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