RT volume estimation

In the proposed method, the predictive painting (Fomel, 2010), based on the prediction operator arising from PWD (Fomel, 2002), is utilized to generate the RT volume. In the predictive painting, after local slopes are estimated, an RT volume is computed by spreading time values of reference traces along dominant event slopes throughout the whole seismic volume.

Using linear operator notation, PWD operation is defined as

$$\mathbf{d} = \mathbf{D}{}\mathbf{s},$$ (5)

where $\mathbf{s} =[\mathbf{s}_1 \,\mathbf{s}_2 \,\cdots\, \mathbf{s}_N]^T$ is a collection of seismic traces and $\mathbf{d}$ denotes the residual from destruction. $\mathbf{D}$ is the destruction operator defined as

$$\mathbf{D}= \begin{bmatrix} \mathbf{I} & 0 & 0 & \cdots & 0 \\ -... ...s & \vdots \\ 0 & \cdots & 0 & -\mathbf{P}_{N-1,N} & \mathbf{I} \end{bmatrix},$$ (6)

where $\mathbf{I}$ denotes the identity matrix and $\mathbf{P}_{i,i+1}$ is an operator that predicts the $(i+1)$-th trace from the $i$-th trace according to the local slope of seismic event or seismic horizon.

By a simple recursion, a trace can be predicted utilizing a distant trace. For example, $\mathbf{P}_{r,k}$, predicting the $k$-th trace from the $r$-th trace, is defined as

$$\mathbf{P}_{r,k}=\mathbf{P}_{k-1,k}\mathbf{P}_{k-2,k-1}\cdots \mathbf{P}_{r+1,r+2}\mathbf{P}_{r,r+1}.$$ (7)

Then, the prediction of trace $\mathbf{s}_r$, given a reference trace $\mathbf{s}_r$ is accomplished by

$$\mathbf{s}_k^{\text{pre}}=\mathbf{P}_{r,k}\mathbf{s}_r.$$ (8)

After determining the prediction operators in equation 6 by PWD, the predictive painting algorithm recursively spreads information from a reference trace to the whole seismic data volume by following local slopes using equation 8. If the information spread by the predictive painting are the time coordinates, an RT volume is easily computed.

Figure 1 shows a synthetic image with several complex geologic structures. Local slopes corresponding to Figure 1 are shown in Figure 2, which are estimated by PWD. Figure 3a is the RT volume calculated by the predictive painting using local slopes in Figure 2. The reference trace of this RT volume is in the center of the image, which is marked as a vertical line in Figure 3a. This single reference RT volume is used to flatten the input seismic image (Figure 1) by unshifting each traces using the information it contains, since RT indicates how much a given trace is shifted with respect to the reference trace. The flattened image is shown in Figure 4a. Another RT volume with multiple references, shown in Figure 3b, is generated by selecting five reference traces and calculating the average of all estimated RT volumes. Figure 4b shows the corresponding flattened image. The comparison between two flattened seismic images (Figure 4a and 4b) indicates that the RT volume with only one reference trace does not necessarily have information about all seismic events since there are discontinuities such as faults and unconformities, while the RT volume computed with more reference traces contains more information.

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Figure 1. Synthetic seismic image from Claerbout (2000).

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Figure 2. Local slopes estimated from Figure 1.

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Figure 3. RT estimated by predictive painting from Figure 1 using (a) a single reference trace and (b) multiple reference traces.

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Figure 4. Flattened images using RT volumes from (a) Figure 3a and (b) Figure 3b.