## 1. Introduction

Satellite altimetry has become an essential tool in oceanography as it provides the only existing near-global, time-continuous measurements of sea surface height (SSH). As with all observational systems, an estimate of the errors in altimeter SSH observations is essential for quantitative use of the data. In general, the data are analyzed directly (e.g., as in frequency-wavenumber spectral estimates) or used to constrain general circulation models in a least squares sense. In most applications, the data are separated into a time-mean sea surface (MSS) and anomalies about that mean. When the MSS is combined with a geoid height surface, one has an estimate of the absolute, time-averaged surface dynamic topography (e.g., Wunsch and Stammer 1998). Temporal anomalies about the surface dynamic topography are independent of the geoid, which has its own error budget, but nonetheless contain various complex corrupting elements.

The immediate motivation for this note is the need to define altimeter errors in a global ocean state estimation system [called Estimating the Circulation and Climate of the Ocean (ECCO; see Stammer et al. 2002; Köhl et al. 2007; Wunsch and Heimbach 2007); the particular calculations will be referred to as the Massachusetts Institute of Technology–Atmospheric and Environmental Research, Inc. (MIT–AER) estimates because the ECCO effort has several independent strands]. A general circulation model is fit explicitly in a weighted least squares sense to the altimetric measurements from several different satellites. As in any least squares problem, the solution can be no better than the accuracy with which data (and model) errors are represented. If underestimating errors, one fits noise, and if overestimating them, one is discarding useful information. The same error field estimates are useful in any more direct use of the data, and elements of the error estimates simultaneously provide useful descriptions of the ocean (its time-dependent variances, wave heights, etc.).

Although the ongoing ECCO calculations use data from the Ocean Topography Experiment (TOPEX)/Poseidon, *Jason-1*, *Geosat Follow-On* (*GFO*), *ERS-1, 2*, and *Envisat* missions, we focus here on understanding of the TOPEX/Poseidon (hereafter TP) and *Jason-1* data, as these provide the basic calibration for all the other altimeters (Le Traon et al. 2003) and have been the subject of the most intense scrutiny. Knowledge of the error budget for TP is summarized in Chelton et al. (2001, hereafter denoted C2001). Their Table 11 contains estimates for various error sources: altimeter radar noise; electromagnetic and skewness biases, which together make up the so-called sea state bias (SSB); ionospheric correction, and wet and dry tropospheric corrections (all are part of the environmental effects that cause signal path delays); and orbit error. C2001 specified the total (mean and time variable) error for each of these terms separately, with the largest being orbit error (2.5 cm), and the combined root-sum-of-squares (rss) error was estimated at 4.1 cm. [Numbers in C2001 refer specifically to the dual-frequency TOPEX altimeter; the single-frequency POSEIDON altimeter is slightly more noisy and has a more uncertain ionospheric correction (Fu et al. 1994), but such a distinction will be ignored here.] Although only globally averaged values are given, their comprehensive discussion of each error source indicates that several of them are expected to have considerable spatial dependence (see also Tsaoussi and Koblinsky 1994), a dependence which is the main focus here as it is of prime importance in global estimation problems.

The launch of *Jason-1* in 2001 and the “tandem mission” with TP, during which the altimeters were flying about 70 s apart in identical orbits and thus sampling essentially the same ocean variability, provided an unusual opportunity for altimeter comparison and cross calibration. These efforts are discussed in six dedicated numbers of *Marine Geodesy* (Vol. 26, Nos. 3–4; Vol. 27, Nos. 1–4). Ménard et al. (2003) show globally averaged errors for *Jason-1* similar to those of TP (C2001), apart from a somewhat lower orbit error (1.5 cm), and a total rss error of 3.3 cm, well within the prelaunch specifications for *Jason-1*. In that sense, the engineering specifications for the missions have been more than met. Sufficiently small pointwise error differences are, however, a necessary, but not sufficient, prescription of errors. If large-scale (geographically correlated) errors exist, even though bounded at any given point, they can give rise to serious estimation errors in any analysis or least squares fit to the observations.

Geographically dependent errors in altimetry have been discussed primarily in the context of time mean (systematic or bias) differences between TP and *Jason-1* (Beckley et al. 2004; Dorandeu et al. 2004; Chambers et al. 2003b), with orbit and SSB errors found to be partly to blame. Han (2004) analyzes root-mean-square (rms) differences but only for a small region off Nova Scotia. Regional error budgets, focusing on time-variable terms, have not been much considered. Exceptions are the studies of Le Traon and Dibarboure (2004) and Whitmer et al. (2004), but the former is based on data that are substantially smoothed in time and space, while the latter is based on crossover analyses that assume high-frequency signals to be part of the error.

In this article, a more direct and simpler approach is used to infer geographical dependence of altimeter noise. Spatial correlations are not explicitly dealt with; we seek to map regions of error significantly larger or smaller than their space–time average. Finding estimates of actual space–time correlations within and between such regions is a far more demanding task than the one described here. The focus is on errors affecting the time-variable SSH, with systematic biases (including geoid problems) treated elsewhere [Stammer et al. (2007) specifically discuss the geoid error]. As will be seen, the story remains incomplete, and this work is in part an attempt to stress the need for more complete modeling of altimeter data uncertainties.

## 2. Time-variable instrument noise

We consider SSH errors intrinsic to the altimeter measurement (i.e., range noise), involving radar noise, SSB errors, uncertainties in ionospheric, wet and dry tropospheric corrections, and orbit error. Our analysis is based on point measurements at 1-Hz sampling rate, which is the basic altimeter dataset. Smoothed products are discussed at the end.

Here we use two different methods to arrive at an approximate determination of spatial dependences in the range measurement errors. In one approach, we take advantage of the TP and *Jason-1* tandem mission. The particular SSH TP and *Jason-1* datasets used here are described in the appendix. Differences between the two inferred SSH estimates can be interpreted as representing differences between instrument errors in the two altimeters. This method provides an easy way of separating time-mean and variable errors, which is convenient, given our need to estimate errors in SSH anomalies. Some assumptions about the partition of errors between instruments are required. Along-track and crossover analyses suggest slightly noisier Jason measurements compared to TP, mostly related to differences at wavelengths shorter than 50 km (Dorandeu et al. 2004). For simplicity, we ignore any small differences in Jason and TP instrument performance and assume an approximate equipartition of error variance.

The standard deviation of the differences, TP minus Jason, based on 1-Hz range estimates, is shown in Fig. 1a. A description of the temporal and spatial characteristics of the variability in the difference series is beyond our scope, but we note that empirical orthogonal function analysis of 10-day TP–Jason maps (not shown) suggests a wide range of space and time scales, including very long (basin scale) patterns indicative of spatially correlated errors. The standard deviation in Fig. 1a is interpreted as representing the combined effects of orbit errors and radar noise and also errors in SSB, ionospheric, and wet tropospheric corrections, which are instrument dependent. Estimates in Fig. 1a are somewhat noisy at the shortest scales, which probably reflect the relatively small amount of data (only 20 orbit repeat cycles or ∼200 days) available for the TP–Jason series. Notice also that the relatively short record duration may affect the separation of time-variable errors from those in the mean, as the stability of the mean is a function of the averaging time.

Errors in the dry tropospheric correction are not included in Fig. 1a, as the same atmospheric pressure field is used in correcting both altimeters. A separate estimate of the uncertainty in the dry tropospheric correction is shown in Fig. 1b. Values correspond to 23% of the errors in the pressure field (C2001), which are estimated based on the standard deviation of the difference between European Centre for Medium-Range Weather Forecasts (ECMWF) and National Centers for Environmental Prediction (NCEP) operational analyses for the years 2001–03, and assuming equipartition of errors between the analyses (Salstein et al. 2006, manuscript submitted to *J. Geophys. Res.*). The difference between analyses likely defines a lower bound on the uncertainty of the pressure fields, because possible common errors are removed by subtraction. Errors in the dry tropospheric correction are <0.5 cm, except in high southern latitudes, and thus small compared to other errors (cf. Fig. 1a). Fu et al. (1994) estimate it as a constant, 0.7 cm, based on a value of 3-cm rms error in the pressure fields. Estimates in Fig. 1b are lower and indicate improvements in the weather center analyses over the last few years; for the early years of TP, these errors are likely larger than the estimates in Fig. 1b (Ponte and Dorandeu 2003; Salstein et al. 2006, manuscript submitted to *J. Geophys. Res.*).

The total error, given in Fig. 1c, mostly resembles the pattern in Fig. 1a, divided by

*S*), which tends to be largest at middle and high latitudes (C2001). An independent method of estimating the errors consists of modeling radar and SSB noise using known relationships with

_{w}*S*. Based on Fig. 2 of Vincent et al. (2003), radar noise

_{w}*r*is linearly related to

_{n}*S*as with

_{w}*S*given in meters and

_{w}*a*∼ 1.21 cm and

*b*∼ 0.18 cm m

^{−1}. Values are appropriate for 1-Hz measurements. The standard deviation of the time-dependent noise is then

*bσ*, where

_{Sw}*σ*denotes the standard deviation of

_{Sw}*S*in meters. Errors in the SSB correction are similarly related to

_{w}*S*. C2001 suggest an uncertainty in SSB correction of approximately 1% of

_{w}*S*. These relations are expected to apply to both TP and Jason (Vincent et al. 2003).

_{w}Assuming radar and SSB noise are uncorrelated, their combined effect amounts to ∼1.02*σ _{Sw}*. Values of

*σ*have been calculated by C2001 from the first 7 yr of TP data (cf. their Fig. 36). To this estimate, one can add uncertainties in the dry tropospheric correction from Fig. 1b. There are no readily available spatial-dependence estimates for the remaining terms (wet tropospheric and ionospheric corrections, orbit error), and we use the values 1.1, 0.5, and 2.5 cm, respectively, from Table 11 of C2001, assuming they mostly represent time-variable errors. [In principle, the ionospheric correction is also related to

_{Sw}*S*(Fu et al. 1994), but it is the smallest term in the error budget and a constant is sufficient for the current purposes.]

_{w}Combining all these effects leads to the rss error shown in Fig. 1d, which exhibits large-scale spatial patterns very similar to those in Fig. 1c. These patterns are largely explained by the dependence of radar and SSB noise on *S _{w}* (C2001). The amplitudes in Figs. 1c,d are also quite consistent, although there is a tendency for larger errors in Fig. 1d at most latitudes. Note that contributions from the wet tropospheric and ionospheric correction errors, as well as orbit errors, may have been overestimated in Fig. 1d, given that numbers in C2001 represent total errors. Any spatial variability in these terms, not accounted for here, may also contribute to the differences with Fig. 1c. There is also an indication that because of recent improvements in precision orbit determination, current orbit errors are somewhat smaller than those quoted in C2001 (Ménard et al. 2003). In any case, the resulting error estimates from the two different methods are in reasonable agreement.

## 3. Representation errors in state estimation

This section will be of interest both to those engaged in state estimation using altimetric data with general circulation models and also for anyone attempting to use altimetric data in isolation—models are necessarily used in any data analysis even if not explicitly noticed (e.g., an assumption that data are dominated by mesoscale eddies, rather than internal tides, is a model). Because model developers do not provide explicit forms for the errors incurred in their numerics, it is usually more convenient to rewrite model errors as data errors (e.g., any physical phenomenon appearing in the data but not in the model is a “representation” error). A quantitative description of representation errors is very difficult, as it involves knowledge of data statistics and model characteristics and is specific to each particular problem. In the MIT–AER ECCO 1° × 1° model forced by wind stress and heat and freshwater fluxes (e.g., Wunsch and Heimbach 2007), we attempt to account for three major representation errors of altimetric data, namely, those related to ocean tides and atmospheric pressure-driven signals, which are of general interest to all altimeter data users, and unresolved eddy variability.

### a. Tides

Altimeter data are routinely corrected for tides and thus errors from this term are present insofar as the tide models are not perfect. A typical value for deep ocean tide model errors is 2-cm rms with part of it likely related to omission of internal tides (Shum et al. 1997). The latter will diminish strongly in the 1° × 1° averages. Large-scale errors cannot be excluded, however. For example, orbit errors are known to include tidal components, some minor tides are omitted, etc. With absent adequate information, we use a constant value of 1 cm for the representation error in deep-sea tides. (Tide errors over shallow water are certainly much higher. A crude spatially variable estimate can be obtained from differencing the two tide models available with the altimeter datasets, but the results are not displayed here.)

### b. Atmospheric pressure load

Altimeter data are similarly corrected for pressure-driven signals using a simple inverted barometer (IB) correction (∼1 cm hPa^{−1}; C2001). Errors in pressure fields and omission of any dynamic signals both contribute here. For the IB error, as for the dry tropospheric correction, we base again our lower-bound estimate on the standard deviation of differences between ECMWF and NCEP pressure fields. Thus, IB errors are approximately 4 times larger than the values for the dry tropospheric correction shown in Fig. 1b.

An estimate of dynamic signals resulting from atmospheric loading can be obtained from the modeling experiments of Ponte and Vinogradov (2007). A 1-yr run of a simplified version of the ECCO general circulation model forced by pressure forcing alone gives the standard deviation values in Fig. 2a, which are similar to previous estimates from purely barotropic models (Ponte 1993). Results in Fig. 2a do not include effects of the mean (climatological) *S*_{1} and *S*_{2} air tide forcing, which were removed from the forcing fields (Ponte and Vinogradov 2007). These effects should be accounted for as part of the tide model correction, but *S*_{1} is not included in early versions of those corrections (being considered in most recent data releases only). An amplitude estimate for the latter tide (Ponte and Vinogradov 2007) is shown in Fig. 2b. Results are very similar to those of Ray and Egbert (2004). The combined error from pressure-forced dynamic signals, based on Figs. 2a,b, is shown in Fig. 2c. Apart from shallow regions, typical values are 2–3 cm, which are larger than our assumed tide errors.

### c. Mesoscale eddies

The remaining term in the time-dependent error is a consequence of unresolved eddy variability in the model and, as already noted, can be regarded as either a data or a model error. As will be seen, it is likely the dominant time-dependent error for present purposes, but unfortunately it is difficult to estimate, in part because there is no clear operational definition of “eddy.” For simplicity, the following procedure was used. First, the “signal” variance in the altimeter data was estimated by subtracting the computed error variance (instrument noise from Fig. 1d plus representation errors related to tides and atmospheric pressure load as discussed above) from the observed TP and *Jason-1* altimeter variance based on 1-Hz measurements. A percentage of such “signal” variance is associated with unresolved eddies. An estimate of that percentage was obtained by comparing the variance from an eddy resolving (1/8°) version of the ECCO model (D. Menememlis 2005, personal communication) to that from the 1° MIT–AER ECCO runs. The percentage of eddy variance was calculated as 100 × (*σ*^{2}_{1/8°} − *σ*^{2}_{1°})/*σ*^{2}_{1/8°}, where the *σ*^{2}_{j} refer to the variances of the two models. (The two models differ in a number of ways apart from their grid resolutions—in particular, the eddy-resolving runs are not constrained to any ocean data—but the assumption that the difference in variance is mostly due to eddy effects seems a reasonable one.)

The resulting estimate for the eddy error term (Fig. 2d) can reach values >30 cm in western boundary regions with vigorous current systems and is also enhanced along the path of the Antarctic Circumpolar Current. Values in quiet interior regions (Tropics, eastern basins) are mostly <5 cm but not zero. The total representation error (tides + pressure signals + eddies; not shown) is dominated by the eddy term as expected and closely resembles Fig. 2d. For eddy-resolving modeling efforts, the situation would be somewhat different, with other representation errors becoming more important.

## 4. Summary remarks

The combination of instrument noise and representation errors yields the estimate of total error shown in Fig. 3. Here, we have taken the instrument noise based on *S _{w}* (Fig. 1d), but results are similar when using estimates based on the TP–Jason analysis (Fig. lc). In addition, the correlation between errors in dry tropospheric correction and IB effect has been neglected. Representation errors dominate the total budget and thus the spatial pattern in Fig. 3 closely follows that of Fig. 2d. Instrument errors contribute to raise the noise levels mainly in interior regions where our estimate of eddy errors is comparable to that of instrument noise. Estimates in Fig. 3 are probably relatively robust in boundary current regions, whereas levels in the interior may be too high. The underlying issue is whether eddies can be as important everywhere as the 1/8° to 1° model comparison seems to suggest.

Lower error values will result when considering data averaged along track over 1° boxes, as is used in the MIT–AER ECCO estimation procedure, instead of 1-Hz measurements. One can calibrate the errors in Fig. 3 by comparing the standard deviation of TP or *Jason-1* data based on 1-Hz data with that based on along-track averaged data. For averaging over 2° boxes (1° could be tried but there would be many boxes without data), typical reduction in TP standard deviation is 10%–20%, which can be assumed for simplicity to result from proportional decreases in signal and noise variance. For the estimated error adjusted for the effects of data smoothing, one can reduce the values in Fig. 3 by equivalent amounts.

A more involved step-by-step calibration could also be used. For instrument noise, changes would involve calculating TP–Jason difference based on along-track averages or reducing components that are likely to be attenuated by the averaging (radar noise, SSB, possibly wet tropospheric correction, ionospheric correction errors already estimated based on ∼100-km arcs). For representation errors, one could use altimeter standard deviations based on averaged along-track data for the calculation of eddy term.

The results in Fig. 3 are the basis for the errors currently used in the MIT–AER ECCO estimates to control the model/altimeter data misfits. As it is dominated by the eddy error term, the same figure represents an estimate of the mesoscale eddy variability in the ocean, which is currently not resolved in the ECCO model. The present results are not, and cannot be, definitive as they are dependent upon elements of the altimetric system noise that are poorly known. They are presented here at this time as being our best global estimates (which we use) in the hope that they may be helpful to others and that they may stimulate further work on this important dataset.

## Acknowledgments

We thank L.-L. Fu (JPL) for suggestions on how to address the spatial structure of altimeter errors, D. Chelton (OSU) for the significant wave height data, K. Case (JPL) for clarifications on the altimeter dataset, D. Menemenlis (JPL) for the output of the 1/8° model, and C. King (MIT) for help with data processing. Supported by NASA through the Jet Propulsion Laboratory *Jason-1* program and by the National Ocean Partnership Program (NOPP) of NASA, NSF, and NOAA.

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## APPENDIX

### Altimeter Data

The SSH altimeter series used are the TP and *Jason-1* Along Track Gridded Sea Surface Height Anomaly products provided by the Physical Oceanography Distributed Active Archive Center (DAAC) at the Jet Propulsion Laboratory. Both series are generated from the geophysical data record (GDR), using the respective GDR orbits, and given at 1-Hz sampling. TP and *Jason-1* anomalies are defined relative to the same MSS and both series are corrected for tides using the same model (GOT99.2b). All geophysical corrections available on the GDR have been applied, with the exception of the SSB correction for the case of TP, which is replaced as recommended in Chambers et al. (2003a). In addition, the ionospheric correction used is based on values smoothed over approximately 100 km along track, centered on the location of the data value. The smoothing reduces the uncertainty in the ionospheric correction and is consistent with commonly quoted errors for that correction (Chelton et al. 2001). Further information on the data processing of the TP and *Jason-1* Along Track Gridded Sea Surface Height Anomaly products can be found in the documentation available at the original archival site.

During the first 20 cycles of *Jason-1*, both altimeters flew along the same ground track separated in time by ∼70 s. Data from this period were used to construct the difference series used in Fig. 1a. Given the described data processing, corrections for tides, the inverted barometer effect, and path delay by the dry troposphere are essentially the same in TP and *Jason-1* series, as they are based on the same tide and atmospheric pressure models. The differences in the TP and *Jason-1* anomaly series at the same along-track points are thus taken to represent mostly radar noise, other instrument errors related to the ionospheric, wet tropospheric, and SSB corrections, and orbit error. The results in Fig. 1a represent the averaged instrument noise without considering possible differences between ascending and descending tracks, which are known to affect the calculation of the bias between TP and *Jason-1* (S. Philipps et al. 2006, personal communication; see also http://www.jason.oceanobs.com/html/swt/posters2006_fr.html).

A number of reprocessed TP and *Jason-1* datasets became available in early 2006, after our original data analysis was performed. Judging from preliminary comparisons discussed at the Venice meeting, the reprocessed data should lead to reduced TP and Jason differences, but large changes from the standard deviation values in Fig. 1a are not expected (e.g., see again the poster by Phillips et al.). In any case, estimates of the altimeter noise provided here will necessarily have to be updated as improved datasets continue to be produced.