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

Zhang and Claerbout (2010) proposed a new method for blind deconvolution to overcome the minimum-phase assumption, called ``bidirectional deconvolution''. A seismic data trace can be represented by a convolution of a wavelet with a reflectivity series,

where denotes the seismic data trace, denotes the reflectivity series, and denotes a wavelet.

In conventional blind deconvolution, we assume that
is a minimum-phase wavelet. However this assumption is not applicable for field seismic data. If
denotes a mixed-phase wavelet, it can be represented by a convolution of two parts:
, where
is a minimum-phase wavelet and
is a reversed minimum-phase wavelet; hence
itself is a maximum-phase wavelet. (The superscript r denotes reversed in time.) Thus equation 1 can be rewritten as

If we know the inverse filters and for and , respectively, to satisfy

we can recover the reflectivity series:

where filter is the inverse signal of and filter is the inverse signal of

Now we can use nonlinear inversion to solve this blind deconvolution problem for a mixed-phase wavelet by solving the two equations below alternately:

where both and are minimum-phase signals.

Shen et al. (2011) proposed another method to solve equation 4. Instead of solving for and alternately, they solve and simultaneously. Using this new approach allows us to estimate results with similar waveforms for and , which is a natural characteristic for data with a Ricker-like wavelet. In addition, this new method is faster than previous one. Hence we will use this to perform bidirectional deconvolution.

Since bidirectional deconvolution is a nonlinear problem, it requires that the starting model be close to the true one, and it is highly sensitive to the initial guess for both and . Shen et al. (2011) uses a simple one-spike impulse function for both filters. However, sometimes the true model does not resemble an impulse function. Therefore, we attempt to find a better initial guess.

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

2011-05-24