Velocity picking and slicing

After the velocity continuation process has created a velocity cube in the prestack common-offset migration domain, we can pick the best focusing velocity from that cube. To automatize the velocity picking procedure, I have designed a simple algorithm. The algorithm based on solving the following regularized least-square system:  

$$\left\{\begin{array} {rcl} \bold{W}\,\bold{x} & \approx & ... ...n \bold{D} \bold{x} & \approx & \bold{0} \end{array}\right.\;.$$ (7)

Here $\bold{p}$ are blind maximum-semblance picks (possibly in a predefined fairway), $\bold{x}$ is the estimated velocity picks, $\bold{W}$ is the weighting operator with the weight corresponding to the semblance values at $\bold{p}$, $\bold{D}$ is a roughening operator, and $\epsilon$ is the scalar regularization parameter. The first least-square fitting goal in (7) states that the estimated velocity picks should match the measured picks where the semblance is high enough. The second fitting goal tries to find the smoothest velocity function possible. The least-square solution of problem (7) takes the form  

$$\bold{x} = \left(\bold{W}^2 + \epsilon^2\,\bold{D}^T\,\bold{D}\right)^{-1}\,\bold{W}^2\,\bold{p}\;,$$ (8)

where $\bold{D}^T$ denotes the adjoint operator. In the case of picking a one-dimensional velocity function from a single semblance panel, we can simplify the algorithm by choosing $\bold{D}$ to be the a convolution with the derivative filter (1,-1). It is easy to notice that in this case the inverted matrix in formula (8) has a tridiagonal structure and therefore can be easily inverted with a linear-time algorithm. The regularization parameter $\epsilon$controls the amount of smoothing of the estimated velocity function. Figure shows a velocity spectrum and two automatic picks for different values of $\epsilon$.

 

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Figure 4 Semblance panel (left) and automatic velocity picks for different values of the regularization parameter. Center: $\epsilon=0.01$, right: $\epsilon=0.1$. Higher values of $\epsilon$ lead to smoother velocities.

In the case of picking two- or three-dimensional velocity functions, one could generalize problem (7) by defining $\bold{D}$as a 2-D or 3-D roughening operator. I chose to use a more simplistic approach. I transform system (7) to the form  

$$\left\{\begin{array} {rcl} \bold{W}\,\bold{x} & \approx & ... ... \bold{x} & \approx & \lambda \bold{x_0} \end{array}\right.\;,$$ (9)

where $\bold{x}$ is still one-dimensional, and $\bold{x}_0$ is the estimate from the previous midpoint location. The scalar parameter $\lambda$ controls the amount of lateral continuity in the estimated velocity function. The least-square solution to system (9) takes the form:  

$$\bold{x} = \left(\bold{W}^2 + \epsilon^2\,\bold{D}^T\,\bo... ...\, \left(\bold{W}^2\,\bold{p} + \lambda^2 \bold{x_0}\right)\;,$$ (10)

where $\bold{I}$ denotes the identity matrix. Formula (10) also reduces to an efficient tridiagonal matrix inversion. Figure  shows a result of two-dimensional velocity picking after velocity continuation. I used values of $\epsilon=0.1$and $\lambda=0.1$. The first parameter controls the vertical smoothing of velocities, while the second parameter controls the amount of lateral continuity.

 

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Figure 5 Automatic picks of 2-D RMS velocity after velocity continuation. The contour spacing is 0.1 km/s, starting from 1.5 km/s.

Figure shows the final result of velocity continuation: an image, obtained by slicing through the velocity cube with the picked RMS velocity. Different parts of the image have been properly positioned and focused by the velocity continuation process. One way to further improve the image quality is hybrid migration : demigration to zero-offset, followed by post-stack depth migration Kim et al. (1997). This step requires constructing an interval velocity model from the picked RMS velocities.

 

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Figure 6 Final result of velocity continuation: seismic image, obtained by slicing through the velocity cube.

Without repeating the details of the procedure, Figures  and show picked RMS velocities and the velocity continuation image for the Blake Outer Ridge data, shown in many other papers in this report.

 

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Figure 7 Blake Outer Ridge data. Automatic picks of 2-D RMS velocity after velocity continuation. The contour spacing is 0.01 km/s, starting from 1.5 km/s.

 

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Figure 8 Blake Outer Ridge data. Final result of velocity continuation: seismic image, obtained by slicing through the velocity cube.