Spectral division by shaping regularization

The key step in the SRM is the division between $Y(f,t_2,x)$ and $Y(f,t_1,x)$. The division may be unstable when denominator $Y(f,t_1,x)$ is very small or the spectrum is not smooth. Fomel (2007a) and Du et al. (2010) treated the division of two vectors ( $\mathbf {a}$ and $\mathbf {b}$) as a regularized least-squares minimization problem:

$\displaystyle \min_{\mathbf{c}} \parallel \mathbf{a} -\mathbf{B}\mathbf{c} \parallel_2^2 + \mathbf{R}(\mathbf{c}),$ (9)

where $\mathbf{B}=diag(\mathbf{b})$, $diag$ denotes the diagonal matrix composed of $\mathbf {b}$, $\mathbf{R}$ is the regularization operator. Shaping regularization (Fomel, 2007b) can provide a convenient way for enforcing the smoothness of the solution in equation 9 and the solution of equation 9 has the following form:

$\displaystyle \mathbf{c} = [\lambda^2\mathbf{I} + \mathcal{T}(\mathbf{B}^T\mathbf{B}-\lambda_1^2\mathbf{I})]^{-1}\mathcal{T}\mathbf{B}^T\mathbf{a},$ (10)

where $\mathbf{\mathcal{T}}$ is a two-dimensional smoothing operator, which controls the smoothness along the frequency and space axes. $\lambda$ is a parameter controlling the physical dimensionality and enabling fast convergence when inversion is implemented iteratively. It can be chosen as $\lambda = \Arrowvert\mathbf{B}^T\mathbf{B}\Arrowvert_2$ (Fomel, 2007a).

Here, in order to obtain a stable division between $Y(f,t_2,x)$ and $Y(f,t_1,x)$, we only need to substitute $\mathbf {a}$ and $\mathbf {b}$ with $Y(f,t_2,x)$ and $Y(f,t_1,x)$, where $f\in(f_{min},f_{max})$, respectively.

When the spectral ratio $d(f,x)$ is calculated, we can do the least-squares line-fitting by taking the natural log of each side of equation 8:

$\displaystyle \ln[d(f,x)] = A(x)+B(x)f,$ (11)

where $A(x)$ and $B(x)$ are both varying with the space coordinate $x$, and

$\displaystyle A(x)$ $\displaystyle =\ln(A(t_2,x)/A(t_1,x)),$ (12)
$\displaystyle B(x)$ $\displaystyle =\frac{-\pi(t_2-t_1)}{Q(x)}.$ (13)

The least-squares line-fitting for estimating $A(x)$ and $B(x)$ can be expressed as

$\displaystyle \left(\hat{A}(x),\hat{B}(x) \right) = \displaystyle \arg \min_{\l...
\vdots \\
\end{array}\right] \right\Vert _2^2.$ (14)

After the least-squares line-fitting, Q can be calculated as

$\displaystyle Q(x)=\frac{\pi(t_1-t_2)}{B(x)}.$ (15)

The proposed method requires a time-frequency mapping to obtain $Y(f,t,x)$ for the subsequent calculation. Reine et al. (2009) showed that the time-frequency transform used to calculate the spectrum of the seismic wave affects the accuracy of estimated Q, and the S transform and CWT decrease the variability of the attenuation estimate, specifically at the high and low ends of the spectrum because of a variable window function used in the definition. Because of the success of the S transform in estimating Q as shown in Reine et al. (2009) and Du et al. (2010), we continue to use the S transform as the time-frequency transform in our multi-channel Q estimation framework. In the appendix, we provide a brief review of the S transform.

The proposed method is convenient to implement and its parameters are easy to control. The proposed multi-channel Q estimation method is developed based on the state-of-the-art spectral ratio method framework. The difference between the proposed method and the traditional method lies in the way we calculate the spectral division. The regularized spectral division is an elegant way to avoid the problem of spectral nulls when doing a spectral division conventionally. The shaping regularization method has also been a state-of-the-art algorithm in recent years. A detailed algorithm framework of the iterative shaping regularization can be found in Fomel (2007b). The smoothness constraint is very straightforward to apply, i.e., two smoothing operators along the frequency and space directions individually. The smoothing operators can be a triangle operator or a Gaussian smoothing operator, both of which are very typical in the signal processing literature. In the whole algorithm framework, the only parameters we need to set are the smoothing radii along the frequency and space axes and the frequency range for least-squares line fitting. The larger smoothing radius, the smoother spectral ratio we will get. The frequency range is related with the dominant frequency range of the useful signals. We have mentioned the parameter selection in detail for each example. In practice, one need to adjust these parameters several times. Since there parameters have very straightforward implications, the adjustment of the parameters is still easy to control.