Single-channel spectral ratio method

The seismic wave amplitude spectrum after propagating from traveltime $t_1$ to $t_2$ in a homogeneous attenuating medium are given by:

$$Y(f,t_1) = A(t_1)Y(f,t_0)e^{-\frac{\pi f(t_1-t_0)}{Q}},$$ (1)

$$Y(f,t_2) = A(t_2)Y(f,t_0)e^{-\frac{\pi f(t_2-t_0)}{Q}},$$ (2)

where $f$ is the frequency, $Y(f,t_1)$ and $Y(f,t_2)$ are the seismic wave amplitude spectrum after propagating traveltime $t_1$ and $t_2$, respectively, $Y(f,t_0)$ is the initial amplitude spectrum, and $A(t)$ is a factor independent of frequency. Combining equations 1 and 2, we can obtain:

$$\frac{Y(f,t_2)}{Y(f,t_1)} = \frac{A(t_2)e^{-\frac{\pi ft_2}{Q}}}{A(t_1)e^{-\frac{\pi ft_1}{Q}}} = d(f).$$ (3)

Take a natural log of each side of equation 3, we obtain:

$$\ln[d(f)] = A+Bf,$$ (4)

where $A$ and $B$ are both constants, and $A=\ln(A(t_2)/A(t_1))$, $B=-(\pi(t_2-t_1)/Q)$. Here, $B$ can be estimated by the slope of linear regression line. Then, $Q$ can be estimated by

$$Q=\frac{\pi(t_1-t_2)}{B}.$$ (5)