B. J. Rye, “Power Ratio Estimation in Incoherent Backscatter Lidar: Heterodyne Receiver with Square Law Detection,” J. Climate Appl. Meteorol. 22, 1899–1913 (1983).

[CrossRef]

R. H. Jones, “Maximum Likelihood Fitting of ARMA Models to Time Series with Missing Observations,” Technometrics 22, 389–395 (1980).

[CrossRef]

D. S. Zrnic, “Mean Power Estimation with a Recursive Filter,” IEEE Trans. Aerosp. Electron. Syst. AES-13, 281–289 (1977).

[CrossRef]

R. K. Mehra, “On the Identification of Variances and Adaptive Kalman Filtering,” IEEE Trans. Autom. Control AC-15, 175–184 (1970).

[CrossRef]

J. A. Nelder, R. Mead, “A Simplex Method for Function Minimization,” Comput. J. 7, 308–313 (1965).

[CrossRef]

R. H. Jones, “Maximum Likelihood Fitting of ARMA Models to Time Series with Missing Observations,” Technometrics 22, 389–395 (1980).

[CrossRef]

J. A. Nelder, R. Mead, “A Simplex Method for Function Minimization,” Comput. J. 7, 308–313 (1965).

[CrossRef]

R. K. Mehra, “On the Identification of Variances and Adaptive Kalman Filtering,” IEEE Trans. Autom. Control AC-15, 175–184 (1970).

[CrossRef]

A. P. Sage, J. L. Melsa, Estimation Theory with Applications to Communications and Control (McGraw-Hill, New York, 1971).

J. A. Nelder, R. Mead, “A Simplex Method for Function Minimization,” Comput. J. 7, 308–313 (1965).

[CrossRef]

B. J. Rye, R. M. Hardesty, “Time Series Identification and Kalman Filtering Techniques for Doppler Lidar Velocity Estimation,” Appl. Opt. 28, 879–891 (1989).

[CrossRef]
[PubMed]

B. J. Rye, “A Wavelength Switching Algorithm for Single Laser Differential Absorption Lidar Systems,” Proc. Soc. Photo-Opt. Instrum. Eng. 1062, 267–273 (1989).

B. J. Rye, “Power Ratio Estimation in Incoherent Backscatter Lidar: Heterodyne Receiver with Square Law Detection,” J. Climate Appl. Meteorol. 22, 1899–1913 (1983).

[CrossRef]

A. P. Sage, J. L. Melsa, Estimation Theory with Applications to Communications and Control (McGraw-Hill, New York, 1971).

D. S. Zrnic, “Mean Power Estimation with a Recursive Filter,” IEEE Trans. Aerosp. Electron. Syst. AES-13, 281–289 (1977).

[CrossRef]

J. A. Nelder, R. Mead, “A Simplex Method for Function Minimization,” Comput. J. 7, 308–313 (1965).

[CrossRef]

D. S. Zrnic, “Mean Power Estimation with a Recursive Filter,” IEEE Trans. Aerosp. Electron. Syst. AES-13, 281–289 (1977).

[CrossRef]

R. K. Mehra, “On the Identification of Variances and Adaptive Kalman Filtering,” IEEE Trans. Autom. Control AC-15, 175–184 (1970).

[CrossRef]

B. J. Rye, “Power Ratio Estimation in Incoherent Backscatter Lidar: Heterodyne Receiver with Square Law Detection,” J. Climate Appl. Meteorol. 22, 1899–1913 (1983).

[CrossRef]

B. J. Rye, “A Wavelength Switching Algorithm for Single Laser Differential Absorption Lidar Systems,” Proc. Soc. Photo-Opt. Instrum. Eng. 1062, 267–273 (1989).

R. H. Jones, “Maximum Likelihood Fitting of ARMA Models to Time Series with Missing Observations,” Technometrics 22, 389–395 (1980).

[CrossRef]

Because the approximation described makes the linear filter slightly suboptimal and might arguably lead to results that are prejudiced against it, the process was repeated with the power, rather than the log power, generated using a random walk [using Eqs. (6a) and (2)] and filtered optimally with the constant value for Q1 from the simulation; the log power was then filtered suboptimally using a variable Q1 generated by Eq. (14). The conclusions drawn from the results were unaffected by these changes.

Inspection of Eqs. (13c) and (13d) indicates that, if the variance terms are normalized12 to Q1, the unknowns can be combined to leave only two, Q1m and Q1R. It is believed that physical interpretation calls for knowledge of all three despite the additional computational burden entailed.

A. Gelb, Ed., Applied Optimal Estimation (MIT Press, Cambridge, MA, 1974).

A. P. Sage, J. L. Melsa, Estimation Theory with Applications to Communications and Control (McGraw-Hill, New York, 1971).

An alternative approach to the measurement Eq. (4) is logarithmic transformation of the measurement; using y = ln[Y − U], x = ln[S], w = ln[W], and in the absence of additive noise, we then obtain a linear measurement equation like that of Eq. (2) but with w appearing as the additive noise term. A complication arises if w does not have zero mean. If x is known to be constant and the statistics of w are stationary, the problem is simply one of determining the resulting bias in the average14; otherwise it is necessary to show that the variation of the bias is negligible (e.g., less than other sources of error) over the range of parameters encountered. For differential log ratio measurements of the form x = ln[S1/S2], y = ln[(Y1 − U1)/(Y2 − U2)], the problem is mitigated because w = ln[W1/W2] does have zero mean provided the statistics of W1 and W2 are identical. This removes bias in the absence of extra additive noise and leads7 to relatively small bias provided the variances R1 and R2 of this noise are small or S1/S2 ∼ R1R2.

Strictly P can only be interpreted as the covariance of the estimate for a linear filter with known system model. For nonlinear filters, including adaptive filters designed to determine the properties of an unknown system model, P should be at best regarded as a useful approximation to the covariance matrix; here we use the term estimate covariance matrix for brevity.