An image-focusing semblance functional for velocity analysis

Pitfalls of Minimum Entropy functional

Minimizing the image entropy measured on moving spatial windows is a well-known approach to measuring image focusing. The varimax norm (Wiggins, 1985) is commonly used to measure the "entropy" of an image instead of the conventional entropy functional. The varimax norm is cheaper to evaluate than the conventional entropy functional because it does not require the evaluation of a logarithmic function. A peak in the varimax corresponds to a point of minimum entropy. I computed the varimax for local windows extracted from image ensembles computed by applying residual prestack migration to an initial prestack migration performed with a low velocity.

I define ${\bf R}\left({\bf x},\gamma ,\rho\right)$ as an ensemble of prestack images obtained by residual prestack migration where the parameter $\rho$ is the ratio between the new migration velocity and the migration velocity used for the initial migration. The aperture angle is $\gamma$ and ${\bf x}=\left\{z,x\right\}$ is the vector of spatial coordinates, where $z$ is depth and $x$ is the horizontal location.

I define the image window $\bar{{\bf x}}$ as:

$$\bar{{\bf x}}: \left\{\bar{z}-\Delta z \leq z \leq \bar{z}+ \Delta z, \bar{x}-\Delta x \leq x \leq \bar{x}+ \Delta x\right\},$$ (1)

where $2\Delta z$ is the height of the window and $2\Delta x$ is its width, and $\bar{z}$ and $\bar{x}$ are the coordinates of the window's center.

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Figure 3. (a) Graph of the varimax norm as a function of $\rho$, (b) stacked section for $\rho =1$, (c) stacked section for $\rho =1.025$, and (d) angle-domain common image gather for $\rho =1.025$ at $x=4,700$ meters. [CR]

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Figure 4. (a) Graph of the varimax norm as a function of $\rho$, (b) stacked section for $\rho =1$, (c) stacked section for $\rho =1.04$, and (d) angle-domain common image gather for $\rho =1.04$ at $x=4,700$ meters. [CR]

The varimax norm computed for $\bar{{\bf x}}$ is defined as:

$$E_{{\bf x}}\left(\rho\right)= \frac{ N_{\bar{{\bf x}}}\sum_{\bar{... ...left[\sum_\gamma {\bf R}\left({\bf x},\gamma ,\rho\right)\right]^2\right\}^2 },$$ (2)

where $\sum_{{\bf x}}$ signifies summation over all the image points in $\bar{{\bf x}}$ and $N_{\bar{{\bf x}}}$ is the number of points in $\bar{{\bf x}}$. Notice that the varimax in equation 2 includes stacking over the aperture angle $\gamma$.

For the first data set (Figure 1,) I computed the varimax in equation 2 as a function of $\rho$ in two windows: the first centered on the point diffractor, the second centered on the reflector truncation. Figure 3 shows the following four plots for the point-diffractor window: a) the graph of the varimax norm as a function of $\rho$, b) the stacked section for $\rho =1$; that is, the window of the initial undermigrated section in Figure 1b, c) the stacked section for $\rho =1.025$; that is; for the peak of the curve shown in Figure 3a, and d) the angle-domain common image gather for the same value of $\rho =1.025$ and extracted from the prestack cube at the horizontal location of the point diffractor.

Figure 4 shows analogous plots as the ones shown in the previous figure, but for the reflector-truncation window. Figure 4a shows the graph of the varimax as a function of $\rho$. Figure 4b shows the stacked section for $\rho =1$. Figure 4c shows the stacked section for $\rho =1.04$; that is, for the peak of the curve shown in Figure 4a, whereas Figure 4d shows the angle-domain common image gather for the same value of $\rho =1.04$ and extracted from the prestack cube at the horizontal location of the reflector's truncation.

For both windows, the maximum of the varimax norm corresponds to the value of $\rho$ that best focuses the prestack image and best flattens the angle-domain common image gathers. The semblance peak for the point diffractor is sharper than for the reflector truncation, suggesting that point diffractors provide higher-resolution information on migration velocity than reflectors' truncations.

I also computed the varimax in equation 2 as a function of $\rho$ in two windows of the prestack migrated image corresponding to the sinusoidal reflector (Figure 2.) The first window is centered on the bottom of the syncline and the second centered on the top of the anticline. Figure 5 and Figure 6 show: a) graphs of the varimax as function of $\rho$, b) the stacked sections corresponding the correct values of $\rho$ ($\rho =1.06$ for Figure 5b and $\rho =1.045$ for Figure 6b,) c) the stacked sections corresponding the the varimax peaks ($\rho =.995$ for Figure 5c and $\rho =1.105$ for Figure 6c,) and d) the angle-domain common image gathers extracted at the very bottom of the syncline in Figure 5d and top of the anticline in Figure 6d.

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Figure 5. (a) Graph of the varimax norm as a function of $\rho$, (b) stacked section for $\rho =1.06$, (c) stacked section for $\rho =.995$, and (d) angle-domain common image gather for $\rho =.995$ at $x=4,250$ meters. [CR]

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Figure 6. (a) Graph of the varimax norm as a function of $\rho$, (b) stacked section for $\rho =1.045$, (c) stacked section for $\rho =1.105$, and (d) angle-domain common image gather for $\rho =1.105$ at $x=4,750$ meters. [CR]

For the first window, the peak of the varimax corresponds to a value of $\rho$ that is too low, whereas for the second window the peak of the varimax corresponds to a value of $\rho$ that is too high. The cause of these errors is that the image of concave reflectors can be made more spiky (i.e. lower entropy) by undermigration than by migration with the correct velocity. Similarly, the image of a convex reflector can be made more spiky by overmigration than by migration with the correct velocity. If the varimax norm were used to determine the residual-migration parameter $\rho$ it would lead to images with wrong structure and non-flat common-image gathers. However, the secondary peaks of the varimax norm in both Figure 5 and Figure 6 are approximately located at the correct value of $\rho$. This secondary peaks indicate that there is potentially useful focusing information in the images, but to be practically useful we must devise a method that is not biased by the reflectors' curvature.

An image-focusing semblance functional for velocity analysis