Added clarification about sparse sampling shortcomings.

This commit is contained in:
Jeremy Karst 2026-09-12 22:35:48 -04:00
parent 02b403d630
commit 0fb7af77ce

View file

@ -79,10 +79,17 @@ cells. The ergodic pattern costs about 1.6 times the error here. That cost is th
premium for insurance that pays out on the Fourier-sparse truth, where half the signal varies too premium for insurance that pays out on the Fourier-sparse truth, where half the signal varies too
fast for the sparse grid: there the sparse grid garbles the fast waves into false slow ones fast for the sparse grid: there the sparse grid garbles the fast waves into false slow ones
(this is aliasing, and is destructive; compare the sparse-grid and ergodic columns of (this is aliasing, and is destructive; compare the sparse-grid and ergodic columns of
figure 2). THIS CASE IS MOSTLY THEORETICAL: If we knew that there was no small-scale variation figure 2). THIS CASE IS MOSTLY THEORETICAL: sparse sampling's win here does not carry over to
to capture, the sparse grid would be the right choice outright; in practice that is rarely known real applications, for two reasons. First, the rough scale of variation is always a known prior
before sampling, which is exactly what motivates the signal-agnostic ergodic pattern. See in practice (it is how any sample spacing gets chosen at all), but acting on that prior does not
the "Potential Future Work" section for some related thoughts. lead to a comfortably dense regular grid: any scale at which we knew there were no spatial
frequencies beyond what uniform spacing resolves would be better served by sensing the same area
with fewer, ergodically spaced samples. The regular grid's win in this column is really a
statement that its sample budget was larger than the known field scale required. Second, for
potential fields in particular, variation much finer than the survey scale reflects shallow
sources rather than the deep signal of interest, which is why this ground truth is generated
with a source-depth attenuation term. See the "Potential Future Work" section for some related
thoughts.
![Figure 2](plt1a.png) ![Figure 2](plt1a.png)