FORMULATON

Let ${\bf x}$ denote a data plane in time and offset. Let $\v$ denote a model plane in velocity and vertical time. Let ${\bf B}$ denote a modeling program. In this study I chose ${\bf B'}$ to be the pixel-precise data processing operator (Claerbout [Pixel..., 1990]). Thus the modeling operator is defined to be the transpose conjugate of a well-known processing operator. To a wave theorist this may seem strange, but the goal is to learn how to best handle truncation and aliasing so any reasonable approximation to the kinematics should be fine.

Linear inverse theory suggests an algorithm based on

$$\min_v \quad[ ({\bf x}- {\bf B}\v)'\bold W_{xx}({\bf x}- {\bf B}\v) + \lambda \v ' \bold W_{vv} \v ]$$ (1)

where $\bold W_{xx}$ and $\bold W_{vv}$ are the usual inverse covariance weighting functions and $\lambda$ is a parameter that is presumably explained by the textbook literature (though SEP-65 page 214 shows some pitfalls there.) We'll consider weights later. For the unweighted problem the formal solution is a power series in ${\bf B}{\bf B'}$.In real life, the operator ${\bf B}$ is expensive, especially considering the huge volume of data requiring analysis. I chose to consider only the first two terms of the power series for careful examination to identify the most significant practical effect,  

$$\v {\quad = \quad}(\beta {\bf B'}+ \alpha {\bf B'}{\bf B}{\bf B'}) {\bf x}$$ (2)

where $\alpha$ and $\beta$ have yet to be chosen. In the formula (2) the data is taken to be separable into two parts, the known k and the missing m which is taken to be zero, say

$${\bf x}{\quad = \quad}({\bf x}_k ,{\bf x}_m) {\quad = \quad}({\bf x}_k, 0)$$ (3)