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

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 ?c = 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 |?| < ?c, |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.