QUINC2. Harvest MNPEAK MCDON4 MAZTLN HOLLAS

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1 Orbit analysis for the TOPEX altimeter calibration P.N.A.M. Visser 1 and C.K. Shum University of Texas at Austin Center for Space Research October 1993 Abstract Several orbits have been used in the calibration of the TOPEX altimeter instrument at the Harvest and Lampedusa overies. The TOPEX altimeter bias is estimated to be cm. These values hold for 17 Harvest overies, and for 1-day JGM-2 orbits that were computed using both SLR tracking measurements and satellite altimeter crossover dierences. It was also found that the Harvest and Lampedusa calibration results are in close agreement, although the Lampedusa results are based on only two overies. Furthermore, it was found that the performance of nominal 10-day JGM-2 TOPEX orbits in the TOPEX calibration was almost as good as of the 1-day TOPEX orbits. In addition, a cross calibration between TOPEX, ERS-1 and GEOSAT shows that the TOPEX, ERS-1 and GEOSAT bias estimates are in close agreement with each other. 1 Introduction The altimeter satellite TOPEX/POSEIDON was launched in August The primary objective of the TOPEX/POSEIDON mission is to study global ocean circulation. Part of this study involves an assessment of long-term global sea level change. The study of long-term sea level change requires the combination of altimeter data gathered by an array of altimeter satellites, like GEOS- 3, SEASAT, GEOSAT, ERS-1, and TOPEX/POSEIDON. An absolute reference for all altimeter data is required. In order to achieve this, an accurate calibration of the altimeter instruments is necessary. An important step in the calibration of an altimeter measurement is a precise orbit determination. This report discusses the TOPEX/POSEIDON precise orbit determination close to the Harvest and Lampedusa verication sites. The precise orbits are used to calibrate the TOPEX altimeter instrument. TOPEX/POSEIDON is being tracked by a global network of DORIS Doppler and SLR (Satellite Laser Ranging) stations. In addition, TOPEX/POSEIDON is equipped with a high-precision GPS receiver and serves as a test bed for precise tracking by the Global Positioning System (GPS). TOPEX/POSEIDON orbits have been computed for several tracking combinations, SLRonly, Doppler-only, GPS-only, and SLR and Doppler combined. In some cases, satellite altimeter crossover dierences have also been used in the orbit computation. In general, the orbit is likely to be less constrained at the beginning and end of an orbital arc. Therefore, 1-day orbit arcs have been computed with the middle of the 1-day arc coinciding with the Harvest or Lampedusa overy. The TOPEX calibration based on 1-day orbit arcs will be compared with its calibration based on nominal 10-day orbit arcs to test the previous assumption. An important prerequisite for a precise orbit computation is the availability of a high-accuracy gravity eld model. Besides the JGM-1 and 1 On leave from Delft University of Technology 1

2 empirical tangential drag periodic transverse acceleration periodic normal acceleration 2 per day, estimated 1 per day, estimated 1 per day, estimated Dynamical model geopotential atmospheric drag solar radiation pressure earth radiation pressure dynamic solid earth tide dynamic ocean tide planetary perturbations thermal radiator (Y-bias) JGM-1, JGM-2, JGM-2Tx, GRIM-4C3 DTM model Table 1. TOPEX orbit computation JGM-2 gravity eld models, the JGM-2Tx and GRIM-4C3 have been used to study the sensitivity of the TOPEX calibration to the gravity eld. This report starts with a description of all the dierent TOPEX/POSEIDON orbits used to calibrate the TOPEX altimeter instrument. An accuracy assessment of these orbits will be presented. This will be followed by a description of the method used to incorporate these orbits in the TOPEX altimeter calibration and a discussion of the results. The TOPEX orbit analyses at the Harvest and Lampedusa calibration sites to be described hold for the rst 30 TOPEX repeat cycles. However, for the calibration at Lampedusa only enough tide gauge and altimeter data was available for cycles 4 and 7. For Harvest, this was the case for cycles 2, 3, 5, 7, 8, 10, 11, 13, 15, 17, 18, 19, and 21 through 30. The TOPEX calibration results will be compared with altimeter bias estimates of ERS-1 and GEOSAT. This will be referred to as 'cross-calibration'. 2 TOPEX/POSEIDON orbits The TOPEX/POSEIDON precise orbits have been computed with the UTOPIA program developed at the Center for Space Research (CSR) at the University of Texas (UT). Table 1 gives some information of the TOPEX orbit computation. Besides two empirical drag parameters per day, both 1-cpr (cycle per revolution) transverse and normal acceleration parameters are estimated. Dierent gravity eld models have been applied to study the inuence of gravity modeling on the TOPEX altimeter calibration. Drag and solar radiation models are applied. Perturbations by the planets are taken into account. Both the dynamic eect of solid earth and ocean tides on the TOPEX orbit are modeled. In addition the eect of asymmetric thermal radiation by TOPEX (referred to as Y-bias) is accounted for. The analysis started with the incorporation of 1-day SLR-only orbits using the JGM-1 gravity 2

3 eld model in the estimation of the TOPEX altimeter bias. Some results of this orbit computation are listed in Table 2. For cycles 11 and 21 insucient data was available. The accuracy of the TOPEX calibration is linearly dependent on the radial accuracy of the orbit determination. In order to be able to assess this radial accuracy, the rms of t of altimeter crossover dierences is also listed in Table 2 although in this case these crossover dierences have not been used in the orbit computation. For all results to be discussed, residual altimeter crossover dierences with an absolute value greater than 35 cm are edited. The rms of t of the altimeter crossover dierences is computed by subtracting the computed orbit height above the reference ellipsoid (semi-major axis a e = m, attening = 1./ ) from the altimeter measurements and by applying all available corrections to the altimeter measurements, including instrument corrections, ionospheric and tropospheric delay corrections, solid earth and ocean tide corrections, and an inverse barometric correction. This correction was obtained from the following equations: press = (1000:0 drytrp)=(2:277 (1: + 0:0026 cos(2))) barcor = (?9:948 (press? 1013:3))=10:0 (1) where press is the local atmospheric pressure (mbar), drytrp the dry tropospheric correction (cm), the geodetic latitude (radians), and barcor the inverse barometric correction (cm). No signicant wave height (SWH) correction has been applied. The rms of t for the SLR residuals ranges from 1.2 to 5.1 cm, where this rms of t is dened as the weighted rms of the SLR residuals. In the computation of this rms, all SLR residuals are weighted by a station dependent standard deviation. The rms of t for the crossover residuals are on the order of 10 cm, indicating an upper bound of 7 cm for the rms of the TOPEX radial orbit errors. However, this 10 cm is a sum of radial orbit errors, tide modeling errors, ocean surface variability, and altimeter measurement correction errors. It is therefore expected that the TOPEX radial orbit error is below 7 cm and is on the order of 3-5 cm. However, for cycle 13 at the Harvest overy, the crossover dierence residual rms is equal to 14.2 cm. For the selected 1-day period in cycle 13 only a sparse SLR tracking data set is available accounting for this rather high crossover rms. Besides nominal 1-day SLR-only JGM-1 orbits, 1-day SLR-only JGM-1 orbits have been computed for which the weight of SLR stations that track TOPEX/POSEIDON in the time interval, beginning 15 min before the calibrate site overy and that ends 15 min after, has been increased by a factor 100. In this way, the orbit is 'tied' to the SLR tracking stations positioned close to the calibration site. The question that has to be addressed is whether this approach leads to a more accurate orbit close to the calibration sites. Results of this orbit computation are listed in Table 3. In this case, converged orbits could not be obtained for cycles 11, 17, 19 and 21. The rms of t of both SLR measurements and altimeter crossover dierences has increased signicantly compared to the values listed in Table 2 (in fact, many crossover dierences were edited, because its residual was above the 35-cm threshold). However, it was found that the rms of t of the SLR measurements of the SLR tracking stations close to the calibration sites was decreased, indicating that the overall accuracy of the 1-day orbit may be less, but the accuracy of the orbit close to the calibration sites may be higher. Also shown in Table 3 is the number of SLR tracking stations that have been assigned a higher weight. It may be argued that by increasing the weight of the stations close to Harvest or Lampedusa the orbit computation begins to approach a geometric orbit determination. However, for a geometric orbit determination at least 3 simultaneously tracking SLR stations are necessary. Table 3 shows that this has only been the case for cycles 2, 3, 7, 8,10,24,29 and 30 for the Harvest overy. The SLR tracking stations close to Harvest and Lampedusa are shown in Figures 1 and 2, together with 15 visibility circles. Included also are the TOPEX/POSEIDON ground tracks at 3

4 Cycle number of SLR residuals Crossover residuals number stations nobs rms (cm) nobs rms (cm) 2 (H) (H) (H) (H) (H) (H) (H) (H) (H) (H) (H) (H) (H) (H) (H) (H) (H) (H) (H) (H) (L) (L) Table 2. Statistics of 1-day SLR-only JGM-1 orbits centered at the Harvest (H) and Lampedusa (L) overy 4

5 QUINC2 30 Harvest MNPEAK MCDON4 20 HOLLAS MAZTLN Figure 1. TOPEX/POSEIDON SLR tracking station network at Harvest. Also indicated are the ground track of TOPEX/POSEIDON and 15 visibility circles the Harvest and Lampedusa overies. Comparing Figures 1 and 2 it may be concluded that the geometry of SLR tracking stations at Lampedusa is more favorable for a high-accuracy calibration of the TOPEX altimeter. Moreover, SLR stations are located on both the Eastern and Western sides of the ground track and on both the Northern and Southern side of Lampedusa island. Only 6 SLR tracking stations are adjacent to Harvest and all of them are located on the Western or Northern side of the ground track, except for the SLR station designated by HOLLAS. However, Harvest is located outside the visibility circle of this station. In addition, analyzing the ground track patterns close to Harvest and Lampedusa, it can be seen that altimeter measurements will be available only before the Harvest overy, whereas for Lampedusa they will be available both directly before and after the overy. In the ideal situation, the TOPEX altimeter measures exactly above the tide gauge. However, in reality the altimeter measurements have to be interpolated to this location or to the location along the ground track which is the closest to the tide gauge. For Harvest, positioned close to the land, this becomes more of an extrapolation rather than an interpolation. It is therefore anticipated that a more consistent calibration of TOPEX can be done at Lampedusa. Unfortunately, at the moment only for two Lampedusa overies are all data 5

6 RIGA POTSDM BOROWC RGO WETZL2 ZIMMER GRAZ GRASSE KATSIV SIMEIZ MATERA YIGILC SANFER LAMPED Lamp. HELWAN Figure 2. TOPEX/POSEIDON SLR tracking station network at Lampedusa. Also indicated are the ground track of TOPEX/POSEIDON and 15 visibility circles necessary for a proper calibration available, so that it is dicult to prove the previous assumption. In addition to a network of SLR tracking stations, a network of DORIS Doppler stations is available to track TOPEX/POSEIDON. The Doppler measurement is basically a range-rate measurement or a measurement of the change in range of a satellite to the ground station in a certain time interval. The Doppler measurements have been used to compute 1-day JGM-1 orbits. Results of the orbit computation are listed in Table 4. For cycle 10 at the Harvest overy no DORIS Doppler data was available, and for cycle 7 at the Lampedusa overy not enough DORIS Doppler tracking data was available to support a stable 1-day orbit determination. The rms of t of the Doppler measurements is on the order of 0.55 mm/s and the rms of t of altimeter crossover dierences ranges from 5.8 to 11.3 cm, with the exception of a deviating 19.4 cm for the non-converged orbit for cycle 7 at Lampedusa. Remarkable is the rms of t of 5.8 cm for cycle 8. Comparing these results with the results listed in Table 2, it may be concluded that the radial accuracy of the Doppler-only orbits is comparable to the radial accuracy of the SLR-only orbits. Unfortunately, both the Doppler and SLR tracking is rather sparse for the selected 1-day periods in cycle 11. This may lead to an unreliable estimate of the TOPEX altimeter bias for this cycle. 6

7 Cycle number of high- SLR residuals Crossover residuals number weight stations nobs rms (cm) nobs rms (cm) 2 (H) (H) (H) (H) (H) (H) (H) (H) (H) (H) (H) (H) (H) (H) (H) (H) (H) (H) (L) (L) Table 3. Statistics of 1-day SLR-only JGM-1 orbits centered at the Harvest (H) and Lampedusa (L) overy. The weight of the SLR tracking stations close to the calibration sites was increased 7

8 Cycle number of Doppler residuals Crossover residuals number stations nobs rms (mm/s) nobs rms (cm) 2 (H) (H) (H) (H) (H) (H) (H) (H) (H) (H) (H) (H) (H) (H) (H) (H) (H) (H) (H) (H) (H) (L) Table 4. Statistics of 1-day Doppler-only JGM-1 orbits centered at the Harvest (H) and Lampedusa (L) overy 8

9 An experiment was conducted in which 1-day orbits were computed based on SLR data, with increased weight for the SLR observations close to the Harvest overy, and Doppler data. Results are listed in Table 5. Compared to the orbits without the Doppler data Table 3), the overall rms of the SLR and crossover dierence residuals has improved signicantly, whereas compared to the nominal 1-day SLR-only orbits (Table 2) the rms of the SLR residuals close to the Harvest/Lampedusa overies has been decreased. In addition, a stable orbit could be computed for more cycles. Besides SLR and Doppler tracking, TOPEX/POSEIDON is being tracked by GPS. For cycle 10, a 1-day GPS-only JGM-1 orbit centered at the Harvest overy was computed. This orbit was compared with the 1-day SLR-only JGM-1 orbit. The rms of the radial, along-track and cross-track position dierences is equal to respectively 3.3, 12.6 and 24.2 cm, and the mean to respectively 0.3, 0.0, and 0.6 cm. The rms of t of 71 altimeter crossover dierences is equal to 7.9 cm, compared to 8.6 cm for the SLR-only orbit. In order to overcome a possible sparse tracking by SLR and/or DORIS Doppler stations, orbits have been computed using both SLR data and altimeter crossover dierences. Also orbits have been computed based on all available observations, thus combining SLR measurements, Doppler measurements and altimeter crossover dierences. Results are listed in Tables 6 and 7. Analyzing these Tables, it may be concluded that the incorporation of altimeter crossover dierences in the orbit computation hardly changes the rms of t of the SLR and Doppler measurements. However, if the SLR and/or Doppler tracking is sparse, the inclusion of altimeter crossover dierences in the orbit computation leads to a more stable orbit computation. However, the addition of crossover data to the SLR data alone was not enough to support a stable orbit computation for cycles 11 and 21. Because gravity is the major force driving the TOPEX/POSEIDON orbit, dierent gravity eld models have been used in the orbit computation. Four dierent gravity eld models have been used in computing 1-day orbits based on both SLR measurements and altimeter crossover dierences. Besides JGM-1, JGM-2, JGM-2Tx, and the less-accurate GRIM-4C3 gravity eld models have been used. Comparing the results listed in Tables 6 and 8, it can be concluded that the overall performance of the JGM-2 gravity eld model is a little bit better than of JGM-1. The rms of t of the SLR measurements has slightly decreased. The same is true for the peak values of the rms of t of the altimeter crossover dierences. The overall results using the JGM-2Tx gravity eld model (Table 10) are even better. Analyzing the results listed in Table 11, it may be concluded that the orbits computed with the GRIM-4C3 are of inferior quality, with an rms of t of the SLR measurements as high as 12.6 cm and of the altimeter crossover dierences as high as 18.3 cm, with several crossover dierences edited. Table 9 shows statistics for 1-day JGM-2 orbits using SLR, Doppler and crossover data. Compared to Table 7, the results listed in this Table again show the generally slightly better performance of the JGM-2 gravity eld model as compared to JGM-1. So far, only 1-day orbits have been discussed for which the middle coincides with the Harvest or Lampedusa overy, because it is expected that the orbit is constrained most in the middle of the orbital arc. To be able to conrm the previous assumption, TOPEX altimeter bias estimates have been computed based on nominal 10-day orbits computed at UT/CSR. These orbits are based on both SLR and Doppler measurements and the gravity eld used was JGM-2. Results of these orbit computations are listed in Table 12. The rms of t of the SLR measurements ranges from 3.4 to 6.4 cm. For the Doppler measurements the rms of t ranges from 0.51 to 0.58 mm/s, and for the altimeter crossover dierences from 9.6 to 11.4 cm. Comparing these results with the results of the 1-day JGM-1/JGM-2 orbits listed in Tables 2-10, it may be concluded that the overall accuracy of the 10-day orbits is of the same order of magnitude as the accuracy of the 1-day orbits, with a slight 9

10 Cycle SLR residuals Doppler Crossover number nobs rms (cm) nobs rms (cm) nobs rms (cm) 2 (H) (H) (H) (H) (H) (H) (H) (H) (H) (H) (H) (H) (H) (H) (H) (H) (H) (H) (H) (H) (H) (L) (L) Table 5. Statistics of 1-day SLR/Doppler JGM-1 orbits centered at the Harvest (H) and Lampedusa (L) overy. The weight of the SLR tracking stations close to the calibration sites has been increased 10

11 Cycle SLR residuals Crossover residuals number nobs rms (cm) nobs rms (cm) 2 (H) (H) (H) (H) (H) (H) (H) (H) (H) (H) (H) (H) (H) (H) (H) (H) (H) (H) (H) (H) (L) (L) Table 6. Statistics of 1-day JGM-1 orbits centered at the Harvest (H) and Lampedusa (L) overy using both SLR and altimeter crossover data 11

12 Cycle SLR Doppler Crossover number nobs rms (cm) nobs rms (mm/s) nobs rms (cm) 2 (H) (H) (H) (H) (H) (H) (H) (H) (H) (H) (H) (H) (H) (H) (H) (H) (H) (H) (H) (H) (H) (H) (L) (L) Table 7. Statistics of 1-day JGM-1 orbits centered at the Harvest (H) and Lampedusa (L) overy using both SLR, Doppler, and altimeter crossover data 12

13 Cycle SLR residuals Crossover residuals number nobs rms (cm) nobs rms (cm) 2 (H) (H) (H) (H) (H) (H) (H) (H) (H) (H) (H) (H) (H) (H) (H) (H) (H) (H) (H) (H) (L) (L) Table 8. Statistics of 1-day JGM-2 orbits centered at the Harvest (H) and Lampedusa (L) overy using both SLR and altimeter crossover data 13

14 Cycle SLR Doppler Crossover number nobs rms (cm) nobs rms (mm/s) nobs rms (cm) 2 (H) (H) (H) (H) (H) (H) (H) (H) (H) (H) (H) (H) (H) (H) (H) (H) (H) (H) (H) (H) (H) (H) (L) (L) Table 9. Statistics of 1-day JGM-2 orbits centered at the Harvest (H) and Lampedusa (L) overy using both SLR, Doppler, and altimeter crossover data 14

15 Cycle SLR residuals Crossover residuals number nobs rms (cm) nobs rms (cm) 2 (H) (H) (H) (H) (H) (H) (H) (H) (H) (H) (H) (H) (H) (H) (H) (H) (H) (H) (H) (H) (L) (L) Table 10. Statistics of 1-day JGM-2Tx orbits centered at the Harvest (H) and Lampedusa (L) overy using both SLR and altimeter crossover data 15

16 Cycle SLR residuals Crossover residuals number nobs rms (cm) nobs rms (cm) 2 (H) (H) (H) (H) (H) (H) (H) (H) (H) (H) (H) (H) (H) (H) (H) (H) (H) (H) (H) (H) (L) (L) Table 11. Statistics of 1-day GRIM-4C3 orbits centered at the Harvest (H) and Lampedusa (L) overy using both SLR and altimeter crossover data 16

17 Cycle SLR Doppler Crossover number nobs rms (cm) nobs rms (mm/s) nobs rms (cm) Table 12. Statistics of 10-day JGM-2 orbits using both SLR and Doppler data increase in the rms of t of the SLR measurements. However, the question that has to be answered is which orbits are the most accurate in the radial direction at the Harvest and Lampedusa overies. This question will be addressed in the next Section. 3 TOPEX altimeter bias estimation The altimeter measurements used in the analysis described in this report consist of 1-sec normal points and have been derived from the Geophysical Data Records (GDR) delivered by the Jet Propulsion Laboratory (JPL). In the ideal case, a satellite altimeter measurement is equal to the computed instantaneous distance between the satellite's center of mass and the ocean surface. However, the measurement is subject to many disturbances and corrections which have to be accounted for. The basic altimetry equation reads [Tapley et al., 1982; Zandbergen, 1990]: h = h + h com + h i + h a + h s + h g + h t + h 0? h bias + (2) 17

18 site longitude (deg) latitude (deg) height (m) Lampedusa Harvest Table 13. Geodetic coordinates of the Harvest and Lampedusa tide gauge markers. For Lampedusa several surveys have been done and the height of tide gauge marker can be dierent for each overy The left-hand side h represents the distance between the satellite's center of mass and a suitably selected reference ellipsoid and follows from a precise orbit computation. As indicated before, several orbits have been computed and the inuence of the orbit computation on the estimate for the altimeter bias h bias will be studied. The actual measurement is represented by h, with a value of about 1350 km for TOPEX and a measurement noise level of 2-5 cm. The measurements have to be corrected for the position oset of the altimeter instrument from the satellite's center of mass: h com. This value can be determined accurately before launch. Furthermore, corrections have to be applied to account for instrumental delays: h i. These corrections are on the order of a few decimeters, with sub-decimeter accuracy. The correction term h a represents atmospheric path length delays, i.e. the altimeter measurement has to be reduced by this correction. This term can be divided into a dry tropospheric correction, on the order of 2.3 m with an error of 1-2 cm, a wet tropospheric correction, usually lower than 30 cm with an error of 1-4 cm, and an ionospheric correction, usually lower than 20 cm with an error of 1-3 cm. An additional instrument correction, h s, has to be applied to account for the interaction between the radar pulse and the ocean surface (electromagnetic bias). This correction may be related to the so-called signicant wave height (SWH). This SWH is on the order of a few meters, but in extreme circumstances the SWH may be as large as 20 m. The error of this correction is on the order of 1 % of the SWH, i.e cm. However, on the average the SWH will be below 4 m, thus the error of the correction will be below 4 cm. In fact, it was found that the SWH-value at Harvest ranged from 1.8 to 3.6 m and at Lampedusa from 0.5 to 1.0 m. Thus a 1 % error in the SWH-correction can cause a 0.5 to 3.6 cm error. The geoid height and the heights induced by the solid-earth and ocean tides are represented by respectively h g and h t. Finally, h 0 represents the ocean topography, which is the elevation of the ocean surface above the geoid, as caused primarily by ocean currents. This ocean topography consists of a semi-permanent and a variable part. The sum of h g, h t and h 0, in the following denoted as the instantaneous sea level, has been measured in situ. This was done in the following way. First, the position of GPS markers at Harvest and Lampedusa was determined with high precision by a GPS campaign, and the positions of tide gauge instruments at these sites were determined relative to the GPS markers. This resulted in the geodetic coordinates listed in Table 13. The instantaneous sea level at a Harvest or Lampedusa overy by TOPEX/POSEIDON is obtained by adding the tide gauge reading to the height above the reference ellipsoid of the tide gauge marker. If all these corrections are applied, the remaining altimeter measurement residual gives an estimate of the altimeter bias h bias. The previous description of the altimeter measurement, its corrections and the accuracy assess- 18

19 ment of these corrections hold for a global analysis. In fact, the errors in the altimeter measurement corrections at Harvest and Lampedusa are expected to be smaller [TOPEX/POSEIDON Joint Veri- cation Team, 1992]. The total altimetry error (including tide gauge reading, excluding orbit error) is divided into a xed and variable part. At Harvest, the xed altimetry error is estimated to be 1.8 cm and the variable error 3.6 cm for a single overight. For Lampedusa, no number has been speci- ed for the xed error, the variable altimetry error is estimated to be 2.3/2.9 cm (Summer/Winter). In the calibration, besides altimeter measurement correction errors, errors in the coordinates of the verication site have to be taken into account. For Harvest, the xed error is estimated to be 2.1 cm, mostly GPS survey error, and the variable error 1.2 cm. The variable error is caused by the exibility in the Harvest tower. For Lampedusa, only a xed error equal to the GPS survey error is specied, equal to 0.45 cm. Finally, as indicated in the previous Section, the altimeter measurements have to be interpolated to the position along the ground track which is the closest to the Harvest or Lampedusa calibration sites (Table 13). In the analysis of the IGDR data it was found that the largest distance between the point of closest approach on the ground track and Harvest or Lampedusa was equal to respectively 1080 and 140 m. The slope of the geoid in the relevant direction as computed from the OSU91A geoid [Rapp et al., 1991] was found to be 0.34 and 1.1 cm/km at Harvest and Lampedusa respectively. Thus these slopes can introduce bias estimation errors of less than 4 mm. Summing the previous error estimates, the total calibration errors at Harvest and Lampedusa become 2.8/3.8 (cm) (Fixed/Variable) and 0.45/2.3/2.9 (Fixed/Variable-Summer/Variable- Winter). Thus, the total error (Fixed+Variable) becomes 4.7 (cm) at Harvest and 2.4/3.0 (cm) (Summer/Winter) at Lampedusa, for a single overight. Added to these numbers must be the radial orbit error at Harvest or Lampedusa. After applying all instrument corrections, and ionospheric and tropospheric delay corrections to the altimeter measurements, an altimeter height residual (h) is obtained after subtracting the orbit and geoid heights from these corrected altimeter measurements. The geoid height has been subtracted in order to obtain residuals with a low absolute power. In this way it is expected that the interpolation errors will be small. The altimeter height residual is represented by (sea also equation 2): h = h? h + h com + h i + h a + h s + h g (3) The estimate for the altimeter bias is obtained by rst adding the geoid height (h g ) to the altimeter height residual and then subtracting the height of the tide gauge marker (Table 13) and the tide gauge reading. Altimeter measurements within 60 sec (about 430 km) of the calibration site overy have been selected and used in the interpolation. This means for Harvest a total number of about 60 altimeter measurements, and for Lampedusa about 120. The interpolation applied is based on the concept of least-squares collocation [Moritz, 1980]. The interpolation can be represented by: h int = C st (C tt + D)?1 h (4) where h int is the interpolated altimeter height residual, h the vector of altimeter height residuals computed from the selected altimeter measurements, C st the row of covariances between h int and h, C tt the covariance matrix of h, and D a diagonal matrix whose elements are equal to the measurement standard deviation. The measurement standard deviation was estimated to be 5 cm. The covariances were computed using an exponential covariance function: (h 1 ; h 2 ) = Ae?(=)2 (5) 19

20 where is the distance in location between h 1 and h 2, and the correlation length equal to about 11 km (0.1 ), of the same order of magnitude as the distance of 7 km between two successive altimeter measurements. The amplitude A was chosen equal to 1 m 2 to reect the magnitude of the altimeter height residuals. The previously described method has been applied rst to estimate bias estimates from the Harvest overies using the 10-day JGM-2 SLR/Doppler orbits. For Harvest, two sets of tide gauge readings are available, referred to as NOAA and CU tide gauge measurements and it had to be decided which data set to use for all the other cases. The NOAA tide gauge measurements are 6 min averages, and are obtained by one downward looking acoustic sensor. The NOAA tide gauge value at the overy is obtained by linearly interpolating between the two NOAA 6 min tide gauge values closest to the time of the overy. Instead, the CU tide gauge measurements are listed as about 1 per second data. These measurements can uctuate by more than 30 cm in one second. Therefore, an averaging scheme is applied in which a rectilinear line is tted through 10 min of CU-tide gauge measurements, between 5 minutes before and after the overy. This line is then evaluated at the overy time to deliver the tide gauge value to be used in the calibration. For Lampedusa, only 1 set of 2 min average tide gauge measurements is available. This data sets shows that the Mediterranean close to Lampedusa is much more quiet than the Pacic close to harvest. The tide gauge value at the Lampedusa overies is simply obtained by a linear interpolation between the two tide gauge values closest in time to the overy. Harvest bias estimates using both CU and NOAA tide gauge measurements are listed in Table 14. Indicated are also the mean and rms about mean () of the bias estimates. For cycle 27 no CU tide gauge value was available. Therefore cycle 27 is excluded in the computation of the mean and rms about mean of the bias estimates. Using the NOAA tide gauge measurements leads to a mean bias estimate equal to with an rms about mean of 4.82 cm. Using the CU tide gauge measurements, these values are respectively and 4.33 cm. Considering these numbers, the CU tide gauge values seem to be slightly more reliable than the NOAA values. Especially when looking at the bias estimates for the rst 22 cycles, the CU tide gauge values seem to be more reliable. For cycle 18, the NOAA and CU tide gauge measurements at the overy dier more than 12 cm. If this cycle is not taken into account, the mean bias estimate using CU and NOAA tide gauge measurements becomes respectively and cm, with values for the rms about mean of 4.40 and 4.57 cm. For cycles after 22, the NOAA and CU tide gauge values agree very well, to within -1.8 to 1.0 cm. Therefore, it was decided to use the CU tide gauge values in the following for the Harvest overies, and to use the NOAA tide gauge values for cycle 27 only. The bias estimation procedure has led to TOPEX altimeter bias estimates as listed in Tables 15 and 16. All the estimates listed in Table 15 hold for 1-day JGM-1 orbits. The open spaces or the values between brackets indicate outliers, either caused by problems in the orbit computation (orbits that did not converge) or by other problems, possibly errors in the tide gauge values or errors in the altimeter measurement corrections. These values are not used in the computation of the mean and rms about mean. For the SLR-only (I) orbits the average of the Harvest bias estimates is equal to cm with an rms about mean of 2.99 cm. For Lampedusa these numbers are respectively and 5.49 cm. However, only 2 Lampedusa bias estimates have been made and it is therefore dicult to compute a reliable number for the rms about mean. As shown in Table 15, the mean of the Harvest bias estimates ranges from cm for the SLR-only (I) to for the SLR-only orbits with increased weight of the SLR stations close to the Harvest overy (III). All these values are within twice the computed rms about mean. The lowest rms about mean, 2.72 cm, was found for SLR/Crossover orbits (V). Table 15 further shows that the Harvest bias estimates for orbits designated (III) are deviating, with a relatively high rms about mean of

21 cycle Bias based on Bias based on Dierence NOAA tide gauge (cm) CU tide gauge (cm) (cm) (-21.3) mean mean (excl. cyc. 18) (excl. cyc. 18) Table 14. TOPEX altimeter bias estimates at Harvest based on 10-day SLR/Doppler JGM-2 orbits. Two dierent tide gauge systems are compared 21

22 cm. The Lampedusa mean bias estimates range from for orbits designated (III) to for the Doppler-only orbits. The dierence between the mean of the Lampedusa bias estimates and the mean of the Harvest bias estimates is between 2.3 and 8.5 cm, which is always smaller than 3 times the rms about mean of the Harvest bias estimates. These dierences are also of the same order of magnitude as the estimated error budget for calibrating the TOPEX altimeter at Harvest and Lampedusa. It is therefore concluded that the Harvest results are in close agreement with the Lampedusa results. The values for the mean and rms about mean discussed above hold for dierent values for the number of cycles. Therefore, included in Table 15 are values for the mean and rms about mean for the same 15 cycles for each case. These cycles are 2, 3, 5, 7, 8, 13, 15, 18, 22, 23, 25, 27, 28, 29 and 30. These values are denoted by (H,*). As can be seen in Table 15, the statistics do not change much. The lowest rms about mean is still obtained for the SLR/Crossover orbits (V), 2.81 cm. Table 15 further shows that an approach in which high weights are applied to SLR observations close to the overy does not improve the bias estimation (cases (III) and (IV)). This can be explained by the changing geometry of the SLR tracking stations that actually tracked TOPEX close to the overy. As can be seen in Table 3 the number of these stations varies from 0 to 4. This has as a result a dierent weighting of possible station coordinate errors for each cycle and might introduce additional variability in the bias estimates. In addition, as indicated in the previous Section, all SLR tracking stations are located east of the ground track and this asymmetry might introduce orbit errors. One bias estimate not included in Table 15 is an estimate based on a 1-day GPS-only JGM-1 orbit for cycle 10. This bias estimate is equal to cm, in close agreement with the other bias estimates for cycle 10. With gravity being the major force driving the TOPEX/POSEIDON orbit, dierent gravity eld models were used in the orbit computation. Bias estimates based on the dierent orbits obtained using respectively the JGM-2, JGM-2Tx and GRIM-4C3 gravity eld models are listed in Table 16. Both SLR tracking and altimeter crossover dierences have been used in computing these orbits. The mean of the Harvest bias estimates for these orbits ranges from to -18.2, with a rather high rms about mean for the GRIM-4C3 orbits. It has been shown that with even the use of a less accurate gravity eld model (GRIM-4C3), the mean bias estimate remains about the same. The lowest rms about mean of the bias estimates for the SLR/Crossover orbits holds for the case where the JGM-2 gravity eld model has been used: 2.54 cm, although this value is close to the values using JGM-2Tx and JGM-1 (Table 15), respectively 2.68 and 2.72 cm. The Lampedusa bias estimates are in close agreement with the Harvest bias estimates. Taking only the 15 selected cycles, the rms about mean for the 1-day JGM-2 SLR/Crossover orbits is 2.68 cm, with a value of cm for the mean. Finally, included in Table 16 are bias estimates based on nominal 10-day JGM-2 SLR/Doppler orbits. The rms about mean of the Harvest bias estimates, 2.70 cm, compares very well with the rms about mean for the most accurate 1-day orbits. The mean, cm, is still in close agreement with the other listed values. For Lampedusa, the mean of the bias estimates is cm, within twice the rms about mean of the Harvest estimates. The latter results indicate that only minor improvements in the bias estimates can be expected if instead of the nominal 10-day JGM-2 SLR/Doppler TOPEX orbits 1-day orbits are computed. Most values for the rms about mean of the mean altimeter bias estimates are comparable to (and sometimes even smaller than) the estimates of the variable calibration errors at Harvest, 3.8 (cm), and Lampedusa, 2.3/2.9 (cm) (Summer/Winter). As mentioned before, in the computation of these values possible orbit errors were not taken into account. 22

23 Cycle Bias Bias Bias Bias Bias Bias number I (cm) II (cm) III (cm) IV (cm) V (cm ) VI (cm) 2 (H) (H) (H) (H) (H) (H) (-27.7) { (-31.5) { (-27.7) (-27.7) 11 (H) (-17.9) ( -2.9) (-20.6) (H) (H) (H) (-27.7) (H) (H) (-29.5) (H) (-21.3) (-24.3) (-19.3) (H) (H) (H) ( -9.1) (-11.1) (-12.1) (-11.9) ( -8.9) ( -9.9) 25 (H) (H) ( -6.0) ( -8.0) (-11.0) (-10.0) ( -5.0) ( -7.0) 27 (H) (H) (H) (H) mean (H) (H) mean (H,*) (H,*) (L) (L) (- 0.6) mean (L) (L) (I) SLR-only (II) Doppler-only (III) SLR-only, increased weight adjacent to Harvest (IV) = (III) + Doppler observations (V) SLR and crossover dierences (VI) SLR, Doppler, and crossover dierences Table 15. TOPEX altimeter bias estimates at Harvest (H) and Lampedusa (L) based on 1-day JGM-1 orbits. The values denoted by (H,*) are based on 15 selected cycles. 23

24 Cycle Bias Bias Bias Bias Bias number I (cm) II (cm) III (cm) IV (cm) (V) (cm) 2 (H) (H) (H) (H) (H) (H) (-29.3) (-27.7) (-29.7) (-29.3) (-29.3) 11 (H) (-23.9) (-25.7) (-23.7) (H) (H) (H) (H) (H) (H) (-20.1) (-22.6) (-20.3) (H) (H) (H) (-11.1) ( -9.1) (-12.1) (-11.1) (-11.9) 25 (H) (H) ( -7.0) ( -6.0) ( -7.0) ( -9.0) ( -9.0) 27 (H) (H) (H) (H) mean (H) (H) mean (H,*) (H,*) (L) (L) mean (L) (L) (I) 1-day SLR/Crossover JGM-2 orbit (II) 1-day SLR/Crossover GRIM-4C3 orbit (III) 1-day SLR/Crossover JGM-2Tx orbit (IV) 10-day SLR/Doppler JGM-2 orbit (V) 1-day SLR/Doppler/Crossover JGM-2 orbit Table 16. TOPEX altimeter bias estimates at Harvest (H) and Lampedusa (L). The values denoted by (H,*) are based on 15 selected cycles. 24

25 4 Cross calibration TOPEX/POSEIDON is only one in an array of altimeter satellites. This opens the possibility of a so-called cross calibration between the TOPEX altimeter and other altimeters. The TOPEX bias estimate has been compared with bias estimates of the GEOSAT and ERS-1 altimeters. As a reference, the Harvest TOPEX bias estimate with the lowest rms about mean, 2.54 cm (Table 16), will be used in the cross calibration. The cross calibration has been performed by the following procedure. First, residual sea heights have been computed. These residual sea heights can be represented as: h = h? h + h com + h i + h a + h s + h g + h t + h SST (6) In fact, the residual sea heights can be obtained by subtracting the solid earth and ocean tide corrections from the altimeter residuals (equation 3), and by subtracting the height of the dynamic sea surface topography h SST. This dynamic sea surface topography is computed using the JGM2.SST T model complete to degree and order 25 [CSR, priv. comm.]. The JGM-2 gravity eld model complete to degree and order 70 was used to compute the geoid heights h g. The residual sea heights were used to compute altimeter normal points. Altimeter normal points have been computed from ERS-1 IGDR's, GEOSAT GDR's and TOPEX GDR's. Altimeter normal points were computed by tting a straight line to corrected 1 per second measurements in 10 second windows. A normal point was edited if less than 4 measurements were present in a 10 seconds window or if the rms of t of these measurements to the straight line was greater than 0.15 cm. In addition, normal points were edited if the absolute value of the high-frequency geoid height was larger than 2.0 m. The high-frequency geoid was computed using the OSU91A geoid model above degree 70 [Rapp et al., 1991]. The normal points were corrected using this high-frequency geoid. Normal points were also edited if the geoid gradient was above 5.0 m/degree, the depth of the ocean at the subsatellite point was smaller than 500 m, and if the absolute value of the dierence between the mean surface and the OSU91A geoid was larger than 1.5 m. Finally, normal points were edited if no tide information was available and if the normal point was located in an area with high sea surface variability. In the normal point computation, the best available ERS-1 JGM-2 orbits at CSR were used, with a residual altimeter crossover dierence rms of about 16 cm. For TOPEX, the operational 10-day Doppler/SLR JGM-2 orbit (Table 12) was used, and for GEOSAT the best available CSR TEG-2B orbits were used, with a crossover rms of about 16 cm. Altimeter normal points were computed from day periods of ERS-1 data (04/14/92-04/25/92), from 1 cycle of TOPEX data (cycle 10 from 12/21/92 to 12/31/92), and from 1 cycle of GEOSAT data (cycle 4 from 12/29/86-01/15/87). This resulted in 22,430 altimeter normal points for ERS-1, 38,295 for TOPEX, and 57,210 for GEOSAT. The mean of these altimeter normal points, in the following referred to as global mean, are listed in Table 17. The mean is equal to -56.0, -35.2, and for ERS-1, TOPEX, and GEOSAT, respectively. It must be noted that in the computation of these values the inverse barometric correction was applied. No SWH corrections were applied. An analysis of single satellite ERS-1, TOPEX and GEOSAT altimeter crossover dierences indicate SWH corrections equal to 1.8 %, 0.0 % and 2.3 % respectively. The global mean of the SWH-values on the used ERS-1, TOPEX and GEOSAT data is equal to respectively 2.66, 2.51 and 2.51 m. A column is added to Table 17 showing the eect of these SWH corrections. The mean is now equal to respectively -60.8, -35.2, and cm for ERS-1, TOPEX and GEOSAT. These values are relative to a reference ellipsoid with a semi-major axis equal to m. 25