Lidar System Model for Use With Path Obscurants and Experimental Validation

Transcription

1 Portland State University PDXScolar Electrical and Computer Engineering Faculty Publications and Presentations Electrical and Computer Engineering Lidar System Model for Use Wit Pat Obscurants and Experimental Validation J. W. Giles Portland State University I. N. Bankman R. M. Sova T. R. Morgan Donald D. Duncan Portland State University See next page for additional autors Let us know ow access to tis document benefits you. Follow tis and additional works at: ttp://pdxscolar.library.pdx.edu/ece_fac Part of te Electrical and Computer Engineering Commons Citation Details J. W. Giles, I. N. Bankman, R. M. Sova, T. R. Morgan, D. D. Duncan, J. A. Millard, W. J. Green, and F. J. Marcotte, "Lidar system model for use wit pat obscurants and experimental validation," Appl. Opt. 47, (2008) Tis Article is brougt to you for free and open access. It as been accepted for inclusion in Electrical and Computer Engineering Faculty Publications and Presentations by an autoried administrator of PDXScolar. For more information, please contact

2 Autors J. W. Giles, I. N. Bankman, R. M. Sova, T. R. Morgan, Donald D. Duncan, J. A. Millard, W. J. Green, and F. J. Marcotte Tis article is available at PDXScolar: ttp://pdxscolar.library.pdx.edu/ece_fac/123

3 Lidar system model for use wit pat obscurants and experimental validation J. W. Giles, 1, * I. N. Bankman, 1 R. M. Sova, 1 T. R. Morgan, 2 D. D. Duncan, 3 J. A. Millard, 4 W. J. Green, 5 and F. J. Marcotte 1 1 Te Jons Hopkins University Applied Pysics Laboratory, Jons Hopkins Road, Laurel, Maryland 20723, USA Aalea Dr., Cedar Park, Texas 78613, USA 3 Oregon Healt and Science University, Department of Biomedical Engineering, 3303 Soutwest Bond Avenue, Portland, Oregon 97239, USA 4 Rayteon Missile Systems, P.O. Box 11337, Building 801, Mail Station H15, Tucson, Ariona 85734, USA 5 NASA/Goddard Space Fligt Center, Nortrop Grumman Tecnical Services, Code 429, Building 16W, Room N240R, Greenbelt, Maryland 20771, USA *Corresponding autor: jon.giles@juapl.edu Received 6 May 2008; accepted 19 May 2008; posted 20 June 2008 (Doc. ID 95578); publised 28 July 2008 Wen lidar pulses travel troug a sort pat tat includes a relatively ig concentration of aerosols, scattering penomena can alter te power and temporal properties of te pulses significantly, causing undesirable effects in te received pulse. In many applications te design of te lidar transmitter and receiver must consider adverse environmental aerosol conditions to ensure te desired performance. We present an analytical model of lidar system operation wen te optical pat includes aerosols for use in support of instrument design, simulations, and system evaluation. Te model considers an optical pat terminated wit a solid object, altoug it can also be applied, wit minor modifications, to cases were te expected backscatter occurs from nonsolid objects. Te optical pat aerosols are caracteried by teir attenuation and backscatter coefficients derived by te Mie teory from te concentration and particle sie distribution of te aerosol. Oter inputs include te lidar system parameters and instrument response function, and te model output is te time-resolved received pulse. Te model is demonstrated and experimentally validated wit military fog oil smoke for sort ranges (several meters). Te results are obtained wit a lidar system operating at a wavelengt of 0:905 μm witin and outside te aerosol. Te model goodness of fit is evaluated using te statistical coefficient of determination wose value ranged from 0.88 to 0.99 in tis study Optical Society of America OCIS codes: , , , Introduction Lidar is often used for range finding [1]. A major issue wit te use of tis type of system in obscurants suc as clouds, fog, or battlefield conditions is tat te lidar return from te obscurant can be significant potentially greater tan te return from a solid object /08/ $15.00/ Optical Society of America (ard target). Tis effect can cause confusion wen te lidar is used to detect a ard target in applications suc as smart munitions, sensors, and veicle and maritime vessel collision avoidance in te presence of obscurants suc as clouds, fog, and battlefield smoke [2]. To analye tis penomenon furter, a model as been developed to calculate te actual time-resolved return from te obscurant and te ard target at ranges down to several meters. Te model is able to predict te transimpedance amplifier output waveform used in some lidar systems operating in 1 August 2008 / Vol. 47, No. 22 / APPLIED OPTICS 4085

4 various atmosperic conditions. Te model addresses te situations were te lidar is operating in clear air, were te lidar and te target are witin an obscurant, and wen te lidar is outside te obscurant. An empirically determined impulse response function from a commercial sort-range system tat operates at a wavelengt of 0:905 μm is used in te prediction of te analog output waveform for tis particular system. Computed waveforms sow te effects of backscatter for aerosol fog oil smoke conditions. Experimental validation of te model for a ard target in military fog oil smoke at close range is sown. Te model equations are presented in Section 2. Te modeling of te smoke conditions is presented in Section 3. Te experimental configuration and conditions in te smoke camber are discussed in Section 4. Te results sowing te comparison of te teoretical calculations wit te experimental results wit te fog oil smoke are presented in Section Equations for Modeling te Time-Resolved Lidar Return Waveform Figure 1 sows te case were te sensor is outside and te ard target is witin te obscurant wose Fig. 1. Configuration for te sensor located outside te obscurant. tickness is S, and te distance from te target to te sensor is H wit an angle of θ s between te sensor axis and te target normal. Te concentration of te obscurant can vary from very tin to very tick. As te obscurant becomes ticker, te multiple scattering of te lidar return becomes more significant. For ligt to medium ticknesses, single scattering lidar equations can be used. As te obscurant becomes ticker, te single scattering equations become less valid. For te sort pat lengts of interest in tis work, te single scattering treatment is adequate in most of te cases. As te obscurant becomes ticker, multiple scattering issues must be considered. We start our discussion wit te simpler single scattering treatment. Later (in Subsection 3.C) we present a general discussion of multiple scattering involving te obscurants. In Subsection 5.B we sow tat te single scattering treatment appears to be adequate for te cases treated in tis paper. Te single scattering equations tat describe our situation are given in Subsection 2.A. A. Return Laser Power Considering single scattering, te return laser power from a Lambertian target [3 5] larger tan or equal to te beamwidt for all θ s is given by 8 >< P aerosol ðþ :¼ >: P pk ηoðþπðβ aerosol ðπþþβ clear air ðπþþcτ int D 2 exp 8 cosðθsþ 2 cosðθsþ cτ int 2 P pk ηoðþρ cosðθ s ÞD 2 exp P pk ηoðþπβ clear air ðπþcτ int D 2 exp 8 cosðθsþ 4 2 α clear air cosðθsþ þα ð HþSÞ ext cosðθsþ ii ; if H > ðh SÞ½ is in obscurantš; H cosðθsþ 2 cosðθsþ cτ int 2 H α clear air cosðθsþ þα S extcosðθsþ i 2 ; if ¼ H½ is at targetš; i α clear air cosðθsþ ; oterwise ½ is before obscurantš; ð1þ werep aerosol ðþis teoptical powerreturnedfrom distance (measured from te sensor) of te obscurant target system (W), τ int is te integration time of lidar system (s) laser transmitter pulse þ widt receiver response time, P pk is te peak laser power equaltotepulse energy=τ int ðwþ,ηisteopticaltransmittance of te system (dimensionless), OðÞ is te overlap function (dimensionless), β aerosol ðπþ is te volume scattering function of te aerosol in te backward direction(m 1 sr 1 ),β clear air ðπþistevolumescattering function of clear air in te backward direction (m 1 sr 1 ), ρ is te emisperical reflectance of te extended Lambertian target (dimensionless), c is te speedofligt( m=s 1 ),Distereceiveraperture diameter (m), α clear air is te extinction coefficient of clear air (m 1 ), α ext is te extinction coefficient of te aerosol (m 1 ), and θ s is te angle between te sensor axis and te target normal APPLIED OPTICS / Vol. 47, No. 22 / 1 August 2008

5 If te obscurant covers bot te sensor and te target, te above equations become series waveform is a realistic output waveform for a single pulse. If te response time of te sensor is 8 >< P aerosol ðþ :¼ >: P pk ηoðþπðβ aerosol ðπþþβ clear air ðπþþcτ int D 2 exp 8 cosðθsþ P pk ηoðþρ cosðθ s ÞD 2 exp 2 cosðθsþ cτ int H cosðθsþ α clear air cosðθsþ þα extcosðθsþ ii ; if H > > 0½ is in obscurantš; i α H clear air cosðθsþ þα H extcosðθsþ 2 ; if ¼ H½ is at targetš: ð2þ Tese equations yield te received laser power as a function of distance,, measured from te sensor and equal to H at te ard target. Te cτ int =2 term in te denominator of Eqs. (1) and (2) is te result of considering te output laser pulse from te detector as a square wave wit an effective pulse widt of τ int [3]. To use tese equations witout modification, cτ int =2 sould be small compared wit H and S in Fig. 1. However, a tecnique is sown in Subsection 2.B for using tese equations wit larger values of cτ int =2. Distance can be converted to te elapsed time, t, by substituting ¼ ct=2. Tese equations provide te laser power as a function of te range of te laser pulse leading edge from te sensor or, equivalently, te elapsed time. Terefore from Eqs. (1) and (2), we can calculate te return from a target wen te sensor is outside te aerosol and only te target is witin te obscurant, and also for te case in wic bot te target and te sensor are immersed in te obscurant. Te power received by te sensor operating in or near an obscurant, suc as a cloud, witout a ard target present can also be calculated. B. Instrument Function Effects Te following discussion denotes FðtÞ ¼P aerosol ðtþ as te aerosol target signature function, i.e., impulse response, were P aerosol ðtþ is obtained from P aerosol ðþ using ¼ ct=2 for te cange of variable. If te laser pulse widt is very narrow, say τ 1 ns, and te detector and receiver ave a very sort response time, say, τ R 0:1 ns (receiver bandwidt B ¼ 1=ð2τ R Þ 5 GH), so te instrument response approximates a delta function, Eqs. (1) and (2) can be used to convert directly from te optical power to te actual voltage output of te electronic amplifier (as a function of time). Tis conversion is expressed as F 0 ðtþ ¼FðtÞRG; ð3þ were R is te responsivity of te detector (A=W), and G is te amplifier gain (V=A). Te calculated time somewat longer (i.e., if te laser pulse is somewat wider or te detector response time is somewat longer), tese equations do not represent a realistic output waveform. In tese cases we ten need to determine te effect of lidar system parameters, suc as transmitter pulse widt and detector response time, on te return waveform. Here te time-resolved output of lidar f ðtþ is given by f ðtþ ¼ Z t 0 ¼t 0 Iðt 0 ÞF 0 ðt t 0 Þdt 0 ¼ IðtÞF 0 ðtþ; ð4þ were IðtÞ is te instrument impulse response function, F 0 ðtþ is te function in Eq. (3) above, and denotes convolution. IðtÞ can be calculated using system parameters. Te impulse response can also be determined by pointing te transmitter directly into te receiver in clear air, if feasible, or by directing te return beam into te receiver using a reflective target in clear air. Te instrument impulse response function is ten convolved wit F 0 ðtþ to calculate te timeresolved output of te lidar. If te area under te instrument impulse response function is normalied to unity, te convolution gives te actual voltage output of te system. If tis is not done or te parameters in Eq. (3) are not well known, te actual output is known to witin a scaling constant after taking te convolution. From our estimate of te responsivity of te detector (0.65 to 1:0 A=W) and te amplifier gain (5000 to 10; 000 V=A), we estimate te product RG in Eq. (3)to be 3000 to 10; 000 V=W. Because of tis, after taking te convolution wit te area under te instrument impulse response function normalied to unity, te output is known to witin a factor of 3. From Eq. (3) tis range of values for RG along wit our calculation of FðtÞ gives an estimate of te peak output voltage tat brackets te experimentally observed peak values. As discussed in Subsection 5.A in te experimental analysis, a scaling constant K, wic is 1 August 2008 / Vol. 47, No. 22 / APPLIED OPTICS 4087

6 equal to RG, was selected so te modeled peak output agreed wit te experimental peak receiver output. 3. Modeling of Smoke Conditions Te obscurant used in tis researc was military fog oil smoke since it is easily generated, stable and long lasting, and reasonably nontoxic. A. Particle Sie Distribution Te fog oil smoke particle sie distribution, nðrþ, is generally approximated by a lognormal distribution nðrþ ¼ dn dr ¼ N pffiffiffiffiffi exp ðln r=r gþ 2 r 2π ln σg 2ðln σ g Þ 2 ; ð5þ were r is te particle radius (μm), N is te particle number density (number=m 3 ), r g is te distribution median radius (μm), and σ g is te distribution widt parameter (dimensionless). In tis distribution te natural logaritm of te particle radius rater tan te particle radius itself is normally distributed [6,7]. Te mass median diameter, MMD, often used in te literature [6,7] is given by lnðmmdþ ¼ln 2r g þ 3ðln σ g Þ 2 : ð6þ Te parameters for our lognormal fog oil smoke particle distribution are MMD ¼ 1:26 μm, σ g ¼ 1:4, and r g ¼ 0:45 μm. B. Mie Calculations Because fog oil smoke particles are sperical and teir sie is on te order of te wavelengt of te radiation [8], Mie teory is applicable. Teoretical calculations using code [9] based on te Mie teory [10] were performed to determine te values of te extinction coefficient, α ext, and te backscatter coefficient, β aerosol ðπþ, for various smoke conditions. Te extinction cross section per unit volume calculated from te Mie teory is given by α ext ¼ Z 0 C ext ðrþnðrþdr; ð7þ were C ext is te extinction cross section (m 2 ). We furter write α ext ¼ κ ext C; ð8þ were κ ext is te mass extinction coefficient tat is particle distribution dependent (m 2 =g), C is te aerosol mass concentration [ R πr3 ρnðrþdrðg=m 3 Þ], and ρ is te density of te fog oil droplets comprising te aerosol cloud (0:89 g=cm 3 ). For a lognormal distribution te mass concentration is given by [11] C ¼ πr3 g exp 2 ðln σ gþ 2 ρn: ð9þ Using te value of α ext from Eq. (7) normalied for a number concentration of 1 particle=m 3 by dividing te rigt-and side of Eq. (7) byn ¼ R 0 nðrþdr and te value of C in Eq. (9) for N ¼ 1 particle=m 3,we can solve for κ ext in Eq. (8). Ten, again wit Eq. (8), we can solve for α ext for a given concentration. Similarly te scatter cross section per unit volume calculated from Mie teory is expressed as α sca ¼ Z 0 C sca ðrþnðrþdr; were C sca is te scatter cross section (m 2 ), and α sca ¼ κ sca C; ð10þ ð11þ were κ sca is te mass scatter coefficient. Now we calculate β aerosol ðπþ for te desired lognormal distribution. Te pase function [9] is given by PðθÞ ¼N 4πS 11ðθÞ k 2 α sca ; ð12þ were S 11 ðθþ is te first Mueller matrix element calculated from te Mie teory, and k ¼ 2π=λ; were λ is te wavelengt. Finally wit ð13þ β aerosol ðπþ ¼ PðπÞ 4π α sca ¼ PðπÞ 4π κ scac; ð14þ we calculate β aerosol ðπþ if we know te concentration, C. Te complex index of refraction for fog oil smoke [12] at a wavelengt of 0:905 μm is1:4743 þ i0: Te calculated parameter values for ligt, medium, eavy, and very eavy fog oil smoke wit te lognormal parameters above are sown in Table 1. Tese values may be entered into Eqs. (1) and (2) to calculate te waveforms for te various fog oil smoke conditions. Te definitions of te various smoke densities are similar to tose in Ref. [13]. Te α clear air and β clear air ðπþ for clear air used in tese calculations are 1: m 1 and 1: m 1 -sr 1, respectively. C. Single and Multiple Scattering Te intensity of multiply scattered ligt depends on te scattering medium properties, suc as te sie and distribution of te scattering particles, and te optical dept of te scattering volume given by τ OD ¼ Z Z 0 α ext ðþd: ð15þ Te amount of multiply scattered ligt also depends significantly on te lidar geometry, increasing dramatically wit increasing laser beam divergence, te receiver s field of view, and te distance between te lidar and te scattering volume [14]. If τ OD < 0:8, 4088 APPLIED OPTICS / Vol. 47, No. 22 / 1 August 2008

7 Table 1. Fog Oil Smoke Parameters Calculated from te Mie Teory Smoke Concentration (g=m 3 ) α ext (m 1 ) β aerosol ðπþ (m 1 sr 1 ) Ligt (0.0152) : Medium (0.038) : Heavy (0.076) : Very eavy (0.114) : single scattering prevails; for 0:8 < τ OD 1, tere is only a small contribution from second-order scattering. For larger values of te optical dept, iger orders of multiple scattering sould be considered [14]. For medium smoke and pat lengts of 2 to 4 m, for wic α ext ¼ 0:18 m 1 from Table 1, te optical dept is 0:3 to 0.6, so multiple scattering is not significant. For te eavy smoke for wic α ext ¼ 0:36 m 1 from Table 1, te optical dept is 0.72 to 1.4 for pat lengts of 2 to 4 m, and we may start to see some multiple scattering effects. 4. Measurement of Time-Resolved Lidar Output in Fog Oil Smoke A. Description of Lidar Used in Smoke Tests Te time-resolved return pulses of a sort-range lidar were measured in a fog oil smoke camber in various fog oil smoke conditions. Table 2 sows te major system parameter values of te lidar system (H. N. Burns Engineering Corporation, Orlando, Florida) tat is based on a GaAs laser diode and a silicon PIN diode. Te effects of te transmitter pulse widt, as well as te receiver response time, are caracteried wit te impulse response function for tis sensor as sown in Fig. 2. Tis function was determined experimentally using a planar reflective target placed in a position perpendicular to te laser beam in clear air as discussed in Subsection 2.B. Te curve, as sown in Fig. 2, is normalied to ave a peak value of unity. Te full widt at alf maximum (FWHM) value of te impulse response function is approximately 12 ns. B. Description of Smoke Camber Te smoke camber, sown in Fig. 3, isa12:2 m 2:1 m 2:1 m portable sipping container wit a Plexiglas window. Two fans, one near te rear and one near te front of te camber, were used to elp disperse te smoke evenly. Te sensor and motoried track assembly can be placed inside or outside te smoke camber. Te output voltage of te receiver s transimpedance amplifier was recorded wit a Tektronix TDS 754D digital oscilloscope, wic as a maximum digitiation rate of 2 GH. C. Description of Fog Oil Smoke and Smoke Generator Te smoke was generated by vaporiing military standard grade fuel number 2 (SGF2) fog oil wit a Neutralier model 2760 termal smoke generator (Curtis Dyna-Fog, Ltd., Westfield, Indiana). Measurements of particle sie distribution were made wit a model 100 particle measuring systems forward-scattering spectrum probe [15] (also known as a Knollenberg particle counter) manufactured by Particle Measuring Systems, Inc. of Boulder, Colorado. Tis instrument measures te particle diameters in 15 bins, eac 0:5 μm wide starting wit diameters of 0.5 to 1:0 μm for te first bin, 1.0 to 1:5 μm for te second, up to 7.5 to 8:0 μm for te fifteent bin creating te distribution sown in Fig. 4. Te sie of te measurement bins (as radii) is indicated by te oriontal bars on te grap. For comparison a lognormal distribution is also sown on te grap. Our best estimate of te particle sie distribution from te provider of te smoke generator [12] and tese data is MMD ¼ 1:26 μm, σ g ¼ 1:4, and r g ¼ 0:45 μm. Note tat te number of particles in te upper tail is greater tan tat of te lognormal distribution. D. Measurement of Smoke Concentration and Effects of Particle Sie Distribution Te density of te smoke was determined using a He Ne laser transmissometer (see Fig. 3) wose receiver was equipped wit a copper, a lock-in amplifier, and a PIN diode. In te case of a igly Fig. 2. Instrument response function for te sort-range lidar. Fig. 3. Fog oil smoke camber. Te sensor and motoried track assembly can also be placed inside te smoke camber. 1 August 2008 / Vol. 47, No. 22 / APPLIED OPTICS 4089

8 Wit te known transmittance, pat lengt, and κ ext, te smoke concentration can be determined from Eq. (16). For our configuration a mirror was used to double te transmissometer pat as sown in Fig. 3. Te one-way pat lengt was 6 m. Uncertainties in te smoke concentrations are estimated to be 20%. Te uncertainties in te smoke concentrations are due to te uncertainty in κ ext, estimated to be 15%, wic is caused by te uncertainty in te particle sie distribution and te uncertainty in te transmission measurement, wic is estimated to be 15%. Tese uncertainties combine to give a 20% total uncertainty (root sum of squares). Fig. 4. Particle density versus particle radius for a lognormal distribution wit MMD ¼ 1:26 μm, rg ¼ 0:45 μm, and σ g ¼ 1:4 and te measured distribution. Te oriontal bars sow bin widt (0:25 μm radius). collimated beam, te transmittance troug smoke is expressed by T ¼ expð κext CLÞ; ð16þ were κext is te mass extinction coefficient (m2 =g), C is te aerosol droplet mass concentration (g=m3 ), and L is te total pat lengt in smoke (m). κ ext depends on te sie distribution of particles in te smoke. Figure 5 sows te mass extinction coefficient calculated wit te Mie teory for MMD values of 0.25, 0.5, 1.0, and 1:5 μm, and σ g ¼ 1:4. Te concentration can be determined using te measured transmittance, T, calculated as te ratio of te received laser power in clear air to tat troug a known smoke type and distance. At a wavelengt of 0:6328 μm, κext is 4:46 m2 =g for our fog oil smoke wit MMD ¼ 1:26 μm and σ g ¼ 1:4. Fig. 5. Mass extinction coefficient versus wavelengt for various mass median fog oil smoke diameter particles for a lognormal distribution APPLIED OPTICS / Vol. 47, No. 22 / 1 August Results A. Model and Experimental Comparisons Waveforms at 0:905 μm calculated from te model for te target and sensor immersed in medium smoke and tose measured in te smoke camber are compared in Fig. 6. Te model calculations were made by calculating Paerosol ðtþ wit te sensor system parameters in Table 2, and te integration time of te system, τint, set equal to 1 ns in Eq. (2) and ten convolving te result wit te instrument response function in 1 ns increments as sown in Fig. 2 using Eq. (4) as discussed in Subsection 2.B. After taking te convolution, te output is known to witin a scaling constant because RG in Eq. (3) is only known to witin a factor of 3 for tis system. A scaling constant K of 3000 V=W, wic is equal to RG, was selected so te modeled peak output agreed wit te experimental peak receiver output for te medium smoke case for te sensor target distance of 2:4 m. Te same value of K was also used for te oter medium smoke distances. For te initial calculations in medium smoke, te parameters calculated from te Mie teory are Fig. 6. Model and experimental waveforms for sensor (a) 2.4, (b) 3.0, (c) 3.7, and (d) 4:3 m from target. Te target and te sensor are immersed in medium smoke. Te solid curves are model calculations, and te dased curves are experimental.

9 Table 2. Sensor System Parameters Parameter Value Wavelengt λ ¼ 0:905 μm Pulse energy Q ¼ 0:64 μj Laser pulse widt (FWHM) τ ¼ 8 ns Detector response time (rise time) τ r ¼ 3:5 ns Peak power P pk ¼ Q=ðτ þ τ r Þ¼56 W Pulse repetition frequency 5 kh Amplifier bandwidt B ¼ 140 MH Receiver aperture diameter D ¼ 0:025 m Transmitter beam spread 1 Receiver field of view 8 Overlap function 0:3 m from sensor 0.2 (dimensionless) 0:45 m from sensor 0.5 Beyond 0:6 m from sensor 1 sown in Table 1. We ten find a single medium smoke value of β aerosol ðπþ, α ext, and te target reflectance to obtain te best-fit visual inspection agreement between te model and experimental results for te four medium smoke distances. Tese values were adjusted witin te estimated uncertainties. Target reflectances were measured relative to Spectralon by comparing return signals from te sensor. Te uncertainty in te reflectance measurements for te 0:6 m 0:9 m paper-covered polystyrene board gray and black targets is estimated to be 25%. Tere is an estimated uncertainty of 25% in te value of α ext and β aerosol ðπþ defined by Eqs. (8) and (14) owing to te uncertainty in te smoke concentrations and in te parameters in Eqs. (8) and (14), wic depend on te particle sie distribution. Tere is an estimated 15% uncertainty because of te uncertainty in te particle sie distribution and a 20% uncertainty in te smoke concentration as discussed in Subsection 4.D. Tese uncertainties combine to give a 25% total uncertainty (root sum of squares). Te smoke and target parameter values used to obtain te good-fit model curves are sown in Table 3. Te smoke parameter values calculated from te Mie teory for medium smoke sown in Table 1 and te measured values of te smoke concentration and target reflectance are sown in Table 3 for comparison wit te good-fit model values. Te sensor target distances for te plots sown in Fig. 6 are 2.4, 3.0, 3.7, and 4:3 m. Te fit between model output and experimental data was quantified wit te coefficient of determination R 2 defined as R 2 ¼ 1 RSS=TSS; ð17þ were RSS is te sum of squares of eac residual difference between te model value and te experimental value, TSS ¼ N times te variance of te experimental data values, and N is te number of data points. R 2 was or better for te plots sown in Fig. 6. For te waveforms in Fig. 6, te near return, or first peak, is te return from te smoke and te second is te return from te ard target. Te first peak sape is determined by te instrument properties, including te overlap function and te instrument response function, as well as te smoke properties. We see tat as te sensor moves farter away from te target, te return from te ard target gets weaker, but te return from te smoke remains te same. Te ard target return decreases at longer pat lengts because of (1) te ig attenuation of te fog oil smoke (α ¼ 0:18 m 1 ) and (2) te 2 dependence of te return as sown in Eq. (2). Since te instrument is immersed in omogeneous fog oil smoke, te smoke return in Fig. 6 does not cange as te instrument is moved. Te near field return from te fog oil smoke as te larger amplitude as compared wit te returns from te ard target. Using te same metod and te same value of K from above, te comparisons of te model and experimental waveforms for te sensor in ligt smoke and eavy smoke are sown in Figs. 7 and 8, respectively. In Fig. 7 we again see te return from te smoke at near range followed by te ard target return. Te ard target return (second peak) is more prominent tan in te medium smoke case, because (1) tere is less backscatter from te lower density smoke making te smoke peak smaller, and (2) tere is less attenuation from te smoke making te ard target peak larger. Te coefficients of determination for tese comparisons are or better. For te eavy smoke condition in Fig. 8, te ard target cannot be seen at any distance. Once again, for eac distance from te target in eavy smoke, we are looking at te same tickness of smoke column (for omogeneous smoke) in front of te transmitter detector, so te returns for te various distances all look te same. Te coefficients of determination for tese comparisons are or better. Te measured and calculated parameters used in te model for tese conditions are once again compared wit te goodfit model comparisons in Table 3. Comparisons of te model and experimental waveforms for te sensor outside of medium smoke are Table 3. Parameter Comparison for Modeled and Experimental Waveforms a Sensor Configuration In Ligt Smoke In Medium Smoke In Heavy Smoke Outside Medium Smoke C (gm=m 3 ) 0:0152=0:0152 0:038=0:038 0:076=0:076 0:114=0:114 Target reflectance (%) 15:2=19:0 15:2=19:0 15:2=15:2 4:0=4:0 β aerosol ðπþ (m 1 sr 1 ) ð1:58=1:32þ 10 3 ð4:0=3:3þ 10 3 ð7:9=7:9þ 10 3 ð4:0=3:3þ 10 3 α ext (m 1 ) 0:072=0:073 0:18=0:16 0:36=0:36 0:18=0:16 a Te first number is te calculated or measured value. Te second number is te good-fit value used for te model waveform in Figs August 2008 / Vol. 47, No. 22 / APPLIED OPTICS 4091

10 Fig. 7. Model and experimental waveforms for sensor (a) 2.7, (b) 3.0, (c) 3.7, and (d) 4:3 m from target. Te target and te sensor are immersed in ligt smoke. Te solid curves are model calculations, and te dased curves are experimental. sown in Fig. 9. Te 4% reflectance target is located 7:3 m from te smoke interface. In tis case only returns from te air smoke interface are seen; te target is completely obscured. Te distance to te ard target in tis case as rougly doubled from te previous example (from 4 to now 8 m). Because of range dependence ( 2 ), te signal return will be approximately four times less. Te coefficients of determination for Figs. 9(a) 9(d) are 0.977, 0.936, 0.877, and 0.923, respectively. Te lower signal levels in te experimental curves in Figs. 9(c) and 9(d) reduce te coefficients, owing to lower signal-to-noise ratio. Te measured and calculated parameters used in te model for tese conditions are once again compared wit te good-fit model comparisons in Table 3. Te agreement between te calculated and experimental waveforms sown in Figs. 6 9 is good wit a coefficient of determination of or iger. From Fig. 9. Model and experimental waveforms for sensor (a) 0, (b) 0.3, (c) 0.6, and (d) 1:2 m from medium smoke interface. Te 4% reflectance target is located 7:3 m from te smoke interface. Te solid curves are model calculations, and te dased curves are experimental. Table 3 te calculated and measured parameters agree wit te good-fit model parameters to witin 25% or better. It is believed te discrepancies between te good-fit model parameters and tose calculated wit te Mie teory are due to uncertainty in te smoke concentrations, wic are caused by uncertainty in te transmissometer measurements from wic te smoke concentrations are derived, and uncertainty in te particle sie distribution. Te departure of te particle sie distribution from lognormal at te larger sies is sown in Fig. 4. More particles in tat sie range would lower te calculated backscatter coefficient, tus improving te agreement. Uncertainty in te particle sie distribution also results in uncertainty of te smoke concentration, owing to te dependence of κ ext, te transmissometer extinction coefficient at 0:6328 μm, calculated from te Mie teory based on a particular particle sie distribution. B. Fig. 8. Model and experimental waveforms for sensor (a) 2.4, (b) 3.0, (c) 3.7, and (d) 4:3 m from target. Te target and te sensor are immersed in eavy smoke. Te solid curves are model calculations, and te dased curves are experimental APPLIED OPTICS / Vol. 47, No. 22 / 1 August 2008 Discussion of Results We see good agreement between te experimental results and te single-scattering predictions. Tis is an indication tat for our geometry, te single-scattering treatment is adequate. As stated previously if te optical dept τod < 0:8, single scattering prevails; for 0:8 < τod 1, tere is only a small contribution from second-order scattering. For larger values of te optical dept, iger orders of multiple scattering sould be considered [14]. For medium smoke and pat lengts of 2 to 4 m, te good-fit optical dept is 0:3 to 0.6, so multiple scattering is not significant. Te comparisons sown in Figs. 6 9 do not indicate tat multiple scattering is a major factor. Considering te stretcing of te return pulse as an indicator of multiple scattering [16], we do not see significant pulse stretcing in te experimental data or significant

11 cange in te magnitude of te signal due to te extinction coefficient effectively decreasing, because more potons are received at te detector owing to multiple scattering as compared wit single scattering. Multiple scattering effects do not appear greater tan effects on te waveforms due to uncertainty in te smoke parameters caused by uncertainties in te particle sie distribution and smoke concentrations. It is apparent tat we are operating just below te regime were multiple scattering would need to be included. For future work te effects of multiple scattering need to be included in te treatment. Multiplescattering effects would especially be seen at greater ranges (tan our 4 m). A study migt also be conducted for te effects of multiple scattering as te laser beam divergence and receiver field of view is varied. 6. Conclusions A tecnique is presented for calculating time-resolved lidar return signals at sort ranges for a lidar system eiter inside or outside an obscurant. At sort ranges te return from te obscurant can be significant, potentially greater tan tat from a ard target in te obscurant. Te obscurant in tis work is military fog oil smoke (SGF2) for wic te droplets are sperical and te Mie teory is applicable. Altoug it is sown tat te particle sie distribution is not exactly lognormal, suc a distribution in wic r g ¼ 0:45 μm, σ g ¼ 1:4, and MMD ¼ 1:26 μm is a reasonably good approximation. Te lidar operates at 0:905 μm. Using te Mie teory, calculations of te extinction coefficient, α ext, and te volume scattering function in te backward direction β aerosol ðπþ at te operational wavelengt are sown for ligt, medium, eavy, and very eavy fog oil smoke concentrations. Te ground trut smoke concentration was calculated using a transmissometer operating at 0:6328 μm and applying a mass extinction coefficient based on te Mie teory. Uncertainly in te smoke concentration measurements is estimated to be 20%. Comparisons sow good agreement between te model and experimental waveforms for sort-range sensor distances and for various smoke concentrations. Te model accurately predicts te sape of te waveforms. For a lidar and a low-reflectance target immersed in ligt and medium smoke, two returns can be expected: (1) a backscatter return from te smoke and (2) a return from te ard target. As te density of te smoke increases, only a return from te smoke can be expected because te target is completely obscured due to te attenuation and backscatter of te lidar signal. Tese results sow tat a return from te smoke could generate a false target detection if simple tresolding rules are applied. Te calculated and measured smoke parameters agree wit te good-fit model parameters to witin 25% or better. It is believed tese discrepancies are due to uncertainties in te particle sie distribution and te departure of te distribution from lognormal at te larger sies. Tese, in turn, result in uncertainties in te extinction coefficient used for determining te smoke concentrations and uncertainties in te extinction and backscatter coefficients for te various smoke concentrations. Multiple-scattering effects suc as pulse stretcing or significantly smaller extinction coefficients are not apparent. Te presented model, validated by experimental results, can be used for sort-range calculations if te backscatter and extinction parameters are known for te medium of interest and if te reflectance of te target is known. Tis model can also be used to predict te performance of te current lidar sensor and oter sensors under various atmosperic and battlefield smoke conditions. Te autors gratefully acknowledge te Janney Fellowsip from Te Jons Hopkins University Applied Pysics Laboratory for partial support of tis effort. References 1. R. W. Byren, Laser rangefinders, in Te Infrared and Electro-Optical Systems Handbook, C. S. Fox, ed. (SPIE Optical Engineering, 1993), Vol. 6, Cap. 2, pp M. Kleiman and N. Siloa, Te effect of dense atmosperic environment on te performances of laser radar sensors used for collision avoidance, Proc. SPIE 3707, (1999). 3. R. M. Measures, Laser Remote Sensing (Jon Wiley, 1984). 4. H. N. Burns, C. G. Cristodoulou, and G. D. Boreman, System design of a pulsed laser rangefinder, Opt. Eng. 30, (1991). 5. G. W. Kamerman, Laser radar, in Te Infrared and Electro- Optical Systems Handbook, C. S. Fox, ed. (SPIE Optical Engineering, 1993), Vol. 6, Cap. 1, pp W. C. Hinds, Aerosol Tecnology (Wiley, 1999). 7. K. Willeke and P. A. Baron, Aerosol Measurement (Wiley, 1993). 8. H. R. Carlon, D. H. Anderson, M. E. Milam, T. L. Tarnove, R. H. Frickel, and I. Sindoni, Infrared extinction spectra of some common liquid aerosols, Appl. Opt. 16, (1977). 9. M. E. Tomas and D. D. Duncan, Atmosperic transmission, in Te Infrared and Electro-Optical Systems Handbook, F.G. Smit, ed. (SPIE Optical Engineering, 1993), Vol. 2, Cap. 1, pp D. Deirmendjian, Electromagnetic Scattering on Sperical Polydispersions (Elsevier, 1969). 11. C. S. Zender, Particle sie distributions: teory and application to aerosols, clouds, and soils, (2008), ttp://dust.ess.uci.edu/facts/psd/psd.pdf. 12. D. H. Anderson, Soldier Biological Cemical Command (SBCCOM), Edgewood, Maryland (personal communication, 2001). 13. Joint Tecnical Coordinating Group for Munitions Effectiveness, Smoke and Aerosol Working Group, Smoke and natural aerosol parameters (SNAP) manual, Report No. 61JTCG/ ME-85-2 (1985), p V. A. Kovalev and W. E. Eicinger, Elastic Lidar (Wiley, 2004). 15. R. G. Knollenberg, Tecniques for probing cloud microstructure, in Clouds Teir Formation, Optical Properties, and Effects, P. V. Hobbs and A. Deepak, eds. (Academic, 1981), pp R. E. Walker and J. W. McLean, Lidar equations for turbid media wit pulse stretcing, Appl. 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