RT formulation of prediction and update operators

RT contains the relationship between traces. From RT volume, it is straightforward to obtain the shift of one trace with respect to the reference. Therefore, the trace prediction using RT volumes is formulated as follows:

$$\mathbf{s}_k^{\text{pre}}(t)=\mathbf{s}_r(\mathbf{\tau}_{k,r}(t))$$ (9)

where $\mathbf{s}_k$ and $\mathbf{s}_r$ denote the $k$-th and the $r$-th trace, respectively. $\tau_{k,r}(t)$ is the RT value of the $k$-th trace with the $r$-th trace as the reference trace, indicating the shift of the $k$-th trace with respect to the $r$-th trace. According to equation 9, the prediction of the $k$-th trace from the $r$-th trace is easily accomplished by simply applying forward interpolation to the $r$-th trace using corresponding RT values. In the proposed method, we use equation 9 to define operators $\mathbf{S}$ in equations 3 and 4 to construct corresponding prediction and update operators. In this way, a trace is predicted from a distant trace directly by the RT attribute instead of the recursive computation used in the PWD-seislet transform. Also, because an accurate RT volume stores information about all horizons and structural discontinuities including faults and unconformities, predictions of traces around discontinuities using equation 9 are accurate. Therefore, a better delineation of faults and unconformities can be achieved.

In computing an RT volume, we need to first choose one or multiple reference traces. After computing the RT volume, the RT attribute between any two traces is easily obtained from one RT volume using:

$$\mathbf{\tau}_{k,r}(t)=\mathbf{\tau}^{-1}_{r,l}(\mathbf{\tau}_{k,l}(t)),$$ (10)

where $\mathbf{\tau}_{j,i}(t)$ represents the RT value of the $j$-th trace with the $i$-th trace as the reference trace, and $\mathbf{\tau}^{-1}_{r,l}(t)$ is the time warping (Burnett and Fomel, 2009) of $\tau_{r,l}(t)$. According to equation 10, given the shift information $\tau_{r,l}(t)$ and $\tau_{k,l}(t)$, the shift relationship between the $k$-th trace and the $r$-th trace can be obtained by applying an inverse interpolation to $\tau_{r,l}(t)$ using $\tau_{k,l}(t)$. Then, equation 9 is applied to implement the prediction of the $k$-th trace from the $r$-th trace. Therefore, only one RT volume is needed for the proposed implementation of the seislet transform.

Figure 5 shows an example of implementing equation 10 to compute an RT volume from another one with different reference traces. The RT volume in Figure 5a is obtained by the predictive painting with the 150th trace as the reference trace. This RT volume can be represented by $\mathbf{\tau}_{k,150}(t), k=1,2,\ldots,n$. Figure 5b is the time warping of this RT volume. We extract the 50th trace from Figure 5b, which is denoted by $\mathbf{\tau}^{-1}_{50,150}(t)$. Then, by implementing equation 10, i.e., inverse interpolating $\mathbf{\tau}^{-1}_{50,150}(t)$ using the whole RT volume ( $\mathbf{\tau}_{k,150}(t), k=1,2,\ldots,n$), we can get a new RT volume, as shown in Figure 5c ( $\mathbf{\tau}_{k,50}(t), k=1,2,\ldots,n$). Here, we use the 50th trace as the reference trace. To evaluate the effectiveness of equation 10, RT volumes from Figure 5c and 5d are used to flatten Figure 1. Flattened images are shown in Figure 6. Small differences between Figure 6a and 6b indicate that these two RT volumes (Figure 5c and 5d) contain similar information, which demonstrate that equation 10 can be used to get the relationship between any two traces from one single RT volume.

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Figure 5. (a) The RT volume obtained by the predictive painting and the reference trace is the 150th trace. (b) Time warping of (a). (c) RT volume obtained by equation 10. (d) The RT volume obtained by the predictive painting and the reference trace is the 50th trace.

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Figure 6. Flattened images using RT volumes from (a) Figure 5c and (b) Figure 5d.