REGRIDDING IRREGULARLY SAMPLED TRACES

For regridding irregularly sampled traces, the dip picking stage is the same as the case of regularly sampled traces because slant stack works in both cases. We cannot use the simple Z-transform technique, however, to simulate the prediction error filter for irregularly sampled case.

The filtering operation in a continuous signal is a convolution as follows:

$$y(t) = \int f(\tau)x(t - \tau)d\tau$$ (1)

where y(t) is an filtered output, x(t) is an input and f(t) is a filter. In the case of a discrete signal, the integral operator can be approximated by summation operator as follows

$$y(t_i) = \sum_j f(\tau_j)x(t_i - \tau_j)\Delta\tau_j.$$ (2)

If discrete points are uniformly sampled, $d\tau_j$ is always same so that it can be dropped. But for irregularly sampled signal, the $d\tau_j$ must be considered in the convolution formula because it will vary with sampling intervals. If we write a filtering operation as a form of matrix multiplication, the matrix which corresponds to this operation will have a structure as shown below.

$$\left[ \begin{array} {c} y_1\\ y_2\\ y_3\\ y_4\\ y_5\\ \end{... ...gin{array} {c} x_1\\ x_2\\ x_3\\ x_4\\ x_5\\ \end{array}\right]$$

where $\Delta\tau_i$ represent offset interval between i th trace and i+1 th trace and $\Delta\tau_r$ represent the interval of regularly sampled traces which is used to generate the prediction error filter. The operation described above look like a nonstationary filtering operation. But, the filter itself does not change and the change during the convolution is the sampling point on the filter. Each row of the matrix represents the same filter but the different sampling points as explained in Figure . Figure schematically shows this filtering operation. At each position of the filter shifted to convolve with the irregularly sampled data, all coefficients of the filter should be recalculated from the spectrum of the filter. Here, I make a spectrum by FFT for the prediction error filter simulated as in the first section, and sample at points where the filter meets the irregularly sampled data points during the convolution. Figure shows an example of the real and imaginary parts of the matrix for filtering on irregularly sampled grid for a frequency.

 

irrgrim
Figure 6 The convolution of an irregularly sampled data with a filter which is continuous.

 

nsflt
Figure 7 (a) Real component of the filter for the frequency of an half of the Nyquist. (b) Imaginary component of the filter for the frequency of an half of the Nyquist.