Mesoscale Mapping Capabilities of Multiple-Satellite Altimeter Missions

1208 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME 16

P. Y. L E TRAON AND G. DIBARBOURE CLS Space Oceanography Division, Ramonville St. Agne, France

(Manuscript received 20 April 1998, in ®nal form 11 November 1998)

ABSTRACT The purpose of this paper is to quantify the contribution of merging multiple-satellite altimeter missions to the mesoscale mapping of sea level anomaly (H), and zonal (U) and meridional (V) geostrophic velocities. A space/time suboptimal interpolation method is used to estimate the mean and standard deviation of the H, U, and V mapping errors (as a percentage of signal variance) for different orbit con®gurations. Only existing or planned orbits [TOPEX/Poseidon (T/P), Jason-1, ERS-1/2±ENVISAT, Geosat±GFO] are analyzed. Jason-1 and T/P orbits are assumed to be interleaved. A large number of simulations are performed, including studies of sensitivity to a priori space scales and timescales, noise, and latitude. In all simulations, the Geosat orbit provides the best sea level and velocity mapping for the single-satellite case. In most simulations, the Jason-1±T/P orbit provides the best two-satellite mapping. However, the gain from an optimized two-satellite con®guration (Jason-1 ϩ T/P) compared to a nonoptimized con®guration (T/P ϩ ERS or T/P ϩ Geosat) is small. There is a large improvement when going from one satellite to two satellites. Compared to T/P, the combination of T/P and ERS, for example, reduces the H mean mapping error by a factor of 4 and the standard deviation by a factor of 5. Compared to ERS or even Geosat, the reduction is smaller but still by a factor of more than 2. The H mapping improvement is not as signi®cant when going from two to three or three to four satellites. Compared to the Geosat, ERS, and T/P mean mapping errors, the Jason-1 ϩ T/P mean mapping error is, respectively, reduced by 5%, 9%, and 17% of the signal variance. The reduction in mean mapping error by going from two to three and from three to four satellites is, however, only 1.5% and 0.7% of the signal variance, respectively. These results differ from Greenslade et al. mainly because of the de®nition of resolution adopted in their study. The velocity ®eld mapping is also more demanding in terms of sampling. The U and V mean mapping errors are two to four times larger than the H mapping error. Only a combination of three satellites can actually provide a velocity ®eld mean mapping error below 10% of the signal variance. The mapping of V is also less accurate than the mapping of U but by only 10%±20%, even at low latitudes. These results are con®rmed using model data from the Parallel Ocean Climate Model (POCM). POCM H, U, and V are thus very well reconstructed from along-track altimeter data when at least two satellites are used. The study also shows that the Jason-1± T/P orbit tandem scenario has to be optimized taking into account the other satellites (GFO and ENVISAT). It also con®rms the usually agreed upon main requirement for future altimeter missions: at least two (and preferably three) missions (with one very precise long-term altimeter system to provide a reference for the other missions) are needed.

1. Introduction particular the mesoscale ocean circulation. Several studies have shown that a two-altimeter combination is need- Over the last ®ve years, several altimeter satellites ed for a better representation of the eddy ®eld (e.g., (TOPEX/Poseidon, ERS-1, and ERS-2) have been ¯ying Chassignet et al. 1992; Blayo et al. 1997). More theo- simultaneously. This will recur in the future with the retical analyses of the sampling or resolution capabilities Geosat follow-on (GFO), ENVISAT, and Jason-1 mis- of single- or multiple-altimeter missions have also been sions. While TOPEX/Poseidon (T/P), with its unprec- carried out (e.g., Wunsch 1989; Greenslade et al. 1997). edented accuracy, has provided a new picture of the The results are not always consistent, however, and are ocean, it cannot observe the full spectrum of the sea often sensitive to the method used for assessing the level and ocean circulation variations. Only a combi- contribution of altimeter data. This is particularly true nation of several altimeter missions will resolve the when using data assimilation techniques, whose results main space and timescales of the ocean circulation, in are often representative of the model/assimilation skill rather than the true capability of the observing system. Although the latter techniques should be used ultimately to determine the impact of altimeter data, it is still useful Corresponding author address: Dr. P. Y. Le Traon, CLS Space Oceanography Division, 8-10 rue Hermes, Parc Technologique du and necessary to quantify the contribution of the data Canal, 31526 Ramonville St. Agne, France. themselves. E-mail: [email protected] Recently Greenslade et al. (1997, hereafter GCS97)

᭧ 1999 American Meteorological Society

Unauthenticated | Downloaded 10/02/21 04:15 PM UTC SEPTEMBER 1999 LE TRAON AND DIBARBOURE 1209 analyzed the resolution capability of sea surface height ping error will be realistic only when the a priori co- (SSH) ®elds constructed from single- and multiple-sat- variance functions are well chosen. These functions are ellite altimeter missions. The main criterion was that the rather well known, thanks to statistical analyses of al- expected squared errors over the space±time grid should timeter data (e.g., Le Traon 1991; Stammer 1997). As be homogeneous to within 10% (maximum±minimum they depend on geographical position (e.g., latitude), of errors below 10% of the mean squared error). It was our calculation will not be limited to a single space scale shown that the mapping resolution capabilities of non- and timescale but will use a range of possible values. optimized combined datasets (such as T/P and ERS-1/2) were not signi®cantly better than those of ®elds constructed from T/P alone. The criterion chosen for de- a. Optimal interpolation method ®ning the resolution capability partly explains this un- A space/time suboptimal objective analysis of along- expected conclusion. It requires a very homogeneous track altimeter sea level anomaly (H) data as described mapping error. As a result, only large-scale signals can in Le Traon and Hernandez (1992) or Le Traon et al. be ``resolved'' and GCS97 conclude that mesoscale var- (1998) will be used. This method is similar to a con- iability cannot be mapped with acceptable accuracy with ventional optimal interpolation method (Bretherton et any of the existing or future multiple-altimeter missions. al. 1976) except in the suboptimality. For our applica- Choice of criteria for the assessment of the mapping tion, along-track altimeter data will be selected in a capability is, however, a subjective decision, and we sphere of in¯uence whose radius is equal to the space will argue that mesoscale variability can be rather well scale (L) and timescale (T) of the ocean signal (see the mapped from multiple-altimeter missions. exact de®nitions of L and T below). Along-track data The purpose of this paper is thus to quantify the con- are also undersampled (one point in three) to reduce the tribution of merging one, two, three, and four satellite computational load. Note that results will be similar to altimeter missions to the mesoscale mapping of sea level a full along-track sampling with a measurement noise anomaly (H), and zonal (U) and meridional (V) geo- level variance multiplied by three. We assume here a strophic velocities. It also aims to provide useful results noncorrelated noise of 10% of signal variance for the for the choice of the T/P±Jason-1 tandem mission sce- different satellites (tests will be done with a reduced nario. Optimal interpolation will be used to determine noise level of 2%). the mapping error of the different existing orbit con®g- urations (T/P±Jason-1, ERS-1/2±ENVISAT, Geosat± GFO). We will not take into account the differences in b. Zonal and meridional velocity error budgets and will only analyze the sampling char- We will also analyze the mapping error on the zonal acteristics of the orbits (not the missions). The differ- (U) and meridional (V) geostrophic velocities. This can ences in error budgets could be, however, partly reduced easily be done with the same mapping procedure by prior to or during mapping by using, for example, the modifying the correlation function between the ®eld to more precise mission (T/P and later on Jason-1)asa be mapped (i.e., U or V in place of H) and the obser- reference for the other missions (e.g., Le Traon and Ogor vations (H) (see Le Traon and Hernandez 1992). If we 1998; Le Traon et al. 1998). Results obtained should assume geostrophy, only the H covariance function thus remain valid for missions with different error bud- C(r, t) needs to be known, as the correlation between gets. This also means that the unique contribution of U, V, and H can be directly derived from C (e.g., Le T/P and later on Jason-1 will be to provide a reference Traon and Hernandez 1992). As the covariance between for the other missions (constraint on the large-scale sig- U or V and H is related to the ®rst derivative of C, the nal). The different altimeter missions will then provide estimation of the velocity mapping error is more sen- complementary space/time sampling of the ocean, sitive, however, to the a priori choice of the covariance which is the very subject of this paper. function. This paper is organized as follows. Methods are presented in section 2. Section 3 shows the results and discusses their sensitivity to the a priori choice of space c. Correlation function and time correlation functions, a priori noise, and lati- The following space±time correlation function C(r, t) tude. Model simulations are used in section 4 to verify of the sea level anomaly ®eld already used by Le Traon the results. Main conclusions are given in section 5. and Hernandez (1992) and Le Traon et al. (1998) will be used: 2. Methods C(r, t) ϭ [1 ϩ ar ϩ (1/6)(ar)2 Ϫ (1/6)(ar)3] exp(Ϫar) Optimal interpolation is an effective way of testing ϫ exp(Ϫt2/T2), a sampling strategy (Bretherton et al. 1976). The mapping error depends on the space/time location of the where r is distance, t time, L ϭ 3.34/a the space cor- observation points and on the a priori space and time relation radius (®rst zero crossing of C), and T the tem- covariance function of the signal and noise. The map- poral correlation radius. The nominal values of L and

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T will be set to 150 km and 15 days, respectively, scales complementary sampling, which explains why combin- typical of mesoscale ocean circulation. Tests will be ing two of these satellites enhances the mapping. Com- done with L ranging from 75 to 225 km and T ranging pared with T/P, the combination of T/P and ERS has, from 10 to 20 days. for example, a mean mapping error reduced by a factor of 4 and a standard deviation reduced by a factor of 5. Compared to ERS, the reduction is smaller but still a d. Methods factor of more than 2. Error maps will be calculated on a 10Њϫ10Њ area The two three-satellite con®gurations (T/P ϩ ERS ϩ (between 40Њ±50ЊN) every 5 days over 35 days (eight Geosat and T/P ϩ ERS ϩ Jason-1) have almost the maps) for different satellite orbit con®gurations. The same mapping errors. Compared to the two-satellite con- grid spacing is 0.2Њ. The space and time mean error ®gurations, the mapping error is reduced by about 30%. (expressed as a percentage of signal variance) and the The four-satellite con®guration reduces error slightly standard deviation of the error will then be calculated more again. The main result, however, is that there is for H, U, and V. As an illustration, Fig. 1 shows the a large improvement when going from one to two sat- error maps on H obtained for the Geosat±GFO orbit. ellites. Three or even four satellites are still useful but Tests will also be done on a different latitude band the increase in mapping accuracy from two to three or between 10Њ and 20ЊN to analyze the difference in sam- four is less signi®cant. Compared to the Geosat, ERS, pling according to latitude (in¯uence of intertrack dis- and T/P mean mapping errors, the Jason-1 ϩ T/P mean tance and track inclination). mapping error is, respectively, reduced by 5%, 9%, and Figure 2 shows the tracks of the different satellites 17% of the signal variance. The reduction in mean map- (ERS-1/2 or ENVISAT, T/P, Jason-1, Geosat or GFO). ping error by going from two to three and from three The Jason-1 and T/P orbits are assumed to be inter- to four satellites is, however, only 1.5% and 0.7% of leaved. This is one possible scenario for the tandem the signal variance, respectively. phase of T/P and Jason-1 (Jason-1 will ¯y along the T/P track after the tandem phase). It can be considered, b. Comparison with Greenslade et al. (1997) results however, as an optimal sampling design for a two-satellite con®guration. Another scenario would be to have Contrary to GCS97 results, there is thus a very clear T/P and Jason-1 covering the same ground track, but improvement due to the merging of T/P and ERS and this solution would be less suitable for sampling issues more generally due to the merging of multiple-altimeter (except for some speci®c ocean signals) and will not be missions. Note that the ratio R of the standard deviation analyzed here. of error over the mean mapping error is typically around 0.5 for most of the con®gurations we analyzed. GCS97 require that the ratio maximum±minimum of errors over 3. Results the mean mapping error to be 0.1, which means that a. Sea level anomaly mapping (for Gaussian distribution) they require an R of 0.03. This may be a sensible choice, but, as a result, only Figure 3 gives the sea level anomaly mapping error large-scale signals can be ``resolved.'' To get such a and its standard deviation for the nominal values of L ratio, we need, as in GCS97, to smooth the data, that and T (150 km and 15 days) and for all tested orbit is, to use correlation functions with much larger space con®gurations. Geosat gives the best results for the sin- scales and timescales than the actual (mesoscale) signal. gle-satellite case and provides a good mapping. The GCS97 thus ®nd that the mesoscale signal cannot be mean error is only 10%, which is the a priori along- resolved. One may argue about the resolution of the track error, and the standard deviation of the error is mesoscale signals (see discussion below). Still, it is use- 5%. ERS errors are larger. TOPEX/Poseidon (or Ja- ful to quantify the improvement of the mesoscale map- son-1) has the largest error. This was expected as the ping when merging several altimeter datasets. This is T/P orbit was optimized for the large-scale ocean signal not done in GCS97 but can be extrapolated from results observation and not for mesoscale variability mapping. shown at bottom left corners of their mean RESB and Note that the error structures in these three cases are %RESB ®gures (i.e., at the highest analyzed spatial and very different. Because the repeat period of T/P is short temporal resolutions). There, the comparison of the dif- compared to the a priori timescale T, T/P error is very ferent satellites con®gurations is more in agreement with homogeneous over time but has large spatial variability. our ®ndings. Our conclusions are thus different from ERS errors, and to a lesser extent Geosat, are more those of GCS97 mainly because the contribution of the complex, however, exhibiting both spatial and temporal merging for mesoscale and large-scale mapping is quite variability (see Fig. 1). different. The Jason-1 ϩ T/P combination provides the best According to GCS97 de®nition, none of our con®g- two-satellite con®guration, but there is not much dif- urations (even those with four satellites) can resolve the ference relative to the combinations of T/P ϩ Geosat ocean mesoscale variability. While the paper does not or T/P ϩ ERS. Geosat, T/P, Jason-1, or ERS provide deal with the resolution capability (in that sense, it is

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FIG. 1. Mapping error on sea level anomaly (H) for the Geosat orbit con®guration. There is one map every ®ve days. Units are percentages of signal variance.

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FIG. 2. The study area with the tracks of the different satellite orbits superimposed (T/P is the bold line, Jason-1 is the dotted bold line, ERS±ENVISAT is the dotted line, and Gesoat±GFO is the thin line). less ambitious than the GCS97 approach), we think, Such mapping errors will allow a very good tracking however, that mesoscale signals are rather well or well of most of the eddies since the mapping error will be resolved by merging several altimeter datasets. All the always much smaller than the signal to be mapped. two-satellite con®gurations we analyzed have mean GCS97 requires a ratio R of 0.03, that is, assuming the mapping errors, for example, comparable to that ob- same mean mapping error, a standard deviation of error tained with dense hydrographic arrays (e.g., Hernandez of 0.09%. That is, for this particular example, an error et al. 1995). The mapping errors are always less than between 1.71 and 1.73 cm (which is a very homogenous the a priori along-track error, and the maximum error mapping error). They thus require to have almost ev- is always below 20% of the signal variance. The three erywhere the same mapping error. It is very clear that satellite con®gurations typically have a mean mapping the mesoscale signal will be always better estimated error of 3% of the signal variance and a standard de- along the tracks, but even if the mapping error is in- viation of the error of 1.5% (R ϭ 0.5). This means that creased by a factor of 2 between the tracks, this may for a signal of 10 cm rms, the mapping error will be be acceptable if the error remains below a certain thresh- between 1.2 and 2.1 cm (at one standard deviation). old (compared to the signal variance). As discussed by

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FIG. 3. Mean and standard deviation of sea level anomaly mapping error for single- and multiple-altimeter missions. Units are percentages of signal variance.

GCS97, the mapping error spatial and temporal inho- needed. As a result, the error is more homogeneous but mogeneity can be ignored if the overall mapping error larger than for H. The minimal errors for U and V are is deemed to be acceptably small. GCS97 chose a cri- actually found near crossover points, where sea level terion of 1% for the mean squared mapping error. We gradients are best estimated. The structure of U and V also believe that the spatial and temporal inhomogeneity errors is also different. Near a crossover point, the zonal criterion can be considerably relaxed (but not neces- sea level gradients are best estimated in the north±south sarily ignored) when mean mapping errors are below a direction, which explains the elongated north±south certain threshold. A reasonable and less demandingÐ structure of the V errors. but also subjectiveÐalternative to the GCS97 criterion These main characteristics are found for all tested could thus be to have mapping errors always below 10% orbit con®gurations (Figs. 5 and 6). The structures of of the signal variance, regardless of the inhomogeneity U, V, and H errors are very different; the U and V mean of the mapping error. Our results show that this criterion mapping errors are between two to four times larger is met by all three satellite con®gurations. than for H, and the U and V standard deviations are smaller than the H standard deviation. The ratio of standard deviation to mean mapping error is thus typically c. Zonal and meridional velocity mapping three or four times smaller for U and V compared to H. The structure of the mapping error on U or V and H The ranking of the different con®gurations in terms of is very different. This is exempli®ed in Fig. 4, which sampling is, however, similar. The only slight difference shows the mapping error on H, U, and V for the T/P is that the T/P ϩ Geosat combination has a slightly con®guration. First, the error on U and V is larger (more smaller mean error on V and U compared to the Jason-1 than 40% of the signal variance) than for H (20% of ϩ T/P con®guration. The error on V is always larger signal variance). As expected, estimating a derivative is than the error on U but by 10% only. Calculating the more dif®cult than estimating the signal itself and is two components of the velocity ®eld at crossover points more demanding in terms of sampling (see the appen- yields a V noise variance larger than U noise variance dix). The standard deviation of the error is, however, by a factor of (1/tan␪) 2, where ␪ is the angle between smaller because the error is more evenly distributed in the ground track and the north meridian (which depends space. The along-track data are not suf®cient to estimate on latitude) (e.g., Morrow et al. 1994). For our latitude the velocity ®eld and data from neighboring tracks are band (see also the results for the latitude band between

Unauthenticated | Downloaded 10/02/21 04:15 PM UTC 1214 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME 16 ) variance. V or H, U, for the T/P orbit con®guration. Units are percentages of signal ( V and H, U, . 4. Comparison of the mapping error on IG F

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FIG. 5. Mean and standard deviation of zonal geostrophic velocity (U) mapping error for single- and multiple-altimeter missions. Units are percentages of signal variance.

FIG. 6. Mean and standard deviation of zonal geostrophic velocity (V) mapping error for single- and multiple-altimeter missions. Units are percentages of signal variance

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FIG. 7. Mean of sea level anomaly (H) mapping error for single- and multiple-altimeter missions for different a priori space scales and timescales. Units are percentages of signal variance.

10Њ and 20Њ), this yields a V noise variance almost three generally not modi®ed, although for the case L ϭ 75 times larger than the error on U for the Geosat orbit km, T ϭ 15 days, the T/P ϩ Geosat or T/P ϩ ERS con®guration. The difference in mapping error between con®gurations give better results than the Jason-1 ϩ V and U is much smaller (between 10% and 20%) be- T/P con®guration. The different con®gurations do not cause (i) it represents both the noise and (mainly) the have the same sensitivity, however, to the change in sampling error variances and (ii) the adjacent and cross- correlation function. For example, T/P is virtually in- ing tracks suitably constrain estimates of the zonal sea sensitive to our timescale change, while ERS is more level gradient and thus of the meridional velocity V at sensitive. These characteristics are also found in the any grid point. This also shows that the criterion of a two-satellite con®gurations. satellite orbit with a rather low inclination for a better estimation of the velocity ®eld is not really relevant, in particular for a multisatellite con®guration. e. Sensitivity to noise and latitude Studies of sensitivity to the a priori noise and to the d. Sensitivity to correlation scales latitude were also performed. Figure 8 shows the ratio of mean mapping error with an a priori noise of 2% Four other calculations with different space scales (L) over mean mapping error with an a priori noise of 10%. and timescales (T) were performed to analyze the sen- As expected, the mapping error is reduced when using sitivity of the results to different correlation scales. Two an a priori noise of 2%. The reduction factor is less calculations use different timescales (10 and 20 days) than 2 (although noise is reduced by a factor of 5). This and the same space scale as the nominal case (150 km). factor is larger when more satellites are sampling the The other two use the same timescale as the nominal sea surface and smaller for the velocity ®eld. The dif- case (15 days) and two different space scales (75 and ferences between the one- and two- and two- and three- 225 km). These values cover the most likely ranges of satellite results are thus increased. This is because when mesoscale variability space and timescales (e.g., Le the signal is quite well sampled (e.g., with at least two Traon 1991; Stammer 1997). Results for H mean map- satellites), the percentage of the mapping error due to ping error are given in Fig. 7. The error varies consid- the a priori noise is greater. This also explains why the erably according to the correlation function. For ex- mapping error is reduced less for the velocity ®eld, ample, it is reduced by up to a factor of 6 when going which is more demanding in terms of sampling. The from L ϭ 75 km, T ϭ 15 days to L ϭ 225 km, T ϭ same explanation holds for the larger reduction in error 15 days. The ranking of the different con®gurations is for U compared to V. The Jason-1 ϩ T/P con®guration

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FIG. 8. Ratio of the H mean mapping error with an a priori noise of 2% over the H mean mapping error with an a priori noise of 10%. is also slightly less sensitive than the T/P ϩ ERS or choice of L and T (which are not really realistic in this T/P ϩ Geosat orbit con®guration. With an a priori noise latitude band). This is because the other con®gurations of 2%, the T/P ϩ Geosat orbit con®guration provides clearly do not provide adequate spatial sampling. The a better mapping of the velocity ®eld (U and V) than U and V mapping errors are also proportionally less the Jason-1 ϩ T/P con®guration. This actually was al- sensitive than the H mapping error to latitude, but we ready noted with an a priori noise of 10%, but the effect must remember that the errors on U and V are much is now greater. larger than the error on H. Figure 9 shows the ratio of the H mean mapping error for the latitude band (10Њ±20Њ) over the H mean mapping 4. Veri®cation of results using POCM model data error for the nominal (40Њ±50Њ) latitude band. The mean mapping error is larger in the 10Њ±20Њ band (for the To further illustrate the capability of mesosale signal same a priori space scales and timescales) as the tracks mapping from multiple-altimeter missions, a simulation are farther apart (note, however, that in reality the a using Parallel Ocean Climate Model (POCM) outputs priori space scales are larger, which should reduce the is performed. As in GCS97, the model used is the run effect). The error on the velocity ®eld is also modi®ed 11 of the primitive equation, multilayer model devel- because of the change in the track inclination (the Cor- oped by the Parallel Ocean Program (POP) at Los Al- iolis parameter also is smaller, but this should not impact amos National Laboratory (Dukowicz and Smith 1994). the relative mapping error; see the appendix). The error We chose an area dominated by mesoscale variability on V increases more than the error on U (except for in the Gulf Stream extension between 36Њ and 41ЊN and ERS) because of the change in track inclination. The 65Њ and 55ЊW. The model sea level outputs were ®rst variation in latitude does not affect the different orbit transformed into sea level anomaly data by removing a con®gurations in the same way. The ERS orbit con®g- 3-yr mean. They were then subsampled to obtain sim- uration, in particular, is less sensitive to the change in ulated along-track altimeter datasets for T/P, Jason-1, latitude than T/P and Geosat. This is clearly observed ERS, and GFO. Those simulated datasets were then used for the one- and two-satellite con®gurations. In fact, the to reconstruct the sea level anomaly gridded ®elds using T/P ϩ ERS con®guration is the best two-satellite con- the optimal interpolation method presented in section ®guration for this latitude band and for the nominal 2. The covariance functions are those used for our nom-

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FIG. 9. Ratio of the H mean mapping error for the (10Њ±20Њ) latitude band over the H mean mapping error for the nominal (40Њ±50Њ) latitude band. inal case (L ϭ 150 km, T ϭ 15 days). They are thus and 0.49 cm for T/P, ERS, T/P ϩ ERS, T/P ϩ Jason-1, only approximations of the actual (i.e., here derived and T/P ϩ ERS ϩ GFO, respectively. The signal var- from model ®elds) covariance functions. Such a choice iance is about 50 cm2, which means that the relative is more representative of a real mapping exercise. Note mapping errors (in percentage of signal variance) are that the signal mapping (contrary to the formal error all below 1%, except for T/P. For a statistically more estimation) is not very sensitive to this a priori choice representative estimation, the mapping errors were cal- because the constraint from the data is strong. The noise culated over one year both for the sea level H and the level was set to 1% (i.e., corresponding to near-perfect U and V geostrophic velocities. Results are summarized data). in Tables 1 and 2. They con®rm the results derived from Comparison of the reconstructed ®elds with the ref- the previous example. With two satellites or more, the erence model ®elds allows an estimation of the mapping error on the sea level is less than 1 cm rms or less than error. The main interest of such a simulation is that it 1% of the signal variance. POCM mesoscale ®elds are allows us to visualize the mapping errors, while our thus extremely well resolved in the Gulf Stream exten- formal errors only give a statistical estimation. This can sion with multiple-altimeter missions. More interesting- help to determine whether mapping errors are acceptable ly, the velocity ®elds are also very well reconstructed or not. Figure 10 shows the POCM sea level anomaly with a mapping error of about 1 cm sϪ1 for a signal (relative to a 3-yr mean) for a particular day in the year variance of about 100 cm2 sϪ2 (i.e., a relative error of and the same ®eld but reconstructed from T/P, ERS, T/P about 1% of the signal variance). ϩ ERS, T/P ϩ Jason-1, and T/P ϩ ERS ϩ GFO sim- The mapping capability varies in space and time. ulated along-track data. The POCM signal ranges from Some con®gurations have a mapping error very stable Ϫ24 to ϩ16 cm and corresponds to meanders, rings, or in time (e.g., T/P, T/P ϩ Jason-1), while others have eddies of the Gulf Stream extension. The signal is qual- complex space/time variations of mapping errors (GFO, itatively well recovered with all con®gurations. Only ERS). To complement the previous illustrations, we the T/P case shows clear difference with the reference show the evolution in time of mapping error at a given ®eld. The differences with the reference ®eld are shown location for the different satellite con®gurations. The on Fig. 11. The rms difference is 1.49, 0.66, 0.52, 0.51, point location is situated between two T/P crossovers.

Unauthenticated | Downloaded 10/02/21 04:15 PM UTC SEPTEMBER 1999 LE TRAON AND DIBARBOURE 1219 GFO along-track ϩ ERS ϩ and T/P Jason-1, ϩ ERS, T/P ϩ sampling. Contour interval is 2 cm. . 10. Comparison of POCM sea level anomaly ®eld (reference ®eld) with maps reconstructed from T/P, ERS, T/P IG F

Unauthenticated | Downloaded 10/02/21 04:15 PM UTC 1220 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME 16 GFO. Contour interval is 0.5 cm. ϩ ERS ϩ 1, and T/P Jason- ϩ ERS, T/P ϩ . 11. Differences between the reconstructed maps and the reference ®eld for T/P, ERS, T/P IG F

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FIG. 12. Evolution of the mapping error at a given location (39ЊN, 300.4ЊE) for T/P, ERS, T/P ϩ ERS, T/P ϩ Jason-1, and T/P ϩ ERS ϩ GFO. Units in cm.

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TABLE 1. Root-mean-square mapping error for H, U, and V (dif- TABLE 2. Same as Table 1 but differences are given in percentage ference between POCM sea level anomaly and maps reconstructed of the signal variance. from along-track simulated altimeter data) over one year. Units are in cm for H and in cm sϪ1 for U and V. H (%) U (%) V (%) TP 8.4 13.9 12.8 H (cm) U (cm sϪ1) V (cm sϪ1) ERS 1.5 2.1 3.7 TP 1.9 2.7 3.5 TP ϩ ERS 0.8 1.2 1.6 ERS 0.9 1.2 2.2 TP ϩ Jason-1 0.7 1.1 1.1 TP ϩ ERS 0.7 1.0 1.4 TP ϩ ERS ϩ GFO 0.6 0.9 1.1 TP ϩ Jason-1 0.6 0.9 1.1 TP ϩ ERS ϩ GFO 0.6 0.8 1.2 and velocity mapping for the single-satellite case for all the simulations we performed. The T/P error is, of course, large (3.8 cm rms), ERS R In most simulations the Jason-1 ϩ T/P (interleaved error is smaller (1 cm rms), T/P ϩ Jason-1 and T/P ϩ T/P±Jason-1 tandem orbit scenario) provides the best ERS have similar errors (0.8 and 0.9 cm rms, respec- mapping for the two-satellite case. There are few dif- tively), and the combination of T/P ϩ ERS ϩ GFO has ferences, however, with respect to the T/P ϩ ERS or the smallest error (0.64 cm rms). The two- and three- T/P ϩ Geosat scenario. The T/P ϩ Geosat con®gu- satellite combinations thus have rms errors below 1 cm ration actually performs better for a few simulations, with maximum errors generally below 2 cm. Such map- particularly for the velocity ®eld mapping. The gain ping errors are much lower than the signal, which has provided by an optimized two-satellite (Jason-1 ϩ an rms of 8.5 cm and maximum/minimum values be- T/P) con®guration relative to a nonoptimized con®g- tween Ϫ18 and ϩ15 cm. Thus, the variations in time uration (T/P ϩ ERS or T/P ϩ Geosat) is thus not very of the mapping errors will not be a problem for inter- large. Also, the T/P ϩ ERS or T/P ϩ Geosat con®g- preting the reconstructed ocean signal. These variations urations have the advantages of providing data at are actually more representative of very short timescale higher latitudes. events in the model ®elds rather than time inhomoge- R The conclusions differ from those of Greenslade et neity of the mapping error. Results are different at other al. (1997), mainly because of their de®nition of res- locations and, in particular, the ranking of the different olution (standard deviation of error had to be very con®gurations will change (e.g., at a T/P crossover small compared to mean error). As a result, only the point, T/P will obviously perform better than the com- large-scale signal can be ``resolved.'' Greenslade et bination of Jason-1, ERS, and GFO). They all show, al. (1997) thus show that the T/P ϩ Jason-1 combi- however, that the mapping errors are always suf®ciently nation performs much better than a nonoptimized two- low compared to the ocean signal when at least two satellite combination (such as T/P ϩ ERS) for large- satellites are combined. scale signal mapping. Greenslade et al. (1997) also These simulations thus con®rm the conclusions de- conclude that the mesoscale signal cannot be mapped rived from our formal error analyses. They show that with an acceptable accuracy from present and future the mesoscale signals are well resolved with at least two multiple-altimeter missions. A resolution de®nition is satellites. Mapping errors derived from these simula- a fundamentally subjective choice, however, and our tions are even much smaller than the ones derived from analyses show that the mesoscale signal can be rather our formal error estimates. This can be explained by the well or well mapped from multiple-altimeter missions. large space and timescales of POCM sea level anomaly Although the mapping errors are not homogeneous, ®elds. The very good results on the velocity estimations they remain suf®ciently small relative to the signal. are also probably due to the absence of small-scale sig- They are also a priori known, which should avoid any nals in POCM data. It is clear, however, that the me- misinterpretation of results. soscale variability is not well simulated by POCM (e.g., R There is a large improvement in sea level mapping Stammer et al. 1996). Our previous formal error esti- when going from one satellite to two satellites. Com- mates are thus probably a more reliable estimation of pared to T/P, the combination of T/P and ERS has, the actual mesoscale variability mapping errors. for example, a mean mapping error reduced by a factor of 4 and a standard deviation reduced by a factor of 5. Conclusions 5. Compared to ERS or even Geosat, the reduction is smaller but still a factor of more than 2. The improve- The mesoscale mapping capability of sea level anom- ment in sea level mapping is not as large when going aly, and zonal and meridional velocity by existing or from two to three or three to four satellites. Compared future altimeter missions was analyzed. A large number to the Geosat, ERS, and T/P mean mapping errors, of simulations was performed, including studies of sen- the Jason-1 ϩ T/P mean mapping error is, respec- sitivity to correlation function, a priori noise, and lati- tively, reduced by 5%, 9%, and 17% of the signal tude. The main conclusions are as follows. variance. The reduction in mean mapping error by R The Geosat (or GFO) orbit provides the best sea level going from two to three and from three to four sat-

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ellites is, however, only 1.5% and 0.7% of the signal Hernandez 1992) that, given our choice of C(r, t), the 22 variance, respectively. This also holds if a much H(␴␴Hu ) and U( ) variances are related as follows: smaller a priori noise is assumed (2% instead of 10%). ␴␴22ϭ (2/3)(g/f)2 /(L/3.34)2. (A2) R The velocity ®eld mapping is more demanding, how- uH ever, in terms of sampling. The U and V mean mapping Equations (A1) and (A2) yield 22 2 2 2 errors are two to four times larger than the H mapping ␧uu/␴ ϭ 3/␧ HH␴ [(L/3.34)L1] . (A3) error. Their error structure is also very different from Here, (L/3.34)L should be around unity, which means the H error structure. The contribution from a third 1 the relative mapping error on U (or V) should be typ- satellite is also more signi®cant than for H. Only a ically three times greater than the mapping error on H. combination of three satellites can actually provide a This is only a crude estimation of the expected ratio. In velocity ®eld mapping error below 10% of the signal reality, the situation will be different because of the track variance. Mapping of the meridional velocity is less inclination, the presence of neighboring tracks, and accurate but by only 10% to 20% even at low latitudes. mainly because ␧ cannot be assumed to be uncorre- This suggests that the criterion of a satellite orbit with H lated. a rather low inclination for a better estimation of the velocity ®eld is not really relevant, particularly for a multisatellite con®guration. REFERENCES R These results are con®rmed using POCM model data. Blayo, E., T. Mailly, B. Barnier, P. Brasseur, C. Le Provost, J. M. POCM H, U, and V ®elds in the Gulf Stream extension Molines, and J. Verron, 1997: Complementarity of ERS-1 and are very well reconstructed from along-track altimeter TOPEX/Poseidon altimeter data in estimating the ocean circu- data when at least two satellites are used. This is partly lation: Assimilation into a model of the North Atlantic. J. Geo- phys. Res., 102, 18 573±18 584. due, however, to the large space and timescales of Bretherton, F., R. Davis, and C. Fandry, 1976: A technique for ob- POCM mesoscale variability. jective analysis and design of oceanographic experiments applied to MODE-73. Deep-Sea Res., 23, 559±582. The study also shows that the Jason-1±T/P orbit tan- Chassignet, E. P., W. R. Holland, and A. Capotondi, 1992: Impact of dem scenario should be optimized by taking the other the altimeter orbit on the reproduction of oceanic rings: Appli- satellites (GFO and ENVISAT) into account. It also con- cation to a regional model of the Gulf Stream. Oceanol. Acta, 15, 479±490. ®rms what is generally agreed as being the main re- Dukowicz, J. K., and R. D. Smith, 1994: Implicit free-surface method quirement for future altimeter missions, that is, that at for the Bryan±Cox±Semtner ocean model. J. Geophys. Res., 99, least two (and preferably three) missions are needed 7991±8014. with one very precise long-term altimeter system (such Greenslade, D. J. M., D. B. Chelton, and M. G. Schlax, 1997: The as T/P and later on Jason-1) to provide a reference for midlatitude resolution capability of sea level ®elds constructed from single and multiple satellite altimeter datasets. J. Atmos. the other missions. Oceanic Technol., 14, 849±870. Hernandez, F., P. Y. Le Traon, and R. Morrow, 1995: Mapping me- Acknowledgments. This study was partly supported soscale variability of the Azores current using TOPEX/Poseidon by a CNES contract. We wish to thank Philippe Escudier and ERS-1 altimetry, together with hydrographic and Lagrangian measurements. J. Geophys. Res., 100, 24 995±25 006. and Nicolas Ducet for useful discussions on this work. Le Traon, P. Y., 1991: Time scales of mesoscale variability and their M. Schlax kindly provided us with his own archive of relationship with spatial scales in the North Atlantic. J. Mar. POCM data, which he got from R. Tokmakian. Res., 49, 1±26. , and F. Hernandez, 1992: Mapping of the oceanic mesoscale circulation: Validation of satellite altimetry using surface drift- APPENDIX ers. J. Atmos. Oceanic Technol., 9, 687±698. , and F. Ogor, 1998: ERS-1/2 orbit improvement using TOPEX/ Comparison of Mapping Error on H and U or V Poseidon: The 2-cm challenge. J. Geophys. Res., 103, 8045± 8057. To better understand the relationship between the , F. Nadal, and N. Ducet, 1998: An improved mapping method mapping error on H and U or V, it is instructive to of multisatellite altimeter data. J. Atmos. Oceanic Technol., 15, 522±534. analyze the simple example of a north±south track. In Morrow, R., R. Coleman, J. Church, and D. B. Chelton, 1994: Surface 2 this case,␧u , the variance on the error on u, is simply eddy momentum ¯ux and velocity variances in the Southern given by Ocean from Geosat altimetry. J. Phys. Oceanogr., 24, 2050± 2071. 2222 ␧␧uHϭ 2(g/f) /,L1 (A1) Stammer, D., 1997: Global characteristics of ocean variability estimated from regional TOPEX/Poseidon altimeter measurements. 2 where␧H is the H error variance (assumed to be un- J. Phys. Oceanogr., 27, 1743±1769. correlated) and L is a scale over which the derivative , R. Tokmakian, A. Semtner, and C. Wunsch, 1996: How well 1 does a 1/4Њ global circulation model simulate large-scale oceanic should be calculated. It has to be much smaller than the observations? J. Geophys. Res., 101, 25 779±25 812. scale of the correlation function. Wunsch, C., 1989: Sampling characteristics of satellite orbits. J. At- On the other hand, we can show (e.g., Le Traon and mos. Oceanic Technol., 6, 892±907.

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