detection and discrimination using multiple-wavelength differential absorption lidar

5
Detection and discrimination using multiple-wavelength differential absorption lidar Russell E. Warren A methodology is presented for generalizing two-wavelengthsingle-material differential absorption lidar to multiple wavelengths for use in simultaneous multimaterial detection and discrimination. A key role in the analysis is played by the concentration path length (CL) product covariance matrix ACL, which generalizes the CL variance. Detection statistics for a multiwavelength alarm system are computed using A with a multivariate normal distribution for the estimated CL product values. Off-diagonal elements in A are found to affect significantly the predicted performance of two-material detection systems. I. Introduction Differential absorption lidar (DIAL) with two wave- lengths is a proven technology for the remote sensing of single materials. Pioneering work by Schotland 1 and Byer and Garbuny 2 characterized the uncertainty in the estimated concentration (or CL product) as a function of the absorptivity of the detected material and uncertainties in the lidar measurement at the on- and off-resonance wavelengths. Generalization of the technique to more than two wavelengths would offer potential advantages in improved sensitivity for sin- gle-material detection as well as in the capability for simultaneous detection and discrimination of multiple materials. Achieving a multiple-wavelength capabili- ty, however, requires consideration of several interre- lated problems: the number and specific identity of wavelengths that should be used for a given set of materials; the way in which wavelength returns are to be used to estimate the CL values; the way in which uncertainties in estimated CL val- ues are to be computed; the effect of pulse correlation on CL uncertainties; the way in which detection statistics for a multiple- wavelength alarm system are to be computed. This paper describes an approach for answering the above concerns, including a methodology for estimat- ing the detection/discrimination performance of mul- tiple wavelength DIAL systems. For simplicity, only The author is with SRI International, Electro-Optics Systems Laboratory, 333 Ravenswood Avenue, Menlo Park, California 94025. Received 22 April 1985. 0003-6935/85/213541-05$02.00/0. ©1985 Optical Society of America. column-content (CL) systems are considered; howev- er, range-resolved lidar is conceptually similar, al- though the data processing becomes more involved. Issues are addressed primarily in terms of an arbitrary number of simultaneous materials; while example de- tection statistics are calculated only for one and two materials, the generalization to more than two materi- als is indicated. Section II describes the generalized DIAL signal model and its statistical properties need- ed for the construction of a maximum likelihood (ML) estimator for the solution vector CL of CL product values in Sec. III. Section IV develops a detection model for generalized DIAL using the statistics of CL in a multivariate normal distribution. 11. Generalized DIAL Signal Model For purposes of this paper, the lidar return, pi at wavelength X, i = 1,. . ., M, produced by backscatter from either the natural atmosphere or a topographic target at range L, can be written p = ETVG(L) ex{-2(XPICL + eiL)] + n (1) where ET = transmitted pulse energy, v = speckle modulation factor, G(L) = system- and range-dependent parame- ters, N = number of simultaneous materials, pil = absorptivity of material at wavelength i, CLI = concentration-path length product of material 1, ei = combined extinction coefficient for the natural atmosphere and any interferent materials, and ni = receiver noise. All parameters above except ET, v, and n are consid- 1 November 1985 / Vol. 24, No. 21 / APPLIED OPTICS 3541

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Detection and discrimination using multiple-wavelengthdifferential absorption lidar

Russell E. Warren

A methodology is presented for generalizing two-wavelength single-material differential absorption lidar tomultiple wavelengths for use in simultaneous multimaterial detection and discrimination. A key role in theanalysis is played by the concentration path length (CL) product covariance matrix ACL, which generalizes theCL variance. Detection statistics for a multiwavelength alarm system are computed using A with amultivariate normal distribution for the estimated CL product values. Off-diagonal elements in A arefound to affect significantly the predicted performance of two-material detection systems.

I. Introduction

Differential absorption lidar (DIAL) with two wave-lengths is a proven technology for the remote sensingof single materials. Pioneering work by Schotland1

and Byer and Garbuny2 characterized the uncertaintyin the estimated concentration (or CL product) as afunction of the absorptivity of the detected materialand uncertainties in the lidar measurement at the on-and off-resonance wavelengths. Generalization of thetechnique to more than two wavelengths would offerpotential advantages in improved sensitivity for sin-gle-material detection as well as in the capability forsimultaneous detection and discrimination of multiplematerials. Achieving a multiple-wavelength capabili-ty, however, requires consideration of several interre-lated problems:

the number and specific identity of wavelengths thatshould be used for a given set of materials;

the way in which wavelength returns are to be usedto estimate the CL values;

the way in which uncertainties in estimated CL val-ues are to be computed;

the effect of pulse correlation on CL uncertainties;the way in which detection statistics for a multiple-

wavelength alarm system are to be computed.This paper describes an approach for answering the

above concerns, including a methodology for estimat-ing the detection/discrimination performance of mul-tiple wavelength DIAL systems. For simplicity, only

The author is with SRI International, Electro-Optics SystemsLaboratory, 333 Ravenswood Avenue, Menlo Park, California 94025.

Received 22 April 1985.0003-6935/85/213541-05$02.00/0.© 1985 Optical Society of America.

column-content (CL) systems are considered; howev-er, range-resolved lidar is conceptually similar, al-though the data processing becomes more involved.Issues are addressed primarily in terms of an arbitrarynumber of simultaneous materials; while example de-tection statistics are calculated only for one and twomaterials, the generalization to more than two materi-als is indicated. Section II describes the generalizedDIAL signal model and its statistical properties need-ed for the construction of a maximum likelihood (ML)estimator for the solution vector CL of CL productvalues in Sec. III. Section IV develops a detectionmodel for generalized DIAL using the statistics of CLin a multivariate normal distribution.

11. Generalized DIAL Signal Model

For purposes of this paper, the lidar return, pi atwavelength X, i = 1,. . ., M, produced by backscatterfrom either the natural atmosphere or a topographictarget at range L, can be written

p = ETVG(L) ex{-2(XPICL + eiL)] + n (1)

where ET = transmitted pulse energy,v = speckle modulation factor,

G(L) = system- and range-dependent parame-ters,

N = number of simultaneous materials,pil = absorptivity of material at wavelength i,

CLI = concentration-path length product ofmaterial 1,

ei = combined extinction coefficient for thenatural atmosphere and any interferentmaterials, and

ni = receiver noise.All parameters above except ET, v, and n are consid-

1 November 1985 / Vol. 24, No. 21 / APPLIED OPTICS 3541

ered to be nonfluctuating. ET and v introduce multi-plicative noise while ni is additive. The mean andvariance of ET are denoted by (ET) and aT2 = (E?) -

(ET)2 . The speckle modulation v has unity mean andvariance a,2 = ( 2) - 1. The receiver noise has zeromean and variance an2. Using these parameters, thefirst and second statistical moments of pi are

(2)

(3)

(pi) = (ET)POi,

(p?) = (E2T) (V2)p + 2,

in terms of the nonrandom function

poi = G(L) ex{-2(pCLi +

The variance in pi can be written

and n,, the zero-mean vector representing an m - 1dimensional Gaussian noise process. The covariancematrix for n,, A,, defined as

A, = ([S - (S)I[S - ()]T) = (nn"T), (12)

is estimated by performing a first-order error propaga-tion of the pi fluctuations through Eq. (9). The result-ing matrix has elements

= 1/4 (SNi+ij+ , Si+iJ(A~j=/ SNRi+,SNRj+j SNRi+,SNRj

ij+l + AijSNRiSNRj+l SNRiSNRJ

(4)

I(p) - (pi) 2

= p2, (E)2 a2 + (E ( (a2+1) + a (5)

giving the following expression for the square of thedirect detection signal-to-noise ratio (SNRi):

SNRj2- (Pi) ,p~i

CNR?'

1 +2S [1 + (E T (1 + SNRS2)

where the carrier-to-noise ratio(ET)POi

CNRi =_ an I

and speckle-limited signal-to-noiseSNRS is given approximately as

(6)

SNR = 1/

(7)

UV.

SNR, kaROT (8)

in terms of k = 2r/X, aR, the receiver aperture radius,and OT, the transmitter divergence half-angle. For aT= 0, Eq. (6) reduces to the direct detection SNR modeldeveloped by Shapiro et al.

3 and applied to two-wave-length DIAL by Harney. 4

The term G(L) is assumed to be independent ofwavelength for closely spaced wavelengths but other-wise not in general. For this reason, only pairwiseratios of pi are used to estimate CLi; i.e., we define theobservation vector s having components

Si = /In-IPi

N

I(p P+)CLI + n,, i = 1.M- 1, (9)l=1

where nsi is the noise contribution to si taken to beadditive in the strong signal regime. In matrix nota-tion Eq. (9) becomes

s = KCL + n, (10)

in terms of CL, the M - 1 X N-dimensional matrix Kwhose elements are

(K)i, = - (11)

(13)

in terms of SNRi and the pulse correlation coefficientsbij. By definition /lii = 1, and Aij, i j, may or may notbe nonzero, depending on the closeness of the wave-lengths, the time interval between measuring pi and pj,and target roughness. Increasing any of these factorsdecreases the pulse correlation.

The probability density for the signal vector s condi-tional on CL is taken to be an M-1-dimensional normaldistribution

P,(sCL) = (2 )(M-)/9A 11/2 exp[-/2(s - KCL)TAj(s - KCL)],

(14)

where A, I is the determinant of A,. In the followingsection the signal model described here will be used toconstruct an estimator for CL.

Ill. Maximum Likelihood Estimation of CL

The statistical signal model Eqs. (10)-(14) can beused to provide a maximum likelihood (ML) estimatorCL for the vector of CL-product values CL (Middle-ton5 ) by solving

VCL lnP,(5 CL) cL=L ° 0 (15)

for CL. Using standard matrix manipulations theresult is

CL = (KTAj'K)-IKTA-lS.

The mean of CL is computed as

(CL) fSdsP,(sCL)CL = CL,

showing that CL is an unbiased estimator of CL.CL covariance

AOL ([CL - CLI [CL - CLIT)

is similarly shown to be

AOL = (KTAS-lK)-l

Evaluation of the Cram6r-Rao bound5

AOL > [fSdsP,(sI CL) VCL lnPs(s CL)l 'P

(16)

(17)

The

(18)

(19)

(20)

produces an equality indicating that CL is efficient aswell.

For M = 2, N = 1, Eqs. (16) and (19) generate theusual two-wavelength single-material results:

3542 APPLIED OPTICS / Vol. 24, No. 21 / 1 November 1985

CL = 1 ,) Pn ,

AOCL 2,1 1 _ _ _ _ 12 1-4(PL_2)

2 (N SNRSNR2 + SNR2) = acL- (21)

The implications of the more general formalism formultimaterial detection and discrimination are dis-cussed in the next section.

IV. Statistical Detection Model

The CL covariance matrix ACL described above gen-eralizes the two-wavelength DIAL CL result, Eq. (21),and provides sensitivity estimates for multimaterialmultiwavelength lidar. For designing and character-izing the performance of such systems, however, a sta-tistical detection model is needed that allows compu-tation of probabilities of false alarm and detection forarbitrary combinations of materials and numbers ofwavelengths. An approach is described here that usesthe CL covariance matrix in a multivariate normaldistribution for the probability density of the estimat-ed CLI values. This is probably the simplest formal-ism that includes all the variables of importance to thedetection/discrimination problem.

The probability density of CL, PN(CL), is taken tobe the multivariate normal density:

PN(CL (27 r)N/2 AL 1/

X exp[-1/ 2(fL - CL)TA-1(CL - CL)] (22)

in terms of the inverse and determinant of the CLcovariance matrix, AL- 1 and I AfL respectively. ForN = 1, ACL reduces to the single-material variance CL2

giving the normal density:

P,(CL) L expp 2L CL) 2(23)

V~7r+CL [ 2arCL JFor N = 2, ACL can be written in terms of CL1, CL2 ,and the correlation coefficient p as

/2L-CL, 7CLUCL2PAft = 2 ,(24)

6CLTCL2P aCL2

and Eq. (22) produces the bivariate density:= 1 exp{ 1 [(L - CL1)2

P2(0CL0L2) 2uCL UCLP2 -2(1 P L CL(CL1 - CL,)(CL2 -CL2) (CL2 -CL )2

2 IIa7CL,'YCL, UCL2 j

(25)

The detection model used here follows the Neyman-Pearson approach of Eq. (1) constructing a detectionthreshold using the false-alarm probability PFA and (2)computing the detection probabilities using thisthreshold. A false alarm is said to occur when a detec-tion is made of any material or combination of materi-als when, in fact, nothing is present. In terms of PN,this is equivalent to setting

OL T, ~ CLT~C'P(PFA = 1 dCL ... J dCLNPN(CJICL = 0), (26)

where CLT, ... CLTN are the detection thresholds forthe individual materials. It is convenient to scale theCLT values by their respective uncertainties aCL andto define a single normalized threshold T as

CLT,T= ' 1=1_ .. N.

acLI(27)

Defining T allows a single-threshold value to be associ-ated with a given PFA and removes the ambiguity inspecifying CLTI individually.

For single-material detection, Eq. (26) gives

PFA = 1 1 J dt exp(-t 2 /2)-=Q(TI. (28)

For two materials, Eq. (26) produces

PFA = 2Q(T) - L(TTp)

in terms of the bivariate integral:

L(h,k,p) 1 J dx dy2ir1 _- p2 h a

X ex{- -I 2 (E2- 2Pxy + y 2) 3,

(29)

(30)

as defined by Abramowitz and Stegun.6 Higher di-mensional analogs of Eq. (30) can be constructed andused to express PFA for N > 2.

Figure 1 plots PFA VS T for two-material detectionwith p = 0 and p = +1. The p value of 0 corresponds touncorrelated fluctuations between L1 and CL2, asestimated from Eq. (16); the values +1 represent limit-ing cases of complete correlation and anticorrelation inthe fluctuations, respectively. The curves in the fig-ure show that a threshold T of 3.73 CL standard devi-ations is needed to reduce the false-alarm probabilityper measurement to 10-4 for p = 1, while T = 3.93 isrequired for p < 0.

Detection probabilities are computed by integratingEq. (22) over subsets of the N-dimensional CL detec-tion space corresponding to the desired materials orcombinations of materials. For example, the proba-bility of detecting only a single material k becomes

PDk = JCLri d fL . dCL C. Jc dCLNPN(0L), (31)PD1, [CL k .. (D

where CLT = TL i = 1, . . . , N. Detection proba-bilities for sets consisting of arbitrary combinations ofmaterials are computed analogously by integrating PNup to the CL thresholds for materials not belonging tothe set and from the thresholds to infinity for thoseincluded in the set. In all cases, the integrations aredone for a specified vector CL representing theamount of each material actually present.

For one- and two-material detection, the calcula-tions simplify substantially. Setting R = CL/UcL, it iseasily seen that for N = 1

PD = Q(T-R). (32)

1 November 1985 / Vol. 24, No. 21 / APPLIED OPTICS 3543

99.99

99.9

99.8

50

40-

30-

20-

10

2

0.5-

0.2 -

0.1

0.05-

0.010 1 2 34

NORMALIZED THRESHOLD - T

Fig. 1. Dependence of false-alarm probability on normalized

threshold and CL correlation (two-material detection).

99.99

R2 0

99.9 \ _

99.8

99 <

98

95-

909

90-

70- R4

60L

PFA

Fig. 3. Detection probability for material 1 vs false alarm probabili-

ty for CL correlation = 0 (only material 1 present).

'FA

Fig. 2. Detection probability formaterial 1 vs false alarm probabili-

ty for CL correlation = 1 (only material 1 present).

99.99

R2 0

999 _ R1 - 8\ p -R1

99.8

99

989

95 -~ ~ ~ ~ ~ ~ ~ ~

PFA

Fig. 4. Detection probability for material 1 vs false alarm probabili-

ty for CL correlation -1 (only material 1 present).

3544 APPLIED OPTICS / Vol. 24, No. 21 / 1 November 1985

01

0.1

10-4 10-3 1o2

lo-, 1

'FA

Fig. 5. Detection probability for both materials 1 and 2 vs falsealarm probability (both materials 1 and 2 present).

For N = 2, three probabilities are of interest:PD, probability of detecting only material 1;PD2 probability of detecting only material 2;PD3 probability of detecting both materials 1

and 2.Setting R = CLi/aCLi and T = T - R, i = 1,2, it can beshown that

PD = Q(T1) - L(T1,T2,p), (33)

PD2 = Q(T2) - L(T,T2,p), (34)

PD3 = L(T1,T2,p). (35)

Figures 2-4 plot PD, VS PFA as computed using Eqs.(33) and (29), respectively, for p = 0, 1. The curvesin the figures represent CL values for material 1 inunits of- CL, R,, with R2 = 0. PD, passes through amaximum for a given R, as PFA increases. This is theresult of T decreasing to zero or negative values for

large PFA, which has the effect of increasing PD3 at theexpense of PD,; low or negative thresholds cause thefalse detection of both materials in this model. Forsufficiently small PFA (i.e., sufficiently high Tf, thefigures show higher detection probability for a givenR,, PFA for p = 1 compared with p < 0. Figure 5 showsPD3 plotted against PFA for Rl = R2 for p = 0, ±1. Forall PFA, PD, is largest for p = 1.

V. Conclusions

The multiwavelength multimaterial lidar analysispresented here is a natural generalization of the two-wavelength single-material theory. The most signifi-cant feature of this generalization is the need to consid-er a matrix ACL of CL uncertainty values in place of thesingle-material uCL value. Because of off-diagonal ele-ments in ACL representing correlations in the statisti-cal fluctuations of estimated L values for differentmaterials, detection wavelengths must be chosen care-fully to include the effect of the full CL covariancematrix, not just the diagonal elements.

The statistical detection model developed here indi-cates that detection performance is optimized by find-ing wavelengths that produce high CL correlations. Asimilar analysis for the detection probability of materi-al 1 when only material 2 is present shows that PD, issmallest (in fact 0) when p = 1. Thus high CL correla-tion optimizes discrimination performance as well atleast for two-material detection. A full analyticaltreatment for simultaneous detection of more than twomaterials remains to be completed.

This work was funded under contract DAAK11-82-C-0158 with the U.S. Army Chemical Research andDevelopment Center, Aberdeen Proving Ground, Md.The author gratefully acknowledges the commentsmade by Jeffrey H. Shapiro on the application of MLestimation theory to this problem.

References

1. R. M. Schotland, "Errors in the Lidar Measurement of Atmo-spheric Gases by Differential Absorption," J. Appl. Meteorol. 13,71 (1974).

2. R. L. Byer and M. Garbuny, "Pollutant Detection by AbsorptionUsing Mie Scattering and Topographic Targets as Retroreflec-tors," Appl. Opt. 12, 1496 (1973).

3. J. H. Shapiro, B. A. Capron, and R. C. Harney, "Imaging andTarget Detection with a Heterodyne-Reception Optical Radar,"Appl. Opt. 20, 3292 (1981).

4. R. C. Harney, "Laser prf Considerations in Differential Absorp-tion Lidar Applications," Appl. Opt. 22, 3747 (1983).

5. D. Middleton, An Introduction to Statistical CommunicationTheory (McGraw-Hill, New York, 1960).

6. M. Abramowitz and I. A. Stegun, Eds., Handbook of Mathemati-cal Functions (Dover, New York, 1965).

1 November 1985 / Vol. 24, No. 21 / APPLIED OPTICS 3545