Seismic attenuation coefficient estimation using smoothed pseudo Wigner-Ville distribution

Elastic energy of seismic wave is lost through propagation into the earth. Various factors affect seismic energy. Some of them are frequency independent and recovered in processing steps. But other factors such as intrinsic absorption of medium are frequency dependent and cannot be recovered by processing methods. Lost energy caused by these factors is called attenuation. There are various methods to study the attenuation of seismic energy.
Because of the frequency dependency behaviour of attenuation, it acts as a non-stationary quantity. Attenuation coefficient is usually studied in frequency domain based on power spectrum and statistical methods. Since Fourier transform does not consider the temporal variation of frequency content of seismic data, and due to the dependence of attenuation to frequency we used time-frequency tools in this study. Time-frequency transforms such as short-time Fourier transform, wavelet transform and S-transform are common tools in the processing and interpretation of seismic data. In this study, we used the Wigner-Ville distribution as a time-frequency tool to study the seismic wave attenuation.
Wigner-Ville Distribution: Wigner-Ville distribution (WVD) of a signal is defined as (Boashash, 2003):
(1)
The importance of the WVD is due to its marginal property and high resolution of time and frequency axis. But the existence of cross-term in WVD limited its application in engineering and science. Pseudo WVD (PWVD) and Smoothed Pseudo WVD (SPWVD) are two well-known methods with less cross-term than WVD. PWVD and SPWVD are defined as (Polarikas, 2000):
(2)
(3)
where, is the smoothing window which acts on frequency direction of distribution and is the smoothing window which acts on time direction of distribution. These two methods reduce the cross-term but extend the auto-term. In our study the noiselessness of distribution is more important than resolution. Therefore, we use SPWVD in our study.
Attenuation Estimation: The results of Zhang and Ulrich (2002) are the principle of attenuation estimation based on SPWVD. They show that the effect of attenuation on seismic wavelet can be referred as:
1- Peak frequency shift to lower frequencies.
2-Frequencies above the peak frequency are affected by attenuation more than frequencies which belong below the peak frequency.
The slope of the line which is fitted to WVD at each time for frequency range between peak frequency to half of Nyquist frequency can be used as a attenuation coefficient based on the relation below (Yandong and Xiaodong, 2007):
(4)
where, is the centeroid frequency and is defined as:
(5)
Because centeroid frequency is less sensitive to noise than peak frequency, we used centeroid frequency instead of peak frequency.
We investigated the efficiency of the method on both synthetic and real seismic data. Comparison of results with well logs and inversion of seismic data showed that this method can estimate both the attenuation coefficient and related anomaly location properly.