• AVISO, 1997: AVISO Handbook: Sea Level Anomaly Files. AVI-NT-011-312-CN. 21st ed. Rep., 24 pp.

  • Blanc, F., P. Y. Le Traon, and S. Houry, 1995: Reducing orbit error with an inverse method to estimate the oceanic variability from satellite altimetry. J. Atmos. Oceanic Technol.,12, 150–160.

    • Crossref
    • Export Citation
  • Brankart, J. M., and P. Brasseur, 1996: Optimal analysis of in situ data in the western Mediterranean using statistics and cross-validation. J. Atmos. Oceanic Technol.,13, 477–491.

    • Crossref
    • Export Citation
  • Bretherton, F., R. Davis, and C. Fandry, 1976: A technique for objective analysis and design of oceanographic experiments applied to MODE-73. Deep-Sea Res.,23, 559–582.

    • Crossref
    • Export Citation
  • Chao, Y., and L.-L. Fu, 1995: A comparison between the TOPEX/POSEIDON data and a global ocean general circulation model during 1992–1993. J. Geophys. Res.,100, 24965–24976.

    • Crossref
    • Export Citation
  • De Mey, P., and A. R. Robinson, 1987: Assimilation of altimetric fields in a limited area quasigeostrophic model. J. Phys. Oceanogr.,17, 2280–2293.

    • Crossref
    • Export Citation
  • Fu, L. L., and R. D. Smith, 1996: Global ocean circulation from satellite altimetry and high-resolution computer simulation. Bull. Amer. Meteor. Soc.,77, 2625–2636.

    • Crossref
    • Export Citation
  • ——, E. J. Christensen, C. A. Yamarone Jr., M. Lefebvre, Y. Menard, M. Dorrer, and P. Escudier, 1994: TOPEX/POSEIDON mission overview. J. Geophys. Res.,99, 24369–24381.

    • Crossref
    • Export Citation
  • Fukumori, I., 1995: Assimilation of TOPEX sea level measurements with a reduced-gravity, shallow water model of the tropical Pacific Ocean. J. Geophys. Res.,100, 25027–25039.

    • Crossref
    • Export Citation
  • Hernandez, F, P. Y. Le Traon, and R. Morrow, 1995: Mapping mesoscale variability of the Azores current using TOPEX/POSEIDON and ERS-1 altimetry, together with hydrographic and Lagrangian measurements. J. Geophys. Res.,100, 24995–25006.

    • Crossref
    • Export Citation
  • Le Traon, P. Y., and P. De Mey, 1994: The eddy field associated with the Azores front east of the Mid-Atlantic Ridge as observed by the Geosat altimeter. J. Geophys. Res.,99, 9907–9923.

    • Crossref
    • Export Citation
  • ——, C. Boissier, and P. Gaspar, 1991: Analysis of errors due to polynomial adjustment of altimeter profiles. J. Atmos. Oceanic Technol.,8, 385–396.

  • ——, P. Gaspar, F. Bouyssel, and H. Makhmara, 1995a: Using Topex/Poseidon data to enhance ERS-1 data. J. Atmos. Oceanic Technol.,12, 161–170.

    • Crossref
    • Export Citation
  • ——, ——, F. Ogor, and J. Dorandeu, 1995b: Satellites work in tandem to improve accuracy of data. Eos, Trans. Amer. Geophys. Union,76, 385–389.

    • Crossref
    • Export Citation
  • Mazzega, P., and S. Houry, 1989: An experiment to invert Seasat altimetry for the Mediterranean and Black Sea mean surfaces. Geophys. J.,96, 259–272.

    • Crossref
    • Export Citation
  • ——, M. Berge, A. Cazenave, and P. Schaeffer, 1998: Maps of the mean sea surface and corresponding gravity anomalies from ERS-1 geodetic mission. J. Geophys. Res., in press.

  • Menard, Y., 1983: Observation of eddy fields in the northwestern Atlantic and northwestern Pacific by SEASAT altimeter data. J. Geophys. Res.,88, 1853–1866.

    • Crossref
    • Export Citation
  • Mesias, J. M., and T. Strub, 1995: An inversion method to determine ocean surface currents using irregularly sampled satellite altimetry data. J. Atmos. Oceanic Technol.,12, 831–849.

    • Crossref
    • Export Citation
  • Oschlies, A., and J. Willebrand, 1996: Assimilation of Geosat altimeter data into an eddy-resolving primitive equation model of the North Atlantic Ocean. J. Geophys. Res.,101, 14175–14190.

    • Crossref
    • Export Citation
  • Tai, C. K., 1991: How to observe the gyre to global-scale variability in satellite altimetry: Signal attenuation by orbit error removal. J. Atmos. Oceanic Technol.,8, 271–288.

  • ——, and J. Kuhn, 1995: Orbit and tide error reduction for the first 2 years of TOPEX/POSEIDON data. J. Geophys. Res.,100, 25353–25364.

    • Crossref
    • Export Citation
  • Vasquez, J., V. Zlotnicki, and L. L. Fu, 1990: Sea level variabilities in the Gulf Stream between Cape Hatteras and 50°W: A Geosat study. J. Geophys. Res.,95, 17957–17964.

    • Crossref
    • Export Citation
  • Wunsch, C., and V. Zlotnicki, 1984: The accuracy of altimetric surfaces. Geophys. J. Royal. Astron. Soc.,78, 795–808.

    • Crossref
    • Export Citation
  • View in gallery

    Comparison of sea level mapping using conventional objective analysis method (uncorrelated noise) (upper figure), with objective analysis taking account of long-wavelength error correlated along the tracks (lower figure) when random biases with standard deviation of 5 cm are simulated along the T/P tracks. The relative variance of the long-wavelength error was set at 50%. Units are in centimeters. Contour interval is 2 cm.

  • View in gallery

    (a) Sea level anomaly maps in the Canary Basin for three successive maps: 22 November, 2 December, and 12 December 1992, derived by the conventional objective analysis (COA) method using T/P data. Units are in centimeters. Contour interval is 2 cm. (b) As in Fig. 2a but with objective analysis taking account of long-wavelength error (LWA). (c) Difference between COA and LWA (Fig. 2a minus Fig. 2b).

  • View in gallery

    (a) Sea level anomaly maps in the Canary Basin for three successive maps: 22 November, 2 December, and 12 December 1992, derived by COA using ERS-1 data. Units are in centimeters. Contour interval is 2 cm. (b) As in Fig. 3a but with LWA method. (c) Difference between COA and LWA maps (Fig. 3a minus Fig. 3b).

  • View in gallery

    (a) Sea level anomaly maps in Canary Basin for three successive maps: 22 November, 2 December, and 12 December 1992, derived by COA using combined T/P and ERS-1 data. Units are in centimeters. Contour interval is 2 cm. (b) as in Fig. 4a but with LWA method. (c) Difference between COA and LWA maps (Fig. 4a minus Fig 4b).

  • View in gallery

    (a) Difference between T/P and ERS-1 maps derived by COA. Units are in centimeters. Contour interval is 2 cm. (b) As in Fig. 5a but with LWA method.

  • View in gallery

    (a) Global sea level anomaly derived using COA method, late October 1994 (Julian day 16424), with T/P data. Units are in centimeters. (b) As in Fig. 5a but with LWA method. (c) Difference between COA and LWA maps (Fig. 5a minus Fig. 5b).

  • View in gallery

    (Continued)

All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 371 371 48
PDF Downloads 210 210 36

An Improved Mapping Method of Multisatellite Altimeter Data

View More View Less
  • 1 CLS—Space Oceanography Division, Toulouse, France
© Get Permissions
Full access

Abstract

Objective analysis of altimetric data (sea level anomaly) usually assumes that measurement errors are well represented by a white noise, though there are long-wavelength errors that are correlated over thousands of kilometers along the satellite tracks. These errors are typically 3 cm rms for TOPEX/Poseidon (T/P), which is not negligible in low-energy regions. Analyzing maps produced by conventional objective analysis thus reveals residual long-wavelength errors in the form of tracks on the maps. These errors induce sea level gradients perpendicular to the track and, therefore, high geostrophic velocities that can obscure ocean features. To overcome this problem, an improved objective analysis method that takes into account along-track correlated errors is developed. A specific data selection is used to allow an efficient correction of long-wavelength errors while estimating the oceanic signal. The influence of data selection is analyzed, and the method is first tested with simulated data. The method is then applied to real T/P and ERS-1 data in the Canary Basin (a region typical of low eddy energy regions), and the results are compared to those of a conventional objective analysis method. The correction for the along-track long-wavelength error has a very significant effect. For T/P and ERS-1 separately, the mapping difference between the two methods is about 2 cm rms (20% of the signal variance). The variance of the difference in zonal and meridional velocities is roughly 30% and 60%, respectively, of the velocity signal variance. The effect is larger when T/P and ERS-1 are combined. Correcting the long-wavelength error also considerably improves the consistency between the T/P and ERS-1 datasets. The variance of the difference (T/P–ERS-1) is reduced by a factor of 1.7 for the sea level, 1.6 for zonal velocities, and 2.3 for meridional velocities. The method is finally applied globally to T/P data. It is shown that it is tractable at the global scale and that it provides an improved mapping.

Corresponding author address: Dr. P. Y. Le Traon, CLS—Space Oceanography Division, 18, avenue Edouard Belin, 31055 Toulouse Cedex, France.

Email: Letraon@cls.cnes.fr

Abstract

Objective analysis of altimetric data (sea level anomaly) usually assumes that measurement errors are well represented by a white noise, though there are long-wavelength errors that are correlated over thousands of kilometers along the satellite tracks. These errors are typically 3 cm rms for TOPEX/Poseidon (T/P), which is not negligible in low-energy regions. Analyzing maps produced by conventional objective analysis thus reveals residual long-wavelength errors in the form of tracks on the maps. These errors induce sea level gradients perpendicular to the track and, therefore, high geostrophic velocities that can obscure ocean features. To overcome this problem, an improved objective analysis method that takes into account along-track correlated errors is developed. A specific data selection is used to allow an efficient correction of long-wavelength errors while estimating the oceanic signal. The influence of data selection is analyzed, and the method is first tested with simulated data. The method is then applied to real T/P and ERS-1 data in the Canary Basin (a region typical of low eddy energy regions), and the results are compared to those of a conventional objective analysis method. The correction for the along-track long-wavelength error has a very significant effect. For T/P and ERS-1 separately, the mapping difference between the two methods is about 2 cm rms (20% of the signal variance). The variance of the difference in zonal and meridional velocities is roughly 30% and 60%, respectively, of the velocity signal variance. The effect is larger when T/P and ERS-1 are combined. Correcting the long-wavelength error also considerably improves the consistency between the T/P and ERS-1 datasets. The variance of the difference (T/P–ERS-1) is reduced by a factor of 1.7 for the sea level, 1.6 for zonal velocities, and 2.3 for meridional velocities. The method is finally applied globally to T/P data. It is shown that it is tractable at the global scale and that it provides an improved mapping.

Corresponding author address: Dr. P. Y. Le Traon, CLS—Space Oceanography Division, 18, avenue Edouard Belin, 31055 Toulouse Cedex, France.

Email: Letraon@cls.cnes.fr

1. Introduction

Satellite altimeters provide sea surface height measurements with irregular space–time sampling. Because the geoid is not known with sufficient accuracy, these measurements are used to derive precise estimations of the sea level anomaly relative to a given mean. Gridded sea level anomaly data are generally used for signal analysis, comparison with in situ measurements and models, and assimilation into models (e.g., Chao and Fu 1995; Hernandez et al. 1995). This is particularly true in studies combining data from several satellites. Most of the methods commonly used to map altimeter data (e.g., De Mey and Robinson 1987) are based on the objective analysis (or optimal interpolation) method, first used in oceanography by Bretherton et al. (1976). Alternative approaches, such as successive corrections or spline smoothing functions (e.g., Vasquez et al. 1990; Brankart and Brasseur 1996), are fairly similar to objective analysis.

Objective analysis methods use an a priori statistical knowledge on covariance functions of the signal to be mapped and of the data noise. They usually assume white measurement noise, though the error spectrum on the altimeter measurements is far more complex. In particular, there are long-wavelength errors (e.g., orbit error, residual tidal correction, or inverse barometer errors) that are correlated over thousands of kilometers along the satellite tracks. Before the TOPEX/Poseidon (T/P) altimetric mission, these errors, in particular orbit error, were rather large and had to be removed using polynomial adjustment before analysis (e.g., Menard 1983). With T/P this is no longer the case, as such methods affect the ocean signal and remove, in particular, the large-scale ocean signal (e.g., Le Traon et al. 1991; Tai 1991). Still, some residual long-wavelength errors remain in T/P data, typically on the order of 3 cm rms (e.g., Fu et al. 1994). This is nonnegligible, especially in low-energy areas. Analyzing sea level anomaly maps produced by objective analysis, assuming white noise, thus reveals residual long-wavelength errors in the form of tracks on the maps. These errors induce sea level gradients perpendicular to the track and, therefore, high geostrophic velocities that can obscure ocean features (e.g., Mesias and Strub 1995). This becomes crucial when data from various altimeter satellites are combined, as the tracks are closer together and the gradients steeper.

The aim of this study was to develop an objective analysis method taking account of correlated along-track errors and thus enhance sea level anomaly mapping from one or more altimeter satellites. Objective analysis of altimeter data using correlated error was initially discussed by Wunsch and Zlotnicki (1984) and has led to a series of studies on the application of inverse methods to altimeter data (e.g., Mazzega and Houry 1989). The main objective was to correct for rather large orbit error while mapping the mean sea surface signal or the absolute sea surface topography (i.e., mainly the geoid). The sea level anomaly mapping is a different problem. The signal is much weaker and residual long-wavelength errors (even of a few centimeters) cannot be neglected. The mapping of a variable signal also requires much more efficient methods.

The paper is organized as follows. The objective analysis method is described and tested in section 2. It is applied at a regional scale to T/P and ERS-1 data in section 3 and at a global scale to T/P data in section 4. Conclusions and prospects appear in section 5.

2. Objective analysis

a. The method

The problem is to determine the value of a field θ (here θ is the sea level anomaly relative to a given mean) at a point in time and space, given various measurements of the field unevenly spread over time and space Φobsi, i = 1, . . . , n. The best least squares linear estimator θest(x) is given by (Bretherton et al. 1976)
i1520-0426-15-2-522-e1
with Φobsi = Φi + ɛi, where Φi is the true value and ɛi the measurement error.
Here, A is the covariance matrix for the observations themselves, and C is the covariance vector for the observations and the field to be estimated
i1520-0426-15-2-522-eq1
The associated error variance e2 is given by
i1520-0426-15-2-522-e2
The measurement error is assumed to be uncorrelated with the signal. Here, A and C can also be defined to within a constant factor [in Eq. (1)], and both are generally normalized by the signal variance (e2 is thus given in percentage of signal variance).

Objective analysis has been used in many altimetric applications to map sea level variations from along-track data (e.g., De Mey and Robinson 1987; Hernandez et al. 1995). Given the high number of altimetric measurements, the method is “suboptimal”: only useful data, that is, values close to the point to be estimated, are used. In practice, this comes down to selecting data in a space–time subdomain. More sophisticated sorting methods can also be used if needed (P. De Mey 1995, personal communication).

b. Long-wavelength errors

Various altimetry applications have taken account of correlated noise in objective analysis (or in inverse methods), in particular for estimating orbit error or mapping the geoid (e.g., Wunsch and Zlotnicki 1984; Mazzega and Houry 1989; Blanc et al. 1995; Mazzega et al. 1998). This is done simply by modifying the error covariance 〈ɛiɛj〉.

The objective is to take into account an error correlation, along-track only, for a given cycle. The method will use observations in areas typically 1000–2000 km in diameter, which are of smaller scales than those typical of long-wavelength errors. The long-wavelength error is therefore assumed constant along the tracks, that is, 〈ɛiɛj〉 can be expressed in the following form:

  1. 〈ɛiɛj〉 = δijb2 for points i, j not on the same track or in the same cycle and
  2. 〈ɛiɛj〉 = δijb2 + ELW for points i, j on the same track and in the same cycle,
where b2 is the variance of the white measurement noise and ELW is the variance of the long-wavelength error.

This statistical description of the long-wavelength error should reduce the relative between-track biases in a given area and, therefore, reduce the inconsistencies between adjacent or crossing tracks.

c. Selecting the data

Conventional objective analysis, that is, assuming uncorrelated noise, only needs data in a small subdomain around the point to be estimated. The size of the subdomain corresponds roughly to the correlation scales of the ocean signal, that is, on the order of 100 km and 10 days. If correlated noise is taken into account, larger subdomains are needed to properly distinguish between the ocean signal and this long-wavelength error. Ideally, one would need to select all the data in areas several thousand kilometers across. This is not possible because of the difficulty of inverting the covariance matrices (A). A specific method for selecting data has therefore been developed. It proceeds as follows.

  • Along-track sea level anomaly data are smoothed to reduce measurement noise and subsampled. Only useful observations are kept, and smoothing and subsampling (in effect) create “super observations.”
  • For each point to be estimated, data are initially selected in a large subdomain, with typical radii of 1000–2000 km (space scale of long-wavelength errors) and 10–20 days (timescale of oceanic signal).
  • Among these selected points, all points within a small subdomain with at least the size of the typical space scales of ocean signals are retained. Outside the subdomain, only one in three to five points is kept.

We tested this selection method using T/P data in a 20° × 20° area of the Canary Basin. The smoothing cutoff wavelength was 100 km and subsampling was one point in three (roughly one point every 21 km) above 30°N and 200 km and one point in five below 30°N. The large subdomain radii were 1000 km and 10 days where one point in three was selected; the radius of the small subdomain was 300 km. In practice, we selected 300 to 500 points for analysis.

The following space–time correlation function C(r, t) of the sea level anomaly field was used:
Cr, tarar2ar3art2T2
where r is distance, t is time, L = 3.34/a is the space correlation radius (first zero crossing of C), and T is the temporal correlation radius. Here, L and T are set at 150 km and 20 days, respectively, scales typical of mesoscale circulation in this part of the ocean (e.g., Le Traon and De Mey 1994).

We assumed the variance of the long-wavelength error to be 50% of the signal variance (ELW = 0.5). To test whether the selection mode theoretically corrects long-wavelength error satisfactorily, we compared a posteriori error maps [Eq. (2)] with and without long-wavelength error. If long-wavelength errors were fully corrected, there should be no difference between the maps. The mean difference is 1% of the signal variance, with along-track maxima slightly below 2%. This shows that a lot of the initial error (50%) is reduced.

It is instructive to compare this residual error with that which would be obtained by fitting a bias directly to each track. The adjustment error variance is then on the order of 1/N, where N is the number of independent measurements. This adjustment error was calculated using the Le Traon et al. (1991) method, assuming a 2000-km profile. The error variance is approximately 5%. This is independent of the a priori variance of the orbit error—it represents the fraction of the ocean signal removed during bias fitting (the fraction depends on the assumed spectrum of the ocean signal via the covariance function). Through objective analysis we can therefore reduce the error, roughly by a factor of 3, by using data from neighboring tracks and a priori statistical knowledge of the signal and of the orbit error. Long-wavelength ocean signals are also preserved while they are removed during bias fitting.

A posteriori error analysis provided a means of checking that the selection method was satisfactory. It showed that there is no need to use all the data in the large subdomain. Thus, selecting one point in four rather than one in three increases error by just 0.2% on average. However, discarding data outside the small subdomain increases error by almost 3% on average.

d. Tests with simulated data

To show the improvement that this new objective analysis method should provide, simulated data were generated along T/P tracks in the area. Each track was assigned a random bias, with zero mean and standard deviation of 5 cm, and the ocean signal was set to zero. We then compared the sea level maps derived using conventional objective analysis (COA) and by analysis taking account of the long-wavelength error (LWA). Figure 1 compares the results, with a relative variance in the long-wavelength error of 50% (ELW = 0.5). The reduction in long-wavelength error is very noticeable. The track structure is the same but the signal, which with COA was from −12 to roughly +8 cm, is −1 to +1 cm with LWA. This result agrees, of course, with the a posteriori error estimates given above. Tests were also performed with different a priori long-wavelength errors. The long-wavelength error is slightly less well corrected with ELW = 0.1 (maximum residual error below 3 cm) and slightly better corrected with ELW = 1 (maximum residual error below 1 cm). This is because if the a priori variance of long-wavelength error is weak, the method will only adjust a weak variance signal to preserve as much of the ocean signal as possible. If not, the method is “freer” to adjust the required signal. To derive an optimal estimate, it is thus important to accurately define the ratio of long-wavelength error variance to signal variance. The previous tests show, however, that results are not too sensitive to it since for the three tested ratio (0.1, 0.5, and 1) a good correction was achieved.

e. Non-zero-mean field

In the simulation above, the data were not centered, as the mean of the biases created was set to zero. In practice, with real data, a mean (or large space scale signal) is removed before the analysis and added back after. However, this can be a source of additional error as the mean is sensitive to long-wavelength errors.

Bretherton et al. (1976) analyzed the impact of error on the mean of the field to be estimated on the analysis error. The additional error 〈e2m〉 for the uncertainty on the mean is given by
i1520-0426-15-2-522-e3
Note that here the mean is not the arithmetic mean as it is weighted. Correlated points have less weight than isolated points. With altimeter data, the two means (arithmetic and weighted) are, however, very close as the altimeter data are evenly spread out.

We did tests varying ELW. The extra error term [Eq. (3)], which directly reflects the uncertainty on the mean, increases with ELW. For ELW of 0, 0.2, and 1, its mean is 0.5%, 1.2%, and 4.3%, respectively. These results show that the contribution of the error on the mean has a significant impact, especially when long-wavelength errors increase. This is because it is not possible to distinguish between the mean ocean signal and the mean long-wavelength error.

3. Regional application

The method was then applied to real T/P and ERS-1 data. For the tests we used the T/P and ERS-1 sea level anomaly (SLA) files distributed by AVISO for October 1992 through December 1993 (AVISO 1997). The AVISO ERS-1 data were first corrected using the T/P data as a reference (Le Traon et al. 1995a; Le Traon et al. 1995b). The correction accuracy was 2–3 cm rms, so that the long-wavelength residual errors for ERS-1 are therefore of the same order of magnitude as for T/P. Ocean variability in the Canary Basin area is between 5 and 8 cm rms, that is, ELW is roughly 0.2 (assuming long-wavelength error of 2–3 cm rms). This is typical of low eddy energy regions. The method for selecting data and the correlation function are the same as described in section 2 above.

a. TOPEX/Poseidon

We calculated sea level anomaly maps for a 3-month period (October 1992 through February 1993), that is, 10 maps, one every 10 days. The grid spacing is 0.5° in latitude and in longitude. To compare the COA and LWA methods, maps were calculated using both, taking ELW = 0.2 for LWA method. Over the 3-month period, COA and LWA each indicate a similar decrease in sea level, related to the contraction of surface waters due to heat exchange with the atmosphere. The other major feature is eddy activity around 35°N due to the Azores Current/Front (e.g., Le Traon and De Mey 1994). Satellite tracks appear quite clearly on the COA maps, while LWA corrects the effects quite well. This is well evidenced in Figs. 2a and 2b, which show three successive maps (22 November, 2 December, 12 December) using COA and LWA. Figure 2c shows the difference between the two methods. Satellite tracks are visible with the difference reaching 8 cm for some of them.

Table 1 is a statistical comparison of the COA and LWA maps over the 3-month period. Both SLA and geostrophic velocity calculated from SLA maps by finite differences (i.e., over 0.5°) are compared. The SLA and velocity variances are always higher with COA, and the difference in variance between the methods can be almost completely explained by the variance in the difference (COA − LWA). This shows that the difference is not correlated with the ocean signal and suggests it is almost only due to errors in COA.

The mean standard deviation of the difference in SLA, over the 10 maps, is 1.8 cm, that is, roughly 20% of the signal variance (as given by LWA map). The mean difference is negligible, that is, the method does not introduce bias. The variance of the difference in zonal and meridional velocities is roughly 30% and 60%, respectively, of the velocity signal variance. As expected, the effect on velocities is therefore considerably greater. In particular, while the COA maps seem to indicate marked anisotropy, the LWA maps give very similar variances for zonal and meridional velocities. The correction for the along-track long-wavelength error therefore has a very significant effect.

Note that the variances deduced from the maps are weaker than those deduced from along-track data because when no data is available, for example, between tracks, the estimation given by objective analysis tends toward the mean.

b. ERS-1

Similar figures were produced for ERS-1 using COA (Fig. 3a) and LWA with ELW = 0.2 (Fig. 3b). Figure 3c is the difference between them. The ERS-1 tracks are visible. The correction for the tracks can reach 8 cm. Also note the more localized correction areas. This can be explained by the closeness of the ERS-1 tracks relative to those of T/P: The tracks are 80 km apart at the equator with ERS-1 and 310 km with T/P. The statistics comparing COA and LWA maps over 3 months (Table 2) are similar to those obtained for T/P (Table 1). However, the tilt of the ERS-1 tracks implies less marked differences than T/P for zonal velocities and more marked for meridional velocities. Also, the COA and LWA variances are higher than for T/P, as ERS-1 samples mesoscale structures better.

c. TOPEX/Poseidon and ERS-1

The datasets are now combined over the same period of time. To reduce the total number of data selected, the subdomain was reduced to 750 km, with just one point in four selected in the large subdomain. This makes the number of points selected comparable to the number selected for a single satellite (300–500). Note, however, that the long-wavelength error reduction problem is better constrained with several satellites since there are more crossing tracks than with a single satellite.

The COA (Fig. 4a) brings out the structure of the T/P and ERS-1 tracks, although overlaying ERS-1 and T/P tracks often makes interpretation more difficult. As with the previous tests, LWA (ELW = 0.2) provides a good correction for these effects, and successive maps are more consistent (Fig. 4b). The difference maps (Fig. 4c) are noisier than with a single satellite and can easily be confused with actual circulation signals, for example, mesoscale eddies.

Table 3 shows statistics comparing the COA and LWA maps over the 3-month period. The mean standard deviation of the differences between the maps is 2.2 cm. The correction is therefore larger than when considering a single satellite (mean standard deviation of 1.8 cm). This is due because more tracks, that is, those of two satellites, are corrected in a given area. The correction represents roughly 30% of the signal variance (as given by LWA). The variances of the difference, between the methods, in zonal and meridional velocities represent roughly 25% and 80%, respectively, of the velocity signal variances. The effect on meridional velocities is therefore very significant: Since the combined set of satellite tracks is denser, a small error can be transformed into a large cross-track gradient, that is, essentially a meridional velocity.

d. Comparison between TOPEX/Poseidon and ERS-1

The results derived from T/P and ERS-1 were compared separately. An initial set of maps, covering the same 3-month period, was derived by taking the difference between the ERS-1 and T/P maps obtained with the COA. Figure 5a shows the results for the three maps discussed above. The maps show both T/P and the ERS-1 tracks. They reveal the sampling differences between the satellites, plus their errors. The mean standard deviation of the difference over the 3-month period is 4.1 cm. Figure 5b compares the results obtained with the LWA method (ELW = 0.2). The satellite tracks are no longer visible, and the mean standard deviation of the difference is only 3 cm. The differences are mainly small scale and are probably due mainly to ocean features observed differently by the two satellites.

Similarly, comparing the variances of the differences between the T/P and ERS-1 geostrophic velocity maps shows the improvement due to LWA. With COA, the difference is 36 cm2 s−2 for zonal velocities, and 83 cm2 s−2 for meridional velocities; the corresponding figures for LWA are 22 and 36 cm2 s−2.

The differences between T/P and ERS-1 are therefore smaller after correcting for long-wavelength errors. Most of the remaining difference can presumably be explained by sampling errors. Correcting the long-wavelength error thus considerably improves the consistency between the datasets from the two satellites. The variance of the difference is reduced by a factor of 1.7 for the sea level anomaly, 1.6 for zonal velocities, and 2.3 for meridional velocities.

4. Global application

We conducted a global application of the analysis using T/P data. The main problem in extending objective analysis to a global application is computing time. The LWA requires the inversion of a typically 500 × 500 matrix for each grid point. Also, for each grid point, data selection in the subdomain must be made using a global dataset comprising several million data points. To reduce the time needed to select the data, the initial residual file is therefore split into 60 files corresponding to particular geographical areas. A main grid (spacing 2°) is then defined where the analysis is performed as follows: data selection, inversion of matrix A, and calculation of estimation and error. As the data selected correspond to a fairly large area around the estimation point, they are also used for neighboring grid points on a subgrid (spacing 0.5°). As the same data as for the main grid point are used for all neighboring subgrid points, the matrix A is thus inverted only for the main grid points. This considerably reduces the computing time. Moving to higher resolution will also not be very penalizing.

For the global application we used a T/P sea level anomaly file computed from three years of data (cycle 10 to 121) and using the new JGM-3 orbits and the CSR3.0 tide model. The sea level anomaly files are smoothed (using a Lanczos low-pass filter), subsampled files. The Lanczos filter cutoff wavelength depended on the latitude, taking account of variations in the assumed typical scales of the ocean signal: 300 km equatorward of 10° north–south, 200 km between 10° and 30° north–south, 100 km between 30° and 50° north–south, and 70 km above 50° north–south. Smoothing reduces considerably instrumental noise. The data are then subsampled according to the degree of smoothing: one point in seven at 0° to 10° north–south (roughly every 50 km), one point in five at 10° to 30° north–south (roughly every 35 km), and one point in three poleward of 30° north–south (roughly every 21 km).

The space correlation scales (zero crossing of correlation function) were set as follows:

  • −14° < latitude < 14°:
    i1520-0426-15-2-522-eq3
  • poleward of 14°:
    i1520-0426-15-2-522-eq3a
    where “Lat” is the latitude in degrees.
These scales are intended to represent typical space scales of the sea level anomaly in equatorial and tropical regions, and the reduction in scales of mesoscale eddies with latitude (variation in internal Rossby radius). They are similar to the ones used by Oschlies and Willebrand (1996). The temporal correlation scale was set at 15 days.

The sizes of the small subdomains are equal to the space correlation scales plus 1.5° to take account of the subgrid calculation. The spatial radii of the large subdomains are three times the correlation radii. The selection mode is similar to that described in section 2. In the small subdomains we selected all the points—outside, but in the large subdomain, only one point in three.

The white noise was calculated assuming measurement noise of 3 cm rms and taking account of the reduction due to smoothing (which depends on latitude). To calculate the long-wavelength error, we assumed a uniform error of 3 cm rms. The variance of the white measurement noise and long-wavelength error relative to the ocean signal was then calculated from an a priori estimate of the sea level anomaly variance calculated from ERS-1 data previously corrected using T/P data (Le Traon et al. 1995b). Thus, the relative variance of long-wavelength error (i.e., ELW) can vary between 1% and 2% in high-energy areas, such as the Gulf Stream, and can reach 40% in tropical areas and other low-energy areas.

It takes about 2 h of CPU time to obtain a global map with 0.5° resolution on a DEC Alpha workstation. The long-wavelength calculation takes roughly 10 times as long as a similar calculation with conventional objective analysis.

Figures 6a (COA) and 6b (LWA) show the maps for late October 1994. Figure 6c shows the difference between them. The main ocean signals are identical in both cases. Note the steric effects, seasonal variations in tropical currents, and the signature of a Kelvin wave associated with the 1994 El Niño event (e.g., Fu and Smith 1996). Satellite tracks are visible on the COA figure, especially in the South Pacific and low-energy areas. LWA corrects these well. Figure 6c, the difference map, clearly shows that the corrections are along track and mainly long wavelength. The standard deviation of the difference is 2.8 cm, with the higher values along the tracks. Differences are not larger in high-energy areas, such as the Gulf Stream and Antarctic Circumpolar Current, which show that the method does not affect the ocean signal.

To test the sensitivity of the method to the a priori choice of long-wavelength error variance, two other LWA maps were obtained with an a priori error of 2 and 4 cm rms, respectively. The two maps (not shown) are very similar to the reference LWA map shown on Fig. 6b. Difference with the COA map is slightly reduced for the 2-cm case (2.3 cm rms) and slightly increased for the 4-cm case (3.1 cm rms). The differences between the two LWA maps and the reference LWA map are about 0.5 cm rms. This shows that the method is not too sensitive to the a priori choice of the long-wavelength error variance (see also discussion in section 2d).

5. Conclusions

This improved objective analysis method reduces residual long-wavelength errors in altimetric sea level anomaly maps. It does not significantly affect the ocean signal. Although it is more penalizing in computing time, it can run on powerful workstations, even at a global scale. It is particularly useful for combining data from several satellites, as long-wavelength residual errors can induce major errors in the estimation of the velocity field (especially in the meridional velocity). Tests also show that the method can make T/P and ERS-1 data significantly more consistent with each other.

Note that although this method allows a very effective correction of residual long-wavelength errors, it does not necessarily replace global crossover minimization methods (e.g., Le Traon et al. 1995a; Tai and Kuhn 1995). The two approaches are complementary. Our method only works locally and assumes that the mean over the large subdomain is perfectly known. It cannot thus correct for systematic errors. The latter can only be achieved with global methods. On the other hand, global crossover minimization methods may provide a good correction of orbit error, but they will not be able to correct for all residual long-wavelength errors and thus remove all the inconsistencies between neighboring tracks.

The study results also show that assimilation methods using along-track information should take account of correlated along-track errors. Kalman filter-type methods (e.g., Fukomori 1995) make this possible, although the computational burden is often prohibitive. A less penalizing solution would therefore be to directly assimilate maps obtained by the type of analysis described in this study. The method could also be adapted to estimate directly an along-track long-wavelength correction that could be applied to altimetric data before the assimilation.

Acknowledgments

The software for the global analysis method has been developed by P. Sicard. We wish to thank P. Gaspar, F. Hernandez, and G. Larnicol for useful discussions on this work. The Canary study was supported by SHOM under Contract CLS/SHOM 96.87.015.00.470.29.45 and by the MAST CANIGO project of the European Union (MAS3-CT-960060). The global analysis study is part of the AGORA project, supported by the Environment and Climate program of the European Union.

REFERENCES

  • AVISO, 1997: AVISO Handbook: Sea Level Anomaly Files. AVI-NT-011-312-CN. 21st ed. Rep., 24 pp.

  • Blanc, F., P. Y. Le Traon, and S. Houry, 1995: Reducing orbit error with an inverse method to estimate the oceanic variability from satellite altimetry. J. Atmos. Oceanic Technol.,12, 150–160.

    • Crossref
    • Export Citation
  • Brankart, J. M., and P. Brasseur, 1996: Optimal analysis of in situ data in the western Mediterranean using statistics and cross-validation. J. Atmos. Oceanic Technol.,13, 477–491.

    • Crossref
    • Export Citation
  • Bretherton, F., R. Davis, and C. Fandry, 1976: A technique for objective analysis and design of oceanographic experiments applied to MODE-73. Deep-Sea Res.,23, 559–582.

    • Crossref
    • Export Citation
  • Chao, Y., and L.-L. Fu, 1995: A comparison between the TOPEX/POSEIDON data and a global ocean general circulation model during 1992–1993. J. Geophys. Res.,100, 24965–24976.

    • Crossref
    • Export Citation
  • De Mey, P., and A. R. Robinson, 1987: Assimilation of altimetric fields in a limited area quasigeostrophic model. J. Phys. Oceanogr.,17, 2280–2293.

    • Crossref
    • Export Citation
  • Fu, L. L., and R. D. Smith, 1996: Global ocean circulation from satellite altimetry and high-resolution computer simulation. Bull. Amer. Meteor. Soc.,77, 2625–2636.

    • Crossref
    • Export Citation
  • ——, E. J. Christensen, C. A. Yamarone Jr., M. Lefebvre, Y. Menard, M. Dorrer, and P. Escudier, 1994: TOPEX/POSEIDON mission overview. J. Geophys. Res.,99, 24369–24381.

    • Crossref
    • Export Citation
  • Fukumori, I., 1995: Assimilation of TOPEX sea level measurements with a reduced-gravity, shallow water model of the tropical Pacific Ocean. J. Geophys. Res.,100, 25027–25039.

    • Crossref
    • Export Citation
  • Hernandez, F, P. Y. Le Traon, and R. Morrow, 1995: Mapping mesoscale variability of the Azores current using TOPEX/POSEIDON and ERS-1 altimetry, together with hydrographic and Lagrangian measurements. J. Geophys. Res.,100, 24995–25006.

    • Crossref
    • Export Citation
  • Le Traon, P. Y., and P. De Mey, 1994: The eddy field associated with the Azores front east of the Mid-Atlantic Ridge as observed by the Geosat altimeter. J. Geophys. Res.,99, 9907–9923.

    • Crossref
    • Export Citation
  • ——, C. Boissier, and P. Gaspar, 1991: Analysis of errors due to polynomial adjustment of altimeter profiles. J. Atmos. Oceanic Technol.,8, 385–396.

  • ——, P. Gaspar, F. Bouyssel, and H. Makhmara, 1995a: Using Topex/Poseidon data to enhance ERS-1 data. J. Atmos. Oceanic Technol.,12, 161–170.

    • Crossref
    • Export Citation
  • ——, ——, F. Ogor, and J. Dorandeu, 1995b: Satellites work in tandem to improve accuracy of data. Eos, Trans. Amer. Geophys. Union,76, 385–389.

    • Crossref
    • Export Citation
  • Mazzega, P., and S. Houry, 1989: An experiment to invert Seasat altimetry for the Mediterranean and Black Sea mean surfaces. Geophys. J.,96, 259–272.

    • Crossref
    • Export Citation
  • ——, M. Berge, A. Cazenave, and P. Schaeffer, 1998: Maps of the mean sea surface and corresponding gravity anomalies from ERS-1 geodetic mission. J. Geophys. Res., in press.

  • Menard, Y., 1983: Observation of eddy fields in the northwestern Atlantic and northwestern Pacific by SEASAT altimeter data. J. Geophys. Res.,88, 1853–1866.

    • Crossref
    • Export Citation
  • Mesias, J. M., and T. Strub, 1995: An inversion method to determine ocean surface currents using irregularly sampled satellite altimetry data. J. Atmos. Oceanic Technol.,12, 831–849.

    • Crossref
    • Export Citation
  • Oschlies, A., and J. Willebrand, 1996: Assimilation of Geosat altimeter data into an eddy-resolving primitive equation model of the North Atlantic Ocean. J. Geophys. Res.,101, 14175–14190.

    • Crossref
    • Export Citation
  • Tai, C. K., 1991: How to observe the gyre to global-scale variability in satellite altimetry: Signal attenuation by orbit error removal. J. Atmos. Oceanic Technol.,8, 271–288.

  • ——, and J. Kuhn, 1995: Orbit and tide error reduction for the first 2 years of TOPEX/POSEIDON data. J. Geophys. Res.,100, 25353–25364.

    • Crossref
    • Export Citation
  • Vasquez, J., V. Zlotnicki, and L. L. Fu, 1990: Sea level variabilities in the Gulf Stream between Cape Hatteras and 50°W: A Geosat study. J. Geophys. Res.,95, 17957–17964.

    • Crossref
    • Export Citation
  • Wunsch, C., and V. Zlotnicki, 1984: The accuracy of altimetric surfaces. Geophys. J. Royal. Astron. Soc.,78, 795–808.

    • Crossref
    • Export Citation

Fig. 1.
Fig. 1.

Comparison of sea level mapping using conventional objective analysis method (uncorrelated noise) (upper figure), with objective analysis taking account of long-wavelength error correlated along the tracks (lower figure) when random biases with standard deviation of 5 cm are simulated along the T/P tracks. The relative variance of the long-wavelength error was set at 50%. Units are in centimeters. Contour interval is 2 cm.

Citation: Journal of Atmospheric and Oceanic Technology 15, 2; 10.1175/1520-0426(1998)015<0522:AIMMOM>2.0.CO;2

Fig. 2.
Fig. 2.

(a) Sea level anomaly maps in the Canary Basin for three successive maps: 22 November, 2 December, and 12 December 1992, derived by the conventional objective analysis (COA) method using T/P data. Units are in centimeters. Contour interval is 2 cm. (b) As in Fig. 2a but with objective analysis taking account of long-wavelength error (LWA). (c) Difference between COA and LWA (Fig. 2a minus Fig. 2b).

Citation: Journal of Atmospheric and Oceanic Technology 15, 2; 10.1175/1520-0426(1998)015<0522:AIMMOM>2.0.CO;2

Fig. 3.
Fig. 3.

(a) Sea level anomaly maps in the Canary Basin for three successive maps: 22 November, 2 December, and 12 December 1992, derived by COA using ERS-1 data. Units are in centimeters. Contour interval is 2 cm. (b) As in Fig. 3a but with LWA method. (c) Difference between COA and LWA maps (Fig. 3a minus Fig. 3b).

Citation: Journal of Atmospheric and Oceanic Technology 15, 2; 10.1175/1520-0426(1998)015<0522:AIMMOM>2.0.CO;2

Fig. 4.
Fig. 4.

(a) Sea level anomaly maps in Canary Basin for three successive maps: 22 November, 2 December, and 12 December 1992, derived by COA using combined T/P and ERS-1 data. Units are in centimeters. Contour interval is 2 cm. (b) as in Fig. 4a but with LWA method. (c) Difference between COA and LWA maps (Fig. 4a minus Fig 4b).

Citation: Journal of Atmospheric and Oceanic Technology 15, 2; 10.1175/1520-0426(1998)015<0522:AIMMOM>2.0.CO;2

Fig. 5.
Fig. 5.

(a) Difference between T/P and ERS-1 maps derived by COA. Units are in centimeters. Contour interval is 2 cm. (b) As in Fig. 5a but with LWA method.

Citation: Journal of Atmospheric and Oceanic Technology 15, 2; 10.1175/1520-0426(1998)015<0522:AIMMOM>2.0.CO;2

Fig. 6.
Fig. 6.

(a) Global sea level anomaly derived using COA method, late October 1994 (Julian day 16424), with T/P data. Units are in centimeters. (b) As in Fig. 5a but with LWA method. (c) Difference between COA and LWA maps (Fig. 5a minus Fig. 5b).

Citation: Journal of Atmospheric and Oceanic Technology 15, 2; 10.1175/1520-0426(1998)015<0522:AIMMOM>2.0.CO;2

Fig. 6.
Fig. 6.

(Continued)

Citation: Journal of Atmospheric and Oceanic Technology 15, 2; 10.1175/1520-0426(1998)015<0522:AIMMOM>2.0.CO;2

Table 1.

Variances of sea level anomaly 〈h2〉 (cm2) and of zonal 〈u2〉 and meridional velocities 〈υ2〉 (cm2 s−2) in the Canary Basin, derived after mapping T/P data using COA and LWA. The (COA–LWA) column shows the variances of the differences between the two mappings.

Table 1.
Table 2.

As in Table 1 but for ERS-1.

Table 2.
Table 3.

As in Table 1 but for ERS-1–T/P data.

Table 3.
Save