$\displaystyle \tilde{P}(\Omega,k,v,\beta,v_{bias}) = \hat{P}_0 (\Omega,k)\,e^{-\frac{i k^2 v^2 }{16\Omega} - \beta(v_{bias} - v)^2}\;,$ (7)

and corresponding integral expression describing path-summation migration takes the following form:

$\displaystyle \hat{I_{GPI}}(\Omega,k,v_a,v_b,\beta,v_{bias})$ $\displaystyle = \int^{v_b}_{v_a} \hat{P}_0(k,\Omega)\ e^{-\frac{i k^2 v^2}{16\Omega} - \beta(v_{bias} - v)^2} dv\ $    
  $\displaystyle = \hat{P}_0(\Omega,k)\ \int^{v_b}_{v_a} e^{-\frac{i k^2 v^2}{16\Omega} - \beta(v_{bias} - v)^2} dv$    
  $\displaystyle = \hat{P}_0(\Omega,k) \cdot F_{GPI}(\Omega,k,v_a,v_b,\beta,v_{bias}),$ (8)

where filter $ F_{GPI}(\Omega,k,v_a,v_b,\beta,v_{bias})$ is evaluated analytically as follows:

\begin{multline}
F_{GPI}(\Omega,k,v_a,v_b,\beta,v_{bias}) = \\
\frac{\sqrt{\pi}...
... v_{bias}}{\gamma(\Omega,k,\beta)} \big) \bigg\vert _{v_a}^{v_b}.
\end{multline}

Here, $ \gamma(\Omega,k,\beta)=i\sqrt{ik^2/16\Omega + \beta}$ , $ \beta$ is a parameter controlling contribution of each velocity to the final image and $ v_{bias}$ is a velocity bias. By choosing a constant value of $ v_{bias}$ for the whole section between $ v_a$ and $ v_b$ we can enhance the apexes of summed hyperbolas and decrease tails’ contribution to the final image. Since in a section, velocity varies and $ v_{bias}$ is constant, over- or undermigrated diffraction curves may get emphasized at locations where the true velocity is higher or lower than $ v_{bias}$ respectively. This is a possible drawback of the Gaussian weighting approach.

We analyze a response of a single diffraction hyperbola modelled with 1.5 [km/s] velocity (Figure 1a) to both path-summation and Gaussian weighted path-summation migrations. Synthetic model spectrum in $ \Omega -k$ domain is shown in Figure 1b. Path-summation migration filter magnitude as well as its phase spectrum are shown in Figures 2a and 2c. It can be interpreted as a particular form of dip filtering. Wave-numbers in the vicinity of zero appear to be preserved and either remain stationary or exhibit small phaze-shift values. This area corresponds to the diffraction hyperbola apex remaining stationary under time-migration velocity perturbation. Non-zero wave-number events have lower corresponding filter coefficients, which taper out to the spectrum boundaries. Figure 3c shows the result of path-summation migration applied to diffraction hyperbola. Several lobes with non-zero wavenumber values appear to be noticeable. Corresponding phase-shift values (Figure 2c) are not zero and lead to displacement of the corresponding events on the path-summation migration image (Figure 3c). These events are associated with tail artifacts.

Gaussian weighting scheme spectrum magnitude and phase are shown in Figures 2b and 2d. The result of corresponding path-summation migration in $ \Omega -k$ domain is shown in Figure 3b. Gaussian weighting scheme tapers non-stationary lobes and, therefore, suppresses the tails. Figure 3d corresponds to the Gaussian weighting scheme for path-summation migration. The tail artifacts are successfully eliminated.

Proposed expressions for direct and analytical path-integral evaluation in one step allow us to drastically reduce computational costs. Following sections illustrate the applicability and the effectiveness of the proposed techniques.

a-data a-fft-mag
a-data,a-fft-mag
Figure 1.
(a) Diffraction hyperbola modelled with 1.5 km/s velocity; (b) amplitude spectrum of (a) in $ \Omega -k$ domain.
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a-erfi-fft-mag a-gaussian-erfi-mag a-erfi-fft-phase a-gaussian-erfi-phase
a-erfi-fft-mag,a-gaussian-erfi-mag,a-erfi-fft-phase,a-gaussian-erfi-phase
Figure 2.
Path-summation migration filter (formula 6): (a) amplitude spectrum; (c) phase-spectrum; Gaussian weighted path-summation migration filter (formula 8): (b) amplitude spectrum; (d) phase spectrum.
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a-pi-fft-mag a-gpi-fft-mag a-path-integral a-pi-gaussian
a-pi-fft-mag,a-gpi-fft-mag,a-path-integral,a-pi-gaussian
Figure 3.
Unweighted path-summation migration of a diffraction hyperbola (v = 1.5 km/s) (formulae (4) and (6)): (a) $ \Omega -k$ amplitude spectrum; (c) image; path-summation migration with Gaussian weighting (formulae 8 and 9): (b) $ \Omega -k$ amplitude spectrum; (d) image.
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Next: Field data examples Up: Method Previous: Method

2017-04-20