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On the Spectral Ranges that Are Resolved by a Single Satellite in Exact-Repeat Sampling Mode

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  • 1 Laboratory for Satellite Altimetry, NOAA, Silver Spring, Maryland
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Abstract

The sampling problem for satellite data in exact-repeat orbit configuration is treated in this paper. Specifically, for exact-repeat satellite sampling, the author seeks to solve the problem equivalent to finding the Nyquist frequency and wavenumbers for a textbook regular grid that frame a resolved spectral range within which all properly separated (assuming the data coverage is not infinite) spectral components can be distinguished (i.e., resolved) from each other, while an inside component is still indistinguishable from an infinite number of spectral components outside the range (i.e., aliasing). It is shown that there are multitudes of spectral ranges that are resolved with various degrees of uncertainty by the data: the suitable choice depends on the phenomena one wishes to observe and the noises one endeavors to avoid.

The problem is idealized for applications to regions of limited latitudinal extent, so that straight lines represent satellite ground tracks well. Let X and Y be the east–west and north–south separations of parallel tracks, let T be the repeat period, and let k, l, and ω be the nonangular wavenumbers and frequency with k in the east–west direction. The spectral range that is perfectly resolved (as if the data were placed on a regular space–time grid) covers (−1/X, 1/X) in k, (−1/Y, 1/Y) in l, and [0, 1/2T) in ω. There are other spectral ranges that extend either the spatial or temporal resolution with increased uncertainty beyond the above-mentioned perfectly resolved range.

The idealized problem is solved in stages, progressing from the 1D, to the 2D, to the fully 3D problem. The process is aided by a discovery that has implications that go beyond the scope of this paper. It is found that from a multidimensional regular grid one is free to introduce misalignments to all dimensions except one without incurring any penalty in spectral resolution. That is, the misaligned grid is equivalent to a perfectly aligned grid in spectral resolution. (However, the misalignment does induce far more complicated aliasing.) Thus, nonsimultaneous observations are equivalent to simultaneous ones, hence reducing the 3D problem to the 2D one. One 2D grid (the crossover grid) is equivalent to not just one but two regular 2D grids. These results are all verified numerically. An analytic proof is provided for the equivalency theorem.

Corresponding author address: Dr. Chang-Kou Tai, Laboratory for Satellite Altimetry, NOAA, NODC, E/OC2, 1315 East–West Highway, Silver Spring, MD 20910.

Email: tai@duck.grdl.noaa.gov

Abstract

The sampling problem for satellite data in exact-repeat orbit configuration is treated in this paper. Specifically, for exact-repeat satellite sampling, the author seeks to solve the problem equivalent to finding the Nyquist frequency and wavenumbers for a textbook regular grid that frame a resolved spectral range within which all properly separated (assuming the data coverage is not infinite) spectral components can be distinguished (i.e., resolved) from each other, while an inside component is still indistinguishable from an infinite number of spectral components outside the range (i.e., aliasing). It is shown that there are multitudes of spectral ranges that are resolved with various degrees of uncertainty by the data: the suitable choice depends on the phenomena one wishes to observe and the noises one endeavors to avoid.

The problem is idealized for applications to regions of limited latitudinal extent, so that straight lines represent satellite ground tracks well. Let X and Y be the east–west and north–south separations of parallel tracks, let T be the repeat period, and let k, l, and ω be the nonangular wavenumbers and frequency with k in the east–west direction. The spectral range that is perfectly resolved (as if the data were placed on a regular space–time grid) covers (−1/X, 1/X) in k, (−1/Y, 1/Y) in l, and [0, 1/2T) in ω. There are other spectral ranges that extend either the spatial or temporal resolution with increased uncertainty beyond the above-mentioned perfectly resolved range.

The idealized problem is solved in stages, progressing from the 1D, to the 2D, to the fully 3D problem. The process is aided by a discovery that has implications that go beyond the scope of this paper. It is found that from a multidimensional regular grid one is free to introduce misalignments to all dimensions except one without incurring any penalty in spectral resolution. That is, the misaligned grid is equivalent to a perfectly aligned grid in spectral resolution. (However, the misalignment does induce far more complicated aliasing.) Thus, nonsimultaneous observations are equivalent to simultaneous ones, hence reducing the 3D problem to the 2D one. One 2D grid (the crossover grid) is equivalent to not just one but two regular 2D grids. These results are all verified numerically. An analytic proof is provided for the equivalency theorem.

Corresponding author address: Dr. Chang-Kou Tai, Laboratory for Satellite Altimetry, NOAA, NODC, E/OC2, 1315 East–West Highway, Silver Spring, MD 20910.

Email: tai@duck.grdl.noaa.gov

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