Model parameterization by spline

An accurate estimation of the gradient of the slowness field is necessary if we are to trace rays in a velocity model with high velocity contrast. Therefore, I have chosen to use cubic B-splines Inoue (1986) to parametrize the slowness model, which enables me to calculate the slowness gradient at an arbitrary point in the model. Another reason for using splines is that a wide variety of slowness models can be represented by a few spline coefficients. This means that the number of parameters in an optimization scheme that inverts for slowness can be reduced considerably.

The spline representation of a two-dimensional slowness model is  

$$w (x_1,x_2) = \sum^N_{i=1} \sum^M_{j=1} c_{ij} F_i(x_1)G_j(x_2),$$ (40)

with N and M the number of spline knots in the x₁- and x₂-direction. F_i and G_j are the spline functions at the ith knot in the x₁-direction and the jth knot in the x₂-direction, respectively.