Ambient seismic noise eikonal tomography for near-surface imaging at Valhall

Appendix: Bicubic Spline reguarization

Bicubic spline interpolation is method of interpolation and regularization that relies on fitting the data by a set of weighted Green's functions for cubic cplines (Sandwell, 1987). It is intuitively comparable to bending a metal plate to fit through desired points, by applying and positioning different weights at positions along the plate. The Green's function for a cubic spline with forcing at $\delta(\mathbf{x})$ satisfies

$$\nabla^4 G(\mathbf{x}) = \delta(\mathbf{x}).$$ (12)

To fit $N$ datapoints using $N$ forcing functions weighted by $w_j$ , we have the system

$$\nabla^4 d_i(\mathbf{x}) = \sum_{j=1}^N w_j\delta(\mathbf{x}_i-\mathbf{x}_j).$$ (13)

Using the defined Green's function, we have the system

$$d_i(\mathbf{x})=\sum_{j=1}^N w_j G(\mathbf{x}_i-\mathbf{x}_j),$$ (14)

or in matrix notation

$$\mathbf{d}=\mathbf{G}\mathbf{w},$$ (15)

where $G_{ij}=G(\mathbf{x}_i-\mathbf{x}_j)$ is a kernel with Green's functions. The system is solved using an f90 library that performs LU decomposition (Press et al., 1986; Moreau, 2011). Green's function solutions for cubic splines in various dimensions have been derived and are summarized by Wessel (2009). This paper uses the two dimensional solution

$$G(\mathbf{x}_i-\mathbf{x}_j) = r^2\left(\mathrm{ln}r -1\right),$$ (16)

where $r=\left\vert\mathbf{x}_i-\mathbf{x}_j\right\vert$ .

Ambient seismic noise eikonal tomography for near-surface imaging at Valhall