Effect of pulse-pair correlation on differential

absorption lidar

Russell E. Warren

Calculations are presented of the effect of pulse-pair correlation on the detection statistics of alarm systems

using differential absorption lidar (DIAL) for the remote sensing of toxic gases. This experimentally ob-

served correlation is found to have a significant beneficial effect on the performance of such systems. The

calculations are performed for both coherent- and direct-detection DIAL systems using a statistical detec-

tion model that assumes a bivariate normal distribution for the on- and off-resonance returns.

1. Introduction

Remote sensing of toxic gases is being actively in-vestigated with systems employing differential ab-sorption lidar (DIAL). Previous studies1' 2 have usedthe rms concentration-path length product uncertaintyOCL (Ref. 3) for the material under consideration as thefigure of merit for comparing system performance.Although useful for that purpose, the design and eval-uation of possible future alarm systems require the in-clusion of false alarm rates, detection probabilities, andthreshold settings, which are not easily related to OrCL.

In addition to these parameters, a consistent statisticaldetection model should include pulse-pair correlationfor the case of a dual-laser system. Experimental re-sults4 have shown the presence of correlation betweenon- and off-resonance returns for dual-laser DIAL de-tection for small (<50-,usec) interpulse time separations.In this paper, system performance will be shown to begreatly affected by this correlation.

This paper presents a statistical model for a DIAL-based alarm system that relates lidar signal-to-noisemodels to the detection parameters of interest to asystems designer. Although direct (energy) detectionsystems have the advantage of proved technology,previous calculations1 have indicated that heterodyneor coherent-detection systems could offer significantsensitivity advantages over direct detection if high pulserepetition frequency (prf) lasers are used. For thisreason, both coherent- and direct-detection DIALsystems are considered here.

Section II of this paper summarizes the DIAL sig-nal-to-noise models developed previously. Section IIIdescribes the statistical detection model developed forcharacterizing the performance of an alarm system

The author is with SRI International, 333 Ravenswood Avenue,

Menlo Park, California 94025.Received 29 January 1985.

0003-6935/85/213472-04$02.00/0.© 1985 Optical Society of America.

using DIAL. In Section IV, the models are applied todirect and coherent DIAL detection, and conclusionsand recommendations are given in Sec. V.

11. Lidar System Models

In the case of direct detection, the lidar system per-formance is characterized by the multiple pulse sig-nal-to-noise ratio1 5:

SNRD [ NCNRD 1/2

1 + CNRDiSNRrad(1)

where N is the number of averaged pulse pairs, CNRDis the so-called carrier-to-noise ratio, which can bewritten'

CNRD = pafv)2 A -I2 r2 exp[-4kL],

CND prf I ADI 4L2J (2)

where e is the system optical efficiency, Pay is the aver-age transmitter power for each of the two lasers, prf isthe laser pulse repetition frequency, D* is the direct-detection detectivity, AD is the area of the detector, -ris the effective pulse width, D is the receiver aperturediameter, L is the target range, r is the target reflec-tivity, and k is the atmospheric extinction coefficient.SNRsat is the signal-to-noise ratio in the strong-signallimit and is determined by speckle and atmosphericturbulence effects. For the moderate ranges (<5 km)of interest here, SNRsat is well approximated by thespeckle-only result:

SNRsat D/DT, (3)

where DT is the transmitter beam diameter. Theanalogous signal-to-noise ratio for coherent detectionis given by5

N CNRH/2 1/2SNRH =

1 + CNRH/2SNR2t + 1/2CNRHI

where the corresponding carrier-to-noise ratio isCPav D

2

CNRH = /eff - r exp(-2kL)hvprf 4L2

(4)

(5)

in terms of the quantum energy hv and effective het-erodyne efficiency lqeff, that includes the quantum ef-ficiency and the coherence loss due to speckle and tur-bulence. For coherent detection, SNRsat = 1 for a dif-

3472 APPLIED OPTICS / Vol. 24, No. 21 / 1 November 1985

fuse target. With the notation used here, the columncontent rms CL uncertainty is given as2

1

d` ApSNRwhere Ap is the differential absorptivity, and SNIgiven by Eq. (1) or (4).6

Ill. DIAL Statistical Detection ModelAlthough the rms CL uncertainty TCL given by ]

(6) provides an approximate basis for comparing systperformance, it suffers from the objections stated ab4for use in specifying the performance of an alarm stem. To construct a detection model more suitablea two-wavelength alarm system, we make the followassumptions:

(A) Only a single toxic gas or class of spectrasimilar gases is present at one time.

(B) A wavelength selection has been made such t]interferences from the natural atmosphere, as wellfrom anthropogenic sources such as smokes, canneglected.

With these assumptions, the only source of statistiuncertainty is the system noise, as characterized by Isignal-to-noise ratio Eqs. (1) and (4). Letpo, andjrepresent the on- and off-resonance return signals fr(an N-pulse average. The inferred CL productthen7

CL -ln II -2Ap \P riI

The detection model is based on a Neyman-Pearsthreshold test; namely, we wish to construct a threshvalue CLo using SNR together with the desired faltalarm probability, such that if CL, as computed fr(Eq. (7), is greater than CLO, we say that toxic gaspresent, otherwise not. Because Eq. (7) is a monotoirelation between CL and Poff/Pon, it is more conveniEto use the signal ratio T Poff/Pon as the detection sitistic. This is equivalent to using CL because

probability that CL > CLo probICL ) CLo = prob{T To,

where

To - exp(2ApCLo).

(6)

1P(PonPoff) = = exp[-I(Pon - (Pon))2

-2.t(p 0 n- (Pon))(Poff- (Poff))

+ (poff - (off)) 2J/2(1 - y2)C2].Use of a bivariate normal density is justified when eitherthe number of received photons/pulse is large or whenthe number of averaged pulses is greater than 4; bothconditions are usually satisfied in practice. SettingSNRon = (Pon)/(, SNRoff = (poff)/lo, it is shown in theAppendix that substituting Eq. (11) into Eq. (10) pro-duces

prob{T > To} = 1/2 1 - erf (ToSNR0 o - SNR+ff

in terms of the error function, erf.Equation (12) can be used to compute the false-alarm

probability, PFA, by recognizing that the condition fora false alarm is T > T when no agent is present; i.e.,when SNRoff = SNRon. This gives

P A= 1/2 1- erf I (To_-_________ . (13)[ (d+/To--2ToL + 1);(3

Similarly, Eq. (12) as it stands gives the detectionprobability PD for a given choice of To, A, SNRon,off.

Rather than using Eqs. (12) and (13) to compute PDand PFA, it is usually of greater interest to input PD andPFA and solve for To and the minimum detectable CL.The latter can be defined by the relation

(12)

SNRon = exp(-2ApCLmin)SNRoff.

Solving Eq. (13) for To giveson = 1 + 1/262 + [1 + 1/4'32]1/2)ld where

3e-~~~~~~~~~~~~-om Ad _~~ 2(1A) /2 a_[Q (PFA) 2

is [1 - |SNRoffnlic and Q-1 is the inverse of the functionent'a- Qf - : dt exp(_t2/2).

(8)

To compute To, let P(pon,p off) represent the proba-bility density for po, and poff; i.e.,

probtpon , Pon and poff Poff)

= f d.- dPoffP(ponPpff). (9)

Then

probIT To = 4' dp n 4To dpoffP(po.,p~ff). (10)

To make the calculation specific, we assume P(pon,poff)is well approximated by a bivariate normal density 8 withmeans (Pon) and (poff), standard deviations , andcorrelation coefficient :

(14)

(15)

(16)

(17)

Abramowitz and Stegun8 provide a rational polynomi-nal approximation to Q-1. Substituting Eq. (14) intoEq. (12) and solving for CLmin give

Crnin = - In TOI. (18)2p [1 + (To - 1)Q'-(PD)/Q '(PFA)jEquations (15)-(18) provide a means of setting the de-tection threshold and estimating the detection sensi-tivity for a two-wavelength DIAL-based alarmsystem.

IV. Comparison of Direct- and Coherent-DetectionDIAL

The detection sensitivity model of Sec. III was usedto compare the direct and coherent system performancefor column content detection in terms of the parameterCLmin using SNR models of Sec. II and the parameterslisted in Table I. A prf of 30 Hz was chosen for the di-rect-detection system to provide sensitivity of betterthan 50 mg/M2 at near ranges with = 0 while remain-ing relatively flat at the longer ranges. Higher prf

1 November 1985 / Vol. 24, No. 21 / APPLIED OPTICS 3473

(I11)

Table 1. Parameters used in Direct/Coherent Detection Comparison

Parameter Symbol Direct-detection value Coherent-detection value

Transmitter average Pav 0.4 W 0.4 W

powerOptical efficiency 0.14 0.14

Pulse repetition prf 30 Hz 30 kHzfrequency

Receiver diameter D 12.7 cm 12.7 cm

Transmitter diameter DT 1 cm 12.7 cm

Target diffuse r 0.05 0.05

reflectivityAtmospheric k 0.5 dB/km 0.5 dB/km

extinctionDetectivity D* 2 X 1010 cm Hz'1/2 /WDetector area AD 1O-4 Cm

2

Pulse duration r 700 nsec 700 nsec

Differential AP 1.17 10-3 m2 /mg 1.17 X 10-3 m2 /mg

absorptivityHeterodyne 7leff - 0.4

efficiencyNumber of integrated N 30 3 X 104

pulsesProbability of PD 0.95 0.95

detectionProbability of false PFA 2.777 X 10-4 2.777 X 10-4

alarm

200

2

0

0I-zJ7

n

10090807060

50

40

30

20

ic0 1 2 3

RANGE - km

COHERENT DETECTION

- DIRECT DETECTION

I I IL I

_~~~~~~~~~~~

Fig. 1. Signal-to-noise ratio dependence on range for direct- and

coherent-detection systems.

values would improve near-field sensitivity, but thesensitivity would deterioriate more rapidly with range,assuming the average power 0.4 W is held constant. Forthe coherent system, the highest prf of 30 kHz was usedthat would allow ranging to 5 km without interferingwith the next pulse pair. Figure 1 plots the signal-to-noise ratios for the direct- and coherent-detection sys-tems vs range computed using Eqs. (1) and (4) with theparameters listed in Table I.

Figure 2 plots CLmin vs range for pulse-pair correla-tions of 0, 0.2, 0.4, 0.6, 0.8, 0.99, assuming 1-sec inte-gration time, detection probability of 0.95, and a false-alarm rate of 1/h. The effect of pulse-pair correlationis to provide approximately a factor of 1.5 sensitivity

120

100

CN4E

-J

UE

E._

80

60

40

20

50 1 2 3

Range -- km

Fig. 2. Minimum detectable CL vs range for direct detection.

enhancement for ,u = 0.5 compared with Az = 0. Pulsecorrelations of 0.5 were observed 4 for direct detectionfrom diffuse targets using short time delay dual-laserDIAL. Assuming A = 0.5 indicates that the direct-detection system should exceed 50-mg/M2 sensitivityup to 5 km with a 1-sec integration time.

Figure 3 plots the detection sensitivity of the coher-ent-detection system for the same pulse-pair correlationvalues used in the direct calculation. Killinger et al. 4

measured only gi = 0.2 for coherent detection of diffusetargets with unequal wavelengths, so less correlationenhancement is anticipated in this case. Nevertheless,the detection sensitivity assuming ,u = 0.2 is almosttwice the direct-detection sensitivity with bt = 0.5 for thesame integration time and detection probabilities.

3474 APPLIED OPTICS / Vol. 24, No. 21 / 1 November 1985

I IT _ I I

4

24where

CN

E

E

-JUE

.2

20

16

12

8

4

00 1 2 3

Range -- km4

yo = To(x + (Pon)/Vc) - (pff)/V¶. (A2)

Changing variables in the second integral to t = (y -yux)/ 1 - /,t,

prob{T >i To) = - dx exp(-x 2 ) dt exp+-t2),

(A3)where

A = (To -u),

B = (To (Pon) - (Poff))/\/o1\ _- (A4)

By changing variables to polar coordinates x = p coso,t = p sink, Eq. (A3) can be brought into the form

probfT > To =-g3 do 3' dpp exp(-p 2 )7r o s/sin5

Fig. 3. Minimum detectable CL vs range for coherent detection.

V. Conclusions and RecommendationsPulse-pair correlation is expected to have a significant

beneficial effect on DIAL performance capability,particularly for direct-detection systems. The en-hanced sensitivity could be achieved with dual-lasersystems with short pulse delays (<50 iusec), such as theGMB and ARB systems currently under developmentat SRI. Measurements are currently under way withthese systems of the pulse-pair correlation for varioustarget configurations.

Compact direct-detection DIAL appears capable ofachieving a sensitivity of 50 mg/M2 up to 5 km withreasonable detection probability and false-alarm ratesof 0.95 and 1/h, respectively.

Compact coherent-detection DIAL shows greatersensitivity than the corresponding direct system. Be-cause of this advantage, it is recommended that con-sideration be given to exploiting this technology in fu-ture remote alarm systems, particularly those for whichsize and power consumption are important factors.

This work was funded by contract DAAK11-82-C-0158 with the U.S. Army Chemical Research and De-velopment Center, Aberdeen Proving Ground, Md.

This paper was presented at the OSA Topical Meet-ing on Remote Sensing of the Atmosphere, InclineVillage, Nev., 15-18 Jan. 1985.

Appendix: Evaluation of the Probability DensityIntegral

Setting x = (Pon - (Pn))/V/20, Y = (Poff(poff)/x/ao in Eq. (11), Eq. (10) can be written

prob{T > To = - I dx exp(-x2

) 3' dy

X exp[-(y - ftx) 2 /(l - ,2)], (Al)

1 pr/2

dO exp(-s 2 /sin 2 0),r with

B To(P 0n) - (Poff)v/1+A 2 V0rVTo-2Toju + 1

The last integral can be shown to give

(A5)

(A6)

probIT > To} = 1/2[1 - erfs],

which yields Eq. (12) on setting SNRon = (Pon)/U,SNRoff = (Poff)/0.

References1. R. C. Harney, "Laser prf Considerations in Differential Absorption

Lidar Applications," Appl. Opt. 22, 3747 (1983).2. P. Brockman, R. V. Hess, and C. H. Bair, "CO 2 DIAL Sensitivity

Studies for Measurements of Atmospheric Trace Gases," in Op-tical and Laser Remote Sensing, D. K. Killinger and A. Mooradian,Eds. (Springer, New York, 1983).

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

4. D. K. Killinger, N. Menyuk, and W. E. DeFeo, "ExperimentalComparison of Heterodyne and Direct Detection for Pulsed Dif-ferential Absorption CO 2 Lidar," Appl. Opt. 22, 682 (1983).

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

6. Equation (6) assumes the on- and off-resonance signal-to-noiseratios are approximately equal as they would be for weak toxic gasabsorption and equal atmospheric extinction. It also neglectspulse correlation which would introduce an additional factor

7. In Eq. (7) it is intended that the N-pulse averaging takes placebefore ratioing or taking the logarithm. Computer simulationshave shown that averaging the ratios of the individual pulsesproduces a biased estimator for CL while averaging the differencesof the logarithms increases the variance in the estimator.

8. M. Abramowitz and I. A. Stegun, Eds., Handbook of MathematicalFunctions (Dover, New York, 1965).

1 November 1985 / Vol. 24, No. 21 / APPLIED OPTICS 3475

) _ ~~~~~Pulse-Pair Correlations

0.0

. ~~~~~~~~~~~0.20.4

0.6

0.8

0.99

l l l I