Stable absorption compensation with lateral constraint

Abstract

The presence of seismic absorption distorts seismic record and reduces seismogram resolution, which can be partially compensated by application of absorption compensation algorithms. Conventional absorption compensation techniques are based on 1D forward model with each seismic trace being compensated independently. Therefore, the 2D results combined by each compensation trace may be noisy and discontinuity. To eliminate this issue, we extend the 1D forward model to the 2D forward system and further add an additional lateral constraint to the compensation algorithm for enforcing the lateral continuity of the compensated section. Solving the proposed laterally constrained absorption compensation (LCAC) problem, we simultaneously obtain the multiple compensated traces with lateral smoother transition and higher signal-to-noise ratio (S/N). We testify the effectiveness of the proposed method by applying both synthetic and field data. Synthetic data examples demonstrate the superior performance of the LCAC algorithm in terms of improving algorithmic stability and protecting lateral continuity. The field data tests further indicate its ability to not only improve seismic resolution, but also inhibit the amplification of high-frequency noise.

Appendix: Derivation of Eq. 5

As we all know, the convolution in the time domain is equal to the product in the frequency domain, and then, the frequency domain expressions of Eq. (1) and Eq. (2) are

$$\begin{aligned} S_0(\omega ) = W(\omega ) R(\omega ), \end{aligned}$$

(A.1)

and

$$\begin{aligned} S(\omega ) = {{\hat{W}}}(\omega ,\tau ) R(\omega ), \end{aligned}$$

(A.2)

where $S_0(\omega )$ and $S(\omega )$ are, respectively, the non-attenuated and attenuated seismic signals in the frequency domain, $R(\omega )$ is the frequency domain reflectivity, and ${{\hat{W}}}(\omega ,\tau )$ is the time-varying wavelet in the frequency domain. According to Eq. (3), the frequency domain time-varying wavelet can be written by:

$$\begin{aligned} {{\hat{W}}}(\omega ,\tau ) = W(\omega ) A(\omega ,\tau ), \end{aligned}$$

(A.3)

Substituting Eq. (A.3) back into Eq. (A.2), we have,

$$\begin{aligned} S(\omega ) = W(\omega ) A(\omega ,\tau ) R(\omega ) = A(\omega ,\tau ) S_0(\omega ), \end{aligned}$$

(A.4)

By transforming Eq. (A.4) into the time domain, we obtain Eq. (5),

$$\begin{aligned} s(t) = a(t,\tau ) \otimes s_0(t), \end{aligned}$$

(A.5)

where $a(t,\tau )=\int _0^\infty {A(\omega ,\tau ) e^{i\omega t} d\omega }$ is the inverse Fourier transform of frequency domain Q-filtering function.