Time series identification and Kalman filtering techniques for Doppler lidar velocity estimation

Barry J. Rye; R. Michael Hardesty

doi:10.1364/AO.28.000879

Applied Optics
Vol. 28,
Issue 5,
pp. 879-891
(1989)
•https://doi.org/10.1364/AO.28.000879

Time series identification and Kalman filtering techniques for Doppler lidar velocity estimation

Barry J. Rye and R. Michael Hardesty

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Barry J. Rye and R. Michael Hardesty, "Time series identification and Kalman filtering techniques for Doppler lidar velocity estimation," Appl. Opt. 28, 879-891 (1989)

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Abstract

The use of recursive techniques based on Kalman filter algorithms for identification of time series system models for Doppler lidar returns and the subsequent filtering and smoothing of measured data is explored. The form of possible stochastic system models is reviewed, and reiterative maximum likelihood and innovation spectral tests are used for identification. It is found that a random walk model is adequate for the returns here, and possible explanations for this are considered. Examples are given to illustrate the extension of our method to real-time applications and on-line outlier rejection.

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Signal model	Sample number
Signal model	1	2	3	4	5
R/Q = 1
Constant	7307(32)	6602(32)	6972(32)	6611(32)	7363(32)
(0,1,0)	6055(5)	6056(5)	6085(3)	6064(5)	6071(10)
(1,1,0)	6055(3)	6056(4)	6085(2)	6064(4)	6071(10)
(0,1,1)	6055(3)	6056(4)	6085(2)	6064(4)	6071(8)
(1,1,1)	6023(3)	6017(0)	6048(1)	6024(1)	6022(2)

R/Q = 100
Constant	5438(11)	5363(4)	5350(3)	5387(1)	5381(21)
(0,1,0)	5399(4)	5363(4)	5335(4)	5383(3)	5320(3)
(1,1,0)	5399(4)	5363(3)	5335(3)	5383(1)	5320(3)
(0,1,1)	5399(4)	5363(3)	5335(3)	5383(1)	5320(3)
(1,1,1)	5398(4)	5357(2)	5228(3)	5380(1)	5318(3)

	Sample number
	1	2	3	4	5
Mean difference	0.07	0.07	0.03	0.05	0.05
Root meansquare difference	0.2	0.19	0.19	0.18	0.18
Filter estimate	0.14	0.13	0.12	0.14	0.13

m	Noise ratio	Model
m	Noise ratio	ARMA (0,1,0)	ARMA (1,1,0)	ARMA (2,1,0)	ARMA (1,1,1)
5/3	0.1	5309 ± 43	5313 ± 43	—	5306 ± 43
	0.3	5404 ± 43	5407 ± 44	—	5401 ± 43
	1	6015 ± 41	6016 ± 42	—	6013 ± 41

8/3	0.1	5353 ± 83	5060 ± 57	5050 ± 56	—
	0.3	5469 ± 73	5284 ± 50	5272 ± 50	—
	1	6202 ± 44	6157 ± 40	6144 ± 40	—

Parameter	Values
Range (km)	3	6	9	12
SNR (dB)	15.1	3.8	−6	−17.3
Estimated rms velocity error m s⁻¹)	0.613	0.92	6.58	23.1

Signal model	Sample number
Signal model	1	2	3	4	5
Constant	5920(3)	5920(7)	6155(32)	5710(29)	6185(32)
(0,1,0)	5914(1)	5898(0)	5659(5)	5639(4)	6031(3)
(1,1,0)	5913(1)	5897(0)	5657(4)	5639(2)	6029(2)
(1,1,1)	5910(1)	5897(3)	5657(3)	5639(2)	6028(2)

	Sample number
	1	2	3	4	5	6	7	8	9	10	11	12
Range 3 km
Data outliers	5	4	7	8	11	86	8	44	4	110	13	5
Q^1/2 (0.01 m/s)	1.8	2.3	2.1	2.8	5.6	1.9	3.7	3.1	4.1	3.3	5.4	1.5
R^1/2 (m/s)	1.3	1.3	1.3	1.1	1.1	1.2	1.0	1.4	1.4	1.1	1.0	1.2
ACF outliers	1	2	2	0	3	0	2	0	3	0	0	3
Range 6 km
Data outliers	12	17	24	20	19	92	22	40	32	101	25	78
Q^1/2 (0.01 m/s)	1.9	3.6	2.9	2.4	6.9	0.0	0.7	2.6	0.0	3.4	1.7	1.6
R^1/2 (m/s)	1.6	1.6	1.5	1.4	1.5	1.5	1.3	1.4	1.6	1.6	1.6	1.7
ACF outliers	1	2	1	3	1	0	0	0	0	0	3	0
Range 9 km
Data outliers	—	—	22	32	33	19	51	41	27	0	22	—
Q^1/2 (0.01 m/s)	—	—	10	7.8	10	5.8	0.0	0.0	0.0	0.0	4.0	—
R^1/2 (m/s)	—	—	7.3	5.9	5.9	7.1	5.7	6.2	6.0	9.0	6.8	—
ACF outliers	—	—	1	1	2	0	0	1	3	0	2
Range 12 km
Data outliers	0	0	0	0	0	0	0	0	0	0	0	0
Q^1/2 (0.01 m/s)	1.8	0	25.0	9.8	6.5	6.3	0	3.6	0	0	9.7	0
R^1/2 (m/s)	12	12	12	12	12	12	11	11	12	12	11	12
ACF outliers	3	2	5	3	5	3	3	1	4	3	1	2

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Applied Optics