Discretizing the problem

Claerbout approximates the derivatives of equation (1) with 2x2 finite difference stencils. Assuming that the grid spacing in both the t and x directions are unity:  

$$\frac{\partial}{\partial x} \approx 0.5* \left[\begin{arra... ...eft[\begin{array} {rr} -1 & -1 \\ 1 & 1 \end{array}\right].$$ (4)

By convolving together these first-order stencils, we can construct appropriate finite-difference stencils to approximate the second-order differential operators of equation (3):

$$\begin{eqnarray} \frac{\partial}{\partial x}*\frac{\partial}{\partial x} = \fr... ...-1 & -2 & -1 \\ 2 & 4 & 2 \\ -1 & -2 & -1 \\ \end{array}\right]\end{eqnarray}$$ (5) (6) (7)

The stencils of equations (5)-(7) are convolved with the data, $\bf u$. For simplicity, we can define the following notation:  

$$\frac{\partial^2}{\partial x^2}*\bold u = \bold D_{xx} ; \hs... ....2in} \frac{\partial^2}{\partial t^2}*\bold u = \bold D_{tt} ,$$ (8)

and rewrite equation (3) in matrix form:  

$$\bold r = \left[\begin{array} {rrr} \bold D_{xx} & \bold ... ...n{array} {c} 1 \\ p_1+p_2 \\ p_1 p_2 \\ \end{array}\right].$$ (9)

The vector $\bf r$ has the same dimension as the data, $\bf u$.If the data consists only of plane waves with slopes p₁ and p₂, then equation (9) predicts values of $\bf u$ from nearby values of $\bf u$. If the data's slopes change in time and space, however, equation (9) is valid only across local ``patches'' of the data. We can rewrite equation (9) to reflect this fact:  

$$\bold r = \left[\begin{array} {ccc} \bold D^1_{xx} & \bol... ...in{array} {c} 1 \\ p_1+p_2 \\ p_1 p_2 \\ \end{array}\right]$$ (10)

Equation (10) denotes the convolution of the respective finite-difference stencils over a data patch of size n, where n may be as large as the entire data, or as small as $3 \times 3$

While it is tempting to make a change of variables (a=p₁+p₂, b=p₁ p₂) and treat equation (10) as a linear relationship, I have found that this approach produces trivial coupled estimates of the true slopes. This problem is inherently nonlinear.