power ratio estimation in incoherent backscatter lidar: direct detection with gaussian noise

Power ratio estimation in incoherent backscatter lidar:direct detection with Gaussian noise

Barry J. Rye

Properties of small sample estimators for the return signal power ratio or log ratio in direct detectionincoherent backscatter lidar systems are analyzed. As for heterodyne receivers it is usually preferable to forman estimator from the logarithmic difference of the sample averages rather than their ratio. Calculatedvalues of bias and noise figures are confirmed using simulated data based on constant signal models andcompared with the estimates obtained from nonlinear Kalman filters. The latter generally provide the leastbias at high noise levels at the cost of greater computational complexity.

1. Introduction

In an earlier paper 1 the preprocessing of short-term(or small sample) estimates of the optical power ratioor log ratio required in some differential lidar measure-ments (e.g., depolarization, differential absorption)were considered for a heterodyne receiver with squarelaw detection. Such preprocessing is desirable to re-move the effects of fluctuations in lidar equation pa-rameters common to the two measurement channels;ratio and log ratio estimation are related as an estima-tor optimized for one is also optimized for the other.Problems arise partly because of the nonlinear relationbetween these two functions and the measurementvariables, i.e., the return powers, and partly because ofthe need to reject data when forming short-term esti-mators should the measurements be negative or belowsome other threshold. The consequent bias and ex-cess noise were evaluated for simple estimators formedby either ratioing or differencing the logarithm of rawmeasurement sample averages; these were, respective-ly, referred to as Fand z estimators, following standardterminology used in the statistics of chi-square distrib-uted processes (e.g., speckle in incoherent backscatterlidar returns) in the absence of any equivalent of mea-surement noise. Here the analysis is first extended tosome direct detection systems.

When this work was done the author was with University of Hull,North Humberside HU6 7RX, U.K.; he is now with CooperativeInstitute for Research in Environmental Sciences, University ofColorado/NOAA, Boulder, Colorado 80309-0216.

Received 11 May 1988.0003-6935/89/173639-08$02.00/0.© 1989 Optical Society of America.

The calculations make use of moments of the fre-quency function of the noise sources and can thereforeonly be employed directly if the return signal powersare constant. This model is unlikely to apply to manyatmospheric measurements and if it was then the useof a short-term estimator would be superfluous. Morerealistically, the ratio or log ratio being estimatedshould be represented by a stochastic model of thetypes reviewed recently in relation to Kalman filterprocessing schemes.2 The role of constant signal cal-culations in the context of fluctuating data would thenbe to delineate the regime in which the bias of a givenshort-term estimator lies within some acceptable limit,thus opening the way for use of linear filters based on asuitable time-dependent model for longer term pro-cessing.

The problem of processing direct detection DIALsystem returns has been considered by Warren,3 whoeffectively uses the z estimator and assumes a constantlarge signal, together with additive noise havingGaussian statistics; multiplicative noise, such asspeckle, and bias at low signal levels are not thereforetaken into account. The calculation is extended tosystems with more than two wavelength channels anda maximum likelihood (ML) estimator of 'a vectormade up of z components is formed. With these as-sumptions the linear Kalman filter should lead to thesame algorithm, as for Gaussian statistics it also pro-vides an ML estimate; by making the role of the systemstructure explicit the Kalman algorithm is readily ex-tended to include a time-dependent model for the zcomponents in the analysis.

An alternative method is to model the entire process(including the return powers) and form the ratio or logratio estimate using a nonlinear Kalman filter. Multi-plicative noise can be directly included as a further

1 September 1989 / Vol. 28, No. 17 / APPLIED OPTICS 3639

nonlinear term.4 The extension of this approach totake into account multiple measurement channels, in-terferents, and time-varying (including nonstation-ary) signal models is computationally straightforwardif laborious. In view of the potential and flexibility ofthis approach the second goal here is to demonstratethis technique in comparison with z and F estimation.To keep the discussion brief we restrict ourselves totwo measurement channels and the constant signalmodel. Because of the nonlinearity of the filter thispart of the paper relies on simulated data.

As to the assumptions used here, in the infrared andfor some UV/visible systems measurement noise willarise predominantly from background or electronicsources; shot noise and data collection using photoncounting techniques are not included. In generalspeckle within the signal itself is not a significant com-ponent of the noise when using direct detection al-though in the infrared the combination of relativelylarge coherence and small photodetector etendues canmake it so. Speckle is therefore included in the limits:

(i) That the speckle count is sufficiently high thatits statistics can be assumed to be Gaussian, like that ofthe measurement noise; the overall noise is then alsoGaussian, but it is useful to discriminate the twosources to exhibit the biasing effect of multiplicativenoise on the ratio estimator even at negligible measure-ment noise levels.

(ii) That a point detector is used, so speckle statis-tics are negative exponential. That is included tocover direct detection receivers using single-mode(high resolution) interferometers and as useful experi-ments are done in the infrared with direct detection tosimulate heterodyne systems.

The layout of the first part of this paper closelyfollows that of Ref. 1. In Sec. II the estimation prob-lem is considered; the parameters needed for charac-terizing the bias and excess noise of the short-termestimators are summarized and an account given of thestructure of extended nonlinear Kalman filters for es-timation of ratio and log ratio state vector componentsin two-channel systems. Results of the short-termestimator calculations are given in Sec. III, togetherwith a comparison based on simulated data of their usewith that of the filter for log ratio estimation. Some ofthe results presented have been previously used inconference publications. 5 -7

II. Estimators

A. Return Power

In direct detection receivers the photodetector out-put can be regarded as the sum of the signal andmeasurement noise. If we exclude the effects of atmo-spheric propagation fluctuations the statistics of thesignal to be considered are determined by shot noiseand speckle. The approach taken by Goodman8 is toregard the shot noise as an additive noise within eachspeckle fluctuation. As all these processes are addi-tive and independent it follows that the probabilitydensity or frequency function (FRF) and the variance

of the detector output are given, respectively, by theconvolution and the sum of those of the noise sourcesseparately. Taking the variances first, that of themeasurement noise current IN is written as

var(IN) = (IN2 ) = N2.

The variance due to speckle isvar(I)speCk1l = (Is)2 /M = S2/M,

(1)

(2)

where m is the effective speckle count and S = (IS) isthe mean signal current. Since I, - i7P, where 77 is thephotodetector quantum efficiency and P, is the opticalpower, and the shot noise power is likewise -7PhvBwhere hv is the photon energy and B is the receiveroutput bandwidth, it follows that the shot noise vari-ance is

var(I)shot noise = S /n, (3)

where the generated carrier (or photoelectron) countwithin the detector response time is n = qPj/(hvB); ifthe detector performance is impaired because of re-combination (g - r) or some source of excess noise nmight be redefined as an effective count. Addingthese variances for the total output current I we find

nvar(I) = var(1)/S2= 1/62 + 1/n + 1/m, (4)

where 3 = SIN is the current signal-to-measurementnoise ratio.

The comparable expression for the heterodyne re-ceiver is somewhat simpler as the predominant back-ground and shot noise terms are linked to the localoscillator, leading to the factorization:

nvar(I) = m(l/n + 1/M)2 = (1/m)(1 + 1/5)2

(see, e.g., Ref. 1). In Eq. (4) each term has a differentdependence on the signal level S(ccn). To shorten thispaper and simplify the comparison with Ref. 1 the shotnoise term 1/n in the direct detection case is thereforeneglected; if shot noise were to be included it would bedesirable to extend the discussion to include photoncounting as well as analog data acquisition.

Equation (4) may also be compared with other nota-tion used previously. If the measurement noise iswritten as the sum of the shot noise of a backgroundflux with carrier count nb spread over m' specklecounts and an independent dark noise count na,

var(I) = n + n2 /m + na + nb + nb2/Mn

in agreement with the formulation of Elbaum andDiamant.9 If the measurement noise terms are omit-ted we obtain the well-known expression for photo-count variance in the presence of a quasithermalsource, var(I) = n + n2/m; while if in Eq. (4) the shotnoise term is omitted, the notational changes

SNR = 1/62 = (1/62 + 1/m)l,

(SNR)sat = M,

CNR = 62

result in the form used by Shapiro10 for infrared sys-

3640 APPLIED OPTICS / Vol. 28, No. 17 / 1 September 1989

tems. It might be useful to note that the signal-to-measurement noise ratio is what is normally referredto below, i.e., 6 and not 85,

B. Small Sample Power Ratio and Log Ratio

As in Ref. 1 we are concerned with the moments ofestimators that might be formed by averaging returnsfrom a small data sample taken over a sufficientlyshort term that atmospheric parameters can be as-sumed to be constant. Averaging within the returnfrom a single pulse (which is unavoidable in directdetection as the return is integrated over the rangeresolution bandwidth) is clearly acceptable; averagingover longer time scales may be possible if atmosphericconditions are relatively stable. Parameters such asthe speckle count m and the signal-to-noise ratios havetheir usual meanings in the former case but wouldotherwise have to be interpreted in terms of averagedquantities.

1. Frequency FunctionsFor constant signal in the presence of Gaussian noise

the FRF of the detector output is

P = 1 exp[-(I - S) 2 /(2N2)J.(2ir)'1 2N

To normalize the parameters as in Ref. 1 this is rewrit-ten in terms of and the ratio

I I/NS a

The conditional FRF of c for a given value of is thengiven by

p(I) = (S) exp[-I/S], I > 0,and the convolution yields

(2r)12SN exp[-I/S] exp[-(I - F)1(2N)]dI,

or

p(wl) = /2 exp[-(o - 1/52)] erfc[-6(w - 1/62)/21/21,

(9)

(10)

where the complementary error function

erfc[xj = (2/(7r)/ 2 ) J exp[-t 2jdt

and use has been made of standard integrals fromGradshteyn and Ryzhik.11

2. Bias and Noise Figure SpecificationFollowing Ref. 1 the ratio to be determined is regard-

ed as the ratio of the signal-to-measurement noiseratios (or ratio of the SINs) in the two channels (sub-scripts 1 and 2)

(11)'P = 51/52-

For given noise statistics this function characterizesthe problem better than, e.g., the otherwise simplerratio S/S 2 of the returns. Likewise the log ratio ischaracterized by X = n(p). Since direct detectionimplies a square-law detector which was assumed as analternative to other options for heterodyning in Ref. 1,we can make use of the same short-term estimators asbefore. The conditional ratio estimator for a given S/N 81 in the numerator channel and S/N ratio p istherefore

p() = (2 1/ exp[-t5(w - 1)2/21.(R*161,p) = P(W11W2161,P),

(6)

To include speckle the signal statistics have to beincorporated. Goodman8 made use of the gammaFRF which in the present notation is

p(I) = (m/S)ImY-1 exp[-mI/S]/r(m), I > 0,

or

p(wl) = mmwm-l exp[-mw]/r(m), w > . (7)

For integer m this is identical with the chi-squarefunction used in Ref. 1. Computation of the overallFRF by convolution of Eqs. (6) and (7) entails a nu-merical integration but the result can be simplified inthe limit of high speckle count (m > 10) when the FRFwill approximate a Gaussian with mean S and variancegiven by Eq. (3), i.e.,

PW = ~ 1 e[ -(I S) 2 1[2r(N2 + S2/m)]L/2 2(N2 + S2/mrJ

or

p(wjb) = ,/(27r)1/2 exp[-62(w - 1)2/21. (8)

For a point detector on the other hand the signal is'exponentially distributed [m = 1 in Eq. (7)], so

and the conditional log ratio estimator

(L*I61,p) = X + (ln(W1/W2)). (13)

As before the fractional bias of either estimatorwhen used for finding the ratio is approximately equal(for small bias) to its absolute bias when estimating thelog ratio; for the ratio estimator these biases are givenby

BR = ( 1/W261,P) - 1,

and for the log ratio estimator they are

BL = (n(wj)jk) - (n(w2)I(61/p)).

(14a)

(14b)

Although ratio estimators are biased at all signal-to-measurement noise ratios in the presence of speckle(see Ref. 1), the limiting bias (at large S/N) is noteliminated in Eq. (14a) as in a practical situation thespeckle count may not be known so any correctionfactor would be difficult to apply. For the limitingcase of m = 1 the ratio estimator has indeterminatebias in the zero measurement noise limit so it is notevaluated.

Noise figures are defined for the two estimatorsagain following Ref. 1 using

NR = 10 logj0j[nvar(R*6jp)]/e 2 j, (15a)


(12)

NjL = 10 logloi[nvar(L*ISp)I/e 2}1

where

= 2/r + 1/82 + 1/62 (16)

obtained from Eq. (3) as the normalized variance in theratio (or as the absolute variance in the log ratio) foruncorrelated measurement noise. The figures charac-terize the excess noise arising from use of the smallsampling technique.

In evaluating the above expressions using the FRF ofSec. II.B.1 [modified in Eq. (17) below] it is assumedthat speckle as well as the other measurement noisesources are correlated in neither time nor wavelength.The moment integrals, calculated analytically in Ref.1, have here been computed numerically.

3. Negative DataUse of the small sample estimators (R*151,p) and

(L*151 ,p) is complicated by the existence of negativedata, i.e., nonzero values of p(cWl,2 161 ,2) at Wv1,2 < = 0.

Negative samples have to be rejected as their loga-rithm is undetermined and negative or infinite ratioscannot be used. Suppose that the minimum value of cto avoid rejection is set to be Wmin. Further conse-quences are then as follows:

(i) The expressions for the FRFp(iv) above have tobe replaced by

pwIS) {co(X ert) > = mine (17)

where

a = 1/ (l)d

(and subscripts 1 and 2 on c, 6, and a have beenomitted). For the log ratio estimator Cvmin can be zero.For the ratio estimator however the moment integralsof the denominator (i.e., in (1/2), (1/X2)) are diver-gent necessitating a cutoff cvmin > 0. Results below areplotted for cutoff factors f = Imin/N = Cvminb of 0.1 and0.01 in both the numerator and the denominator.

(ii) The effect of data rejection is akin to that of anincrease in measurement noise, in the sense that itincreases the time taken to complete a measurement.While this is difficult to evaluate in a general sense, fora stationary system with uncorrelated measurementnoise the factor by which the noise power is effectivelyincreased equals that by which the measurement isprolonged. If a fraction f of the data is rejected, thisfactor is 1/(1 - f) and the noise figure would be in-creased by -10 loglo(l - f). In a two-channel systemwith independent noise sources the overall fraction ofdata rejected is f = fi + f2 - f&f2 , so using fl,2 = 1 - (1/al,2) the effective increase in the noise figure becomes

Neff(dB) = 10 logl 0(ala 2 ). (18)

C.Nonlinear Kalman Filter

Within the nonlinear Kalman filter algorithm allunknown parameters are included in a state vector;

Y1

Fig. 1. Block diagrams of two channel models for estimation of (i)

constant ratio xi and (it) constant log ratio xi from measurementsYI,2 in the presence of constant reference signal X2, speckle Sl,2, and

additive measurement noise V1,2.

estimates of all components of this vector are updatedusing weightings dependent on the nonlinear relation-ship between the parameters and the measurements.It is not necessary to reject any data other than outli-ers.

The models on which filters were based here aredepicted in Fig. 1. The variable to be estimated (theratio or log ratio) is represented by the first componentof the state vector x = [Xl,X 2 ,sl,S 2]T (where the super-script T indicates the transpose) and the referencesignal is the second. These elements xi and x2 aremodeled as unknown constants. Speckle terms s, andS2 are treated as state vector components and modeledas described in Ref. 4. The measurements are madeup of a nonlinear combination of these elements writ-ten

y(k) = h[x(k)] + v(k), (19)

where h(x) = [x 2 sl,x 2 s2] T, ratio estimation= [exp(xi + x 2)sl, exp(x2 )s2 ]T, log ratio es-

timation, andV = [V1 ,V2] T is the measurement noise.

As pointed out before,' in some applications (e.g.,single wavelength depolarization measurements) thetwo speckle terms might be correlated; this is readilyincluded in the model, but it is not taken into accounthere. Our measurement Eq. (19) also expresses theassumption that there is no difference between the twosignal channels other than those produced by the pa-rameter to be estimated and speckle; further differ-


Y1

(15b)

ences would have to be modeled by introducing eithercorrelation (for stationary signals xi and x2) or a fur-ther state vector component.

The estimate x was generated using the discretemodel form of the extended Kalman filter.12 At eachstep the system vector propagation (prediction) is de-scribed by il, 2 (k) = l, 2(k - 1) (constant signal) and91,2(k) = 1. The variances of the noise terms W1,2 (=1/m) and v1,2 were assumed known and these values werealso used to initialize diagonal terms in the covariancematrix. When used with the simulated data samplesdescribed below the state vector estimates were initial-ized using values obtained by the action of the z esti-mator on the first sixteen data points.

Ill. Results

As in Ref. 1, cross sections of the dependence ofestimator bias and noise figures as a function of theratio to be determined at two values of the numeratorS/N and vice versa are presented. Comment has beenminimized as the data are largely self-explanatory.Minimum limiting values of S/N that can be toleratedif the fractional bias is to be kept below either 1% or10% are summarized in Table I.

For the simulation studies used when investigatingthe Kalman filter approach, a miniensemble of sixteensamples each containing 500 data point pairs was pre-pared to the model of Fig. 1(ii). Chi-square and notGaussian statistics were employed for generating thespeckle. The simulated data were also processed us-ing the z and F estimator methods to check valuescomputed from the analytic expressions in Sec. II.B.The bias determined in all cases is the bias in theestimate of the log ratio; simulations based on themodel of Fig. 1(i), from which the (fractional) bias ofthe estimate of the ratio itself was determined, havealso been studied, but the results do not differ appre-ciably from those presented here. The data are pre-

Table I. Limiting S/N In the Ratio Numerator 51 for the Absolute Bias tobe Below Bn,,,n, Determined for F and z Estimators

(i) B = 0.01

m p &.(dB)ratio(F) logratio(z)

averaged 0.1 4.0 8.5averaged 0.9 9.7 5.0

10 0.1 5.0- 9.010 0.9 9.0- 1.81 0.1 - 23.01 0.9 - 15.0

(ii) Bn- = 0.1

m A . (dB)ratio(F) logratio(z)

averaged 0.1 1.9 3.8averaged 0.9 5.2 < -3.0

10 0.1 2.4- -0.510 0.9 6.0' < -3.01 0.1 - 10.91 0.9 - < -3.0

(*) Bias determined here as difference between estimate

and the limiting bias at large S/N; if the latter isnot known then the values in the Table are overly

optimistic.

sented in Sec. B dealing with the variation of bias onthe numerator S/N.

A. Dependence on the Ratio

The property of the z estimator, that its bias is zero ifthe ratio p of the S/N is unity, is seen in Fig. 2(i). Aconsequence is that the variation of the bias tends torestricted compared with that of the F estimator [Fig.2(ii)]; the z estimator tends therefore to have the betterperformance.

The effect of different values of the positive cutoffapplied to avoid ratioing problems with the F estima-tor is seen from curves 2 and 5 in Fig. 2(ii). Results arevery sensitive to choice of cutoff in general only if thebias is sufficiently serious to warrant rejection of thisestimator.

The noise figures in Figs. 2(iii)-(v) include the ex-cess values obtained by adding Neff using Eq. (19).The curves show that it is possible for NRL to benegative, as it is an additional property of data rejec-tion that the FRF p'(WO) of Eq. (17) is narrowed com-pared withp(1I6) so its normalized variance is reduced.This is especially the case at low S/N. Unfortunatelywe might expect this result to be sensitive to noisestatistics and significant noise figure reduction tendsto occur where bias is unacceptable.

B. Dependence on the Numerator S/N

As in Ref. 1 the bias and noise figures of the log ratioestimators for a pair of values c, given 81 and CW2 given 62= 51/p can be found from universal curves of

(ln(wl)) = (nI) - nS,

(In2(W16)) - (ln(wIb))2 = (In2I) - (nS) 2.

(20)

(21)

The former can be regarded as the fractional biasobtained if the estimator exp[(ln(co16))] were inadvi-sedly to be used to estimate the mean signal level andthe latter the normalized variance of this estimate.These curves are plotted in Fig. 3 as a function of the(given) value of d. Both BL and N7L approach zero asthe noise level decreases.

Curves for the noise figures of both short-term esti-mators are given in Fig. 4. Broadly the noise firstincreases as 61 is reduced, to a maximum of 2-3 dB forthe z estimator, and then diminishes again as discussedabove.

The results of the bias calculations for both F and zestimators, and those of the simulation studies includ-ing use of the extended Kalman filter, are broughttogether in Figs. 5. For the simulations a sample ofsixteen estimates was obtained by processing themembers of the miniensemble; the average of this sam-ple and its standard deviation (the SDSE) are shown,the latter as error bars indicating the spread in valuesobtained and therefore the output noise level.

In all cases the z estimator and the extended Kalmanfilter produce estimates better than those of the Festimator. For the first the simulation studies con-firm the calculations, while the filter generates bias


BR

01

30

20

1.0

0.0

-10

-20

-30

NL(dB)(m ) I I . . I I 1

1 A\\\\I

_ | t tellls \ ,X,~~~~~~~~~~0.1 10

p

N, (dB)160

140

120

100

80

60

40

20

0.0

-20

p

1.0

p

Fig.2. BiasandnoisefiguresofestimatorsasafunctionofSNR. (i) Bias of thez estimator: 1,1/m = 0,61 = OdB; 2,rm = 10, 51 = OdB; 3,1/m

= 0,6 = 6dB;4,m = 10,6 = 6dB. (ii)BiasoftheFestimator: 1,1/m = 0,51 = OdB,f =0.1;2,1/m= ,Ai =OdB,f =0.01;3,m = 10,1 = OdB,f

= 0.1;4,1/m = 0,1 = 6dB,f =0.1;5,1/m = 0,1 = 6dB,f =0.01;6,rm = 10,81 = 6dB,f =0.1. (iii) NoisefiguresNL(-) andNL+Neff(---)

of the z estimator: 1,1/m = 0,61 = OdB; 2, m = 10,61 = OdB; 3,1/m = 0,61 = 6dB; 4, m = 10,61 = 6dB. (iv) Noise figuresNR (-) and NR + Neff- -) of the Festimator: 1,1/m = 0, = O dB,f = 0.1; 2,1/m = 0, 6 = O dB,f = 0.01; 3, m = 10,61 = O dB,f= 0.1. (v) Noise figuresNR (-)

'and NR + Neff (---)of the F estimator: 1, 1/m = 0,61 =6 dB, f =0.1; 2, 1/m =0,61 = 6 dB, f = 0.01; 3, m = 10,61 = 6 dB, f = 0.1.

B, N. (dB)0 4 I~) | , , ,30 (j -o: -- 0; 2, m = 10; 3, = 10 () N s f

20 -0 3 10

0 2 1.

-20

0 1 -2-0 ~~~~~~~~~~~~~~~~~~Fig. 3. Generalized curves for computing bias

0 0 /3- and noise figures of the z estimator as a function1,3 ~ ~ -- 0of numerator S/N ,Si. (i Bias curve: 1, 1/rn =

-0.1 2-50 0; 2, rn = 10; 3, rn = 100. (ii) Noise figure curves-n__________________ for Ni. (-) and NL +Nff (-- -) as in (i).

6(dB) 6(dB)

that is both insignificant in terms of the sample devi-ations and well within a 10% limit (see Table I).

There is some discrepancy between the calculatedand simulation results for the F estimator. For 1/m =0 [Figs. 5(i) and (ii)], the simulation produces lowerbias than the calculation at low S/N. It is likely thatthis is a consequence of the instability of F in this

regime (and, for example its sensitivity to roundingerrors) as found also in Ref. 1. The difference betweenthe results at high S/N for finite speckle [Figs. 5(iii)and (iv)] probably arises as a result of the assumptionof Gaussian statistics for speckle within the calcula-tions above; the residual bias at high S/N is expectedfor chi-square speckle to be 1/(m - 1) = 0.11 (Ref. 1) as


0 12

010

0.08

006

O M5

0 00

-0-02

-006a1

-40 00

N6 16B) N6 (d0)2.0 - 40 ~~~~~~~~~~~~~~~~~~~~~~~~~~~14-0

10

0.0~~~~~~~~~~~~~~~~~~~~~~~~~~~~~0

0.0 60

- 1.0 6.~~~~~~~~~~ 0 -- -

-2-0 ~ ~ ~ ~ ~ ~ ~ ~ -2

-022.

4 . 10.0 ____________________- 40 -2.0 0.0 2.0 4.0 6 .0 8.0 10.0 -4-0 -2-0 00 20 4 0 60 6.0 10 0 -4-0 -2 0 0-0 2 0 4,0 6-0 80 10-0

6 (dB) 6(du 6 (dB)

Fig. 4. Noise figures for NLF (-) and NLF + Neff (---) of estimators as a function of numerator S/N i. (i) z estimator: 1, 1/m = 0, p =0.9;2,1/m = O,p = 0.1;3,m = 10,p = 0.9;4,m = 10,p = 0.1. ()Festimatorl/m = 0: l,p = 0.9,f= 0.1;2,p = 0.1,f= 0.1;3,p = 0.9,f= 0.01;4,p =

0.1, f = 0.01. (iii) F estimator m = 10: 1, p =0.9,f = 0.1; 2, p = 0.1, f = 0.1; 3, p = 0.9, f = 0.01.

NR (dB)

B

-2 0 2 4 6 8 10

6(d B)

B

-2 0 2 4 6 8 10

C(dB)

obtained using the simulations, whereas the calcula-tions give 0.2. Further data (not shown) using simu-lations with Gaussian speckle give good agreement,and the Gaussian assumption yields the expected re-sults if m > 20. The performance of the z estimatordoes not appear to be affected by the approximationeven for m = 10 and this point was not pursued further.

Data for m = 1 are given for the z estimator (but notfor the ratio estimator for the reason cited above) inFig. 6 and are also included in Table I. The calculatedbias is large except at high 61 or p very close to unity.The filter continues to be satisfactory, except for alarge increase in its output noise level as indicated inthe SDSE at large 81; the existence of this effect wasconfirmed in further simulations, the magnitude of theSDSE being about constant at higher 61, as would beexpected since the predominant noise in this regime isthe speckle. A possible cause of this is the strong

B B

0 .9

.8

07

0.6

0.5

0.4

0.3

0.2

0.1

-0

-4

6 (dB)-2 6 6 10

6 (dB)

Fig. 5. Bias of estimators as a function of numerator S/N a,. Calculated values for F(- -- )and z (-) estimators are compared with results obtained from simulations using F (circles), z(triangles), and filter (crosses) algorithms; bars indicate standard deviation of sampleestimates for the latter where these are significant: (i) 1/m = 0, p = 0.1; (ii) 1/m = 0, p = 0.9;

(iii) m = 10, p = 0.1; (iv) = 10, p = 0.9.

asymmetry of the exponential speckle FRF, which becomes more apparent as measurement noise decreases.The calculated noise figures for the z estimator aregiven for m = 1 in Fig. 6(iii), and it would appear thatthis is one regime where the z estimator offers betterperformance than the nonlinear filter. Use of an iter-ated extended Kalman filter,13 which has been triedwith order of iteration up to 3, has however been foundto consistently halve the SDSE obtained using thefilter in this regime and bring its noise performanceinto line with that of the z estimator without impairingthe bias; a detailed study of this effect, which is com-plex and appears to depend on both 61 and p, is beyondthe scope of this paper.

IV. Summary

The general properties of the short term F and zestimators are similar to those for heterodyning con-


% ' I ' ' '

\a+ s~

V

1 .31 .21 .1

0.90.60.7

0.60.60.4

0.30.20.1

-0.1-0.2

0.9

0.8

0.7

06

0.5

0.4

0.3

0.2

0.1

-0 .1

-0 .2

L -' --.. (iv)

i i

\ , . . . . m? ' '\ ' ' ' ' ' ~~~~~~~~~~~~~~~(iii)]

N i\I i IIi

1I .,

N, (d B) N, (deg)

;

B09t

0.607

04 i

0.3-

02

0.)

-3 -2 0 2 4 6 a 10

6 (dB)

B02

0 1 ,I

010

0.06

0.020 02

0 H- 002- 004

-006 jjl

-2 0 2 4 6 a 10

6 (dB)

NL (d B)

I v '::: 2'I I~~~~~~~~~~~~~~~ii2I4 6

2 4 6 3(I (dB)

Fig. 6. Bias and noise figures for exponentially distributed speckle noise as a function of numerator S/N 61. Calculated values for the z

estimator are presented as lines and values obtained from simulations as in Fig. 5: (i) bias, p = 0.1; (ii) bias, p = 0.9; (iii) noise figures NL (-)

and NL +Neff(- - -): 1, p= 0.1; 2, p= 0.9.

sidered previously. The comparison as to the bias isperhaps conveniently made between Figs. 5 here andFigs. 6 and 7 in Ref. 1. The curves are broadly of thesame form but numerically the bias is somewhat lessfor heterodyning; it would appear that the rectificationof measurement noise occurring in that case might bemarginally helpful. Here as before the z estimator issuperior and acceptable over a wide range of 81 where pis about unity (see Table I); if this cannot be ensuredcaution must be used if the front-end current (or volt-age) signal-to-noise ratio is less than -10 dB. Whilethe F estimator appears better at low p its performanceis less reliable in the sense that (i) a wider range of biasis encountered, (ii) bias persists at high signal-to-noiseratio in the presence of speckle, and (iii) there areproblems arising from sensitivity to the low level signalcutoff necessary to avoid effects of divergence in thedenominator moment integrals; in practice this ap-pears in excess noise figures showing that the F estima-tor is in general less efficient than the z estimator forwhich the integrals are well-behaved.

In comparison with the third approach, use of thenonlinear Kalman filter, the short-term estimatorshave the advantage that they do not call for a statisticalmodel for the fluctuations of those lidar equation pa-rameters that depend on atmospheric conditions.Models for the quantity being measured and atmo-spheric fluctuations not common to both channels arestill required. While this greatly simplifies the algo-rithm-for the unoptimized implementations fromwhich the above results were obtained the nonlinearfilter took about 10 (without speckle) and 100 (includ-ing speckle) times longer to run than either the z or Festimators-the filter is much more powerful, containsinformation (e.g., on reference signal power and speck-le statistics) that has not been exploited here, does notreject any good data, and can be adapted to take ac-count of more complex system models, including, forexample, correlated noise sources, interferents, andnonstationarity. It can also be used to control systemsin which the different wavelengths are not transmittedsimultaneously. In general the results obtained herefor constant signals favor the filter at high noise levels,although as for all processing techniques involving

nonlinearities they should be rechecked using realisticsimulation of any data that appears not to fit thismodel.

References1. B. J. Rye, "Power Ratio Estimation in Incoherent Backscatter

Lidar: Heterodyne Receiver with Square Law Detection," J.Climate Appl. Meteorol. 22, 1899-1913 (1983).

2. B. J. Rye and R. M. Hardesty, "Time Series Identification andKalman Filtering Techniques for Doppler Lidar Velocity Esti-mation," Appl. Opt. 28, 879-891 (1989).

3. R. E. Warren, "Detection and Discrimination Using Multiple-Wavelength Differential Absorption Lidar," Appl. Opt. 24,3541-3545 (1985).

4. B. J. Rye and R. M. Hardesty, "Nonlinear Kalman FilteringTechniques for Incoherent Backscatter Lidar: Return Powerand Log Power Estimation," Appl. Opt. 28, to be published(1989).

5. J. W. van Dijk, J. F. Kusters, A. Layfield, and B. J. Rye, "The Use

of TEA and Multiatmosphere CO2 Lasers in Active RemoteSensing," in Proceedings, ESA Workshop on Space Laser Ap-plications and Technology, ESA SP-202, Les Diablerets, 225-30(1984).

6. A. Layfield, B. J. Rye, and J. W. van Dijk, "Application of

Optimal Estimation Techniques in DIAL," in Proceedings,Third International Topical Meeting on Coherent Lidar:Technology and Applications, Malvern (1985).

7. A. Layfield and B. J. Rye, "Software Filtering of DifferentialAbsorption Lidar Returns," in Proceedings, Workshop on DIALData Collection and Analysis, Virginia Beach (Nov.1985), to bepublished as a NASA report.

8. J. W. Goodman, "Some Effects of Target-Induced Scintillationon Optical Radar Performance," Proc. IEEE 53, 1688-1700(1965).

9. M. Elbaum and P. Diamant, "Signal-to-Noise Ratio in Photo-counting Images of Rough Objects in Partially Coherent Light,"

Appl. Opt. 15, 2268-2275 (1976).10. J. H. Shapiro, "Target Detection with a Direct-Reception Opti-

cal Radar," MIT Lincoln Laboratory Report TST-27 (1978).11. T. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series

and Products, translated by A. Jeffrey (Academic, New York,

1965).12. A. P. Sage and J. L. Melsa, Estimation Theory with Applica-

tions to Communications and Control (McGraw-Hill, NewYork, 1971).

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


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power ratio estimation in incoherent backscatter lidar: direct detection with gaussian noise

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