An approximation of the inverse Ricker wavelet as an initial guess for bidirectional deconvolution

Next: Examples Up: Approximation of the inverse Previous: Separation

## Avoiding the singularity

In the Z domain, the half-Ricker wavelet has two zeros, which makes it impossible to directly invert. Therefore, we modify the formula with two new factors to avoid errors caused by dividing by zero:

 (11)

For simplicity, we use the same value for and , .

Figures 3(a) and 3(b) show the fourth-order Ricker wavelet in the time and frequency domains with different values.

Figures 4(a) and 4(b) show the fourth-order half-Ricker wavelet in the time and frequency domains with different values.

ricker-with-rho,ricker-with-rho-freq
Figure 3.
The fourth-order Ricker wavelet with different values: (a) in the time domain; (b) in the frequency domain.

half-ricker-with-rho,half-ricker-with-rho-freq
Figure 4.
The fourth-order half-Ricker wavelet with different values: (a) in the time domain; (b) in the frequency domain.

Since we have and in the Z domain ( and ), we have two methods to recover time-domain filter coefficients. The first one is to use to get frequency domain filters and then transfer them into the time domain by an inverse Fourier transform. Another method is to use a Taylor expansion to get time-domain filter coefficients. In theory, the two methods should be the same. For less accuracy problem, we always use the frequency method in this paper.

Finally we obtain the approximate inverse Ricker wavelet. Figures 5(a) and 5(b) show the inverse fourth-order causal half-Ricker wavelet ( ) in the time and frequency domains with different values. The inverse anti-causal half-Ricker wavelet ( ) can be generated by reversing along the time axis. and will be our initial guesses for bidirectional deconvolution.

freq-domain-a,freq-domain-a-freq
Figure 5.
The inverse fourth-order causal half-Ricker wavelet ( ) with different values: (a) in the time domains ; (b) in the frequency domain.

To test how good our approximation of the inverse Ricker wavelet is, we convolve our fourth-order approximate Ricker wavelet (with no factor) with our inverse Ricker wavelet (equation 12).

 test resultRicker (12)

If our inversion is good, the output should be a spike in the time domain and a flat spectrum in the frequency domain.

Figures 6(a) and 6(b) show the test result in the time domain and the frequency domain with different values. We see for we get a good result. In particular, the frequency spectra are flatten as increases.

freq-domain-result,freq-domain-result-freq
Figure 6.
The test results with different values: (a) in the time domain; (b) in the frequency domain.

Large value can yield even better results. For large , the frequency spectra are almost flat, except for two zero-value points (at 0 frequency and the Nyquist frequency). Figures 7(a) and 7(b) show these results.

freq-domain-9-99-result,freq-domain-9-99-result-freq
Figure 7.
The test results with large values: (a) in the time domain; (b) in the frequency domain.

However, for our goal (to use the approximate inverse Ricker wavelet as an initial guess for bidirectional deconvolution), we do not need such large values. With large values, the filter will be long and can easily to lead to an unstable result in our deconvolution scheme.

We now have an approximate inverse of the Ricker wavelet separated into two symmetric parts. This matches our requirements for initial guesses for filters and . In the next section we will test the approximation using synthetic and field data.

 An approximation of the inverse Ricker wavelet as an initial guess for bidirectional deconvolution

Next: Examples Up: Approximation of the inverse Previous: Separation

2011-05-24