singular value decomposition applied on altimeter waveforms

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


• 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


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


• 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


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


700m10km


• Determination of the truncature threshold– Investigations on the frequential spectrum of the SLA residuals

(SLAProducts – SLASVD)



700m10km





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


Raw waveformsFiltered waveforms


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


700m10km

- SLA Prod- SLA SVD


SLA variationSLA spectrum – Ku band

8 cm at 20 Hz

7 cm at 20 Hz


700m10km

Results on Sea Level Anomaly

Bias : SLASVD = SLAProd – 4 mm (except for small waves)



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


700m10km

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


Small impact on residuals (smaller on the leading adge; higher on the trailing edge) SVD

Impact on residuals (threshold of 90 %)


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 = 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 = 0.11)

• Observed gains at 20 Hz are reduced at 1 Hz.• The 20 Hz denoising allows to increase the number of elementary measurements


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

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


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.


Thank you !

No impact of the SVD on mispointing angle

singular value decomposition applied on altimeter waveforms

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