Estimating the Smoothing Radius

To estimate the triangle smoothing radius for matching two seismic datasets, we utilize the Gauss-Newton approach to solving non-linear least-squares problems (Lawson and Hanson, 1995). We define a triangle smoothing operator with radius $\mathbf{R}$ applied to data $\mathbf{d}$ as $\mathbf{S}_R[\mathbf{d}]$ and a triangle smoothing derivative operator as $\mathbf{S}_R'[\mathbf{d}]$. Given the original data $\mathbf{d}_{input}$ and the smoothed data $\mathbf{d}_{output}$, we define the Taylor expansion

$$\mathbf{S}_R[\mathbf{d}_{input}] \approx \mathbf{S}_{R_0}[\mathbf{d}_{input}] + \mathbf{S}_{R_0}'[\mathbf{d}_{input}](\mathbf{R}-\mathbf{R}_0),$$ (13)

where $\mathbf{R}_0$ is the first guess for the radius and $\mathbf{R}$ is the best estimate for the radius. Noting that $\mathbf{S}_R[\mathbf{d}_{input}] \approx \mathbf{d}_{output}$, we rearrange equation 13 to solve for $R$:

$$\mathbf{R} \approx \mathbf{R}_0 + \left(\mathbf{S}_{R_0}'[\mathbf{d}_{input}]\right)^{-1}\left(\mathbf{d}_{output} - \mathbf{d}_{input}\right)\;.$$ (14)

We can repeat this approach and solve for the radius iteratively, where the radius at the $i_{th}$ iteration is given by

$$\mathbf{R}_{i+1} = \mathbf{R}_i + \left(\mathbf{S}_{R_i}'[\mathbf... ...^{-1}\left(\mathbf{d}_{output} - \mathbf{S}_{R_i}[\mathbf{d}_{input}]\right)\;.$$ (15)

The proposed method in theory converges with a rate approaching quadratic, although convergence is not guaranteed if the initial guess is far from the true value (Lawson and Hanson, 1995). The method is directly extended to solve for a non-stationary triangle smoothing radius given that both triangle smoothing and its derivative are non-stationary. Note that for stability of the solution, we must take care in performing the division in equation 15. We implement smooth division which treats division as inversion and regularizes the inversion in equation 15 using shaping regularization (Fomel, 2007b). The shaping regularization is controlled by its own smoothness radius and reduces the radius estimation to stationary least-squares estimate when the smoothing radius for shaping is set to be very large.