Comparison between the Kalman and the Non-Linear Least-Squares Estimators in Low Signal-to-Noise Ratio Lidar Inversion.
ABSTRACT This works departs from previously published results of the authors and focus on joint estimation and time evolution of the atmospheric backscatter profile and a range-independent lidar ratio by means of 1) adaptive extended Kalman filtering (EKF) and 2) non-linear least-squares (NLSQ), under moderate-to-low signal-to-noise ratios (SNR<100 at the starting sounding range). A Rayleigh/Mie atmosphere and a calibrated lidar system are considered. Performance parameters studied are data sufficiency, tracking of the optical parameter time fluctuations, inversion errors, power estimation, and noise impact. The EKF inversion solution is, in turn, compared with Klett's method as a reference. Finally, it is shown that the EKF outweighs the NSLQ in noisy environments.
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ABSTRACT: A simple analytical method is presented that shows some potential for application to the problem of extracting attenuation and backscatter coefficients in an inhomogeneous atmosphere from the return signal of a monostatic single-wavelength lidar system. The method assumes the validity of the single-scattering lidar equation and a power law relationship between backscatter and attenuation. For optical depths greater than unity the inversion method can be applied in principle using only information contained in the signal itself. In contrast to a well-known related analytical inversion solution, the new solution form is shown to be stable with respect to perturbations in the signal, the postulated relationship between backscatter and attenuation, and the assumed or estimated boundary value of attenuation.Applied Optics 01/1981; 20(2):211-20. · 1.69 Impact Factor
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ABSTRACT: Recursive estimation of nonlinear functions of the return power in a lidar system entails use of a nonlinear filter. This also permits processing of returns in the presence of multiplicative noise (speckle). The use of the extended Kalman filter is assessed here for estimation of return power, log power, and speckle noise (which is regarded as a system rather than a measurement component), using coherent lidar returns and tested with simulated data. Reiterative processing of data samples using system models comprising a random walk signal together with an uncorrelated speckle term leads to self-consistent estimation of the parameters.Applied Optics 09/1989; 28(18):3908-17. · 1.69 Impact Factor
- 01/1997; Wiley New York.
COMPARISON BETWEEN THE KALMAN AND THE NON-LINEAR LEAST-SQUARES
ESTIMATORS IN LOW SIGNAL-TO-NOISE RATIO LIDAR INVERSION
Francesc Rocadenbosch, Michaël Sicard, Adolfo Comerón, M. N. Md. Reba, and Adriano Camps
Universitat Politècnica de Catalunya (UPC), Dep. of Signal Theory and Communications (TSC),
Remote Sensing Lab (RSLAB), C/Jordi Girona, 1-3, D4-016, 08034 Barcelona, SPAIN.
Email: firstname.lastname@example.org, Phone: +34 93-401-60-85, Fax: +34 93-401-72-32
The comparatively recent application of low-power high-repetition-rate diode light sources to low-cost backscatter lidars
as compared to classical high-energy low-repetition rate laser sources opens the research framework of low signal-to-
noise ratio (SNR) inversion methods. Thus, micro-pulse lidars usually operate with energies in the 5-40 ?J range and
repetition rates in the kHz region to achieve, e.g., 30-60-s time resolution and 30-to-75-m spatial resolution using photon
counting detection. In comparison to classic 0.1-1-J energy, 10-50-Hz repetition-rate laser sources with the same
resolutions, this represents a 40-80 dB reduction in the SNR (this figure being defined at the receiver’s voltage output).
Independent inversion of the optical atmospheric parameters of interest, namely the aerosol extinction, the aerosol
backscatter, and the lidar ratio can only be tackled by combining at least one elastic and one inelastic Raman channel,
multiple zenith-angle elastic returns under the assumption of a homogeneously horizontally stratified atmosphere or by
means of a HSRL (High Spectral Resolution Lidar).
In the single-scatter elastic lidar equation, range-dependent inversion of the sought-after optical parameters requires both
the introduction of “a priori” correlation hypotheses between the extinction and the backscatter profiles such as the
assumption of a linear/power-law dependency between the optical parameters (Klett’s method, ) or the assumption of
a range-dependent aerosol lidar ratio (Klett-Fernald-Sasano’s (KFS) method ), plus a boundary calibration.
Besides, a temperature/pressure balloon-borne measurement (or a US-standard atmospheric model) is used to separate
the molecular from the total optical components inverted by the KFS method.
This paper focus on joint estimation of a range-independent lidar ratio and time evolution of the atmospheric backscatter
profile by means of 1) adaptive inversion based on extended Kalman filtering (EKF)  and 2) non-linear least-squares
estimation (NLSQ) , under moderate-to-low signal-to-noise ratios (typically below 20 dB). This preliminary
comparative study follows a computer intensive simulation approach and assumes a moderately turbid atmosphere (i.e., a
one component atmosphere) so that the aerosol component can be assumed dominant along the inversion range (the
boundary layer). The paper discusses on the estimation problem in terms of data sufficiency, noise impact, and model
parameters (in the case of the EKF), with reference to previously published works . The former point of data
sufficiency, i.e., the classical question of how to retrieve two unknowns (the backscatter and the lidar ratio in this case)
from one single equation (the observable noisy lidar power), is solved here by introducing the concept of data decimation
in the backscatter estimates. This means that the backscatter coefficient is estimated in less inversion cells than all the
available power-measurement cells (observation cells), following a 1-to-M ratio, so as to guarantee data sufficiency.
Consequently, the inversion resolution is reduced by the same factor M as compared to the (raw-data) observation
Performance parameters taken into account are: errors in the sought-after optical estimates, tracking capability on the
atmospheric backscatter time fluctuations, observable power de-noising capability, and initial user-error variance
reduction. Whenever possible, the time-animated examples shown for both estimators are, in turn, compared with the
single-profile time-averaged Klett’s solution (or its variants) as reference. In the case of Klett’s inversion, time/spatial
averaging of the whole time-animated set of simulated input lidar records is necessary to boost the signal-to-noise ratio
to suitable levels apt for inversion.
Finally, a preliminary real data example is presented, where obviously the filter’s model is just a rough reasonable
approximation to the true blind atmospheric one.
Keywords: lidar, inversion, Kalman filter, least-squares, control theory.
The authors wish to acknowledge the following entities for partially supporting this research work and lidar systems
developed at UPC: European Commission under the EARLINET-ASOS (EU Coordination Action) contract nº 025991
(RICA), and (EU Specific Support Action) contract nº 011863 (RIDS): “Technology development programme towards a
European Extremely Large Telescope”; MCYT (Spanish Ministry of Science and Technology) and FEDER funds under
the projects TEC2006-07850/TCM, Complementary Actions CGL2006-26149-E/CLI, CTM2006-27154-E/TECNO, and
Special Action REN2002-12784-E; MITYC (Spanish Ministry of Industry, Tourism and Commerce) under the PROFIT
project, CIT-020400-2005-56. MCYT is also thanked for the Ramón y Cajal position hold by Dr. M. Sicard, and Local
Government of Catalonia (Generalitat de Catalunya/AGAUR) for Mr. Md. Reba’s predoctoral fellowship.
We also thank the fruitful discussions about the problem provided by Dr. Gregori Vázquez, Dep. Signal Theory and
Communications (TSC), Universitat Politècnica de Catalunya (UPC).
J.D. Klett, "Stable analytical inversion solution for processing lidar returns," Appl. Opt. 20, 211-220 (1985).
F.G. Fernald, “Analysis of Atmospheric Lidar Observations: Some Comments,” Appl. Opt. 23, 652-3 (1984).
J.D. Klett, "Lidar Inversion with variable backscatter/extinction ratios," Appl. Opt. 24, 1638-1643 (1985).
R.G. Brown, P.Y.C. Hwang, Introduction to Random Signals and Applied Kalman Filtering (Wiley, New York, 1992).
R.J. Barlow, “Least Squares”, Chap.6 in Statistics: A Guide To The Use Of Statistical Methods In The Physical Sciences,
(Wiley, New York, 1989).
F. Rocadenbosch, G. Vázquez, A. Comerón, “Adaptive Filter Solution For Processing Lidar Returns: Optical Parameter
Estimation,” Appl. Opt., 37, 7019-7034 (1998).
B.J. Rye and R.M. Hardesty, "Nonlinear Kalman filtering techniques for incoherent backscatter LIDAR: Return Power and Log
Power Estimation," Appl. Opt. 28, 3908-3917 (1989).
D.G. Lainiotis, P. Papaparaskeva, G. Kothapalli, K. Plataniotis., "Adaptive Filter Applications to LIDAR: Return Power and Log
Power Estimation," IEEE Trans. Geosci. Remote Sensing 34, 886-891 (1996).