Page 1 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 representing the 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 index The rank-k matrix Ak represents a filtered signal noise signal + noise S= 1 VS 1 +….+ k VS k +….. r VS r S= U V* Computing the Singular Value 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 Waveforms classification Estimation Process (retracking) SVD Class 1Class 2Class 3Class 4Class 5Class 6 Class 12Class 23Class 13Class 24Class 15Class 0 Class 21Class 35Class 16Class 99 Brown echosPeak echosVery noisy echosLinear echosPeak at the end of the echos Very large peak echos Brown + Peaky echos CS 32 Brown + strong decreasing plateau ?? Doubt Brown + leading edge perturbation Peaky + Noise Brown + Peak echos Leading at the end + noise Brown + Peaky + linear variation Brown + increasing leading edge Igarss 2011 – Vancouver – Pierre THIBAUT
Page 5 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
Page 6 Parameters used for the SVD filtering Determination of the truncature threshold –Investigations on the frequential spectrum of the SLA residuals (SLA Products – SLA SVD ) 96 % threshold - SLA Products - SLA SVD - Residuals Igarss 2011 – Vancouver – Pierre THIBAUT 700m10km
Page 7 92 % threshold - SLA Products - SLA SVD - Residuals Determination of the truncature threshold –Investigations on the frequential spectrum of the SLA residuals (SLA Products – SLA SVD ) Parameters used for the SVD filtering Igarss 2011 – Vancouver – Pierre THIBAUT 700m10km
Page 8 90 % threshold - SLA Products - SLA SVD - Residuals Determination of the truncature threshold –Investigations on the frequential spectrum of the SLA residuals (SLA Products – SLA SVD ) Parameters used for the SVD filtering Igarss 2011 – Vancouver – Pierre THIBAUT 700m10km
Page 9 88 % threshold - SLA Products - SLA SVD - Residuals Determination of the truncature threshold –Investigations on the frequential spectrum of the SLA residuals (SLA Products – SLA SVD ) Parameters used for the SVD filtering Igarss 2011 – Vancouver – Pierre THIBAUT 700m10km
Page % threshold - SLA Products - SLA SVD - Residuals Determination of the truncature threshold –Investigations on the frequential spectrum of the SLA residuals (SLA Products – SLA SVD ) Parameters used for the SVD filtering Igarss 2011 – Vancouver – Pierre THIBAUT 700m10km
Page % threshold - SLA Products - SLA SVD - Residuals Determination of the truncature threshold –Investigations on the frequential spectrum of the SLA residuals (SLA Products – SLA SVD ) Parameters used for the SVD filtering Igarss 2011 – Vancouver – Pierre THIBAUT 700m10km
J1 Ku band raw waveformsJ1 Ku band filtered waveforms gate index Waveform index Truncated Singular Value Decomposition Noise Reduction
Truncated Singular Value Decomposition for Noise Reduction Raw waveforms Filtered 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
Page 15 - SLA Prod - SLA SVD Impact on range (threshold of 90 %) SLA variation SLA spectrum – Ku band 8 cm at 20 Hz 7 cm at 20 Hz Igarss 2011 – Vancouver – Pierre THIBAUT 700m10km
Page 16 Results on Sea Level Anomaly Bias : SLA SVD = SLA Prod – 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
Page 18 SWH Bias : SWH SVD = SWH Prod - 3 cm (Ku) Two SWH populations appear in the SWH distribution SWH SVD SWH 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
Page 19 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
Page 20 Performances on noise level Synthesis of the noise levels obtained (for 90%) 20 Hz1 Hz KuC C Range 7.07 cm (products = 8.02) 16.6 cm (products = 20.14) 2.64 cm (products = 2.81) 4.6 cm (products = 5.34) SWH cm (products = 50.7) 100 cm (products = 120) 11 cm (products = 14) cm (products = 32.73) Sigma 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
Page 21 Applications on a track that corsses the Alghulas current (South Africa) Jason-2 pass 96 has been processed : SVD (84%)+MLE4. SLA SWH Igarss 2011 – Vancouver – Pierre THIBAUT
Page 22 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 Hz SLA 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