filter length, cost, and accuracy

For 3D tilted TI media, the explicit correction operator is a 2D convolution operator. For a medium with lateral variation, a table of the convolution coefficients c_nx,ny are calculated before the wavefield extrapolation. Long filters can extrapolate high-angle energy accurately. However, it is too expensive to run a 2D convolution filter as long as 19 points in both the x and y directions. Furthermore, it is not practical to store such a big table in the memory. By the weighted least-square method, we can shorten the filter length at the price of losing accuracy for the high-angle energy.

We test a 2D example to check how the length of a filter affects its accuracy. The medium is homogeneous, in which the P-wave velocity in the direction parallel to the symmetry axis is 2000m/s, $\varepsilon=0.4$, $\delta=0.2$ and $\varphi=\frac{\pi}{6}$. The frequency is 45.0 Hz.

Let k_x^max be the beginning wavenumber for the evanescent energy. We assign a weight of 1 to the wavenumbers smaller than k_x^max and a weight of 0.001 to the wavenumbers bigger than k_x^max. In Figure , the phase for the even part of the 19-point filter example is very close to the true operator. In this model, the beginning wavenumber for the evanescent energy k_x^max is 0.15. In Figure , the phase curve for the even part of the 5-point filter oscillates around the true operator. The 5-point filter is not accurate even for the low wavenumber energy. If our aim is to guarantee the accuracy of the low-angle (low-wavenumber) energy, we can assign big weights to the low-angle energy but small weights to the high-angle energy. We can also smooth the amplitude and phase of the high angle-energy. Now we assign a weight of 1 to the wavenumbers smaller than $\frac{5}{6}k_x^{max}$ and a weight of 0.001 to the wavenumbers bigger than $\frac{5}{6}k_x^{max}$.In this model, $\frac{5}{6}k_x^{max}$ is 0.125. Figure shows the phase curve of the 5-point filter after we change the weighting policy. The new 5-point filter is very close to the true operator at the low wavenumbers (smaller than 0.12) but has a big error at the high wavenumbers.

 

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Figure 2 Comparison between the phase curves for the even part of the 19-point filter and the true operator. The continuous curve is the phase of the true operator and the dashed line is the phase of the 19-point filter.

 

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Figure 3 Comparison between the phase curves for the even part of the 5-point filter and the true operator. The continuous curve is the phase of the true operator and the dashed line is the phase of the 5-point filter.

 

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Figure 4 Comparison between the phase curves for the even part of the new 5-point filter and the true operator. The continuous curve is the phase of the true operator and the dashed line is the phase of the new 5-point filter.

Though we lose the accuracy of high-angle energy when shorten the filter when we shorten the filter, we greatly improve the efficiency of our algorithm. If we use the 5-point filter in both inline and crossline directions in 3D wavefield extrapolation, the correction operator is a $9\times9$ 2D filter, while it is a $37\times37$ 2D filter if we use the 19-point filter. Therefore using the 5-pointer filter, computation cost for the convolution in 3D wavefield extrapolation is about $\frac{1}{16}$ the cost using the 19-pointer filter. Furthermore, when the media is not homogeneous, searching for the coefficients of a filter in the coefficient table plays an important role in 3D. The total size of the coefficient table of the 5-point filter is about $\frac{1}{16}$ the size of the 19-point filter in 3D. If we build the table with 100 discrete $\omega/V_{P0}$s, 10 discrete $\varepsilon$s and 10 discrete $\delta$s, the size of table is about 8 Megabyte for the 5-point filter and is about 128 Megabyte for the 19-point filter. The speed of searching in a 8 Megabyte is much faster than that in a 128 Megabyte table. By shortening the filter, we can greatly reduce the cost for the explicit correction operator in 3D wavefield extrapolation.

We lose the accuracy of high-angle energy when we shorten the length of the filter. But we can apply plane-wave decomposition and tilted coordinates Shan and Biondi (2004a) to make the wavefield-extrapolation direction close to the direction of wave propagation. By doing this, we can get good accuracy for the high-angle energy even with a less accurate operator.