Altimeter s Capability of Reconstructing Realistic Eddy Fields Using Space-Time Optimum Interpolation

Transcription

1 Journal of Oceanography, Vol. 59, pp. 765 to 781, 2003 Altimeter s Capability of Reconstructing Realistic Eddy Fields Using Space-Time Optimum Interpolation TSURANE KURAGANO* and MASAFUMI KAMACHI Oceanographic Research Department, Meteorological Research Institute, Tsukuba , Japan (Received 22 August 2002; in revised form 7 March 2003; accepted 14 March 2003) Sea surface height anomaly maps of realistic eddy activity were obtained by applying space-time optimum interpolation to altimeter data. Analysis error and rate of reconstructing eddy signals were investigated by taking account of: 1) dependency on orbit configurations of single and multiple altimeters; 2) dependency on space-time scales of realistic, dominant eddies; and 3) effect of space-time scales of eddy propagation. Large-scale sea surface height anomalies are subtracted from altimeter data by applying an along-track filter to allow easy handling of eddy signals. The space-time scales of the first-guess error in the optimum interpolation are statistically evaluated by fitting a space-time anisotropic Gaussian function to space-time-distributed correlation coefficients of sea surface height using the TOPEX data. The results of the optimum interpolation clarify the followings: 1) ERS has a better capability of reconstructing eddy signals than TOPEX. Comparison of maps from multi-altimeter data shows that TOPEX+ERS has a better capability than Jason-1+TOPEX in lower latitudes and vice versa in higher latitudes, though the differences are small. 2) The small space-time scale yields a low reconstruction rate in marginal seas and alongside the equator. The persistent timescale is large, and westward propagation is dominant in the subtropical and subarctic regions, where the reconstruction rates are high. 3) The optimum interpolation, taking account of eddy propagation, provides higher reconstruction rates than that taking no account of the propagation. The effect of propagation on the optimum interpolation is greater when it is applied to single-altimeter data than to multi-altimeter data. Keywords: Altimeter, sea surface height, optimum interpolation, resolution capability, space-time scales, orbit, eddy, TOPEX/ POSEIDON, ERS, Jason Introduction Sea surface height (SSH) variations are caused by various ocean phenomena, which have intrinsic space and time scales with different dynamics. Altimeter measurements can capture the phenomena with larger scales than the altimeter s sampling interval in time and space, such as seasonal steric variability, inter-annual or decadal variability interacting with atmospheric conditions, and the mean sea level change concerning to the global warming. Meso-scale eddies also play a significant role in transporting mass and heat in the ocean, but the altimeter cannot resolve them as well as it resolves the large-scale variability. * Corresponding author. kuragano@met.kishou.go.jp Present address: Office of Marine Prediction, Japan Meteorological Agency, Tokyo , Japan. Copyright The Oceanographic Society of Japan. A statistical inversion from the altimetric SSH into vertical temperature and salinity structures is one of methods for assimilating altimeter data. The Japan Meteorological Agency adopts such an inversion method for its operational ocean assimilation system (Kuragano et al., 2000). The various phenomena associated with SSH have different vertical structures. It is therefore better to divide the SSH according to the intrinsic dynamics before applying the inversion. However, the altimeter has a critical resolving capability in representing eddy scale SSH. Many previous studies have stated that the eddy resolving capability depends on an altimeter s orbit configuration. The pioneering study by Chelton and Schlax (1994) studied the resolution capability of the altimeter using the Geosat orbit, which has a 17-day interval and 1.4 spacing between two parallel tracks. They concluded that the spatial and temporal resolving scales are 3 and 30 days and Geosat is not suitable for mapping eddy fields using an objective interpolation method. Greenslade et al. (1997) compared the resolving capabilities of Geosat, 765

2 ERS and TOPEX/POSEIDON using a similar method to that used by Chelton and Schlax (1994), obtaining similar results to those reported by Chelton and Schlax (1994): the space and time resolution capabilities for ERS, Geosat and TOPEX/POSEIDON are (4.25, 35-day), (3.25, 50- day) and (6.75, 25-day), respectively. Greenslade et al. (1997) discussed the relationship between possible resolution scale and the orbit configurations, and were not much concerned with realistic eddy scales. Le Traon and Dibarboure (1999) investigated an altimeter s capability of resolving realistic eddies dominating the ocean, using the orbits of Geosat, ERS, TOPEX/POSEIDON and Jason- 1. Le Traon et al. (2001) also investigated these capabilities using a model SSH. Both papers concluded that Geosat has the best capability for resolving eddies and TOPEX/POSEIDON is the worst, when the single-altimeter analyses are compared. They also concluded that multi-altimeter analyses are much better than single-altimeter ones. In particular, Jason-1+TOPEX has a better capability than TOPEX+ERS. Ducet et al. (2000) investigated the effect of merging ERS-1 and -2 data and TOPEX/POSEIDON in their objective analysis, which showed that mean mapping error variance in the mapped SSH from TOPEX/POSEIDON alone is 25% of signal variance and is reduced to 10% by merging two altimeter data. These studies adopted an objective interpolation of the along-track SSH into grid points, and evaluated the analysis error of the grid point values (GPVs). They assumed that the SSH variability has isotropic directional correlations in the space-time domain. However, the realistic, dominant eddies propagate mostly westward and have a long lifetime, as shown by Jacobs et al. (2001). Kuragano and Kamachi (2000) (KK2000 hereafter) schematically showed that the propagating SSH has a correlation scale that is stretched in a tilted direction from the time axis, and the timescale in the tilted direction is longer than that of a fixed point. They also demonstrated a space-time optimum interpolation (3D_OI) using the tilted correlation coefficient of the first-guess error, revealing a smaller analysis error and smoother movement of eddies than that from a space OI (2D_OI using correlation only on the horizontal plane). De Mey and Ménard (1989) had previously proposed a 3D_OI, taking account of the phase speed of the eddy, and studied eddy movement from the GEOS-3 data. The above studies of resolution capability also apply to 3D_OI, but isotropic functions were adopted for the first-guess error covariances (Le Traon and Diarboure, 1999; Ducet et al., 2000; Le Traon et al., 2001). Chelton and Schlax (1994) and Greenslade et al. (1997) used the Loess smoother to calculate GPVs, and their weight functions also had an isotropic form. The objective of this paper is to evaluate how much the realistic, dominant eddy can be reconstructed by the 3D_OI applied to single- and multi-satellite data; TOPEX, ERS, TOPEX+ERS and Jason-1+TOPEX. It is different from previous studies in that the propagation of eddy is taken into account in the first-guess error statistics. We evaluate our OI for each orbit (including multi-satellite orbits). Geophysical distribution of the analysis error is investigated in relation to eddy characteristics and orbit configurations. We have performed the OI for each orbit using real along-track SSH data. The analysis error is evaluated in the form of the OI theory as a statistical value. The real SSH data have observation errors and true values, and these are unknown. It is impossible to calculate the analysis error by direct comparison between analyzed SSH and true SSH. However, proper observation error and firstguess error, which can be calculated by the method of KK2000, will derive statistical analysis error precisely. An approach of Le Traon et al. (2001), in which model SSH fields are sampled and reconstructed, would be more convincing because analysis error would be calculated directly. The model fields, however, do not present the exact, real ocean. For example, eddy kinetic energy in the Kuroshio Extension is less than that given by TOPEX, the location of the Kuroshio Extension has some bias, and the Oyashio intrusion tends to be weak. The model errors may be reduced by, for example, a highresolution model, but we have not developed it yet. We therefore do not adopt such approach here. Altimeter data, the methods by which they are processed, and the 3D_OI are explained in Section 2. The analyzed SSH fields in the Kuroshio Extension region are discussed in Section 3. Global distributions of reconstruction rate of realistic and dominant eddies are compared as between the single and multiple altimeters in Section 4. The global reconstruction rate is discussed using synthetic space-time scales in terms of a relation between space-time scale and orbit in Section 5. The synthetic scales also compensate for the shortage of the realistic and dominant eddy. The term realistic and dominant eddy as used in this paper does not necessarily mean real closed eddies. It is detected from along-track SSH with a filtering method, and sometimes has a longitudinally large feature affected by a zonal current structure. We summarize the results in Section 6. For the application of the OI, space-time scales of the first-guess error and observation errors are evaluated from both TOPEX and ERS data, and these are explained in Appendices A and B. 2. Data and Method 2.1 Altimeter data The altimeter measures SSH along the satellite 766 T. Kuragano and M. Kamachi

3 Fig. 1. Ground tracks of TOPEX (solid line) and ERS (dashed line) in the western North Pacific. ground track. Repeating the measurement along the same ground tracks, the altimeter provides temporal variations of SSH. The repeat period and spacing of the measurement depend on the orbit configuration. The repeat period and spacing are almost inversely proportional to each other. TOPEX/POSEIDON, launched in 1992, repeats 254 tracks between 66 S and 66 N in days. Ground tracks over the western North Pacific are shown by solid lines in Fig. 1. The measurement in one repeat period is called a cycle. TOPEX/POSEIDON has two altimeter sensors, TOPEX and POSIEDON. TOPEX works for most of the period, and POSEIDON works for one cycle in approximately 10 cycles. There is a little bias between them, so that the TOPEX altimeter data alone are used for the present analysis. ERS-1 adopts several orbit configurations. It adopts a 35-day-repeat orbit from April 1992 to December 1993 and from March 1995 to May 1996, revolving the earth 501 times in 35 days (ground tracks are shown by dashed lines in Fig. 1). ERS-2 has adopted the same 35-day-repeat orbit from the start in April Jason-1 was launched in December It follows the same orbit as TOPEX/POSEIDON. Jason-1 and TOPEX have been operated in tandem phase since August One candidate for the tandem-orbit configuration is to place TOPEX/POSEIDON between two Jason- 1 tracks. Since the geophysical data record (GDR) of Jason-1 have not yet been released, the resolving capability of the two combined orbits is examined assuming a pseudo-observation error for the Jason-1 data. 2.2 Data processing TOPEX data from cycles 11 to 272 (from December 31, 1992 to February 11, 2000) were extracted from the MGDR-B (Benada, 1997). ERS-1 data from cycles 6 to 18 of phase-c (from October 5, 1992 to December 23, 1993), and from cycles 1 to 13 of phase-g (from March 24, 1995 to June 2, 1996), and ERS-2 data from cycles 0 to 46 (from April 29, 1995 to September 24, 2000) were extracted from CORSSH (AVISO, 1997). To apply the collinear method and to save computer resources, 27-km average values were calculated between 66 S and 66 N for TOPEX and between 80 S and 80 N for ERS. The 27-km interval can resolve phenomena of 135 (=5 27) km (see section 2 of KK2000), which is not a dominant wavelength; this is km, as revealed by a forthcoming along-track spectrum analysis. Le Traon (1991) also showed that the typical zero-crossing scale of eddies is at a minimum, about 70 km (280 km in wavelength), in a high latitude region in the Atlantic Ocean. Ducet et al. (2000) performed a 21-km subsampling procedure when applying an OI to the altimeter data for a typical scale of km. Even the interval of 27 km is small enough to resolve eddies. Reduction of the observation error by averaging makes the estimation of spacetime scales more precise (in Subsection 2.3). The reduction of the observation error decreases analysis error in the present OI, which compensates for increases of analysis error by reduction of the number of data. The SSH anomaly (SSHA), which is an anomaly from the mean sea surface, was calculated at each 27 km along each cycle/track using the collinear method. The mean SSH was calculated for the period for the TOPEX data, and December 16, 1992 December 16, 1993 and October 8, 1995 October 7, 1999 for the ERS data. The mean SSH was determined at each position at which the data were obtained for over 80% of the period. If the mean SSH was not determined, no SSHA is available at that position. No ERS-1 data in phase-c are available in the Japan Sea, so that the mean SSH in this region is determined only from ERS-2 data for October 8, 1995 October 7, This shortening of the averaging period causes at most a 1 cm difference in the mean SSH (estimated from the TOPEX data). The SSHA was smoothed along the track using the following function, ) h o ()= l R R o( ) h l+ s exp s / L ds R, 2 2 exp s / L ds R 2 2 ( ) ( ) () 1 where ) h o is the smoothed SSHA, h o the SSHA, l the coordinate along the satellite track, s the dummy variable along the track, L the cutoff scale for the smoothing (4L represents the cutoff wavelength), and R is selected as equal to 3L. The length of 400 km was selected for the cutoff scale L, corresponding to the cutoff wavelength of 1600 km. Altimeter s Capability of Reconstructing Realistic Eddy Fields Using Space-Time Optimum Interpolation 767

4 Fig. 2. Wavenumber spectra of SSHA along tracks. Solid line shows a spectrum of SSHA. Thin and dashed lines show spectra of smoothed and residual SSHAs, respectively. (a) Average spectra for total tracks of TOPEX (59 tracks for each cycle). (b) Average spectra for total tracks of ERS (178 tracks for each cycle). Bold straight lines show k 0.8, k 3 and k 2.5. Figure 2 shows averaged wavenumber spectra of the along-track SSHA. The spectrum has a slight depression at wavelengths km when compared with the line of k 0.8. The slight peak in the spectrum at wavelength km corresponds to eddy activity. Wavenumber spectra for the smoothed SSHA and the residual SSHA, h o (l) ) h o (l), are also shown in Fig. 2. The residual SSHA conserves the spectrum peak at wavelengths km. KK2000 estimated the space-time scales from TOPEX/POSEIDON data without scale decomposition. The estimated spatial scales are <240 km (the corresponding wavelength is <1200 km: wavelength corresponds to about 5 times e-folding scale) in the latitudinal direction in the regions where eddy activities are dominant, such as the mid-latitudes of the western North Pacific and the North Atlantic, and mid- and high-latitudes in the Southern Hemisphere. The estimated spatial scales are >400 km (the corresponding wavelength being >2000 km) in the latitudinal direction in the regions where seasonal steric variability is dominant, such as the eastern North Pacific and subarctic of the North Pacific and the North Atlantic. The estimated eddy scales are contaminated by steric variabilities, and the estimated scales of steric variabilities are contaminated by eddy activities. The actual scales of eddies (steric variabilities) are smaller (larger) than the estimated values given in KK2000. The selected filter conserves variability of wavelength <1000 km and eliminates those of wavelength >2000 km. The residual SSHA still includes large-scale variability, because the complete cutoff cannot be performed due to the equation itself and track lengths limited by land masses. The residual SSHA, however, is the value at which the large-scale SSHA is considerably reduced. Therefore the OI in the following sections must detect the eddy signal better by applying it to the residual SSHA than to the original SSHA. Moreover, studies of in situ data imply that the cutoff wavelength of 1600 km is appropriate to eliminate large-scale variability. Colosi and Barnett (1990) showed that large-scale SST variability has an e-folding scale of 1600 km in the Southern Hemisphere using COADS and TOGA drifter data. They eliminated eddy scale variability by averaging over 4 latitude 4 longitude, which has a similar effect to the above smoothing. 2.3 Space-time OI method The OI was applied to the residual SSHA in the space-time domain. The first guess is set to zero SSHA. The decorrelation space and time scales and the RMS of the first-guess error are shown in Fig. 3. These are evaluated using the method of KK2000 applied to the residual SSHA (see Appendix A). The RMS observation error is also shown in Fig. 4 (see Appendix B). The observation error is assumed to have no correlation between the data on different tracks and on different cycles, even in the same track. The observation error has a correlation only between the data on the same track of the same cycle (the method of evaluating the correlation coefficient along track is also shown in Appendix B). The SSHAs are interpolated into every grid point. Data are selected in the area where the first-guess error has a correlation coefficient >exp( 1) with the grid point. The statistical propagation of SSHA is taken account by applying the space-time scale shown in Fig. 3. In order to evaluate the advantage of the present 3D_OI, we perform another type of three-dimensional OI, 768 T. Kuragano and M. Kamachi

5 Fig. 3. Statistical e-folding scales of the first-guess error estimated from the residual SSHA of TOPEX. (a) Longitudinal scales, (b) latitudinal scales, (c) timescales at fixed points, (d) persistent timescales, i.e., lifetimes of eddy signals, (e) phase speeds for longitudinal direction, (f) those for latitudinal direction, and (g) RMS variability of the first-guess error. Altimeter s Capability of Reconstructing Realistic Eddy Fields Using Space-Time Optimum Interpolation 769

6 e N p = σ 1 w µ, ( 2) g g p i gi i= 1 Fig. 4. RMS variability of the residual SSHA error (cm). (a) RMS variability of random noise in the SSHA among different TOPEX along-track data, and (b) that among different ERS along-track data. in which the decorrelation timescale of 15 days is adopted, the propagation of SSHA is not accounted for, and the same spatial scale as the present OI is adopted. The firstguess error for this OI is explained in Appendix A. We abbreviate the present 3D_OI to Advanced OI (AD_OI) and the other to Non-Propagation OI (NP_OI). ERS-1 and -2 were in tandem phase during April 1995 May The ERS-2 data alone are used to avoid redundancy of the tracks if a mapping time is in the tandem phase. The observation error of Jason-1 is expected to be less than TOPEX/POSEIDON (2.5 cm RMS), being the total of orbit error and instrumental noise. Jason-1 data are not available yet, so that only the analysis error for this tandem orbit is tested. The observation error of Jason- 1 was assumed to take the same values as that of TOPEX. 2.4 Evaluating reconstruction rate The resolution capability is evaluated by the analysis error from the OIs for TOPEX, ERS, TOPEX+ERS and Jason-1+TOPEX, respectively. The analysis error e g of the gridded value calculated by the OI is p where σ g is the RMS variability of the first-guess error at the grid point, i the data number, w i the weight coefficient for the data i, µ gi the correlation coefficient of the p first-guess error between the grid point and data point i, and N the total number of data adopted for the OI. The analysis error value is influenced by the magnitude of the first-guess error, so that it is difficult to evaluate geographical differences in reconstruction rate from the analysis error. The value of i N p = 1 wiµ gi = ( σ g ) p 2 eg σg p 2 2 in Eq. (2) is a variance ratio of the GPV to the first-guess error, which indicates how much the rate of the true SSHA is reconstructed. When the value is unity, the analysis error equals zero, indicating a completely successful reconstruction. When the value is zero, the analysis error is the same as the first-guess error and the GPV takes the same value as the first guess. This case indicates that the data do not contribute to determining the GPV at all. The reconstruction rate is discussed in Sections 4 and Eddy Representation in the Kuroshio Extension Region Figure 5 shows the SSHA maps according to AD_OI and Fig. 6 shows the analysis error maps for TOPEX, ERS, TOPEX+ERS and Jason-1+TOPEX (SSHA is not available for the last one). Though the maps in Figs. 5 and 6 are instantaneous ones, a time series of the analysis (not shown) shows that the error fields in other times are not so different from those given in Fig. 6. The SSHA maps show quite similar characteristics at a length scale of ~100 km. Differences are recognized in smaller scale structures. The TOPEX SSHA includes smaller structures than the ERS and TOPEX+ERS SSHAs: contours of zero anomaly are smoother in the ERS and TOPEX+ERS SSHAs than those in the TOPEX SSHA, and eddies enveloped by an oval are represented as one eddy in the ERS SSHA. The analysis error map from TOPEX shows a strong track pattern: there is a small error along the tracks and a local maximum error at each center of the diamond area enclosed by the nearby tracks. Though ERS has a greater observation error than TOPEX, the analysis error from ERS is generally smaller than that from TOPEX, and does not show the strong track pattern. The analysis error map from TOPEX+ERS also shows the TOPEX track pattern, but the values are less than those from TOPEX or ERS alone. The analysis error map from Jason-1+TOPEX still shows the track pattern, but not as strongly as that from TOPEX alone. The differences are more pronounced in time-longitude sections 770 T. Kuragano and M. Kamachi

7 Fig. 5. Comparison of SSHA in the western North Pacific. The SSHA maps are based on interpolated values at grids obtained by the AD_OI applied to along-track altimeter data. Analysis date is January 3, 1996, and unit is cm. (a) SSHA fields from TOPEX, (b) from ERS, and (c) from TOPEX+ERS. See text for explanations of black oval lines. Fig. 6. Comparison of analysis error in the western North Pacific. (a) Analysis error from TOPEX, (b) from ERS, and (c) from TOPEX+ERS; these are estimated for SSHAs in Fig. 5. (d) Analysis error is estimated for Jason-1+TOPEX, where Jason-1 is assumed to have the same observation error as TOPEX. Unit is cm. Altimeter s Capability of Reconstructing Realistic Eddy Fields Using Space-Time Optimum Interpolation 771

8 Fig. 7. Time-longitude section of the SSHA based on the same GPV as Fig. 5. Latitude is N, and unit is cm. (a) SSHA section from TOPEX, (b) analysis error for (a), (c) SSHA section from ERS, (d) analysis error for (c), (e) SSHA section from TOPEX + ERS, and (f) analysis error for (e). 772 T. Kuragano and M. Kamachi Fig. 8. As Fig. 7 but obtained by NP_OI.

9 (Fig. 7). SSHA sections show a westward propagating anomaly to the east of 145 E, related to the Rossby wave propagation. This pattern is contaminated by eastward propagating signals to the west of 145 E, where SSHAs are advected by the Kuroshio. TOPEX has a stripe-like analysis error structure restricted to the track locations, and the analyzed SSHA seems to have a small structure, especially to the east of 165 E, where the amplitudes of SSHA are small (Figs. 7(a) and (b)). ERS and TOPEX+ERS do not have such a stripe-like error structure and small SSHA structure. The small-scale SSHA structures in the TOPEX SSHA are due to the wide spacing of the TOPEX tracks. SSHAs on time-longitude section at N using NP_OI are not as smooth as those found by AD_OI (Figs. 8(a), (c) and (e)). The TOPEX analysis has a large analysis error emerging over the whole longitude once in several cycles, corresponding to a missing cycle in the TOPEX observation (POSEIDON s cycle) (Fig. 8(b)). The ERS analysis has a westward propagating error signal, and the greater error emerges when the observation is absent (Fig. 8(d)). The TOPEX+ERS data make analysis error smaller than single altimeter analysis, but SSHA is not as smooth as those found with AD_OI (Fig. 8(e)). 4. Reconstruction Rate in the Global Ocean The global mean analysis errors found by AD_OI are 3.08, 2.68, 2.31 and 2.23 cm for TOPEX, ERS, TOPEX+ERS and Jason-1+TOPEX, respectively. Therefore, the Jason-1+TOPEX orbit may be best for analyzing eddy fields. However, the analysis error value is influenced by the magnitude of the first-guess error, so that the reconstruction rate described in Subsection 2.4 should be evaluated. Figure 9 shows global maps of reconstruction rate for the TOPEX, ERS, TOPEX+ERS and Jason-1+TOPEX. Table 1 shows the average reconstruction rates for the global ocean and four zonal bands. Figure 9 shows that SSHA is not reconstructed well at extremely high latitudes, alongside the equator, in marginal seas and northeast of the North Pacific. In these regions, the space-time scales are generally small. The equatorial Kelvin wave, Rossby wave, Legeckis wave (Legeckis, 1977) and the mixed Rossby-gravity wave (Moore and Philander, 1977) coexist adjacent to the equator. The space-time scales in Fig. 3 are combinations of those waves in this region. However, the space-time scales alongside the equator, where the spatial scale is relatively small and the timescale is quite small, seem to be governed mainly by the Legeckis and mixed Rossby-gravity waves, and the reconstruction rate in this region is low. The reconstruction rate for the TOPEX analysis shows a tracking pattern, except at low latitudes: >90% along the TOPEX tracks and <60% at many midpoints of diamond areas (enveloped by four Fig. 9. Global maps of reconstruction rate for a realistic eddy field obtained from (a) TOPEX, (b) ERS, (c) TOPEX+ERS and (d) Jason-1+TOPEX. The values are averaged for ten analyses (every 5 days from July 17 to August 31, 1996). nearby tracks). The rate at the midpoint is lower than that found by Larnicol et al. (1995) who reported 70% of the reconstruction rate at the midpoint. This is because the present analysis is restricted to the eddy-scale SSHA. Such a tracking pattern is not recognized in the ERS analysis. TOPEX has a larger reconstruction rate in the TROPIC region than ERS, and ERS has 8.9 points larger than TOPEX in the SUBTROPIC and SUBARCTIC regions (Table 1), where the tracking pattern is dominant in the TOPEX analysis. The reconstruction rates from TOPEX+ERS and Jason-1+TOPEX indicate a great improvement over single-altimeter analyses. The globally averaged reconstruction rates increase >5 points from the ERS analysis and (%) Altimeter s Capability of Reconstructing Realistic Eddy Fields Using Space-Time Optimum Interpolation 773

10 Table 1. Reconstruction rates (%). Areas are global and several latitude bands as shown by the column Latitude. AD_OI indicates reconstruction rate of 3D_OI using the present space-time scales for the first-guess error covariance, NP_OI reconstruction rate by 3D_OI but decorrelation timescale is changed to 15 days and propagating character is removed, and ADV means (AD_OI) (NP_OI), which indicates an improvement of AD_OI over NP_OI. Unit: %. >10 points from the TOPEX analysis (Table 1). The tracking pattern remains in the TOPEX+ERS analysis. The reconstruction rate is more homogeneous for the Jason- 1+TOPEX analysis. TOPEX+ERS reconstructs SSHA better than Jason-1+TOPEX in latitudes of <40 and vice versa in latitudes of >40 (Table 1). The differences are below 1 point except in the ARCTIC, though. The reconstruction rates of AD_OI are greater than those of the NP_OI (Table 1). This is due that the larger number of data decide a GPV in AD_OI than in NP_OI. AD_OI can introduce the data in a different location and at a different time, because the first-guess error in AD_OI represents the propagation of SSHA. The advantage of AD_OI over NP_OI is greater for the single altimeter than the multiple altimeter, especially in the SUBTROPIC area, where eddies propagate rapidly (Fig. 3(e)) and the persistent timescales are much greater than 15 days (Fig. 3(d)). The ERS analysis by AD_OI in the SUBTROPIC area has a higher reconstruction rate than the multi-altimeter analyses by NP_OI, indicating a great advantage of AD_OI. The temporal continuity of SSHA (Figs. 7 and 8) also shows the great advantage of AD_OI over NP_OI. Le Traon et al. (2001) demonstrated the advantage of Jason-1+TOPEX (complementary orbits) over TOPEX+ERS (uncomplementary orbits) in their spacetime OI, in which a decorrelation timescale of 15 days is applied. The present study also shows that Jason- 1+TOPEX has higher reconstruction rates than TOPEX+ERS when NP_OI is applied. When AD_OI is applied, TOPEX+ERS has greater reconstruction rates than the Jason-1+TOPEX, however. The mapping errors as a percent of signal variance shown in Plate 3 of Ducet et al. (2000), which is an inverse concept of the reconstruction rate, display a strong tracking pattern. Except for the tracking pattern, the errors have a zonal pattern for TOPEX, because their spacetime scales depend on latitude alone. The tracking pattern for ERS has an undulation for several tracks, while the present reconstruction rate does not. This is because a long persistent timescale is adopted in AD_OI but the timescale in Ducet et al. (2000) is 15 days smaller than the temporal interval of the ERS observation. The areas of >95% reconstruction rate (corresponding to <5% error) is widely spread in the tropical and subtropical oceans for TOPEX and ERS while the areas of <5% error in Ducet et al. (2000) are restricted along tracks in those oceans. 5. Dependence of Reconstruction Rate on the Relation between Space-Time Scale and Orbit The reconstruction rate depends on the space-time scale, latitude and the orbit configurations. To clarify which space-time scale AD_OI can reconstruct and to confirm the realistic reconstruction rate shown in Section 4, the reconstruction rate is tested for various synthetic scales of the first-guess error. The test site was the center of each diamond area (a midpoint between tracks at the same latitude as a cross over point), where the reconstruction rate is expected to be at a minimum in the area. The analyses were performed for every 12 GMT of a 10-day sequence for TOPEX and a 35-day sequence for ERS, and the minimum rate in the analyses was adopted for each test site. Figures 10 and 11 show the reconstruction rate as a function of latitude, changing the first-guess error covariance, i.e., space-time scale and westward propagating phase speed. RMS variability of the observation error is given as half of the first-guess error. The observation error is assumed not to correlate with those for data on different tracks or on different cycles. Half of the observation error is assumed to correlate along track with an e-folding scale of 200 km, and there is no correlation for the rest. Note that this is not a meridional section of the reconstruction rate of Fig. 9 but rather a lower envelope line of the reconstruction rate from diamond to diamond. The tested reconstruction rate of the TOPEX orbit at each midpoint of a diamond area changes regularly along 774 T. Kuragano and M. Kamachi

11 (a) (b) (c) (d) Fig. 10. Reconstruction rate for TOPEX orbit. Spatial scale and persistent timescale (e-folding scales) are changed for each figure, as indicated. Phase speed is selected westward 1 (dotted thin line), 2 (broken thin line), 2.8 (solid thin line), 4 (dotted bold line), 5.7 (broken bold line), 8 cm s 1 (solid bold line). The ratio is calculated at the midpoint of diamond area enclosed by nearby 4 tracks in all different latitudes. The spatial scale is (a) 40 km, (b) 60 km, (c) 80 km and (d) 110 km with the same timescale of 80 days. the meridian (Fig. 10). It is lower in lower latitudes for smaller phase speed, for smaller spatial scale and for smaller timescale (dependence on timescales is not shown in Fig. 10). Figure 10 also indicates the inadequacy of Fig. 11. As Fig. 10 but for ERS orbit. Note (d), which shows the rates for larger phase speeds and larger spatial scale than those in (a) to (c). See text for labels A to F and X to Z. keeping L constant in an OI. Figure 12(a) shows the locations of the TOPEX tracks in a time-longitude section, indicating that more numerous data are used to calculate a GPV when spatial and temporal scales are larger, phase speed is larger, and latitude is higher. The space interval of the TOPEX measurement is larger than the tested spatial scales and the temporal interval is smaller than the tested time scales. The propagating signal can be reconstructed from the data of different cycles, even if it is not Altimeter s Capability of Reconstructing Realistic Eddy Fields Using Space-Time Optimum Interpolation 775

12 Fig. 12. Relative orbit locations in time-longitude section. Closed circles indicate ascending tracks, and open circles descending tracks. (a) Latitude of N of the TOPEX orbit, and the solid and dotted lines show phase speeds of westward 4 cm s 1 and 8 cm s 1. (b) N of the ERS, and the solid and dotted lines show phase speeds of westward 4 cm s 1 and 50 cm s 1. (c) and (d) Latitudes of N and N of the ERS orbit and lines are the same as (b). observed in the analysis time. The OI at the midpoint of a diamond needs observations 35 days after/before the analysis time for a signal propagating with 4 cm/sec, while it needs 17 days for 8 cm/sec (see dotted and solid lines in Fig. 12(a)). The larger reconstruction rate is then accomplished for the larger propagation speed because of the restriction to a persistent time scale. If the spatial scale is extremely small, however, the signal happens to go through a temporal interval of observations at a crossover point and the reconstruction rate decreases (see the rate for 8 cm/sec in Fig. 10(a)). The tested reconstruction rate for the ERS orbit is also higher for a larger space-time scale, such as the TOPEX. However, the along-latitude distribution is not as simple as that for TOPEX. Except for a jagged feature along latitude, the reconstruction rate has a local minimum at different latitudes for different phase speeds. For example, the minimum rate is located on the equator for phase speed 5.7 cm s 1 (A in Fig. 11(b)), at latitude ~30 for 4 cm s 1 (B in Fig. 11(b)), and ~50 for 2.8 cm s 1 (C in Fig. 11(b)). A minimum rate is also recognized for much greater phase speed: it is located on the equator for phase speed 64 cm s 1 (D in Fig. 11(d)), latitude of ~40 for 45 cm s 1 (E in Fig. 11(d)), ~55 for 32 cm s 1 (F in Fig. 11(d)), and so on. Figure 12(c) shows that propagating signals with 4 cm s 1 and 50 cm s 1 pass all crossover points at the same timing in a cycle at the latitude ~34. Therefore, if an eddy is located between space-time sampling points, it cannot be detected well. Spacing between the crossover points becomes closer at higher latitudes, so that the phase speed of an undetectable eddy becomes smaller at higher latitudes. In terms of the small-scale jagged feature, the latitudes N (X in Fig. 11(a)) and N (Y in Fig. 776 T. Kuragano and M. Kamachi

13 11(a)) are where the rate has a local minimum and maximum for phase speed 4 cm s 1. At latitude N, the ascending and descending tracks lie almost over each other (Fig. 12(c)) and the sampling density is therefore substantially one-half of that at latitude N (Fig. 12(b)). The crossover latitudes like those shown in Figs. 12(b) and (c) are arrayed almost alternately around this latitude, which creates the jagged feature of the reconstruction rate. The jag has a smaller amplitude at latitude ~27 N (Z in Fig. 11(a)) where the ascending and descending tracks are interleaved in time (Fig. 12(d)) and the ascending and descending tracks could not overlie each other. The larger spatial scale jagged features are recognized for larger phase speeds (Fig. 11(d)), which are related to the temporal difference of the ascending and descending tracks. The local maxima of the reconstruction rate are at ~9, ~27, ~41, ~51, etc. (Z in Fig. 11(d) for ~27 N), where the ascending and descending tracks are interleaved in time (Fig. 12(d)) and the small-scale jagged feature has a small amplitude for smaller phase speeds (Z in Fig. 11(a)). The ascending and descending tracks pass at similar times at latitudes where the local minima are recognized (Figs. 12(b) and (c)). Therefore, the reconstruction rate from the ERS orbit depends on latitude, i.e., the spatial and temporal relative location of the ascending and descending tracks, while that from the TOPEX orbit is not so dependent. The relative location of the ascending and descending tracks of the TOPEX changes only in time by latitude, and the impact from this change on the reconstruction rate is small due to the large eddy timescales. We now discuss the real reconstruction rate, in consideration of the result of the test and the space-time scale of realistic eddies. The reconstruction rate for TOPEX at lower latitudes <15 is high, except in the coastal regions, because the spatial scales of realistic eddies are sufficiently large (Fig. 9(a)). However, small timescales of SSHA (~10 days) causes a small reconstruction rate along the equator, despite the large spatial scales. The realistic spatial scale and phase speed become smaller, as at higher latitudes (see Fig. 3(e)). Therefore, the reconstruction rate for TOPEX is low at the midpoints of diamond areas in latitudes >20. The reconstruction rate is slightly depressed on a line just north of the Antarctic Circumpolar Current, where the zonal phase speed is zero. The test shows that it is more difficult for TOPEX to detect an eddy propagating at relatively a low phase speed. According to the test, a possible low rate for the ERS related to the low phase speed (<8 cm s 1 ) is restricted to smaller spatial scale variability: the case is the spatial scale 60 (40) km and the phase speed 4 (2.8) cm s 1 at latitudes (~50 ). However, the realistic spatial scale is much larger or the phase speed is smaller at these latitudes (Figs. 3(a) and (e)), so that ERS does not have problems in those latitudes. The extremely low rates for ERS are recognized alongside the equator (Fig. 9(b)), where the westward phase speeds are over 25 cm s 1 (Fig. 3). The ERS orbit configuration means that it is difficult to capture the SSHA signals propagating westward at high speed (Figs. 12(b) (d)). The small first-guess error and the large observation error also cause a low reconstruction rate, as seen in the eastern North Pacific. This is not critical, however, because of the small absolute value of the analysis error. 6. Conclusion We have studied the altimeter s capability of reconstructing SSHA of the realistic, dominant eddies in consideration of the orbit configurations and the first-guess error. The results can be summarized as: 1. AD_OI, which takes account of the statistical propagation of eddy in its first-guess error covariance, provides a higher reconstruction rate for eddy signals than NP_OI, which does not take account of eddy propagation. The temporal continuity of the signals also indicates the advantage of AD_OI. 2. The reconstruction rates from single-altimeter data show that ERS is better at reconstructing eddy signals than TOPEX, and those from multi-altimeter data shows that TOPEX+ERS is better than Jason-1+TOPEX at lower latitudes and vice versa in higher latitudes, but global averaged reconstruction rates are similar to each other. The results from NP_OI show that Jason-1+TOPEX is better than TOPEX+ERS, which is similar to the results obtained by Le Traon et al. (2001). 3. The geographical differences of the space-time scales of realistic eddies yield geographic differences of the reconstruction rates. The small space-time scale yields a low reconstruction rate in marginal seas. The small timescale alongside the equator, where the Legeckis and mixed Rossby-gravity waves make spatial scales fairly small, also causes a low reconstruction rate. The great westward propagation in this region makes the reconstruction rate extremely small for ERS. The persistent timescale is large and the westward propagation is dominant in the subtropical and subarctic regions, where the reconstruction rates are high. 4. The TOPEX orbit yields large, tracking-patterned analysis. Though ERS has a larger observation error than TOPEX, analysis error is smaller and tracking pattern is hardly recognized. The joint orbit of TOPEX+ERS makes analysis error smaller but still yields a TOPEX tracking pattern. Analysis of Jason-1+TOPEX data also has a tracking pattern, but not as strong as that of TOPEX. 5. The dependence of reconstruction rate on the orbit configuration is simple for TOPEX. It is high along track and has a local minimum at each midpoint of a dia- Altimeter s Capability of Reconstructing Realistic Eddy Fields Using Space-Time Optimum Interpolation 777

14 mond area. The minimum value simply depends on latitude and space-time scale: it is lower at lower latitudes, for smaller scale, and for lower propagation speed. The reconstruction rate for ERS depends on the relative location of ascending and descending tracks in the space-time domain: the minimum rate at the midpoint of the diamond area has a jagged feature along latitude. However, realistic eddies have larger space-time scales than those which possibly cause such a problem. Though ERS has a better capability for eddy analysis than TOPEX, the multiple altimeter data is much better than that from a single altimeter. The discussion in Section 5 indicates that the configuration of the tandem orbits should be determined carefully. For a tandem of two TOPEX-orbit altimeters, an interleaved orbit in the longitudinal direction is suitable to detect an eddy signal, and the time difference between the orbits does not matter. For a tandem of two ERS-orbit altimeters, the interleaved orbit in time is suitable for reducing the jagged feature of the reconstruction. The technical requirement for combining altimeters is that the observation error and the first-guess error should be estimated appropriately. The observation error and the first-guess error in the observed SSHA should complement each other. This requires that the space-time scale and the RMS variability of the true SSHA estimated from TOPEX and ERS should have the same value. The present statistical features do not completely coincide with each other, however. This inconsistency is not accounted for in the estimation of the analysis error in Figs. 6(c) and 7(f) and the reconstruction rate in Fig. 9(c). The original targets of the TOPEX/POSEIDON mission are the large-scale ocean circulation and the mean sea level change. The orbit configuration of TOPEX/ POSEIDON was determined not only to match these targets but also to reduce alias error of a tide model, which was a big problem for the previous altimeters. TOPEX/ POSEIDON has contributed to the development of excellent tide models. These models improved the accuracy of sea surface height of TOPEX/POSEIDON and ERS. Moreover, the ERS SSH compiled in the CORSSH is adjusted to the TOPEX SSH for large-scale orbit error elimination (Le Traon et al., 1995; Le Traon and Ogor, 1998). Though the ERS orbit is more appropriate for mapping eddy fields, even taking account of its observation error, such excellent mapping would never have been attained without the TOPEX/POSEIDON mission. A new altimeter has been proposed for the Jason-2 mission to attain high-resolution mapping: a wide swath ocean altimeter (WSOA) with a measurement of 200 km across the track. For the WSOA, the discussion of eddy resolving capability related to the orbit configurations may no longer be a problem. Such densely observed data include various phenomena and afford the possibility to detect much smaller-scale phenomena, which cannot be detected from the present altimeters. The temporal density is the same as TOPEX, however. Therefore AD_OI will be an effective method for detecting the phenomena, even from WSOA data. In order to detect smaller phenomena than those discussed in the present study, observation error and first-guess error should be detected properly from the data. The Gaussian fitting method given in Appendix A may be a powerful tool, even for the WSOA data. Acknowledgements A part of this work is supported by the Category 7 of MEXT RR2002 Project for Sustainable Coexistence of Human, Nature and the Earth. We are grateful to two anonymous reviewers for their helpful comments and advice. The TOPEX/POSEIDON altimeter data were provided by the NASA Physical Oceanography Distributed Active Archive Center at the Jet Propulsion Laboratory, California Institute of Technology. The CORSSH products are supplied by the CLS Space Oceanography Division, Toulouse, France. The ERS products were generated as part of the proposal Joint analysis of ERS-1, ERS- 2 and TOPEX/POSEIDON altimeter data for oceanic circulation studies selected in response to the Announcement of Opportunity for ERS-1/2 by the European Space Agency (Proposal code: A02.f105). Appendix A. Space-Time Scales and RMS Variability of the First-Guess Error Space-time scales and RMS variability of the firstguess error are evaluated from the altimeter data itself using a space-time Gaussian function. As the first guess is set to zero SSHA, a true SSHA is an error of the firstguess error. The scales and variabilities of the first-guess error are those of the true SSHA. They are evaluated from the along-track SSHA, which is denoted as observed SSHA in this appendix. The observed SSHA is the sum of true value and observation error. The mean value of the error at a position for the whole period is assumed to be zero, and the error is assumed to be random noise in a different track or even in the same track in a different cycle. Each of the observed SSHA, true SSHA and observation error is assumed to have the same variance among nearby positions adopted for calculating the correlation function. The true SSHA is assumed to have a correlation function t µ 3 ( x y t)= ( a 1 x + a2y + a3t + a4yt + a5tx + a6xy),, exp, ( A1) as a function of relative position (x, y, t) in space-time domain, where x is the longitudinal distance, y the latitu- 778 T. Kuragano and M. Kamachi

15 Fig. A1. Statistical e-folding scales of the first-guess error estimated from eddy-induced SSHA of ERS. (a) Longitudinal scales, (b) latitudinal scales, (c) timescales at fixed points, (d) persistent timescales, i.e., life times of eddy signals, (e) phase speeds for longitudinal direction, (f) those for latitudinal direction, and (g) RMS variability of the first-guess error. Altimeter s Capability of Reconstructing Realistic Eddy Fields Using Space-Time Optimum Interpolation 779

16 dinal distance, t the time difference, and the a s are the coefficients. From these assumptions, the correlation function of the observed SSHA is expressed as µ 3 ( ) o xyt,, ( ax 1 2 ay 2 2 at 3 2 ayt 4 atx 5 axy 6 a7) ( A2) = exp , where the coefficient a 7 is related to the RMS variability of the observed SSHA (σ o ), and those of the true SSHA (σ t ), expressed as t exp ( a7)= o. σ σ 2 ( A3) The RMS variability of the observation error is expressed as e o t ( σ ) 2 = ( σ ) 2 ( σ ) 2. ( A4 ) At all 2 2 grids covering the global ocean, the above statistics are derived from correlation coefficients of the observed SSHA. The correlation coefficient for the relative location and time, (x, y, t), from a grid was calculated as follows. The observed SSHA in a 2 -latitude 4 -longitude box, the center of which is the grid, is regarded as a datum locating at the grid (see figure 3 in KK2000 for the details). We abbreviate the datum as an origin-side datum. The observed SSHAs at relative locations (x, y, t) from an origin-side datum are adopted as a perimeter-side data. The adopted perimeter-side data are within a relative distance of ±1110 km in the longitudinal direction and ±555 km in the latitudinal direction and a relative period of ±100 days from the origin-side datum. The data pair is not selected if the two data are on the same track in the same cycle. The origin-side datum, with its paired perimeter-side data, is moved to the grid in parallel. The same procedure is conducted for all origin-side data in the 2 4 box. The perimeter-side data are binned to day boxes. The correlation coefficient for each relative location and time from the grid is calculated with all pairs in which the perimeter-side data have the same relative location and time from the origin-side data. The coefficients a 1 to a 7 in Eq. (A2) of the correlation function were determined by fitting to the above correlation coefficients using a least-squares method with an iteration procedure. The correlation coefficient µ 3 t was obtained from Eq. (A1) using the same coefficients a 1 to a 6. The value of σ o was also derived from the observed SSHA; then σ t was obtained from Eq. (A3). Equation (A1) derives the e-folding space and time scales, phase speed of the true SSHA and Eq. (A3) RMS of the true SSHA. Figure 3 shows geographical distributions of those values estimated from TOPEX SSHA, and Fig. A1 from the ERS SSHA for comparison. If the coefficients a 4 and a 5 are set to zero in Eq. (A1), it does not represent propagation of SSHA. If a 3 is set to 1/(15 days) 2, the decorrelation timescale is restricted to 15 days. Equation (A1) with these substitutions is also adopted for the first-guess error scale of the 3D_OI. This OI is referred as non-propagation type OI (NP_OI) in the text, while the OI using the original form of Eq. (A1) is as the advanced OI (AD_OI). Appendix B. Observation Error Figure 4 shows the observation errors estimated from Eq. (A4) from TOPEX and ERS. Observation error was assumed to have no correlation between the data on different tracks, nor on different cycles. However, there may be a correlation only between the data on the same track of the same cycle. A one-dimensional Gaussian function was fitted to the correlation coefficients of the observed SSHA along the track. We can then obtain a correlation function for the total variation of the true SSHA and a correlated error. Eliminating the space-time correlation coefficient estimated by Eq. (A2) from the one-dimensional correlation coefficient along a track derives the correlation coefficient of the observation error along the track. References Archiving, Varidation, and Interpretation of Satellite Oceanographic Data (AVISO) (1997): AVISO user handbook corrected sea surface heights (CORSSHs), 3.0 ed., AVI-NT CN, Toulouse, France, 21 pp. Benada, R. (1997): Merged GDR (TOPEX/POSEIDON) Generation B users handbook, Ver. 2.0, JPL D-11007, Jet Propul. Lab., Pasadena, Calif. Chelton, D. B. and M. G. Schlax (1994): The resolution capacity of an irregularly sampled dataset: With application to Geosat altimeter data. J. Atmos. Ocean. Technol., 11, Colosi, J. and T. Barnett (1990): The characteristic spatial and temporal scales for SLP, SST, and air temperature in the southern hemisphere. J. Appl. Meteor., 29, De Mey, P. and Y. Ménard (1989): Synoptic analysis and dynamical adjustment of GEOS 3 and Seasat altimeter eddy fields in the Northwest Atlantic. J. Geophys. Res., 94, Ducet, N., P.-Y. Le Traon and G. Reverdin (2000): Global highresolution mapping of ocean circulation from TOPEX/ Poseidon and ERS-1 and 2. J. Geophys. Res., 105, Greenslade, D. J. M., D. B. Chelton and M. G. Schlax (1997): The Midlatitude resolution capability of sea level fields constructed from single and multiple satellite altimeter datasets. J. Atmos. Ocean. Technol., 14, T. Kuragano and M. Kamachi