Mixed-phase seismic wavelet estimation by analyzing the zeros of autocorrelation function in Z-domain

A seismic section which is very close to the true model of the earth is the goal of reflection seismology. Because raw seismic data are affected by the various ingredients such as geometry, noise and, etc., we are not able to use them for interpretation. Therefore, a processing flow must be applied on the raw seismic data to remove the effects of undesirable ingredients. Deconvolution is one of the most important stages in seismic data processing removed source signature from seismic trace to improve the temporal resolution. Inverse filtering and spiking deconvolution are usual methods in seismic deconvolution.
If the seismic source wavelet is known, it can be removed from seismic trace, easily. But in most cases, the source signature is unknown and there is only an approximation of the source wavelet autocorrelation function. The usual deconvolution methods work well when the seismic source is of minimum-phase wavelet. Often, the seismic source wavelet is mixed-phase, thus estimation of mixed-phase wavelet is important for deconvolution of seismic data. Moreover, the efficiency of the inversion of seismic data depends on the correlation of the real and synthetic seismic data near the well position. A good estimation of seismic source wavelet can improve the correlation of real and synthetic seismic traces.
In this paper, we estimate the mixed-phase seismic source wavelet by analyzing the zeros of estimated seismic source autocorrelation function. This method is very simple and efficient. We tested the efficiency of mentioned method on both real and synthetic seismic data. It can estimate all of seismic source wavelet type, but given the importance of mixed-phase seismic source wavelet estimation; this paper deals with this issue.
In the most seismic deconvolution methods, the reflection series of the earth is considered as a random series. With this assumption, we can estimate the autocorrelation of source wavelet from seismic trace. Autocorrelation of a signal is equivalent to the convolution of signal with reversed version of it. The reversing effect of a signal simply reciprocates its zeros. Thus, the zeros of autocorrelation function of a signal contain the zeros of original signal combined with zeros of reversed version of it. Let us call the zeros of reversed version of signal as "image" zeros. Because, the image zeros are the complex conjugate of the zeros of original signal, the autocorrelation function is a zero-phase series.
To estimate the original seismic source wavelet, we must find a way to distinguish between the original zeros and its images between the zeros of autocorrelation function. We use a second autocorrelation function calculated from the signal multiplied by exponential, decaying or expanding. Multiplying the original data with an expanding exponential will cause original zeros to move toward the origin along their radials. However, image zeros, being reciprocals of original zeros, will move in the opposite direction away from the origin. Thus, as we observe the movement of the original roots of the autocorrelation function with the expanding exponential multiplication, we see some of the roots move in the expected direction. Therefore, they are the seismic source wavelet roots. Now we can define a process by which the roots of the wavelet Z-transform can be determined uniquely. This will, in turn, define the wavelet phase function.
We tested the efficiency of the proposed algorithm for mixed-phase seismic source wavelet estimation on both synthetic and real seismic data. In the case of synthetic seismic data, first the algorithm was tested on the free noise trace. The estimated mixed-phase wavelet was very similar to that used in the generation of synthetic seismic trace. To investigate the algorithm sensitivity to noise, Gaussian random noise with different signal to noise ratios were added to synthetic trace and mixed-phase wavelet were estimated. We saw that the algorithm can estimate the source wavelet with good accuracy in the presence of random noise with SNR greater than 20 dB. For investigation of efficiency of algorithm on real seismic data, we selected a part of common-midpoint gather with 75 traces and 4 ms sampling interval. We applied the mentioned methods on various traces of CMP.
The theory and obtained results in both synthetic an real seismic data show that: (1) The method has a simple theory, (2) The uncomplicated theory of this method caused it to have low cost of computations and (3) The method has a good efficiency in existence of noise. Therefore, autocorrelation function zeros analysis method for seismic source wavelet estimation is a suitable an efficient algorithm.