The Method

Claerbout (1992) casts the problem of slope estimation as a univariate optimization problem, based on the observation that the partial differential equation  

$$\left( \frac{\partial}{\partial x} + p \frac{\partial}{\partial t} \right) u(t,x)$$ (1)

is zero-valued if the wavefield u(t,x) consists only of plane waves with time slope, or ``stepout,'' p. Claerbout also notes that a cascade of two PDEs annihilates data consisting of plane waves with two slopes, p₁ and p₂. The analog to equation (1) is:  

$$\left( \frac{\partial}{\partial x} + p_1 \frac{\partial}{\pa... ...}{\partial x} + p_2 \frac{\partial}{\partial t} \right) u(t,x),$$ (2)

or after expansion,  

$$\left( \frac{\partial^2}{\partial x^2} + (p_1+p_2) \frac{\pa... ... t} + p_1 p_2 \frac{\partial^2}{\partial t^2} \right) u(t,x).$$ (3)