nonlinear kalman filtering techniques for incoherent backscatter lidar: return power and log power...

Nonlinear Kalman filtering techniques for incoherentbackscatter lidar: return power andlog power estimation

Barry J. Rye and R. Michael Hardesty

Recursive estimation of nonlinear functions of the return power in a lidar system entails use of a nonlinearfilter. This also permits processing of returns in the presence of multiplicative noise (speckle). The use ofthe extended Kalman filter is assessed here for estimation of return power, log power, and speckle noise (whichis regarded as a system rather than a measurement component), using coherent lidar returns and tested withsimulated data. Reiterative processing of data samples using system models comprising a random walksignal together with an uncorrelated speckle term leads to self-consistent estimation of the parameters.

1. Introductionc

In an earlier paper' we considered stochastic modelidentification and adaptive processing in the contextof atmospheric Doppler lidar returns. One applica-tion of the Kalman filtering approach developed thereis estimation of a variable unknown only through anonlinear measurement equation. This is of consider-able importance for lidar signal analysis because non-linearities occur (1) when estimating a variable that is anonlinear function of the measured parameter(s), suchas the return-power ratio in polarization studies, or thelogarithm of the return power in differential absorp-tion lidar (DIAL), and (2) when there is a significantcontribution in the return from speckle, which is amultiplicative noise source. Here we consider estima-tion of return power and the logarithm of return power(or log power) for an incoherent backscatter lidar sys-tem in which speckle is present.

The processing problem considered is that of recur-sive estimation of a parameter, given data in a time

R. M. Hardesty is with NOAA Wave Propagation Laboratory,Boulder, Colorado 80303. When this work was done B. J. Rye waswith University of Hull, North Humberside HU6 7RX, U.K.; he isnow with Cooperative Institute for Research in Environmental Sci-ences, University of Colorado/NOAA, Boulder, Colorado 80309-0449.

Received 7 November 1988.

series format. To illustrate the method, we again use aset of time series of returns obtained using the Nation-al Oceanic and Atmospheric Administration WavePropagation Laboratory coherent lidar. Since this is asingle laser system without an automated wavelengthtuning facility, on-line differential absorption was notconsidered directly, but the generalization to this fromlog power estimation is straightforward. Likewise, thetreatment is readily extended to returns obtained us-ing direct detection; the main problems arise not fromthe mode of detection but from the need for an adap-tive and recursive method for handling the nonlinear-ity in the presence of temporal -fluctuations of themeasured parameter.

The modeling implicit in the Kalman filter algo-rithm leaves room for a degree of initiative and creativ-ity on the part of the system designer. We seek here todescribe and demonstrate some methods wherebythese algorithms may be applied to lidar measure-ments. In Sec. II we consider the form of the measure-ment equation and some possible lidar system modelsin the presence of both additive and multiplicative(speckle) noise giving particular reference to estima-tion of the log power. Results obtained using simulat-ed data relevant to both direct detection and coherentlidars are described in Sec III.A. It is usual in theseapplications that the parameters of the filter are ini-tially unknown, so estimates of these, as well as of thesignal, have to be formed; reiterative procedures em-ployed for this, somewhat similar to those used previ-ously, are discussed with reference to further simulat-ed data in Sec. III.B. The techniques developed areapplied to real data sequences in Sec. IV.

3908 APPLIED OPTICS / Vol. 28, No. 18 / 15 September 1989

(b)

Y(k)

w,(k)

1 + w2(k) W2(k) 1+w2 (k)

w2(k)

Fig. 1. Block diagrams of various measurement models: (a) a linear measurement equation [Eq. (2)]; (b) the system variable is the logarithmof the measurement variable [Eq. (3)]; (c) the system variables are a random walk and multiplicative noise modeled using the linearapproximation [Eqs. (6) and (7a)]; z-1 is the backward shift operator, i.e., z-'[x(k)] = x(k - 1); (d) the logarithm of the first system componentis modeled as a random walk, and the second is multiplicative noise [Eqs. (6) and (7b)]; (e) as (d), except that the multiplicative noise is mod-

eled using the exponential approximation.

y(k)

II. Filter Equations

A. Measurement Equation

The kth discrete measurement of a return signalpower S(k) in the presence of an additive noise V(k)can be written in general terms as

Y(k) = S(k) + V(k) + U, (1)

where Y(k) is the raw measured quantity and U repre-sents any offset. The first problem is to transform thisequation into the form used within the Kalman filteralgorithm. 2 ' 3

In a coherent lidar such an equation can be appliedto the output of the signal processing system followingthe photomixing and detection stages. In what fol-lows we assume square law detection. An offset Uthen occurs because of rectification of noise at thedetector input that originates in the photomixer andintermediate stages (this should be dominated by localoscillator noise in an ideal heterodyne system). If thisnoise power is known, the offset can be calculated anddeducted from the measured power. The measure-ment y(k) to be used within the Kalman filter algo-rithms can then be expressed directly as y(k) = Y(k) -U(k). In a direct detection system, offsets should not(in principle) be present, so U(k) = 0 and y(k) = Y(k).

If the parameter xl(k) to be estimated is the signalpower, i.e., x 1(x) = S(k), and the additive noise V(k) isthe zero mean term v(k), the measurement equationtakes the simple linear form

y(k) = xl(k) + v(k), (2)

as used previously1 for Doppler velocity estimation.Equations (1) and (2) are depicted in Fig. 1(a). Ifhowever we wish to estimate the absorbance of thelidar channel, which is proportional to the logarithm ofthe return power, the measurement equation becomesnonlinear. Writing the absorbance as xl(k) = ln[S(k)],the equation becomes

y(k) = exp[x1 (k)] + v(k) (3)

[Fig. 1(b)]. Evaluation of an estimate xei(k) of x(k)entails minimizing some cost function,2 e.g., in Eq. (3),if the measurements are equally weighted, the sum ofthe squares of the terms y(k)-exp[xi(k)]. The non-linear Kalman filter equations contain the algorithmfor dealing with problems of this type; similar consid-erations apply if the parameter to be estimated is someother nonlinear function of the measurement(s), suchas a ratio.

The treatment of multiplicative noise is somewhatmore complicated. The measurement equation can bewritten generally as

Y(k) = S(k)W(k) + V(k) + U, (4a)

where W(k) is the multiplicative noise. For estimat-ing the return power the assumptions made earlierconcerning the offset and additive noise lead to theequation

y(k) = xI(k)W(k) + v(k). (4b)

15 September 1989 / Vol. 28, No. 18 / APPLIED OPTICS 3909

w2(k)

It remains to find an expression for W(k) that plausi-bly models speckle. Spectrally, W(k) can be expectedto be uncorrelated from pulse to pulse for atmosphericbackscatter. Statistically the density function may benonsymmetrical, although for high-order speckle it isapproximately Gaussian, consistent with the centrallimit theorem. In the linear filtering problem, thestatistics of the noise terms do not enter into the Kal-man algorithm; but the latter leads in general to aminimum variance estimate, that is also a maximumlikelihood estimate if the noise terms are symmetrical-ly distributed. In nonlinear filtering the optimality ofthe algorithm is not well defined, except in those limit-ing cases where it approaches the linear filter. 2 Apragmatic viewpoint must then be adopted, and per-formance tested by simulations; such tests would in-clude realistic noise statistics (see Sec. III).

The properties of W(k) needed here are that (1) itmust have a mean value (W(k)) = 1, because, e.g., ifxl(k) were constant in Eq. (4b), the mean value of they(k) would equal x1, (2) it should be positive, and (3) itshould be expressible in terms of a zero mean noisesequence w(k) for incorporation in the Kalman filtermodel equations. Properties (1) and (3) are satisfiedby substitution of the linear relation W(k) = 1 + w(k),and properties (2) and (3) are satisfied by use of theexponential form W(k) = exp[w(k)]. Each of thesemodels is clearly suboptimal, but this is not criticalsince it can easily be shown that they both lead to thesame processing algorithm within the first-order filterapproximation used in Sec. II.C. The linear formula-tion is used from this point to exemplify the method.

The measurement equation therefore has the form

y(k) = x(k)[1 + w(k)] + v(k), xl = S (5a)

for estimation of return power and

y(k) = exp[x1 (k)][1 + w(k)] + v(k), x = n[S] (5b)

for estimation of log power.4

B. System Equations

A further difference between speckle and additivenoise is that it is intuitively preferable to regard theformer as a system rather than a measurement vari-able; unlike measurement noise, it disappears if thesignal is blocked off. We therefore bring the unknownx1(k) together with the noise term W(k) in Eq. (4a) asthe two components of a system state vector x(k) =[X1(k)x2(k)]T (where the notation xT indicates thetranspose of x) and identify x2(k) with W(k).

The 2(k) values are stationary and can be modeledusing the linear formulation for multiplicative noiseindicated above. A suitable model must also be foundfor the time dependence of x(k). As discussed previ-ouslyl there is no a priori reason for choosing a con-stant signal or any other deterministic model whenconsidering atmospheric measurements, and the sim-plest stochastic generalization of the constant signalmodel is the random walk (RW). We therefore arriveat the two-system model equations:

xl(k) = x(k - 1) + wl(k),

x 2(k) = 1 + W2 (k),

(6a)

(6b)

where the w1,2(k) are independent zero-mean whitenoise sequences, w2 (k) replacing the w(k) in Eq. (5).Use of a white noise model for w2(k) rests on theassumption that speckle is decorrelated betweensuccessive measurements. The RW model Eq. (6a)has also been used by Warren.5 Here its suitability isimplicitly checked by derivation, from the data, of anonzero value for the variance Qi of the wl(k) (Secs.III.B.5 and IV.C) and by statistical tests performed onthe outputs of the filters (see Sec. III.B.3).

In this notation the measurement Eqs.(5a) and (5b)become, respectively,

y(k) = xl(k)x 2(k) + v(k), X1 = S. (7a)

(7b)y(k) = exp[x1 (k)]x2 (k) + v(k), x, = ln[S].

Equations (6) and (7a) are depicted in Fig. 1(c), andEqs. (6) and (7b) in Fig. 1(d); Fig. 1(e) is similar to Fig.1(d) except that the exponential model is used for thespeckle.

C. Filter Algorithm

Casting these equations in the standard form for thenonlinear Kalman filter2 3 we summarize Eq. (6) as

x(k) = f [x(k - 1)] + w(k), (8)

wherethevectorsf(x) = [xi,1]Tandw= [wI,w2]T. Themeasurement equation is expressed in general as

y(k) = h[x(k)] + v(k). (9)

The texts cited indicate a number of algorithms fordealing with such problems. Of these, the extendedKalman filter is the simplest. In the discrete formula-tion3 the predicted estimates xp for the system vari-ables and its covariance matrix6 P p at the kth step arepropagated using

xp(k) = f[Xe(k - 1)],

Pp(k) = F(k)P(k - 1)FT(k) + Q(k),

(l0a)

(lOb)

where xe = [Xel,Xe2]T is the vector containing the esti-mates of x and x 2 , F(k) = df/dx evaluated at x = xp(k),and Q(k) is the system noise covariance matrix. Thevariance V(k) of the innovation sequence is found as

V(k) = H(k)Pp(k)HT(k) + R(k), (lOc)

where H(k) = dh/dx also evaluated at x = xp(k), andR(k) is the covariance matrix of the measurementnoise. The Kalman gain

K(k) = Pp(k)HT(k)V(k)-l

is then used to form the updated estimates

xe(k) = x(k) + K(k)e(k),

P(k) = [I - K(k)H(k)]Pp(k),

(lOd)

(lOe)

(lOf)

where the innovation written in its general form is e(k)= y(k) - h[xp(k)] and I is the identity matrix.


These equations are greatly simplified here becausethe measurement equation is scalar and the systemequations linear. Equations (7) show that h[x(k)] =h[x(k)]; i.e., h is a scalar given by

h[x(k)] x1(k)x2 (k), xl = S.x~k)] = exp[x1(k)]x2(k), xl = n[S].

estimate of the speckle state variable x2 can also beobtained from Eq. (lOe) giving

K 2(k) = H2(k)/[mV(k)],

Xe 2 (k) = 1 + K 2(k)e(k).

(13g)

(13h)

(11)

It then follows that H(k) = H(k) = [Hi(k),H2(k)],where

h(k) = xei(k - 1), Hl(k) = 1, H2 (k) = xei(k - 1), xl = S, (12a)

h(k) = Hl(k) = H2 (k) = exp[xi(k - 1)], x = ln[S],

and we have written h(k) = h[xp(k)]. Likewise e(k),V(k), and R(k) are also found to be scalars e(k), V(k),and R(k), the last being the variance of the measure-ment noise sequence v(k). Comparison of Eqs. (6) and(lOa) shows that F(k) = FT(k) = 1 while in Eq. (lOd)the gain K(k) becomes a vector K(k) = [KI(k),K 2 (k)]T.

We assume wi and w2 are uncorrelated and havevariances Qi and Q2, so Q has components Q11 = Q1, Q22

= Q2, and Q12 = Q21 = 0. If the postdetector signalfrom a coherent lidar is sampled discretely at suffi-ciently long intervals that the speckle is uncorrelated,the normalized or fractional variance of the signal fluc-tuations due to speckle is 1/m, where m is the numberof independent speckle elements sampled. 7 Becausethis is a useful number for characterizing the system,we retain this notation even though uncorrelated dis-crete sampling of the real data below was not used,making m an effective speckle count and in generalnoninteger. The variance Q2 of the noise term W2 inthe system Eq. (6b) is therefore 1/m. Q is unknown inmeteorological measurements and to be determined bymaking the filter adaptive. The detector output alsocontains a rectified noise offset N, knowledge of whichis crucial for determining the accuracy of the estimate.

If we make these substitutions, the algorithm takeson the simple form

y(k) = Y(k) - N, (13a)

e(k) = y(k) - h(k), (13b)

V(k) = H1 (k) 2 [P(k - 1) + Ql] + H2 (k)2 /m + R, (13c)

K1(k) = [P(k - 1) + QJ]H1(k)/V(k), (13d)

xei(k) = xei(k - 1) + Kj(k)e(k), (13e)

P(k) = [1 - K1(k)H1(k)][P(k - 1) + QJ, (13f)

where P(k) = P(k), the other components of thismatrix not being required. These equations can beused for determination of either the return signal pow-er or log power, provided the appropriate values areused for h, H1 , and H2 from Eq. (12). Initial valuesXe(O) and P(O) are conveniently calculated in practiceusing the average and variance of early values withinthe data sample, the large P(O) obtained in this wayquantifying uncertainty in the initial estimate. An

Ill. Examples Using Simulated Data

A. Log Power Estimation Without Speckle

We first justify the introduction of nonlinear filtersby exemplifying the problems that may arise if a non-linear function of the measurement variable is estimat-ed following use of a linear filter. To simplify thediscussion we consider estimation of the log powergiven measurements contaminated by additive noisebut without multiplicative noise; thus speckle is ne-glected, but the example still has application to directdetection systems. The computations were carriedout using programs written in TURBO PASCAL on avariety of IBM and IBM-compatible microcomputers.

Signal sequences, each containing 1000 points, weregenerated using Eq. (6a), and measurement time seriesusing the nonlinear Eq. (3). For processing, the algo-rithm of Eqs. (13a)-(13f) was used, together with ei-ther Eq. (12a) for power estimation (making the equa-tions identical to those of the linear Kalman filter forthis problem) or Eq. (12b) for log power estimation; ineach case N (offset) and H2 (speckle) were set equal tozero. To obtain an optimum filter for estimating logpower, Qi could be set to the value used in generatingthe time series; let us call this constant valueQl[ln(xl)]. Clearly an approximation for the drivingnoise Qi[xi] of a RW system model had to be used whendesigning an (approximately) optimum linear filter forestimating power.8 The relation that the fractionalvariance of a signal is to first order equal to the vari-ance of the natural logarithm of the signal, i.e.,

Ql[xl]/x'- Qjln(xi)], (14)

was exploited by setting Ql[xl] at each increment k tothe variable value Q,[xl] = Ql[ln(xl)]xl(k)2. The mea-surement noise variance used in generating the timeseries was assigned to R in both cases. Initial valueswere set by averaging the first thirty data points forxei(0) and by equating P(O) to R in each case.

A typical data set is shown in Figs. 2(a) and (b). Theestimated standard deviation P(k)1/2 obtained fromthe nonlinear filter for log power is shown in Fig. 2(c);unlike that for the nonadaptive linear filter, this doesnot decrease monotonically to a limiting value but is arandom variable depending on the data set.2 Thedifference between the estimate of the log power signaland its true value in the simulation is shown in Fig. 2(d)normalized to P(k)112; this plot suggests that the latteris indeed a reasonable measure of the standard devi-ation because the signal estimate is almost alwayswithin d3P(k)112 of the true value. The linear filterlikewise gives an estimate of power within three esti-mated standard deviations of the true value (notshown). However, in Fig. 2(e) we show the differencebetween the estimate of log power, obtained using the


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Fig. 2. Comparison of (1) the logarithm of a power estimate that is determined from a linear filter and (2) the log power estimate obtained di-rectly from a nonlinear filter: (a) log power simulated as a random walk with Q = 1.1 10-4 (arbitrary units); (b) simulated powermeasurement data degraded by additive noise; the variance of the latter is R = 105; (c) estimated standard deviation (square root of covarianceestimate) obtained from the nonlinear filter; (d) difference between the nonlinear filter estimate for the log power and the true (simulated) logpower [Fig. 2(a)], as a fraction of the standard deviation estimate from the nonlinear filter [Fig. 2(c)]; (e) difference between the logarithm of

the linear filter estimate for the power and the true log power, as a fraction of the standard deviation estimate from the nonlinear filter.

linear filter, and the true value, normalized to the samevalues of P(k)1/2 as in Fig. 2(d) (i.e., those from thenonlinear filter); this log power estimate is obtained bytaking the natural logarithm of the power estimategiven by the linear filter. It is apparent that althoughthe signal level does not change by more than 30%throughout the time series, the estimate from the lin-

ear filter has a bias potentially much greater than itsstatistical error. Comparison of Figs. 2(a) and (e)shows that the bias crudely follows the profile of the logpower signal. This example confirms the needstressed in Sec. I for optimized nonlinear recursivefiltering if the signal is not constant and if the measure-ment is nonlinear.


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Fig. 3. Evaluation of signal covariance estimates obtained using synthetic data generated using a random walk model for the power signal,

and multiplicative noise generated with the linear model [see Table I and Eq. (6)]: (a) simulated signal including multiplicative noise; (b) esti-mated standard deviation of log power; (c) difference between simulated data and estimated values as a fraction of the estimated standard de-

viation.

B. General Case: Inclusion of Multiplicative Noise andEstimation of Model Parameters

1. Model

The simulation used in Sec. III.A was extended toinclude a multiplicative noise, so that the signal wasdescribed by Eq. (6a) and the measurements by Eq.(7b). To simulate low-order speckle, x2 (k) was gener-ated in Eq. (7b) as noise with chi-square statistics; allother noise terms in the simulation were Gaussian. Asimulated signal, generated with the same randomnumber sequences as for Fig. 2, is shown in Fig. 3(a);the parameters Q1, m, and R used (Table I) were simi-lar to those found for short range (3-km) returns (Sec.IV.D).

2. OptimizationThe parameters m, R, and of course Q, of the real

data in Sec. IV were unknown for the purposes of thisstudy, so the present simulation was used to develop afiltering algorithm for estimation of these parametersas well as the signal.9 Reiterated passes were madethrough the data to jointly optimize signal and param-eter estimates. As before,' the optimization process

made use of an adaptive simplex routine, in whichoptimization is based on the outcome of a single testafter each pass. The routine is straightforward toimplement on microcomputers and was originally de-rived10 because of its suitability for statistical applica-tions. In contrast to our previous work, however, thetests were not applied to the likelihood function, as ourgoal here was not to compare signal models. Insteadthe test was simply that for whiteness of the innovationsequence"; this readily leads to a method that can beused in real time.' Despite the asymmetry of theirdensity function (see Sec. IV below), the innovationsare a white noise sequence with zero mean. In thereiterative approach here, the average of the innova-tion sequence autocovariance function terms over anumber of lags was minimized within the simplex rou-tine; this number was eventually selected to be sixteenas this appeared to be large enough to give stableresults without excessive computation.

3. DiagnosticsThe diagnostics for innovation sequence whiteness

presented in the tables below are (i) the innovationautocorrelation function (ACF) outlier count, an outli-er being defined as an ACF value at a particular lag


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that lies outside the 5% confidence points,": 1.96/n1/2, and (ii) the mean innovation, the mean of theinnovation sequence normalized to its 5% confidencelevel: 1.96[((e(k)2) - (e(k))2)/n]'/ 2, where n is thedata sample size. For each diagnostic the ACF wasexamined over thirty-two lags.

4. Data OutliersAdvantage can be taken of the existence within the

Kalman filter algorithm of an estimate of the innova-tion variance V(k) [Eq. (13c)], which is used here asbefore' to reject data with innovations more than threestandard deviations from the predicted value. Thiscriterion is satisfactory if the innovation is zero meanand Gaussian. A problem arises here from the asym-metry of the speckle density function, so that a morestringent criterion is needed for negative innovationsthat for positive. For a chi-square density function oforder (defined as 2m) equal to 14, the probability ofdata being within three standard deviations above themean is 99.3%; the probability of being within twostandard deviations below the mean is approximatelythe same at 99.7%. We therefore adopted the follow-ing criteria for the outlier rejection:

e(k)2 > 9V(k), e(k) > 0, (15a)

e(k) 2 > 4V(k), e(k) < 0. (15b)

These criteria are of course somewhat arbitrary andare only intended to automatically reject data that arelikely to be spurious. Rejected data were treated asmissing, and the filter estimates were propagated butnot updated., 2

5. ResultsTable I shows the parameters and diagnostics aris-

ing from processing the 1000-point data set derivedfrom the signal in Fig. 3(a). The parameters seemreasonably well determined, except perhaps Q. Theestimated standard deviation of the log power is indi-cated in Fig. 3(b). These values correspond approxi-mately [using Eq. (14)] to fractional standard devi-ations (FSD) of a power estimate of between 7 and 12%and may be compared with a fractional deviation (1/M)"/2 arising from the simulated speckle of 41%. Thegraph in Fig. 3(c) shows that the estimate is self-consis-tent; the differences between the simulated signal andthe estimate are again within three estimated standarddeviations throughout the sample.

IV. Analysis of Real Data

A. Experimental Data

The data used in this study were preprocessed, beingobtained as zero-moment estimates from the Dopplerhardware processer during the same lidar runs as theDoppler data considered in Ref. 1 and described there-in.

Because only a single photodetector was used, thespeckle count m depends on the degree of range inte-gration within a range resolution cell and therefore onthe return signal Doppler bandwidth, which could varywith range and throughout the time series. The localoscillator power however is stabilized, so that the offsetN and the measurement noise variance R may be con-sidered constant. N had been removed from the data,but R remained to be determined.

B. Parameter SelectionAlthough optimization techniques have been shown

capable of determining the three unknown parametersm, R, and Q given no a priori information in thecontext of simulated data, we preferred to place con-straints on the values of m and R when dealing withreal data. Approximate values for R were found fromthe long range (12-km) returns where the signal issmall. Likewise approximate values for m were ob-tained from estimates of the fractional noise in thereturns at short range (3 km) where measurementnoise is small. These were found simply by averagingthe data, so the values could not be used directly unlessthe signal was believed to be constant. Within theoptimization algorithm, m and R were each con-strained to a range of values containing these averages.For R this range lay between 200 and 300 (squaredarbitrary units in which the return power is expressed,see Tables I-III), and for m, between 5 and 8. Withineach 1000-point data set, m and R were modeled asconstants to be determined.

C. Results

The optimization algorithm and diagnostic checkswere those described in Sec. III.B.

1. Single Data SampleTo illustrate results obtained with the technique,

Figs. 4(a)-(c) show values and diagnostic checks forboth power and log power estimates using a single-

Table 1. Nonlinear Filter Output Using a Smulated Data Sequencea

DiagnosticsInnovations

ACF DataParameters Normalized outlier Outlier

Values Q1 m R mean count count

Simulation 1000 6.0 240 - - -Filter Output 1496 6.9 249 -0.96 1 12

a Comparison of true (simulation) parameters with filter output and filter performance diagnostics.Other output from the filter is shown in Fig. 3. The units for the system and measurement noisevariances Q and R are arbitrary.


return data sample of 1000 points from a range of 3 km.Details of the parameters obtained are given in TableII in a format similar to that of Table I. A perceptiveinternal reviewer has noted that the number of dataoutliers for this particular sample is somewhat belowwhat would be expected.

The raw data are superimposed on the power esti-mate in Fig. 4(a), and-the log power estimate is shownin Fig. 4(b). Estimates from the filters of the FSD inthe power and the standard deviation of the log powerare each -7-8%; as for the simulation (Sec. III.B.5)these uncertainties are attributable to a combinationof speckle and atmospheric fluctuations. A conse-quence of the high estimate uncertainty is that, unlikethe simulation of Fig. 2, the natural logarithm of thepower estimate does not in this case differ from the logpower estimate by more than a standard deviation(now shown); if the noise level is sufficiently high theoutput is insensitive to subtleties of the processing.

Figure 4(c) is a histogram showing the frequencydistribution of the power filter estimates of the secondstate variable, the speckle. This is compared withwhat is expected for a chi-square distributed variableof order 2m 14, chosen to approximate the data of

Table II. Agreement is reasonable, and there is evi-dence in the experimental data of the asymmetry char-acteristic of low-order (<20) chi-square density func-tions.

2. Summary of Results for the Entire Data SetRuns similar to those leading to Fig. 4 were carried

out on 12,000-data point sets obtained at ranges of 3, 6,and 9 km; to observe the variability of the data, eachset was subdivided into 1000-point samples and thesamples were treated in the same way as described inthe previous section.

Table III summarizes the estimated parameters anddiagnostics. To save space, only the output of thefilter estimating power is presented here; that from thelog power filter is similar. The mean of the innovationsequence and the number of outliers in the innovationACF confirm the adequacy of the random walk model.

For the returns from a given range, the Q' values arethe most variable, their minima indicating that some ofthe data are relatively constant, so that the returnpower can be determined with good precision; for somesamples on the other hand the power varies signifi-cantly.' Although m and R of course remain relatively

Table 11. Nonlinear Filter Output Using a Real Data Sequencea

DiagnosticsInnovations

ACF DataFilter Parameters Normalized outlier Outlier

estimate Q1 m R mean count count

Power 1.2E + 03 6.8 240 -0.197 1 1Log power 2.4E - 04 7.1 240 -0.201 1 1

a Filter output parameters and performance diagnostics. Other output from the filter is shown inFig. 4. The units for the system and measurement noise variance Qi and R are arbitary.

Table Ill. Nonlinear Filter Output Obtained Using a Set of Real Data Sequencesa

DiagnosticsACF Precision

Parameters Normalized outlier FSDValues Q1 m R mean count estimate

3-km RangeMin 9 5.0 241 -0.87 0 0.04Max 1388 7.8 292 1.80 4 0.08Mean 537 6.2 254 0.16 2.0 0.06Standard deviation 407 1.0 14 0.65 1.4 -

6-km RangeMin 0.01 5.1 252 -2.00 0 0.04Max 55 6.0 294 0.74 3 0.12Mean 15 5.6 274 -0.37 1.1 0.09Standard deviation 18 0.2 11 0.64 0.9

9-km RangeMin 0.01 5.4 203 -1.36 0 0.07Max 1.11 7.8 270 0.93 3 0.22Mean 0.33 6.6 229 -0.05 0.9 0.14Standard deviation 0.32 0.9 25 0.53 1.1 -

a A summary of the parameters, diagnostics, and covariance obtained from the filter estimatingsignal power using data from several ranges. The units for the system and measurement noisevariances Qi and R are arbitrary; the covariance estimate is expressed as the fractional standarddeviation (FSD) of the power estimate.


0.2 0.4 0.6 0.8(ThauUnds)

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40

30

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Fig. 4. Data sample from a range of 3 km; processing power and log power estimates: (a) measured return superimposed over the estimatedreturn signal power obtained from the filter for power; (b) log power estimate from the filter for log power; (c) histogram of the power filter esti-mates of the speckle and chi-square density function of order 14; the latter is normalized to make the areas under the two curves identical.

independent of the range r, the magnitude of Q, de-creases. Because these are far field returns, a reduc-tion in signal power at least as fast as I/r2 at constantbackscatter would be expected. If the atmosphericvariability was independent of range, Q would de-crease at least as fast as 1/r4; the data support thisexpectation, if allowance is made for the variability ofQ'.

The FSD of the power estimate for each sample isvariable, as mentioned in Sec. III.A, so the values forTable III were obtained from an arbitrary point nearthe center of the sample in each case. We find that themean FSD changes only from 5.5% to 14% between r =3 km and r = 9 km despite a (greater than) ninefoldreduction in signal level. The explanation is that thislidar is not optimized for return power measurement,and the major noise component for r < 9 km is thespeckle, which has an FSD that is range independent.

V. Conclusions

We have extended the methods previously applied'to lidar data in the context of linear systems and mea-surements to estimation of return power and log powerin the presence of speckle. The improvement ob-tained by using a nonlinear filter, when the measure-

ment is nonlinear and the signal not constant, has beendemonstrated by the comparison given in Sec. III.A;failure to take proper account of the nonlinearity, byuse of a linear filter, led to significant bias. Nonlinearmeasurement equations characterize a number of ex-isting and potential lidar applications. The presenttechniques permit construction and evaluation of algo-rithms to estimate time varying unknowns in applica-tions having arbitrary nonlinearities in either the mea-surement or system equations.

As recommended in our earlier paper applying linearfiltering,' a stochastic signal model has been used totake account of the unpredictable time variations thatshould be assumed a priori for atmospheric returns.There are further consequences of this assumption inthe design of atmospheric lidars; in applications whereit is required to monitor a return as closely as possible,it would be expected that the data rate should (inprinciple) be increased to the point where the filteredspeckle and measurement noise levels each match thesignal fluctuation. It might be useful for this purposeto quantify and record the fractional fluctuation level(monitored here for the log power signal by Q) indifferent atmospheric conditions and at different laserwavelengths.


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As far as processing is concerned, plausible esti-mates of the precision of the output have been shownto be available, without which it is impossible to evalu-ate any processing technique. In general the signallevel obtained with these data appears to fluctuatemore slowly than did the Doppler data. A smoothedrandom walk signal model may be more appropriatethan the random walk, but this has not been explored;from a pragmatic point of view, it is sufficient that theresults obtained with the simpler model appear to beself-consistent according to the diagnostic checks.

Although determination of the model parametershas been successfully accomplished, here the desire todetermine three such parameters (m, R, and Q1) hascomplicated the computational problem. Given thatthe signal properties are statistical, it is the more im-portant that those lidar system parameters known tobe constant should be independently measured to op-timize estimation of the signal. If this were the case, itwould be straightforward to process the output of asingle-channel lidar in a single pass, i.e., in (perhapsquasi-) real time using a method based on the innova-tion variance.',' 3

Power and log power estimation have been exam-ined primarily but estimates are also obtained of thereturn speckle, which is modeled as the second statevariable in the signal. These reveal the statistics ofthis source; in principle, and with an appropriatelyaugmented model, the correlation properties of thisoutput that are of interest (e.g., in propagation studiesand for measuring crosswinds using hard targets)could also be examined. For such applications signalvariations (measured by Q1) and speckle fluctuationsmay be each much greater than the measurement noise(R) as was the case for the data in Figs. 3 and 4. In thelimit that R is negligible the processing problem thenapproaches one of the observer type, 2 in which esti-mates of state variables are formed, given measure-ments that are free of measurement noise, but insuffi-cient at a given instant for unambiguous estimation.This example further illustrates the wealth of process-ing techniques for lidar applications that remain to beexploited within the Kalman filter approach.

We are grateful to S. Alejandro for the collaborationwhereby one of us (BJR) was able to undertake muchof this work during short-term appointments at theCooperative Institute for Research in EnvironmentalScience of the University of Colorado, supported bythe Air Force Geophysics Laboratory.

References1. B. J. Rye and R. M. Hardesty, "Time Series Identification and

Kalman Filtering Techniques for Doppler Lidar Velocity Esti-mation," Appl. Opt. 28, 879-891 (1989).

2. A. Gelb, Ed., Applied Optimal Estimation (MIT Press, Cam-bridge, MA, 1974).

3. A. P. Sage and J. L. Melsa, Estimation Theory with Applica-tions to Communications and Control (McGraw-Hill, NewYork, 1971).

4. An alternative approach to the measurement Eq. (4) is logarith-mic 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) butwith w appearing as the additive noise term. A complicationarises if w does not have zero mean. If x is known to be constantand the statistics of w are stationary, the problem is simply oneof determining the resulting bias in the average'4 ; otherwise it isnecessary to show that the variation of the bias is negligible (e.g.,less than other sources of error) over the range of parametersencountered. For differential log ratio measurements of theform x = n[S1/S2 ], y = ln[(Y, - U,)/(Y 2 - U2 )], the problem ismitigated because w = ln[W,/W 2 ] does have zero mean providedthe statistics of Wi and W2 are identical. This removes bias inthe absence of extra additive noise and leads7 to relatively smallbias provided the variances R, and R2 of this noise are small orS11S2 - Rl/R 2.

5. R. E. Warren, "Adaptive Kalman-Bucy Filter for DifferentialAbsorption Lidar Time Series Data," Appl. Opt. 26, 4755-4760(1987).

6. Strictly P can only be interpreted as the covariance of the

estimate for a linear filter with known system model. Fornonlinear filters, including adaptive filters designed to deter-mine the properties of an unknown system model, P should be atbest regarded as a useful approximation to the covariance ma-trix; here we use the term estimate covariance matrix for brevity.

7. B. J. Rye, "Power Ratio Estimation in Incoherent BackscatterLidar: Heterodyne Receiver with Square Law Detection," J.Climate Appl. Meteorol. 22, 1899-1913 (1983).

8. Because the approximation described makes the linear filterslightly suboptimal and might arguably lead to results that areprejudiced against it, the process was repeated with the power,rather than the log power, generated using a random walk [usingEqs. (6a) and (2)] and filtered optimally with the constant valuefor Q, from the simulation; the log power was then filteredsuboptimally using a variable Q, generated by Eq. (14). Theconclusions drawn from the results were unaffected by thesechanges.

9. Inspection of Eqs. (13c) and (13d) indicates that, if the varianceterms are normalized' 2 to Q1, the unknowns can be combined to

leave only two, Qlm and Q1R. It is believed that physical

interpretation calls for knowledge of all three despite the addi-tional computational burden entailed.

10. J. A. Nelder and R. Mead, "A Simplex Method for FunctionMinimization," Comput. J. 7, 308-313 (1965).

11. R. K. Mehra, "On the Identification of Variances and AdaptiveKalman Filtering," IEEE Trans. Autom. Control AC-15, 175-184 (1970).

12. R. H. Jones, "Maximum Likelihood Fitting of ARMA Models to

Time Series with Missing Observations," Technometrics 22,389-395 (1980).

13. B. J. Rye, "A Wavelength Switching Algorithm for Single LaserDifferential Absorption Lidar Systems," Proc. Soc. Photo-Opt.Instrum. Eng. 1062, 267-273 (1989).

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

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