Aliasing of Sea Level Sampled by a Single Exact-Repeat Altimetric Satellite or a Coordinated Constellation of Satellites: Analytic Aliasing Formulas

Chang-Kou Tai NOAA/NESDIS, Camp Springs, Maryland

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Abstract

The aliasing problem for exact-repeat altimetric satellite sampling is solved analytically by the least squares method. To make the problem tractable, the latitudinal extent of the problem needs to be moderate for the satellite ground tracks to appear as two sets of parallel straight lines, and the along-track sampling is assumed to be dense enough to resolve any along-track features of interest. The aliasing formulas thus derived confirm the previously discovered resolving power, which is characterized by the Nyquist frequency ωc = π/T (where T is the repeat period of the satellite) and by (in local Cartesian coordinates) the zonal and meridional Nyquist wavenumbers kc = 2π/X and lc = 2π/Y, respectively (where X and Y are the east–west and north–south separations between adjacent parallel ground tracks). There are three major differences with the textbook aliasing. First, instead of the one-to-one correspondence, an outside spectral component is usually aliased into more than one inside spectral components (i.e., those inside the resolved spectral range with |ω| < ωc, |k| < kc, and |l| < lc). Second, instead of power conservation, the aliased components have less power than their corresponding outside spectral components. Third, not all outside components are aliased into the resolved range. Rather, only outside components inside certain well-defined regions in the spectral space are aliased inside. Numerical confirmation of these formulas has been achieved. Moreover, the soundness of these formulas is demonstrated through real examples of tidal aliasing. Furthermore, it is shown that these results can be generalized easily to the case with a coordinated constellation of satellites. The least squares methodology yields the optimal solution, that is, the best fitting, as well as yielding the least aliasing. However, the usual practice is to smooth (i.e., low-pass filter) the altimeter data onto a regular space–time grid. The framework for computing the aliasing of smoothed altimeter data is provided. The smoothing approach produces two major differences with the least squares results. First, the one-to-one correspondence of aliasing is mostly restored. Second, and more important, smoothing reduces the effective Nyquist wavenumbers to π/X = kc/2 and π/Y = lc/2, respectively (i.e., the resolved spectral space is reduced to a quarter of the size that is obtained by the least squares methodology). Ironically, the more one tries to filter out the small-scale features, the more precise the above statement becomes. However, like the least squares results, only outside components inside certain well-defined regions in the spectral space are aliased inside, and this occurs with less power. How much less depends on the characteristics of the smoother.

Corresponding author address: Chang-Kou Tai, NOAA/NESDIS, E/RA3, WWB, Rm. 601, 5200 Auth Rd., Camp Springs, MD 20746. Email: ck.tai@noaa.gov

Abstract

The aliasing problem for exact-repeat altimetric satellite sampling is solved analytically by the least squares method. To make the problem tractable, the latitudinal extent of the problem needs to be moderate for the satellite ground tracks to appear as two sets of parallel straight lines, and the along-track sampling is assumed to be dense enough to resolve any along-track features of interest. The aliasing formulas thus derived confirm the previously discovered resolving power, which is characterized by the Nyquist frequency ωc = π/T (where T is the repeat period of the satellite) and by (in local Cartesian coordinates) the zonal and meridional Nyquist wavenumbers kc = 2π/X and lc = 2π/Y, respectively (where X and Y are the east–west and north–south separations between adjacent parallel ground tracks). There are three major differences with the textbook aliasing. First, instead of the one-to-one correspondence, an outside spectral component is usually aliased into more than one inside spectral components (i.e., those inside the resolved spectral range with |ω| < ωc, |k| < kc, and |l| < lc). Second, instead of power conservation, the aliased components have less power than their corresponding outside spectral components. Third, not all outside components are aliased into the resolved range. Rather, only outside components inside certain well-defined regions in the spectral space are aliased inside. Numerical confirmation of these formulas has been achieved. Moreover, the soundness of these formulas is demonstrated through real examples of tidal aliasing. Furthermore, it is shown that these results can be generalized easily to the case with a coordinated constellation of satellites. The least squares methodology yields the optimal solution, that is, the best fitting, as well as yielding the least aliasing. However, the usual practice is to smooth (i.e., low-pass filter) the altimeter data onto a regular space–time grid. The framework for computing the aliasing of smoothed altimeter data is provided. The smoothing approach produces two major differences with the least squares results. First, the one-to-one correspondence of aliasing is mostly restored. Second, and more important, smoothing reduces the effective Nyquist wavenumbers to π/X = kc/2 and π/Y = lc/2, respectively (i.e., the resolved spectral space is reduced to a quarter of the size that is obtained by the least squares methodology). Ironically, the more one tries to filter out the small-scale features, the more precise the above statement becomes. However, like the least squares results, only outside components inside certain well-defined regions in the spectral space are aliased inside, and this occurs with less power. How much less depends on the characteristics of the smoother.

Corresponding author address: Chang-Kou Tai, NOAA/NESDIS, E/RA3, WWB, Rm. 601, 5200 Auth Rd., Camp Springs, MD 20746. Email: ck.tai@noaa.gov

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  • Chen, G., and Lin H. , 2000: The effect of temporal aliasing in satellite altimetry. Photogramm. Eng. Remote Sens, 66 , 639644.

  • Ducet, N., Le Traon P. Y. , and Reverdin G. , 2000: Global high-resolution mapping of ocean circulation from TOPEX/Poseidon and ERS-1 and -2. J. Geophys. Res, 105 , 1947719498.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Greenslade, D. J. M., Chelton D. B. , and Schlax M. G. , 1997: The midlatitude resolution capability of sea level fields constructed from single and multiple satellite altimeter datasets. J. Atmos. Oceanic Technol, 14 , 849870.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Jacobs, G. A., Born G. H. , Parke M. E. , and Allen P. C. , 1992: The global structure of the annual and semiannual sea surface height variability from Geosat altimeter data. J. Geophys. Res, 97 , 1781317828.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Le Traon, P. Y., and Dibarboure G. , 1999: Mesoscale mapping capabilities of multiple-satellite altimeter missions. J. Atmos. Oceanic Technol, 16 , 12081223.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Parke, M. E., Stewart R. H. , Farless D. L. , and Cartwright D. E. , 1987: On the choice of orbits for an altimetric satellite to study ocean circulation and tides. J. Geophys. Res, 92 , 1169311707.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Schlax, M. G., and Chelton D. B. , 1994: Aliased tidal errors in TOPEX/POSEIDON sea surface height data. J. Geophys. Res, 99 , 2476124775. Corrigendum. 101 , 18541.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Stammer, D., Wunsch C. , and Ponte R. M. , 2000: De-aliasing of global high frequency barotropic motions in altimeter observations. Geophys. Res. Lett, 27 , 11751178.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Tai, C-K., 1995: On the spatial and temporal resolving power of satellite data in repeat sampling configuration. NOAA Tech. Memo. OES 17, 21 pp.

  • Tai, C-K., 1998: On the spectral ranges that are resolved by a single satellite in exact-repeat sampling mode. J. Atmos. Oceanic Technol, 15 , 14591470.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Tai, C-K., 1999: Aliasing of satellite altimeter data in exact-repeat sampling mode: Analytic formulas for the mid-point grid. NOAA Tech. Rep. NESDIS 91, 15 pp.

  • Tai, C-K., 2001: The resolving power of a single exact-repeat altimetric satellite or a coordinated constellation of satellites: The definitive answer and data compression. NOAA Tech. Rep. NESDIS 100, 8 pp.

  • Tai, C-K., 2002: Analytic formulas for the aliasing of sea level sampled by a single exact-repeat altimetric satellite or a coordinated constellation of satellites. NOAA Tech. Rep. NESDIS 108, 30 pp.

  • Tai, C-K., 2004: The resolving power of a single exact-repeat altimetric satellite or a coordinated constellation of satellites. J. Atmos. Oceanic Technol, 21 , 810818.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Tierney, C., Wahr J. , Bryan F. , and Zlotnicki V. , 2000: Short-period oceanic circulation: Implications for satellite altimetry. Geophys. Res. Lett, 27 , 12551258.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Wunsch, C., 1989: Sampling characteristics of satellite orbits. J. Atmos. Oceanic Technol, 6 , 891907.

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