Assessment of optimally filtered recent geodetic mean dynamic topographies


  • F. Siegismund

    Corresponding author
    1. Institut für Meereskunde, Centrum für Erdsystemforschung und Nachhaltigkeit, Universität Hamburg, Hamburg, Germany
    • Corresponding author: F. Siegismund, Institut für Meereskunde, Centrum für Erdsystemforschung und Nachhaltigkeit, Universität Hamburg, Hamburg, Germany. (

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[1] Recent geoids from the Gravity Recovery and Climate Experiment (GRACE) and the Gravity field and steady state Ocean Circulation Explorer satellite mission (GOCE) contain useful short-scale information for the construction of a geodetic ocean mean dynamic topography (MDT). The geodetic MDT is obtained from subtracting the geoid from a mean sea surface (MSS) as measured by satellite altimetry. A gainful use of the MDT and an adequate assessment needs an optimal filtering. This is accomplished here by defining a cutoff length scale dmax for the geoid and applying a Gaussian filter with half-width radius r on the MDT. A series of MDTs (GRACE, GOCE, and combined satellite-only (GOCO) solutions) is tested, using different sets of filter parameters dmax and r. Optimal global and regional dependent filter parameters are estimated. To find optimal parameters and to assess the resulting MDTs, the geostrophic surface currents induced by the filtered geodetic MDT are compared to corrected near-surface currents obtained from the Global Drifter Program (GDP). The global optimal cutoff degree and order (d/o) dmax (half-width radius r of the spatial Gaussian filter) is 160 (1.1°) for GRACE; 180 (1.1–1.2°) for 1st releases of GOCE (time- and space-wise methods) and GOCO models; and 210 (1.0 degree) for 2nd and 3rd releases of GOCE and GOCO models. The cutoff d/o is generally larger (smaller) and the filter length smaller (larger) for regions with strong, small-scale (slow, broad scale) currents. The smallest deviations from the drifter data are obtained with the GOCO03s geoid model, although deviations of other models are only slightly higher.

1 Introduction

[2] The mean dynamic topography (MDT) is defined as the time mean deviation of the height of the geometrical ocean surface from a reference equipotential surface of gravity, the geoid. Its gradients determine, applying the geostrophic approximation, the large-scale ocean surface circulation on a time scale of a few days and longer and, in combination with the density field, also the deep circulation. Applying satellite data, the calculation of the MDT ζ is performed by

display math(1)

with h the mean sea surface (MSS) height of the ocean surface, as measured from satellite altimetry, and N the geoid height. Both surfaces have to be defined on the same reference ellipsoid and tide system.

[3] However, careful calculation of the MDT is required when using equation (1), since ζ is a small difference of two nearly identical surfaces, with values 2 orders of magnitude smaller than those of h and N. Two issues need particular attention: First, h and N are provided in different resolution and different representation. h is offered in high resolution (usually 1/30 degree) as a gridded map, while N is provided as coefficients of a spherical harmonic series in a much lower resolution of up to maximum degree and order (d/o) 250 (80 km) for recent satellite-only geoid models. The difference of h and N will thus contain the small-scale geoid signal omitted in N. To avoid this, a common resolution needs to be defined, and both models have to be adapted to this resolution. Since noise in the geoid is growing with resolution, while signal strength is diminishing following Kaula's rule [1966], the optimum cutoff d/o is not necessarily the maximum d/o the geoid model is provided with. Second, after application of equation (1), the resulting MDT is still noisy due to the commission error of the scales resolved in both the geoid and the MSS, and thus a spatial filtering of the MDT is necessary. Together, the cutoff d/o of the geoid model (and the MSS) and the parameters of the spatial filter are called the filter parameters in the following. Haines et al. [2011] provide a comprehensive discussion over recent techniques of MDT determination.

[4] Through the advent of the Gravity field and steady state Ocean Circulation Explorer satellite mission (GOCE) (launched in March 2009), considerable improvements in the determination of the short scales in the geoid height field are expected, with the target of 1–2 cm accuracy at a length scale of 100 km [Johannessen et al., 2003]. The opening question for this paper was thus to assess the quality of geodetic MDTs that are based on recent geoid models. Only few publications are currently available on this problem [Bingham et al., 2011; Knudsen et al., 2011]. The assessment of MDTs, however, is closely related to the choice of filter parameters discussed above. Optimal parameters have to be found before a meaningful assessment can be performed. Thus, this paper is dedicated to both the determination of optimal filter parameters and the assessment of the optimized geodetic MDTs.

[5] The optimal filter parameters will depend on the region of interest. Along steep slopes of topography and especially for the western boundary currents such as the Gulf Stream or the Kuroshio, high resolution is needed to reproduce the small-scale currents, and so reducing the scales of the omission errors is necessary despite the corresponding increase in commission errors at the smaller scales. In the central ocean basins, where the circulation is weak and large scale, the commission error is the main source of error, and thus a large-scale filter might be the best choice. For some global applications, a globally consistent filtering is recommended, while for regional studies, filtering can be optimized for the region of interest. Thus, in this paper, optimal filter parameters for global MDT solutions are provided, but also regional dependence is analyzed.

[6] A number of geodetic MDTs, based on geoids computed from Gravity Recovery and Climate Experiment (GRACE) and GOCE gravimetric data, are investigated. Two of the models (GOCO02s, Goiginger et al. [2011], and GOCO3, Mayer-Gürr et al. [2012]) include in addition input data from the Challenging Minisatellite Payload (CHAMP) mission and Satellite Laser Ranging (SLR). A complete list of MDT models is provided in Table 1. The filter parameters consist of the choice for the maximum d/o of the geoid model and the length scale of the spatial filter applied. The spatial filter is a simple Gaussian kernel. The evaluation is performed by comparison to velocity measurements available from near-surface (15 m depth) drifters provided by the Global Drifter Program (GDP, Lumpkin and Pazos [2007]).

Table 1. Optimum Cutoff Degrees and Filter Lengths for Recent Geodetic MDT Modelsa
MDTGeoid Modeldmaxr (deg)RMS (mm s−1)
  1. aThe minimized global mean RMS differences of geostrophic surface currents to the corrected GDP data set is also displayed.
GRACEITG-GRACE2010s, Mayer-Gürr et al. [2010]1601.17.4
TIM-1GOCE, 1st release, time-wise method, Pail et al. [2010a]1801.27.6
TIM-2GOCE, 2nd release, time-wise method, Pail et al. [2011]2101.07.3
TIM-3GOCE, 3rd release, time-wise method, Pail et al. [2011]2101.07.2
DIR-1GOCE, 1st release, direct method, Bruinsma et al. [2010]2401.27.5
DIR-2GOCE, 2nd release, direct method, Bruinsma et al. [2010]2101.07.3
DIR-3GOCE, 3rd release, direct method, Bruinsma et al. [2010]2101.17.3
SPW-1GOCE, 1st release, space-wise method, Migliaccio et al. [2010]1801.27.7
SPW-2GOCE, 2nd release, space-wise method, Migliaccio et al. [2011]1801.17.4
GOCO1combined model (GOCE, GRACE), Pail et al. [2010b]1801.17.3
GOCO2Combined model (GOCE, GRACE, CHAMP, SLR),2101.07.1

Goiginger et al. [2011]

GOCO3Combined model (GOCE, GRACE, CHAMP, SLR),2101.07.1

Mayer-Gürr et al. [2012]


[7] The validation part of this study carries forward the study by Knudsen et al. [2011]. They used very early results from GOCE to produce an MDT and compared this and a GRACE MDT with the MDT produced by Maximenko et al. [2009]. Their method for computing the MDT was more straightforward than what is done here by using the MSS at full resolution and the geoids in the resolution they are provided with, although they state (see also their Figure 4) that the inconsistency in resolution of MSS and geoid causes an omission error from true small-scale signals contained in the MSS but missing in the geoid model. They show that this inconsistency is a major part of the full error in their filtered GRACE MDT but that this omission error diminishes with resolution of the geoid.

[8] As already stated, in the present study, consistency of MSS and geoid is provided by reducing the resolution of the MSS, following the spectral method described by Bingham et al. [2008]. This approach is in particular necessary for the regional mapping of optimal filter parameters described below, since there the geoid is not used in full resolution for the major part of the ocean and also not for the global solutions.

[9] In contrast to the study of Knudsen et al. [2011], we compare the geostrophic surface currents obtained from the geodetic MDTs to GDP drifter data. These data are corrected to estimate the geostrophic surface currents. It has to be mentioned here that spurious trends in the observed drifter currents were reported [e.g., Rio et al., 2011; Grodsky et al., 2011] presumably caused by a bias in the observations themselves. Other unbiased observational errors and additional errors in the corrections applied in this study might in total result in a considerable uncertainty in the estimated geostrophic currents. However, comparison to the drifter data set is preferred here because of its expected independence to the geodetic MDTs due to the completely different observation strategy. This allows for a determination of optimal filter parameters and a relative assessment of geodetic MDTs among themselves. It has, however, to be noted, that the RMS differences reported might considerably overestimate the true errors of the geostrophic surface velocities obtained from the geodetic MDTs.

[10] The paper is structured as follows: In section 2, the computation of the MDT is explained. The correction of the drifter data is described, which is necessary to estimate the temporal mean geostrophic surface currents. In section 3, the commission error of (unfiltered) recent geoid models is estimated. In section 4, global MDT solutions are determined and assessed, while in section 5, the regional dependence of optimal filter parameters is investigated, including a subsection on selected regions of specific interest. Concluding remarks are provided in section 6.

2 Methods

2.1 Geodetic Mean Dynamic Topography

[11] The computation of the (unfiltered) MDT follows equation (1) and is performed on a 15′ × 15′ grid. Two sources of errors have to be considered: The omission error in the low-resolution geoid model that causes a short-scale geoid signal from the MSS to remain in the MDT after applying equation (1), if not treated properly, and the commission error of both the geoid and the MSS.

[12] The inconsistency in resolution of the geoid and the MSS is treated here by reducing the resolution of the MSS to the resolution of the geoid. This is done by transformation of the MSS to spherical harmonic coefficients, cutting off all coefficients beyond a selected cutoff d/o dmax of the geoid model, and transforming back to physical space. dmax might be smaller than the maximum d/o the geoid model is provided with. The MDT omission error is determined by dmax and can only be reduced by adding small-scale information from external sources, as with the remove-restore approach suggested by Haines et al. [2011].

[13] The MDT commission error resulting from application of equation (1) is erased for scales shorter than those defined by dmax. To reduce the commission error on the longer scales, a spatial filter is applied. The filter uses a Gaussian kernel with a half-width radius r (and is truncated at 3r). Both parameters, dmax and r, depend on the MDT model and the region considered.

[14] The geoid models included in this study are listed in Table 1. The MSS applied, MSS-CNES-CLS11 [Schaeffer et al., 2010], was produced by CLS Space Oceanography Division and distributed by Aviso, with support from Cnes ( Before application in equation (1), it has been re-referenced to the period of interest (1993–2009). For this purpose, the updated merged sea level anomaly (SLA) product as provided by Aviso has been applied.

[15] The geostrophic surface currents are calculated from the filtered MDT math formula as

display math(2)

with g the acceleration of gravity, f the Coriolis parameter, k the vertical unit vector, and ∇ the horizontal gradient operator.

2.2 Mean Geostrophic Currents From Drifters

[16] The drifter data applied in our study are the 6-hourly data set as provided by the Global Drifter Program (GDP, Lumpkin and Pazos [2007]). All available data until September 2010, made up of more than 11 million entries, are used. To estimate the time average surface geostrophic circulation, as can be drawn from the MDT maps, a number of corrections are necessary. In a first step, zonal and meridional wind speed from the European Centre for Medium-Range Weather Forecasts interim re-analysis project [Dee et al., 2011] is attached to the drifter data set by linear interpolation to the positions of the drifters. Slip of the surface buoys of the drifters is estimated to 0.7 cm s−1 per 10 m s−1 wind speed [Niiler and Paduan, 1995] and subtracted from the drifter velocities. The extended data set is filtered for inertial currents by applying a running mean over two inertial periods for all components: the drifter velocities and positions, as well as the wind speeds.

[17] To yield the time mean currents, firstly the time variable part of the geostrophic currents is estimated and subtracted. For this purpose, the updated merged SLA product as provided by Aviso is applied. The SLA data are re-referenced to the period of interest (1993–2009), and geostrophic velocity anomalies are calculated from central SLA differences. Thus, zonal velocities are placed centered between two adjacent points in meridional direction, while meridional velocities are centered between two adjacent grid points in zonal direction. Both components of the geostrophic velocity anomalies are linearly interpolated to the position and time of the drifter observations.

[18] The estimation of the wind-driven (Ekman) currents follows the work of Rio and Hernandez [2003] and Ralph and Niiler [1999], applying a linear relationship between wind speed |Uw| and Ekman current, e.g.,

display math(3)

and fitting the parameters α and c by least squares fitting of Ue to Ekman current estimates from the drifter velocities, which are received after subtraction of the geostrophic currents. To determine the geostrophic currents, the time mean currents have to be added to the already calculated anomalies. For that purpose, the GOCO1-MDT (but with DTU10 as MSS, see Andersen [2010]) is applied, and the calculation of geostrophic currents and interpolation are performed as described for the SLA data.

[19] The parameter fit is performed in 5° × 5° boxes. The parameters are finally smoothed by averaging over a block of 3 × 3 boxes and applying the result to the center box.

[20] Using equation (3)) and inserting the fitted parameters and the wind speeds, the Ekman currents are calculated and subtracted from the drifter velocities. To allow for a convenient use of the data set in the assessment of geostrophic currents computed from the geodetic MDTs, the drifter velocities are binned to 15′ × 15′ boxes and filtered applying a Gaussian kernel with a 0.4° half-width radius (and truncated at 1.2°). The smoothing reduces small-scale noise of the drifter data but does not limit the applicability, since the filter length is considerably shorter than the maximum resolution of the geoids involved (80 km). Figure 1 displays the time mean geostrophic currents estimated from the drifter data.

Figure 1.

Absolute velocity [m s1] of near-surface geostrophic currents for the period 1993–2009 estimated from the drifter data. The red encircled region (55°30′–41°15′W, 26°30′–30°15′N) is applied in section 3, while the black encircled regions (Gulf Stream: 85°–60°W, 20°–40°N; Kuroshio: 120°–155°E, 15°–45°N; Agulhas Current: 15°–55°E, 5°–45°S; Tropical North Pacific: 180°–100°W, 5°–15°N) and the section crossing the Gulf Stream (72°W35°N–74°W40°N) are used in section 5.2.

3 Geoid Commission Error

[21] The commission error of a geoid model and its dependence on resolution determines the ability of the model to reveal an MDT pattern with a given spectral characteristic, provided the sufficient quality of the MSS model. However, estimation of the commission error by comparison with observations in the physical domain, as is accomplished in this study, is complicated by two sources of errors. First, the observations contain errors. These are the errors in both the original drifter data and the applied corrections. Second, since the comparison is done in physical space, the “true” spectral density is unknown, and the unresolved scales in the MDT cause an omission error of unknown size.

[22] To minimize both error sources, a small region in the tropical North Atlantic is selected (26°30′N–30°15′N, 55°30′W–41°15′W, see red box in Figure 1), where the geostrophic surface currents are expected to be rather low. The drifters reveal an area average speed of 5 mm s−1 for both vector components and small-scale velocities of approximately, and smaller than, 3 cm s−1. The small “true” MDT signal will be neglected in the following, and the commission error is thus estimated by the signal strength of the unfiltered MDT model. According to the formal error estimates provided with the geoid models, the commission error has predominantly a latitudinal dependence, with higher errors near the equator and lower errors in polar regions (not shown), following the ground track density of the satellite, but no distinct short-scale structure. Thus, the commission error for the selected region might give a reasonable estimate of the global mean, although the error might, due to the low latitude considered, be somewhat overestimated.

[23] The standard deviation of the unfiltered GRACE and the MDT models based on available GOCE release-3 geoid models (TIM-3 and DIR-3), dependent on the selected maximum d/o dmax, is displayed in Figure 2. Since the region is rather small (400 km × 1600 km), we concentrate on the short scales (dmax ≥ 100) that are resolved. Up to dmax = 150, the standard deviation of all three models match almost perfectly with values around 5 cm, probably because of a common error source in the MSS model and from a true, although small, MDT signal. Starting from dmax = 160, the standard deviation of the GRACE model diverges from the other two, with a strong increase of 15 cm for its maximum available resolution at dmax = 180. The GOCE models show in comparison only an increase of less than 1 cm for dmax = 180 and a 6 and 8 cm increase for their maximum available resolution at dmax =240 and dmax =250 for DIR-3 and TIM-3, respectively.

Figure 2.

Statistics of MDT and geostrophic currents for the tropical North Atlantic region (see Figure 1). (left) Standard deviation of the unfiltered MDT (m), dependent on the maximum d/o, for the models indicated in the legend. The square root of the a priori error degree variances of the corresponding geoid models, accumulated over degree, is given as dotted lines. (right) RMS of absolute geostrophic velocities, dependent on the maximum d/o of the MDT model.

[24] These commission error estimates for the geodetic MDTs are compared with the formal error degree variances of the geoid models, accumulated over degree (dotted lines in Figure 2). Keeping in mind that errors in the MSS and a true, although small, slope in the MDT might add to the standard deviation of the geodetic MDT, the formal error estimates seem reasonable, although they might be slightly overestimated for both the GRACE and the TIM-3 model.

[25] The error in the geostrophic currents, as caused by the commission error just discussed, is displayed in the right panel of Figure 2. The characteristics are similar to the standard deviation of the MDTs for all models. The strong signals of 80 cm s−1 for GRACE at dmax = 180, 50 cm s−1 for DIR-3 at dmax = 240, and 60 cm s−1 for TIM-3 at dmax = 250 show the apparent need of filtering the geodetic MDT models, as is discussed in the following sections. The optimum filter parameters will be a trade-off of reduced commission error through completely removing (reducing) small-scale errors by defining the cutoff d/o dmax (filter length r) and the omission error caused by the deleted (reduced) small-scale information.

4 Global Solution

[26] Before analyzing the regional dependence of optimal filter parameters, a global optimal filter length and cutoff d/o for each of the MDT models listed in Table 1 is determined. This global solution is later used as a reference for the regional dependent solutions. To find the optimum solution, the pair of filter parameters dmax, r is searched, which, applied to the geodetic MDT, minimizes the area-weighted RMS difference of geostrophic velocities, calculated from the filtered MDT, to the drifter velocities. The minimum RMS is found by simply stepping through different cutoff d/o dmax, from 100 to the maximum d/o of the geoid model, with a step size of 10, and through different filter lengths r, from 0.5°–2.0° with a step size of 0.1°. The region considered for the optimization is restricted to 60°N–60°S, since the polar regions are only sparsely covered by drifter data.

[27] The RMS depending on the spatial filter length for dmax =160, 210, and 240 is displayed in Figure 3. The minimum RMS is generally found around 1.0 degree for all models and all cutoff d/o but declines for recent models and with increasing dmax. The improvements in recent models compared to the GOCE release-1 model and GRACE are clearly visible in lower RMS values for GOCO3 and the GOCE release-3 models. The GOCE release-1 models SPW-1 and TIM-1 behave almost identically in terms of RMS differences to the drifter data for both dmax =160 and dmax =210 (both models are not available for dmax =240), while the DIR-1 model behaves differently for dmax =210 with lower RMS differences. The probable reason for this behavior is that for DIR-1, the EIGEN-51C [Bruinsma et al., 2010] is applied as background model, while in TIM-1 only, GOCE data are used, and for SPW-1, additional gravity field models are used as reference only but do not enter directly into the solution. The lowest RMS is 7.1 cm s−1 when applying a 1.0° filter for the GOCO3 model truncated at dmax =210. However, the TIM-3 solution is only slightly worse, which is not surprising, since the short-scale information in GOCO3 is based on that model. GOCO3 and TIM-3 are very close on the scales considered here, while the release-1 models differ considerably for dmax =210.

Figure 3.

Global RMS of absolute vectorial differences between geostrophic currents of selected geodetic MDT products and corrected drifter velocities as a function of the filter length r for different truncations of the geoid model. (left) dmax =160, (middle) dmax = 210, and (right) dmax = 240.

[28] The optimum global filter parameters for the different recent geoid models are displayed in Table 1. The optimum cutoff d/o dmax is 160 for GRACE; 180 for TIM-1, SPW-1, SPW-2, and GOCO1; and 210 for all release-2 and -3 GOCE and GOCO models, despite SPW-2. DIR-1 behaves differently with dmax =240, probably again due to its dependence on the EIGEN-51C background model. Overall, there is a clear indication that GOCE provides information on a higher spatial resolution than GRACE and that the larger amount of gravimetric data in the GOCE second release models allows to extract information at even shorter scales. However, for all models investigated, despite DIR-1, the optimum cutoff d/o is lower than the maximum d/o the models are provided with. Corresponding to the larger dmax, the filter lengths are shorter for the recent models and shortest for TIM-2, TIM-3, DIR-2, GOCO2, and GOCO3 with 1.0°. The optimal filter length for TIM-1 (1.2°) is, however, longer than for GRACE and GOCO1 (1.1°). The DIR-1 model is again outstanding with a filter length of 1.2°. This result is close to the optimal filter length of 140 km found by Knudsen et al. [2011].

[29] The difference in RMS between the different MDT models is rather low, with the smallest value of 7.1 cm s−1 for GOCO3 and the highest value of 7.6 cm s−1 for the TIM-1 model. It has to be noted, however, that the RMS contains also errors of the drifter velocities. Assuming that these errors are uncorrelated with the errors of the geostrophic currents, the difference of the mean squares 7.62 − 7.12 cm2 s−2 = 7.4 cm2 s−2 provides the difference in error variance for TIM-1 compared to GOCO3.

5 Regional Dependent Filter Parameters

5.1 Global Mapping

[30] To analyze the regional dependence of optimal filter parameters, 20° × 20° boxes are defined, and for each box the optimal filter parameters are allocated to the center of that box. This procedure is performed on a 1° × 1° grid for the GRACE and GOCO3 MDT models.

[31] The patterns for both the cutoff d/o dmax and the filter length r are similar for the two models but with generally smaller cutoff d/o and slightly larger filter lengths for GRACE (see Figure 4). dmax is generally higher in the vicinity of the western boundary currents than for the central ocean basins and near the east coasts, with values of 200 and above for GOCO3 and close to the maximum d/o of 180 for GRACE. This result is expected because of the strong signal strength on short scales in these regions. Consequently, all the regions along the western ocean margins have filter lengths at or below 1.0° for both MDT models. The shortest filter lengths are observed in the Gulf Stream region, declining to 0.6° near the US coast. However, to reduce the strong commission error in the short scales of the geoid, the cutoff d/o is here only around 180 for GOCO3 and 150 for GRACE.

Figure 4.

Optimal regional (top) cutoff d/o dmax, (center) filter length r [degree], and RMS of absolute vectorial differences to the corrected drifter velocities for (left) GRACE and (right) GOCO3.

[32] On the other hand, for the tropical and subtropical regions of the Southern Hemisphere, the cutoff d/o is 150 and less, accompanied by a filter length of 1.5° and longer. The probable reason for the strong filtering is that in those regions the currents are slow and large scale (the equatorial band 5°S–5°N is not considered); thus, the filter reduces noise in the MDT derived geostrophic currents, but the signal is still resolved. The situation is different for the Northern Hemisphere tropics, where strong equatorial currents exist, and hence dmax is larger and r is lower. Due to the reduced filtering, the errors remain higher here, as seen in the bottom panels of Figure 4.

[33] In the Southern Hemisphere, north of the Antarctic Circumpolar Current in the Indian and Pacific Oceans, and in the North Pacific reside patches with the maximum filter length of 2°. There, the MDT is flat on the large scale, and the short-scale geostrophic currents do not correlate with the drifter currents. A possible explanation for the large filter scale is that the true MDT is close to a curvature free surface in those regions. If the geodetic MDT is reproducing the true MDT plus some small-scale noise, a large-scale isotropic spatial filter will not disturb the true MDT but can effectively reduce the noise. Since the RMS differences between geostrophic currents from the MDTs to those from the drifter data are not outstanding in those regions, there is no indication for especially strong errors in either the drifter data or the geodetic MDTs.

[34] Along the western coastlines and in the Indian and Pacific sector of the Southern Ocean, discrepancies between current velocities from the geodetic MDTs and those from the drifter data exceed 10 cm s−1. For the western boundaries, the geodetic MDTs do not resolve the small-scale structure of the currents, although errors are somewhat smaller for the GOCO3 model. In the Southern Ocean, not much drifter data are available, and a considerable part of the discrepancies might originate from errors in the drifter velocities.

5.2 Specific Regions

[35] Table 2 provides optimal filter parameters and resulting RMS values for the four selected regions displayed in Figure 1. The RMS for the global solution is provided in parentheses, respectively. The regional optimized filter parameters are overall rather consistent for all the MDT models for a selected region, despite the lower cutoff d/o values for the GRACE model.

Table 2. Optimum Cutoff Degrees and Filter Lengths for Selected Regionsa
MDTdmax, r, RMS
Gulf StreamKuroshioAgulhasTropical North Pacific
  1. aThe regional mean RMS differences of geostrophic surface currents with respect to the corrected GDP data set are also displayed. RMS differences for the global optimized MDT models (see Table 1) are given in parentheses.
GRACE150, 0.7, 11.0 (12.8)150, 0.9, 8.0 ( 8.4)150, 1.0, 11.1 (11.3)150, 1.1, 5.2 ( 5.2)
TIM-1180, 0.8, 10.7 (13.3)220, 1.0, 8.0 ( 8.8)220, 1.0, 10.7 (11.8)130, 1.3, 6.1 ( 6.2)
TIM-2180, 0.7, 9.7 (11.7)210, 0.9, 7.4 ( 7.8)230, 0.9, 10.0 (10.4)160, 1.1, 5.5 ( 5.9)
TIM-3190, 0.7, 9.4 (11.7)210, 0.8, 7.2 ( 7.7)230, 0.9, 9.7 (10.3)160, 1.1, 5.2 ( 5.3)
DIR-1220, 0.7, 10.4 (13.3)220, 0.9, 7.7 ( 8.6)240, 0.9, 10.6 (11.7)130, 1.3, 5.8 ( 6.0)
DIR-2180, 0.7, 9.7 (11.6)240, 0.9, 7.5 ( 7.8)220, 0.9, 10.1 (10.5)160, 1.1, 5.4 ( 5.7)
DIR-3180, 0.7, 10.0 (12.5)230, 0.9, 7.4 ( 8.2)230, 0.9, 9.8 (11.0)160, 1.1, 5.2 ( 5.2)
SPW-1180, 0.8, 10.9 (13.3)180, 0.9, 8.2 ( 8.8)210, 0.8, 10.8 (11.9)130, 1.3, 6.2 ( 6.4)
SPW-2180, 0.7, 10.0 (12.7)180, 0.9, 7.7 ( 8.3)190, 0.9, 10.5 (11.2)160, 1.2, 5.7 ( 5.8)
GOCO1180, 0.8, 10.2 (12.6)180, 0.9, 7.7 ( 8.3)220, 1.0, 10.5 (11.2)160, 1.1, 5.0 ( 5.1)
GOCO2180, 0.7, 9.5 (11.6)250, 0.9, 7.3 ( 7.7)230, 0.9, 9.9 (10.4)160, 1.1, 5.0 ( 5.1)
GOCO3180, 0.7, 9.3 (11.6)250, 0.9, 7.2 ( 7.7)230, 0.9, 9.7 (10.3)180, 1.0, 5.0 ( 5.0)

[36] In the Gulf Stream region, the cutoff d/o is 180 for all GOCO and all but one GOCE model. The only exception is the DIR-1 MDT model (dmax =220), possibly caused by the influence of the high-resolution combined background model (EIGEN-51C), which might reduce the commission error on the short scales. Optimal filter lengths have values between 0.7° and 0.8° for all MDT models. The effect of regional filtering is clearly visible in the RMS difference to the drifter velocities for all MDT models, with a reduction of about 20% compared to the global solution, respectively. Also, the application of recent (release-3) geoid models improves the MDT models by about 10% compared to the release-1 models.

[37] For the Kuroshio and the Agulhas Current, both the cutoff degree and the spatial filter length are generally larger. For part of the MDTs, the full available resolution of the corresponding geoid model is optimal (DIR-2, GOCO2, GOCO3), with the highest resolution for the GOCO models (dmax =250). Improvements through regional filtering are significant for both regions but smaller than for the Gulf Stream (5–10%); the same holds for the improvements when applying recent geoid models (again 5–10%).

[38] The circulation pattern in the Tropical North Pacific is rather large scale and thus different from the other three regions. Thus, for this region the filter length is larger and the cutoff d/o is lower, and the GRACE model is closer to the drifter data than the pure GOCE models and only slightly worse compared to the GOCO solutions. The GOCO3 model has lowest RMS values for all regions, although for the Tropical North Pacific the difference to the next best model (GOCO2) is lower than 1 mm s−1 (in terms of RMS).

[39] To test the ability to reproduce the strength of intense swift currents, the GRACE and the GOCO3 MDTs were optimally filtered for the Gulf Stream region. This time, in addition to the global and regional (for the region displayed in Figure 1) optimized filtering, a “core restricted” filtering is performed, restricting the least squares optimization to the smaller region, where the current speed exceeds 10 cm s−1.

[40] With this condition, the filter length reduces from 0.7° in the regional solution for both MDT models to 0.5° (0.4°) for GOCO3 (GRACE), while the cutoff d/o slightly increases for GOCO3 (from 180 to 190) and remains unchanged for GRACE (150). With these filter parameters, both MDT models reproduce 80–90% of the peak velocities of the Florida Current, as observed by the mapped drifter velocities, all the way from the Florida Strait to Cape Hatteras (see Figure 5). Generally, the peak velocities increase more than 50% from the global to the core restricted solutions for both geoid models. Still in the regions of highest currents, for the Florida Strait and in the vicinity of Cape Hatteras, the velocities are strongly underestimated in all three solutions for both geoid models. Although peak velocities are generally slightly higher in the GOCO3 solutions, the major impact on the reproduction of the strength of the Gulf Stream comes from the choice of filter parameters, not from the choice of the geoid model. The better representation of the small-scale structure of the Gulf Stream when using a weaker filter is accompanied by a higher overall noise level. This noise level is, however, somewhat lower for the GOCO3 model.

Figure 5.

Surface geostrophic current from (a, b, c, e, f, g) the GOCO3 and (d, h) the GRACE MDT model in the Gulf Stream region. Absolute velocity [m s1] (in Figures 5a–5d) and absolute vector difference [m s1] (in Figures 5e–5h) to corrected GDP drifter currents for global (Figures 5a and 5e), regional (Figures 5b and 5f), and core restricted filter parameters (Figures 5c, 5d, 5g, and 5h). For the latter, the filter optimization is restricted to the region where the current speed exceeds 10 cm s1.

[41] The mapped drifter velocities used for the comparisons so far have reduced resolution due to the mapping and the spatial filtering, and velocities of small-scale currents are thus underestimated. However, the velocities drawn from the geodetic MDTs have still smoother patterns and lower peak velocities, and a relative quality assessment of the different MDT solutions against drifter data should remain unaffected.

[42] To give, however, a more accurate estimate of the absolute flow field, Figure 6 shows, as an example, the flow crossing a section over the Gulf Stream just downstream of Cape Hatteras (see Figure 1) with a different mapping of the observed velocities. Here, the observed speed is directly estimated from the corrected speed of drifters crossing the section. Nineteen adjacent drifter velocity observations are averaged. However, since more water crosses the section when velocity is large, the latter would be overestimated when computing the mean. Instead, a reciprocal velocity weighting is applied, as suggested by Maximenko [2004]. The observed average speed of the Gulf Stream core, defined here between km 150 and 200, is 1.02 m s−1. Only about 40% of this value is reproduced by the global solution, slightly less than 60% by the regional solutions, and about 70% by the core restricted solutions, with no significant impact from the choice of the geoid model. The 2-D transports as integrated currents over the section are almost identical for the two geoid models and the three filter strategies (Figure 6, bottom). They run close to the transports obtained from the drifters giving no indication for a bias or a long lengthscale error in either model.

Figure 6.

Geostrophic surface flow across the Gulf Stream section (see Figure 1) from corrected drifter data and the GOCO3 and GRACE MDT models applying global, regional, and core restricted filter parameters. (Top) Currents [m s1] and (bottom) surface transport [m2 s1], obtained from integrating the surface velocity starting at km 50.

6 Conclusions

[43] In this study, a number of geodetic MDTs have been analyzed applying recent satellite-only geoid models. Optimal global and regional filter parameters, as well as an assessment of the filtered MDTs, have been provided, based on comparisons to GDP drifter data. The main findings are as follows:

  1. [44] The global optimal cutoff d/o, defining the maximum d/o considered in the calculation of the MDT, is lowest for the GRACE geoid model (dmax =160) and highest for the recent third release GOCE and satellite-only combined (GOCO03s) solutions (dmax =210). Most of these last-mentioned models also have the shortest global optimal filter length (r = 1.0°) and hence retain more small-scale information than the other models.

  2. [45] Generally, for the regionally optimized solutions, filter lengths are shorter and maximum d/o higher near the western ocean margins with its strong swift current regimes giving high signal to noise ratio, while longer filter lengths and lower resolution is optimal for the central ocean basins, where mean currents are slow and large scale.

  3. [46] For the western intensified currents and the Agulhas Current, the GOCE and GOCO models perform slightly better than GRACE. For the Tropical North Pacific, with its larger scale signal and amplification of errors because of large 1/f near the equator (see equation (2))), GRACE has a lower RMS than the pure GOCE models. Only the GOCO models perform better. The GOCO03s geoid model has the best overall performance.

  4. [47] For all geoid models, except for DIR-1, the global optimal cutoff d/o is lower than the maximum d/o the models are provided with, meaning that using the full resolution causes a commission error increase that outweighs the omission error decrease. This is also true for the regionally optimized solutions despite part of the western intensified currents along the Eastern coasts. This does, however, not necessarily mean that no useful information is contained in the shortest scales of the geoid models. A more sophisticated filtering, taking into account the error covariances of the geoid models, might be necessary to separate the small-scale signal from the noise.


[48] The computations of the geodetic MDTs and the corresponding geodetic surface currents were performed with the GOCE User Toolbox (GUT), provided by the European Space Agency (ESA) and available at MSS-CNES-CLS11 was produced by CLS Space Oceanography Division and distributed by Aviso, with support from Cnes ( The SLA products were produced by Ssalto/Duacs and distributed by Aviso, with support from Cnes ( Support for the research was provided by the BMBF funded Geotechnology program REAL-GOCE. The study is a contribution to The Cluster of Excellence “Integrated Climate System Analysis and Prediction” (CliSAP) of the University of Hamburg, funded by the German Science Foundation (DFG).