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The Resolving Power of a Single Exact-Repeat Altimetric Satellite or a Coordinated Constellation of Satellites

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  • 1 NOAA/NESDIS/ORA, Camp Springs, Maryland
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Abstract

It is proved that the midpoint grid, which is composed of samples obtained at ground track locations midway between crossover points (thus a subset of the full sampling), has the same resolving power as the full set; that is, they resolve the same three-dimensional spectral space. The resolving power of the midpoint grid is characterized by the Nyquist frequency ωc = π/T (where T is the repeat period of the exact-repeat satellite), and by (in local Cartesian coordinates) the zonal and meridional Nyquist wavenumber kc = 2π/X and lc = 2π/Y, respectively (where X and Y are the east–west and north–south separation between adjacent parallel ground tracks). (Note that the term “Nyquist” is used here for lack of a better term. It retains its textbook meaning for delimiting the resolved spectral range. However, it may not have the same textbook meaning as far as aliasing is concerned.) Here, this result is rederived in simplified terms so that it is understood better. First, it is shown through the sampling theorem that even though samples of the real midpoint grid are not taken simultaneously, they resolve the same frequency range as that of a hypothetical midpoint grid, wherein samples are taken simultaneously at time t = nT (where n is an integer), hence sharing the same Nyquist frequency as cited above. This also reduces the three-dimensional problem to a two-dimensional one. The spatial part of the midpoint grid is a textbook regular grid with zonal and meridional sampling interval of X/2 and Y/2, respectively. As such, the Nyquist wavenumbers are exactly as those stated above. Now with the resolving power of the midpoint grid clearly understood, it can be proved that the midpoint grid provides the maximum resolving power. Putting the origin of the xy coordinate system on one of the crossover points, the proof comes in the demonstration that cos(2πx/X) is indistinguishable from cos(2πy/Y) along track; that is, along-track samples are unable to distinguish the cosine part of the spectral component k = ±kc, l = 0 from that of the spectral component k = 0, l = ±lc. Thus any spatial spectral range containing both of these spectral components will be unresolved. In other words, the midpoint grid already has the maximum spatial resolution that can be attained by the complete along-track samples. Moreover, it is shown that there is a corresponding midpoint grid for a coordinated constellation of satellites, extending the theory for a single satellite to multiple ones. In addition, the proof implies that not all along-track samples are needed for three-dimensional analysis. Tremendous saving can be achieved through data compression. The best ways to achieve this while still retaining the aliasing-reduction benefits offered by the extra observations are discussed.

Corresponding author address: Dr. Chang-Kou Tai, E/RA3, NOAA/NESDIS/ORA, 5200 Auth Road, Camp Springs, MD 20746. Email: ck.tai@noaa.gov

Abstract

It is proved that the midpoint grid, which is composed of samples obtained at ground track locations midway between crossover points (thus a subset of the full sampling), has the same resolving power as the full set; that is, they resolve the same three-dimensional spectral space. The resolving power of the midpoint grid is characterized by the Nyquist frequency ωc = π/T (where T is the repeat period of the exact-repeat satellite), and by (in local Cartesian coordinates) the zonal and meridional Nyquist wavenumber kc = 2π/X and lc = 2π/Y, respectively (where X and Y are the east–west and north–south separation between adjacent parallel ground tracks). (Note that the term “Nyquist” is used here for lack of a better term. It retains its textbook meaning for delimiting the resolved spectral range. However, it may not have the same textbook meaning as far as aliasing is concerned.) Here, this result is rederived in simplified terms so that it is understood better. First, it is shown through the sampling theorem that even though samples of the real midpoint grid are not taken simultaneously, they resolve the same frequency range as that of a hypothetical midpoint grid, wherein samples are taken simultaneously at time t = nT (where n is an integer), hence sharing the same Nyquist frequency as cited above. This also reduces the three-dimensional problem to a two-dimensional one. The spatial part of the midpoint grid is a textbook regular grid with zonal and meridional sampling interval of X/2 and Y/2, respectively. As such, the Nyquist wavenumbers are exactly as those stated above. Now with the resolving power of the midpoint grid clearly understood, it can be proved that the midpoint grid provides the maximum resolving power. Putting the origin of the xy coordinate system on one of the crossover points, the proof comes in the demonstration that cos(2πx/X) is indistinguishable from cos(2πy/Y) along track; that is, along-track samples are unable to distinguish the cosine part of the spectral component k = ±kc, l = 0 from that of the spectral component k = 0, l = ±lc. Thus any spatial spectral range containing both of these spectral components will be unresolved. In other words, the midpoint grid already has the maximum spatial resolution that can be attained by the complete along-track samples. Moreover, it is shown that there is a corresponding midpoint grid for a coordinated constellation of satellites, extending the theory for a single satellite to multiple ones. In addition, the proof implies that not all along-track samples are needed for three-dimensional analysis. Tremendous saving can be achieved through data compression. The best ways to achieve this while still retaining the aliasing-reduction benefits offered by the extra observations are discussed.

Corresponding author address: Dr. Chang-Kou Tai, E/RA3, NOAA/NESDIS/ORA, 5200 Auth Road, Camp Springs, MD 20746. Email: ck.tai@noaa.gov

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