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Applied Optics

Applied Optics


  • Editor: James C. Wyant
  • Vol. 47, Iss. 24 — Aug. 20, 2008
  • pp: 4309–4320

Simultaneous estimation of aerosol cloud concentration and spectral backscatter from multiple-wavelength lidar data

Russell E. Warren, Richard G. Vanderbeek, Avishai Ben-David, and Jeffrey L. Ahl  »View Author Affiliations

Applied Optics, Vol. 47, Issue 24, pp. 4309-4320 (2008)

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We present a sequential algorithm for estimating both concentration dependence on range and time and backscatter coefficient spectral dependence of optically thin localized atmospheric aerosols using data from rapidly tuned lidar. The range dependence of the aerosol is modeled as an expansion of the concentration in an orthonormal basis set whose coefficients carry the time dependence. Two estimators are run in parallel: a Kalman filter for the concentration range and time dependence and a maximum-likelihood estimator for the aerosol backscatter wavelength and time dependence. These two estimators exchange information continuously over the data-processing stream. The state model parameters of the Kalman filter are also estimated sequentially together with the concentration and backscatter. Lidar data collected prior to the aerosol release are used to estimate the ambient lidar return. The approach is illustrated on atmospheric backscatter long-wave infrared ( CO 2 ) lidar data.

© 2008 Optical Society of America

OCIS Codes
(010.1100) Atmospheric and oceanic optics : Aerosol detection
(010.1120) Atmospheric and oceanic optics : Air pollution monitoring
(010.3640) Atmospheric and oceanic optics : Lidar
(280.1415) Remote sensing and sensors : Biological sensing and sensors
(010.1350) Atmospheric and oceanic optics : Backscattering

ToC Category:
Atmospheric and Oceanic Optics

Original Manuscript: May 2, 2008
Revised Manuscript: July 2, 2008
Manuscript Accepted: July 9, 2008
Published: August 12, 2008

Russell E. Warren, Richard G. Vanderbeek, Avishai Ben-David, and Jeffrey L. Ahl, "Simultaneous estimation of aerosol cloud concentration and spectral backscatter from multiple-wavelength lidar data," Appl. Opt. 47, 4309-4320 (2008)

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  1. The text Elastic Lidar, by V. A. Kovalev and W. E. Eichinger (Wiley, 2004) has an extensive treatment and bibliography of lidar analysis methods for aerosols distributed continuously over extended paths. [CrossRef]
  2. We are assuming a single aerosol material here. Generalization to more than one material present at a given time and range is currently being investigated.
  3. A note on units. From Eqs. we see that the physical units of αC must be those of β, 1/(m sr), for example. Since concentration is typically expressed in units of mg/m3, this requires the units of α to be m2/(mg sr). Setting g=1 of course means that these units no longer apply and, consequently, only relative backscatter and concentration can be estimated by this choice. This is described in more detail in Section .
  4. B. R. Frieden, Probability, Statistical Optics, and Data Testing (Springer-Verlag, 1983).
  5. A Gaussian model for Sg is not the only choice, of course. Using a step function with width rg would give Sg(f)=σg2rgsinc⁡(frg). This model has many more ripples than the Gaussian and provides less noise damping for a given rg.
  6. Even with the choice of maximum likelihood, there is the decision of whether the estimates should be computed independently for each time step or be processed by a recursive algorithm, such as recursive least squares. We have adopted the former choice here since the estimates are to be used to train and test classifiers for which independent training samples are required. Actual deployment may be better served by a recursive algorithm.
  7. A related concern is the huge amount of background data needed to construct suitable sample estimates of the range-covariance matrices. This is not compatible with the limited amount of background data typically available.
  8. See S. Mallat, Wavelet Tour of Signal Processing, Section 9.1.3 and Appendix A6, for an excellent discussion of the KL expansion (Academic, 1998).
  9. If our covariance function, given by Eq. , had circular periodicity, Mallat shows that the Fourier basis would be the KL basis. We cannot claim this, but for target aerosols not close to either end of the sampling range, there is little error in making this assumption.
  10. See, for example, F. L. Lewis, Optimal Estimation (Wiley, 1986).
  11. A. P. Dempster, N. M. Laird, and D. B. Rubin, “Maximum likelihood from incomplete data via the EM algorithm (with discussion),” J. R. Stat. Soc. Ser. B. Methodol. 39, 1-38 (1977).
  12. R. H. Shumway and D. S. Stoffer, Time Series Analysis and Its Applications (Springer-Verlag, 2000), Chap. 4.
  13. S. M. Kay, Fundamentals of Statistical Signal Processing: Estimation Theory (Prentice Hall, 1993) has very readable treatments of ML and related topics.

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