Plane waves in three dimensions

PEFs overcome spatial aliasing of difference operators

The problem I found with finite-difference representations of differential operators is that they are susceptible to spatial aliasing. Even before they encounter spatial aliasing, they are susceptible to accuracy problems known in finite-difference wave propagation as ``frequency dispersion.'' The aliasing problem can be avoided by the use of spatial prediction operators such as

$$\begin{array}{cc} \cdot &a \\ \cdot &b \\ 1 &c \\ \cdot &d \\ \cdot &e \end{array}$$ (1)

where the vertical axis is time; the horizontal axis is space; and the ``$\cdot$ ''s are zeros. Another possibility is the 2-D whitening filter

$$\begin{array}{cc} f &a \\ g &b \\ 1 &c \\ \cdot &d \\ \cdot &e \end{array}$$ (2)

Imagine all the coefficients vanished but $d=-1$ and the given 1. Such filters would annihilate the appropriately sloping plane wave. Slopes that are not exact integers are also approximately extinguishable, because the adjustable filter coefficients can interpolate in time. Filters like (2) do the operation $\partial_x + p_x \partial_t$ , which is a component of the gradient in the plane of the wavefront, and they include a temporal deconvolution aspect and a spatial coherency aspect. My experience shows that the operators (1) and (2) behave significantly differently in practice, and I am not prepared to fully explain the difference, but it seems to be similar to the gapping of one-dimensional filters.

You might find it alarming that your teacher is not fully prepared to explain the difference between a volume and two planes, but please remember that we are talking about the factorization of the volumetric spectrum. Spectral matrices are well known to have structure, but books on theory typically handle them as simply $\lambda \bold I$ . Anyway, wherever you see an $\bold A$ in a three-dimensional context, you may wonder whether it should be interpreted as a cubic filter that takes one volume to another, or as two planar filters that take one volume to two volumes as shown in Figure 2.

rayab3Doper Figure 2. An inline 2-D PEF and a crossline 2-D PEF both applied throughout the volume. To find each filter, minimize each output power independently.

Plane waves in three dimensions