It involves application of the asynoptic form of the space-time transform, 
followed by an application of the synoptic form of the inverse space-time 
transform (Salby, 1982c). This procedure, in effect, unravels the mixed, 
space-time dependence of asynoptic measurements , while preserving, fully, 
the information content of the combined data. 
2. Synoptic Retrieval Theorem 
Consider an evolving field, y containing no spectral contribution 
outside the regions allowed by both asynoptic sampling and equivalent (same 
number of nodes), twice-daily synoptic sampling. Then given combined, 
asynoptic observations of y over an integer number of days, its space-time 
spectrum, y, can be uniquely constructed (Salby, 1982c). The resolvable 
wavenumbers and frequencies where Y is determinable, overlap with those 
corresponding to twice-daily, synoptic observations. A two-dimensional 
extension of the Sampling Theorem (see B4th, 1974) uniquely relates the 
space-time spectrum to the uniformly incremented synoptic series. Therefore, 
the synoptic sequence (x.t) follows from the asynoptic sequence "uniquely." 
Theorem 
Let y(A,t) contain no spectral contribution outside the region 
permitted by both combined asynoptic and equivalent, twice-daily synoptic 
sampling. Then, corresponding to a combined asynoptic sequence of observations 
of y, over an integer number of days, there exists one, and only one, 
twice-daily, synoptic sequence of alias-free maps. 
The proof may be found in Salby (1982b). 
3. Fast Fourier Synoptic Mapping 
The uniqueness between alias-free asynoptic measurements and 
synoptic sequences, established in the Synoptic Retrieval Theorem, follows 
from their one-to-one correspondence with their common space-time spectra. 
In particular, the synoptic behavior and space-time spectra are uniquely 
related by the Sampling Theorem. This holds regardless of how the space- 
time spectrum is obtained. Therefore, once the space-time spectrum is 
constructed, e.g. by the asynoptic form of the space-time transform (Salby, 
1982a,c), the synoptic evolution may be recovered by inverting the space- 
time transform synoptically. Some components resolvable in asynoptic data, 
however, are not resolvable in twice-daily, synoptic observations, and vice 
versa (see Fig. 1). Consequently, some features resolvable in comb i ned 
asynoptic data, cannot be retrieved in a twice-daily synoptic sequence. 
Fast Fourier Synoptic Mapping (FFSM), in short, involves two FFT's 
of the asynoptic data: one of the ascending sequence and one of the de- 
scending sequence. Space-time spectra then follow analytically via the 
Asynoptic Sampling Theorem (Salby, 1982a). Finally, the synoptic behavior 
is recovered by a double FFT along wavenumber and frequency over the spectra 
permitted in both types of data (see Fig. 1). The reader is referred to 
Salby (1982b) for the details. 
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