Parameter estimation for peaky altimetric waveforms (1)
(1)
(1)
J.-Y. Tourneret , J. Severini , C. Mailhes
(2)
and P. Thibaut
(1)
University of Toulouse, IRIT-ENSEEIHT-T´eSA, Toulouse, France Collecte Localisation Satellite (CLS), Ramonville Saint-Agne, France
(2)
1. Introduction
2. Problem formulation
Altimetric waveforms contaminated by land returns, by summation of backscattered signals... 300 250
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where c is the light speed and σp2 is a known parameter.
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Model for peaky altimetric waveforms [2] Superposition of a Brown echo and a sum of Gaussian peaks
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Simplified Brown’s model parametrized by Pu, τ and σc, !# " ασc2 2 kTs − τ − ασc Pu −α kTs−τ − 2 √ e 1 + erf xk = (1) 2 2σc where xk = x(kTs) the kth data sample, Ts the sampling period, erf(.) the Gaussian error function, α a known parameter. – Pu : amplitude – τ : epoch – σc related to the significant wave height SW H as follows 2 SW H 2 2 σc = σp + 2c
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3. Maximum likelihood estimation
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Examples of altered altimetric waveforms.
x ek = xk + pk
CNES/PISTACH project [4]
Peaks parametrized by – amplitudes A = (A1, ..., Aq ), – location parameters t = (t1, ..., tq ), – widths related to σ 2 = (σ12, ..., σq2 )
No closed-form expressions even for the MLE of θ B ☞ Iterative algorithms. MLE for Brown’s model (currently used) Quasi-Newton based recursive algorithm [1]
−1 θ B(n + 1) = θ B(n) − µn BB T BD. where B = µn = 1.
1 ∂xk xk ∂θB,i
i=1,...3 , D =
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xk −yk and xk k=1,...,N
MLE for Brown + Peak model using the following quasi-Newton T e eT e e BB BP e eT e eT
µn = 1.
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PB
!−1
e B e P
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PP e = 1 ∂pk e = 1 ∂xk , P where B x ek ∂θB,i i=1,...3 x ek ∂θp,i i=1,...3q and
yk = x ek nk , k = 1, ..., N. where nk is the multiplicative speckle noise.
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θ(n + 1) = θ(n) − µn
Real altimetric waveforms from PISTACH project
4. Simulation results
Maximum likelihood estimator (MLE) Obtained by differentiating the log-likelihood log f y; θ B; θ p with respect to the unknown parameters [3] N L−1 N X LN L Y yk yk f y; θB; θ p = exp −L N L x ek x e [Γ (L)] k k=1 k=1
T MLE of θ = θ TB , θ Tp recursion
Observed altimetric waveforms
☞ Particular interest in waveforms corrupted by the presence of peaks on their trailing edge. ☞ Specific attention to a single peak in this paper.
with θB = (Pu, τ, σc)T the Brown model parameter vector and θ p = (A, t, σ 2)T the peak parameter vector.
(2)
where xk has been defined in (1), and pk is the sum of q peaks " # q X (kTs − ti)2 Ai exp − pk = 2 2σ i i=1
How to estimate the corresponding altimeter parameters ?
T Unknown parameter vector : θ = θ TB , θ Tp
k=1,...,N
k=1,...,N
5. Conclusions
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Synthetic altimetric waveforms defined in (2) with different peak characteristics (position, amplitude, width) and Pu = 160 τ = 32 SW H = 4.
Summary
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MLE (black) 150
– Estimation strategy based on MLE principle for Brown + peak model.
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MLE and MLE−peak on a peaky echo
MLE and MLE−peak on a Brown model
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MLE (in black)
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MLE−peak (red)
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Classes 6 (left) and 10 (right).
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MLE (in black) superimposed with MLE−peak (in read)
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Generalization
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MLE (black)
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– Multiple Gaussian peaks and other shapes of peaks.
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MLE and MLE−peak on a peaky echo
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MLE and MLE−peak on a peaky echo
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MLE (in black)
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– Validation on simulated and real signals : this model improves parameter estimation for waveforms corrupted by peaks.
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– Validation using additional real datasets.
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Classes 13 (left) and 9 (right).
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MLE−peak (in red)
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MLE−peak (in red)
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References
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MLE (black) MLE and MLE−peak on a peaky echo
MLE and MLE−peak on a peaky echo
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1. Dumont, J.P. (1985). Estimation optimale des param`etres altim´etriques des signaux radar Pos´eidon. PhD thesis, Institut National Polytechnique (INP), Toulouse, France.
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MLE−peak (in red)
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Classes 13 (left) and 4 (right).
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MLE−peak (red)
Examples of simulated peaky altimetric waveforms.
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Pu MLE 156.45 MLE-peak 156.45 MLE 167.84 MLE-peak 157.82 MLE 162.17 MLE-peak 160.18
τ 31.99 31.98 32.36 32.11 32.2 32.16
SWH 4.26 4.25 4.57 3.81 3.58 3.53
Pu 167.65 159.34 169.44 161.87 177.93 158.33
τ 31.99 31.83 32.39 32.17 32.41 31.96
Corresponding estimated parameters.
SWH 3.71 3.49 5.08 4.58 4.82 3.8
MLE: algorithm has diverged
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2. G´omez-Enri, J., S. Vignudelli, G.D. Quartly, C.P. Gommenginger, P. Cipollini, P.G. Challenor and J. Benveniste (2009). Modeling ENVISAT RA-2 waveforms in the coastal zone : case-study of calm water contamination. 3rd Coastal Altimetry Workshop, Frascati (Italy).
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3. Kay S.M. (1993). Fundamentals of Statistical Signal Processing : Estimation Theory. Prentice Hall, Englewood Cliffs.
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MLE−peak (red)
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Classes 3 (left) and 5 (right). MLE and MLE-peak for different waveforms from PISTACH project.
4. Thibaut, P. and J.C. Poisson (2008). Waveform Processing in PISTACH project. 2nd Coastal Altimetry Workshop, Pisa (Italy). 5. Walsh, E.J. (1982). Pulse-to-pulse correlation in satellite radar altimeters. Radio Science, 17(4) :786-800.
