An image-focusing semblance functional for velocity analysis

Measuring image coherency across structural dips

As an alternative to minimizing entropy, in this paper I propose to measure image focusing by maximizing coherency along both the structural-dip axes and the aperture/azimuth axes. The simultaneous use of dips and aperture angles is discussed in the next section. In this section, I show that measuring coherency along the structural dips does provide information on image focusing and I illustrate the concept by using the same two 2D synthetic data sets shown above. I will also demonstrate that maximizing coherency only along the structural dips may lead us to similar problems as the minimization of entropy.

To measure coherency along the structural dip $\alpha$, I first create the dip-decomposed prestack image ${\bf R}\left({\bf x},\gamma ,\alpha ,\rho\right)$ by residual prestack migration, and then I compute the following semblance functional:

$$S_\alpha \left({\bf x},\rho\right)= \frac{ \left[\sum_\alpha \sum... ... \left[\sum_\gamma {\bf R}\left({\bf x},\gamma ,\alpha ,\rho\right)\right]^2 },$$ (3)

where $N_{\alpha }$ is the number of dips to be included in the computation. Notice that, as for the varimax in equation 2, semblance along structural dips is computed after stacking over the aperture angle $\gamma$.

The determination of the dip summation range at each image location and for each value of the parameter $\rho$ is a practical problem of the proposed method. For the examples shown in this paper I determined the summation ranges for both $\alpha$ and $\gamma$ by applying an amplitude thresholding criterion based on both local and global amplitude maxima measured from the images. To improve the smoothness of the semblance spectra, I averaged the evaluation of equation 3, and of all the other semblance functionals introduced in this paper, over spatial windows extending along both the $z$ and $x$ directions.

Dips-4700-diffr-overn Figure 7. a) Dip-decomposed stack image of the diffractor-point window as a function of the dip angle extracted at $x=4,700$ meters and $\rho =1.025$), and (b) semblance $\rho$-spectrum computed at $x=4,700$ meters. [CR]

Dips-4700-trunc-overn Figure 8. a) Dip-decomposed stack image of the reflector-truncation window as a function of the dip angle extracted at $x=4,700$ meters and $\rho =1.04$), and (b) semblance $\rho$-spectrum computed at $x=4,700$ meters. [CR]

Figure 7a shows the dip-decomposed stack image of the diffractor-point window as a function of the dip angle $\alpha$ extracted from ${\bf R}\left({\bf x},\alpha ,\rho\right)$ at the point-diffractor's horizontal position and for $\rho =1.025$; that is, the correct value of $\rho$. The image is consistent as a function of dips, with the exception of an image artifact caused by interference with the image from the planar reflectors below the point diffractor. Figure 7b shows the semblance computed by applying equation 3 at the horizontal position of the point diffractor. It has a sharp peak for $\rho =1.025$. The dip-coherency analysis has thus the potential to provide accurate velocity information.

Figure 8a shows the dip-decomposed stack image of the reflector-truncation window as a function of the dip angle $\alpha$ at extracted from ${\bf R}\left({\bf x},\alpha ,\rho\right)$ at the horizontal position of the reflector's truncation for $\rho =1.04$; that is, the correct value of $\rho$. The dip-decomposed image is strongly peaked at $\alpha =-15^{o}$; that is the dip of the reflector. The event is weak away from $\alpha =-15^{o}$; and much weaker than the point-diffractor event shown in Figure 7a. Furthermore, polarity of the event switches at $\alpha =-15^{o}$. At the transition corresponding to the reflector dip, the image is actually rotated by 45 degrees. To compute a higher-quality semblance spectrum, I zeroed the image at $\alpha =-15^{o}$ and split the computation of the numerator in equation 3 between dips larger than 15 degrees and dips smaller than 15 degrees; that is I computed the following modified semblance functional:

$$S_{\bar{\alpha}}\left({\bf x},\rho\right)= \frac{ \left[ \sum_{\a... ...left[\sum_{\gamma }{\bf R}\left({\bf x},\gamma ,\alpha ,\rho\right)\right]^2 },$$ (4)

where $\bar{\alpha}$ is the structural dip of the truncated reflector. The need to identify a reflector truncation and to estimate the local dip of the reflector is potentially a practical problem with using dip coherency to extract velocity information from reflector's truncations.

The semblance spectrum shown in Figure 8b was computed by applying equation 4 with $\bar{\alpha}=-15^{o}$. The semblance peak is at the correct value of $\rho =1.04$ but it is much broader than the peak corresponding to the point diffractor shown in Figure 7b. As noted when comparing Figure 3a with Figure 4a, the velocity information provided by focusing analysis of reflectors' truncations seems to be more difficult to use than the one provided by point diffractors.

The computation of the dip spectra for the data set with sinusoidal reflector illustrates the limitations and potential dangers of relying on dip-only spectra when continuous reflectors have a strong curvature. Figures 9a and 9b show the image decomposed according to structural dips for the bottom of the syncline window for two different values of $\rho$: $\rho =.995$ for Figure 9a, and $\rho =1.06$ for Figure 9b (same values of $\rho$ as for Figure 5c and Figure 5b, respectively.) The image is flat as a function of the dip angle for the wrong value of $\rho$ and is frowning for the correct value of $\rho$. Consequently the dip spectrum shown in Figure 9c peaks at a low value of $\rho$ and would mislead velocity estimation.

The analysis of Figure 10 leads to similar conclusions. In this case the image is flat for a higher value of $\rho$ ( $\rho =1.105$) than the correct one ( $\rho =1.045$), for which the image is actually smiling. The semblance spectrum is also biased toward higher values of $\rho$.

Dips-4250-overn Figure 9. a) Dip-decomposed stack image of the bottom of the syncline window as a function of the dip angle extracted at $x=4,250$ meters and $\rho =.995$), (b) dip-decomposed stack image for $\rho =1.06$), and (c) semblance $\rho$-spectrum computed at $x=4,250$ meters. [CR]

Dips-4750-overn Figure 10. a) Dip-decomposed stack image of the top of the anticline window as a function of the dip angle extracted at $x=4,750$ meters and $\rho =1.105$), (b) dip-decomposed stack image for $\rho =1.045$), and (c) semblance $\rho$-spectrum computed at $x=4,750$ meters. [CR]

An image-focusing semblance functional for velocity analysis