This representation of the space-time spectrum is not an estimate, but rather 
an exact form. It is free of distortion and ambiguity. Mers 
r. 
In practice, evaluation of the transform, whether it be calculated RE, 
synoptically over the rectangle D, or asynoptically over the strip D', must frequ 
be performed discretely. The discrete method that the transform is evalu- plane 
ated, is a direct consequence of the nature of the sampling. Synoptic frequ 
observations lend themselves to the discrete form of the synoptic transform withi 
(5.2). Asynoptic observations, on the other hand, are directly amenable to 
the discrete form of the asynoptic space-time transform (9.1). kis 
4. Spectral Resolution and Aliasing. S 
There are two important limitations in determining the space-time orbit 
spectrum Y, regardless of whether it be evaluated synoptically or asynop- - recta 
tically. The first of these is finite spectral resolution, and is a con- alias 
sequence of the finite sample length T. It can be shown (Salby, 1982a), that seen. 
the spectral resolution for asynoptic sampling is 2m/T, identical to the case withi 
for synoptic sampling. 
The second limitation is aliasing and is determined by the discrete 
nature of the sampling in space and time. Aliasing proceeds in the transform 
plane along lines parallel to the sampling. The aliases are separated 2 
Nyquist units corresponding to each direction. For the asynoptic case, 
wavenumber-frequency components influence one another along lines parallel to 
the k.,k, axes. Because of the uniform spacing of the observations along the 
s coofdihate (Fig. 4), the Nyquist wavenumber for the ke component follows e 
simply as 
TV 
Get ES an 
TT c 
0 
Data along the r coordinate, on the other hand, are in general not . 
equispaced (cf. Fig. 4). Rather, ascending ^ descending loci are separated 
by Arq, while descending ^ ascending loci are Ar, units apart, where 
2m - AX! Fra 
ar, = ——âd (11.1) 
"1 + c 
data . 
1 that 
AA 
ar, = —ad_ (11.2) Poihe 
+ c latit 
By coi 
samp] 
! z the r 
and ; Mad. = Mad} lc lat, 4 . (12) ‚50% 0 
AX), is just the "instantaneous" separation between ascending and descending 
trüfersals of the orbit. Only for nadir data, and only at the equator, are a 
ascending and descending loci equispaced along r. This turns out to have N 
rather important implications to the aliasing properties of the data. ene. 
tamin 
where 
160 
uum s ccu MA MAO ERU 
e: —L On ici a Nise