TOPEX & Jason Retracking
OSTST ‘07Retracking and SSB Splinter
TOPEX and Jason Retracking
Ernesto Rodriguez, Phil Callahan, Ted Lungu
March 13, 2007
Jet Propulsion LaboratoryCalifornia Institute of Technology
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TOPEX & Jason Retracking
OSTST Retracking & SSB Splinter – Overview
• Discussion – Goal: Allow studies of global and regional variations
using the whole TOPEX + Jason time series to determine sea level changes to a few tenths of a mm per year
– Recommend approaches for final processing for Jason (reprocessing?), TOPEX RGDR, in particular, the SSB for final cross-calibration see proposal for options
• Would like OSTST recommendation
– Estimate error structure of Jason and TOPEX data
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TOPEX & Jason Retracking
Retracking TOPEX & Jason – Outline
• Identical software used for both – Avg Jason WF to TOPEX structure (10/frame, 64 bins) – Software has skewness fixed to 0 or solves (cannot set
specific value)
• No significant changes to TOPEX retracking since Mar ’06 (LSE & MAP)
• Jason Changes since Mar ’06: Using WF weighting, slightly revised PTR
• Tests on Jason simulated Waveforms • Results to Date
– Greatly improved agreement between CNES/JPL on Jason data
– MAP not providing expected benefits – has lower noise but has bias, SWH dependence
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TOPEX & Jason Retracking
Retracking Progress
• Retracked 2 yr TOPEX Alt-B and produced RGDRs with improved orbits – LSE skewness absorbs WF leakages so much reduced N/S
Asc/Des (“Quadrant”) difference, but still some • MAP skewness much smaller so large variations with SWH
– Need to assess waveform residuals to correction for leakages, OR rely on empirical correction
• Worked issues with CNES on differences of MLE4, LSE, MAP– Processed large set of simulated data, numerous PTRs
– Found no anomalies in Jason waveform residuals
– However, MLE4 only agrees with LSE when solve for skewness, not fixed skewness. MAP has SWH dependence
– Similar results found from simulated WF
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TOPEX & Jason Retracking
June ’06 CNES Simulated Waveform Results
• New simulations with 10,000 pts: – SWH = 2, 4, 6, 8; Attitude = 0, 0.1;
Skewness = 0, 0.1
• Some findings – LSE had small SWH bias at higher SWH – MAP height std dev a factor of 2-3
smaller – MAP std dev on other parameters was
negligible – Solving for skewness prevented height
changes of a few mm at higher SWH – Skewness was recovered, but LSE std
dev ~ 0.1 – Additional terms in Gaussian expansion
of PTR generaly had expected effect, although somewhat larger than expected – showed the need to extend PTR to farther sidelobes
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TOPEX & Jason Retracking
Jason Features (Cycles 19-21) Jason LSE SWH, solving for skewness Jason LSE-GDR Range Correction
Des
Asc
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TOPEX & Jason Retracking
TOPEX Waveform Contamination EvidenceTOPEX Skewness Jason Skewness Cyc 19-21 (avg = 0.06)
Des
Asc
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TOPEX & Jason Retracking
Height Differences
-20 Range difference (mm) 30
As with Jason, LSE and MAP retrackers exhibit a SWH dependence difference.
In order to make TOPEX and Jason compatible at the 1cm level, the waveform leakage contamination be mitigated.
TOPEX LSE-GDR (toward) Vs Att / SWH
TOPEX LSE-MAP(toward) vs Att / SWH
0 Range difference (mm) 40
Jason LSE-MAP vs Att / SWH
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TOPEX & Jason Retracking
Jason: Are LSE and MAP Biases Consistent?Skewness vs no Skewness Estimation
When skewness is not estimated, the mean difference between LSE and MAP increases. The SWH dependence is similar, but not identical
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TOPEX & Jason Retracking
Track Point Difference Statistical Results
• Examined mean SSH differences using different retracking methods and behavior of the residuals after subtracting the mean differences for Jason cycles 7-21
• SSH surfaces examined:– Topex GDR
– JPL Topex LSE and MAP retracking
– Jason GDR
– JPL Jason LSE and MAP retracking
• Topex SSH constructed with improved acceleration correction and new orbits and media corrections
• Jason and Topex data interpolated to a common grid and differenced for coincident passes
• Retracking compared against the SGDR retrack estimateCNES dh = -[ku_range + ku_range_20Hz - (ku_tracker20Hz +
total_instr_correction)]
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TOPEX & Jason Retracking
GDR Differences
There appears to be a discontinuity at the equator which is different for ascending and descending passes
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TOPEX & Jason Retracking
JPL Topex LSE vs Jason GDR
Equatorial discontinuity present and more marked
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TOPEX & Jason Retracking
JPL Topex LSE vs JPL Jason LSE
Equatorial discontinuity present, notice change in bias value = 8mm
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TOPEX & Jason Retracking
Retracking Conclusions
• TOPEX retracking must use LSE solving for skewness – Residual Quadrant bias has SWH dependence, so needs
correction like dSSH(q) = a0(q) + a1(q) * SWH
• Jason LSE does not have major SWH dependence, but must solve for skewness– Avg skewness ~0.06
• Check of software have not found any problems in MAP implementation, so behavior is not fully understood – Since MAP is weighted and uses a priori information, it is
more likely to be biased. However, MLE4 is unweighted …
• Jason data seem very sensitive to small changes in retracking setup
TOPEX & Jason Retracking
Backup / Previous Material
OSTST ‘07
TOPEX and Jason Retracking
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TOPEX & Jason Retracking
Retracking Algorithm Comparison
MLE3 MLE4 JPL LSE JPL MAP
Estimation Type
Least Squares + correction table
Least Squares + correction table
Least squares
Maximum a Posteriori
Attitude estimated? No Yes Yes YesSkewness Estimated No No Yes YesWeighted No No No Yes
Estimation frequency 20 Hz 20 Hz
10 Hz heights, 1 Hz other parameters
10 Hz heights, 1 Hz other parameters
Point Traget Response 1 Gaussian 1 Gaussian
Sum of Gaussians
Sum of Gaussians
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TOPEX & Jason Retracking
Topex: Are LSE and MAP Biases Consistent?
There appears to be a SWH dependent bias between MAP and LSE, but no apparent discontinuity at the equator
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TOPEX & Jason Retracking
Jason: Are LSE and MAP Biases Consistent?
There appears to be a SWH dependent bias between MAP and LSE, as in Topex. However, differences seem to be larger
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TOPEX & Jason Retracking
JPL Topex MAP vs Jason GDR
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TOPEX & Jason Retracking
JPL Topex MAP vs JPL Jason MAP
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TOPEX & Jason Retracking
TOPEX Waveform Artifacts
Averaging Time: 40 secondsDue to onboard signal leakages, TOPEX waveforms are contaminated by spurious signals which appear in the leading edge and are hard to model.
Rodriguez and Martin (JGR, 1994) estimated height biases of ~+/-1 cm which were geographically dependent by comparing with LSE retracking.
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TOPEX & Jason Retracking
Maximum a Posteriori Retracking:a 3rd Generation Retracking Scheme
• 1st Generation retracking (Rodriguez and Martin, JGR 94):– Decomposition of the PTR into sum of Gaussians – Arbitrary attitude angle (expansion to higher order terms) – Linearized least squares estimation, including Skewness
• 2nd Generation retracking (Callahan and Rodriguez, MG 04)– Added iterative estimation of parameters until retracker fully
converged
• 3rd Generation retracking: Maximum a Posteriori (MAP) – 1st and 2nd generation retrackers operated on 1 second frames
without constraints – Retracker unbiased, but noisy and retrieved parameters could be
highly correlated – MAP estimation constrains the parameter space for the inversion
using a priori knowledge (data are still estimated from 1 sec frames)
• Attitude varies slowly, SWH correlation distance ~100 km and known to better than 60cm, Track Point known to better than 20 cm, |skewness|<1
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TOPEX & Jason Retracking
Retracking Algorithms
Maximum Likelihood Estimator (MLE) Minimizes:
Maximum a Posteriori (MAP) Minimizes:
Where x is the data, a are the parameters to be estimated, A are the parameter a priori values, i are the measurement errors and n measures the prior confidence level. Setting the priors and their confidence levels is the trick!Prior Values: smooth LSE SWH and attitude data over an extent < 80 km relative to centerPrior Uncertainties: Root Squares Sum residual values in smoothing window with conservative estimate of minimum uncertainty of SWH and attitude variance. Use 1.5 as uncertainty on the skewness, and infinite variances (no priors) on the other parameters, including height.
log(p(x | a)) x i M(a) 2
i2
i1
Ndata
log(p(x | a)p(a)) x i M(a) 2
i2
i1
Ndata
an An 2
n2
n1
Nparams
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TOPEX & Jason Retracking
MAP Retracking Simulation Results Least squares estimation MAP STD: SWH = 60cm, Skew = 1, Att = 0.02deg, Height = 20cm
Sigma0 SWH Skew Attitude Height STD Sigma0 SWH Skew Attitude Height STDSigma0 1.00 -0.06 0.10 -0.79 -0.03 0.10 Sigma0 1.00 -0.13 -0.07 -0.51 0.00 0.05SWH [cm] -0.06 1.00 -0.70 -0.27 -0.89 8.60 SWH [cm] -0.13 1.00 0.46 0.10 -0.45 0.93Skew 0.10 -0.70 1.00 0.15 0.69 0.28 Skew -0.07 0.46 1.00 -0.03 -0.40 0.02Attitude [deg] -0.79 -0.27 0.15 1.00 0.38 0.08 Attitude [deg] -0.51 0.10 -0.03 1.00 0.08 0.00Height [cm] -0.03 -0.89 0.69 0.28 1.00 1.80 Height [cm] 0.00 -0.45 -0.40 0.08 1.00 0.59
Maximum Likelihood (weighted least squares) MAP STD: SWH = 30cm, Skew = 1, Att = 0.01deg, Height = 10cmSigma0 SWH Skew Attitude Height STD Sigma0 SWH Skew Attitude Height STD
Sigma0 1.00 -0.01 -0.01 -0.80 -0.11 0.09 Sigma0 1.00 -0.09 -0.07 -0.51 -0.05 0.05SWH [cm] -0.01 1.00 -0.88 -0.27 -0.87 7.27 SWH [cm] -0.09 1.00 0.59 0.10 -0.27 0.31Skew -0.01 -0.88 1.00 0.21 0.63 0.11 Skew -0.07 0.59 1.00 0.00 -0.35 0.02Attitude [deg] -0.80 -0.27 0.21 1.00 0.41 0.08 Attitude [deg] -0.51 0.10 0.00 1.00 0.12 0.00Height [cm] -0.11 -0.87 0.63 0.41 1.00 1.46 Height [cm] -0.05 -0.27 -0.35 0.12 1.00 0.55
• By putting very moderate constraints on the retrieved parameters, the estimated parameters are almost completely uncorrelated.• Even better, the estimation noise drops by a factor of 3 for height and an order of magnitude for SWH and skewness!• If one uses the SGDR to set a priori constraints, the increased burden on computation is small or negative (faster convergence). However, biases must be quantified.•To remove biases, one can retrack using least squares, derive priors, and retrack again with MAP. Computation doubles (still feasible, though).
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