TOPEX & Jason Retracking OSTST ‘07 Retracking and SSB Splinter TOPEX and Jason Retracking Ernesto...

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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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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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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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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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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Jason Features (Cycles 19-21) Jason LSE SWH, solving for skewness Jason LSE-GDR Range Correction

Des

Asc

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TOPEX Waveform Contamination EvidenceTOPEX Skewness Jason Skewness Cyc 19-21 (avg = 0.06)

Des

Asc

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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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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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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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GDR Differences

There appears to be a discontinuity at the equator which is different for ascending and descending passes

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JPL Topex LSE vs Jason GDR

Equatorial discontinuity present and more marked

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JPL Topex LSE vs JPL Jason LSE

Equatorial discontinuity present, notice change in bias value = 8mm

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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


Backup / Previous Material

OSTST ‘07

TOPEX and Jason Retracking

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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: 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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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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JPL Topex MAP vs Jason GDR

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JPL Topex MAP vs JPL Jason MAP

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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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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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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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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).

TOPEX & Jason Retracking OSTST ‘07 Retracking and SSB Splinter TOPEX and Jason Retracking Ernesto...

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