lidar aerosol backscatter measurements: systematic, modeling, and calibration error considerations

Lidar aerosol backscatter measurements: systematic,modeling, and calibration error considerations

Michael J. Kavaya and Robert T. Menzies

Sources of systematic, modeling, and calibration errors that affect the interpretation and calibration of lidaraerosol backscatter data are discussed. The treatment pertains primarily to ground-based pulsed CO2 lidarsthat probe the troposphere and are calibrated using hard calibration targets. However, a large part of theanalysis is relevant to other types of lidar system such as lidars operating at other wavelengths; cw focused

lidars; airborne or earth-orbiting lidars; lidars measuring other regions of the atmosphere; lidars measuringnonaerosol elastic or inelastic backscatter; and lidars employing other calibration techniques.

1. Introduction

The remote measurement of atmospheric aerosolbackscatter coefficients using lidar is well suited forthe present need to expand the statistical data base ofbackscatter coefficients over broad ranges of latitude,longitude, altitude, time, and weather conditions.Quantitative measurements in the 9-11-m spectralregion are important for various CO2 lidar remote sens-ing goals. These goals include improved understand-ing of aerosol particles and their transport in the atmo-sphere, use of the DIAL technique to measureatmospheric species concentrations and distributionsof pollutant gases, and assessing the feasibility of anearth-orbiting CO2 lidar which would use aerosol back-scatter to globally measure atmospheric winds. Glob-al wind field measurements, with 1-km vertical resolu-tion in the troposphere, would greatly enhance 12-hour-5-day weather forecasts.1 Since the returnsignal from a given range using infrared coherent lidartechniques is directly proportional to the aerosol back-scatter coefficient in that region, uncertainties in theknowledge of the backscatter coefficient imply corre-sponding uncertainties in required laser transmitterpulse energy and receiver telescope diameter of anearth-orbiting lidar.

We have been operating a ground-based pulsed CO2lidar over the past three years at JPL in order tomeasure the tropospheric aerosol backscatter coeffi-cient in the 9-11-,um region. This system is wave-

Both authors were with the California Institute of Technology, JetPropulsion Laboratory, 4800 Oak Grove Drive, Pasadena, California91109, when this work was done; M. J. Kavaya is now with CoherentTechnologies, Inc., P.O. Box 7488, Boulder, Colorado 80306-0403.

Received 10 May 1985.0003-6935/85/213444-10$02.00/0.© 1985 Optical Society of America.

length tunable using an injection-locked TEA laserwith a Littrow-mount reflection grating in order toproduce tunable single-frequency pulses for coherentdetection. Both the injection and local oscillator la-sers are grating tunable and frequency stabilized.Aerosol backscatter profiles at both 9.25 and 10.6 ,umhave been measured using this lidar system at JPL on anumber of days, beginning in February 1983.2

Since quantitative values of the backscatter coeffi-cient are desired, a technique for accurately calibratingthe backscatter data has been developed which uses alarge, flat surface as a lidar calibration target.3 Thecalibration technique employs the addition of numer-ous individual shots of the lidar aimed (1) horizontallyat the calibration target, (2) horizontally through theatmosphere (boundary layer), and (3) verticallythrough the atmosphere. In addition to the returnsignal, a transmitted pulse power signal is recorded forpulse energy normalization. Thus each lidar pulseresults in two digitized records which are related totransmitted power vs time and received (backscat-tered) power vs time. These data records must beappropriately acquired, processed, averaged, and thencombined with the measured or modeled characteris-tics of the lidar system and parameters of the atmo-sphere and calibration target in order to correctly cal-culate the desired aerosol backscatter coefficientprofile. The exact processing steps and their orderwill necessarily depend on the type of lidar system, theoperating wavelength, the receiver electronics, and thedesired measurement outputs. Some of the steps en-countered with our coherent CO2 aerosol backscatterlidar system are shown in Fig. 1. Several of these stepsmay not be appropriate in certain cases whereas addi-tional steps may be necessary for other lidar systems,such as DIAL systems.4 Equally important consider-ations include the choice of processing steps which are

3444 APPLIED OPTICS / Vol. 24, No. 21 / 1 November 1985

DIGITIZE SIGNAL

SUBTRACT DC OFFSET

NONLINEAR RECEIVERELEMENT

Vout 9 VN -1 I vot

NONLINEAR DETECTION

VIN o\/P- VIN

NORMALIZE TOPULSE ENERGY

CORRECT FORPULSE PROFILE

SIGNAL AVERAGE DIVIDE BY TELESCOPEN SHOTS OVERLAP FUNCTION

SMOOTH DATA

CALCULATE RMS NOISE

REMOVE ATMOSPHERICEXTINCTION

MULTI PLY BY R2

Fig. 1. Possible steps in lidar aerosol backscatter data processing.

implemented for each pulse rather than the averageddata and the specific order of the processing steps.

In this paper we discuss the sources of systematic,modeling, and calibration errors encountered in lidaraerosol backscatter measurements, their interaction,and their relationship to the many steps in data pro-cessing and to the lidar receiver and data acquisitionhardware characteristics. In Sec. II the lidar equa-tions for backscatter from the atmospheric aerosol andfrom a hard target are reviewed. Sources of systemat-ic error that arise from the data acquisition and dataprocessing steps are then discussed in Secs. III and IV,respectively. Section V contains a discussion of errorswhich arise due to incorrect modeling of the measure-ment hardware and/or environment. Conclusions andrecommendations appear in Sec. VI.

The phenomena of speckle and turbulence can sig-nificantly affect the statistics of lidar backscatter sig-nals. These effects are described in the literature5 -8and will not be treated here since the topics in thispaper can, for the most part, be considered separately.When significant, the effects of neglecting speckleand/or turbulence will be pointed out in the discussion.

1. Lidar Backscatter Signals

Consider a pulsed infrared lidar system with collin-ear or near-collinear (side-by-side) geometry for thetransmitter and receiver telescopes and assume thatboth telescopes subtend very small solid angles at theranges of interest. Let the transmitted pulse start attime t = 0 and end at time t = p with a power profile ofPtb(t) (W), as shown in Fig. 2(a). The received power(W) due to aerosol backscatter is then given by

Pb(t) = I Ptb t -- ) (R) n O(R)kCt-t/ c/ R2

R -X exp -2 |' ab(R )dR ]dR, (1)

where A is the effective receiver area, R is the line-of-sight range from the telescopes, iq is the system's opti-cal efficiency, O(R) is the range-dependent telescope

overlap function, ab(R) (m-1) is the total extinctioncoefficient of the atmosphere along the optical path,

(R) ( 2 m-3 sr-' = m- 1 sr-') is the aerosol volumebackscatter coefficient defined as the fraction of inci-dent energy scattered in the backward direction perunit solid angle per unit atmospheric length, andwhere the integration over R indicates that the re-ceived power at time t is due to contributions from aslab of atmosphere of thickness Tp/2 centered at Rb =c(t/2--rp/4). Other sources of received power at timet, such as multiple scattering, Rayleigh scattering, res-onant fluorescence, fluorescence, and Raman scatter-ing are neglected. (Rayleigh scattering and associatedextinction must be included for near-IR wavelengths,but at 10-ym wavelength its neglect is a good approxi-mation.) The probability density function of Pb(t)will be determined by speckle and turbulence effectsand by the detection process of the receiver (hetero-dyne or direct). Thus Pb(t) is understood to be theexpected value. For large turbulence, the expectedvalue will be lowered and a turbulence factor should beadded to Eq. (1). Any effect of the laser pulse itself onthe atmospheric parameters is also neglected. Theseassumptions, if untrue, could introduce systematic er-rors in determining (R) in Eq. (1). Monochromatictransmitted radiation is assumed. If the receiver em-ploys heterodyne detection, /3(R) in Eq. (1) refers onlyto the backscattered radiation with polarization paral-lel to the polarization of the local oscillator. Figure2(b) depicts the aerosol backscatter signal.

From Eq. (1) we see that the finite pulse durationcauses a bias against the measurement of low values of3(R) that might occur over spatial distances smallerthan cTp/2 in addition to the bias against short-livedlow /3(R) values due to temporal averaging over themeasurement time. This bias must be consideredwhen using backscatter data to estimate the probabili-ty of encountering low values of :(R) on various spatialand temporal scales. The smoothing function thatoperates on $3(R) is shaped by the range-dependentterms in Eq. (1). If we now assume that both /3(R) andO(R) are slowly varying compared to the spatial dis-tance cTp/2 (300 m for a 2 -,sec pulse), Eq. (1) becomes

Pb(t) = (Rb)Aij(Rb) 2 Ptb(t -2R)R-21C(tTp)/2 C

x exp [-2 JR ab(R')dR'] dR. (2)

A further simplification of Eq. (2), which is commonlyused, is to assume that the entire integrand exceptPtb(t) is constant over the integration range, yielding

A [Rb ()d'c 1)Pb(Rb) = fl(Rb) 0O(Rb) exp [ 2E b 2 | Pb(t)dt.

(3)

This simplification can lead to systematic errors in thecalculated value of /3(R) and will be discussed later.The integral of Ptb(t) in Eq. (3) is simply the pulseenergy Etb (J)-

We now consider the same pulsed lidar system di-

1 November 1985 / Vol. 24, No. 21 / APPLIED OPTICS 3445

p b(t)

ps ()

(a)

) rp t

b2/R

o t

(c)Rs RANGE TO TARGEc

o 20 2 ts - +pC C

Fig. 2. Pulsed lidar temporal profile examples for (a) the transmit-

ted laser pulse power, (b) the backscattered power from the atmo-

spheric aerosol, and (c) the backscattered power from a hard target.

rected at a calibration target that is at a range R, fromthe telescopes. Let the target be larger than both thelaser spot size at R, and the receiver field of view at R3,and let the polar angle between the optical axis and theaverage surface normal of the target be 0. If the pulsein Fig. 2(a) [Pt,(t) in this hard-target case] is transmit-ted, the received power is given by

P8 (t) = Pt - Cp R2

x exp [-2 a(R)dR] (4)

where a,(R) (m-1) is again total extinction and is mostlikely different from ab(R) due to the different atmo-spheric paths employed, p*(sr-') is the target parame-ter defined as the reflected power per steradian towardthe receiver divided by the incident power, and aerosolbackscatter is added to the earlier list of neglectedsources of received power at time t. [Determining thecorrect value of p* to use in Eq. (4) is critical to accu-rately determining /3(R) and has many pitfalls associ-ated with it. This process is described in detail else-where.39] Note that the received-power profile isidentical to the transmitted-power profile but is de-layed by 2R,/c sec. This is depicted in Fig. 2(c). Onceagain, Pj(t) is understood to be the expected value inthe presence of speckle fluctuations, and a turbulencefactor should be added to Eq. (4) if a high level ofturbulence exists along the optical path, which mightbe expected for long horizontal paths to hard targets.7

Let us examine a very simple case in which one lidarpulse is fired at the atmosphere and one pulse is firedat the calibration target. Comparing Eqs. (3) and (4),we see that the integral of the target return profilegiven by Eq. (4) must be calculated. This time inte-gral of P8(t), I, (J), is given by

2R,

Ps(t)dt = p* A 2 10(R,) exp -2 Ia(R')dR' Ets

C (5)

Combining Eqs. (3) and (5) we see that

(b)

/3(R,) xPb( Its x p* x O) X 2 X RbflR)=Etb IS, O(Rb) c R

XR

exp [-2 J, as(R')dR'1

exp [-2 J:" b(R ) dR] (6)

where we let the atmospheric and target pulses havetransmitted energies Etb and Et,, respectively. Evenin this simple example in which fluctuations of thereturned signal intensity due to speckle and atmo-spheric turbulence are ignored and in which an ade-quate signal-to-noise ratio is assumed from one lidarpulse, it is clear that the lidar pulse energies, the targetparameter p*, the telescope overlap function O(R), andthe extinction profiles as(R) and ab(R) must be wellknown. In addition, the assumptions and simplifica-tions leading to Eqs. (3) and (5) cannot be ignored.

M1. Data Acquisition Errors

The discussion leading to Eq. (6) assumed that thetransmitted-pulse profiles Ptb(t) and Pt,(t), the aerosolbackscatter signal Pb(t), and the target backscattersignal P,(t) were all known in their fundamental unitsof watts. In reality, detectors are used to convert theoptical fields into a voltage (or current), followed bypreamplifiers, amplifiers, filters, etc. Finally, thisvoltage is recorded for later processing. We may lumpall the characteristics of this chain of components (orreceiver) into a single function that operates on theoptical signal power: V(t) = F[P(t)], where V(t) (V) isthe recorded signal available for processing and is notnecessarily a linear function of P(t). We may factorthe function F into various typical subfunctions, suchas the gain G of the receiver and the largest timeconstant X of the receiver, which represents the slowestelement in the component chain. The recorded volt-ages from the transmitted pulse (t), the aerosol back-scatter (b), and the target backscatter (s) are thengiven by

Vt(t) = Gt X F[P(t)] * I-exp(-t/Tt) X H(t)]

Vb(t) = Gb X Fb[Pb(t) * - exp(-t/b) X H(t)]

V5(t) = Gs X F[P5(t)] * [- exp(-t/T,5 ) X H(t)]

(7a)

(7b)

(7c)

where the symbol * stands for the convolution opera-tion and H(t) is the Heaviside unit step function.lWe are assuming that the slowest element of the re-ceiver is much slower than the next slowest elementand that this element can be approximated by a simpleRC low-pass filter (r = RC).

Strictly, the first four terms on the right-hand side ofEq. (6) must be replaced by appropriate expressionsusing Eq. (7). Equation (7a) must be applied twice tosolve for Ptb(t) and Pt,(t), which are then integrated to


PtQl)

obtain Etb and Et,. Equation (7b) must be solved forPb(t) and Eq. (7c) must be solved for P5 (t), which isthen substituted into Eq. (5) to obtain I,. Fortunate-ly, several simplifications are possible. Since the con-volution operator in Eq. (7a) does not affect the areaunder Vt(t),10 and only the integral of Vt(t) is needed,the convolution in Eq. (7a) may be neglected, providedthat the function Ft is linear. (The assumption thatthe function Ft is linear is reasonable since linear de-tectors are usually used to monitor pulse power.) Wewill make this assumption and thus Eq. (7a) reduces to

Vtb(t) = Gtptb(t),

Vt,(t) = GtPts(t)

(8a)

(8b)

Note that it is also assumed that gain Gt is constant.If only a small portion of the pulse cross section isimaged onto the pulse-power detector, pulse-to-pulsevariations in the pulse spatial profile may cause varia-tions in the fraction of pulse power incident on thedetector. In effect, a pulse-to-pulse (or time) varia-tion in G results. A similar effect may arise frominterference between the front and rear reflections offa beam splitter used to monitor the pulse. Imaging theentire pulse cross section onto the detector may bedifficult with the typically small-area, high-speed de-tectors. This approach may also damage the detectoror cause nonlinear operation. One possible solution tothis dilemma may be to use an integrating sphere witha low throughput valve for the pulse-power monitor.

It is not quite as reasonable to eliminate the convolu-tions in Eqs. (7b) and (7c), since often a lidar receiver isnot linear in received power. Two common nonlinearexamples of the functions Fb and F are V\P (see Fig. 1)or lnP. (A heterodyne receiver with a linear videodetector will yield a voltage proportional to the squareroot of received power. It is common to use a logarith-mic amplifier on signals proportional to received pow-er or even proportional to the square root of receivedpower.) If Pb(t) does not have frequency componentsof interest greater than 1 /rb, the convolution in Eq. (7b)may be eliminated. Although Eq. (5) is used to inte-grate P(t), the convolution in Eq. (7c) may not beignored unless F is linear or is very small. [Eventhough P(t) is identical to Pt(t) except for a delay, theneglect of the convolutions in Eqs. (7a) and (7c) de-pends on the linearity of Ft and F, respectively.] Nev-ertheless, to keep the expressions tractable, we willneglect the convolutions in Eqs. (7b) and (7c), thuspossibly introducing a systematic source of error incalculating (R). It is usually the case that Fb = F =Fr, where the subscript r refers to the lidar receiver.Often the gain of the receiver is adjusted betweenfiring at the atmosphere and firing at the calibrationtarget due to the much larger signal from the target.In addition, the gains Gb and Gs may depend on thereceived powers Pb and P and on the bandwidth of thereceived power due to a nonlinear and/or dispersivecomponent. If a receiver component is nonlinear, i.e.,Vout = g(Vi.) (see Fig. 1), this nonlinear dependencemust be inverted before signal averaging. The band-

width of P5(t) may differ from the bandwidth of Pb(t)due to speckle, turbulence, and target effects. Theseeffects on the receiver gain can introduce errors if notaccounted for. We will ignore these effects in whatfollows and simplify Eqs. (7b) and (7c) to

Vb(t) = Gb X Fr{Pb(t)1 (9a)

V(t) = G, x Fr{Ps(t)1 -

Combining Eqs. (5), (8), and (9) we find that Eq. (6)becomes

fl(Rb) Fr {Vb(t)1Gb1 X _R,

fJ' Vtb(t)dt Wc C

O(R) ) 2 = 2,- _X-

(9b)

J|P Vt,(t)dt

X p*

exp [2 :I cs(R')dR' (e [ f° I ,I (10)

exp [-2 fo ab(R')dRII

where the factors G in Eqs. (8) have canceled andFr-1{xJ represents the inverse operation of FrIx}.

At this point we observe a major difference betweenthe linear, square-root, and logarithmic lidar receiverfunctions Fr. As stated earlier, gains Gb and G areoften purposely made different due to the larger returnfrom calibration targets than from the atmosphericaerosol. In practice, it is much easier to know the ratioof these gains GS/Gb than it is to know both their valuesindependently. From Eq. (10) it is clear that the ratioG/Gb is sufficient to calculate (R) for linear andsquare-root lidar receivers, where Fr-ix} = x and x2,respectively. However, for a logarithmic receiver, Fr-1{x = exp(x) and the actual values of Gb and G arerequired to solve for (R). This suggests that logarith-mic receivers are relatively difficult to calibrate ifquantitative lidar aerosol backscatter data are to beobtained using the hard-target calibration technique.

Also, it has been assumed in this section that asufficiently fast transient digitizer is used to recordVtb(t), V(t), Vb(t), and V(t). Another type of A-Dconverter that is often used is an integrating A-Dconverter. These devices usually have a variable gatewidth and integrate the voltage waveform during thegate period. It is clear that an integrating A-D con-verter is acceptable to record the pulse energy signals,provided that the detector is linear in power. FromEq. (10), however, it is apparent that the use of anintegrating A-D converter would be inappropriate forintegrating V(t), and systematic errors would result ifthe function F is nonlinear in power.

IV. Data Processing Errors

We have derived an expression for (R), given by Eq.(10) for the case of a single lidar pulse directed into theatmosphere and a single lidar pulse directed at a cali-bration target. The energy of each pulse was used tonormalize the return signals. It was assumed that thesignal-to-noise ratios were adequate and that both


FPF,-'JV,,(t)1GJdt

speckle and atmospheric turbulence effects were notpresent. In reality, the received-power profiles fromboth the calibration target and the atmospheric aero-sol will fluctuate from pulse to pulse due to speckle andatmospheric turbulence. These sources of signal fluc-tuation and the desire to improve low signal-to-noiseratios usually make it mandatory to signal-average theresults of many lidar pulses.

We now consider a measurement consisting of Nb

pulses fired into the atmosphere and N, pulses fired atthe calibration target. [Of course, the extinction pro-files ab(R) and a,(R), as well as /(R) itself, may changeduring the measurement time, due to temporalchanges of of the atmosphere or due to spatial move-ment of the lidar system, thus introducing a loss oftemporal or spatial resolution.] Equation (10) be-comes

-35 - vx- } -4

-10

-50 5 GAMMA DENSITY FUNCTION

20 - I4 4 4 8M 10 12 14 16 18 2

order~~ ~~~ M.1-

= 25 - 1.5 . \p0I8-30 -06

-40~~~~~~~2-45

-50 0 2 4 6 8 10 12 14 16 18 20

M

Fig. 3. Calculated error in the estimated mean value of a random

variable x due to averaging the functions \Vx and lnx vs M where the

probability density function of x is the gamma density function oforder M.

F,-7'<Fr(X)>} = Fr- { J Fr(X)P(X)dX}

</(Rb)> =

I1 F [V,5(t)lGb]

Nb E= JP V(t)dt

2R .

N + RW Fr 1 [V,,j(t)1Gj dt1 C

., j=1 Vt j(t)dtf.

Rb2s 2

O(R,):

exp [-2 i: as(R')dR']

exp [ -2 Jf ab(R')dR']

2C

(11)

where < > indicates an ensemble average and thesuperscripts i and j on Vb(t), Vtb(t), V5 (t), and Vt,(t)represent the pulse-to-pulse variation in the recordedwaveforms. Often, however, Eq. (11) is not used toprocess signal-averaged lidar data. Instead, the re-corded voltage waveforms Vbi(t) and Vj(t) are aver-aged and only afterward is the inverse operator Fr'-applied. If Fr is a linear function, both methods areidentical.

It is interesting to calculate the error caused by usingthe latter (incorrect) signal processing method de-scribed above in typical lidar aerosol and target back-scatter conditions, when Fr is a nonlinear function suchas VP or lnP. Let Xi be an independent randomvariable with a probability density function p(x) andmean value <x>. Let Y = (1/N) [X 1 + X2 + . . . XN]have probability density p(y). By the central limittheorem, for large N p(y) will be a Gaussian densityfunction with the mean value <y> = <x>. This meanvalue is given by

<y> = <x> J xp(x)dx (12)

and represents the correct signal-averaged result givenbyEq. (11). We now consider afunction ofx,Fr(x). Ifwe signal-average Fr(x), we again obtain a Gaussiandensity function with a mean value equal to <Fr(x)>.Following the incorrect technique, we now perform theinverse operation

(13)

The ratio of Eq. (13) to Eq. (12) yields the error causedby the incorrect signal-averaging technique, and thiserror obviously depends on p(x) and Fr(x).

We may approximate the probability density func-tion of the returned power in a typical lidar system dueto speckle and atmospheric turbulence by the gammadensity function of order M.8 This function is showninset in Fig. 3 and ranges from an exponential for M = 1to a Gaussianlike function (but with x > 0 only) forlarge M. The exponential (M = 1) form is predictedfor fully developed speckle with no aperture averaging.The ratio of Eq. (13) to Eq. (12) has been calculated forthe gamma density function and for FrIx} = Vx andlnx. For FrIxI = Vx the ratio is

<\/x> 2 r I1X3X5X ... x(2M-1) 2

<x> ML (M - )! X2M I (14)

which approaches 1 for large M. The percentage erroris plotted in Fig. 3 and reaches a maximum of -22%when M = 1. Thus /3(R) will be underestimated by22%. When Fr{x) = lnx, the ratio is given by

exp<lnx>I = (0.561)M-1 [exp 'I n

n=l

(15)

which also approaches 1 for large M. As seen in Fig. 3,the error in this case is greater than for FrIx} = Vx andreaches a maximun of -44% for M = 1. Zrnic'l ad-dressed the closely related problem of comparing /X(linear) and lnX (logarithmic) radar receivers with X(quadratic) receivers. He treated only the M = 1exponential power distribution (Rayleigh amplitudedistribution) case and solved for both the error andstandard deviation of the power estimate as a functionof the number of samples N. For large N his resultsare identical to our M = 1 results. He found that forlarge N, a linear receiver requires 1.09 times moresamples than a quadratic receiver to achieve the samestandard deviation, and the logarithmic receiver re-quires 1.64 times more samples.

It is clear that, when the receiver function Fr isnonlinear, the correct signal-averaging technique of


Eq. (11) shouldbeused. Onlyifp(x) andFrfx}arewellknown could the expressions of Eqs. (12) and (13) beused to calculate a correction factor for the derivedvalues of /3(R) when using the incorrect technique.

A second source of data processing systematic errormay be due to the technique of pulse energy normaliza-tion. Equation (11) indicates pulse-by-pulse normal-ization. Often, however, normalization by the pulseenergy is not done on a pulse-by-pulse basis, but ratherthe signal voltages and pulse energies are summedseparately and the sums are divided at the end of themeasurement. We may generalize either the numera-tor or denominator of Eq. (11) and use a more conve-nient notation. Let the desired ensemble average,including the pulse-by-pulse normalization, be givenby

N K (16)

where Ki is the return signal and Ej is the pulse energyof the ith pulse. Let S' be the incorrectly calculatedensemble average as described above:

NN EZKi

S' =. *-1(17)

i=1

If all the pulse energies Ei equal E0, a constant, S' = S,and either technique is satisfactory. We now assumethat E = E + e, where E0 is the average pulse energyand ei represents the deviation of the ith pulse energyfrom Eo. Since it is obvious that S' - S for large pulse-energy fluctuations, we will assume the deviations aresmall, i.e., ei << E. If we insert the expression for Eiinto Eqs. (16) and (17), the fractional error is given by

NE Kici 2

E Kr + ° EO * (18)

The error goes to 0 as ei/Eo approaches 0. The size ofthe error will increase to the extent that K and ei arecorrelated. However, speckle and atmospheric turbu-lence will tend to reduce this correlation. (Thus thecorrelation may be range dependent.) Staehr et al. 4

discuss this source of error for DIAL measurementsand report a maximum error of 10% in SO2 concentra-tions integrated over a plume at a distance of 900 m forlaser energy fluctuations of 10%.

V. Modeling Errors

This section deals with systematic errors that arisefrom incorrectly modeling system or atmosphericcharacteristics. We begin with the assumption thatthe transmitted pulse has a rectangular temporal pro-file.

Equation (2) is an exact expression for the aerosolbackscatter power profile Pb(t), provided that theaerosol backscatter coefficient (R) and the telescope

overlap function O(R) are slowly varying with respectto the spatial distance crp/2. In going from Eq. (2) toEq. (3), both the R-2 and exponential terms were re-moved from the integrand in order that the remainingintegral would reduce to the transmitted pulse energy,an easily measured quantity. All terms removed fromthe integral in Eq. (1) were assigned their values at themidpoint of the integration range Rb. The rationalefor this simplification often starts with the assumptionthat the transmitted pulse, Ptb(t), is rectangular withpower Po, duration rp, and therefore energy Prb.Thus Ptb is removed immediately from the integral inEq. (2) and replaced with the constant Po. At thispoint the range of integration is assumed to be narrowenough that the values at the midpoint, Rb, are used forthe remainder of the integrand. However, the R-2 andexponential terms in the integrand combine to form aweighting function for the laser pulse profile Ptb(t).This function favors the tail of the laser pulse (smallerR) in a complex way that depends on R, ab(R), and rp.We have evaluated the percentage error in Eq. (2) thatresults from assuming that the pulse shape is rectangu-lar with power Po when the actual pulse shape consistsof two rectangular sections of unequal power and dura-tion, as shown in Fig. 4. Keeping the total pulse dura-tion constant at rp, we let the initial pulse power be Pfor rp/a sec and then provide for a tail with power P1/bfor the remainder of the pulse. This bilevel pulseprofile is a good approximation to the output profile ofmany pulsed lasers that exhibit an initial gain-switched spike followed by a lower-power longer-last-ing tail. The pulse energy is fixed at P-rp by letting

P = Pab O (19)

If a = b = 1, Eq. (19) reduces to the rectangular pulse.Figure 4 shows the resultant percentage error vs rangefor several values of the parameters (a,b), for the totalpulse duration rp, and for a constant attenuation coef-ficient ab(R) = 0.125km-1. Itisclearthatthepercent-

t.AS

0 4 8 12 16 20 24 28 32 36 4070 I I I

60 \ o = 0.5 , P 1 -1 )0\ -b = 0125 kml PULSE ENERGY = P1p

50 T t/2 b

tb'7 fy dR p1

30

0 1.2 2.4 3.6 4.8 6.0RANGE, km

Fig. 4. Calculated error vs range due to the assumption of a rectan-gular transmitted-pulse profile when the actual profile is bilevel

with duration Tp and parameters a and b.


age error can be very large, especially for large values of(a,b), the error increases with increasing pulse dura-tion, and the error decreases with increasing range.Calculations using other values for ab show that thepercentage error is not strongly dependent on ab . Forexample, with t = 5 jisec, rp = 4 ,gsec, and (a,b) = (8,8),where the error is large, letting ab = 0.4, 0.125, and 0km-1 yielded percentage errors of 72%, 68%, and 66%,respectively. These results show that /3(R) will beoverestimated by a range-dependent factor if Eq. (3) isused. Only by using Eq. (2) or correcting by the range-dependent factor can this source of error be eliminat-ed. Even if Eq. (2) is used to solve for /(R) by deter-mining the pulse profile Ptb(t) and the atmosphericextinction profile ab(R), the return power profile Pb(t)is not linearly proportional to pulse energy, makingnormalization to pulse energy very difficult. Only ifthe transmitted-pulse profile of each pulse were con-stant, so that Ptb(t) could be written as the product of apulse-energy term and a pulse-profile term, could Eq.(2) be used to eliminate this pulse-profile source oferror. Of course, both /3(R) and O(R) may be varyingsignificantly over distances of crp/2 and thus Eq. (1)should be used, making the determination of /3(R) verydifficult, since the profiles /3(R) and ab(R) are bothunknowns in the measurement.

A second potential modeling error is the lidar sys-tem's telescope overlap function. The range-depen-dent telescope overlap function O(R) was introducedin Eqs. (1) and (4) as an important term that affects thereceived-power profile from both the atmosphericaerosol and hard targets. Essentially, O(R) is definedas the fraction of the transmitted pulse energy that iswithin the receiver's field-of-view, is imaged onto thephotodetector, and contributes to the detected signal.Obviously, factors that must be considered are thephysical separation of the transmitting and receivingtelescopes, their coalignment or misalignment, therange-dependent spatial power profile of the transmit-ted beam, and the effective range-dependent spatialprofile of the receiving telescope. (Pulse-to-pulsevariations in the transmitted spatial power profile oraim direction would therefore produce pulse-to-pulsefluctuations in the telescope overlap function.) Theeffective spatial profile of the receiver is due to manyfactors, including any central obstruction in the tele-scope, the finite size and shape of the detector element,sensitivity variations over the detector area, the varia-tion in the position of the focal plane with range, anddiffraction. For systems employing heterodyne detec-tion, the local oscillator's spatial profile, polarization,and alignment at the detector plane must be consid-ered.

Often the telescope overlap function is neglected inthe lidar equation, especially when coaxial lidar sys-tems are being analyzed. However, all the factorscontributing to O(R) for side-by-side lidar systemsapply to coaxial systems except that the physical sepa-ration of transmitter and receiver is very small. Sas-sen and Dodd1 2 have analyzed the behavior of O(R) forboth Gaussian and uniform transmitted-pulse pro-

2 0.5

2 0.4us=^N 4,0.3

0.2

8A 0. 1

1H

ioP(26) CALIBRATION TARGET IX, I ( 010. 6 RANGE - - - X ) ( )

g / / (-0.1, 0\ ~~55 tLRAD 3840 m

I g / ~~~~-6 (01 ) 55 FRAD

>~~~~~~~~~~ - --02 0) - 0 - A -_ -

(-0.4, 0) 219 RAD 960 m

12 14 _ .A A A0 2 4 6 8 10

RANGE, km

Fig. 5. Modeled telescope overlap function of the JPL lidar system

vs range at the 1OP(20) C0 2 laser line as a function of variousdetector positions (in millimeters) with respect to the optical axis

and lying in the receiver focal plane.

files, and for various values of transmitter and receiverdivergence, and transmitter-receiver misalignment.They have shown that O(R) is strongly dependent onalignment, especially for typical lidar systems whichemploy narrow transmitter and receiver beamwidths.Referring to Eq. (1), we see that experimentally deter-mining O(R) at a given wavelength requires that a veryshort rectangular laser pulse be fired into an atmo-sphere with well-known and slowly varying profiles/(R) and ab(R). Since this experiment would be diffi-cult, a model for O(R) is usually used instead.

We have modeled the overlap function for the side-by-side telescope lidar system that is being used atJPL. Figure 5 shows the modeled overlap function vsrange at the 1OP(20) CO2 laser wavelength (10.59 tom).The indicated parameters (X,Y) represent the posi-tion (misalignment) of the detector element in milli-meters with respect to the receiver optical axis, whichis parallel to the transmitted pulse, and lying in thefocal plane (R = a)) of the receiver. Therefore, (X,Y)= (0,0) represents exact centering of the detector onthe optical axis and (X,Y) = (-0.1,0) represents a 100-,m movement of the detector from the centered posi-tion. A 0.1-mm displacement is equivalent to a 55-,rad angular misalignment. The centers of the 15-cmdiam transmitter and receiver telescopes in our lidarsystem are displaced by 21 cm in the X direction. Forour sign convention, negative values of X result in thecrossing of the transmitter and receiver optical axes atsome positive range from the telescopes. This range isindicated in Fig. 5 for the appropriate curves. Themodel includes the effects of the finite detector size,the variation of the image plane with range, and thetransmitted intensity profile due to our positive-branch unstable resonator TEA CO2 laser cavity withan annular output coupler (magnification -2.2).13 Weassume uniform LO power on the detector and uniformdetector response.

It is clear from Fig. 5 that the overlap function variessignificantly with very small detector misalignments.Not only is it important to know O(R) when reducingthe aerosol backscatter power profile, but it is especial-ly important when using a hard target for calibration.A common range used with our calibration target is 2


12 14 10 Id ZU

km, which is indicated in Fig. 5. The calculation of 3at any range R will involve the ratio of the telescopeoverlap function at the target's range, Rs, to its value atR as shown in Eq. (11). It is clear from the curves inFig. 5 that assuming an incorrect telescope overlapfunction (or neglecting it) can cause significant errorsin calculating 3(R). For example, when the system isperfectly aligned, i.e., when (X,Y) = (0,0), the ratio ofthe overlap function's value at 2 km to its value at 20km is -0.28. However, if the system were misalignedby only 220 Arad, e.g., (X,Y) = (-0.4,0), the same ratiois -4.8. Misalignment by 220 Arad corresponds to adisplacement of 44 cm at 2-km range. If correct align-ment was mistakenly assumed, the value of (20 km)would be underestimated by a factor of 16.8. Ofcourse, if a different calibration technique was used(no hard target) and the telescope overlap function wasneglected [i.e., (R) = 1 was assumed], for (X,Y) =(0,0) or (-0.4,0), the value of (20 km) would be under-estimated by a factor of 1.85 or 28.7, respectively.This source of systematic error depends on the tele-scope geometry, the transmitter and receiver charac-teristics, the calibration technique, the distance to thecalibration target, etc. and is potentially the largestsource of error. It is clear that the telescope overlapfunction of each lidar system should be modeled asaccurately as possible at each wavelength used and themodel should then be checked experimentally.

A third error source is due to incorrectly modelinga.(R) and ab(R), the atmospheric extinction profilesfor the optical paths from the lidar to the hard targetand aerosol particles, respectively. Ideally, both pro-files would be accurately measured at the time of eachlidar measurement and at the transmitted wavelength.It is clear from Eq. (2) that the lidar signal does notreadily yield the atmospheric extinction profile when/(R) and ab(R) are both unknowns. An analyticalrelationship between /3 and a is not possible at 10 ,umsince the extinction is mainly molecular while thebackscatter is due to the aerosol particles. However, ifthe pulse duration -rp is sufficiently short, if the overlapfunction O(R) is well known, and if both /3(R) and ab(R)can be assumed to be independent of R, the lidar signalin Eq. (2) can be used to determine the extinctioncoefficient. These conditions may be reasonable forhorizontal paths. The extinction profile a(R) may beconsidered to be constant if a horizontal path betweenthe lidar and the calibration target is employed. Fur-thermore, by aiming the lidar so that it just misses thetarget and therefore obtaining backscatter from amuch longer yet similar horizontal path, the value ofa%(R) = a may be determined. (For a ground-basedlidar, such as the coherent CO2 lidar at JPL, the hori-zontal path lies in the boundary layer and a is calledthe boundary layer total attenuation BL. We havefound that accurately obtaining aBL from horizontalaerosol data requires careful base line subtractionsince finding the slope of the logarithm of the data isnecessary. A slant-path measurement through theboundary layer may also be employed to determine aBLand the height of the boundary layer.)

These steps are shown in Fig. 6 in relation to theoverall calibration process used at JPL to reduce thelidar backscatter data and calculate vertical and hori-zontal /3 profiles from vertical and horizontal aerosolbackscatter data and from horizontal hard-targetbackscatter data.2 9

Obtaining the extinction profile ab(R) for the verti-cal aerosol backscatter data is much more difficultthan obtaining as(R). Equation (2) may not be used,since both ab(R) and /3(R) will depend on R (i.e., heightZ). Direct measurement of extinction vs height at thetransmitted wavelength and at the time of each mea-surement would be very difficult. At CO2 laser wave-lengths the total extinction consists of molecular ex-tinction and aerosol extinction. If the CO2 line ischosen to avoid absorption by trace species such asozone, the molecular extinction will be due primarily toCO2 and water vapor. The extinction due to these twomolecules may be calculated from altitude profiles ofatmospheric temperature, pressure, and relative hu-midity. These profiles may be measured directly us-ing radiosonde ascent probes. An alternative to theinconvenience and cost of ascent probes is to use amodel for the atmospheric temperature, pressure, andrelative humidity that is appropriate to the measure-ment location and season. As shown in Fig. 6, thislatter technique is used at JPL. The molecular atten-uation vs altitude above the boundary layer is calculat-ed using the mid-latitude-summer model profiles fortemperature and water vapor,14 the CO2 and H20 ab-sorption line parameters from the 1982 AFGL atmo-spheric absorption line parameter compilation,15 andthe water-vapor continuum absorption parametersgiven in LOWTRAN 516

We have investigated the altitude dependence of theerror in that results from using various incorrectatmospheric attenuation models. Data from an actualbackscatter profile were used and the results are shownin Fig. 7. (A more general treatment of modelingerrors can be found in Ref. 9.) Our lidar altitude was-0.4 km and the assumed correct atmospheric modelwas a mid-latitude-summer model with a 1.5-kmboundary layer altitude, ZBL, and a 0.39-km'1 bound-ary layer attenuation. (The 0.39-kmboundary layerattenuation represents a worst-case situation, withtypical values occurring in the 0.15-0.25-km-1 range.)Figure 7 gives the percentage error in (Z) resultingfrom using four incorrect atmospheric models: (1) amid-latitude-winter model above the boundary layer,(2) a tropical model above the boundary layer, (3) amid-latitude-summer model with a boundary layeraltitude of 1 km, and (4) a mid-latitude-summer modelwith a 2-km boundary layer altitude. The effects ofusing an incorrect value of ZBL are independent ofaltitude. The error due to assuming ZBL = 2 km is tentimes the error due to assuming ZBL = 1 km. The useof incorrect atmospheric models above the boundarylayer results in an altitude-dependent error in /3. Inour example, the error increases with increasing alti-tude, reaching as much as 30% at 8 km.

It is clear from these results that an incorrect atmo-


Fig. 6. Flow diagram depicting the main components in the lidar calibration and d profile computation.

spheric attenuation model can easily lead to +30%errors in the calculated values of /. Correct modelingof the boundary layer is important for upward-point-ing lidars that pass through a significant boundarylayer. Of course, the boundary layer effect on down-ward-pointing lidar data is much less significant.

- MI DLATITUDEblWINTER MODEL

6

- LIDAR ALTITUDE

--I-30 -20 -10

TROPICALMOD

BOUNDARY LAYERALTITUDE - 2 km

I I

0 +10ERROR, %

+20 +30 +40

Fig. 7. Plot of error vs altitude in calculated /3 values resulting fromthe assumption of incorrect atmospheric models or boundary layeraltitude. The nominal backscatter profile used actual 21 July 1983backscatter data at the 1OP(20) CO2 laser line; a mid-latitude-sum-mer temperature, pressure, and humidity profile; and a boundary

layer altitude of 1.5 km.

VI. Conclusions and Recommendations

Several possible sources of systematic error in inter-preting and calibrating lidar aerosol backscatter datahave been presented in this paper. In addition to thedifficulties posed by the physics of the measurement,other areas of concern that were discussed include dataacquisition, data processing, hard-target calibration ofthe data, and modeling of the telescope overlap func-tion and the atmospheric attenuation. In each case,examples of the resulting error due to incorrect datareduction, data interpretation, calibration, or model-ing were given. The largest potential source of errorwas due to incorrectly modeling (or neglecting) thetelescope overlap function. Although the other poten-tial error sources were smaller, they each must beconsidered in relation to the desired measurement ac-curacy of the backscatter coefficient.

For the specific case of heterodyne detection CO2lidar measurements of aerosol backscatter coeffi-cients, we recommend that


6

5

Use

(1) a hard-target calibration technique be used witha well-characterized target reflectance parameter;

(2) the transmitted pulse be as close to temporallyrectangular as is feasible with as short a duration as isconsistent with other considerations such as signalstrength, longitudinal spatial speckle averaging, andthe measurement accuracy of wind-produced Dopplershifts;

(3) sufficiently fast flash-converter transient digi-tizers be used to record the transmitted power andbackscatter signals;

(4) a pulse-by-pulse subtraction of any dc offset,inversion of a nonlinear receiver element function, in-version of the V\P heterodyne detection process, andnormalization to transmitted pulse energy be imple-mented;

(5) the signal-averaged data be corrected for thetelescope overlap function, the transmitted pulse pro-file, and atmospheric extinction effects; and

(6) the various electronic, optical, and environmen-tal characteristics necessary for steps (4) and (5) bemeasured or modeled as accurately as possible.

References

1. M. Halem and R. Dlouhy, "Satellite Meteorology/Remote Sens-ing Applications," Proc. AMS 1, 272 (1984).

2. R. T. Menzies, M. J. Kavaya, P. H. Flamant, and D. A. Haner,"Atmospheric Aerosol Backscatter Measurements Using a Tun-able Coherent CO2 Lidar," Appl. Opt. 23, 2510 (1984).

3. M. J. Kavaya, R. T. Menzies, D. A. Haner, U. P. Oppenheim, andP. H. Flamant, "Target Reflectance Measurements for Calibra-tion of Lidar Atmospheric Backscatter Data," Appl. Opt. 22,2619 (1983).

4. W. Staehr, W. Lahmann, and C. Weitkamp, "Range-ResolvedDifferential Absorption Lidar: Optimization of Range and Sen-sitivity," Appl. Opt. 24, 1950 (1985).

5. B. J. Rye, "Refractive-Turbulence Contribution to IncoherentBackscatter Heterodyne Lidar Returns," J. Opt. Soc. Am. 71,687 (1981).

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

7. J. Y. Wang, "Heterodyne Laser Radar SNR from a DiffuseTarget Containing Multiple Glints," Appl. Opt 21, 464 (1982).

8. V. S. Rao Gudimetla and J. F. Holmes, "Probability DensityFunction of the Intensity for a Laser-Generated Speckle FieldAfter Propagation Through the Turbulent Atmosphere," J. Opt.Soc. Am. 72, 1213 (1982).

9. M. J. Kavaya and R. T. Menzies, "Aerosol Backscatter LidarCalibration and Data Interpretation," Publication 84-6, JetPropulsion Laboratory, California Institute of Technology, Pas-adena, Calif. (1984).

10. R. N. Bracewell, The Fourier Transform and Its Applications(McGraw-Hill, New York, 1978).

11. D. S. Zrnic, "Moments of Estimated Input Power for FiniteSample Averages of Radar Receiver Outputs," IEEE Trans.Aerosp. Electron. Syst. AES-II, 109 (1975).

12. K. Sassen and G. C. Dodd, "Lidar Crossover Function andMisalignment Effects," Appl. Opt. 21, 3126 (1982).

13. W. F. Krupke and W. R. Sooy, "Properties of an UnstableConfocal Resonator CO2 Laser System," IEEE J. QuantumElectron. QE-5, 575 (1969).

14. R. A. McClatchey, R. W. Fenn, J. E. A. Selby, F. E. Volz, and J. S.Garing, "Optical Properties of the Atmosphere" AFCRL-72-0497 (AFCRL, Hanscom Air Force Base, Bedford, Mass., Aug.1972).

15. L. S. Rothman et al., "AFGL Atmospheric Absorption LineParameters Compilation: 1982 Edition," Appl. Opt. 22, 2247(1983).

16. F. X. Kneizys et al., "Atmospheric Transmittance/Radiance:Computer Code LOWTRAN 5," AFGL-TR-80-0067 (AFGL,Hanscom Air Force Base, Bedford, Mass., Feb. 1980).

The authors wish to thank the reviewers for severalsuggested improvements to this paper.

This paper is based on one presented at the OSATopical Meeting on Optical Remote Sensing of theAtmosphere, Incline Village, Nev., 15-18 Jan. 1985.

The research described in this paper was carried outby the Jet Propulsion Laboratory, California Instituteof Technology, under contract with the National Aero-nautics and Space Administration.


lidar aerosol backscatter measurements: systematic, modeling, and calibration error considerations

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