## Abstract

Knowledge of the ocean dynamic topography, defined as the height of the sea surface above its rest-state (the geoid), would allow oceanographers to study the absolute circulation of the ocean and determine the associated geostrophic surface currents that help to regulate the Earth's climate. Here a novel approach to computing a mean dynamic topography (MDT), together with an error field, is presented for the northern North Atlantic. The method uses an ensemble of MDTs, each of which has been produced by the assimilation of hydrographic data into a numerical ocean model, to form a composite MDT, and uses the spread within the ensemble as a measure of the error on this MDT. The r.m.s. error for the composite MDT is 3.2 cm, and for the associated geostrophic currents the r.m.s. error is 2.5 cm s^{−1}. Taylor diagrams are used to compare the composite MDT with several MDTs produced by a variety of alternative methods. Of these, the composite MDT is found to agree remarkably well with an MDT based on the GRACE geoid GGM01C. It is shown how the composite MDT and its error field are useful validation products against which other MDTs and their error fields can be compared.

## 1. Introduction

A key concept in oceanography is that of dynamic topography, which is defined as the height of the sea surface relative to the particular gravitational equipotential surface known as the geoid. If the ocean were at rest, then its surface would coincide with the geoid and so the dynamic topography would be zero. However, wind and buoyancy (heat and freshwater) forcing result in a non-zero dynamic topography and a corresponding circulation of the ocean. Hence the dynamic topography can be used to diagnose the ocean circulation, and, in particular, it can tell us about currents that redistribute heat and thus regulate the Earth's climate.

For some time accurate measurements of the sea surface relative to a reference ellipsoid have been provided by satellite altimeter missions, such as TOPEX/POSEIDON. However, limitations in geoid determination mean that direct reliable measurement of the dynamic topography has yet to be achieved. Presently, oceanographers can obtain the time-dependent part of the dynamic topography by using altimeter repeat track sampling to remove both the geoid and the time—mean dynamic topography (MDT). This method provides useful information on how the ocean circulation is changing during the reference mean time period, but, on its own, tells us nothing about the absolute circulation of the ocean during this period. This can be resolved by adding the anomalous dynamic topography to an MDT for the same time period, which must be obtained by alternative methods.

Various methods have been used to calculate the MDT from *in situ* ocean data. The most straightforward of these uses climatologies of temperature and salinity, based on measurement profiles made over many decades (Levitus & Boyer 1994; Levitus *et al*. 1994), to compute dynamic height relative to an assumed level of no motion. A refinement of this method uses an inverse model with certain dynamical constraints to get the barotropic signal (Legrand *et al*. 1998). Neither method, however, can represent a uniform time average due to the inhomogeneity of hydrography data. Niiler *et al*. (2003) have derived an MDT from a 10-year set of near-surface drifter velocities corrected for temporal bias using altimeter data. Rio & Hernandez (2004) have produced the most sophisticated melding of ocean observations to produce an MDT without the use of a model. They use hydrographic and drifter data to calculate the instantaneous dynamic topography at many locations over the period 1993–1999. Coincident altimeter data were then used to convert this to local estimates of the MDT, which were then used to correct a first guess based on the direct mean sea surface (MSS) minus geoid method and the Levitus climatology. A formal error is obtained based on the data sampling, but the errors in the first guess are still not well determined.

While pure data studies are useful, for the purpose of forming an MDT for a specific time period, they will always suffer from non-uniform spatial and temporal sampling. This is particularly true of hydrography data, where the distribution of data is extremely sparse. By forcing a dynamically evolving Ocean General Circulation Model (OGCM) with realistic winds, and fluxes of heat and freshwater, an MDT with uniform sampling is readily produced for any required time period. Inherent biases in OGCMs can be reduced by the assimilation of altimetric and *in situ* data into the model. This may offer the most effective way of combining observations with a physical understanding of the ocean, and it has the further advantage of enabling the relationship between the internal state of the ocean and the dynamic topography to be examined. Several groups have now performed such data assimilation experiments. Although this method offers the potential to estimate an error field for the calculated MDT, in practise, the computation costs involved are prohibitive (Stammer *et al*. 2002).

This paper describes a new approach to obtaining an MDT together with an error estimate from OGCMs, by combining a number of MDTs to form a composite, and converting the spread between the MDTs into a formal error estimate. The study region is the northern North Atlantic from 79W to 13E and 41N to 79N, and encompasses the EU-funded GOCINA project region (see http://www.gocina.dk/). In §2 we shall outline the method used to produce the composite MDT and error field. In §3 we describe the oceanographic features of the MDT and error field. In §4 we compare our MDT with MDTs produced by the other methods described above. Finally, in §5 we discuss the implications of our findings.

## 2. Material and methods

Our approach to obtaining an OGCM estimate of the MDT, together with an associated error estimate, is to form a composite MDT (CMDT) from an ensemble of OGCM MDTs (see table 1), each of which is derived from a data assimilation experiment with realistic forcing and the assimilation of *in situ* hydrography data. The spread within the ensemble of MDTs is then used to compute the uncertainty, or error, associated with the composite MDT.

The two OCCAM MDTs come from two runs of the OCCAM model, which differed with respect to the forcing used: the first run was forced with wind stresses from the European Centre for Medium-Range Weather Forecasts (ECMWF), while surface temperature was relaxed to Reynolds & Smith (1994) data and surface salinity was relaxed to a monthly climatology from Levitus & Boyer (1994). The relaxation time-scale of one month produced only a weak buoyancy forcing (Fox & Haines 2003). The second run used wind stresses and buoyancy fluxes calculated from the National Centers for Environmental Prediction reanalysis, in addition to a much weaker relaxation of surface temperature and salinity to Levitus.

### (a) Standardizing the MDTs

The model MDTs do not refer to a common time period, so to ensure that the composite MDT refers to a well-defined time period, and that the error estimate does not reflect differences between the MDTs that are due to temporal variability, it is necessary to adjust them to a common time period. For this study we have chosen the time period of 1993–2001, as agreed upon in the GOCINA project. Using altimetric sea-level anomalies, *μ*, each MDT in table 1, *η*, is adjusted to the reference period as follows:(2.1)where the subscript R denotes the reference time period, and the subscript A denotes the actual time period to which the MDT refers. The overbar denotes the temporal mean over the time period. In fact, because the differences between the MDTs are generally larger than the corrections required to standardize the time period, the temporal adjustments do not affect the result greatly.

The resolution, and therefore the resolvable scales, of the model MDTs differ. Therefore, to ensure that each of the model MDTs contributes equally to all spatial scales contained in the composite MDT, and to ensure that the error estimate is only based on variability that can be represented by all of the models, each MDT was low-pass filtered with a 1×1 degree box-car filter to leave only those spatial scales that are resolvable by the members of the ensemble with the coarsest resolution. Finally, a constant offset was removed from each MDT, so that its spatial mean over the domain was zero.

After making these adjustments, the CMDT (see figure 1*a*) was formed from the mean of the MDTs produced by the models in table 1, the oceanographic features of which are described in §3. Additionally, the geostrophic currents derived from the gradients of each MDT were combined to form a composite velocity field.

Although a weighting scheme based on some property of the models such as resolution or amount of data assimilated can be envisioned, here each of the MDTs is given equal weighting in the composite because, as yet, the precise sensitivity of a model MDT to any of these factors is unclear.

### (b) Uncertainty estimation

It is desirable to have an estimate of the error on an MDT, but this is generally not feasible for a model MDT. The basis of our technique is to assume that each member, *η*_{n}, of the ensemble gives an independent and unbiased estimate of the true MDT, *m*, such that *η*_{n}=*m*+*ϵ*_{n}, where *ϵ*_{n} is a random error coming from a normally distributed population. Then we can take the CMDT, as an unbiased estimate of the true MDT, *m*, and assign confidence intervals to this estimate. Since we only have a small sample size (*n*<30), and we only know the sample standard deviation *s* (not the true standard deviation *σ*), we shall employ the variable(2.2)which has a Student's *t*-distribution with *N*−1 degrees of freedom (Emery & Thomson 2001). Using this variable enables us to calculate the confidence intervals for the composite MDT as follows:(2.3)where 1−*α* represents the degree of confidence we can have in our estimate of *m*. For a particular combination of *α* and *N*, the value of *t* can be obtained from a look-up table. For this study *N*=5, and we shall use *α*=0.32, corresponding to a 1*σ* confidence interval. We then take this interval as the error estimate for the composite MDT (see figure 1*b*), since under the assumptions made above we can assert with 68% confidence that the true MDT lies in this interval. This method was also used to calculate an error estimate (see figure 1*c*) for the geostrophic current speeds derived from the MDT gradients.

An attempt was made to assess the assumption of normality on which the method rests by using the method of L-moments (Hosking 1990). The strength of this method is that, in comparison with more conventional techniques, it allows more robust inferences to be made from a small sample regarding the underlying probability distribution. If the null hypothesis is that the sample comes from a normally distributed population, then by the method of L-moments we find that the null hypothesis can be rejected at the 95% significance level for 23% of the points within the domain. With our small sample size, however, it is difficult to make a more definite statement regarding the normality of the ensemble.

Even assuming normality, the method of deriving the CMDT and error field described here cannot account for systematic bias in the set of MDTs that may be present due to inherent limitations shared by all of the ocean models used to produce the MDTs. For instance, assuming that the position of the boundary currents becomes more accurate with increasing resolution, then all of the models in the set are on the ‘wrong side’ of the ideal resolution, thus biasing the set of MDTs. Also, towards high latitudes the model interpolation of the increasingly sparce hydrographic data could bias the set of MDTs in these regions. Comparison with MDTs derived by independent methods is therefore necessary to highlight any systematic bias present in the CMDT (see §4).

## 3. The composite MDT and errors

In broad terms, the CMDT in figure 1a represents all of the expected features of the ocean circulation in the North Atlantic and Nordic Seas. There are two distinct parts to the sub-polar circulation: a dominant gyre south of Greenland and a less intense circulation in the Nordic Seas. The northern part of the sub-tropical gyre circulation is just visible at the southern boundary of the region.

The major current systems of the region are apparent in the CMDT. The most significant of these, the North Atlantic Current (NAC), acts as a front separating the warm sub-tropical waters from the cold sub-polar waters. Part of the NAC feeds the Shelf-Edge Current along the west coast of the British Isles, which in turn supplies the Norwegian current via the Faeroe-Shetland channel. These currents are important since they form the route by which relatively warm water travels northward to heat the high-latitude regions. The East Greenland Current, which takes cold water southward between Greenland and Iceland, and where mean current speeds can reach 50 cm s^{−1}, is also clearly represented by the CMDT.

In the majority of the open ocean, the uncertainty in the CMDT is less than 3 cm, showing that the five MDTs used in this study agree remarkably well. There are, however, significant regions of uncertainty, or disagreement, between the models. Over a large part of the main sub-polar gyre, the error field is 5–6 cm, and reaches 7–8 cm in the Labrador Sea. Along the Greenland coast, and extending into the northern part of the sub-polar gyre, the errors are 4–5 cm. The reason for this is an underestimation of the strength of the sub-polar gyre circulation by the lower resolution models ECCO and ECMWF relative to the higher resolution models. There is also uncertainty in the NAC region, particularly at 40W, 50N where it exceeds 8 cm. This is due to differences in how the NAC is deflected by the Grand Banks in each of the models.

In terms of geostrophic surface currents, the greatest uncertainty in the composite lies in the narrow boundary current regions where the current speeds are greatest, while over most of the open ocean the uncertainty rarely exceeds 1 cm s^{−1}. Again this reflects the fact that the lower resolution models underestimate the strength of boundary currents relative to the higher-resolution models. Lower-resolution models also tend to do less well with regard to the representation and positioning of the finer features of the NAC systems, such as the branch point where the Gulf-stream meets the NAC, and the Northwest Corner of the NAC off Newfoundland. This is apparent in the increased errors in both the CMDT and the associated surface currents in the southwestern corner of the domain (south of 50N and west of 30W).

## 4. Intercomparisons

As was previously pointed out, there are a number of approaches to obtaining an estimate of the MDT. In this section, we compare MDTs obtained by some of these methods and interpret the differences between them. Our analysis includes several MDTs based directly on *in situ* data: an MDT based purely on climatological data (Levitus & Boyer 1994; Levitus *et al*. 1994); an MDT by Legrand *et al*. (1998), the RIO03 MDT by Rio & Hernandez (2004), both produced by inverse methods; and an MDT produced from drifter data (Niiler *et al*. 2003).

The analysis also includes two MDTs produced by the direct method of subtracting the geoids EGM96 (Lemoine *et al*. 1998) and GGM01C (Tapley *et al*. 2004) from the MSS CLSMSS01 (Hernandez & Schaeffer 2001). Until recently, EGM96 represented the state of the art in geoid determination. Its longest wavelengths were computed from satellite data and *in situ* gravimeter measurements provided the short wavelengths. The more recent GRACE geoid used here is also a blend of satellite and terrestrial data. However, according to Tapley *et al*. (2004), GRACE has provided more than an order of magnitude improvement in the long-wavelength satellite determined component of the Earth's gravity field.

### (a) Taylor diagram intercomparisons

There are four statistics that are particularly useful when comparing two spatial fields: the standard deviations of the two fields, *σ*_{1} and *σ*_{2}; the correlation between the fields, *R*; and the centred pattern r.m.s. difference between the two fields, *E*. It can be shown that these quantities are related as follows:(4.1)This equation has the same form as the cosine rule for triangles (see figure 2). Taylor diagrams (Taylor 2001) exploit this equivalence to provide a convenient way of comparing many fields with a chosen reference field on a single diagram. On such diagrams each field is represented by a point, where the radial distance of the point from the origin is proportional to the standard deviation of the field, and the angle subtended by the *x*-axis and a line connecting the point and the origin represents the degree of correlation between the field and the reference field. The correlation of the reference field with itself is one and therefore it appears on the *x*-axis. Finally, according to equation (4.1) the distance between the point representing the field and the point representing the reference field is inversely proportional to the overall similarity of the two fields.

A Taylor diagram intercomparison of the MDTs described above is shown in figure 3*a*. The CMDT is taken as the reference field against which the individual MDTs are compared. The OGCM MDTs (black symbols) are clustered together close to the CMDT on the Taylor diagram showing that they agree well, both with each other and with the CMDT. Their standard deviations range from 19.1 cm for the ECCO MDT to 23.2 cm for the OCCAMv1 MDT, compared with a standard deviation of 20.5 cm for the CMDT. The smaller standard deviation for the lower resolution models ECCO and ECMWF reflects the fact they underestimate the strength of the sub-polar gyre circulation relative to the higher resolution models. Correlations with the CMDT range from 0.93 for the ECMWF MDT to 0.98 for the OCCAMv1 MDT.

Excluding OCCAMv2 from the composite reduces the correlation between OCCAMv1 and the composite by less than 0.01, and OCCAMv1 remains the most well correlated with the revised composite. A similar result holds when OCCAMv2 is excluded. This demonstrates that differences between the two OCCAM MDTs are comparable to their differences relative to the other MDTs from which the CMDT is formed.

Of the MDTs produced by other methods, the MDT derived from the CLS MSS and the GGM01C geoid has the highest correlation (0.95) with the CMDT, and it also has the closest standard deviation (21.0 cm). The Niiler MDT is also similar to the CMDT, with a correlation of just less than 0.95 and a standard deviation of 23.4 cm. This is an encouraging result, given the completely independent means by which these MDTs have been produced. In contrast to the MDT based on GGM01C, the other geodetically derived MDT, based on EGM96, agrees less well with the CMDT, demonstrating that GRACE represents an improvement in geoid determination over the open ocean. The RIO03 MDT has the poorest correlation with the CMDT, and the reasons for this are discussed below. Since the LeGrand MDT is a modification—by inverse modelling—of the same data that the Levitus MDT is based on, their relative positions on the Taylor diagram perhaps indicates an improvement brought by the method. However, as the LeGrand MDT is only defined to 70N the statistics computed for this smaller area are less significant.

While it is not strictly the case that the distance between any two fields on a Taylor diagram is equal to their mutual centred r.m.s. difference—this is only true for the distance between each field and the reference field—it was found that, in general, the relative positions of any two MDTs on the diagrams does reflect their similarity. For instance, using RIO03 as the reference field only increased the correlation score of the Levitus MDT (0.80–0.93), which is the closest MDT to RIO03 in figure 3*a*. With RIO03 as the reference field the correlations of the Niiler and GRACE MDTs decreased from 0.95 to 0.90 and 0.88, respectively, confirming that the CMDT is more similar to both of these independent estimates of the North Atlantic MDT than is RIO03.

A similar Taylor diagram intercomparison of the geostrophic surface currents associated with the MDTs is shown in figure 3*b*, and in this case there is less agreement among the various fields, both in terms of their standard deviation and in terms of their correlation with the reference field. Whereas the standard deviation of each MDT is dominated by the amplitude of its long-wavelength component, in this case the standard deviation is more sensitive to the short-wavelength variability, or roughness, of the MDT.

The correlation between current speed fields derived from the OGCM MDTs and the CMDT range from 0.57 for the ECMWF MDT to 0.95 for the OCCAMv1 MDT, and the standard deviations range from 4.8 cm s^{−1} for ECMWF to 10.6 cm s^{−1} for the FOAM MDT. Because the lower resolution models ECCO and ECMWF underestimate the strength of the sub-polar gyre circulation relative to the higher resolution models, they therefore also underestimate the boundary current speeds, and this is reflected in their lower current speed standard deviations. Of the non-OGCM MDTs, it is the CLSMSS01-GGM01C MDT and the Niiler MDT that are again the most closely correlated with the CMDT having correlations of 0.75 and 0.78, respectively. The Niiler current speed field, however, has a greater standard deviation (10.0 cm s^{−1}) than the CLSMSS01-GGM01C field (5.4 cm s^{−1}). Part of this greater short-wavelength variability is due to the fact that the CLSMSS01-GGM01C MDT was smoothed by more than the one degree smoothing applied to the other MDTs. The high current speed standard deviations associated with the RIO03 MDT suggest that it also has greater spatial variability at small scales compared with the CMDT. The low standard deviations for the Levitus and LeGrand current speeds are due to the greater smoothing involved in mapping the climatological data and the longer time period from which the data were taken.

### (b) Cross-validation

Since we have an error field for the CMDT, we can make a more quantitative comparison between it and the RIO03 and GGM01C-CLSMSS01 MDTs, for which error fields are also given, and assess whether the differences between them are consistent with the given errors. For consistency, the magnitude of the difference between two MDTs should be less than the combination of the errors on the two fields. If this relationship is not satisfied, then the error estimate for at least one of the MDTs must be too optimistic.

Figure 4*a*,*b* shows, respectively, the differences between the CMDT and the GGM01C-CLSMSS01 MDT, for which the r.m.s. difference is 8.9 cm, and the CMDT and the RIO03 MDT, for which the r.m.s. difference is 20.1 cm. (Note that scale for figure 4*a* is half that of figure 4*b*.) Over the majority of the domain, the CMDT and the GGM01C-CLSMSS01 MDT agree to within 10 cm, remarkably given the independence of the methods used to produce them. The region of greatest disagreement, between 70N and 80N off the East Greenland coast, is also a region where both the CMDT and the GGM01C-CLSMSS01 MDT have large errors estimates (figure 4*c*); although not large enough to make the two fields consistent (figure 4*e*). The open ocean regions of disagreement in the central basin, where the CMDT is between 10 and 20 cm greater than the GGM01C-CLSMSS01 MDT, coincide with the location of the Mid-Atlantic Ridge, and therefore most likely corresponds to short-wavelength features of the geoid that are present in CLSMSS01 but that GGM01C has failed to capture. Because these omission errors are not included in the error estimate for the GGM01C geoid, the normalized residual (figure 4*e*) ranges from 3 to 7 standard deviations along the Mid-Atlantic Ridge. Nevertheless, over a substantial portion of the domain, the normalized residual is less than one standard deviation.

Given the close agreement between the CMDT and the GGM01C-CLSMSS01 MDT, the large discrepancies between the CMDT and the RIO03 MDT must clearly be due to an error in the RIO03 MDT. And since this error is not included in the given error estimate for RIO03 (figure 4*d*), the two fields are highly inconsistent by the measure defined above (figure 4*f*). Further analysis suggests that although the CMDT and RIO03 MDTs differ greatly in both the North and South of the domain, it is in fact disagreement at high latitudes, with the RIO03 MDT greatly overestimating the strength of the sub-polar gyre north of Iceland, which ‘contaminates’ the whole domain when the offset over the domain is removed from the MDT. This error is probably due to an error in the first guess MDT, based on Levitus climatology and the EIGEN-2 geoid, used in the inverse technique to produce the RIO03 MDT. In fact, initial analysis of a more recent RIO05 MDT (Rio *et al*. 2006), which uses the EIGEN–GRACE02S geoid (and which only became available after the analysis reported here was complete), has a much less pronounced sub-polar gyre north of Iceland than does RIO03, and is therefore in better agreement with the CMDT at higher latitudes.

## 5. Discussion

One of the primary aims of the EU GOCINA project is to demonstrate how independent measurements of the geoid, the MSS and the MDT, can be rigorously combined to yield an optimum estimate of each quantity. Essential to this process is that each quantity has an associated error field. While, due to the nature of the measurement techniques, errors are readily available for a MSS and can be obtained for a geoid, this is not the case for an MDT produced by an OGCM. In this paper we have described a method for deriving an MDT based on assimilation of *in situ* data into OGCMs, which has the advantage that an error estimate for this MDT is readily obtained.

A limiting factor in this study has been the small number of suitable MDTs and so our statistical analysis must be treated with caution. We cannot fully justify the assumption of normality on which the method rests, and it may be the case that the set of MDTs is biased. Nevertheless, we feel our simple compositing approach to obtaining an MDT and error field has merit since it does not incur the computational costs of an approach based on formal data assimilation theory, which, in any case, does not account for model bias and would be impossible to implement for high-resolution ocean models. Furthermore, this ensemble method will become more reliable as the number of MDTs derived from data assimilation experiments grows.

The lack of suitable MDTs also restricted the scales that we could represent in the composite MDT, since we required that each of the ensemble members contributed equally to the composite and that the error estimate was based on variability that each of the models could represent. With the availability of more higher resolution models we could partition each MDT into wavelength bands, and then form composites, and error fields, for each band. These could then be further combined into a single MDT and error field.

Although the lower resolution models did underestimate the strength of the sub-polar gyre relative to the higher resolution models, the generally small error field on the composite MDT shows that the agreement between OGCMs with data assimilation is generally good. Our analysis revealed particularly close agreement between the composite MDT, the MDT based on the GRACE geoid, and the Niiler MDT based on surface drifters. This level of consensus among MDTs produced by very different methods gives us confidence that we are close to the ‘truth’, at least for scales of one degree and greater. Conversely, our analysis has shown that the RIO03 MDT is likely to contain long-wavelength errors and the formal error estimate for this field is probably overly optimistic by tens of centimetres due to unrepresented errors in the EIGEN-2 geoid.

Future improvements in geoid determination are expected from the GOCE satellite mission planned for launch in 2007. One of the key objectives of GOCE is to better determine the mean ocean circulation and mean transports of mass and heat. Reducing uncertainties in these transports is particularly important in the North Atlantic where critical controls on the thermohaline circulation exist. It is hoped that our analysis of the MDT in this region, together with further investigations, underway as part of the GOCINA project, into the relationship between dynamic topography and ocean transports of heat and mass for this region, will form the basis for validation and utilization of the future GOCE product.

## Acknowledgments

This work was supported by grants from the NERC (NER/A/S/2000/01001) and from the EU GOCINA project. We would like to acknowledge valuable discussions with other members of the GOCINA project and particularly with Dr Jamie Kettle at an early stage of this work. The altimeter products were produced by Salto/Duacs as part of the Environment and Climate EU Enact project (EVK2-CT2001-00117) and distributed by Aviso, with support from CNES.

## Footnotes

One contribution of 20 to a Theme Issue ‘Sea level science’.

- © 2006 The Royal Society