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.

1. Introduction

The peculiar spatial and temporal distribution of satellite observations has presented great difficulties in determining their resolving power. That is, what maximum range, centered on the origin in the frequency–wavenumber space, is resolved by the data in the sense that spectral components within the range are sufficiently distinguishable from each other? (The spectral components are evenly distributed throughout the range and properly separated from each other, assuming the data coverage is of limited extent. This will be discussed further in the next paragraph.) In textbook cases, observations are placed on a regular space–time grid such that the Nyquist frequency and wavenumbers are readily determined. In the real world, this is rarely the case. In cases in which regular sampling is physically possible, the fiscal constraints often prevent it. Or, as in the case of the satellite, the sampling characteristics are constrained by the satellite orbit. Of particular interest is a class of satellites that repeat their ground tracks after a fixed period, thus sampling with a set (albeit irregular) pattern. These include all of the altimetric satellites that measure sea level from space. The intelligent choice of a repeat orbit configuration depends on our ability to answer two questions. First, what is the inherent resolving power of a configuration? Second, what are the approximate frequency and wavenumber spectra of the phenomena we wish to observe as well as the approximate frequency and wavenumber spectra of the noises we wish to avoid? This paper will try to answer the former, and it will discuss the latter only in passing.

The word “resolution,” depending on its usage in the data or spectral domain, unfortunately conveys two different meanings. When it is used in the data domain, it means bandwidth, which is the meaning we use in this paper. When it is applied in the spectral domain, it means how close two spectral components can get before being confused with each other, a situation that is determined by the size of the data domain (i.e., the record length in 1D time series). Since the data coverage is unlikely to be infinite, the spectral components should be properly separated lest there be energy leakage between them.

There have been two studies on the subject. Wunsch (1989, W89 hereafter) has tried to answer exactly the same question posed here, employing least squares to explore the resolving power. The least squares approach can be used to exhaust all possibilities to evaluate the maximum resolving power. However, without sufficient theoretical guidance, the cost of an exhaustive search is prohibitive. Thus, by pursuing a clearly (in retrospect) unresolvable spectral range, W89’s results are rendered meaningless. Chelton and Schlax (1994) have taken a different approach by smoothing the irregular data onto a regular space–time grid and then examining the spectral characteristics of the linear smoother (the so-called equivalent transfer function) to determine what range is well resolved after the smoothing operation. But the range clearly depends on the adopted regular space–time grid as well as on the adopted smoother with its associated parameters. The question posed here is how to find the maximum resolving power of a repeat-orbit configuration. Here we shall adopt W89’s approach but pepper it with more theoretical developments.

The data outage associated with land and data loss can severely complicate the issue and is not treated here. The idealized problem with no data gaps is solved in stages. First, in section 2, the least squares approach of W89 is formulated. Then, in section 3, the relevant 1D problem (for either space or time) is shown to involve two alternating sampling intervals. The meaning of Nyquist frequency (or wavenumber) undergoes a significant change when data are not spaced equally. Next, we demonstrate, in section 4, the important result that observations taken at different times but at regular time intervals (e.g., the exact-repeat sampling by satellites) are equivalent to simultaneous observations in terms of spectral resolution, thus simplifying the 3D problem to a 2D one. It is shown that of the three dimensions only one needs to be aligned originally. We then solve the idealized 2D problem wherein satellite tracks are straight lines in section 5. The meaning of Nyquist wavenumbers undergoes an even more dramatic change (see sections 5b and 5d). Finally, in section 6, the idealized 3D problem is solved using the results obtained for 1D and 2D problems. We conclude in section 7.

2. The least squares approach of W89

The objective is to pose the question as a least squares problem and then to examine the uncertainty of the least squares solution to see whether spectral components within a spectral range are well determined (i.e., whether they can be distinguished from each other). Fitting a multidimensional Fourier series to the data, one can write the equation as

 
Ab + v = h,
(1)

where b is a vector composed of the Fourier coefficients to be determined, h is a vector composed of data, v is a vector that represents the noises, and A is a matrix whose elements are the sine and cosine terms of the Fourier series. It can be shown that the linear optimal estimator for b is (assuming b and v are uncorrelated)

 
= (Rbb−1 + ATRvv−1A)−1ATRvv−1h,
(2)

where the superscripts T and −1 denote the transpose and inverse, respectively, and Rbb = E(bTb) and Rvv = E(vTv) are the correlation matrices. The error of the estimator is represented by

 
Ree = (Rbb−1 + ATRvv−1A)−1,
(3)

where e = b and Ree = E(eTe).

The simplification for the formidable-looking Eqs. (2) and (3) comes from the damped least squares (Lawson and Hanson 1974) or tapered least squares, as referred to by W89, where it is assumed that Rbb = σb2I and Rvv = συ2I, with I being the identity matrix and with σb, συ being scalar constants. Then Eqs. (2) and (3) become

 
= (σ2I + ATA)−1ATh,
(4)
 
Ree = συ2(σ2I + ATA)−1,
(5)

or as normalized error correlation matrix

 
Ree/σb2 = σ2(σ2I + ATA)−1,
(6)

where σ = συ/σb is the noise-to-signal ratio.

In real physical problems, the damped least squares represents a gross simplification. Yet it is perfect for studying the resolving power because a well-resolved (or definitely unresolved) spectral range should not depend on the specific signal or noise. Just as the textbook results can be applied to all signals and noises, as far as this paper is concerned, the satellite observations may be any phenomena. It may even be a mission to other planets. For those not-so-well-resolved spectral ranges (there are many examples in the rest of the paper), the problem does become signal and noise specific, and the damped least squares result should be viewed as a demonstration of possible uncertainties.

For the rest of the paper, Eq. (6) is used to examine the resolving power. Specifically, the diagonal terms give the normalized uncertainty of each spectral component with 50% or greater signaling that is completely unresolved within the spectral range. The off-diagonal terms give the correlation between spectral components. A well-resolved component has little correlation with others, while an unresolved component shows high correlation with components from which it cannot be distinguished. There is only one requirement for the noise-to-signal ratio σ: it cannot be so large that it makes every spectral component uncertain. Otherwise, the exact level is really irrelevant.

3. One-dimensional problem

At crossovers, one has a time series that contains two fixed but alternating sampling intervals. Only rarely are the two sampling intervals identical (e.g., the 1.5 day at crossovers on the equator for Seasat’s 3-day repeat). Similarly, along a fixed latitude, one gets a spatial series with alternating sampling intervals. This is also approximately true along a fixed longitude. This recurring 1D problem has significant implications. A thorough understanding is provided here.

Fitting the Fourier series to the data, the maximum number of unknowns obtainable is the number of data. When data are equally spaced, the maximum range gives the Nyquist frequency. But when data are irregularly spaced, the maximum range may not be obtainable. For example, if two data points are placed too close to each other, they cannot possibly give independent information for the range under consideration. In addition, the least squares solution is more prone to noise with irregularly spaced data. We solve two cases: one for time series at crossovers, the other for spatial series along fixed latitudes. The difference is in the number of data points. For the former, the number is 147 to model about 2 years (73 repeat cycles) of TOPEX/Poseidon (T/P), whereas 21 is used to model an east–west span of about 30° in longitude for T/P. (Note that the number of data points are deliberately kept odd to avoid the confusion at the highest frequency resulting from unevenly spaced data.) Of course, one is free to interpret 21 as about 10.5 repeat cycles and 147 as 147 tracks for any satellite. Besides, these disparate numbers provide a hint of how cases may vary according to the total number of data points.

a. Time series with two sampling intervals

Thus, with 147 data points, we solve for 147 unknowns (Fourier sine and cosine coefficients). The nonangular frequencies are integer multiples of the fundamental frequency, 1/(73.5 × T), where T is the repeat period of the satellite (e.g., 10 days for T/P). We use σ = 1 for this case, but the total noise variance versus the total signal variance is 1 to 74.

The solid line in Fig. 1 displays E, the average of all diagonal elements in Eq. (6), as a function of d, the smaller sampling interval normalized by the average sampling interval, which varies from 0.01 to 1 (where d = 1 means the data are equally spaced). It shows very little uncertainty when d = 1 to E close to 0.5 (i.e., unresolved) when d is small. The interpretation is straightforward. When d is extremely small, the data points are bunched together in pairs, essentially giving only 74 independent observations as far as the 147 unknowns being sought here are concerned. Thus, all spectral terms that would be indistinguishable from each other if we had only 74 equally spaced data become completely confused with each other (as E = 0.5 indicates). Between the two extremes, the resolution also lies in between. The individual normalized uncertainties are fairly uniform and thus close to E, the average. The uncertainty also depends on the size of the noise. The dashed line in Fig. 1 shows E versus d if σ = 2 (i.e., the error variance is quadrupled).

Fig. 1.

Results of the damped least squares formal error analysis for fitting Fourier series to time (or space) series having two alternating sampling intervals. The variable E equaling 0.5 or higher indicates that the average spectral term is completely unresolved. The solid (dashed) line is for the case with 147 data points and an a priori noise-to-signal ratio of 1 (2). The dotted line is for the case with 21 data points and a noise-to-signal ratio of 0.39.

Fig. 1.

Results of the damped least squares formal error analysis for fitting Fourier series to time (or space) series having two alternating sampling intervals. The variable E equaling 0.5 or higher indicates that the average spectral term is completely unresolved. The solid (dashed) line is for the case with 147 data points and an a priori noise-to-signal ratio of 1 (2). The dotted line is for the case with 21 data points and a noise-to-signal ratio of 0.39.

The sea level time series at a crossover can be used to deduce certain tidal constituents (e.g., Tai and Kuhn 1995). However, it does not imply that a spectral range extending as far as the tidal frequencies (e.g., S2 tide’s twice per day) is resolved by the crossover time series. It simply means that one can obtain spectral solutions (and incur aliasing) at a set of a few chosen frequencies by fitting them to the data, but the set does not constitute a spectral range (as defined in this paper) in which the spectral components have to be evenly distributed throughout the range.

b. Spatial series with two sampling intervals

In this case, one has 21 data points and 21 unknowns. To keep the total noise variance to total signal ratio at 1 to 74, σ is changed to 0.39. The dotted line in Fig. 1 shows E versus d. It is barely distinguishable from the solid line except for d < 0.1. That is, the basic conclusions hold even though the number of data points is much reduced.

c. Noise amplification

The fuzziest issue is the effect of the noise. To get a feel for the problem, we model a 2-yr time series at crossovers for T/P with high-frequency noise at M2, K1, S2, and O1 tidal frequencies. The noise amplification (i.e., the power of the aliased result divided by the power of the high-frequency noise) is displayed as a function of d for these four tidal frequencies in Fig. 2, where it shows that the vulnerability to noise is not limited to small d. Note that the noise amplification has been obtained using the damped least squares with σ = 1, without which the results would have been much worse for small d.

Fig. 2.

Fitting time series of 147 data points with the Fourier series with 147 unknowns at T/P crossovers [i.e., trying to resolve up to 1/(10 days)], the noise at M2 (solid), K1 (dashed), S2 (dotted), and O1 (chain dashed) tidal frequencies is aliased and amplified.

Fig. 2.

Fitting time series of 147 data points with the Fourier series with 147 unknowns at T/P crossovers [i.e., trying to resolve up to 1/(10 days)], the noise at M2 (solid), K1 (dashed), S2 (dotted), and O1 (chain dashed) tidal frequencies is aliased and amplified.

It is clear that trying to extend the maximum frequency to 1/(10 days) in the T/P case, leaves us vulnerable to noise. But what if we settle for 1/(20 days)? Should not the extra set of observations help reduce aliasing? The answer is yes, but not always (see also Schlax and Chelton 1994; Schrama and Ray 1994; Tai and Kuhn 1995). Figure 3 demonstrates this. Take, for example, the S2 tide, whose period is 12 h. If the time separation between the ascending and descending observations is an integer multiple of 12 h, there is little reduction in aliasing for S2. (Note that the exact separations that result in absolutely no reduction can be shown to be close to but not exact integer multiples of 12 h.) But if the separation is 6 h plus an integer multiple of 12 h, there is very little aliasing from S2. In the former case, the two sets of observations are sampling the same S2 phases, whereas in the latter, the two sets of the observations sample phases 180° apart. It can be shown for all satellites that the time separation tends to be close to integer days near the turning latitudes (66° for T/P) but close to integer days plus 12 h near the equator. The transition from the equator to the turning latitudes is fairly smooth. Thus, based on time series at crossovers, S2 aliasing is more serious near the equator or turning latitudes, and K1 (period 23.93447 h) aliasing is more prevalent near the turning latitudes.

Fig. 3.

Same as Fig. 2 except that the Fourier series has 73 unknowns [i.e., trying to resolveonly up to 1/(20 days)]. There is no noise amplification, and the aliasing is reduced in many cases.

Fig. 3.

Same as Fig. 2 except that the Fourier series has 73 unknowns [i.e., trying to resolveonly up to 1/(20 days)]. There is no noise amplification, and the aliasing is reduced in many cases.

d. Summary

Irregular sampling does not allow a clear-cut Nyquist frequency. The maximum spectral range that can be resolved depends on the problem’s tolerance for uncertainty and noise. For our particular irregular sampling with two alternating sampling intervals, the maximum resolved frequency (wavenumber) is at least 0.5/(small interval plus large interval). Whether it can be doubled to 0.5/(the average interval) is clear for the extremes but fuzzy in the transition. The noise amplification examples should make one wary of trying to extend the resolution too far using irregularly spaced data, as this will be attempted in sections 5 and 6.

4. Equivalency of regular grid and misaligned grid in spectral resolution

a. Heuristic argument and numerical verification

Wunsch (1989) asserts that there is no interesting 2D problem and proceeds directly to the 3D problem. This is the key point at which this study departs from W89. Let us take a cue from the multidimensional Fourier transform, which is done one dimension at a time. Although the observations at different locations are not simultaneous, the observations at a fixed location are regular in time, allowing easy frequency decomposition at the location. Thus, the temporal misalignment has no impact on the 3D Fourier transform in terms of spectral resolution. The same principle applies to the spatial misalignment in one of the remaining spatial dimensions. For example, let us do a 3D transform in x, y, and t. Suppose the data are aligned in x but not in y and t, even though the sampling is equally spaced in all three dimensions. The transform is done first in t at each fixed x and y. Then we transform in y at each fixed x. The transform is finished easily in x. In the following, this is verified numerically.

We adopt the following 3D grid: xm = m, yn = y0 + n, and ti = t0 + i, where m, n, i = 0, 1, 2, . . . , 10 and y0 and t0 lie in the interval [0, 1); that is, 0 ⩽ y0 and t0 < 1. Note that when y0 = t0 = 0, it is a regular grid. Now let us keep y0 = 0 but let t0 vary with x and y, which is the case of nonsimultaneous observations on a regular spatial grid. To the 3D grid, the following 3D Fourier series is fitted:

 
formula

where h represents the data at grid points, while c and s represent the unknowns. The indices vary as follows:j = 0, 1, . . . , 5 and k, l = −5, −4, . . . , 4, 5 for j > 0, whereas for j = 0, k = −5, −4, . . . , 4, 5 for l = 1, 2, . . . , 5 and k = 0, 1, . . . , 5 for l = 0 with s000 = 0. There are 1331 data points for 1331 unknowns. In Eq. (7) we adopt the convention that the frequencies are nonnegative, and with the minus sign in front of j, the plain wave propagates in the direction of the wavenumber vector. The damped least squares error analysis [i.e., Eq. (6)] confirms our conjecture. A variety of patterns for t0 are tried, culminating in a trial with random t0. Yet all cases yield exactly the same error as the case with t0 = 0 (i.e., simultaneous observations). That is, there is absolutely no penalty on spectral resolution for not having synchronized observations. Furthermore, instead of keeping y0 = 0, we randomize y0 as well. The result is the same as having y0 = t0 = 0. We get the same conclusion with a 2D grid. We have not verified this beyond 3D, but it is clear that the conclusion should hold for 4D and beyond.

b. Analytic proof

The numerical experiments indicate that misalignment induces no change in Eq. (6) for the many cases examined. The result is clearly general, and we should be able to prove it analytically. For this purpose, we use a 2D grid: xm = mΔx and yn = (n + fmy, where m = 1, 2, . . . , M and n = 1, 2, . . . , N; and if the misalignment fm = 0, it is a regular grid. To this grid a 2D Fourier series is fitted:

 
formula

where k = −K,K + 1, . . . , K − 1, K for l = 1, 2, . . . , L; k = 0, 1, . . . , K for l = 0; and M = 2K + 1, N = 2L + 1, s00 = 0. There are MN unknowns for MN grid points.

Now the objective is to show that fm induces no change in Eq. (6), which is true if the matrix ATA in Eq. (6) is unchanged. Let us first examine the element in ATA that corresponds to the unknowns ckl and ckl: that is,

 
formula

where of the four terms involving summation over n, the first two always vanish (i.e., the first two rows vanish), except when l = l′ = 0, and the last two also vanish, except when l = l′. For the exceptions, fm is multiplied by zero, thus inducing no change. One can easily come to the same conclusion for elements in ATA corresponding to ckl and skl as well as skl and skl. Thus, the misalignment induces no change in ATA and therefore no change in Eq. (6).

The interested reader may verify that all of the off-diagonal elements (i.e., when kk′ or ll′) of ATA are zeros and the diagonal elements (i.e., when k = k′ and l = l′) are MN/2 except for the element corresponding to k = k′ = l = l′ = 0, which is MN. The error correlation matrix [i.e., Eq. (6)] is always diagonal, indicating perfect resolution because there is no correlation between determined spectral components. The generalization to 3D and higher follows the same route.

c. Implications

Hence, starting from a multidimensional regular grid, misalignments (even random ones) along all dimensions except one cause no degradation of resolution; that is, these misaligned grids resolve exactly the same spectral space as the regular grid. Thus, conceptually, we can realign grids to make them regular. Most of the spatial grids to be considered in the next section are either regular or misaligned regular. But even if the spatial grid is not regular or misaligned regular, the 3D problem is reduced to a 2D one by the 3D Fourier transform argument.

However, the misalignment does induce much more complicated aliasing than the straightforward case with the regular grid. The aliasing depends on the exact nature of misalignment; thus, it has to be handled in a case-by-case basis. This is beyond the scope of this paper.

5. Idealized 2D problem

We shall restrict ourselves to 2D spectral ranges on the kl plane that are rectangles with sides parallel to the k, l axes. Tai (1995) discusses in more depth the shape of the range and the relation between the 2D range and the along-track sampling (section 5d).

a. Idealized ground tracks

Figure 4 plots an idealized track configuration with two sets of equally spaced parallel straight lines, which intersect to form identical diamonds. The configuration is completely specified with two numbers: the east–west separation between the two corners of the diamond (called X) and the north–south separation of the other two corners (called Y). Note that X and Y are simply the east–west and north–south separations between adjacent parallel tracks. For regions with limited north–south extent, the idealized tracks approximate well the real tracks on a latitude–longitude plot. As the center of the region moves latitudinally, X in longitude remains the same, while Y in latitude changes with the appropriate regional idealization. Take T/P, for example: Y is slightly less than 3X around the equator; Y = 2X around 28° latitude; Y = X near 47° and then Y shrinks fast with Y = X/2 near 57°; Y = X/3 near 61°; Y = X/4 near 63°; and Y = X/5 near 64°; etc. In terms of distance, X changes with latitude too: X equals 2.835° longitude for T/P, 1.475° for the Geosat 17-day repeat, 0.719° for the ERS-1 35-day repeat, and 8.372° for the Seasat 3-day repeat.

Fig. 4.

The idealized satellite ground tracks. The crossover grid is represented by points marked with △ or ▿ (see section 5b, which shows these points can be realigned into two regular grids). The midpoint grid is represented by points marked with •, which form a regular grid already (section 5c). The quarter-point grid is represented by points marked by ○, which cannot be realigned into any regular grid but nonetheless can undergo 2D Fourier analysis.

Fig. 4.

The idealized satellite ground tracks. The crossover grid is represented by points marked with △ or ▿ (see section 5b, which shows these points can be realigned into two regular grids). The midpoint grid is represented by points marked with •, which form a regular grid already (section 5c). The quarter-point grid is represented by points marked by ○, which cannot be realigned into any regular grid but nonetheless can undergo 2D Fourier analysis.

Note that for regions with substantial north–south extent, the Fourier harmonic analysis (to which we have restricted ourselves in this study) is no longer valid. Analysis should be based on spherical harmonics, and land cannot be ignored.

b. Two-dimensional ranges resolved by the crossovers: The crossover grid

The crossovers are special because they may have temporal resolution up to 1/T or at least have less aliasing if resolution up to 1/2T is sought (see sections 3a and 3c). See also Tai (1996) in which the extra set of sea level time series at crossovers is used to reduce aliasing (especially the tidal aliasing) in estimating the global frequency spectra. These crossover points form two independent regular grids with sampling intervals of X and Y (those marked by △ and those by ▿ in Fig. 4), each resolving the same spectral range, (−1/2X, 1/2X) in k and (−1/2Y, 1/2Y) in l, with an area of 1/XY. This is the problem of “nonsimultaneous” observations in x and y. We can realign (see section 4c) from the △ grid locations to those aligned with the ▿ grid.

There are two ways to accomplish this, resulting in two different regular grids. From △’s along fixed y’s, one can realign them to x’s that are aligned with ▿’s. Denoting these by ▴ in Fig. 4, we see that these and the ▿’s form a regular grid with intervals of X and Y/2, resolving the range, (−1/2X, 1/2X) in k and (−1/Y, 1/Y) in l, with an area of 2/XY. On the other hand, one can also realign the △’s along fixed x’s to y’s aligned with ▿’s. Denoting these by ▾ in Fig. 4, then they and the ▿’s form a grid with intervals X/2 and Y, resolving the range, (−1/X, 1/X) in k and (−1/2Y, 1/2Y) in l, also with an area of 2/XY. Numerical verification with the crossover points using these two spectral ranges shows that each range results in the perfect resolution, as if the crossovers were located on a regular grid.

Thus, one has the ultimate in ambiguity: the same data points actually support two equally well-resolved ranges. If we want to keep resolution in k and l as close as possible, the former is preferred in the Tropics and midlatitudes, while the latter is preferred in high latitudes. As if this is not confusing enough, one may pose the question, is any rectangular range, (−1/Lx, 1/Lx) in k and (−1/Ly, 1/Ly) in l, with LxLy = 2XY while Lx ⩽ 2X and Ly ⩽ 2Y, equally well resolved? Numerical verification shows that while the two ranges produced by realignment give as perfect a resolution as a regular grid, any other ranges cause degradation. As stated in section 4c, it is really the realignment that works here.

c. The 2D range resolved by midpoints between crossover points: The midpoint grid

These points (which are called the “midpoints” hereafter) are marked by • in Fig. 4, where they form a regular grid with intervals of X/2 and Y/2, resolving the range, (−1/X, 1/X) in k and (−1/Y, 1/Y) in l, with an area of 4/XY. The midpoint grid gives twice the spatial resolution that the crossover grid does.

d. Two-dimensional ranges resolved by points midway between crossover points and midpoints: The quarter-point grid

This represents an attempt to extend the resolution in one (but not both) direction at the cost of more uncertainty. The points (which are called “quarter points” hereafter) are marked by ○ in Fig. 4. They are obviously not on a regular grid, nor can they be realigned onto a regular grid. However, we can perform the 2D Fourier decomposition one dimension at a time as well. This can be done in two ways. First, for spatial series along fixed y’s, the two sampling intervals are X/4 and 3X/4, from which 1D Fourier series can be obtained as described in section 3. From Fig. 1, the uncertainty is reasonably small (for d = 0.5) for these 1D Fourier series with resolution of (−1/X, 1/X) in k. These series are available along y’s that are Y/4 apart. Thus, further transform in y resolves the range (−2/Y, 2/Y) in l, resulting in a resolved spectral area of 8/XY, which is twice the spatial resolution of the midpoints, but at the cost of more uncertainty. Second, following the same logic but switching the roles of x and y resolves the range, (−2/X, 2/X) in k and (−1/Y, 1/Y) in l, also with an area of 8/XY. As in section 5b, the method preferred depends on which direction we want more resolution.

The damped least squares error analysis is performed to verify these conjectures. A nondimensionalized grid corresponding to Fig. 4 is used with one set of crossovers at x, y = 0, 1, 2, . . . , 10 and another set of crossovers at x, y = 0.5, 1.5, 2.5, . . . , 9.5. The midpoints are midway between the crossovers, while the quarter points are midway between the midpoints and crossovers. To this grid, a 2D Fourier series is fitted:

 
formula

where k and l vary in the way specified in Eq. (8) and where s00 = 0. There are 1421 data points and (2K + 1) × (2L + 1) unknowns.

The midpoints form a 20 × 20 regular grid. Thus, the spectral range with K = L = 9 represents the well-resolved range by the midpoints. The upper panel of Fig. 5 shows the normalized uncertainty in percent for this range using σ = 1 in the damped least squares. This range is indeed well resolved, with the average uncertainty of 0.86% and no uncertainty greater than 1.25%. As discussed previously, the quarter points are equivalent to a 20 × 40 or a 40 × 20 regular grid with increased uncertainty. The middle panel of Fig. 5 shows the normalized uncertainty for the corresponding range with K = 9 and L = 19. (One can obtain the equivalent result with K = 19, L = 9.) As anticipated, the average normalized uncertainty increases to 4.39%, with the highest uncertainty at 8.62%. In addition, the vulnerability to noise amplification could be much worse, as demonstrated in section 3c. To illustrate once more the importance of our interpretation of Fourier transform with one dimension at a time, the bottom panel of Fig. 5 shows the uncertainty for the range K = L = 13. Note the number of unknown sought in this case is 729 (i.e., less than the 741 unknowns sought in the case with K = 19 and L = 9). Yet the average uncertainty has increased to 8.54%. And more importantly, the largest uncertainty is over 30%, spreading over significant portions of the range and indicating that the range is not well resolved at all.

Fig. 5.

(Top panel) The normalized uncertainty, (error variance)/(signal variance) in percent, in the well-resolved spectral range by the midpoint grid. (Middle panel) The normalized uncertainty in the extended spectral range (the resolution in y is doubled in this case) by the quarter-point grid. Or, equivalently, the resolution in x can be doubled. (Bottom panel) The normalized uncertainty in a range with extended resolutions in both directions. Even though fewer unknowns are sought with this range, the uncertainty is greatly increased. Except for the origin (i.e., the mean), the normalized uncertainty shown here is the average of those of the sine and cosine coefficients.

Fig. 5.

(Top panel) The normalized uncertainty, (error variance)/(signal variance) in percent, in the well-resolved spectral range by the midpoint grid. (Middle panel) The normalized uncertainty in the extended spectral range (the resolution in y is doubled in this case) by the quarter-point grid. Or, equivalently, the resolution in x can be doubled. (Bottom panel) The normalized uncertainty in a range with extended resolutions in both directions. Even though fewer unknowns are sought with this range, the uncertainty is greatly increased. Except for the origin (i.e., the mean), the normalized uncertainty shown here is the average of those of the sine and cosine coefficients.

We can extend the logic to get fairly high resolution along one of the directions if the problem demands it and can tolerate the increased uncertainty.

e. Along-track low-pass filtering to reduce the aliasing

One may pose the question, what happens to all of the unmarked data points? Are they wasted? The answer is no. Sections 5b, 5c, and 5d more or less show the maximum resolving power of the 2D dataset; smaller scales cause aliasing, which is where the rest of the data contribute. Along-track low-pass filtering reduces not only spatial aliasing but also temporal aliasing from high-frequency small-scale phenomena: for example, instrumental noise. The notable exceptions are the so-called basin modes and the tide error. The simplest filtering can be achieved by combining the along-track data partitioned evenly around points with the same mark. For example, we obtained the midpoint data by averaging data between crossovers. It should be emphasized that the aliasing is not eliminated by any stretch of the imagination. It is merely reduced.

f. Sacrificing the resolution in one direction to enhance it in the other direction

In the Tropics, the ocean signals of interest have much larger zonal scales than meridional scales, where unfortunately Y is substantially larger than X. Thus, Tai et al. (1989) initiated the use of grids that incorporate data over cells that are elongated in the zonal direction, such as their 10° longitude by 2° latitude grid. Of course, the tactics are sensible only for signals with disparate scales.

6. Idealized 3D problem

By combining the results of the preceding three sections, we are now in a position to answer the question posed in the introduction. As in the 2D case, there are a multitude of resolved ranges, each appropriate under the right circumstances. These are discussed one by one in the following. We have restricted ourselves to rectangular ranges in two dimensions. Here in three dimensions, we shall restrict the shape to that of a cuboid [see Tai (1995) for elaboration; also note that if the three sides of a cuboid are equal, it becomes a cube]. However, it is too optimistic to expect no new complications to result from the additional dimension. This produces some speculations on additional resolved ranges with high temporal but low spatial resolution.

a. Three-dimensional spectral ranges resolved by the crossover grid

If one is content with temporal resolution up to 1/2T with less temporal aliasing, the data resolve two 3D ranges, with the 2D ranges in section 5b complemented by the ω range of [0, 1/2T), resulting in 3D spectral volume of 1/XYT with no more increase in uncertainty than regular grids. If, on the other hand, temporal resolution up to 1/T is sought, the uncertainty is surely increased (see sections 3a and 3c). The question is by how much. Here we compute the uncertainty numerically for two T/P cases.

We adopt the following nondimensional grid: x = 0, 1, 2, . . . , 10 and y = 0, 1, 2, . . . , 4 for one set of crossovers and x = 0.5, 1.5, 2.5, . . . , 10.5 and y = 0.5, 1.5, 2.5, . . . , 4.5 for the other set of crossovers. At each crossover, t = t0, t0 + 1, t0 + 2, . . . , t0 + 6, where |t0| < 1 is the ascending or descending time in the beginning cycle (seven repeat cycles are used here). For a pair of parallel tracks, the eastern one is sampled three days later than the western one. Thus, to specify the temporal sampling, all we have to do is specify the ascending and descending times for one crossover point, say, the point x = y = 0. Two cases are examined with y = 0 corresponding to 2° and 43°N, respectively.

To this grid, the Fourier series specified by Eq. (7) is fitted. From section 5b, we know the spatial grid is equivalent to an 11 × 10 regular grid. Thus, we choose K = 5, L = 4. If we are content with resolution up to 1/2T (i.e., with J = 3), the damped least squares error analysis yields for both latitudes the same result as if the spatial grid were a regular 11 × 10 grid and the temporal sampling were synchronized. With σ = 3, the uncertainty is 0.58% for j = k = l = 0, and 1.16% for the rest. Now let us extend the resolution to 1/T (i.e., with J = 6). The results are shown in Fig. 6. The top panel displays the uncertainty for j = 0 for both cases (i.e., for y = 0 corresponding to 2° or 43°N), where it shows that the uncertainty of the temporal mean when J = 6 remains the same as when J = 3. For j > 0, the uncertainty varies only slightly with different j and is indistinguishable as displayed. Thus, only one uncertainty plot is shown for j = 1, 2, . . . , 6. The middle panel shows the uncertainty for j > 0 of the 2°N case, whereas the bottom panel shows the corresponding 43°N case. The uncertainty is indeed heightened, though not as bad as anticipated. The vulnerability to noise amplification could be much worse, though (see section 3c).

Fig. 6.

The normalized uncertainty, (error variance)/(signal variance) in percent, in the 3D spectral range resolved by the crossover grid. (Top panel) ω = 0 (i.e., the temporal mean), which is the same for two simulated T/P crossover grids with the southernmost latitude at 2° or 43°N. (Middle panel) ω ≠ 0 for the simulated T/P crossover grid with its southernmost latitude at 2°N. (Bottom panel) ω ≠ 0 for the simulated T/P crossover grid with its southernmost latitude at 43°N. The uncertainties for nonzero frequencies differ only slightly and would be indistinguishable if shown. Except for the origin (i.e., the overall mean), the normalized uncertainty shown here is the average of those of the sine and cosine coefficients.

Fig. 6.

The normalized uncertainty, (error variance)/(signal variance) in percent, in the 3D spectral range resolved by the crossover grid. (Top panel) ω = 0 (i.e., the temporal mean), which is the same for two simulated T/P crossover grids with the southernmost latitude at 2° or 43°N. (Middle panel) ω ≠ 0 for the simulated T/P crossover grid with its southernmost latitude at 2°N. (Bottom panel) ω ≠ 0 for the simulated T/P crossover grid with its southernmost latitude at 43°N. The uncertainties for nonzero frequencies differ only slightly and would be indistinguishable if shown. Except for the origin (i.e., the overall mean), the normalized uncertainty shown here is the average of those of the sine and cosine coefficients.

b. Three-dimensional range resolved by the midpoint grid

The 2D range described in section 5c is supplemented by [0, 1/2T) in ω, totaling 2/XYT in volume. This range gives the largest resolved spectral space with no greater uncertainty than a regular grid.

c. Three-dimensional ranges resolved by the quarter-point grid

Complementing [0, 1/2T) in ω to the 2D ranges depicted in section 5d, we obtain two 3D ranges, each with resolved spectral space of 4/XYT; that is, they are twice as large as the midpoint range but at the cost of increased uncertainty.

d. Three-dimensional ranges with high temporal but low spatial resolution

The following heuristic example shows that we can sacrifice spatial resolution to gain more temporal resolution. In the extreme case, if only the spatial mean is sought, then (ignoring aliasing for the moment) the mean is observed every second except when the satellite is over land. It is a resolved range in the sense that no two components in the range are confused with each other. Yet the choice is obviously not sensible. This most narrow spatial range makes noise out of all other spatial scales, entailing the need for severe smoothing. We can obtain a reasonable estimate for the mean only by averaging data that are more or less evenly distributed over the globe. An entire repeat cycle is a good choice, but then we end up with no better temporal resolution than any of the cases considered so far. The compromise is one day.

The 1-day tracks more or less cover the whole globe but they do not repeat themselves every day. The biggest longitudinal excursion from the first 1-day tracks in each repeat cycle is 14.17° for T/P, 11.80° for the Geosat 17-day repeat, and 12.22° for the ERS-1 35-day repeat. If we are content with resolving spatial scales much larger than these excursions, the fact that the patterns do not repeat exactly becomes less important. Note that the so-called exact repeat means only repeating within 1 km of the nominal track at the equator.

Satellites have near-repeat subcycles. For example, the T/P, the Geosat 17-day repeat, and the ERS-1 35-day repeat all nearly repeat every 3 days; that is, in 3 days, an evenly distributed ground-track pattern is produced and then the whole pattern is shifted to the east (west) by 2.835° or 1.475° (1.437°) longitude in the following 3 days for the T/P or Geosat (ERS-1). In addition, ERS-1 has a 16-day near-repeat subcycle, wherein the 16-day pattern is shifted to the east by 0.719° longitude in the following 16 days. However, these near-repeat subcycles drift off continuously in one direction, causing the biggest excursion of the patterns to be no better than the 1-day pattern.

These high temporal but low spatial resolution resolved spectral ranges may be useful for analyzing barotropic Rossby waves and gravity waves of basin scales, as well as large-scale environmental correction errors—for example, atmospheric pressure–related corrections and ionospheric correction. We have not verified these conjectures numerically. Thus, these are better viewed as speculations for now, to be proved or disproved later.

7. Conclusions

Wunsch (1989) fits 17 days of Geosat or T/P sampling (real or simulated) in the North Atlantic (32°–46°N, 280°–315°E) with a Fourier series in the 3D range: (−1/6°, 1/6°) in k and l, [0, 1/(4.25 days)] in ω. In retrospect, the sampling is clearly not capable of resolving this high-frequency range as confirmed by W89’s results. (See W89, Figs. 9, 13, and 15, in which substantial parts of the spectral range have uncertainties greater than 50%—i.e., unresolved.) As discussed in section 6d, in order to obtain temporal resolution that high, the spatial resolution has to be sacrificed. In any rate, it is not the special high-frequency range that is the most meaningful, it is the ranges described in sections 6a,6b, and 6c, which have the maximum resolved spectral space. Chelton and Schlax (1994) conclude that the Geosat orbit configuration is capable of resolving scales of about 3° in latitude and longitude by about 30 days. Considering that Y is about 3° in midlatitudes and X is 1.475° and T is 17 days, they come fairly close, although they still miss the perfectly resolved range (the midpoint grid) by a factor of 2.

The problem is by no means solved. The aliasing for a specific satellite and specific grid is yet to be determined. But at least we have taken the first step. One has to know what spectral ranges are resolved before one can even contemplate aliasing because aliasing by definition is the energy leaking from outside the range to inside the range. The impact of missing data, land, or large latitudinal extent is not yet clear. We still have to determine the 3D spectrum of the signal and noise before we can determine what satellite sampling configuration is the most advantageous.

Acknowledgments

John Kuhn performed the plotting. The work was supported in part under NASA’s TOPEX Altimetry Research in Ocean Circulation Program. The painstaking efforts of three reviewers resulted in a much improved paper over the original (i.e., Tai 1995). In particular, the analytic proof of the equivalency theorem presented here was entirely inspired by a derivation furnished by one of the anonymous reviewers. Another reviewer not only reviewed the original and the revised manuscripts but also the first reviews of the other two reviewers, the author’s responses to these reviews, one reviewer’s second review, and the author’s response to the second review. This reviewer deserves thanks for a truly Herculean effort. Dudley Chelton, the associate editor, handled the review process in a thoroughly professional manner: tough, fair, uncompromising, yet encouraging whenever possible.

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Footnotes

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