Inverse Interpolation and Data Regularization

In the notation of Claerbout (1999), inverse interpolation amounts to a least-squares solution of the system

$$\begin{eqnarray} \bold{L m} & \approx & \bold{d}\;; \ \epsilon \bold{A m} & \approx & \bold{0}\;, \end{eqnarray}$$ (15) (16)

where $\bold{d}$ is a vector of known data f(x_i) at irregular locations x_i, $\bold{m}$ is a vector of unknown function values f(n) at a regular grid n, $\bold{L}$ is a linear interpolation operator of the general form (1), $\bold{A}$ is an appropriate regularization (model styling) operator, and $\epsilon$ is a scaling parameter. In the case of B-spline interpolation, the forward interpolation operator $\bold{L}$ becomes a cascade of two operators: recursive deconvolution $\bold{B}^{-1}$, which converts the model vector $\bold{m}$ to the vector of spline coefficients $\bold{c}$, and a spline basis construction operator $\bold{F}$.System (15-16) transforms to

$$\begin{eqnarray} \bold{F B^{-1} m} & \approx & \bold{d}\;; \ \epsilon \bold{A m} & \approx & \bold{0}\;. \end{eqnarray}$$ (17) (18)

We can rewrite (17-18) in the form that involves only spline coefficients:

$$\begin{eqnarray} \bold{W c} & \approx & \bold{d}\;; \ \epsilon \bold{A B c} & \approx & \bold{0}\;. \end{eqnarray}$$ (19) (20)

After we find a solution of system (19-20), the model $\bold{m}$ will be reconstructed by the simple convolution  

$$\bold{m = B c}\;.$$ (21)

This approach resembles a more general method of model preconditioning Fomel (1997a).

The inconvenient part of system (19-20) is the complex regularization operator $\bold{A B}$. Is it possible to avoid the cascade of $\bold{B}$ and $\bold{A}$ and to construct a regularization operator directly applicable to the spline coefficients $\bold{c}$? In the following subsection, I develop a method for constructing spline regularization operators from differential equations.