Dealiased interpolation via one-step local slope estimation based on velocity-slope transformation

The estimation of the local slope in highly under-sampled data is a challenging problem. Instead of estimating the local slope by using the plane-wave destruction (PWD) algorithm Fomel (2002) directly, we first calculate the NMO velocity by using simple NMO-correction based velocity analysis and then transform the NMO velocity to local slope Liu and Liu (2013) by using

$$p(t,x)=\frac{x}{t(x)v_n^2(t_0,x)},$$ (4)

where $t_0$ is the zero-offset traveltime, $t(x)$ is the traveltime recorded at offset $x$, $v_n(t_0,x)$ is the NMO velocity, and $p(t,x)=dt/dx$ is the local slope. It is salient that $p(t,x)>0$ when $x>0$, as shown by the example in Fig. 1. The detailed velocity-slope transformation was introduced in Liu and Liu (2013). It should be mentioned that such velocity-slope transformation was used previously by Liu et al. (2015b) and Gan et al. (2016b) for signal and noise separation. We use the velocity-slope transformation here for iterative interpolation based on seislet thresholding. We admit here that since equation 4 is derived from prestack data, the iterative interpolation approach based on the velocity-slope transformation cannot be applied to poststack seismic data processing, where the data structure can be arbitrarily complex and local slope can vary rapidly. Although prestack data can also be complex, we limit our algorithm in a simpler case, where we assume that the prestack data structure is relatively simple and do not have negative slope. As far as we know, this assumption is satisfied in most situations.

The seislet transform has found successful applications in noise attenuation Fomel and Liu (2010); Chen et al. (2014b). However, the successful application of the seislet transform in iterative interpolation, especially in the industry, is not reported often. One of the drawbacks that impede the wide application of seislet based interpolation is the efficiency. The seislet transform itself does not slow down the efficiency too much. However, the slope estimation that is required by the seislet transform is much slower. The efficiency of seislet transform is about 2-4 times slower than the fast Fourier transform, and is about 4-8 times slower than the fast wavelet transform Fomel and Liu (2010). In order to accelerate the process, the slope estimation is commonly estimated every several iterations.

In Gan et al. (2015), the slope estimation is iterated every 5 iterations. Even though, the computational cost is still much heavier than the widely used Fourier transform. In this letter, the one-step slope estimation from velocity-slope transformation (shown in equation 4) can greatly improve the efficiency by reducing numerous cost in iterative slope estimation. According to the performance of the synthetic example in the letter, the more accurate slope can even make the finally reconstructed data more accurate. Thus, the utilization of velocity-slope transformation in seislet-based interpolation could be of a huge influence on promoting the wide application of seislet based interpolation approach in the industry.