range estimation of passive infraredtargets through the atmosphere

Range estimation of passive infraredtargets through the atmosphere

Hoonkyung ChoJoohwan ChunDoochun SeoSeokweon Choi

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Range estimation of passive infrared targets through theatmosphere

Hoonkyung ChoJoohwan ChunKorea Advanced Institute of Science and

TechnologyDepartment of Electrical EngineeringDaejeon, Republic of KoreaE-mail: [email protected]

Doochun SeoSeokweon ChoiKorea Aerospace Research Institute169-84, Gwahak-ro, Yuseong-guDaejeon, Republic of Korea

Abstract. Target range estimation is traditionally based on radar andactive sonar systems in modern combat systems. However, jamming sig-nals tremendously degrade the performance of such active sensor devi-ces. We introduce a simple target range estimation method and thefundamental limits of the proposed method based on the atmospherepropagation model. Since passive infrared (IR) sensors measure IR sig-nals radiating from objects in different wavelengths, this method hasrobustness against electromagnetic jamming. The measured target radi-ance of each wavelength at the IR sensor depends on the emissive prop-erties of target material and various attenuation factors (i.e., the distancebetween sensor and target and atmosphere environment parameters).MODTRAN is a tool that models atmospheric propagation of electromag-netic radiation. Based on the results from MODTRAN and atmospherepropagation-based modeling, the target range can be estimated. To ana-lyze the proposed methods performance statistically, we use maximumlikelihood estimation (MLE) and evaluate the Cramer-Rao lower bound(CRLB) via the probability density function of measured radiance. Wealso compare CRLB and the variance of MLE using Monte-Carlo simula-tion. 2013 Society of Photo-Optical Instrumentation Engineers (SPIE) [DOI: 10.1117/1.OE.52.4.046402]

Subject terms: range estimation; atmospheric attenuation; MODTRAN; atmospherictransmittance; maximum likelihood estimation; Cramer-Rao lower bound; passiveinfrared sensor; distance estimation.

Paper 121129PRR received Aug. 7, 2012; revised manuscript received Mar. 12,2013; accepted for publication Mar. 12, 2013; published online Apr. 22, 2013.

1 IntroductionThe infrared (IR) sensor, which measures light intensityimpinging on it, has traditionally been used to detect a targetor to image an object located at a remote distance. To analyzethe detection performance or the signal-to-noise ratio (SNR)of the IR image, researchers1,2 have investigated the relation-ship among the target range, the target radiance, and theSNR, where the SNR (or the detection probability) is themain parameter of interest, and the target range is usuallytreated as a known input parameter. In this paper, we treatthe target range as a parameter to be estimated and thereceived radiance of the target at the sensor as measurement.Therefore, we use an IR sensor to measure the distance pas-sively, unlike previous approaches,35 where the range wasestimated using the reflected IR signal with an illuminatingIR source.

To estimate the target range from the received IR radianceat the sensor, we need to know the emitted radiance at thetarget, as well as the atmospheric propagation information,i.e., transmittance, sky radiance, and path radiance. The emit-ted radiance is a function of temperature and material proper-ties, such as the emissivity of the target surface. In this paper,target temperature is assumed to be known. For a rapidlyapproaching target, however, target temperature can alsobe estimated using target aerodynamic information by solv-ing the differential equation6,7 that governs heat conduction,convection, and radiation. The heat source could be the

irradiance on the target due to solar and sky radiance, aswell as the frictional dissipation.

Modeling the transmittance, sky radiance, and path radi-ance is difficult, because they are dependent on numeroustime-varying atmospheric parameters that are not easyto determine. A usual practice is to utilize the widelyaccepted MODTRAN8 software, which models the atmos-phere as a number of homogenous layers that can be solvedindividually.

In this paper, we shall derive the maximum likelihoodestimate (MLE) d^ of the range and the Cramer-Rao lowerbound (CRLB)the lowest possible mean squared error forthe passive range estimation problem. Our analysis indicatesthat the passive ranging gives reasonably accurate range esti-mation when target temperature, range, and emissivity aremore than 500 K, 2000 m, and 0.7, respectively.

The measurement model used by MODTRAN isexplained in Sec. 2. The CRLB for the passive range estimateis derived in Sec. 3, and simulation study is shown in Sec. 4.Our conclusion is made in Sec. 5.

2 Measurement Model

2.1 Atmospheric Transmittance

The atmosphere consists of innumerable aerosol and gas par-ticles that scatter and absorb radiation. The atmospherictransmittance ; d, where d is the range from the sensorto the target, is the capacity of the atmosphere to transmitelectromagnetic energy, which is modeled as a dth powerof the transmittance per unit distance , so that0091-3286/2013/$25.00 2013 SPIE

Optical Engineering 046402-1 April 2013/Vol. 52(4)

Optical Engineering 52(4), 046402 (April 2013)


; d d; e

k;where and k are the scattering component and theabsorptive component, respectively. If light travels throughatmosphere of thickness d, the resulting radiance valuewill be attenuated by the factor of ; d. Typical atmos-pheric transmittances obtained with MODTRAN 4.0 forvarying distances are illustrated in Fig. 1.

2.2 Radiance Measurement Model

Assuming there is no scattering into the line of sight, thereceived radiance at the sensor would have three spectralradiance components: the self emission LT, the reflectedambient radiance LAE, and the path radiance Lpath. Ifthe IR sensor is responsive in a band of spectrum min; max,then the total radiance power that the sensor would receivewill be

L Z

max

min

LT LAE Lpathd: (1)

Each of the three contributions will be explained indetail below.

2.2.1 Self-emission

Black body radiation is a type of electromagnetic radiationwithin or surrounding a body in thermodynamic equilibrium

with its environment. The radiation has a specific spectrumand intensity that depends only on the bodys temperature.9

The spectral radiance of an ideal black body with a temper-ature TT follows Plancks black body radiation law; that is,

Lbb; T c15

1

ec2T 1

Wm2m;

where c1 3.7411 108 W m4m2, c2 1.4388104 mK, and l is the wavelength in micrometers. A targetwith temperature TT can be modeled approximately as a graybody having the spectral radiance Lbb; TT, where is the spectral emissivity of the target. If the target is locatedin the range d from the sensor, then the received spectral radi-ance at the sensor will be

LT dLbb; TT: (2)

Therefore,

LT Z

max

min

LTd XKk1

kdkLk; TTk: (3)

The summation approximation in Eq. (3) is more useful,because MODTRAN can provide transmittance only in dis-crete steps of spectrum. The general attenuation trend ofblack body radiation as a function of range is depicted inFig. 2. Note that d can be found accurately from Eq. (2)if LT , k, and TT are known.

3 3.2 3.4 3.6 3.8 4 4.2 4.4 4.6 4.8 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Wavelength (m)

Tran

smitt

ance

Transmittance of the MODTRAN output for various target ranges

d = 100md = 250md = 500md = 1000m

Fig. 1 Typical atmospheric transmittance obtained with MODTRAN 4.0 for different target sensor distances, d 100, 250, 500, and 1000 m.


Cho et al.: Range estimation of passive infrared targets through the atmosphere


2.2.2 Reflected ambient radiance

Another contribution is the ambient down-welling radiationdue to the atmosphere that reaches the target surface and isreflected toward the sensor. Assuming the atmosphere is astack of up to N semi-transparent layers, the down-wellingradiation L;n of the nth layer toward the target can beexpressed as

L;n; Lbb; Tn1 ;nYn1j1

;j;

where Tn is the temperature of the nth layer, ;n transmittance through the nth layer, and 1 ;n emissivity of the nth layer by Kirchhoffs law, which linksthe layer emission to transmission. Therefore, the total down-welling irradiance E is given by

E Z2L; cos di;

where L; is the total down-welling spectral radiancegiven as L;

PNn1 L;n;.

The light reflection behavior at a particular surface ischaracterized by the bidirectional reflectance distributionfunction (BRDF), which is denoted as rBRDF;i;r,where i ; is the incident angle, and r 0; 0is the reflection angle. It is the ratio of the reflected radianceexiting along the reflect angle wr to the irradiance incident onthe surface from the direction wi. Therefore,

Lbb; TAE ErBRDF;i;r

; (4)

where 1 is the spectral reflectivity of the tar-get, and TAE is the average downward environmental temper-ature. Therefore, the radiance that reach the sensor will be

LAE dLbb; TAE: (5)

2.2.3 Path radiance

Finally, we need to consider the path radiance emitted by theatmosphere in the path between the target and the IR sensor.With Kirchhoffs law, the path radiance can be written as

Lbb; TATM L 0; 01 d ; (6)

where

L 0; 0 XNn1

L;n 0; 0;

L;n 0; 0 Lbb; Tn1 ;nYn1j1

;j;(7)

and ;n is the transmittance from the target to the sensorthrough the nth layer. Therefore, the radiance that reachesthe sensor will be

Fig. 2 Radiance-to-distance curve in only the surface-emitted case, 1 2 3 to 5 mm, 3 to 4 mm, and 4 to 5 mm.




Lpath 1 dLbb; TATM:

3 Maximum Likelihood Estimation andPerformance Analysis

We shall show how to find the MLE d^ from the noise-corrupted radiance measurement L. Two types of noise(or uncertainty) are considered in this paper: the thermalnoise10,11 vthm generated in the sensor circuitry and the uncer-tainty in the emissivity . We assume that both areGaussian distributed; that is,

L L vthm; vthm N0; 2thm;

v; v N0; 2:

Therefore, the measured radiance L is given by

L Z

max

min

fdLbb; TT

1 dLbb; TAE 1 dLbb; TATMdg vthm: (8)

To get the range estimate d^ from ~L, we first need to findthe probability density function (PDF), pLjd. Note thatthe transformation from ve and vthm to ~L is linear, whichimplies that p ~Ljd is Gaussian, because ve and vthm areGaussian. Therefore, p ~Ljd is completely determined byits mean ~L E ~Ljd and variance 2 varLjd, whichare given by

L Z

max

min

fdLbb; TT dLbb; TAE

1 dLbb; TATMgd; (9)

2 2thm 02 ;

02

2fZ

max

min

dLbb; TT Lbb; TAEdg2

; (10)

assuming ve and vthm are uncorrelated. Note that 02 is afunction of d. Therefore,

pLjd 122thm 02

p exp

L L222thm

02

; (11)

and

ln pLjd 12ln22thm 02

L L222thm

02

: (12)

Now the MLE d^ is the one that maximizes pLjd, whichcan be found by setting the derivative of ln pLjd to bezero; that is,

ln pLjdd

0

2thm 02 0d

L L2thm 02

Ld

L L2

2thm 02 2 0

0d

0; (13)

where

0d

2Z

max

min

lndLbb; TT Lbb; TAEd;

(14)

Ld

lnL Z

max

min

Lbb; TATMd: (15)

Note that Eq. (13) is not linear and must be solvednumerically. However, if 2 0, then Eq. (13) reduces to

L L2thm

0; i:e:; L L;

from which d^ can be found easily using Eq. (9).Now, using Eq. (13) with Eqs. (14) and (15), we can

obtain the CRLB by

CRLB 1E

d ln pLjd

2. (16)

4 Simulation ResultsWe calculate the variance of MLE solution by Monte-Carlosimulation. We set the emissivity e 0.6, 0.7, 0.8, 0.9, andthe temperature TT 450, 500, 550, and 600 K, for variousstandard deviations in thermal noise sthm 0.5, 1.0, 1.5, 2.0,and in emissivity noise se 0.005, 0.010, 0.015, 0.020.

4.1 Effect of Thermal Noise

The average and variance of maximum likelihood estimatordML is evaluated 20,000 times in Monte-Carlo simulation viavarious standard deviations in thermal noise, temperature,and emissivity. The results of Monte-Carlo simulationshown in Figs. 35 are converged to CRLB, because themaximum likelihood estimator of the distance is efficient.

From Fig. 3, we see that the CRLB and the standarddeviation of ML estimated distance to true distance curvefor various thermal noise standard deviations. As the stan-dard deviation in thermal noise sthm increases, the standarddeviation of estimated distance also increases, and the detec-tion performance decreases. As Figs. 4 and 5 show, theintense Ldetector causes better detection performance. In sum-mary, as sthm increases, and the target temperature and emis-sivity decrease, the standard deviation of estimated distanceand detection performance increases. Passive ranging givesreasonably accurate range estimation within a margin oferror of 10% with a 95% confidence interval when the targettemperature, range, standard deviation in thermal noise, andemissivity are more than 500 K, 2000 m, 1.0, and 0.7,respectively.




0 500 1000 1500 2000 2500 3000 3500 4000 4500 50000

50

100

150

200

250

300

Distance (m)

Stan

dard

dev

iatio

n of

estim

ated

dist

ance

Random noise in measured radiance data (thermal = 1, = 0.8)

CR bound (TT = 450K)MC (TT = 450K)CR bound (TT = 500K)MC (TT = 500K)CR bound (TT = 550K)MC (TT = 550K)CR bound (TT = 600K)MC (TT = 600K)

Fig. 4 CRLB and standard deviation of ML estimated distance to distance curve in the thermal noise case, TT 450, 500, 550, and 600 K.

0 500 1000 1500 2000 2500 3000 3500 4000 4500 50000

50

100

150

200

250

300

350

400

450

500

550

600

Distance (m)

Stan

dard

dev

iatio

n of

estim

ated

dist

ance

Random noise in measured radiance data (TT = 450 K, = 0.8)

CR bound ( thermal = 0.5)MC ( thermal = 0.5)CR bound ( thermal = 1)MC ( thermal = 1)CR bound ( thermal = 1.5)MC ( thermal = 1.5)CR bound ( thermal = 2.0)MC ( thermal = 2.0)

Fig. 3 CRLB and standard deviation of ML estimated distance to distance curve in the thermal noise case, sthermal 0.5, 1.0, 1.5, and 2.0.




0 500 1000 1500 2000 2500 3000 3500 4000 4500 50000

50

100

150

200

250

300

Distance (m)

Stan

dard

de

via

tion

o

f est

ima

ted

dist

ance

Random noise in emissivity (TT = 450 K, = 0.8)

CR bound ( = 0.005)MC ( = 0.005)CR bound ( = 0.01)MC ( = 0.01)CR bound ( = 0.015)MC ( = 0.015)CR bound ( = 0.02)MC ( = 0.02)

Fig. 6 CRLB and standard deviation of ML estimated distance to distance curve in the emissivity uncertainty case, se 0.005, 0.01, 0.015, and0.02.

0 500 1000 1500 2000 2500 3000 3500 4000 4500 50000

50

100

150

200

250

300

350

400

Distance (m)

Stan

dard

dev

iatio

n of

estim

ated

dist

ance

Random noise in measured radiance data (thermal = 1, T =T 450 K)

= 0.6)MC ( =0.6)CR bound (

CR bound (

= 0.7)MC ( = 0.7)CR bound ( = 0.8)MC ( = 0.8)CR bound ( = 0.9)MC ( = 0.9)

Fig. 5 CRLB and standard deviation of ML estimated distance to distance curve in the thermal noise case, e 0.6, 0.7, 0.8, and 0.9.




0 500 1000 1500 2000 2500 3000 3500 4000 4500 50000

50

100

150

Distance (m)

Stan

dard

de

via

tion

of e

stim

ate

d di

stan

ce

Random noise in emissivity ( = 0.01, T = 450 K) T

CR bound ( = 0.6)MC ( = 0.6)CR bound ( = 0.7)MC ( = 0.7)CR bound ( = 0.8)MC ( = 0.8)CR bound ( = 0.9)MC ( = 0.9)

Fig. 8 CRLB and standard deviation of ML estimated distance to distance curve in the emissivity uncertainty case, e 0.6, 0.7, 0.8,and 0.9.

0 500 1000 1500 2000 2500 3000 3500 4000 4500 50000

50

100

150

Distance (m)

Stan

dard

dev

iatio

n of

est

imat

ed d

istan

ce

Random noise in emissivity ( = 0.01, = 0.8)

CR bound (TT = 450K)MC (TT = 450K)CR bound (

CR bound (

T = 500K)T

T

T

T

MC (TT = 500K)T = 550K)

MC (TT = 550K)CR bound ( T = 600K)MC bound ( T = 600K)

Fig. 7 CRLB and standard deviation of ML estimated distance to distance curve in the emissivity uncertainty case, TT 450, 500, 550,and 600 K.




4.2 Effect of Emissivity Uncertainty

The average and variance of maximum likelihood estimatordML is also evaluated 20,000 times in Monte-Carlo simula-tion via various standard deviations in emissivity noise, tar-get temperature, and emissivity. Although the mean andvariance of the PDF is a nonlinear equation of the distanced in Eq. (11), the results of Monte-Carlo simulation shown inFigs. 68 are converged to CRLB, and the maximum like-lihood estimator of the distance is efficient. To evaluate theMLE solution dML, we differentiate the log-likelihood func-tion and set it equal to zero. Unlike the case of only thermalnoise, it cannot be obtained from a simple linear equationand closed form. In Fig. 6, we see that the CRLB and thestandard deviation of ML estimated distance to true distancecurve for various emissivity noise standard deviations. As thestandard deviation in emissivity noise se increases, the stan-dard deviation of estimated distance also increases, and thedetection performance decreases. By comparing Figs. 4 and5 with Figs. 7 and 8, we can see that target temperature andemissivity are insensitive to standard deviation of estimateddistance in the case of emissivity noise. In short, as seincreases, the target temperature and emissivity decrease,and the standard deviation of estimated distance increases.Passive ranging gives reasonably accurate range estimationwithin a margin of error of 10% with a 95% confidence inter-val when the target temperature, range, standard deviation inemissivity noise, and emissivity are more than 500 K,2000 m, 0.02, and 0.7, respectively.

5 ConclusionIn this paper, we propose the range estimation method usingmaximum likelihood estimation based on the atmospherictransmittance and atmosphere propagation-based radiancemodel. The fundamental limits of target range estimationare computed for various standard deviations in thermaland emissivity noise, target temperature, and emissivity.To analyze the performance of the proposed method, weevaluate the CRLB and Monte-Carlo simulation results.Because the solution of maximum likelihood estimation is

an efficient estimator, the variance of the estimated targetrange is converged to the evaluated CRLB. In contrastwith the case of thermal noise, the temperature and emissiv-ity are insensitive parameters to the detection performance inthe case of emissivity noise. Although there are many moreuncertainties in the real environment, we find out the funda-mental limits evaluated from CRLB are very dependent onthermal noise and emissivity uncertainty.

AcknowledgmentsThis work was supported in part by the contracts ADDIIRC, 2011, and National Research Foundation of Korea(NRF) Grant funded by the Korean government(No. 2012M4A2026724).

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Biographies and photographs of the authors are not available.




range estimation of passive infraredtargets through the atmosphere

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