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1.
TOPEX/Poseidon is a well known success, with the operational altimeter (TOPEX) and the experimental one (Poseidon-1), providing data of unprecedented quality. However, there are two major differences between the TOPEX and Poseidon-1 radar altimeters on board TOPEX/Poseidon. The first is related to the estimated range noise; the second is linked to the sea-state bias (SSB) model estimates. Since the recent launch of the Jason-1 radar altimeter (also called Poseidon-2), we have been cross-comparing these three systems to better characterize each of them. Analyzing standard user products, we have found that Jason-1 is behaving like Poseidon-1 and thus shows the same observed differences when compared with TOPEX. A comparative analysis of their features was performed, starting from the on-board acquisition of the ocean return and ending with the ground generation of the high level accuracy oceanographic product. The results lead us to believe that the sources for these differences lie in both the waveform tracking processing and the presence or abscence of a retracking procedure whether on-board or on ground. Because Poseidon-1 and Jason-1 waveforms are retracked while TOPEX waveforms are not in the products distributed to the users, we have applied the same ground retracking algorithm to the waveforms of the three radar altimeters to get consistent data sets. The analysis of the outputs has shown that: (a) the noise level for the three radar altimeters is definitively the same, and (b) the source of the relative SSB between Jason-1 and TOPEX lies in the different behavior of the on-board tracking softwares.  相似文献   

2.
Poseidon-2 is the dual frequency radar altimeter embarked on the CNES/NASA oceanographic satellite Jason-1 that was launched on 7 December 2001. The primary objective of the Jason-1 mission is to continue the high accuracy time series of altimeter measurements that began with TOPEX in 1992. To achieve this goal, it is necessary to improve each component of the ground processing continually. Among these components are the look-up correction tables that are used to correct the estimations (range, significant waveheight, and sigma naught) issued from the retracking algorithms (on-board and ground). Look-up tables were first computed taking into account the prelaunch characteristics of the altimeter. They have to be updated to take into account better all the in-flight characteristics of the altimeter and all the updated ground algorithms that can impact the estimation process. The aim of this article is to describe the radar altimeter simulator of performances that has been used to compute look-up tables, to display the freshly computed look-up tables, and to discuss the consequences of these new corrections on the products provided to the users. The updated look-up correction tables allow improvement of SWH estimation, in particular with respect to TOPEX SWH data. It is also shown that no range dependency on SWH has to be looked for in these tables, and that the on-board TOPEX and Poseidon-2 tracking systems may contain the differences explaining the relative sea state bias between both altimeters.  相似文献   

3.
Jason-1 Altimeter Ground Processing Look-Up Correction Tables   总被引：1，自引：0，他引：1  
Poseidon-2 is the dual frequency radar altimeter embarked on the CNES/NASA oceanographic satellite Jason-1 that was launched on 7 December 2001. The primary objective of the Jason-1 mission is to continue the high accuracy time series of altimeter measurements that began with TOPEX in 1992. To achieve this goal, it is necessary to improve each component of the ground processing continually. Among these components are the look-up correction tables that are used to correct the estimations (range, significant waveheight, and sigma naught) issued from the retracking algorithms (on-board and ground). Look-up tables were first computed taking into account the prelaunch characteristics of the altimeter. They have to be updated to take into account better all the in-flight characteristics of the altimeter and all the updated ground algorithms that can impact the estimation process. The aim of this article is to describe the radar altimeter simulator of performances that has been used to compute look-up tables, to display the freshly computed look-up tables, and to discuss the consequences of these new corrections on the products provided to the users. The updated look-up correction tables allow improvement of SWH estimation, in particular with respect to TOPEX SWH data. It is also shown that no range dependency on SWH has to be looked for in these tables, and that the on-board TOPEX and Poseidon-2 tracking systems may contain the differences explaining the relative sea state bias between both altimeters.  相似文献   

4.
《Marine Geodesy》2013,36(3-4):187-199
The Jason-1 satellite altimeter mission represents a first step towards operational oceanography from satellite altimeter missions. An operational data product, the Operational Sensor Data Record (OSDR), provides measurements from the on-board altimeter and radiometer within 3–5 h of real time. This data product is a wind and wave product that is aimed towards near-real–time meteorological applications. A higher accuracy and more detailed data product, the Interim Geophysical Data Record (IGDR), that is better suited to detailed scientific studies of ocean topography, is available no sooner than 2–3 days from real time. The measurements reported on the OSDR primarily differ from those on the IGDR in that the OSDR reports measurements derived from on-board processing of the altimeter waveforms, while ground retracking of the waveforms is performed for the IGDR. The altimeter-derived measurements on the OSDR are validated through a statistical evaluation of the differences between data on the OSDR and IGDR. In doing so, the impact of ground retracking of the altimeter waveforms is also illustrated.  相似文献   

5.
The Jason-1 satellite altimeter mission represents a first step towards operational oceanography from satellite altimeter missions. An operational data product, the Operational Sensor Data Record (OSDR), provides measurements from the on-board altimeter and radiometer within 3-5 h of real time. This data product is a wind and wave product that is aimed towards near-real-time meteorological applications. A higher accuracy and more detailed data product, the Interim Geophysical Data Record (IGDR), that is better suited to detailed scientific studies of ocean topography, is available no sooner than 2-3 days from real time. The measurements reported on the OSDR primarily differ from those on the IGDR in that the OSDR reports measurements derived from on-board processing of the altimeter waveforms, while ground retracking of the waveforms is performed for the IGDR. The altimeter-derived measurements on the OSDR are validated through a statistical evaluation of the differences between data on the OSDR and IGDR. In doing so, the impact of ground retracking of the altimeter waveforms is also illustrated.  相似文献   

6.
《Marine Geodesy》2013,36(3-4):367-382
The verification phase of the Jason-1 satellite altimeter mission presents a unique opportunity for comparing near-simultaneous, independent satellite measurements. Here we examine simultaneous significant wave height measurements by the Jason-1 and TOPEX/Poseidon altimeters. These data are also compared with in situ measurements from deep-ocean buoys and with predicted wave heights from the Wave Watch III operational model. The rms difference between Jason and TOPEX wave heights is 28 cm, and this can be lowered by half through improved outlier editing and filtering of high-frequency noise. Noise is slightly larger in the Jason dataset, exceeding TOPEX by about 7 cm rms at frequencies above 0.05 Hz, which is the frequency at which the coherence between TOPEX and Jason measurements drops to zero. Jason wave heights are more prone to outliers, especially during periods of moderate to high backscatter. Buoy comparisons confirm previous reports that TOPEX wave heights are roughly 5% smaller than buoy measurements for waves between 2 and 5 m; Jason heights in general are 3% smaller than TOPEX. Spurious dips in the TOPEX density function for 3- and 6-m waves, a problem that has existed since the beginning of the mission, can be solved by waveform retracking.  相似文献   

7.
《Marine Geodesy》2013,36(3-4):355-366
Sea surface slope computed from along-track Jason-1 and TOPEX/POSEIDON (T/P) altimeter data at ocean mesoscale wavelengths are compared to determine the equivalent 1 Hz instrument height noise of the Poseidon-2 and TOPEX altimeters. This geophysical evaluation shows that the Ku-band 1-Hz range noise for both instruments is better than 1.7 cm at 2 m significant wave heights (H1/3), exceeding error budget requirements for both missions. Furthermore, we show that the quality of these instruments allows optimal filtering of the 1-Hz along-track sea surface height data for sea surface slopes that can be used to calculate cross track geostrophic velocity anomalies at the baroclinic Rossby radius of deformation to better than 5 cm/sec precision along 87.5% of the satellite ground track between 2 and 60 degrees absolute latitude over the deep abyssal ocean (depths greater than 1000 m). This level of precision will facilitate scientific studies of surface geostrophic velocity variability using data from the Jason-1 and T/P Tandem Mission.  相似文献   

8.
After two years of verification and validation activities of the Jason-1 altimeter data, it appears that all the mission specifications are completely fulfilled. Performances of all instruments embarked onboard the platform meet all the requirements of the mission. However, the star tracker system has shown some occasional abnormal behavior leading to mispointing angles out of the range of Jason-1 system specification which states that the altimeter antenna shall be pointed to the nadir direction with an accuracy below 0.2 degree (3 sigma). This article discusses the platform attitude angle and its consequences on the altimetric estimates. We propose improvements of the Jason-1 retracking process to better account for attitude effects.

The first star tracker anomalies for the Jason-1 mission were detected in April 2002. The Poseidon-2 algorithms were specified assuming an antenna off-nadir angle smaller than 0.3 degree. For higher values, the current method to estimate the ocean parameters is known to be inaccurate. Thus, the algorithm has to be reviewed, and more specifically, the present altimeter echo model has to be modified to meet the desired instrument performance.

Therefore, we derive a second order analytical model of the altimeter echo to take into account attitude angles up to 0.8 degree, and consequently, we adapt the retracking algorithm. This new model is tested on theoretical simulated data using a maximum likelihood estimation. Biases and noise performance characteristics are computed for the different estimated parameters. They are compared to the ones obtained with the current algorithm. This new method provides highly improved estimations for high attitude angles. It is statistically validated on real data by applying it on several cycles of Poseidon-2 raw measurements. The results are found to be consistent with those obtained from simulations. They also fully agree with the TOPEX estimates when flying along the same ground track. Finally, the estimates are also in agreement with the ones available in the current I/GDR (Intermediate Geophysical Data Record) products when mispointing lies in the mission specifications.  相似文献   

9.
After two years of verification and validation activities of the Jason-1 altimeter data, it appears that all the mission specifications are completely fulfilled. Performances of all instruments embarked onboard the platform meet all the requirements of the mission. However, the star tracker system has shown some occasional abnormal behavior leading to mispointing angles out of the range of Jason-1 system specification which states that the altimeter antenna shall be pointed to the nadir direction with an accuracy below 0.2 degree (3 sigma). This article discusses the platform attitude angle and its consequences on the altimetric estimates. We propose improvements of the Jason-1 retracking process to better account for attitude effects.

The first star tracker anomalies for the Jason-1 mission were detected in April 2002. The Poseidon-2 algorithms were specified assuming an antenna off-nadir angle smaller than 0.3 degree. For higher values, the current method to estimate the ocean parameters is known to be inaccurate. Thus, the algorithm has to be reviewed, and more specifically, the present altimeter echo model has to be modified to meet the desired instrument performance.

Therefore, we derive a second order analytical model of the altimeter echo to take into account attitude angles up to 0.8 degree, and consequently, we adapt the retracking algorithm. This new model is tested on theoretical simulated data using a maximum likelihood estimation. Biases and noise performance characteristics are computed for the different estimated parameters. They are compared to the ones obtained with the current algorithm. This new method provides highly improved estimations for high attitude angles. It is statistically validated on real data by applying it on several cycles of Poseidon-2 raw measurements. The results are found to be consistent with those obtained from simulations. They also fully agree with the TOPEX estimates when flying along the same ground track. Finally, the estimates are also in agreement with the ones available in the current I/GDR (Intermediate Geophysical Data Record) products when mispointing lies in the mission specifications.  相似文献   

10.
Sea surface slope computed from along-track Jason-1 and TOPEX/POSEIDON (T/P) altimeter data at ocean mesoscale wavelengths are compared to determine the equivalent 1 Hz instrument height noise of the Poseidon-2 and TOPEX altimeters. This geophysical evaluation shows that the Ku-band 1-Hz range noise for both instruments is better than 1.7 cm at 2 m significant wave heights (H1/3), exceeding error budget requirements for both missions. Furthermore, we show that the quality of these instruments allows optimal filtering of the 1-Hz along-track sea surface height data for sea surface slopes that can be used to calculate cross track geostrophic velocity anomalies at the baroclinic Rossby radius of deformation to better than 5 cm/sec precision along 87.5% of the satellite ground track between 2 and 60 degrees absolute latitude over the deep abyssal ocean (depths greater than 1000 m). This level of precision will facilitate scientific studies of surface geostrophic velocity variability using data from the Jason-1 and T/P Tandem Mission.  相似文献   

11.
《Marine Geodesy》2013,36(3-4):261-284
The double geodetic Corsica site, which includes Ajaccio-Aspretto and Cape Senetosa (40 km south Ajaccio) in the western Mediterranean area, has been chosen to permit the absolute calibration of radar altimeters. It has been developed since 1998 at Cape Senetosa and, in addition to the use of classical tide gauges, a GPS buoy is deployed every 10 days under the satellites ground track (10 km off shore) since 2000. The 2002 absolute calibration campaign made from January to September in Corsica revealed the necessity of deploying different geodetic techniques on a dedicated site to reach an accuracy level of a few mm: in particular, the French Transportable Laser Ranging System (FTLRS) for accurate orbit determination, and various geodetic equipment as well as a local marine geoid, for monitoring the local sea level and mean sea level. TOPEX/Poseidon altimeter calibration has been performed from cycle 208 to 365 using M-GDR products, whereas Jason-1 altimeter calibration used cycles from 1 to 45 using I-GDR products. For Jason-1, improved estimates of sea-state bias and columnar atmospheric wet path delay as well as the most precise orbits available have been used. The goal of this article is to give synthetic results of the analysis of the different error sources for the tandem phase and for the whole studied period, as geophysical corrections, orbits and reference frame, sea level, and finally altimeter biases. Results are at the millimeter level when considering one year of continuous monitoring; they show a great consistency between both satellites with biases of 6 ± 3 mm (ALT-B) and 120 ± 7 mm, respectively, for TOPEX/Poseidon and Jason-1.  相似文献   

12.
The double geodetic Corsica site, which includes Ajaccio-Aspretto and Cape Senetosa (40 km south Ajaccio) in the western Mediterranean area, has been chosen to permit the absolute calibration of radar altimeters. It has been developed since 1998 at Cape Senetosa and, in addition to the use of classical tide gauges, a GPS buoy is deployed every 10 days under the satellites ground track (10 km off shore) since 2000. The 2002 absolute calibration campaign made from January to September in Corsica revealed the necessity of deploying different geodetic techniques on a dedicated site to reach an accuracy level of a few mm: in particular, the French Transportable Laser Ranging System (FTLRS) for accurate orbit determination, and various geodetic equipment as well as a local marine geoid, for monitoring the local sea level and mean sea level. TOPEX/Poseidon altimeter calibration has been performed from cycle 208 to 365 using M-GDR products, whereas Jason-1 altimeter calibration used cycles from 1 to 45 using I-GDR products. For Jason-1, improved estimates of sea-state bias and columnar atmospheric wet path delay as well as the most precise orbits available have been used. The goal of this article is to give synthetic results of the analysis of the different error sources for the tandem phase and for the whole studied period, as geophysical corrections, orbits and reference frame, sea level, and finally altimeter biases. Results are at the millimeter level when considering one year of continuous monitoring; they show a great consistency between both satellites with biases of 6 ± 3 mm (ALT-B) and 120 ± 7 mm, respectively, for TOPEX/Poseidon and Jason-1.  相似文献   

13.
The Jason-1 sea state bias (SSB) is analyzed in depth from the first year of GDR products. Compared to previous missions, this work benefits from two aspects of the empirical determination of the SSB from the altimetric data themselves. First, from a methodological point of view, a nonparametric technique (NP) has been developed and largely tested on TOPEX/Poseidon 1, GFO and Envisat data. The NP estimator has proven to be a useful tool in the SSB estimation, and it is now mature enough to be used for a refined analysis. On the other hand, the SSB can be extracted from three different data sets (crossovers, collinear data, and residuals) with different characteristics. It is then possible to cross calibrate various estimations of the SSB models and to determine the most accurate one. A systematic comparison is made between these different estimates for the Jason-1 altimeter. The collinear and crossover data sets yield very similar estimates despite their difference of spatial and temporal distributions. These SSB models assure consistency with the TOPEX mission when comparing Jason-1 and TOPEX residuals during the tandem phase. Thanks to the present work, the impact of the short wavelengths filtering on the SSB estimation is evidenced. More generally, our understanding of potential errors affecting the sea surface height and their impact onto the SSB estimation is also improved.  相似文献   

14.
The Jason-1 sea state bias (SSB) is analyzed in depth from the first year of GDR products. Compared to previous missions, this work benefits from two aspects of the empirical determination of the SSB from the altimetric data themselves. First, from a methodological point of view, a nonparametric technique (NP) has been developed and largely tested on TOPEX/Poseidon 1, GFO and Envisat data. The NP estimator has proven to be a useful tool in the SSB estimation, and it is now mature enough to be used for a refined analysis. On the other hand, the SSB can be extracted from three different data sets (crossovers, collinear data, and residuals) with different characteristics. It is then possible to cross calibrate various estimations of the SSB models and to determine the most accurate one. A systematic comparison is made between these different estimates for the Jason-1 altimeter. The collinear and crossover data sets yield very similar estimates despite their difference of spatial and temporal distributions. These SSB models assure consistency with the TOPEX mission when comparing Jason-1 and TOPEX residuals during the tandem phase. Thanks to the present work, the impact of the short wavelengths filtering on the SSB estimation is evidenced. More generally, our understanding of potential errors affecting the sea surface height and their impact onto the SSB estimation is also improved.  相似文献   

15.
The Jason-1 satellite was launched on 7 December 2001 with the primary objective of continuing the high accuracy time series of altimeter measurements that began with the TOPEX/Poseidon mission in 1992. To achieve this goal, it is necessary to validate the performance of the Jason-1 measurement system, and to verify that its error budget is at least at the same level as that of the TOPEX/Poseidon mission. The article reviews the main components of the Jason-1 altimetric error budget from instrument characterization to the geophysical use of the data. Using the Interim Geophysical Data Records (16DR) that were distributed to the Jason-1 Science Working Team during the verification phase of the mission, it is shown that the Jason-1 mission is performing well enough to continue studies of the large-scale features of the ocean, and especially to continue time series of mean sea-level variations with an accuracy comparable to TOPEX/Poseidon.  相似文献   

16.
The verification phase of the Jason-1 satellite altimeter mission presents a unique opportunity for comparing near-simultaneous, independent satellite measurements. Here we examine simultaneous significant wave height measurements by the Jason-1 and TOPEX/Poseidon altimeters. These data are also compared with in situ measurements from deep-ocean buoys and with predicted wave heights from the Wave Watch III operational model. The rms difference between Jason and TOPEX wave heights is 28 cm, and this can be lowered by half through improved outlier editing and filtering of high-frequency noise. Noise is slightly larger in the Jason dataset, exceeding TOPEX by about 7 cm rms at frequencies above 0.05 Hz, which is the frequency at which the coherence between TOPEX and Jason measurements drops to zero. Jason wave heights are more prone to outliers, especially during periods of moderate to high backscatter. Buoy comparisons confirm previous reports that TOPEX wave heights are roughly 5% smaller than buoy measurements for waves between 2 and 5 m; Jason heights in general are 3% smaller than TOPEX. Spurious dips in the TOPEX density function for 3- and 6-m waves, a problem that has existed since the beginning of the mission, can be solved by waveform retracking.  相似文献   

17.
《Marine Geodesy》2013,36(3-4):399-421
The Jason-1 radar altimeter satellite, launched on December 7, 2001 is the follow on to the highly successful TOPEX/Poseidon (T/P) mission and will continue the time series of centimeter level ocean topography measurements. Orbit error is a major component in the overall error budget of all altimeter satellite missions. Jason-1 is no exception and has set a 1-cm radial orbit accuracy goal, which represents a factor of two improvement over what is currently being achieved for T/P. The challenge to precision orbit determination (POD) is both achieving the 1-cm radial orbit accuracy and evaluating the performance of the 1-cm orbit. There is reason to hope such an improvement is possible. The early years of T/P showed that GPS tracking data collected by an on-board receiver holds great promise for precise orbit determination. In the years following the T/P launch there have been several enhancements to GPS, improving its POD capability. In addition, Jason-1 carries aboard an enhanced GPS receiver and significantly improved SLR and DORIS tracking systems along with the altimeter itself. In this article we demonstrate the 1-cm radial orbit accuracy goal has been achieved using GPS data alone in a reduced dynamic solution. It is also shown that adding SLR data to the GPS-based solutions improves the orbits even further. In order to assess the performance of these orbits it is necessary to process all of the available tracking data (GPS, SLR, DORIS, and altimeter crossover differences) as either dependent or independent of the orbit solutions. It was also necessary to compute orbit solutions using various combinations of the four available tracking data in order to independently assess the orbit performance. Towards this end, we have greatly improved orbits determined solely from SLR+DORIS data by applying the reduced dynamic solution strategy. In addition, we have computed reduced dynamic orbits based on SLR, DORIS, and crossover data that are a significant improvement over the SLR- and DORIS-based dynamic solutions. These solutions provide the best performing orbits for independent validation of the GPS-based reduced dynamic orbits. The application of the 1-cm orbit will significantly improve the resolution of the altimeter measurement, making possible further strides in radar altimeter remote sensing.  相似文献   

18.
Retracking of Jason-1 Data   总被引：1，自引：0，他引：1  
We present the results of retracking 18 cycles (15 from the Jason-TOPEX collinear period) of Jason-1 data. We used the retracking method of Rodriguez which simultaneously solves for all relevant waveform parameters using a 26 Gaussian model of the altimeter point target response. We find significant differences from the Jason-1 Project retracking in the key parameters of range and significant wave height (SWH) in the second version of the Project SGDRs. The differences from the Jason-1 data have a strong dependence on off-nadir angle and some dependence on SWH. The dependence of range on SWH is what is called sea state bias. The retracking technique also estimates surface skewness. For Jason-1 with its very clean waveforms we make the first direct estimates of the skewness effect on altimeter data. We believe that the differences found here and thus in overall sea surface height are the result of the standard project processing using a single Gaussian approximation to the Point Target Response (PTR) and not solving simultaneously for off nadir angle. We believe that the relatively large sea state bias errors estimated empirically for Jason-1 during the cal/val phase result from sensitivity of quantities, particularly SWH, in project GDRs to off nadir angle. The TOPEX-Jason-1 bias can be determined only when a full retracking of Jason-1 is done for the collinear period.  相似文献   

19.
The Jason-1 radar altimeter satellite, launched on December 7, 2001 is the follow on to the highly successful TOPEX/Poseidon (T/P) mission and will continue the time series of centimeter level ocean topography measurements. Orbit error is a major component in the overall error budget of all altimeter satellite missions. Jason-1 is no exception and has set a 1-cm radial orbit accuracy goal, which represents a factor of two improvement over what is currently being achieved for T/P. The challenge to precision orbit determination (POD) is both achieving the 1-cm radial orbit accuracy and evaluating the performance of the 1-cm orbit. There is reason to hope such an improvement is possible. The early years of T/P showed that GPS tracking data collected by an on-board receiver holds great promise for precise orbit determination. In the years following the T/P launch there have been several enhancements to GPS, improving its POD capability. In addition, Jason-1 carries aboard an enhanced GPS receiver and significantly improved SLR and DORIS tracking systems along with the altimeter itself. In this article we demonstrate the 1-cm radial orbit accuracy goal has been achieved using GPS data alone in a reduced dynamic solution. It is also shown that adding SLR data to the GPS-based solutions improves the orbits even further. In order to assess the performance of these orbits it is necessary to process all of the available tracking data (GPS, SLR, DORIS, and altimeter crossover differences) as either dependent or independent of the orbit solutions. It was also necessary to compute orbit solutions using various combinations of the four available tracking data in order to independently assess the orbit performance. Towards this end, we have greatly improved orbits determined solely from SLR+DORIS data by applying the reduced dynamic solution strategy. In addition, we have computed reduced dynamic orbits based on SLR, DORIS, and crossover data that are a significant improvement over the SLR- and DORIS-based dynamic solutions. These solutions provide the best performing orbits for independent validation of the GPS-based reduced dynamic orbits. The application of the 1-cm orbit will significantly improve the resolution of the altimeter measurement, making possible further strides in radar altimeter remote sensing.  相似文献   

20.
We present the results of retracking 18 cycles (15 from the Jason-TOPEX collinear period) of Jason-1 data. We used the retracking method of Rodriguez which simultaneously solves for all relevant waveform parameters using a 26 Gaussian model of the altimeter point target response. We find significant differences from the Jason-1 Project retracking in the key parameters of range and significant wave height (SWH) in the second version of the Project SGDRs. The differences from the Jason-1 data have a strong dependence on off-nadir angle and some dependence on SWH. The dependence of range on SWH is what is called sea state bias. The retracking technique also estimates surface skewness. For Jason-1 with its very clean waveforms we make the first direct estimates of the skewness effect on altimeter data. We believe that the differences found here and thus in overall sea surface height are the result of the standard project processing using a single Gaussian approximation to the Point Target Response (PTR) and not solving simultaneously for off nadir angle. We believe that the relatively large sea state bias errors estimated empirically for Jason-1 during the cal/val phase result from sensitivity of quantities, particularly SWH, in project GDRs to off nadir angle. The TOPEX-Jason-1 bias can be determined only when a full retracking of Jason-1 is done for the collinear period.  相似文献   

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