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 Knowledge of variability in ocean bottom pressure (pb) at periods < 60 days is essential for minimizing aliasing in satellite gravity missions. We assess how well we know such rapid, non-tidal pb signals by analyzing in-situ bottom pressure recorder (BPR) data and available global estimates from two very different modeling approaches. Estimated pb variance is generally lower than that measured by the BPRs, implying the presence of correlated model errors. Deriving uncertainties from differencing the model estimates can thus severely underestimate the aliasing errors. Removing estimated series from BPR data tends to reduce the variance by up to ∼5 cm2 but residual variance is still ∼5–20 cm2 and not negligible relative to expected variance in climate pb signals. The residual pb variability can be correlated over hundreds of kilometers. Results indicate the need to improve estimates of rapid pb variability in order to minimize aliasing noise in current and future satellite-based pb observations.
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 Bottom pressure (pb) is a fundamental measure of the ocean state. Monthly and longer period variations in pb, and therefore ocean mass, are intrinsically linked to ocean circulation changes, which influence the global heat budget and climate. Changes in pb are also a major component of the sea level budget, together with density-related ocean volume variations. Despite its importance, knowledge of pb variability on climate time scales is lacking. The best, most direct measurements of pb from in-situ bottom pressure recorders (BPRs) are very sparsely located and usually not continuous for more than a year. Model-based estimates of pb have been available for some time, yet without extensive data for comparison their quality is largely unknown. The launch of the Gravity Recovery and Climate Experiment (GRACE) mission in 2002 brought a revolutionary capability to map pb globally at monthly intervals, but improving the signal-to-noise ratio of these observations remains a challenge.
 Aliasing errors are of particular relevance for satellite missions such as GRACE. Any pb variations with periods less than twice the GRACE mapping interval (∼30 days) are aliased in the records. Studies of BPR data reveal that a large part of non-tidal pb variance can be associated with submonthly periods [Gille and Hughes, 2001]. Aliasing is thus an important error source for GRACE and may in fact be the limiting factor for the accuracy of the planned GRACE follow-on mission as instrument errors are reduced [Wiese et al., 2009]. In essence, the better estimates we have of the rapid pb variations, the better we can solve for the long period pb variations. This makes knowledge of the high frequency pb variability essential for maximizing the success of current and future gravity missions in trying to determine pb changes on climate time scales.
 Current de-aliasing of non-tidal pb variability in GRACE uses the Ocean Model for Circulation and Tides (OMCT) [Thomas, 2002; Flechtner, 2007], but estimates of the errors in OMCT, required to predict the aliasing error [Thompson et al., 2004], are not easy to obtain. More generally, very few studies [Hirose et al., 2001; Kanzow et al., 2005] have examined the quality of available model estimates of rapid pb variability. In this regard, “ground truth” provided by BPR point measurements can be useful, given the relatively large decorrelation scales of BPR signals compared to the model grids [e.g., Hughes et al., 2007; Park et al., 2008]. In this work, we present results from a joint analysis of pb estimates from (1) OMCT, (2) data-constrained solutions produced by the Estimating the Circulation and Climate of the Ocean (ECCO) project [Wunsch et al., 2009], and (3) an independent BPR dataset. The 3-way comparison, focusing on periods < 60 days, provides useful insight on the quality of the available pb estimates and the need for improving de-aliasing procedures for GRACE and future gravity missions.
2. Time Series of pb
 The OMCT and ECCO estimates of pb analyzed here are based on two very different models and methodologies. In the case of ECCO, solutions are obtained using an iterative optimization scheme by which a general circulation model is fitted, in a least squares sense, to most available in situ and satellite data (e.g., Argo floats, altimetry, SST), through adjustments in its forcing fields and initial conditions [Wunsch et al., 2009]. For this study, we use version 2, iteration 216 (v2.216) described in detail by Wunsch et al. . The horizontal grid is 1° × 1°, extending between 80°S–80°N, and there are 23 layers in the vertical. Daily estimates are available from 1992 through 2006. The OMCT fields we analyze have been generated by the GRACE project for use as a de-aliasing background model [Flechtner, 2007]. OMCT is forced by atmospheric surface fluxes of momentum, heat and freshwater, as well as by pressure. In contrast with ECCO, no data constraints are involved. The OMCT spatial coverage is global with a grid spacing of 1.875° and 13 layers in the vertical, and output is produced every 6 hours, which we then average into daily values. Output is available from 1976 through 2009. To avoid issues of conservation of mass, we subtract the time-varying spatial mean of pb from both ECCO and OMCT estimates. (All pb estimates in the paper are given in equivalent centimeters of water.)
 We compare the ECCO and OMCT daily pb averages to 3 different BPR datasets: DART (Deep-ocean Assessment and Reporting of Tsunamis) stations available from NOAA (http://nctr.pmel.noaa.gov/Dart/dart_ref.html), and GLOUP (Global Undersea Pressure) and ACCLAIM (Antarctic Circumpolar Current Levels by Altimetry and Island Measurements), both maintained by the Permanent Service for Mean Sea Level (http://www.psmsl.org/links/programmes). All BPR sites are shown in Figure 1. The measurements have been averaged into daily values from higher frequency series. Some of the GLOUP stations have multiple readings, indicating some redundancy, and are therefore averaged into single series. Comparing these multiple readings, we estimate that the BPR error variance is generally < 1 cm2 or <10% of the total variance. Some BPR station locations have been re-occupied with gaps in the time coverage. We have considered these records to be a single, longer time series, with any offsets removed before concatenation. The BPR series have been de-tided using standard least squares harmonic fitting of the tidal constituents, aside from the long period tides in the DART and ACCLAIM data which were removed using the FES2004 tide model [Lyard et al., 2006].
 The OMCT, ECCO, and BPR daily pb series are all filtered to remove variability at periods > 60 days [Mann, 2004], including long term drifts that are known to affect the BPR sensors. Two BPR stations with large, unphysical variations, indicating possible bad pressure sensors, were not used. In addition, for a station very near the Antarctic coast, we found the ECCO variance to be more than 3 times higher than the data variance, indicating possible problems with topography resolution near the coast. This station is also not used in our analyses.
 As OMCT and ECCO output are both widely used, it is important to check their consistency. Both are based on well developed models and estimation methods, yet their high frequency pb fields are noticeably different. Both solutions show the same basic spatial pattern of pb variance (Figures 1a and 1b), with the higher values in the Southern Ocean, the North Pacific, and in shallow coastal areas. However, OMCT tends to have substantially higher variances than ECCO, especially in the Southern Ocean (Figures 1a and 1b). Exceptions to this are certain coastal areas, where ECCO variances tend to be higher.
 A map of the variance of the difference OMCT−ECCO (Figure 1c) gives an approximation of the sum of the error variances in OMCT and ECCO estimates, assuming uncorrelated errors. Those regions with higher overall variance tend to have higher differences, i.e., the Southern Ocean, the North Pacific, and coastal regions. The variances of the OMCT−ECCO series are of the same order of magnitude as the ECCO or OMCT variances for many regions, indicating low signal-to-noise ratios. Correlations between OMCT and ECCO (Figure 1d) are for the most part positive, except for the Mediterranean Sea, but vary significantly with location. Overall, the correlations are quite weak over large regions of the ocean, particularly in the tropics and Indian Ocean. Comparisons of OMCT and ECCO using different filtering cutoff periods as well as spectral analysis (not shown) indicate that >50% of the variance in Figure 1c corresponds to OMCT and ECCO differences at periods < 10 days over most of the oceans. Thus, the estimated pb fields seem to differ the most at the shortest periods analyzed.
 The differences between OMCT and ECCO are significant and pose some questions about how well we can estimate high frequency pb variability and our ability to de-alias data from satellite gravity missions. The BPR data provide the best available measurements of pb for comparison, although at a very limited number of sites (Figure 1a). The high frequency pb variances for the BPR station data are shown in Figure 2a, along with the corresponding ECCO and OMCT variances calculated over the same time ranges. The BPR variances tend to be higher, for some stations many times higher, except when compared to OMCT in the Indian Ocean and for a couple of stations in the South Atlantic. The average BPR variance is 10.8 cm2, compared to 2.9 cm2 and 5.6 cm2 for ECCO and OMCT, respectively.
 A key question for the efficiency of de-aliasing procedures is how much residual variance remains in the BPR time series after the OMCT or ECCO estimates are removed. If we consider the BPR estimates to be “truth”, this residual variance is a measure of the incurred aliasing error variance. Figure 2b shows the original BPR variances together with the residual variances for BPR−ECCO and BPR−OMCT series. The original BPR variance tends to be in the range of 5–25 cm2, depending on location. In general, removing either OMCT or ECCO only reduces the variance by up to 5 cm2 and the residual variance is still quite high. ECCO tends to explain more variance than OMCT, even though OMCT tends to have higher variance than ECCO, which compares more favorably with the high BPR variances. The mean variance explained by OMCT over all the BPR stations is 1.3 cm2, whereas the mean for ECCO is 2.4 cm2. In any case, this is only a small fraction of the total variance in the BPR data—around 15% and 25% on average for OMCT and ECCO, respectively.
 Variances of the OMCT−ECCO series, also shown in Figure 2b, are mostly smaller (many times smaller for several South Atlantic and Pacific sites) than those of the respective BPR−ECCO and BPR−OMCT series. Results are consistent with the tendency for underestimation of pb variance by both OMCT and ECCO, relative to the BPR variance. As variability in both OMCT and ECCO is biased low, the presence of correlated errors is implied. In such case, the variances of the OMCT−ECCO difference series can only provide a lower bound estimate on the combined OMCT and ECCO error variances. The results in Figure 2b indicate that, in many regions, the variances of OMCT−ECCO in Figure 1c may considerably underestimate the underlying uncertainty in both estimates.
 While the BPR data variance explained by OMCT and ECCO is quite low, the correlations between them can be significant (Figure 2c). The BPR station correlation with OMCT and ECCO tends to be higher where the correlation between OMCT and ECCO is also higher, indicating that where OMCT and ECCO are in better agreement, they have indeed lower errors. Higher correlations are generally associated with higher percent variance explained. High correlations, however, do not necessarily imply negligible variance residuals, as seen for example for some of the Indian Ocean stations. In these regions, differences in magnitude of variability might be a primary factor, and simple scaling factors applied to OMCT or ECCO might lead to a better fit of the BPR data. More generally, discrepancies between BPR and the estimated pb series involve more than different magnitudes, as seen for example in the low correlations observed for many South Atlantic stations.
 Given the focus of GRACE on scales of a few hundred kilometers and longer, aliasing errors with similar large scales gain particular importance. The spatial decorrelation scales of residual series BPR−OMCT and BPR−ECCO are indicative of the nature of the incurred aliasing errors. Correlations of those pb residuals among stations within a region (not shown) are indeed significant over hundreds of kilometers. Thus, over the necessarily limited areas analyzed here, ECCO and OMCT estimates of high-frequency pb variability are very likely to contain large-scale errors, which are potentially important for GRACE data processing.
 In practice, aliasing errors arise from a non-trivial sampling in space and time along the GRACE orbital track, and their calculation would require a complex orbital simulation using global maps of the errors in non-tidal pb estimates [e.g., Han et al., 2004]. Although a full assessment of these issues is beyond our scope, in Figure 3 we compare the average geoid degree amplitude spectra of the ECCO−OMCT differences to the GRACE errors [e.g., Wahr et al., 2004]. The ECCO−OMCT variances are greater than the GRACE error variances up to degree 8, indicating that the GRACE solutions could be sensitive to non-tidal aliasing errors at large scales. Thompson et al.  compared differences between baroclinic and barotropic versions of the same ocean model with GRACE errors and reached similar conclusions. Error variances in non-tidal estimates could be 2 to 4 times larger than what the OMCT−ECCO series suggests, however, as seen from the comparisons of OMCT−ECCO and BPR residual variances (Figure 2). In the latter case, significant aliasing error effects could extend to degrees 20–25. Relative to the formal GRACE (pre-launch) errors, which might represent a baseline for future gravity missions, results in Figure 3 suggest that non-tidal aliasing errors could dominate the uncertainty at the largest scales.
4. Summary and Discussion
 The difficulty in estimating high frequency pb fields is apparent in the relatively poor agreement found in the OMCT and ECCO solutions (Figure 1). The magnitudes of pb variability are generally higher for OMCT, and the correlations between OMCT and ECCO fields tend to be weak over broad regions (e.g., Indian and Atlantic Oceans, tropics). Two main factors seem to contribute to these results. First, pb dynamics is very dependent on bottom topography [e.g., Hirose et al., 2001], whose effects are differently represented in OMCT and ECCO, given their different vertical and horizontal resolutions (13 vs. 23 vertical layers; 1.875° vs. 1° grid spacing). Second, rapid pb signals can be sensitive to details of the forcing, which is quite different in OMCT and ECCO (section 2). In particular, the inclusion of atmospheric pressure loading in OMCT is one reason why its variance is considerably higher than that in ECCO, as pressure forcing can dominate wind driving at sub-weekly time scales [Ponte, 1994]. Consistent with this inference, a large part of the variance in ECCO−OMCT series resides at these shortest periods. With important model errors associated with the representation of bottom topography, forcing, and other factors such as frictional dissipation [e.g., Hirose et al., 2001], the use of data fitting and optimization methods, such as those employed in the ECCO estimates, becomes an important step towards mitigating potential model deficiencies.
 The comparisons with independent BPR measurements (Figure 2) give some indication of the quality of the available OMCT and ECCO pb estimates, although the sparseness of the available BPR sites limits the generality of our conclusions. The BPR data variance tends to be higher than that of OMCT and ECCO. These findings are generally consistent with other studies [Hirose et al., 2001; Kanzow et al., 2005] and confirm a general tendency for underestimating observed rapid pb variability, which is especially clear in regions with strong eddy variability, such as near the Gulf Stream and the Drake Passage. Thus, some of the BPR variance may be associated with eddies and other short-scale, localized effects, which are not expected to be represented in relatively coarse resolution models. (Note, however, that the spatial decorrelation scales of residuals BPR−OMCT, BPR−ECCO point to deficient pb estimates on scales of hundreds of kilometers.) Other reasons for the weak model-based estimates of pb variability may involve excessive dissipation or weak atmospheric forcing. Efforts to investigate these questions in detail are currently ongoing and will be reported elsewhere.
 A common way of estimating aliasing errors in GRACE data processing is to assume that pb errors can be approximated by differencing two models [Thompson et al., 2004; Han et al., 2004]. From the present analysis, such estimates are likely to be a lower bound on the uncertainties because of the probable presence of correlated errors. Comparisons with independent BPR data, as done here, provide one way of calibrating such uncertainty estimates, although at a very restricted number of locations where data is available. Given the relation between high frequency variability in pb and sea level [Vinogradova et al., 2007], tide gauge and altimeter data may also be used in these efforts. In the absence of ground truth data, one should avoid severely underestimating the uncertainty by examining a range of values: inflating the error variance estimates based on model differences by at least a factor of two seems warranted. This level of error is greater than the GRACE calibrated error up to approximately degree 15 (Figure 3).
 For the limited sites examined in this study, ECCO tends to explain more BPR data variance than OMCT, but removal of either series from the BPR measurements leaves ∼5–20 cm2 of unexplained high frequency variance. In fact, this potentially aliased variance is comparable or larger than the expected signal variance at the monthly and longer time scales of interest for climate. Thus, efforts to improve the quality of high frequency pb estimates remain an important issue for GRACE, and more so for future gravity missions. Such improvements will involve higher model resolution, better forcing fields and sub-grid scale parameterizations, as well as the use of ECCO-like optimization methods that can make effective use of all available data for the purposes of estimating the rapid pb variability of interest.
 All BPR data were initially compiled and processed by N. Vinogradova (AER). The long-period tide model codes were provided by R. Ray (GSFC). We acknowledge the support of NASA's GRACE project (NNH08CD19C) and the National Oceanographic Partnership Program through its sponsorship of the ECCO project.
 The Editor thanks one anonymous review for their assistance in evaluating this paper.