Detection and discrimination using multiple-wavelength differential absorption lidar

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<ul><li><p>Detection and discrimination using multiple-wavelengthdifferential absorption lidar</p><p>Russell E. Warren</p><p>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.</p><p>I. IntroductionDifferential absorption lidar (DIAL) with two wave-</p><p>lengths is a proven technology for the remote sensingof single materials. Pioneering work by Schotland1and 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:</p><p>the number and specific identity of wavelengths thatshould be used for a given set of materials;</p><p>the way in which wavelength returns are to be usedto estimate the CL values;</p><p>the way in which uncertainties in estimated CL val-ues are to be computed;</p><p>the effect of pulse correlation on CL uncertainties;the way in which detection statistics for a multiple-</p><p>wavelength alarm system are to be computed.This paper describes an approach for answering the</p><p>above concerns, including a methodology for estimat-ing the detection/discrimination performance of mul-tiple wavelength DIAL systems. For simplicity, only</p><p>The author is with SRI International, Electro-Optics SystemsLaboratory, 333 Ravenswood Avenue, Menlo Park, California 94025.</p><p>Received 22 April 1985.0003-6935/85/213541-05$02.00/0. 1985 Optical Society of America.</p><p>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.</p><p>11. Generalized DIAL Signal ModelFor purposes of this paper, the lidar return, pi at</p><p>wavelength X, i = 1,. . ., M, produced by backscatterfrom either the natural atmosphere or a topographictarget at range L, can be written</p><p>p = ETVG(L) ex{-2(XPICL + eiL)] + n (1)</p><p>where ET = transmitted pulse energy,v = speckle modulation factor,</p><p>G(L) = system- and range-dependent parame-ters,</p><p>N = number of simultaneous materials,pil = absorptivity of material at wavelength i,</p><p>CLI = concentration-path length product ofmaterial 1,</p><p>ei = combined extinction coefficient for thenatural atmosphere and any interferentmaterials, and</p><p>ni = receiver noise.All parameters above except ET, v, and n are consid-</p><p>1 November 1985 / Vol. 24, No. 21 / APPLIED OPTICS 3541</p></li><li><p>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</p><p>(2)</p><p>(3)(pi) = (ET)POi,</p><p>(p?) = (E2T) (V2)p + 2,in terms of the nonrandom function</p><p>poi = G(L) ex{-2(pCLi +</p><p>The variance in pi can be written</p><p>and n,, the zero-mean vector representing an m - 1dimensional Gaussian noise process. The covariancematrix for n,, A,, defined as</p><p>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</p><p>= 1/4 (SNi+ij+ , Si+iJ(A~j=/ SNRi+,SNRj+j SNRi+,SNRj</p><p>ij+l + AijSNRiSNRj+l SNRiSNRJ(4)</p><p>I(p) - (pi) 2</p><p>= p2, (E)2 a2 + (E ( (a2+1) + a (5)</p><p>giving the following expression for the square of thedirect detection signal-to-noise ratio (SNRi):</p><p>SNRj2- (Pi) ,p~i</p><p>CNR?'</p><p>1 +2S [1 + (E T (1 + SNRS2)</p><p>where the carrier-to-noise ratio(ET)POi</p><p>CNRi =_ an I</p><p>and speckle-limited signal-to-noiseSNRS is given approximately as</p><p>(6)</p><p>SNR = 1/</p><p>(7)</p><p>UV.</p><p>SNR, kaROT (8)</p><p>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</p><p>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</p><p>Si = /In-IPi</p><p>NI(p P+)CLI + n,, i = 1.M- 1, (9)l=1</p><p>where nsi is the noise contribution to si taken to beadditive in the strong signal regime. In matrix nota-tion Eq. (9) becomes</p><p>s = KCL + n, (10)</p><p>in terms of CL, the M - 1 X N-dimensional matrix Kwhose elements are</p><p>(K)i, = - (11)</p><p>(13)</p><p>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.</p><p>The probability density for the signal vector s condi-tional on CL is taken to be an M-1-dimensional normaldistribution</p><p>P,(sCL) = (2 )(M-)/9A 11/2 exp[-/2(s - KCL)TAj(s - KCL)],(14)</p><p>where A, I is the determinant of A,. In the followingsection the signal model described here will be used toconstruct an estimator for CL.</p><p>Ill. Maximum Likelihood Estimation of CL</p><p>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</p><p>VCL lnP,(5 CL) cL=L 0 (15)</p><p>for CL. Using standard matrix manipulations theresult is</p><p>CL = (KTAj'K)-IKTA-lS.</p><p>The mean of CL is computed as(CL) fSdsP,(sCL)CL = CL,</p><p>showing that CL is an unbiased estimator of CL.CL covariance</p><p>AOL ([CL - CLI [CL - CLIT)is similarly shown to be</p><p>AOL = (KTAS-lK)-l</p><p>Evaluation of the Cram6r-Rao bound5</p><p>AOL &gt; [fSdsP,(sI CL) VCL lnPs(s CL)l 'P</p><p>(16)</p><p>(17)</p><p>The</p><p>(18)</p><p>(19)</p><p>(20)</p><p>produces an equality indicating that CL is efficient aswell.</p><p>For M = 2, N = 1, Eqs. (16) and (19) generate theusual two-wavelength single-material results:</p><p>3542 APPLIED OPTICS / Vol. 24, No. 21 / 1 November 1985</p></li><li><p>CL = 1 ,) Pn ,</p><p>AOCL 2,1 1 _ _ _ _ 12 1</p><p>-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.</p><p>IV. Statistical Detection ModelThe CL covariance matrix ACL described above gen-</p><p>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.</p><p>The probability density of CL, PN(CL), is taken tobe the multivariate normal density:</p><p>PN(CL (27 r)N/2 AL 1/X exp[-1/ 2(fL - CL)TA-1(CL - CL)] (22)</p><p>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 CL2giving the normal density:</p><p>P,(CL) L expp 2L CL) 2 (23)V~7r+CL [ 2arCL J</p><p>For N = 2, ACL can be written in terms of CL1, CL2 ,and the correlation coefficient p as</p><p>/2L-CL, 7CLUCL2PAft = 2 ,(24)</p><p>6CLTCL2P aCL2and Eq. (22) produces the bivariate density:</p><p>= 1 exp{ 1 [(L - CL1)2P2(0CL0L2) 2uCL UCLP2 </p><p>-2(1 P L CL(CL1 - CL,)(CL2 -CL2) (CL2 -CL )22 II</p><p>a7CL,'YCL, UCL2 j</p><p>(25)The detection model used here follows the Neyman-</p><p>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</p><p>OL T, ~ CLT~C'P(PFA = 1 dCL ... J dCLNPN(CJICL = 0), (26)</p><p>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</p><p>CLT,T= ' 1=1_ .. N.acLI (27)</p><p>Defining T allows a single-threshold value to be associ-ated with a given PFA and removes the ambiguity inspecifying CLTI individually.</p><p>For single-material detection, Eq. (26) givesPFA = 1 1 J dt exp(-t 2 /2)-=Q(TI. (28)</p><p>For two materials, Eq. (26) producesPFA = 2Q(T) - L(TTp)</p><p>in terms of the bivariate integral:</p><p>L(h,k,p) 1 J dx dy2ir1 _- p2 h a</p><p>X ex{- -I 2 (E2- 2Pxy + y 2) 3,</p><p>(29)</p><p>(30)</p><p>as defined by Abramowitz and Stegun.6 Higher di-mensional analogs of Eq. (30) can be constructed andused to express PFA for N &gt; 2.</p><p>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 &lt; 0.</p><p>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</p><p>PDk = JCLri d fL . dCL C. Jc dCLNPN(0L), (31)PD1, [CL k .. (Dwhere 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.</p><p>For one- and two-material detection, the calcula-tions simplify substantially. Setting R = CL/UcL, it iseasily seen that for N = 1</p><p>PD = Q(T-R). (32)</p><p>1 November 1985 / Vol. 24, No. 21 / APPLIED OPTICS 3543</p></li><li><p>99.99</p><p>99.9</p><p>99.8</p><p>50</p><p>40-</p><p>30-</p><p>20-</p><p>10</p><p>2</p><p>0.5-</p><p>0.2 -0.1 </p><p>0.05-</p><p>0.010 1 2 34</p><p>NORMALIZED THRESHOLD - T</p><p>Fig. 1. Dependence of false-alarm probability on normalizedthreshold and CL correlation (two-material detection).</p><p>99.99</p><p>R2 0</p><p>99.9 \ _</p><p>99.8 </p><p>99</p></li></ul>