Abstract

The lidar signal-to-noise ratio decreases rapidly with an increase in range, which severely affects the retrieval accuracy and the effective measure range of a lidar based on the Fernald method. To avoid this issue, an alternative approach is proposed to simultaneously retrieve lidar data accurately and obtain a de-noised signal as a by-product by combining the ensemble Kalman filter and the Fernald method. The dynamical model of the new algorithm is generated according to the lidar equation to forecast backscatter coefficients. In this paper, we use the ensemble sizes as 60 and the factor δ1/2 as 1.2 after being weighed against the accuracy and the time cost based on the performance function we define. The retrieval and de-noising results of both simulated and real signals demonstrate that our method is practical and effective. An extensive application of our method can be useful for the long-term determining of the aerosol optical properties.

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References

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2012 (1)

2011 (3)

2010 (1)

2008 (1)

P. Sakov and P. R. Oke, “A deterministic formulation of the ensemble Kalman filter: an alternative to ensemble square root filters,” Tellus, Ser. A, Dyn. Meterol. Oceanogr.60(2), 361–371 (2008).
[CrossRef]

2005 (1)

2004 (1)

H. T. Fang and D. S. Huang, “Noise reduction in lidar signal based on discrete wavelet transform,” Opt. Commun.233(1-3), 67–76 (2004).
[CrossRef]

2003 (2)

V. A. Kovalev, “Stable near-end solution of the lidar equation for clear atmospheres,” Appl. Opt.42(3), 585–591 (2003).
[CrossRef] [PubMed]

G. Evensen, “The ensemble Kalman filter: Theoretical formulation and practical implementation,” Ocean Dyn.53(4), 343–367 (2003).
[CrossRef]

1999 (2)

J. L. Anderson and S. L. Anderson, “A Monte Carlo implementation of the nonlinear filtering problem to produce ensemble assimilations and forecasts,” Mon. Weather Rev.127(12), 2741–2758 (1999).
[CrossRef]

F. Rocadenbosch, C. Soriano, A. Comerón, and J. M. Baldasano, “Lidar inversion of atmospheric backscatter and extinction-to-backscatter ratios by use of a Kalman filter,” Appl. Opt.38(15), 3175–3189 (1999).
[CrossRef] [PubMed]

1996 (1)

1994 (2)

R. N. Miller, M. Ghil, and F. Gauthiez, “Advanced data assimilation in strongly nonlinear dynamical systems,” J. Atmos. Sci.51(8), 1037–1056 (1994).
[CrossRef]

G. Evensen, “Sequential data assimilation with a nonlinear quasi-geostrophic model using Monte Carlo methods to forecast error statistics,” J. Geophys. Res.99(C5), 143–162 (1994).
[CrossRef]

1984 (1)

1981 (1)

1966 (1)

R. Collis, “Lidar: a new atmospheric probe,” Q. J. R. Meteorol. Soc.92(392), 220–230 (1966).
[CrossRef]

1960 (1)

R. E. Kalman, “A new approach to linear filtering and prediction problems,” J. Basic Eng.82(1), 35–45 (1960).
[CrossRef]

Anderson, J. L.

J. L. Anderson and S. L. Anderson, “A Monte Carlo implementation of the nonlinear filtering problem to produce ensemble assimilations and forecasts,” Mon. Weather Rev.127(12), 2741–2758 (1999).
[CrossRef]

Anderson, S. L.

J. L. Anderson and S. L. Anderson, “A Monte Carlo implementation of the nonlinear filtering problem to produce ensemble assimilations and forecasts,” Mon. Weather Rev.127(12), 2741–2758 (1999).
[CrossRef]

Baldasano, J. M.

Collis, R.

R. Collis, “Lidar: a new atmospheric probe,” Q. J. R. Meteorol. Soc.92(392), 220–230 (1966).
[CrossRef]

Comerón, A.

Evensen, G.

G. Evensen, “The ensemble Kalman filter: Theoretical formulation and practical implementation,” Ocean Dyn.53(4), 343–367 (2003).
[CrossRef]

G. Evensen, “Sequential data assimilation with a nonlinear quasi-geostrophic model using Monte Carlo methods to forecast error statistics,” J. Geophys. Res.99(C5), 143–162 (1994).
[CrossRef]

Fang, H. T.

H. T. Fang, D. S. Huang, and Y. H. Wu, “Antinoise approximation of the lidar signal with wavelet neural networks,” Appl. Opt.44(6), 1077–1083 (2005).
[CrossRef] [PubMed]

H. T. Fang and D. S. Huang, “Noise reduction in lidar signal based on discrete wavelet transform,” Opt. Commun.233(1-3), 67–76 (2004).
[CrossRef]

Feiyue, M.

Fernald, F. G.

Gauthiez, F.

R. N. Miller, M. Ghil, and F. Gauthiez, “Advanced data assimilation in strongly nonlinear dynamical systems,” J. Atmos. Sci.51(8), 1037–1056 (1994).
[CrossRef]

Ghil, M.

R. N. Miller, M. Ghil, and F. Gauthiez, “Advanced data assimilation in strongly nonlinear dynamical systems,” J. Atmos. Sci.51(8), 1037–1056 (1994).
[CrossRef]

Gong, W.

Huang, D. S.

H. T. Fang, D. S. Huang, and Y. H. Wu, “Antinoise approximation of the lidar signal with wavelet neural networks,” Appl. Opt.44(6), 1077–1083 (2005).
[CrossRef] [PubMed]

H. T. Fang and D. S. Huang, “Noise reduction in lidar signal based on discrete wavelet transform,” Opt. Commun.233(1-3), 67–76 (2004).
[CrossRef]

Kalman, R. E.

R. E. Kalman, “A new approach to linear filtering and prediction problems,” J. Basic Eng.82(1), 35–45 (1960).
[CrossRef]

Klett, J. D.

Kovalev, V. A.

Li, J.

W. Gong, J. Li, F. Mao, and J. Zhang, “Comparison of simultaneous signals obtained from a dual-field-of-view lidar and its application to noise reduction based on empirical mode decomposition,” Chin. Opt. Lett.9(5), 050101–050104 (2011).
[CrossRef]

W. Gong, F. Mao, and J. Li, “OFLID: Simple method of overlap factor calculation with laser intensity distribution for biaxial lidar,” Opt. Commun.284(12), 2966–2971 (2011).
[CrossRef]

Mao, F.

Miller, R. N.

R. N. Miller, M. Ghil, and F. Gauthiez, “Advanced data assimilation in strongly nonlinear dynamical systems,” J. Atmos. Sci.51(8), 1037–1056 (1994).
[CrossRef]

Oke, P. R.

P. Sakov and P. R. Oke, “A deterministic formulation of the ensemble Kalman filter: an alternative to ensemble square root filters,” Tellus, Ser. A, Dyn. Meterol. Oceanogr.60(2), 361–371 (2008).
[CrossRef]

Reba, M. N.

Rocadenbosch, F.

Sakov, P.

P. Sakov and P. R. Oke, “A deterministic formulation of the ensemble Kalman filter: an alternative to ensemble square root filters,” Tellus, Ser. A, Dyn. Meterol. Oceanogr.60(2), 361–371 (2008).
[CrossRef]

Sasano, Y.

Sicard, M.

Soriano, C.

Wei, G.

Wu, Y. H.

Yingying, M.

Zhang, J.

Zhu, Z.

Appl. Opt. (8)

Chin. Opt. Lett. (1)

J. Atmos. Sci. (1)

R. N. Miller, M. Ghil, and F. Gauthiez, “Advanced data assimilation in strongly nonlinear dynamical systems,” J. Atmos. Sci.51(8), 1037–1056 (1994).
[CrossRef]

J. Basic Eng. (1)

R. E. Kalman, “A new approach to linear filtering and prediction problems,” J. Basic Eng.82(1), 35–45 (1960).
[CrossRef]

J. Geophys. Res. (1)

G. Evensen, “Sequential data assimilation with a nonlinear quasi-geostrophic model using Monte Carlo methods to forecast error statistics,” J. Geophys. Res.99(C5), 143–162 (1994).
[CrossRef]

Mon. Weather Rev. (1)

J. L. Anderson and S. L. Anderson, “A Monte Carlo implementation of the nonlinear filtering problem to produce ensemble assimilations and forecasts,” Mon. Weather Rev.127(12), 2741–2758 (1999).
[CrossRef]

Ocean Dyn. (1)

G. Evensen, “The ensemble Kalman filter: Theoretical formulation and practical implementation,” Ocean Dyn.53(4), 343–367 (2003).
[CrossRef]

Opt. Commun. (2)

H. T. Fang and D. S. Huang, “Noise reduction in lidar signal based on discrete wavelet transform,” Opt. Commun.233(1-3), 67–76 (2004).
[CrossRef]

W. Gong, F. Mao, and J. Li, “OFLID: Simple method of overlap factor calculation with laser intensity distribution for biaxial lidar,” Opt. Commun.284(12), 2966–2971 (2011).
[CrossRef]

Opt. Lett. (1)

Q. J. R. Meteorol. Soc. (1)

R. Collis, “Lidar: a new atmospheric probe,” Q. J. R. Meteorol. Soc.92(392), 220–230 (1966).
[CrossRef]

Tellus, Ser. A, Dyn. Meterol. Oceanogr. (1)

P. Sakov and P. R. Oke, “A deterministic formulation of the ensemble Kalman filter: an alternative to ensemble square root filters,” Tellus, Ser. A, Dyn. Meterol. Oceanogr.60(2), 361–371 (2008).
[CrossRef]

Other (1)

V. A. Kovalev and W. E. Eichinger, Elastic lidar: theory, practice, and analysis methods (Wiley-Interscience, 2004).

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Figures (7)

Fig. 1
Fig. 1

Flow of the retrieval and de-noising algorithm

Fig. 2
Fig. 2

(a) The true, noise contaminated and de-noised normalized range-corrected signals, ln[X(r)], respectively. (b) The true aerosol backscatter coefficient, and the aerosol backscatter coefficients retrieved by the normal Fernald method and our method with a known boundary point and lidar ratio, respectively.

Fig. 3
Fig. 3

Variations of the performance function along with the change of N and δ1/2.

Fig. 4
Fig. 4

(a) The mean and standard deviation of the noise contaminated and the de-noised normalized range-corrected signals, ln[X(r)], respectively. (b) the mean and standard deviation of the aerosol backscatter coefficients retrieved by the normal Fernald method and our method. Note that the error bars in both (a) and (b) are shown with an interval of 40 rang bins to avoid overlay.

Fig. 5
Fig. 5

(a) and (b) The sequential aerosol backscatter coefficients retrieved by the normal Fernald method and our method, respectively, (c) the mean and standard deviation of the normalized range-corrected signals, ln[X(r)], of the observations and the de-noised ones, respectively. (d) the mean and standard deviation of the aerosol backscatter coefficients retrieved by the normal Fernald method and our method. Note that the error bars of (c) and (d) are shown with an interval of 40 rang bins to avoid overlay. Furthermore, parts of the error bars above 9 km in (d) are hidden because negative values cannot be shown in the logarithmic axis.

Fig. 6
Fig. 6

(a)-(f) Six cases of results retrieved by the normal Fernald method and our method, as well as results of 64 minutes averaged signals, which are centered on the one-minute signals, retrieved by the normal Fernald method.

Fig. 7
Fig. 7

(a)-(f) are enlarged subsection views of Figs. 6(a)-6(f), respectively.

Equations (18)

Equations on this page are rendered with MathJax. Learn more.

P(r)= C r 2 G( r )[ β 1 (r)+ β 2 (r) ]exp{ 2 0 r [ α 1 (r)+ α 2 (r) ]dr }+e( r ).
X ratio (i)= X(i) X(i-1) = β 1 (i)+ β 2 (i) β 1 (i-1)+ β 2 (i-1) exp{ 2[ α 1 (i)+ α 2 (i) ]Δr },
X en j (i1)= X en j (i)/ X ratio (i).
X k =M( X k1 )+ ε k ,
Y k =H( X k )+ e k ,
X k f,(i) =M( X k1 a,(i) ),
X k f ¯ = 1 N i=1 N X k f,(i) ,
P k f = 1 N1 i=1 N [ X k f,(i) X f ¯ ] [ X k f,(i) X f ¯ ] T ,
P k f H T = 1 N1 i=1 N [ X k f,(i) X f ¯ ] [H( X k f,(i) )H( X f ¯ )] T ,
H P k f H T = 1 N1 i=1 N [H( X k f,(i) )H( X f ¯ )] [H( X k f,(i) )H( X f ¯ )] T ,
K k = P k f H T [H P k f H T + R k ] 1 ,
X k a,(i) = X k f,(i) + K k [ y k H( X k f,(i) )],
X k a ¯ = 1 N i=1 N X k a,(i) ,
P k a = 1 N1 i=1 N [ X k a,(i) X f ¯ ] [ x k a,(i) X f ¯ ] T ,
{ A a,(i) = X a,(i) X a ¯ X new a,(i) = X a ¯ + δ 1/2 A a,(i) .
β 1 ( i1 )= X( i1 )exp[ A( i1 ) ] X( i ) β( i ) + S 1 { X( i )+X( i1 )exp[ A( i1 ) ] }Δr β 2 ( i1 ),
β 1 ( i+1 )= X( i+1 )exp[ A( i ) ] X( i ) β( i ) S 1 { X( i )+X( i+1 )exp[ A( i ) ] }Δr β 2 ( i+1 ),
F(N, δ 1/2 )= i=1 i= i max { [ β 1 ,R (i) β 1 (i) ] / β 1 (i) } 2 .

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