Four global ocean tide models are compared in terms of their contribution to Gravity Recovery and Climate Experiment (GRACE) satellite-to-satellite tracking residuals. The residuals are computed relative to a comprehensive model of Earth's time-varying gravity, including allowance for mass motions in the atmosphere, ocean, terrestrial hydrology, and mantle, in addition to tides. For each analyzed tide model, 4 years of GRACE range rate data are processed. Range and range acceleration residuals are tidally analyzed by geographic location. All four global tide models are shown to be error prone in various ways, leaving tidally coherent residuals especially in polar regions but also in some lower-latitude regions. Considerable power in the solar semidiurnal S2 tide in low latitudes suggests errors in our adopted model of atmospheric tides, which is based on 3 hourly European Centre for Medium-Range Weather Forecasts operational analyses. Anomalies in the μ2 tidal constituent over some shallow seas suggest the presence of unmodeled nonlinear compound tides, in this case 2MS2. Similarly, anomalies in the nonlinear M4 tide are seen if this constituent is omitted from the models. Errors in assumed seawater density may be contributing to some residuals.
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 Over the past 2 decades knowledge of global ocean tides has advanced markedly, thanks in great part to satellite altimetry. These advances have been documented by many types of tests, including comparisons of tidal models against independent tide estimates extracted from tide gauge and bottom pressure measurements [e.g., Andersen et al., 1995], variance reduction tests with independent satellite altimeter data [e.g., Dorandeu et al., 2000], tidal loading tests with gravimeter or other geodetic data [e.g., Llubes and Mazzega, 1997; Baker and Bos, 2003], comparisons of model tidal currents against independent point measurements at moorings and line integral measurements along long acoustic paths [Dusahw et al., 1997], and comparisons of low-degree spherical harmonic coefficients with those estimated from independent satellite laser ranging data, including the geophysically important degree 2 terms [Ray et al., 2001].
 In this paper we examine a new satellite-based test that offers a unique and complementary perspective. The two Gravity Recovery and Climate Experiment (GRACE) satellites [Tapley et al., 2004], orbiting at an altitude of ∼480 km, form a highly sensitive detector of mass changes in the Earth system. This is accomplished by precisely monitoring the relative distance between the two satellites [Wolff, 1969] by “dual-one way range” [MacArthur and Posner, 1985; Dunn et al., 2003] using K band and Ka band frequencies. In our case the surface mass changes of interest are those caused by oceanic tides.
 As a test of tide models, GRACE offers some advantages over previous surface gravimeter tests [e.g., Baker and Bos, 2003], because it can (to within its inherent spatial resolution) more clearly delineate which regions of the ocean are causing observed anomalies in tidal gravity. It is especially valuable for its superior coverage of the data-sparse regions of the polar oceans. Moreover, its extreme sensitivity can potentially give useful new tidal information even over lower-latitude oceans that are well covered by satellite altimetry. On the other hand, the GRACE spatial resolution limits its ability to detect very short-wavelength tidal errors in models; such errors are known to be significant in shallow seas and near-coastal waters for all currently existing global models.
 It is already well established that GRACE is sensitive to mass fluctuations associated with ocean tides [Ray et al., 2003; Han et al., 2004]. Further, it is sensitive to errors in the tide models currently adopted to remove tidal effects from the GRACE ranging measurements [Han et al., 2005; Schrama et al., 2007; Moore and King, 2008]. In fact, residual tide effects can impact GRACE's ability to resolve nontidal mass motions in the Earth system, which provides an important motivation for tide model improvement.
 Initial efforts have begun to extract tidal signals from GRACE in order to improve the long-wavelength components of tide models, either globally or regionally [Han et al., 2005, 2007]. Inversions of GRACE data to determine such corrections are extremely promising, but, like any inversion of gravity data, issues involving ill posedness, resolution, and regularization must be addressed. We here avoid such complexities by analyzing only the basic intersatellite ranging residuals with respect to various prior models. This limits us to a qualitative analysis of tide models, as opposed to quantitative estimation in terms of sea-surface elevations. The results, however, allow straightforward intercomparison of tidal models in terms of their measurement residuals, and they shed considerable light on deficiencies in all considered models.
Figure 1 is a reminder to the reader of how the fundamental GRACE ranging signals appear in response to a fairly localized mass anomaly. This calculation is based on a simple scenario of two low-low satellites flying in a perfectly circular orbit of radius R with no nonconservative forces. In that case the range-rate between the two satellites is approximately proportional to the induced potential difference at the two satellite positions (R, θ1, ϕ) and (R, θ2, ϕ) [Wolff, 1969]
where R(θ2 − θ1) is the satellite separation distance, G is the gravitational constant and M the mass of the earth. If we express the mass anomaly of Figure 1 in terms of sea-surface elevation (10 cm in this example), expanded as a spherical harmonic series
where ρw is the mean density of seawater, a is the radius of the earth, and k′n are loading Love numbers. If ζ represents a tidal oscillation, then znm is a function of time with tidal periodicity.
 As Figure 1 shows, perturbations in range and range acceleration tend toward maximum amplitude directly over a causative body, although with possible side lobes. Range-rate signals are always offset with a zero crossing over the causative body. For this reason most (but not all) of our analyses of GRACE residuals are done in terms of range and range acceleration. As seen below, both types of signals require filtering to isolate local anomalies, so the information in each tends to be complementary and not necessarily redundant.
3. GRACE Data Processing
 We begin with Level 1B GRACE data produced at the Jet Propulsion Laboratory [Case et al., 2004]. These data sets include filtered accelerometer, range, and range-rate measurements every 5 s along satellite arcs. We calibrate the accelerometry as described by Luthcke et al. , adjusting bias, scale, and orbital resonance parameters in a least squares fit to both the range-rate data and a “reduced dynamic” orbit solution, with weighting appropriate to noise levels in each data type.
3.1. Adopted Models
 We have endeavored to make the satellite force models needed for these calculations as complete as possible, one benefit being that small differences in tide models should then be more easily isolated. Static gravity is modeled with the GGM02C model [Tapley et al., 2005]. Time-variable gravity is modeled from multiple sources, as follows.
 Variable atmospheric mass is modeled with 3 hourly operational analysis outputs from the European Centre for Medium-Range Weather Forecasting (ECMWF) [Klinker et al., 2000]. The 3 hourly data are linearly interpolated to the epoch of interest. These data, in contrast to the classical 6 hourly sampling, allows for adequate sampling of the diurnal, semidiurnal, and terdiurnal atmospheric tides. Note that the ECMWF spatial resolution increased in February 2006, from about 80 (T255) to 25 km (T799), so the later data may show some level of improved realism.
3.1.2. Oceans (Nontidal)
 Variable oceanic mass is modeled with 6-hour outputs from the Toulouse Unstructured Grid Ocean model (T-UGO), which is run in barotropic mode and is forced by 6 hourly ECMWF surface pressures and winds [Carrère and Lyard, 2003]. Mean S1 and S2 tides, adjusted by least squares fitting over the 1997–2007 period, have been removed to avoid double counting with the radiational components of the ocean tide models.
3.1.3. Land Hydrology
 Variable terrestrial water storage on land is modeled with outputs from the Global Land Data Assimilation System (GLDAS), using the Noah land-surface model [Rodell et al., 2004]. These data are available every 3 hours on a global 0.25° grid. We mask out regions of ice sheets and major mountain glaciers because of a lack of dynamic ice modeling in GLDAS.
3.1.4. Mantle Rebound
 Secular changes in mass owing to glacial isostatic adjustment are modeled by using ICE-5G (VM2) of Peltier . Our models for mass variations due to tides are described in detail in section 4.
 All these sources of time-varying gravity are expanded in spherical harmonics and used as adjustments to static GGM02C Stokes geopotential coefficients. These are then used to compute satellite-to-satellite range rates, parameterized as time-varying baseline changes between the two satellites [Rowlands et al., 2002]. The modeled range rates form the basis for computing our measurement residuals.
Figure 2a shows real GRACE residuals along an arc of roughly one complete orbital revolution. Although some level of noise is apparent, signals with amplitudes well below 0.5 μm s−1 can easily be discerned in the time series. Figures 2b and 2d show corresponding range and range-rate-rate residuals, formed by simple trapezoidal integration and finite differences, respectively. As is well known, integration inflates low-frequency errors and differentiation inflates high-frequency errors, which is plainly evident in the two curves. We therefore high-pass filter the former and low-pass filter the latter. The filtered time series (Figures 2c and 2e) form the basis for the remainder of our analysis.
 Our nominal low-cut filter has a cutoff frequency of 5 × 10−4 Hz; the high-cut filter cutoff is 6 × 10−3 Hz and is combined with an 11-point median filter. The effect on the spectrum of the range-rate time series is shown in Figure 3. No claim of optimality can be made for these filters. They were designed subjectively. However, the original range-rate spectrum appears abnormally blue at high frequencies, so the high-frequency cutoff seems reasonable. Placement of the low-frequency cutoff is less easily determined; it may be removing true signal, although most of that signal is unlikely to be tidal since it is of hemispheric wavelengths and longer.
 One undesirable consequence of filtering range residuals is that, given the nature of high-pass filters, any given range anomaly is likely to develop small side lobes, similar to that already expected for range acceleration (Figure 1). Evidence of this is noted below.
4. Tide Models
 In this work we compare four different global tide models by running through the complete data-processing scheme outlined in section 3 and analyzing the resulting signals in the ranging residuals. The selected models are by no means a comprehensive compilation of all available models, but they do include those that have been most commonly employed by groups processing GRACE data. The models examined are GOT00.2, GOT4.7 (both successive updates to Ray ), FES2004 [Lyard et al., 2006], and TPXO7.1 (an update to that presented by Egbert and Erofeeva ). The most recent releases of GRACE Level 2 data (e.g., release 4) by the project teams have employed the FES2004 model, Luthcke et al.  used GOT00, and Luthcke et al.  used GOT4.7.
 The released tide models are given in terms of gridded global arrays of amplitude and phase for eight (sometimes more) major tidal constituents. We here account for 16 additional minor tides by inference from the given major tidal admittances, a procedure which is generally acceptable for deep ocean tides [Munk and Cartwright, 1966] if allowance is made for radiational effects in a few constituents [Cartwright and Ray, 1994; Arbic, 2005]. These inference methods are invalid in some shallow seas or under floating ice shelves [e.g., Pedley et al., 1986] where nonlinear interactions can appear, but these tend to be shorter-wavelength waves less observable at GRACE altitudes (but see discussion below). Lunar constituents are, of course, adjusted for nodal and perigee modulations where appropriate. Our method for accounting for minor tidal lines is in essence equivalent to the convolution methodology of Desai and Yuan , although the mathematical forms of our admittance interpolations are slightly different (e.g., piecewise linear versus sinusoidal) and the implementations are considerably different; largest differences will occur when admittances are extrapolated to the edges of tidal bands, for constituents like 2Q1 and OO1, but these constituents are very small.
 Most released tide models include the long-period constituents Mf and Mm (FES2004 includes the termonthly Mt and quarter monthly MSq as well). The remainder of the long-period tidal band is here assumed to maintain an equilibrium response with its forcing, consistent with mass conservation, loading, and self-attraction [e.g., Agnew and Farrell, 1978].
 To compute the gravitational effects of tides we expand all major and minor constituents in terms of spherical harmonics of the form (2–3). Major tides are expanded to degree 70, which prelaunch calculations suggested was sufficient [e.g., Ray et al., 2003]. (Note, however, that M. Watkins (personal communication, 2007) has seen evidence of small tidal effects in GRACE through degree 90.) We expand minor diurnal and semidiurnal tides to degree 50 and long-period tides to degree 20 or 30, depending on amplitude.
 Atmospheric tides are incorporated automatically by our use of the ECMWF model for general atmospheric mass variability, as noted in section 3. In the past [e.g., Luthcke et al., 2006] we have used a (monthly) mean climatology of atmospheric tides derived from 6 hourly ECMWF data [Ray and Ponte, 2003]. In principle the use of direct 3 hourly ECMWF is preferable for the following reasons: (1) there is no longer need to make assumptions about semidiurnal phase propagation to overcome the inadequate 6-hour temporal sampling [Van den Dool et al., 1997]; (2) temporal variability not captured in climatological means can be accounted for; and (3) lunar atmospheric tides are also incorporated, not because ECMWF models the gravitational tidal forcing of the atmosphere but rather because the lunar tides leak in via assimilation of surface barometer observations [Hsu and Hoskins, 1989]. Below we show evidence that this air-tide modeling is not perfect.
 For completeness we note that the Earth's body tides are modeled according to McCarthy and Petit [2003, section 6.1], with frequency-dependent anelastic Love numbers for the degree 2 components of the tidal potential and a frequency-independent elastic Love number for the degree 3 components.
 These results are based on 4 years of GRACE satellite-to-satellite ranging data, from April 2003 to April 2007. The data have been processed four times through our entire processing scheme, beginning with accelerometry calibration, keeping all geophysical modeling fixed except the ocean tide model.
5.1. Statistics of Residuals
 As a simple global summary Table 1 tabulates the final root-mean-square (RMS) statistics of range-rate residuals for each adopted tide model, divided out by year. It is rather surprising how close are the results, with models sometimes in agreement to four significant figures. As a test for selecting which model may be preferable, Table 1 is thus not so useful, although it does suggest that GOT00.2 can now (rightly) be considered obsolete.
Values are given in μm s−1. Values given in bold mark the smallest RMS in each year.
5.2. Tidal Analysis of Residuals
 A far more enlightening analysis of the residuals is obtained by treating them as almost a form of satellite altimetry to be geographically binned and tidally analyzed. This exploits the fact that anomalies in range and range acceleration tend to be localized over causative bodies. This section discusses the results of such analyses when binning data into small overlapping regions of size 5° (longitude) by 2° (latitude) and harmonically analyzing the data at a number of tidal frequencies.
 Like any satellite in its altitude range, GRACE aliases tidal signals into longer periods, which range anywhere from days to years [Ray and Luthcke, 2006]. Any successful tidal analysis depends on noise remaining incoherent at tidal periods. When aliasing occurs this becomes less likely, because some alias periods can be long compared with the time series length or they may fall near periods of large nontidal variability (e.g., seasonal). For GRACE a number of aliased tidal constituents fall into one or the other of these categories. They are generally solar constituents, including the partly solar K1 and K2.
 Thus, to interpret the following results it is helpful to recall some aspects of tidal aliasing in GRACE [Ray and Luthcke, 2006, section 2]. Nontidal variability occurring at tidal alias frequencies will tend to corrupt analyses of semidiurnal constituents more than diurnal constituents. For the former, on account of the Earth's two-sided tidal bulge, ascending and descending GRACE arcs are nearly in phase, with nearly identical sampling of any aliased energy. In contrast, for diurnal tides ascending and descending arcs are nearly out of phase [see Ray and Luthcke, 2006, Figure 1] thus tending to break correlations with (unaliased) nontidal energy. Table 2 lists the alias periods of the most significant semidiurnal tides [see Ray and Luthcke, 2006]. At 3.7 years, K2, we find, is so dominated by nontidal energy that any true tidal signals in our results are difficult to recognize. We find, as expected, diurnal constituents to be less affected, with one exception. The diurnal K1 is a special case, because its alias period is so long (7.5 years) that our 4-year analysis results for it are nonsensical. We therefore begin discussion with the second largest diurnal tide.
Table 2. Alias Periods for Semidiurnal Tides as Sampled by GRACE
Figures 4 and 5 show results for the principal lunar diurnal tide O1 in (high-pass filtered) residuals of intersatellite range and (low-pass filtered) residuals of range acceleration, respectively. Locations in these maps having significant amplitude (relative to background noise) are without exception locations of known tidal modeling problems, especially in polar seas where satellite altimetry is lacking. The GOT00 model displays large errors in the Arctic Ocean north of Siberia, but these errors are evidently much reduced in the other three models. Both the Bering Sea and Okhotsk Sea are regions having relatively large O1 tidal amplitudes yet are somewhat poorly sampled by altimetry owing to persistent ice contamination; errors in both seas are therefore not unexpected. All four models display errors near Antarctica, especially in the Ross and Weddell Seas. FES2004 looks superior in this region, although FES2004 anomalies are more noticeable in range rate rate (Figure 5) than in range (Figure 4). Also apparent in Figures 4 and 5 is a suggestion of side lobes in the vicinities of large anomalies. This appears most pronounced in GOT4.7 and TPXO.7 north of the Ross and Weddell Seas.
Figure 6 shows similar results for the principal semidiurnal solar constituent S2. The results appear much noisier, with significant power evident in lower latitudes, especially in a pronounced band along the equator. The source of this band is almost surely errors in the S2 atmospheric tide, which peaks along the equator. The same air-tide model is used in all four panels of Figure 6, and indeed the band appears similar, but not identical, in all (TPXO.7 appears most different). The band is not due to errors from mismodeling the ocean's response to the air tide, because that response is highly dynamic, not confined to the tropics, and, in fact, quite similar to the ocean's response to the S2 tidal potential [Arbic, 2005]. The band could arise if modelers assimilate satellite altimeter data while mistakenly applying an “inverted barometer” correction with S2 air tides included, but that has not been done in either GOT or TPXO models (we cannot comment on FES). An unusual aspect of this band is its nearly complete disappearance over Africa, but not South America; however, part of the signal over South America may well arise from true 160-day variability in hydrology that is unaccounted for by GLDAS (160 days is the S2 alias in GRACE, see Table 2) and not necessarily from S2. In any event, this band of S2 power in the residuals could contribute to the 160-day oscillations seen in GRACE solutions for the zonal J2 gravity coefficient (e.g., Chen and Wilson , who attribute the J2 oscillations to ocean-tide errors).
Figure 6 also shows a significant S2 error off northwest Australia in FES2004, but not in the other models. This has been previously reported by others [e.g., Schrama et al., 2007; Moore and King, 2008]. That region off Australia experiences very large semidiurnal tides (Figure 7), with S2 amplitudes exceeding one meter along a long section of coastline. As a complementary and independent analysis Figure 8 shows vector differences in this region between S2 tides estimated point by point along Topex/Poseidon tracks [e.g., Carrère et al., 2004] and three of our four tidal models. Large vector discrepancies are again seen in FES2004, especially in the southeastern Timor Sea just west of Darwin where differences with Topex/Poseidon data reach one meter. With two independent space-geodetic systems showing consistent discrepancies, it seems clear that FES2004 is in error in this region but only for the S2 constituent (and possibly K2).
Figure 9 shows the principal lunar constituent M2. As before, anomalies in polar regions are large, although FES2004 is clearly superior around Antarctica and TPXO7 appears best in the Ross Sea area. TPXO7 also appears best in Hudson Bay, although it still has noticeable anomalies in Hudson Strait, an area of inordinately large tides. In addition to the relatively large anomalies in polar regions, there are a number of interesting lower-latitude anomalies. All show problems near New Zealand, a region of rapid M2 phase changes, and southern Patagonia, a region of large, short-wavelength tides. FES2004 shows a relatively large anomaly near the Australian Great Barrier Reef; TPXO7 somewhat less so. GOT4.7 is anomalous off Maine and Nova Scotia; FES2004 not at all. Figure 9 clearly highlights our conclusion, no global tide model is without problems for processing GRACE data.
 One of the most surprising features of Figure 9 is that all four models show relatively large anomalies in the eastern North Atlantic Ocean. While this is a region of fairly large M2 amplitudes, it is also well covered by high-quality satellite altimetry. One naturally wonders if some systematic error, either in GRACE or in the altimetry, could account for this. One possibility stems from equation (3), where the ocean's density has been taken as constant. More accurate would be to allow the spherical harmonic coefficients znm to represent elevation × density. The error committed by assuming constant seawater density can be evaluated by computing
where H is the amplitude of M2, is the mean density of the ocean water column at any location, and 1031 is the constant mean seawater density assumed by our orbit determination codes. Figure 10 shows the quantity (4) geographically, where we have evaluated from the World Ocean Atlas. The largest density errors do include a region around Britain and extending west of Spain, so it is possible that these do contribute to the observed M2 residuals, but they cannot explain the residuals over the much larger region of the eastern North Atlantic depicted in Figure 9.
 A preliminary inversion of the GRACE M2 tidal residuals to determine adjustments to sea surface elevations, which at this point is still too uncertain to publish, does nonetheless suggest one possible explanation. The M2 errors may be a series of fairly localized errors that merge together owing to GRACE's limited spatial resolving power. Our inversion places large errors off southeastern Greenland, which is perennially ice covered, off southern Ireland, and off the Faeroe Islands. Further investigation is clearly warranted.
 In addition to the constituents already described we have solved for a number of others. Space precludes a detailed discussion, but in Figure 11 we show some of the more interesting ones for the GOT4.7 model. Figure 11a for constituent S1 clearly shows errors in both ocean and air tides. The latter must account for the anomalies over South America, similar to those found for S2. The anomaly in the Gulf of Alaska, however, is undoubtedly oceanic in origin, since this is a region of some of the largest S1 ocean tides [Ray and Egbert, 2004, Figure 3].
 The interpretation of Figure 11b, showing constituent R2, is less clear-cut. The frequencies of R2 and T2 are 1 cpy away from S2, so both constituents pick up seasonal variations in the latter, which are significant in atmospheric tides. This is possibly the source of anomalies over South America and Africa. However, these anomalies are more likely due to mismodeled hydrological changes at the R2 alias of 288 days (Table 2). Examination of T2 (not shown) shows features similar to Figure 11b but with less power, which might be expected if aliasing is the source, because the T2 alias period of 112 days is much shorter than the R2 alias. On the other hand, the R2 air tide induces a significant radiational component in the R2 ocean tide, which could be the source of some of the ocean anomalies. Our admittance methods for inferring R2 (and T2) from S2 ignore this radiational component.
 Finally, Figure 11c shows the constituent μ2, with anomalies evident in a number of shallow seas, especially near Indonesia. This suggests that the errors may actually be induced by the nonlinear compound tide 2MS2, whose frequency coincides with μ2. Supporting this interpretation is the fact that a similar diagram for 2N2 shows almost no anomalies, yet 2N2 is only marginally weaker than the linear μ2. We find no significant anomalies for the nonlinear overtide M4 in GOT4.7, but unlike 2MS2, M4 is included in the released GOT4.7 model. It is not included in GOT00, however, and M4 range acceleration residuals for GOT00 (not shown) are very evident on the Patagonian Shelf, a region of very strong (reaching 30 cm) M4 amplitudes. We find no evidence of any significant anomalies in the third degree M3 constituent, even though it has isolated shelf resonances of 5–10 cm amplitude [e.g., Huthnance, 1980] which are not included in any of our tested models.
 None of the four global ocean tide models examined in this paper can be considered perfect for use in processing GRACE data, since each generates tidally coherent residuals in GRACE's fundamental satellite-to-satellite measurements. Each tide model has flaws in certain regions or certain constituents, and all of them require improvement in polar regions.
 Moreover, the atmospheric tide model used here is also found to be far from perfect. More work is needed to analyze the tides in the 3-hour ECMWF product that we adopted here and to compare these with the tidal fields used by other GRACE processing teams. Even more useful for gauging accuracy would be comparisons with surface barometer data [e.g., Ray, 2001]. Unfortunately, there are hardly any good barometer stations over the region of South America where Figure 11a suggests possible air-tide errors [Ray, 2001]. Mismodeling temporal variability in air tides could possibly account for our results for R2 (which acts to modulate S2 at the annual cycle). Variability at shorter and longer periods is also known to occur, and more work is needed to understand the errors induced by such effects.
 If the residuals seen in the μ2 constituent (Figure 11) are indeed due to the presence of the unmodeled compound 2MS2 tide, then it implies a need to reexamine the methods used in shallow seas for inferring minor tides from major tide admittances in the presence of nonlinearity. The latest global models already include the overtide M4, and these results suggest extending such work to other nonlinear constituents. Local tide models that include compound tides already exist in many regions of the world, of course, and these could be readily adopted in GRACE modeling after appropriate testing.
 While we have tested four commonly employed global models, new models continue to appear, for example, the recently released EOT08a model [Savcenko and Bosch, 2008], which we have not yet examined. Savcenko and Bosch start with the FES2004 model and compute adjustments to it on the basis of data from multiple satellite altimeters. Presumably their adjustments rectify problems such as those seen in Figure 8, but owing to a lack of data they cannot correct the large errors in polar regions.
 At this stage it seems appropriate to exploit some of the local modeling efforts that have been undertaken in polar regions [e.g., Padman et al., 2008], on the basis of a comprehensive approach of correcting bathymetry and ice-grounding geometry to improve realism of the hydrodynamic modeling and incorporating a variety of old and new data for constraints, such as ICESat laser altimetry. Note that Padman et al. use TPXO7 at open boundaries, so the local and global models could be immediately merged to produce a consistent, revised, global data set. In addition, of course, GRACE itself provides new long-wavelength constraints on polar tides [Han et al., 2007], which can be incorporated into global models by appropriate assimilation methods.
 We thank D. Rowlands and S. Bettadpur for useful discussions. This work was supported by the National Aeronautics and Space Administration's GRACE project.