Li et al 2020b - Revision history

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← Older revision		Revision as of 08:01, 19 November 2020
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−	~~==A New Method of Satellite Radar Altimeter Waveform Retracking Based on Waveform Derivative ==~~
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−	~~Zhen Li<sup>1</sup>, Xin Liu<sup>12</sup>, Jinyun Guo<sup>1</sup>, Jiajia Yuan<sup>1</sup>, Yupeng Niu<sup>1</sup>, Bing Ji<sup>3</sup>~~
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−	~~1 College of Geomatics, Shandong University of Science and Technology, Qingdao, Shandong 266590, China~~
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−	~~2 corresponding author: [mailto:xinliu1969@126.com xinliu1969@126.com]~~
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−	~~3 Department of Navigation, Naval University of Engineering, Wuhan, 430022, Hubei, China~~
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	==Abstract==		==Abstract==

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 ==Abstract==
-Waveform retracking for precise sea surface height (SSH) is an important method to improve the quality of satellite altimeter data. Combining the physical explanation of the function fitting and the high adaptability of empirical statistical methods, we effort to provide a comprehensive method for processing the waveforms over the open ocean and coastal area. The new retracking method that uses the maximum slope of leading edge to determine midpoint is proposed, the midpoint is determined by zero of second derivative of theoretical model. The unknown parameters are estimated based on the function fitting. Then combined with the advantage of empirical method, the leading edge midpoint is redetermined by interpolating the estimated midpoint on the measured waveform. The new method is validated by comparison with crossover discrepancies, geoid heights and tide gauges. The RMS of crossover discrepancy obtained by new method is 0.107 m, which is smaller than 0.192 m, 0.124 m, 0.121 m, 0.114 m, 0.112 m obtained by Ice-1,Threshold 50%, <math>5-\beta </math>, MLE3 and MLE4 retrackers. The STD and improvement percentage of the differences between SSHs obtained by new method and geoid heights are also better than the results of single kind of retracker. Comparison with tide gauge records, the STDs difference of height anomaly obtained by new method are 0.183 m and 0.269 m at different regions, which are smaller than 0.239 m, 0.220 m, 0.195 m and 0.303 m, 0.278 m, 0.272 m obtained by Ice-1, MLE4 and Threshold 50% retrackers, respectively. Therefore, the new method can recover more reliable SSH in the open ocean and coastal area.
+Waveform retracking for precise sea surface height (SSH) is an important method to improve the quality of satellite altimeter data. Combining the physical explanation of the function fitting and the high adaptability of empirical statistical methods, we effort to provide a comprehensive method for processing the waveforms over the open ocean and coastal area. The new retracking method that uses the maximum slope of leading edge to determine midpoint is proposed, the midpoint is determined by zero of second derivative of theoretical model. The unknown parameters are estimated based on the function fitting. Then combined with the advantage of empirical method, the leading edge midpoint is redetermined by interpolating the estimated midpoint on the measured waveform. The new method is validated by comparison with crossover discrepancies, geoid heights and tide gauges. The RMS of crossover discrepancy obtained by new method is 0.107 m, which is smaller than 0.192 m, 0.124 m, 0.121 m, 0.114 m, 0.112 m obtained by Ice-1,Threshold 50%, 5-<math>\beta </math>, MLE3 and MLE4 retrackers. The STD and improvement percentage of the differences between SSHs obtained by new method and geoid heights are also better than the results of single kind of retracker. Comparison with tide gauge records, the STDs difference of height anomaly obtained by new method are 0.183 m and 0.269 m at different regions, which are smaller than 0.239 m, 0.220 m, 0.195 m and 0.303 m, 0.278 m, 0.272 m obtained by Ice-1, MLE4 and Threshold 50% retrackers, respectively. Therefore, the new method can recover more reliable SSH in the open ocean and coastal area.
 '''Keywords''': Satellite altimeter, waveform retracking, derivative, leading edge slope, crossover discrepancy, tide gauge

← Older revision		Revision as of 12:57, 7 October 2020
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−	The initial values of the unknown parameters are also determined by the OCOG algorithm. The unknown parameters are estimated until iterative convergence (the convergence criterion is similar to Equation 11). The waveform leading edge midpoint is computed via Equation (4). The midpoint obtained by this way is called the second wave derivative retracker (SWDR) in this paper.	+	The initial values of the unknown parameters are also determined by the OCOG algorithm. The unknown parameters are estimated until iterative convergence (the convergence criterion is similar to Equation (11)). The waveform leading edge midpoint is computed via Equation (4). The midpoint obtained by this way is called the second wave derivative retracker (SWDR) in this paper.

	Similar to Equation (12), the leading edge midpoint is redetermined. The leading edge midpoint power value is obtained by Equation (1), which is linearly interpolated to the adjacent power value of the leading edge of measured waveform to redetermine the midpoint. The midpoint obtained by this way is called the second leading edge interpolation retracker (SLEIR).		Similar to Equation (12), the leading edge midpoint is redetermined. The leading edge midpoint power value is obtained by Equation (1), which is linearly interpolated to the adjacent power value of the leading edge of measured waveform to redetermine the midpoint. The midpoint obtained by this way is called the second leading edge interpolation retracker (SLEIR).

← Older revision		Revision as of 12:55, 7 October 2020
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−	where <math display="inline"> \bf L</math> is a vector that indicates the difference between the measured waveform and the estimated waveform by Equation (1), <math display="inline"> {\bf X}(dA_0,dt_0,d\sigma_c</math> is the correction vector of unknown parameters, and <math display="inline"> \bf A</math> is the partial derivative matrix as	+	where <math display="inline"> \bf L</math> is a vector that indicates the difference between the measured waveform and the estimated waveform by Equation (1), <math display="inline"> {\bf X}(dA_0,dt_0,d\sigma_c)</math> is the correction vector of unknown parameters, and <math display="inline"> \bf A</math> is the partial derivative matrix as

	{\| class="formulaSCP" style="width: 100%; text-align: center;"		{\| class="formulaSCP" style="width: 100%; text-align: center;"

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 where <math display="inline">{t}_{{m}} </math> is the leading edge midpoint, <math display="inline">{\tau }</math>the nominal tracking gate (<math display="inline">{\tau }</math>=32 gate of Jason-2 altimeter wavefrom), <math display="inline">{\Delta }{G}_{{a}} </math> the sampling interval of a gate (1 gate=3.125 ns of Jason-2 altimeter).
-The waveform data of cycles 001 to 200 are processed. For assessing the performance of the new retracker, we compared the retracked SSH from Threshold 50%, OCOG and 5-β retrackers, as well as the Ice-1 and MLE4 ( or Ocean) retrackers from SGDRs. Also, we obtained the ALES [19] retracked SSH data ([http://openadb.dgfi.tum.de/ http://openadb.dgfi.tum.de/]), which is widely recognized as a high-precision SSH products [22,32]. The retracked SSHs and the crossover discrepancies of SSHs in each cycle were computed. Some data of cycles are missing (Such as cycles 174, 175, 190, 191), and the gross error (<math display="inline">>1</math> m) of the crossover discrepancies of SSHs are removed. There are 1502 crossover discrepancies in total. The statistics of crossover discrepancies of SSHs obtained by several retrackers are listed in [[#tab-1|Table 1]]. The Unretracked represents the raw SSHs without waveform retracking.
+The waveform data of cycles 001 to 200 are processed. For assessing the performance of the new retracker, we compared the retracked SSH from Threshold 50%, OCOG and <math>5-\beta </math> retrackers, as well as the Ice-1 and MLE4 (or Ocean) retrackers from SGDRs. Also, we obtained the ALES [19] retracked SSH data ([http://openadb.dgfi.tum.de/ http://openadb.dgfi.tum.de/]), which is widely recognized as a high-precision SSH products [22,32]. The retracked SSHs and the crossover discrepancies of SSHs in each cycle were computed. Some data of cycles are missing (Such as cycles 174, 175, 190, 191), and the gross error (<math display="inline">>1</math> m) of the crossover discrepancies of SSHs are removed. There are 1502 crossover discrepancies in total. The statistics of crossover discrepancies of SSHs obtained by several retrackers are listed in [[#tab-1|Table 1]]. The Unretracked represents the raw SSHs without waveform retracking.
 By analyzing the RMS of the crossover discrepancies of SSHs, it is found that the results of several retracking methods are smaller than unretracked results, indicating that the waveform retracking can reduce the RMS of crossover discrepancies of SSH and improve the quality of satellite altimeter data. The RMSs obtained by FWDR, FLEIR, SWDR and SLEIR are 0.108 m, 0.107 m, 0.114 m and 0.119 m, which are smaller than 0.124 m, 0.177 m and 0.121 m obtained by Threshold 50%, OCOG and 5-<math display="inline">\beta</math> methods, respectively, indicating that the new methods can get more accurate results than the common methods in this region. The RMSs obtained by new mthods, which are less than 0.192 m, and 0.112 m obtained by Ice-1 and MLE4 methods from SGDR data, indicating that it can provide more accurate results than SGDRs (retracked SSH by MLE4, Ice-1 retracker) in the open ocean. In addition, the RMSs obtained by FWDR and FLEIR are 0.108 m and 0.107 m, which are smaller than 0.114 m obtained by MLE3 retracker, indicating that the midpoint determined by two parameters (<math display="inline">{t}_{{0}} </math>,<math display="inline">{\sigma }_{{c}}^{} </math>) is more accurate than one parameter (<math display="inline">{t}_{{0}} </math>).
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 Furthermore, we comparison with the retracked SSH by ALES retraker. The RMS obtained by FLEIR is 0.107 m, which is slightly smaller than 0.105 m obtained by ALES products. The difference may be caused by different correction models, such as sea state bias correction and ocean tide correction. This also shown that the retracked SSH by new methods is reliable compared with the SSH obtained by ALES products.
 ===4.2 Comparison with geoid heights ===

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 |}
-where <math display="inline">{H}_{{alt}} </math> is the altitude of Jason-2 satellite, <math display="inline">{Tracker}</math> is the range between satellite and reflective surface (partial instrumental corrections included, i.e. distance antenna-COG, USO drift correction, internal path correction), <math display="inline">{Corr}_{{doppler}} </math> is the Doppler correction, <math display="inline">{Corr}_{{model}} </math> is the modeled instrumental correction, <math display="inline">{Corr}_{{system\_bias}} </math> is the system bias of instrument, <math display="inline">{Corr}_{{Dry}} </math> is the dry tropospheric correction which is calculated from the atmospheric pressure and tide published by the European Centre for Medium-range Weather Forecasts (ECMWF), <math display="inline">{Corr}_{{Wet}} </math> is the wet tropospheric correction, which is calculated from the data measured by the microwave radiometer carried by the satellite, <math display="inline">{Corr}{}_{{Iono}} </math> is the ionospheric delay corrected using Dual-frequency, <math display="inline">{Corr}{}_{{SSB}} </math> is the sea state bias correction, which is calculated by empirical fitting of significant wave height and wind speed, <math display="inline">{Corr}{}_{{Ib}} </math> (inverted barometer correction) and <math display="inline">{Corr}_{{hf}} </math> (high frequency atmospheric pressure loading correction) are the dynamic atmospheric corrections, <math display="inline">{H}_{{ocean}} </math>is the geocentric ocean tide height correction, obtained by GOT4.10 model, <math display="inline">{H}_{{solid}} </math>is the solid earth tide height correction, and <math display="inline">{H}_{{pole}} </math> is the pole tide height correction. These above corrections are available in the SGDRs.
+where <math display="inline">{H}_{{alt}} </math> is the altitude of Jason-2 satellite, <math display="inline">{Tracker}</math> is the range between satellite and reflective surface (partial instrumental corrections included, i.e. distance antenna-COG, USO drift correction, internal path correction), <math display="inline">{Corr}_{{doppler}} </math> is the Doppler correction, <math display="inline">{Corr}_{{model}} </math> is the modeled instrumental correction, <math display="inline">{Corr}_{{system\_bias}} </math> is the system bias of instrument, <math display="inline">{Corr}_{{Dry}} </math> is the dry tropospheric correction which is calculated from the atmospheric pressure and tide published by the European Centre for Medium-range Weather Forecasts (ECMWF), <math display="inline">{Corr}_{{Wet}} </math> is the wet tropospheric correction, which is calculated from the data measured by the microwave radiometer carried by the satellite, <math display="inline">{Corr}{}_{{Iono}} </math> is the ionospheric delay corrected using Dual-frequency, <math display="inline">{Corr}{}_{{SSB}} </math> is the sea state bias correction, which is calculated by empirical fitting of significant wave height and wind speed, <math display="inline">{Corr}{}_{{Ib}} </math> (inverted barometer correction) and <math display="inline">{Corr}_{{hf}} </math> (high frequency atmospheric pressure loading correction) are the dynamic atmospheric corrections, <math display="inline">{H}_{{ocean}} </math> is the geocentric ocean tide height correction, obtained by GOT4.10 model, <math display="inline">{H}_{{solid}} </math> is the solid earth tide height correction, and <math display="inline">{H}_{{pole}} </math> is the pole tide height correction. These above corrections are available in the SGDRs.
 The range correction  <math display="inline">{R}_{{retracking}} </math> of waveform retracking is given by

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 {| style="text-align: center; margin:auto;"
 |-
-| style="text-align: center;" | <math>\frac{\partial {W}^{{'}} }{\partial {\sigma }_{{c}} } =\frac{{A}_{{0}} }{{2}} {exp(-}\frac{{4}}{{\gamma }} {sin}^{{2}} {\xi )exp(}-{v)}\left\{\left[-{a}\left[{1}+{erf(u)}\right]+\frac{\sqrt{{2}} }{\sqrt{{\pi }} {\sigma }_{{c}} } {exp(}-{u}^{{2}} {)}\right]\frac{-\partial {v}}{\partial {\sigma }_{{c}} } +</math>
+| style="text-align: center;" | <math>\frac{\partial {W}^{{'}} }{\partial {\sigma }_{{c}} } =\frac{{A}_{{0}} }{{2}} {exp(-}\frac{{4}}{{\gamma }} {sin}^{{2}} {\xi )exp(}-{v)}\left\{\left[-{a}\left[{1}+{erf(u)}\right]+\frac{\sqrt{{2}} }{\sqrt{{\pi }} {\sigma }_{{c}} } {exp(}-{u}^{{2}} {)}\right]\frac{-\partial {v}}{\partial {\sigma }_{{c}} } +\right.</math>
 |-
-| style="text-align: center;" | <math>\right.\left.\left[\frac{-{2a}}{\sqrt{{\pi }} } {exp(}-{u}^{{2}} {)}-\frac{{2}\sqrt{{2}} {u}}{\sqrt{{\pi }} {\sigma }_{{c}} } \right]\frac{\partial {u}}{\partial {\sigma }_{{c}} } \right\} </math>
+| style="text-align: center;" | <math>\left.\left[\frac{-{2a}}{\sqrt{{\pi }} } {exp(}-{u}^{{2}} {)}-\frac{{2}\sqrt{{2}} {u}}{\sqrt{{\pi }} {\sigma }_{{c}} } \right]\frac{\partial {u}}{\partial {\sigma }_{{c}} } \right\} </math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (17)