Seismic data interpolation without iteration using $t$ -$x$ -$y$ streaming prediction filter with varying smoothness

discussion

The extension of SPF to higher dimensions is straightforward, hence more space constrains have to be applied to the algorithm. For three spatial dimensions, we use spatial axes $x$ , $y$ and $z$ , and the new filter coefficients by changing equation 3 show as given below:

$$\begin{aligned}\widehat{\mathbf{a}}(t,x,y,z)=&\arg\min_{\mathbf... ...f{E}_n\mathbf{\bar{a}_n}(t,x,y,z)\parallel_{2}^{2}. \end{aligned}$$

The least-squares solution of equation 14 is

$$\mathbf{a}(t,x,y,z)=[\mathbf{d}(t,x,y,z)\mathbf{d}(t,x,y,z)^{T}+\... ...mathbf{I}]^{-1} [d(t,x,y,z)\mathbf{d}(t,x,y,z)+\xi^2\mathbf{\bar{a}}(t,x,y,z)],$$ (15)

where

$$\begin{aligned}\mathbf{\bar{a}}(t,x,y,z)&=\frac{\xi_t^2\mathbf{... ...\xi^2},\\ \xi^2&=\xi_t^2+\xi_x^2+\xi_y^2+\xi_z^2.\\ \end{aligned}$$

Thus, the difference is the increased storage of the $z$ -axis filter coefficients.

For low-amplitude events in field data, an AGC could be applied to the data before filter estimation to help ensure that low amplitude events are given equal attention in the SPF estimation. In practice, the prestack 3D traces could be described in 5D space ($x_s$ , $y_s$ , $x_r$ , $y_r$ , $t$ ), where ($x_s$ , $y_s$ ) and ($x_r$ , $y_r$ ) are the source and receiver coordinates. We can extract 3D seismic gathers from 5D space and use $t$ -$x$ -$y$ SPF to interpolate the prestack 3D traces. Considering the spatial similarity of seismic events, it is recommended to implement the proposed method in the cmp-cmpline-offset-azimuth domain. Theoretically, any two spatial dimensions can be extracted from four spatial dimensions for interpolation; however, the accuracy of interpolated result depends on the number of missing traces and complexity in the 3D seismic gathers.

Seismic data interpolation without iteration using $t$ -$x$ -$y$ streaming prediction filter with varying smoothness