**Authors**

**Abstract**

In exploration seismology quality factor is widely used as a seismic attribute to identify anomalies related to attenuation, especially those caused by hydrocarbon. Previous studies have indicated that seismic energy loss known as attenuation is greater for high frequency components of seismic data compared to the low frequency components. Here the continuous wavelet transform is used to study the attenuation of seismic data and to calculate the energy density at different scales. The results show that the energy loss at low scales is more than that of the high scales. The method is also used for determination of the anomalies related to energy attenuation due to the presence of hydrocarbon. The results indicated that using modified complex Morlet wavelet needs fewer computations than the regular complex Morlet wavelet. We investigated the efficiency of the method on both synthetic and real seismic data and the results are compared to the results obtained from inversion of seismic data to acoustic impedance using the Hampson-Russell software. The results showed an acceptable correlation. We also found that regular complex Morlet wavelet is more sensitive to the presence of noise than the modified complex Morlet wavelet.

Continuous Wavelet Transform: The time domain continuous wavelet transform (CWT) of a signal can be defined as:

(1)

where, denotes the complex conjugate, is scale, is time shift and is the mother wavelet. Shifted and scaled version of the mother wavelet can be computed as:

(2)

We can define the frequency domain CWT as:

(3)

where, is angular frequency, and are the Fourier transform of and mother wavelet, respectively (Poularikas, 2000). Since the Morlet’s wavelet is similar to the seismic source wavelet, we used the complex Morlet wavelet and a modified version of it as the mother wavelet in our study (Li et. al., 2006).

Energy Attenuation Density Equation: Considering a plane wave propagating in the anelastic medium, assuming that the quality factor is constant, its propagating equation is defined as (Aki and Richard, 1980):

(4)

where, is angular frequency, is propagating distance and is phase velocity.

The energy density at any angular frequency is by definition:

(5)

By introducing Eq. (4) to Eq. (5) and calculating the frequency domain CWT, assuming that and the wavelet domain energy density of signal can be obtained as:

(6)

Equation (6) shows that the energy of a signal in the wavelet domain is a function of the quality factor and scale factor as well as travel time . The larger is, the more slowly the energy attenuates; The smaller is, the faster the energy attenuates. The smaller the scale, the less the energy involved in the signal, because high scales correspond to low frequencies, and low scales correspond to high frequencies.

Discussion: This paper derives an energy attenuation formula for seismic waves in the wavelet-scale domain from wavelet theory and the seismic propagation equation in the anelastic medium. To investigate the efficiency of this method, we tested the method on both synthetic and real seismic data. The results showed an acceptable correlation. We also found that regular complex Morlet wavelet is more sensitive to the presence of noise than the modified complex Morlet wavelet. Also, the results indicated that using the modified complex Morlet wavelet needs fewer computations than the regular complex Morlet wavelet.

Continuous Wavelet Transform: The time domain continuous wavelet transform (CWT) of a signal can be defined as:

(1)

where, denotes the complex conjugate, is scale, is time shift and is the mother wavelet. Shifted and scaled version of the mother wavelet can be computed as:

(2)

We can define the frequency domain CWT as:

(3)

where, is angular frequency, and are the Fourier transform of and mother wavelet, respectively (Poularikas, 2000). Since the Morlet’s wavelet is similar to the seismic source wavelet, we used the complex Morlet wavelet and a modified version of it as the mother wavelet in our study (Li et. al., 2006).

Energy Attenuation Density Equation: Considering a plane wave propagating in the anelastic medium, assuming that the quality factor is constant, its propagating equation is defined as (Aki and Richard, 1980):

(4)

where, is angular frequency, is propagating distance and is phase velocity.

The energy density at any angular frequency is by definition:

(5)

By introducing Eq. (4) to Eq. (5) and calculating the frequency domain CWT, assuming that and the wavelet domain energy density of signal can be obtained as:

(6)

Equation (6) shows that the energy of a signal in the wavelet domain is a function of the quality factor and scale factor as well as travel time . The larger is, the more slowly the energy attenuates; The smaller is, the faster the energy attenuates. The smaller the scale, the less the energy involved in the signal, because high scales correspond to low frequencies, and low scales correspond to high frequencies.

Discussion: This paper derives an energy attenuation formula for seismic waves in the wavelet-scale domain from wavelet theory and the seismic propagation equation in the anelastic medium. To investigate the efficiency of this method, we tested the method on both synthetic and real seismic data. The results showed an acceptable correlation. We also found that regular complex Morlet wavelet is more sensitive to the presence of noise than the modified complex Morlet wavelet. Also, the results indicated that using the modified complex Morlet wavelet needs fewer computations than the regular complex Morlet wavelet.

**Keywords**