Plane wave prediction in 3-D

Estimating plane waves

It may seem difficult to estimate the plane slope $p_x$ for a Lax-Wendroff filter of the form (10) because $p_x$ appears non-linearly in the filter coefficients. However, using the analytical form of the filter, we can easily linearize it with respect to the plane slope and set up a simple iterative scheme:

$$p_x^{(k+1)} = p_x^{(k)} + \Delta p_x^{(k)}\;,$$ (11)

where $k$ stands for the iteration count, and $\Delta p_x^{(k)}$ is found from the linearized equation

$$\left(\mathbf{A'}_x \mathbf{U}\right) \Delta p_x = - \mathbf{A}_x \mathbf{U}\;,$$ (12)

where $\mathbf{A'}_x$ is the derivative of $\mathbf{A}_x$ with respect to $p_x$. To avoid unstable division by zero when solving equation (12) for $\Delta p_x$, Adding a regularization equation

$$\epsilon \nabla \Delta p_x \approx 0\;,$$ (13)

where $\epsilon$ is a small scalar regularization parameter, I solve system (12-13) in the least-square sense to obtain a smooth slope variation $\Delta p_x$ at each iteration. In practice, iteration process (11) quickly converges to a stable estimate of $p_x$.

Plane wave prediction in 3-D