Sampling Theory for Asynoptic Satellite Observations. Part I: Space-Time Spectra, Resolution, and Aliasing

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Abstract

The information content of asynoptic satellite data is evaluated for nadir sonde and limb scan observations. Orbital sampling patterns are shown to uniquely determine the space-time spectrum, within well-defined sampling limitations. The latter turn out to be a hybrid of wavenumber and frequency in the same manner that the observations are a mixture of space and time. Space-time spectra thus computed are correct throughout the allowed region of wavenumber and frequency. Complexities such as orbital tilt and day-to-day drift of the nodes, are completely accounted for.

The allowed region of spectra, which defines the information content, is a rectangle in Fourier space, rotated relative to the wavenumber, frequency axes. This rotation is a consequence of the lack of simultaneity in the observations. For “single-node” data, the aliasing limitations correspond approximately to a maximum wavenumber of half the orbital frequency (orbits per day) and frequency extrema of ±0.5 cpd. Definition of the sampling restrictions for “combined-node” (ascending + descending) data is complicated by the introduction of additional aliasing. The latter, which arises from irregular sampling, ultimately from orbital tilt, is serious at middle and high latitudes, where ascending and descending nodes converge. This additional contamination is inherent to combined asynoptic data. It results from the unequal spacing between ascending and descending nodes. Consequently, spectra calculated via the transform, or other form of projection, must be restricted to frequencies less than 0.5 cpd, thus not fully utilizing the combined observations.

The additional contamination can be completely eliminated by replacing the transform along the coordinate of irregular sampling, with explicit evaluation of the Fourier components. Then the region of allowed spectra can be extended to frequencies of ±1.0 cpd, i.e., a doubling in resolution over single-node data. The allowed region is shown to be analogous to that of “twice-daily,” synoptic sampling at equispaced points, equal in number to approximately the orbital frequency. The resulting formulas constitute the Asynoptic Sampling Theorem: uniquely relating a combined asynoptic data set to its space-time spectrum.

Abstract

The information content of asynoptic satellite data is evaluated for nadir sonde and limb scan observations. Orbital sampling patterns are shown to uniquely determine the space-time spectrum, within well-defined sampling limitations. The latter turn out to be a hybrid of wavenumber and frequency in the same manner that the observations are a mixture of space and time. Space-time spectra thus computed are correct throughout the allowed region of wavenumber and frequency. Complexities such as orbital tilt and day-to-day drift of the nodes, are completely accounted for.

The allowed region of spectra, which defines the information content, is a rectangle in Fourier space, rotated relative to the wavenumber, frequency axes. This rotation is a consequence of the lack of simultaneity in the observations. For “single-node” data, the aliasing limitations correspond approximately to a maximum wavenumber of half the orbital frequency (orbits per day) and frequency extrema of ±0.5 cpd. Definition of the sampling restrictions for “combined-node” (ascending + descending) data is complicated by the introduction of additional aliasing. The latter, which arises from irregular sampling, ultimately from orbital tilt, is serious at middle and high latitudes, where ascending and descending nodes converge. This additional contamination is inherent to combined asynoptic data. It results from the unequal spacing between ascending and descending nodes. Consequently, spectra calculated via the transform, or other form of projection, must be restricted to frequencies less than 0.5 cpd, thus not fully utilizing the combined observations.

The additional contamination can be completely eliminated by replacing the transform along the coordinate of irregular sampling, with explicit evaluation of the Fourier components. Then the region of allowed spectra can be extended to frequencies of ±1.0 cpd, i.e., a doubling in resolution over single-node data. The allowed region is shown to be analogous to that of “twice-daily,” synoptic sampling at equispaced points, equal in number to approximately the orbital frequency. The resulting formulas constitute the Asynoptic Sampling Theorem: uniquely relating a combined asynoptic data set to its space-time spectrum.

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