1. Introduction

Global state estimation (data assimilation) has become a near-routine method for producing dynamically and statistically self-consistent syntheses of circulation models and oceanic data (Stammer et al. 2002a, 2003, 2004; Köhl et al. 2007; Wunsch and Heimbach 2006). Their methodology is a form of constrained least squares and, as in any least squares method aiming at a minimum variance estimate, one of the essential ingredients in carrying out such calculations is a quantitative description of the error structure of the data being used. Solutions that are forced closer to observations than is warranted are modeling noise; solutions not close enough to the observations are discarding useful information. Prior error information and residuals of all data types used in such calculations therefore have to be examined in great detail since any inconsistency carries important information about unaccounted data and/or model errors.

Here, we examine the time-mean dynamic topography component of the state estimates as used in the work of the Estimating the Circulation and Climate of the Ocean (ECCO) Consortium (Stammer et al. 2002b) and its German partner (GECCO). Our specific goals are threefold: 1) to determine the impact of the time-mean dynamic topography on the estimated solution; 2) to determine the impact of more accurate geoid fields on a hypothetically converged solution; and 3) to discuss the error information of the dynamic topography required for such an enterprise, beyond just the geoid error covariance. The approach taken is to compare optimization results obtained by using the different geoids along with all of the other data going into the ECCO/GECCO estimates. The paper is an extension of a previous study by Köhl et al. (2007, hereafter KEA07). Those authors describe a first ECCO synthesis on a global 1° geographical grid for the period 1992 through 2002 that was carried out using the first geoid estimate derived from the Gravity Recovery and Climate Experiment (GRACE) geoid (Tapley et al. 2003). Subsequently, Wunsch and Heimbach (2006) carried that estimate forward through 2004.

This article can also be understood as an attempt to extend the previous analyses of the impact of improved geoid estimates on the problem of determining the ocean circulation (Ganachaud et al. 1997; LeGrand et al. 1998; Rio and Hernandez 2004; among others). These earlier papers focused, from practical necessity, on the nominally steady-state ocean circulation. Wunsch and Stammer (2003) then discussed the potential of ocean state estimates as consistency tests and highlighted that the combination of geodetic and oceanographic knowledge permits a better estimate of the geoid as well as the ocean circulation. Here we extend the previous discussion to a fully time-dependent system. As discussed below, we have progressed to the point where the lack of understanding of data and model errors seriously inhibit the possibility of making improved estimates of elements of the ocean circulation. Jayne (2006), in a study related to this one, studies the impact of the GRACE geoid on ocean circulation estimates, but confined to the North Atlantic.

2. Uncertainties in dynamic topography fields

Existing ECCO state estimation efforts are constrained by many data types and their respective error estimates. Among those is the time-mean surface dynamic topography, which is calculated as the difference between time-mean altimetric observations of the sea surface height (SSH) minus a geoid height model. Accordingly, the error covariance for the surface dynamic topography has to take into account errors in both the geoid and the altimetric observations. In the past, geoid uncertainties were considered to dominate the errors of the dynamic topography. However, with ever-decreasing geoid height errors, the uncertainties in the time-mean and time-varying altimetric data are gaining importance, to the point that they may already dominate the error budget of the surface dynamic topography field. Geoid height errors include commission and omission errors alike. The latter are significant on small spatial scales not resolved with a given degree/order, which are important for capturing energetic ocean circulation structures. Because of this effect, one has to smooth the model sea surface height so that it does not contain signals on scales smaller than available in a geoid model provided to a predescribed degree and order.

The other important contribution to the error of the dynamic surface topography comes from altimeter errors. Theses are often conveniently separated into those of the time-mean and those of the temporal anomalies. Unfortunately, the separation is not an absolute one, as the mean continues to be estimated over ever-increasing time intervals and there is no known frequency below which the time variations in the errors vanish. Chelton et al. (2001) summarize best estimates of the time-mean and total errors as of that time. According to their Table 11, the root-mean-square (RMS) altimetric errors are about 4 cm, but results suggest strong regional variations in this and the time-mean value, the latter perhaps reaching over 10 cm in the Southern Ocean (due to errors in wind and wave observations there). A definitive discussion is impossible at the present time, and the lack of proper information on the background time-mean altimetric error must be borne in mind in the following discussion of the geoid component.

An updated discussion of the time-variable errors was provided recently by Ponte et al. (2007). Their comparisons between the independent Ocean Topography Experiment (TOPEX)/Poseidon (T/P) and Jason-1 altimetric missions, when they were in identical orbits, show that point-by-point, the data are consistent within the mission specifications of about 3-cm RMS error, but large-scale dependences exist in the data differences, and these are both poorly known and capable of introducing major errors into oceanic state estimates. The analysis reveals errors reaching 4 cm at mid- and high latitudes and lower ones present in low latitudes.

3. Methodology

Our methodology is to fit a general circulation model (GCM) to a very large set of regional and global data using constrained least squares. For this purpose, Lagrangian multipliers are used and the fit is an iterative one [see Wunsch (2006) for a summary]. The underlying model, data, and methodology are described by Stammer et al. (2002a, 2003). KEA07 describe in detail the specific model setup and datasets used as well as the optimization procedure and results.

In brief, the ECCO GCM is based on the Massachusetts Institute of Technology (MIT) GCM described by Adcroft et al. (2002), coupled to a surface mixed layer model (Large et al. 1994) and using the eddy parameterization scheme of Gent and McWilliams (1990). In our present experiments, the horizontal model resolution is 1° over ±80° in latitude with 23 levels in the vertical, and the estimation period covers the years 1992 through 2002. In experiment optimization 2, the difference T/P − GRACE − University of Texas GRACE geoid model (GGM01c; Tapley et al. 2003) was used to constrain the time-mean dynamic surface topography, using a spatially uniform error of 4.5 cm. A schematic of the data constraints is displayed in Fig. 1; they include several satellite datasets [altimetry from T/P, European Remote Sensing Satellites (ERS-1 and -2), scatterometer data, and Reynolds and microwave SST fields], time-mean surface drifter velocities, in situ hydrographic temperature and salinity profiles, as well as hydrographic sections.

In an earlier calculation (called optimization 1 hereafter), the difference T/P minus the EGM96 geoid (Lemoine et al. 1997) was used to impose constraints on the time-average absolute circulation (see Köhl et al. 2002). A diagonal covariance matrix was used with values taken from the diagonal of the EGM96 error covariance matrix (cf. Lu et al. 2002). An estimate of the resulting 11-yr time-mean dynamic topography field is displayed in Fig. 2a. On the large scale, the results resemble the observed dynamic topography associated with the basin-scale gyre structure of the ocean. However, differences on subbasin scales loom large. Figure 2b shows the associated residuals relative to the imposed mean dynamic topography of the combined missions (data are all T/P) in the model referred to in EGM96. Values are on the order of ±20 cm and reach ±50 cm near steep topography. These differences contradict the estimated prior geoid error fields and a goal here is to understand those residuals in terms of model and data errors alike. For that purpose, the combination of optimization 1 and optimization 2 is used to investigate the following question: does the near-optimized model fit better to the time-mean altimetry based on GRACE than it does to EGM96 assuming that the GRACE error estimates are approximately correct?

In experiment optimization 3, we then used the difference T/P − GRACE − GGM02C (Tapley et al. 2005) as a constraint on the time-mean dynamic surface topography, but with an artificially increased accuracy for the mean dynamic topography field. The combination of optimization 1 and optimization 3 will be used to address the question raised by Ganachaud et al. (1997) of the extent to which a much improved geoid estimate would carry information about the ocean circulation not already contained in other data and in the model physics themselves. While we force the model very strongly toward the GRACE-based dynamic topography, resulting inconsistencies with prior in situ and geoid data—or the lack thereof—will permit the determination of whether the new solution is plausible or not.

4. Impact of the GRACE geoid

To investigate the reason for the large SSH residuals relative to the EGM96-based dynamic topography, optimization 2 was performed in which all constraints remain the same except that the time-mean T/P dynamic topography field minus GGM01c was imposed over the entire 11-yr period. As advised by GRACE, a geographically uniform error value of 4.5 cm was employed along the covariance matrix diagonal. In simple terms, this error can be understood as being composed of roughly 3-cm RMS GRACE error plus 3.5-cm RMS mean T/P time-mean error (the latter is very optimistic, especially over the Southern Ocean as discussed above, but only the sum of the errors affects the results). Optimization 2 was run in parallel to the last six iterations of optimization 1.

Figure 3a shows the contributions to the total objective (or misfit or cost) function that result from individual datasets and remain as residuals at the end of the experiment optimization 2. Each column is labeled according to the dataset it represents as explained in detail in the figure caption and in Table 1. Each column was normalized by the corresponding number of data points used as constraints and divided by the prior error variance; the square root is taken subsequently for plotting. In this form, a fully converged solution would have an expected value of one for which each column and misfits would be distributed in a χ² distribution without spatial structures. The figure reveals that the total misfit is close to the ideal number of one but that the solution is not yet fully converged in the sense that individual columns do deviate from the ideal value of 1. The same holds for optimization 1 and optimization 3 as well. An alternative interpretation could be that prior error statistics are not compatible with the model and the data—a subject of ongoing research. Nevertheless, large-scale features in the estimate appear to be relatively insensitive to the continuing slow improvements established during additional iterations, allowing the use of the existing solutions for the present investigation.

Figure 3c shows the fractional changes in the RMS model data misfits of optimization 2 minus the results from optimization 1 normalized by results from optimization 1; that is, negative numbers in the figure indicate the fractional improvements in the model data misfits imposed through the use of the GRACE geoid height. From the figure it follows that

1. Replacing EGM96 by the GRACE geoid model and reducing the squared error of the dynamic topography to (4.5 cm)² while leaving all other optimization parameters unchanged produces a new solution with its initial conditions significantly closer to the Levitus and Boyer (1994) and Levitus et al. (1994) climatological temperature (T) and salinity (S) fields. This result can be seen from columns 3 and 4 of Fig. 3c, showing up to 50% less deviations from the Levitus temperature and salinity fields than were found in optimization 1. This improvement could be a coincidence because the climatology itself has large and poorly understood errors (see, e.g., Forget and Wunsch 2007). However, it also suggests that the T/P–GRACE dynamic topography is more in line with estimates based on in situ data.

2. Because of the smaller adjustment to Levitus initial temperature and salinity conditions, the model also stays dynamically more balanced, as suggested by the reduction in the vertical velocity drift during optimization 2, as can be seen from the third to last column of Fig. 3c. Misfits between the monthly mean climatological hydrographic fields from optimization 2 and climatological fields are also slightly smaller (columns 1 and 2 of Fig. 3c). Again, both can be interpreted as indications of a dynamically more consistent T/P–GGM01c dynamic topography (i.e., more skill in the GRACE geoid field).

3. The time-mean misfit of the dynamic topography increases with the use of the GRACE geoid. This increase is primarily due to the reduced prior error of the new geoid (i.e., because the misfit terms are divided by a smaller number). Nonetheless, the absolute (unweighted) misfit decreases as demonstrated in Fig. 4a.

These residuals are much smaller than those found from the differences of optimization 1 from the EGM96-based SSH field (cf. Fig. 2b). The dynamic topography misfit in optimization 2 now shows residuals on the order of ±10 cm but still reaches ±40 cm on relatively short spatial scales near steep topography, especially over the Southern Ocean. The implication is that the 4.5-cm RMS error for the time-mean dynamic topography remains unrealistically optimistic in those regions and that a full error covariance budget for the dynamic sea surface height is required—one that would have strong spatial variability and accounts for commission and omission errors in the geoid and errors in the T/P data and the model. It will be very important to analyze in detail the residuals between the GRACE-based dynamic topography and the model results work that is underway currently and which also needs to address model errors in detail.

Figure 4b, showing the differences in the estimated time-mean dynamic topography obtained from optimization 2 minus that of optimization 1, illustrates that adjustments in the dynamic topography forced in optimization 2 relative to those in optimization 1 are small. Large-scale changes are typically only on the order of 1 cm or less and arise from changes in temperature and salinity (not shown) on the order of ±1° and ±0.1 psu, respectively (depending on the depth), bringing the second optimization closer to the hydrographic climatology. Maximum differences of ±5 cm are reached only in two regions.

Two implications follow from the figure: 1) the previous solution was not significantly overconstrained by the errors in the EGM96 model. Only in a few places the GRACE-based SSH solution moves away from it, notably in the Southern Ocean south of New Zealand, in the western North Pacific north of the Kuroshio Extension, and in the subpolar North Atlantic. Adjustments in all other regions are only minor; and 2) because of those small overall changes in SSH, no additional knowledge about the ocean circulation or ocean transports was extracted from the new GRACE geoid field relative to what we know from in situ and altimeter data.

Goals of ocean state estimation include the identification of inconsistencies in the geoid model and its prior error description (like with any other dataset) and to help improve them. Can we identify EGM96 errors from the existing optimizations (i.e., can the residuals, shown in Fig. 2b, be identified as uncertainties in the EGM96 geoid model)? To some level this question can be addressed by investigating the differences between the EGM96 and the hypothetically much more accurate GRACE geoid models, shown in Fig. 5. A visual inspection reveals that many of the structures shown in Fig. 2b are actually also present in the geoid difference field with comparable amplitude. The figure therefore suggests that a significant fraction of the SSH misfit obtained from optimization 1 represents uncertainties in the EGM96 geoid model not accounted for by its error variance and that the ocean state estimation procedure provided additional information for the improvement of this geoid model, except it so far does not provide a full geoid error covariance matrix.

One might wonder if we can also interpret the residuals resulting from optimization 2 (Fig. 4) as uncertainties in the GGM01c model by comparing the two GRACE geoid models GGM01c and GGM02c. However, differences between those two solutions reside to first order on small spatial scales and do not resemble the residuals we encounter in optimization 2. To answer the above question to what extent Fig. 4 resembles GGM01c errors or if, rather, residuals reveal T/P or model errors would need to await the new Gravity and Ocean Circulation Explorer (GOCE) geoid, which is supposed to be a much more accurate geoid than GRACE on scales relevant for basin-scale and regional ocean circulation. Until then, ocean information has to be used to answer this question.

5. Impact of a hypothetically accurate GOCE geoid

We have seen that the impact of a new GRACE geoid on the ocean state estimate is quite small and does not lead to a significant modification of the previous ocean circulation solution. However, because the underlying state estimate is not yet fully converged in the sense that all model data residuals diminished to statistically acceptable levels implies that the contribution of the misfit of the mean dynamic topography to the total cost function remains relatively small and therefore relatively insignificant. The question therefore arises if this is so because the information content in the geoid relative to other ocean data is comparatively low as suggested by Ganachaud et al. (1997) and LeGrand et al. (1998), or whether it is merely the consequence of the failure to bring the system to the asymptotic, fully optimized state (i.e., all misfit terms are much larger than a final state would warrant, or whether the failure to take model errors properly into account poses a principal problem with the given optimization setup, e.g., spatial resolution).

In an attempt to investigate these possibilities, optimization 2 was continued as optimization 3 after replacing GGM01c with GGM02c, with the geoid height error kept fixed, but now with the weights of all remaining misfit terms reduced significantly to simulate a converged solution for which then the mean dynamic topography based on new geoid models can make a sizable contribution to the cost function. To this end, we artificially force the model to the time-mean dynamic topography by increasing the relative errors on the nongeoid misfit terms by a factor of 12 during 10 further iterations, and then by a factor of 50 during a final 9 iterations.

Resulting contributions to the objective function are displayed in Fig. 3b. To compare this figure with results from optimization 2, we use the error weights of optimization 2 in both Figs. 3a and 3b. Figure 3d shows changes in the RMS misfits of optimization 3 minus the results from optimization 1, again normalized by optimization 1. Under present circumstances, we observe a clear decrease of the misfit in the time mean SSH by 50%. But as compared to optimization 2, complex adjustments take place in the absolute misfits of the remaining data to accomplish this success. For example, there is improvement and degradation of monthly mean temperature and salinity misfits (by a few percent), averaging nearly to zero (columns 1 and 2). Sea surface salinity (SSS) misfits are improved (column 16), but SST degrades (column 14). Also noteworthy is the substantial increase in the misfit of the initial temperature and salinity conditions relative to the hydrographic climatology, as well as the drift term in the vertical velocity measuring the dynamical imbalance of the initial conditions. We also note a slight increase in the misfit of all time-mean forcing fields from the NCEP first-guess conditions. While overall the misfit of time-mean components increases, most of the residuals in time-varying datasets diminish. This is especially true for the misfits to all vertical profile data, such as expendable bathythermographs (XBTs), ARGO, and CTD data.

All these changes are difficult to interpret, as they depend upon the detailed nature of the real errors in the GRACE geoid as well as inaccuracies in the remaining data types, neither of which are well known. In the following we will first discuss changes in control terms and discuss their consistency with prior information. We will subsequently analyze changes in the flow field imposed by the geoid field and the associated changes in the control terms.

6. Discussion

The comparison of the oceanographic implications of the EGM96 versus GRACE geoids for estimates of the ocean circulation supports the inference that the GRACE geoid has a significantly improved skill (as compared to EGM96) on spatial scales of 500 km and larger, implying that the GRACE geoid is closer to dynamical consistency with the GCM and other data than is EGM96. Our conclusion therefore is similar to the recent results from Birol et al. (2005) in that existing geoids do show skill in improving ocean circulation estimates on the large scale.

The apparent improvement has oceanographic and climate consequences consistent with the earlier inferences based upon steady-state assumptions. In particular, the GRACE geoid appears to be more consistent with temperature and salinity climatologies than is the older geoid, although this improvement is only weak and not definitive, given issues concerning the climatology accuracies. Nevertheless, the new geoid requires smaller adjustments to the initial model temperature and salinity conditions (from Levitus and Boyer 1994; Levitus et al. 1994). Although at present levels of accuracy, the GRACE geoid height estimates, while apparently an improvement over EGM96, do not lead to qualitative adjustment in the estimated ocean circulation, the optimization itself produces a new, and presumably, improved geoid estimate, but its full use requires an estimated error covariance, which is not yet available.

The implications of a much improved geoid (beyond that now available from GRACE) on a converged solution can be explored by driving the model arbitrarily close to the existing geoid. Forcing such a fit produces structures in the inferred ocean circulation that are significantly different from those estimated using a realistic error estimate. This fit requires only small changes in the meteorological fields—ones that stay well within their error bars. The only region with unreasonable changes in control terms appears to be the Southern Ocean (besides boundary regions). There is no physical reason to reject these circulation structures over all other regions—they are meteorologically and oceanographically reasonable at the current levels of in situ observational error. However, there is no obvious independent test of their reality available. We seemingly have reached a stage with combined altimetry/geodesy in which further geoid improvement could significantly improve estimates of the ocean circulation. Whether such further improvement relative to the errors of the other data types is possible is unclear at the present time.

Because of the specifics of the experimental setup, we cannot attach any significance to specific adjustments of the estimated circulation (i.e., we do not attach any significance to each detail in the changes because none of the solutions are fully converged). Rather, the main results lie with the estimates of the magnitudes of the inferred changes and their relation to prior information. In essence, the implications of this experiment are that if the accuracy of the GRACE-based dynamic surface topography were significantly better than now estimated, that quantitatively improved ocean circulation estimates would be obtained. If one simply assumed that the present GRACE geoid estimate were much more accurate than its authors believe, testing of the result would confront parallel and existing questions about the accuracy of the other data types (e.g., Forget and Wunsch 2007, for hydrographic data errors). However, remaining residuals near the boundaries and in the ACC indicate, particularly, the continued importance of the omission errors of the GRACE geoid—the missing short scales—as well as model uncertainties. But it may prove possible (e.g., Wunsch and Stammer 2003) to test the integral properties of such solutions against observations of the earth’s rotation and polar motion in conjunction with adequate atmospheric models. It will be difficult but critical to confirm estimated changes in the circulation by independent means such as ocean time series or other independent in situ data.

However, for an optimal use of future precise geoids, improved error information is required. The error budget of existing time-mean dynamic topography estimates may now be dominated by residual errors in time-mean altimetric corrections and these need to be better understood jointly with model errors before present and future geoid estimates can be fully used in ocean studies, especially in the Southern Ocean. A critical issue remains in the need to better discriminate between errors in the altimetric and geoid estimates, neither of which are well understood at the levels of accuracy now apparently being achieved by ECCO-like estimation procedures. The results here suggest that a spatially uniform error in the existing GRACE geoid, as provided by the project, is qualitatively incorrect in many locations. As discussed above, the many elements of the altimetric error also appear to have strong spatial variability and the separation of time-mean and time-varying errors remains incomplete. As the measurement errors decrease, we have to think about including model errors when assimilating mean SSH data.