As shown in the previous section, random undersampling
according to a discrete uniform distribution creates favorable
recovery conditions for a reconstruction procedure that promotes
sparsity in the Fourier domain. However, a global transform such as
the Fourier transform does not typically permit a sparse
representation for complex seismic wavefields
(Hennenfent and Herrmann, 2006). It requires a more local transform,
e.g., windowed Fourier (Zwartjes and Sacchi, 2007) or curvelet
(Herrmann and Hennenfent, 2007) transform. In this case, problems arise with
gaps in the data that are larger than the spatio-temporal extent of
the transform elements (Trad et al., 2005). Consequently,
undersampling schemes with no control on the size of the maximum gap,
e.g., random undersampling according to a discrete uniform
distribution, become less attractive. The term gap refers here to the
interval between two adjacent acquired traces minus the interval
associated with the fine interpolation grid, such that adequate
sampling has gaps of zero. We present an undersampling scheme that
has, under some specific conditions, an anti-aliasing effect, yet
offering control on the size of the maximum gap.
Simply denoise: wavefield reconstruction via jittered
undersampling
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