Abstract

The temporal aliasing formulas are derived for the Tandem Mission of Jason-1 and the Ocean Topography Experiment (TOPEX)/Poseidon. Previously, aliasing formulas were derived for a single satellite or a constellation of coordinated satellites, wherein the coordination is such that the collective samplings appear as if they were carried out by a single satellite. In this vein, Jason-1 and TOPEX/Poseidon are coordinated spatially but not temporally. The problem is treated as a general problem about the temporal phasing between any two satellites that are coordinated spatially so that the Tandem Mission is just one special case, whereas the temporally coordinated case is another special case. The present results do agree with the formulas for a constellation of coordinated satellites when the temporal phasing yields temporal coordination, as they should. The benefit of temporal coordination shows itself as having a higher spatial resolution for temporally aliased features. The physical implication is twofold. First, a high-frequency and low-wavenumber feature (e.g., the barotropic Rossby waves) has a better chance of being aliased as a low-frequency and higher-wavenumber feature in a perfectly coordinated mission than it has in the Tandem Mission. Second, in a perfectly coordinated mission, a high-frequency and high-wavenumber feature could be aliased as a low-frequency and high-wavenumber feature rather than as a low-frequency and low-wavenumber feature in the Tandem Mission. Despite the extreme mathematical complexity, the physical case is rather intuitive. Namely, the two satellites need temporal coordination to work as one in fending off temporal aliasing. Without it, the two satellites behave as two independent satellites, thus each reverting to their original (i.e., lower) spatial resolution capability in dealing with temporal aliasing.

1. Introduction

The Tandem Mission of the Jason-1 (Jason hereafter) and the Ocean Topography Experiment (TOPEX)/Poseidon (T/P hereafter) satellites was designed to enhance the spatial resolution by interleaving the ground tracks of Jason and T/P, thus doubling the spatial resolution both in the zonal and meridional directions and remedying one of T/P’s weaknesses—the lack of spatial resolution to resolve more of the mesoscale motions so they would not be aliased into larger-scale motions. After the initial verification phase (in which Jason and T/P followed the same track separated by a short distance), T/P was moved westward by 1.42° longitude to trace out the new interleaving tracks. Their equatorial crossing times are separated by about 7 min, with Jason leading, which means simultaneous measurements at the same latitude for all practical purposes. With the eventual demise of T/P in October 2005, the Tandem Mission lasted a little over 3 years.

As shown in Tai (2004, hereafter T04), the zonal and meridional Nyquist wavenumbers are kc = 2π/X and lc = 2π/Y for Jason or T/P individually, where X and Y are the zonal and meridional separations respectively between adjacent parallel ground tracks of Jason (or T/P), whereas the zonal and meridional Nyquist wavenumbers for the Tandem Mission are doubled to kce = 4π/X = 2kc and lce = 4π/Y = 2lc (where a superscript e has been used to denote parameters relevant to the Tandem Mission) because the separations are now X/2 and Y/2, respectively, between adjacent parallel tracks. The Nyquist frequency for the Tandem Mission as well as for Jason or T/P individually is ωce = π/T = ωc, where T is the repeat period of both Jason and T/P. It was initially surmised that one would have complete freedom in setting the temporal phasing between Jason and TP because that would not seem to impact the temporal resolution. Thus it was decided that Jason and T/P would stay side by side, separated zonally by X/2 at all times because this arrangement would make the computation of zonal gradients of sea level over X/2 easier by having measurements at the same time. (Note that a reviewer has kindly asked the author to change the term “easier” to “possible” in the preceding sentence. The author deems this such a pivotal concept as to warrant the addition of the following paragraph for further elaboration.)

Following T04, the reason the Nyquist period for the 10-day repeat is 20 days (even though the observations are not made simultaneously at 10-day intervals) is the result of the one-dimensional sampling theorem, which states that any band-limited time series with a bandwidth not exceeding the Nyquist can be reproduced exactly (i.e., resolved) anywhere on the time axis from the set of observations at constant intervals. Thus, let us introduce a hypothetical time scheme that all the observations are made at times that are (say) at the middle of each repeat cycle. Then at each sampling location, there are two time series, one representing the real altimetric observations, the other representing the hypothetical time scheme. The sampling theorem implies that knowing one time series is equivalent to knowing the other time series because one can derive one time series from the other as long as the series are band-limited. The errors that incur in the process are due to the aliasing (i.e., the bandwidth exceeding the Nyquist). So the conclusion is that the two time schemes have the same resolving power, that is, the same Nyquist frequency. Hence, any timing scheme essentially resolves the same band-limited signal without error. Of course, in the real world, there is the issue of aliasing to consider. It is the contention of the present paper that the timing scheme we propose is the optimal scheme. That is, it reduces aliasing to the minimum. The principles outlined above are not just some theoretical musings but have been applied to a real case in recovering the 25-day oscillations in the Argentine Basin (Tai and Fu 2005).

Subsequently, the aliasing formulas for a single satellite or a constellation of coordinated satellites were derived in Tai (2006, hereafter T06), where the coordination is such that the multisatellite samplings appear to be carried out by one single satellite. To illustrate this point further, let us consider the aliasing of a single exact-repeat altimetric satellite. The aliasing formulas depend on four constant parameters, namely, the previously defined terms of T, X, Y, and the new term, τ, which is the sampling time difference between adjacent parallel ground tracks; for example, the 3-day separation for Jason, T/P, and Geosat, and the 16-day separation for ERS-1 and -2. Hence, to make a constellation of satellites appear as if they were a single satellite, they have to be coordinated to give a new set of constant parameters, that is, Tc, Xc, Yc, and τc (where a superscript c has been used to designate parameters of the perfectly coordinated constellation). For the Tandem Mission, Te = T, Xe = X/2, and Ye = Y/2, while τe is not a constant (rather, it takes on two values: 0 or 3 days, and alternates between them). That is, the Tandem Mission is coordinated spatially but not temporally. It would have been easy to make the Tandem Mission a coordinated one. For Jason or T/P, τ = 3 days with the eastern track trailing in time for any adjacent pairs (i.e., if one defines τ as the sampling time of the eastern track minus that of the western track), but one is equally justified to say the eastern track is leading by 7 days because the 10-day repeat period, that is, τ = −7 days. The Tandem Mission would have been a perfectly coordinated one if we had adjusted the temporal phasing between Jason and T/P so that τc = 3/2 = 1.5 days, or τc = −7/2 = −3.5 days. (Note that the more precise numbers for the so-called 10-day repeat and 3-day separation are 9.9156 and 2.9669 days, respectively, for Jason and T/P.)

Not being a perfectly coordinated mission means that new temporal aliasing formulas need to be derived rather than having the advantage of having known temporal aliasing formulas. This would not be a serious complication were this the only drawback. It turns out that the perfectly coordinated mission has higher spatial resolution for the temporally aliased features, hence less aliasing into the large-scale sector. The physical implication of the lower spatial resolution capability of the Tandem Mission for temporally aliased features are as follows. First, a high-frequency and low-wavenumber feature (e.g., the barotropic Rossby waves) has a better chance of being aliased as a low-frequency and low-wavenumber feature in the Tandem Mission than it has in a perfectly coordinated mission. Second, in the Tandem Mission, a high-frequency and high-wavenumber feature has a better chance of being aliased as a low-frequency and low-wavenumber feature than in a perfectly coordinated mission.

The rest of the paper is devoted to deriving the temporal aliasing formulas for the Tandem Mission and illustrating the missed opportunity to have a perfectly coordinated mission (from a sampling point of view), that is, one that incurs the least amount of aliasing. The formulas and their derivations are extremely complicated. For most readers, the details are likely to be of little interest. Thus we have taken the step of relegating most of the mathematical developments to the appendix, while reserving the main text for the exposition of the physical significance, hoping that the lesson learned from the Tandem Mission will benefit future tandem missions; for example, in the future, Jason-2 may overlap Jason-1, and, even further in the future, Jason-3 may overlap Jason-2. Just like the aliasing formulas for a single satellite or a perfectly coordinated constellation of satellites, there are two sets of formulas depending on whether the data are processed by the least squares (LS) approach or by the smoothing approach. We shall present both sets of temporal aliasing formulas, first under the LS approach, then the smoothing approach.

2. Methodology

The peculiar sampling patterns of the exact-repeat sampling are characterized by four parameters discussed in the introduction, namely, X, Y, T, and τ. T04 has shown that the resolving power is characterized by a resolved spectral space 𝗥c, where the wavenumbers |k| < kc = 2π/X, |l| < lc = 2π/Y, and the frequencies |ω| < ωc = π/T.

T06 has shown that the spatial aliasing occurs in the following manner. A spectral component outside Rc, called α, may be indistinguishable from up to two spectral components inside Rc, called β and γ along the ascending or descending tracks. For this to occur, α has to reside in special regions in the spectral space. Section 5 (or 8) of T06 gives the LS (or smoothing) solutions to the spatial aliasing. The temporal aliasing involves one extra step. First, pure temporal aliasing causes α to appear as α′ (because of the constant sampling time difference τ between adjacent parallel tracks) along the ascending or descending tracks (see section 3 of T06), and α′ may or may not be inside Rc already. Then α′ may be indistinguishable from up to two spectral components inside Rc along the ascending or descending tracks as well. Again, for this to occur, α has to reside in special regions in the spectral space. Section 6 (or 8) of T06 gives the LS (or smoothing) solutions to the temporal aliasing.

The mechanism of aliasing as described in the previous paragraph is along the ascending tracks or descending tracks separately. Thus, to isolate this mechanism for data along the ascending or descending tracks separately, T06 has taken advantage of the linear nature of both the LS and the smoothing approaches and separated the data into the ascending part and the descending part, respectively. The ascending (or descending) part is composed of ascending (or descending) data along the ascending (or descending) tracks and zero data along the descending (or ascending) tracks [i.e., data that are zeros along the descending (or ascending) tracks]. Clearly, the data are the sum of these ascending and descending parts so defined. Hence, the LS or the smoothing approach can be applied separately to the ascending and descending parts of the data, respectively. We will call the resulting aliasing as the contribution from the ascending (or descending) tracks. Of course, the total aliasing is the sum of these two contributions.

In the following, we will apply these two principles to derive the temporal aliasing formulas for the Tandem Mission. Because the temporal aliasing involves the extra step of getting from α to α′ as described above, we will first do this in section 3.

3. The temporal aliasing when observed by ascending or descending tracks only

In appendixes A and B, we derive how a high-frequency spectral component α appears as αs along Jason tracks but as αp along T/P tracks (where a superscript of s or p denotes parameters pertaining to Jason or T/P); αs and αp differ by a phase of Δψ, which is a function of Δt (i.e., the difference of equatorial crossing times between the two satellites). It is shown that if Δt = 1.5 or −3.5 days (i.e., temporally coordinated), Δψ is an integral multiple of 2π (i.e., αs = αp). Otherwise αs and αp are two different waves.

4. The temporal aliasing formulas under the least squares approach

As described in section 2, the next step is to find whether and how αs and αp are aliased into the resolved spectral range, Rce, of the Tandem Mission (recall that a superscript e is used to denote all things pertaining to the Tandem Mission). The complication is that αs and αp each can be indistinguishable from up to four spectral components that are inside Rce, compared with the situation of only up to two spectral components (i.e., β and γ) described in section 2.

The details are too exhaustive to be included in the main text. We have therefore relegated them to appendixes CH. Appendix C describes the contribution from the descending tracks of Jason (see section 2 for the definition of contribution). Appendix D gives an example of how the results of appendix C are derived. Appendix E lists the contribution from the descending tracks of T/P with the help from appendix B. Appendix F adds up the combined contribution of the descending tracks of Jason and T/P from appendixes C and E, whereas we give the corresponding results for a perfectly coordinated mission in appendix G, where the results are the same as those derived in T06, with kc and lc replaced by 2kc and 2lc.

Comparing appendix F (Tandem results) with appendix G (results of the perfectly coordinated mission), it is clear that as far as the temporal aliasing is concerned, the effective Nyquist wavenumbers revert back to the single-satellite case because Jason and T/P have to be treated individually in the Tandem Mission rather than collectively in the perfectly coordinated case. The end result is that there are many more spectral components, into which the temporal aliasing can take place, and many more chances to masquerade as large-scale motions.

Appendix H gives the combined contribution from the ascending tracks of Jason and T/P for the Tandem Mission. One gets the entire effect of temporal aliasing for the Tandem Mission by combining the contributions from the descending tracks (appendix F) with those from the ascending tracks (appendix H).

Despite the complexity of these results as documented in these appendixes, the physical cause is rather easy to comprehend. One needs the temporal coordination for the two satellites to work as one in order to double the spatial resolution both in the zonal and meridional directions for temporal aliasing. Without the temporal coordination, the two satellites do not work together to enhance the spatial resolution for temporal aliasing but revert to their original spatial resolution, hence resulting in more chances for the temporal aliasing to be aliased into the largest spatial scales.

5. The temporal aliasing formulas if the data are smoothed onto a regular grid

Most altimetric analyses smooth the data onto a variety of regular space–time grids by a variety of smoothers (i.e., low-pass filters). T06 has made the assertion that the LS methodology produces the least aliasing, but the explanation is brief. Here we present a fuller explanation, which distinguishes two sources of aliasing, namely, the inherent aliasing and the artificial aliasing. By the inherent aliasing, we mean the aliasing that arises from the sampling’s inability to distinguish two spectral components. Thus the inherent aliasing is unavoidable no matter what method is used to process the data. The artificial aliasing, on the other hand, refers to the inefficiencies of the method deployed to treat the data. In the LS approach, all the aliasing arises from the sampling’s inability to distinguish two spectral components. There is no artificial aliasing in the LS approach. On the other hand, the smoothing strategy induces a lot of artificial aliasing in addition to the inherent aliasing (see elaborations below). Despite its superior quality, it is much harder to apply the LS methodology. In practice, almost all investigations have adopted the smoothing approach.

One can get a better grasp of the situation if we contrast the altimetric sampling with the textbook ideal case where data are available continuously, so filtering can be regarded as a convolution operation. Then the Fourier transform of the filtered data is the multiplication of the Fourier transform of the data with the Fourier transform of the filter [e.g., the convolution theorem in Bracewell (1986, 108–112)]. However, when a well-defined filter (i.e., one with a known Fourier transform) is applied to the altimetric sampling (i.e., along-track altimeter data repeating at the repeat period), the result is anything but clear-cut. A spectral component when filtered in the textbook ideal case would only have its magnitude changed but remain at the same frequency and wavenumbers. Yet, in the real case, the filtering causes leakage into many other frequencies and wavenumbers. This leakage is in fact what causes the artificial aliasing, even when the optimal smoother is adopted.

The amount of artificial aliasing that is incurred in the smoothing approach, of course, depends on the exact nature of the smoother that is employed. However, two sources of artificial aliasing can be readily identified. The first and more dominant source is the leakage from the unresolved high-frequency and/or high-wavenumber spectral components. An ideal smoother would smooth away all the high-frequency and/or high-wavenumber spectral components that are not resolved by the sampling (aside from those spectral components that induce the inherent aliasing, which no smoothing can remove) lest their residuals on the regular grid become the artificial aliasing. The residual of a high-frequency and/or high-wavenumber spectral component becomes a combination of many low-frequency and low-wavenumber spectral components. Nevertheless, one should always strive to use a smoother that would be capable of removing the unresolved spectral components if the data were continuous. Failing to do so will cause even more artificial aliasing. The second and lesser source is the leakage from the resolved spectral components into other components in the resolved spectral range.

a. The smoothing approach for a single satellite

Notwithstanding the wide variety of smoothers and regular grids being utilized, it is possible to set out the common framework for computing the aliasing when smoothing is utilized, from which one can deduce some common characteristics of the aliasing under smoothing. The smoothing operation (low-pass filtering) can be written as

 
formula

where ĥ(xi,yi,ti) is the smoothed data on a regular space–time grid of (xi,yi,ti), h(xj,yj,tj) is the along-track raw data at (xj,yj,tj), Aij is the smoothing coefficient, and the summation is effectively only over the raw data inside some sphere of influence around (xi,yi,ti) appropriate for the search radius of the smoother. That is, if a raw data point j is outside the sphere of influence of the smoothed data point i, then Aij = 0. Moreover, one can separate the raw data according to ascending or descending data. And (1) can be rewritten as

 
formula

where (Aa)ij = Aij if j is an ascending point, otherwise (Aa)ij = 0; likewise, (Ad)ij = Aij if j is a descending point, otherwise (Ad)ij = 0. Symbolically, we can rewrite (1) and (2) as

 
formula

where ĥ and h are the smoothed and raw data vectors, respectively; and 𝗔, 𝗔a, and 𝗔d are matrixes representing the smoothing operation as well as its ascending and descending parts, respectively.

Clearly the smoothing operation is a linear operator. Thus one is free to consider individual spectral components independently. To be a well-designed low-pass filter, it must pass without distortion the truly large-scale and long-period spectral components. Thus substituting the spatial and temporal means of h and ĥ (which must be the same for an unbiased smoother) in (1) and (2) leads to the following formula, which is true for all i:

 
formula

If the sphere of influence is large enough (coupled with a well-chosen regular grid) to have roughly equivalent distributions of ascending and descending points, then (4) becomes

 
formula

At the risk of being simplistic, we can characterize the smoother (low-pass filter) by two sets of spectral limits: First, let ko, lo, and ωo be the spectral limits beyond which the high-wavenumber and/or high-frequency spectral components (if they are not aliased inside, i.e., if they cannot masquerade as low-wavenumber and low-frequency components along one set of parallel tracks) would be smoothed out by the smoother if data were continuous. Second, let k1, l1, and ω1 be the spectral limits within which the low-wavenmber and low-frequency spectral components would not be distorted by the smoother if the data were continuous. To put these statements into mathematical formulation, let us define 3 three-dimensional spectral ranges (i.e., cuboids, in the jargon of three-dimensional geometry): Ro for spectral components with |k| < ko, |l| < lo, and |ω| < ωo; R1 for spectral components with |k| < k1, |l| < l1, and |ω| < ω1; and Rc with |k| < kc, |l| < lc, and |ω| < ωc (where kc, lc, and ωc are the zonal and meridional Nyquist wavenumber and Nyquist frequency, respectively, for a single satellite). In the following, we will ignore the artificial aliasing for the purpose of deriving the formulas for the inherent aliasing.

Let α be a spectral component outside Ro and let hα be the raw data vector representing α. If α is distinguishable from all spectral components inside Ro along the ascending (and/or descending) tracks, then

 
formula

where 0 is the null vector on the regular grid. Also, let β be a spectral component inside R1 and let hβ and ĥβ be the data vectors representing the spectral component β on the raw and regular grids, respectively, before smoothing; then

 
formula

Moreover, if (5) is true,

 
formula

It is clear that k1 < ko, l1 < lo and ω1 < ωo (i.e., R1 lies inside Ro). The more one smoothes, the smaller Ro and R1 become. It is also clearly desirable to have kokc, lolc and ωoωc (i.e., Ro should be no larger than Rc) lest any remnants from unresolved spectral components linger on after smoothing. Because Ro also determines the density of the regular grid (note that it makes no sense to have a regular grid whose Nyquist frequency and wavenumbers are greater than ωo, ko, and lo when the smoother tries to smooth out any high-frequency and/or high-wavenumber spectral terms outside Ro in the raw data), the regular grid should be no denser than the regular grid with Δx = X/2, Δy = Y/2, and Δt = T, because to do otherwise is not only a waste of resources but also misleading by giving the impression that higher resolution has been achieved.

b. The smoothing approach for the Tandem Mission

First, with two satellites, (1)(5) can be rewritten as

 
formula

where Aijs = Aije if j is a Jason data point, Aijs = 0 otherwise; similarly, Aijp = Aije if j is a T/P data point, Aijp = 0 otherwise (recall that a superscript s, p, or e is used to denote the parameters pertaining to Jason, T/P, or the Tandem Mission). And

 
formula

where we have segregated the smoothing coefficients further into ascending and descending parts, as in (2). Symbolically, we can rewrite (1)e and (2)e as

 
formula

where the symbols should be self-explanatory if one compares them with those in (3). Similarly, for A to be an unbiased estimator, the following equation must be true:

 
formula

By the same token, if the sphere of influence is large enough (coupled with a well-chosen regular grid) to have roughly equivalent distributions of ascending and descending points from both Jason and T/P, then (4)e becomes

 
formula

Then, there is the matter of choosing the proper smoother for the Tandem Mission. The Nyquist frequency and wavenumbers for the Tandem Mission are ωce = π/T, kce = 4π/X, and lce = 4π/Y. Clearly the purpose of the mission is to enhance the spatial resolution, which it apparently achieved when there was no temporal aliasing. But as far as the temporal aliasing is concerned, the spatial resolution reverts back to the single-satellite case (see section 4). It would be safe to use the smoother that is appropriate for a single satellite so as to avoid the worsening of the temporal aliasing. But it would be maddening to give up on the newly gained spatial resolution when there is no temporal aliasing. Thus we will use the smoother that is compatible with the Nyquist frequencies and wavenumbers of the Tandem Mission and suffer the consequences in the temporal aliasing. Hence, we will choose a Roe such that koekce, loelce, and ωoeωce, and (6)(8) are modified as follows.

Let α be a spectral component outside Roe and let hα be the raw data vector representing α. If α is distinguishable from all spectral components inside Roe along the ascending (and/or descending) tracks of Jason and T/P, then

 
formula

Also, let β be a spectral component inside R1e and let hβ and ĥβ be the data vectors representing the spectral component β on the raw and regular grids, respectively, before smoothing; then

 
formula

Moreover, if (5)e is true,

 
formula

c. Contributions from the descending tracks of Jason and T/P

In appendix F, it is shown where α = cos(kx + lyωt + ψ) with |ω| > ωc must lie in the spectral space for the temporal aliasing to occur along the descending tracks of Jason and T/P. Appendix F also shows that α can be indistinguishable from up to four spectral components in Rce, called βs (or βp), along the descending tracks of Jason (or T/P). For any of the pairs of βs and βp (which only differ from each other by a phase), the temporal aliasing can be written as

 
formula

However, it is advantageous to use the pair that has the largest spatial scales because if they lie inside R1e and (5)e is true, then (9) can be simplified to

 
formula

d. Contributions from the ascending tracks of Jason and T/P

Similarly, it is shown in appendix H where α = cos(kx + lyωt + ψ) with |ω| > ωc must lie in the spectral space for the temporal aliasing to occur along the ascending tracks of Jason and T/P. Appendix H also shows that α can be indistinguishable from up to four spectral components in Rce, called βs (or βp), along the ascending tracks of Jason (or T/P). For any of the pairs of βs and βp (which only differ from each other by a phase), the temporal aliasing can be written as

 
formula

However, it is advantageous to use the pair that has the largest spatial scales because if they lie inside R1e and (5)e is true, then (11) can be simplified to

 
formula

Again, as in section 4, the effective Nyquist wavenumbers for the temporal aliasing are kc and lc instead of kce = 2kc and lce = 2lc, thus providing more opportunities for high-frequency small-scale motions to masquerade as low-frequency large-scale motions.

6. Discussion

The temporal aliasing formulas for the Tandem Mission are derived here. The lack of temporal coordination between Jason and T/P makes the formulas far more complicated than those of a perfectly coordinated mission. More significantly, the lack of temporal coordination causes the effective Nyquist wavenumber for temporal aliasing to revert back to the original Nyquist wavenumbers of Jason or T/P when acting independently because Jason and T/P are acting independently on a temporal basis, thus reducing both the zonal and meridional spatial resolution by half for temporal aliasing and affording more chances for high-frequency features to be mistaken as low-frequency and low-wavenumber features rather than low-frequency and higher-wavenumber features in a perfectly coordinated mission. The lesson is that if there should be another chance to have a Tandem-like mission, it needs to be a perfectly coordinated mission to get the full benefits of the enhanced spatial resolution.

Acknowledgments

Two anonymous reviewers have made many constructive suggestions and are much appreciated. The views, opinions, and findings contained in this report are those of the author and should not be construed as an official National Oceanic and Atmospheric Administration or U. S. Government position, policy, or decision.

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The resolving power of a single exact-repeat altimetric satellite or a coordinated constellation of satellites.
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Tai
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On the aliasing of sea level sampled by a single exact-repeat altimetric satellite or a coordinated constellation of satellites: Analytic aliasing formulas.
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APPENDIX A

The Temporal Aliasing When Observed by Ascending or Descending Tracks Only

As explained in T06 (see his sections 2 and 3), the temporal aliasing along parallel tracks of an exact-repeat satellite induces not only an aliased frequency, ω′, but also an additional phase variation (resulting from the constant sampling time difference τ between adjacent tracks) that is the same as the phase variation of a plane wave propagating in the direction perpendicular to the tracks, thus inducing an additional wavenumber vector perpendicular to the tracks.

Because the Tandem Mission is not coordinated temporally (such that Jason and TP can masquerade as a single satellite), the temporal aliasing gives rise to two distinct spectral components with the same aliased frequency ω′ [see Eq. (1) of T06 for how ω′ is determined] and the same additional wavenumber vector but with differing phases. That is, rather than jointly producing one propagating wave, Jason and T/P produce two waves.

T06 has derived the temporally aliased spectral components for a single satellite or a constellation of coordinated satellites. For the Tandem Mission, there are two complications. First, the additional wavenumber vector in T06 has been selected to have the minimum absolute value. In light of the enhanced spatial resolution of the Tandem Mission, it may no longer be appropriate. Thus, Eqs. (5) and (6) of T06 need to be modified. We rewrite the temporal-aliasing-induced additional wavenumber vector for the Tandem Mission as (recalling that a superscript e denotes parameters of the Tandem Mission)

 
formula
 
formula

where kω and lω are as defined in T06, and j = 0 or ±1 (which value is appropriate will be determined below). (Note that when j = ±1, we are essentially adding 2π to or subtracting 2π from the phase variation induced by the temporal aliasing.) Second, the formulas in T06 are derived by placing the spatial origin on one of the crossovers. Here we choose to put the origin on a Jason crossover. Hence, the formulas for T/P require a spatial coordinate transformation, which is examined in detail in appendix B.

Thus, from T06, a spectral component α = cos(kx + lyωt + ψ) with |ω| > ωc appears as αs along Jason tracks (recall that a superscript s denotes parameters specific to Jason) as follows:

 
formula

or

 
formula

where (defining J = <ω/(2ωc)> as in T06, where <w> gives the nearest integer to w)

 
formula

and

 
formula

where tas and tds are the ascending and descending times of Jason at x = 0 and y = 0.

Meanwhile, using a spatial coordinate system (xp,y) with its origin on the T/P crossover just to the west of the Jason crossover at x = 0 and y = 0, i.e., xp = x + X/2 (recall that a superscript p denotes parameters specific to T/P). From T06, the same spectral component [but in the (xp,y) coordinates native to T/P], that is, α′ = cos(kxp + lyωt + ψp), appears as αp along TP tracks (noting that ψ = ψp+ kX/2) as follows:

 
formula

or

 
formula

where

 
formula

and

 
formula

where tap and tdp are the ascending and descending times of T/P at xp = 0 and y = 0. When (A7) and (A8) are transformed to the (x,y) coordinate system, αp becomes

 
formula

or

 
formula

The phase difference between the two waves [i.e., the phase of (A11) minus the phase of (A3), or the phase of (A12) minus the phase of (A4)] is

 
formula

where Δt = tastap = tdstdp. For the Tandem Mission, Δte = 0, thus leaving a phase difference of

 
formula

and making (A3) and (A11) [or (A4) and (A12)] two different waves.

However, for a perfectly coordinated mission, Δt = τ/2 and (A13) becomes (recall that a superscript c denotes parameters pertaining to the perfectly coordinated mission)

 
formula

where Eq. (5) of T06 and the fact that kc = 2π/X and ωc = π/T have been used. Because for a perfectly coordinated mission (A3) and (A11) [or (A4) and (A12)] remain the same wave, the integer in the square brackets in (A15) must be even. Now j can be determined as follows:

 
formula
 
formula
 
formula

The reader can verify that (A16)(A18) can be derived directly from the definitions in Eqs. (5) and (6) of T06 but using X/2 and Y/2 as the separations of adjacent parallel tracks.

For the Tandem Mission, the differences between the two waves [i.e., (A3) minus (A11) and (A4) minus (A12)] are, respectively,

 
formula
 
formula

And the sums of (A3) and (A11) or (A4) and (A12) are, respectively,

 
formula
 
formula

APPENDIX B

Spatial Coordinate Transformation for TP Aliasing Formulas

The aliasing formulas for a single exact-repeat satellite or a coordinated constellation of satellites were derived by placing the origin of the spatial coordinates on one of the crossovers. For the Tandem Mission, we place the origin on one of the Jason crossovers. Hence none of the crossovers of T/P coincide with the spatial origin. This causes the aliasing formulas for T/P to undergo some phase shifts.

For simplicity, let us put the origin of (x,y) on a Jason crossover and put the origin of another coordinate system, say (xp,y) (i.e., using the superscript p to denote parameters native to T/P), on the T/P crossover that is the closest to the (x,y) origin but to the west of it, that is, xp = x + X/2. Then a spectral component appears as cos(kx + ly + ϕ) under the (x,y) coordinates and as cos(kxp + ly + ϕp) under the (xp,y) coordinates; thus,

 
formula

From appendix A of T06, cos(kxp + ly + ϕp) and cos(kxp + ly + ϕp) are indistinguishable along T/P descending (or ascending) tracks if kk′ = nkc and ll′ = ± nlc (where in ±, the minus sign applies for ascending tracks). When these two spectral components are transformed to the (x,y) coordinates using (B1), they become cos(kx + ly+ ϕ + kX/2) and cos(kx + ly + ϕ + kX/2), where the second term can be written as cos(kx + ly + ϕ + kX/2 + nkcX/2) = cos(kx + ly + ϕ + kX/2 + ) = (−1)ncos(kx + ly + ϕ + kX/2) by noting that kc = 2π/X and kk′ = nkc.

That is, under the (x,y) coordinates, cos(kx + ly+ ψ) and (−1)ncos(kx + ly + ψ) are indistinguishable along T/P descending (or ascending) tracks if kk′ = nkc and ll′ = ±nlc (where in ±, the minus sign applies for ascending tracks). Thus not having the freedom to place the spatial origin on one of the crossovers of T/P introduces this somewhat annoying (−1)n factor into all of the aliasing formulas pertaining to T/P under the (x,y) coordinates.

APPENDIX C

LS Contribution from the Descending Tracks of Jason

The LS methodology for finding the aliasing formulas is explained in detail in T06 (see his section 4). In section 2 of this paper, we have given a sketch of how it is done for a single satellite. The complications for the Tandem Mission are that there are now four sets of tracks to consider, and the resolved spatial spectral range is quadrupled in size. These derivations are quite involved, an example of which is given in appendix D, while we present the results here.

In T06, it is shown that the aliasing formulas are different depending on which quadrant of the resolved spatial spectral range the aliased spectral components lie in. Here the size of the resolved spatial spectral range of Rce (where |k| < 2kc and |l |< 2lc) is quadrupled from the spatial spectral range of Rc. Let us call the two-dimensional resolved spectral surface with a fixed frequency (in this case ω′) by the name of S. It turns out that S needs to be divided into 16 subregions with each quadrant further divided into four equal parts in size, which resembles a chessboard with 16 squares, the center of which is at k = ikc/2 and l = jkc/2; here i and j take on the values of ±1 and ±3. For this purpose, we invent a new function, «w», which gives the value of m + ½ if mw < m + 1 (where m is a positive integer or zero) or if m < wm + 1 (where m is a negative integer). Thus for a spectral component (k,l) in S one determines to which subregion it belongs by «k/kc»kc and «l/lc»lc, which gives the center of the subregions. We label the subregion by the double subscripts as 2«k/kc» and 2«l/lc», where 2 is inserted to avoid having fractional values for the subscripts, that is, the subregion Sij has its center at k = ikc/2 and l = jlc/2.

For the equation numbers in the following subsections, we use the same number but add a superscript s, p, e, or c for corresponding equations specific to Jason, T/P, the Tandem Mission, or the perfectly coordinated mission, respectively. For the contributions from descending tracks (which will be presented first), we will present the results in more detail to illustrate the intricacies: first the contribution from Jason descending tracks followed by its counterpart from T/P, then the combined contribution from both Jason and T/P (i.e., the Tandem Mission), and finally the combined contribution if Jason and T/P were perfectly coordinated. For the contributions from the ascending tracks, rather than replicating the details one more time, we will simply present the Tandem Mission results.

A spectral component cos(kx + lyωt + ψ) with |ω| > ωc, letting k + ke = k′ + nkc, l + le = l′ + nlc (where n is an integer and |k′| < 2kc and |l′| < 2lc, and see appendix A to get ke and le) and defining k″ = k′ ± 2kc if ± k′ < 0 and l″ = l′ ± 2lc if ± l′ < 0 [note that k′ and k″ (or l′ and l″) are of the opposite signs with each other, and their absolute values add up to 2kc (or 2lc)], is aliased as follows:

  • (a) If kl′ ≠ 0 and (k,′l′) falls in the subregions of S11 or S−3 – 3 (S33 or S−1 – 1), while we define k′″ = k′ ± kc, l′″ = l′ ± lc, and k″″ = k″ ± kc, l″″ = l″ ± lc (where the minus sign applies when it falls in S33, or S−1 – 1), then the aliasing is 
    formula
  • (b) If kl′ ≠ 0 and (k′,l′) falls in the subregions of S31 or S−1 – 3 (S13 or S−3 – 1), while we define k′″ = (k′ + k″)/2, l′″ = (l′ + l″)/2, and k″″ = k′″ ± 2kc if ±k′″ < 0, l″″ = l′″ ± 2lc if ±l′″ < 0, then the aliasing is 
    formula
  • (c) If kl′ ≠ 0 and (k′,l′) falls in the subregions of S1 – 1 or S−11, while we define k′″ = k′ + kc, l′″ = l′ + lc, and k″″ = k′ − kc, l″″ = l′ − lc, then the aliasing is 
    formula
  • (d) If kl′ ≠ 0 and (k′,l′) falls in the subregions of S1 – 3 or S−31(S3–1 or S−13), while we define k′″ = k′ ± kc, l′″ = l′ ± lc (where the minus sign applies when it falls in S3 – 1 or S−13), and k″″ = k′″ ± 2kc if ±k′″ < 0, l″″ = l′″ ± 2lc if ±l′″ < 0, then the aliasing is 
    formula
  • (e) If kl′ ≠ 0 and (k′,l′) falls in the subregions of S3 – 3 or S33, then the aliasing is 
    formula
  • (f) If k′ = 0 and |l′| < lc (or if l′ = 0 and |k′| < kc), while we define k′″ = k′ + kc, l′″ = l′ + lc, and k″″ = k′ − kc, l″″ = l′ − lc, then the aliasing is 
    formula
  • (g) If k′ = 0 and |l′| ≥ lc (or if l′ = 0 and |k′| ≥ kc), while we define k′″ = k′ ± kc, l′″ = l′ ± lc if ±l′ < 0 (or if ±k′ < 0), and k″″ = k′″ ± 2kc if ±k′″ < 0, l″″ = l′″ ± 2lc if ±l′″ < 0, then the aliasing is 
    formula

APPENDIX D

Derivation of Eq. (C1) as an Example of How the Formulas in Section 4 Are Derived

Recalling the terminology of section 2 and from Eq. (A4), a spectral component α = cos(kx + lyωt + ψ) (where |ω| > ωc) appears as αs = cos[(k + ke)x +(l + le)yωt + ψds] along the descending tracks of Jason. The αs is indistinguishable from β′ = cos(kx + lyωt + ψds), β″ = cos(kx + lyωt + ψds), β′″ = cos(k′″x + l′″yωt + ψds), and β″″ = cos(k″″x + l″″yωt + ψds) if k′, l′, k″, l″, k′″, l′″, k″″, and l″″ are as defined in appendix C(a).

Let the temporal aliasing formulas for appendix C(a) be

 
formula

Then the least squares fitting residual is proportional to

 
formula

where the first term of (D2) is the result of fitting (D1) to α′ along the descending tracks of Jason, and the second term of (D2) is the result of fitting (D1) to zeros along the descending tracks of T/P, whereas the third term of (D2) is the result of fitting (D1) to zeros along the ascending tracks of both Jason and T/P.

Letting the partial derivatives of (D2) with respect to q′, q″, q′″, and q″″ vanish leads to the following:

 
formula
 
formula
 
formula
 
formula

the solution of which leads to (C1), that is,

 
formula

Despite the complexity, our approach of treating the problem as a general problem about the temporal phasing between two spatially coordinated satellites does provide and confirm a consistency check (i.e., when the temporal phasing is 1.5 or −3.5 days, the two satellites become temporally coordinated, and the present results should agree with previous results for perfectly coordinated satellites in T06).

APPENDIX E

LS Contribution from the Descending Tracks of T/P

As noted in appendix B, a (−1)n factor is introduced into T/P aliasing formulas because the spatial origin does not fall on one of the T/P crossovers. Moreover, from appendix A, a phase shift Δψ = keX/2 [see (A14)] is introduced into T/P formulas from the Jason formulas so the same high-frequency spectral component appears as two different temporal aliases, one for Jason and one for T/P.

Thus, a spectral component cos(kx + lyωt + ψ) with |ω| > ωc (while defining k′, l′, k″, and l″ as before in appendix C) is aliased as follows:

  • (a) If kl′ ≠ 0 and (k′,l′) falls in the subregions of S11, S−3 – 3, S33, or S−1 – 1, while we define k′″, l″″, k″″, and l″″ as in appendix C(a), then the aliasing is 
    formula
  • (b) If kl′ ≠ 0 and (k′,l′) falls in the subregions of S31, S−1 – 3, S13, or S−3 – 1, while we define k′″, l″″, k″″, and l″″ as in appendix C(b), then the aliasing is 
    formula
  • (c) If kl′ ≠ 0 and (k′,l′) falls in the subregions of S1 – 1 or S−11, while we define k′″, l″″, k″″, and l″″ as in appendix C(c), then the aliasing is 
    formula
  • (d) If kl′ ≠ 0 and (k′,l′) falls in the subregions of S1 – 3, S−31, S3 – 1, or S−13, while we define k′″, l″″, k″″, and l″″ as in appendix C(d), then the aliasing is 
    formula
  • (e) If kl′ ≠ 0 and (k′,l′) falls in the subregions of S3 – 3 or S−33, then the aliasing is 
    formula
  • (f) If k′ = 0 and |l′| < lc (or if l′ = 0 and |k′| < kc), while we define k′″, l″″, k″″, and l″″ as in appendix C(f), then the aliasing is 
    formula
  • (g) If k′ = 0 and |l′| ≥ lc (or l′ = 0 and |k′| ≥ kc), while we define k′″, l″″, k″″, and l″″ as in appendix C(g), then the aliasing is 
    formula

APPENDIX F

The Combined LS Contribution from the Descending Tracks of Jason and T/P

After combining the contributions from Jason and T/P as derived in appendixes C and E, a spectral component cos(kx + lyωt + ψ) with |ω|>ωc (while defining k′, l′, k″, and l″ as before in appendix C) is aliased as follows:

  • (a) If kl′ ≠ 0 and (k′,l′) falls in the subregions of S11, S−3 – 3, S33, or S−1 – 1, while we define k′″, l″″, k″″, and l″″ as in appendix C(a), then the aliasing is 
    formula
  • (b) If kl′ ≠ 0 and (k′,l′) falls in the subregions of S31, S−1 – 3, S13, or S−3 – 1, while we define k′″, l″″, k″″, and l″″ as in appendix C(b), then the aliasing is 
    formula
  • (c) If kl′ ≠ 0 and (k′,l′) falls in the subregions of S1–1 or S−11, while we define k′″, l″″, k″″, and l″″ as in appendix C(c), then the aliasing is: 
    formula
  • (d) If kl′ ≠ 0 and (k′,l′) falls in the subregions of S1 – 3, S−31, S3 – 1, or S−13, while we define k′″, l″″, k″″, and l″″ as in appendix C(d), then the aliasing is 
    formula
  • (e) If kl′ ≠ 0 and (k′,l′) falls in the subregions of S3 – 3 or S−33, then the aliasing is 
    formula
  • (f) If k′ = 0 and |l′| < lc (or if l′ = 0 and |k′| < kc), while we define k′″, l″″, k″″, and l″″ as in appendix C(f), then the aliasing is 
    formula
  • (g) If k′ = 0 and |l′| ≥ lc (or if l′ = 0 and |k′| ≥ kc), while we define k′″, l″″, k″″, and l″″ as in appendix C(g), then the aliasing is 
    formula

Appendix G

The Combined LS Contribution from the Descending Tracks of Jason and T/P If They Were Perfectly Coordinated

If Jason and T/P were perfectly coordinated, Δψ = 2mπ [where m is an integer; see (A15)(A18)]; that is,

 
formula

and

 
formula

Using (G1) and (G2), a spectral component cos(kx + lyωt + ψ)with |ω|>ωc (while defining k′, l′, k″, and ll′ as before in appendix C) would be aliased as follows:

  • (a) If kl′ ≠ 0 and (k′,l′) falls in the subregions of S11, S−3 – 3, S33, or S−1 – 1, while we define k′″, l‘′″, k″″, and l″″ as in appendix C(a), then the aliasing would be 
    formula
  • (b) If kl′ ≠ 0 and (k′,l′) falls in the subregions of S31, S−1 – 3, S13, or S−3 – 1, while we define k′″, l‘′″, k″″, and l″″ as in appendix C(b), then the aliasing is 
    formula
  • (c) If kl′ ≠ 0 and (k′,l′) falls in the subregions of S1 – 1 or S−11, while we define k′″, l‘′″, k″″, and l″″ as in appendix C(c), then the aliasing is 
    formula
  • (d) If kl′ ≠ 0 and (k′,l′) falls in the subregions of S1 – 3, S−31, S3 – 1, or S−13, while we define k′″, l‘′″, k″″, and l″″ as in appendix C(d), then the aliasing is 
    formula
  • (e) If kl′ ≠ 0 and (k′,l′) falls in the subregions of S3 – 3 or S−33, then the aliasing is 
    formula
    and no aliasing occur if n is odd.
  • (f) If k′ = 0 and |l′| < lc (or if l′ = 0 and |k′| < kc), while we define k′″, l‘′″, k″″, and l″″ as in appendix C(f), then the aliasing is 
    formula
  • (g) If k′ = 0 and |l′| ≥ lc (or l′ = 0 and |k′| < kc), while we define while we define k′″, l‘′″, k″″, and l″″ as in appendix C(g), then the aliasing is 
    formula

These temporal aliasing formulas for a perfectly coordinated mission are the same as those in T06 [see his Eqs. (26)–(28)], with kc and lc replaced by 2kc and 2 lc. Thus n has to be even for the elements involving (k′,l′) and (k″, l″) to be relevant, whereas if n is odd, the elements involving (k′″,l′″) and (k″″,l″″) become relevant because k + ke = k′″ − 2mkc and l + le = l′″ − 2mlc, where m is an integer.

Appendix H

The Combined LS Contribution from the Ascending Tracks of Jason and T/P

A spectral component cos(kx + lyωt + ψ)with |ω|>ωc, letting k + ke=k′ + nkc, l − le = l′ − nlc (where n is an integer and |k′| < 2kc and |l′| < 2lc) and defining k″ and l″ as in appendix C, is aliased as follows:

  • (a) If kl′ ≠ 0 and (k′,l′) falls in the subregions of S1 – 1 or S−33, (S3 – 3 or S–11), while we define k′″ = k′ + kc, l′″ = l′ − lc, and k″″ = k″ + kc, l″″ = l″ − lc (k′″ = k′ − kc, l′″ = l′ + lc, and k″″ = k″ − kc, l″″ = l″ + lc if it falls in S3 – 3, or S–11), then the aliasing is 
    formula
  • (b) If kl′ ≠ 0 and (k′,l′) falls in the subregions of S3 – 1 or S−13 (S1 – 3 or S–31), while we define k′″ = (k′ + k″)/2, l′″ = (l′ + l″)/2, and k″″ = k′″ ± 2kc if ±k′″ < 0, l″″ = l′″ ± 2lc if ±l′″ < 0, then the aliasing is 
    formula
  • (c) If kl′ ≠ 0 and (k′,l′) falls in the subregions of S11 or S−1 – 1, while we define k′″ = k′ + kc, l′″ = l′ − lc, and k″″ = k′ − kc, l″″ = l′ + lc, then the aliasing is 
    formula
  • (d) If kl′ ≠ 0 and (k′,l′) falls in the subregions of S13 or S−3 – 1(S31 or S−1 – 3), while we define k′″ = k′ + kc, l′″ = l′ − lc (k′″ = k′ − kc, l″′ = l′ + lc when it falls in S31 or S−1 – 3), and k″″ = k′″ ± 2kc if ±k′″ < 0, l″″ = l′″ ± 2lc if ±l‘′″ < 0, then the aliasing is 
    formula
  • (e) If kl′ ≠ 0 and (k′,l′) falls in the subregions of S33 or S−3 – 3, then the aliasing is 
    formula
  • (f) If k′ = 0 and |l′| < lc (or if l′ = 0 and |k′| < kc), while we define k′″ = k′ + kc, l′″ = l′ − lc, and k″″ = k′ − kc, l″″ = l′ + lc, then the aliasing is 
    formula
  • (g) If k′ = 0 and |l′| ≥ lc (or if l′ = 0 and |k′| < kc), while we define k′″ = k′ + kc, l′″ = l′ − lc if l′ > 0 (or if k′ < 0) and k′″ = k′ − kc, l′″ = l′ + lc if l′ < 0 (or if k′ > 0), and k″″ = k′″ ± 2kc if ±k′″ < 0, l″″ = l′″ ± 2lc if ±l′″ < 0, then the aliasing is 
    formula

As is the case in T06, certain high-frequency spectral components contribute to the final aliasing via both the ascending and descending tracks.

Footnotes

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