Singular Value Decomposition applied on altimeter waveforms
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Singular Value Decomposition applied on altimeter waveforms
P. Thibaut, J.C.Poisson, A.Ollivier : CLS – Toulouse - France
F.Boy, N.Picot : CNES – Toulouse - France
Context of the study
• The altimetry technique is based on the exploitation of high rate waveforms measured by a pulse limited radar (Pulse Repetition Frequency around 2000Hz). These waveforms are corrupted by multiplicative speckle noise.
• In order to be able to provide useful information to the users, waveforms are averaged to reduce their noise level. The estimation process called retracking is performed at a low rate (20Hz) fitting an ocean model (Hayne model) to these waveforms (LSE).
• Then, the 20-Hz estimates are averaged to derive 1Hz values for significant waveheight, range and sigma naught coefficients.
• We propose here to reduce the noise level of the estimations without any artificial along-track spatial correlation introduced by the 1Hz averaging, …
A Singular Value Decomposition is implemented to reduce the noise level of the WFs before the estimation procedure (which is unchanged wrt current procedure)
Igarss 2011 – Vancouver – Pierre THIBAUT
• Classical technique in signal processing sometimes used to « denoise » signals.•Waveforms filtering using SVD was first investigated by A.Ollivier in his PhD Thesis (2006) with very encouraging results on Poseidon and Jason-1 altimeter data•The processing consists in :
Truncated Singular Value Decomposition for Noise Reduction
Taking the S matrix representingthe noisy signal (Wfs matrix)
J1 Ku band raw waveforms
gate
index
Waveform index
S= S + B
Discarding small singular values of S(which mainly represent the additive noise)
J1 Ku band filtered waveforms
gate
index
Waveform indexThe rank-k matrix Ak represents a filtered signal
noisesignal + noise
S= 1VS1+….+kVSk+…..rVSr
S= UV*Computing the SingularValue Decomposition
SVD filtering
S (m,n) : WF matrix (m= 104; n=300)U (m,m), V*(n,n): unit matrices(m,n) : diagonal matrix
Igarss 2011 – Vancouver – Pierre THIBAUT
• matricial method• results are closely linked to the size of the matrix (number of Along-Track Wfs in the matrix) and the homogeneity of the waveform matrix • results are closely linked to rank-k truncation
Truncated Singular Value Decomposition for Noise Reduction
Waveformsclassification
EstimationProcess
(retracking)SVD
Class 1 Class 2 Class 3 Class 4 Class 5 Class 6
Class 12 Class 23 Class 13 Class 24 Class 15 Class 0
Class 21 Class 35 Class 16 Class 99
Brown echos Peak echos Very noisy echos Linear echos Peak at the endof the echos
Very large peakechos
Brown + Peakyechos
CS 32
Brown + strongdecreasing plateau
??
Doubt
Brown + leadingedge perturbation
Peaky + Noise
Brown + Peakechos
Leading at theend + noise
Brown + Peaky+ linear variation
Brown + increasingleading edge
Igarss 2011 – Vancouver – Pierre THIBAUT
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• Selection of the waveforms using a classification method
Class 1
Class 2
Class 12
Class 15
Class 16
Class 3
Parameters used for the SVD noise reduction
Igarss 2011 – Vancouver – Pierre THIBAUT
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Parameters used for the SVD filtering
• Determination of the truncature threshold– Investigations on the frequential spectrum of the SLA residuals (SLAProducts – SLASVD)
96 % threshold- SLA Products- SLA SVD- Residuals
Igarss 2011 – Vancouver – Pierre THIBAUT
700m10km
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92 % threshold- SLA Products- SLA SVD- Residuals
• Determination of the truncature threshold– Investigations on the frequential spectrum of the SLA residuals
(SLAProducts – SLASVD)
Parameters used for the SVD filtering
Igarss 2011 – Vancouver – Pierre THIBAUT
700m10km
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90 % threshold- SLA Products- SLA SVD- Residuals
• Determination of the truncature threshold– Investigations on the frequential spectrum of the SLA residuals (SLAProducts – SLASVD)
Parameters used for the SVD filtering
Igarss 2011 – Vancouver – Pierre THIBAUT
700m10km
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88 % threshold- SLA Products- SLA SVD- Residuals
• Determination of the truncature threshold– Investigations on the frequential spectrum of the SLA residuals (SLAProducts – SLASVD)
Parameters used for the SVD filtering
Igarss 2011 – Vancouver – Pierre THIBAUT
700m10km
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84 % threshold- SLA Products- SLA SVD- Residuals
• Determination of the truncature threshold– Investigations on the frequential spectrum of the SLA residuals (SLAProducts – SLASVD)
Parameters used for the SVD filtering
Igarss 2011 – Vancouver – Pierre THIBAUT
700m10km
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80 % threshold- SLA Products- SLA SVD- Residuals
• Determination of the truncature threshold– Investigations on the frequential spectrum of the SLA residuals (SLAProducts – SLASVD)
Parameters used for the SVD filtering
Igarss 2011 – Vancouver – Pierre THIBAUT
700m10km
J1 Ku band raw waveforms J1 Ku band filtered waveforms
gate index
gate index
Waveform index
Waveform index
Truncated Singular Value Decomposition Noise Reduction
Truncated Singular Value Decomposition for Noise Reduction
Raw waveformsFiltered waveforms
Igarss 2011 – Vancouver – Pierre THIBAUT
Impact on range (threshold of 84 %)
SLA variation
latitude
MLE4
SVD+MLE4
SLA spectrum – Ku band
8 cm at 20 Hz
5 cm at 20 Hz
MLE4
SVD+MLE4
Igarss 2011 – Vancouver – Pierre THIBAUT
700m10km
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- SLA Prod- SLA SVD
Impact on range (threshold of 90 %)
SLA variationSLA spectrum – Ku band
8 cm at 20 Hz
7 cm at 20 Hz
Igarss 2011 – Vancouver – Pierre THIBAUT
700m10km
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Results on Sea Level Anomaly
Bias : SLASVD = SLAProd – 4 mm (except for small waves)
Impact on range (threshold of 90 %)
Igarss 2011 – Vancouver – Pierre THIBAUT
Impact on SWH (threshold of 84%)
SWH variation
latitude
SWH spectrum – Ku band
MLE4
SVD+MLE4
MLE4
SVD+MLE4
54 cm at 20 Hz
12 cm at 20 Hz
Igarss 2011 – Vancouver – Pierre THIBAUT
700m10km
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SWH
Bias : SWHSVD = SWHProd - 3 cm (Ku)
Two SWH populations appear in the SWH distribution
SWH SVDSWH Prod
Impact on SWH (threshold of 90 %)
SWH spectrum – Ku band
54 cm at 20 Hz
40 cm at 20 Hz
Igarss 2011 – Vancouver – Pierre THIBAUT
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Small impact on residuals (smaller on the leading adge; higher on the trailing edge) SVD
Impact on residuals (threshold of 90 %)
Igarss 2011 – Vancouver – Pierre THIBAUT
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Performances on noise level
• Synthesis of the noise levels obtained (for 90%)
20 Hz 1 Hz
Ku C Ku C
Range7.07 cm
(products = 8.02)16.6 cm
(products = 20.14)2.64 cm
(products = 2.81)4.6 cm
(products = 5.34)
SWH38.27 cm
(products = 50.7)100 cm
(products = 120)11 cm
(products = 14)25.35 cm
(products = 32.73)
Sigma00.307 dB
(products = 0.376)0.13 dB
(products = 0.13)0.14 dB
(products = 0.16)0.11 dB
(products = 0.11)
• Observed gains at 20 Hz are reduced at 1 Hz.• The 20 Hz denoising allows to increase the number of elementary measurements
Igarss 2011 – Vancouver – Pierre THIBAUT
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Applications on a track that corsses the Alghulas current (South Africa)
• Jason-2 pass 96 has been processed : SVD (84%)+MLE4.
SLA SWHIgarss 2011 – Vancouver – Pierre THIBAUT
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Applications on a track that corsses the Alghulas current (South Africa)
• Jason-2 pass 96 on 70 cycles has been processed : SVD+MLE4.
SLA 20 HzSLA SVD20 Hz (84 %)
SLA 1 Hz
Igarss 2011 – Vancouver – Pierre THIBAUT
Conclusions
• SVD allows a strong noise reduction on SWH and range.This reduction depends on the rank truncation and can be adapted to the application
• SVD allows a gain in SLA rms measurements by a factor between 1.2(weak waves) to 2 (strong waves).
• SVD allows to pass from a 7 km resolution (corresponding to 1 Hz products) to a 1.2 resolution (6 Hz) with an equivalent noise (precision of the SLA).
• We are testing SVD processing on different zones where small structures now hidden in the noise level for current products, will clearly appear.
Igarss 2011 – Vancouver – Pierre THIBAUT
Thank you !
No impact of the SVD on mispointing angle