Errors in ground-based water-vapor DIAL measurements
due to Doppler-broadened Rayleigh backscattering
Albert Ansmann
The influence of Doppler broadening by backscattering of air molecules on ground-based H2 0 DIAL mea-
surement accuracy is described. An equation for calculating the absorption cross section as a function of
incident and backscattered laser profiles is given. Analysis of the differential absorption lidar theory shows
that measurement errors due to this Doppler broadening arise from the range dependence of the backscatter
line shape varying with Mie to Rayleigh backscatter ratio and temperature. Simulation results show that
great care has to be taken in the analysis of H 2 0 DIAL measurements when layers with high aerosol concen-
tration, clouds, or strong temperature inversion exist.
1. Introduction
This paper presents a theoretical analysis of the in-fluence of Doppler broadening of the Rayleigh back-scattered laser line on the accuracy of ground-basedwater-vapor DIAL measurement. A ground-baseddifferential absorption lidar (DIAL) system is an in-strument of remotely determining profiles of watervapor concentration with high temporal and spatialresolution. To get high accuracy one has to consider allspectroscopic effects in the calculation of the H2 0density using the DIAL approximation.
The first detailed analysis of DIAL measurementuncertainties due to spectroscopic effects was presentedby Schotland.1 However, this investigation was madefor monochromatic laser emission. Subsequent workby Browell et al.,2 Megie,3 and Korb and Weng4 ana-lyzed the effects of nonmonochromaticity of laseremission on measurement accuracy, and in additionMegie and Korb and Weng took into account the Voigtprofile of the absorption lines. Korb and Weng5 thenconsidered methods of treating the problem of Dop-pler-broadened Rayleigh backscattering in DIAL ex-periments. Ismail et al. 6 carried out simulations ofH20 DIAL measurements including the line broaden-ing effect due to Rayleigh backscattering for airborneand spaceborne DIAL systems. This investigation wasmade for a standard case of atmospheric conditions.
The author is with Max-Planck-Institut fur Meteorologie, Bun-
desstrasse 55, D-2000 Hamburg 13, Federal Republic of Germany.Received 20 April 1985.
In the following treatment, the influence of Doppler-broadened Rayleigh backscattering on the accuracy ofground-based DIAL measurements is analyzed forstandard atmosphere conditions and cases where in-homogeneities along the vertical measurement path dueto strong aerosol layers and temperature inversionsexist. We derived the DIAL approximation (same wayas shown by Schotland') starting from a lidar equation,where convolutions with the incident and backscatteredline shapes are performed. This theoretical analysisshowing the reasons for the measurement uncertaintiesdue to this line broadening effect is given in Sec. II.Computer simulation results for Gaussian distributedlaser emission and Voigt-H 2 0-absorption lines arepresented in Sec. III.
II. Derivation of the DIAL Approximation
We start with the lidar equation in the followingform:
PiP(Rj) = dR *li(Rj) -exp 2 yi(r)drJ I~~2I y(rd
{ /iM(Rj) - hi(v) exp [-2 R a(v,r)dr] dvj3i(Rj) [JRo
+ iR(Rj) hi(v) exp [-J R a(vr)dr]dv
Ji (Rj) exp Ro
- i(v,Rj) exp Rj a(v,r)dr d) (1)
where Pi (Rj) is the number of detector charge carriersfor the wave number v produced by incident photonsfrom an element of the atmosphere at the range Rj withthe thickness dR, Po = (J- H -A)/(h - c vi) with thetransmitted energy in joules per pulse, system efficiencyH, received area A, Planck's constant h, and velocity oflight c, fi (Rj) is the total backscatter coefficient forwave number vi and range Rj, 3 with indices i and M or
R is the backscatter coefficient for vi and Mie or Ray-leigh scattering, respectively, yi is the total extinctioncoefficient except for water vapor for vi, a is the ab-sorption coefficient of H20, hi(v) is the normalizedlaser line shape for Ro (lidar basis) centered at vi, andgi(Rj) is the normalized laser line shape after watervapor absorption from Ro to Rj and Doppler broaden-ing by backscattering of air molecules in the range celldR at Rj centered at vi.
In the usual way (see, for example, Schotland'), theDIAL approximation can be obtained from the lidarequation by forming the logarithm of the quotient[P1(R1) P2 (R2 )]/[Pl(R2 ) P2 (R1)]. With Eq. (1) weobtain
In [P(R 1) P2(R2 ) In A(R) + In B(Rl) B2(R2)(R 2 ) * P2 (R1 )j [A2 (R)I [B2 (R1 ) B 1 (R2 )j
MR, = /jMjR,i for on line (i = 1) and off line (i = 2),
under the conditions (given by Schotland') that thevariation of spectral properties of the scattering andabsorbing (other than H20) medium between v and 2and temporal variations of the scattering medium be-tween the sequentially emitted laser pulses at v and 2due to atmospheric transport are negligible, and underthe additional condition that the broadening of the laserline shape hi(v) and of the Doppler-broadened Ray-leigh backscattered laser line shape g(v,R 2) due towater vapor absorption from R to R, and R2 to R isnegligible, respectively, which is true for absorptionlines selected to yield good measurement accuracy andlaser lines with FWHM < 0.025 cm-'.
From Eq. (2) one obtains the DIAL approximation(3), if the terms Ci (R,) are neglected, the exponentialfunctions in A (R) and B (R) and the logarithm func-tions n[Ai(k)] and ln[Bi(R)] in Eq. (2) are expandedin series, and if the water vapor optical thickness in therange cell from R1 to R2, Ai(R), and BR) are suffi-
ciently small so that the series expansion can be trun-cated after the linear term. Using absorption lines se-lected to yield good accuracy, the optical thickness fromRI to R2 due to water vapor absorption over the spectralprofile for on and off lines and Ai(T) and Bi(P) aresufficiently small against 1.
where N(R) and cr (R) represent the water vapor andabsorption cross section for laser line i, respectively,averaged over the range cell dR centered at B = (R1 +R2)/2.
The absorption cross section o(R) is given by
(T(R) = i(v,R) + flm(R hi(v)2 S_ )
+ /R (R2) gi(vR2)dv/3(R2 ) I
(4)
It is evident that the neglect of the terms Ci(Rj)produces DIAL measurement errors. Because ofC2(R,)/B 2(R1 ) 0, the errors are mainly due to theneglect of the on-line term C,(R). Analysis of C,(Rj)shows that uncertainties arise from the range depen-dence of the backscatter line shape, which varies withMie to Rayleigh backscatter ratio and temperature in-fluencing the linewidth of the Doppler-broadenedRayleigh backscattered laser line g(v,R) and the Ray-leigh scatter coefficient. The backscatter line becomeswider with decreasing M/R and increasing tempera-ture. The broadening of the backscatter line with in-creasing range causes C(Rj) > 0 and a negative errorcontribution in the determined DIAL H20 numberdensity; i.e., the DIAL H20 number density is smallerthan the true H20 number density. In the oppositecase, when the backscatter profile narrows with in-creasing range, the error contribution is positive. In thecase of a homogeneous atmosphere along the measure-ment path, the spectrum of backscattering is range in-dependent, and the error contribution is zero becauseof C(Rl) = 0. These effects are shown in Figs. 1 and2.
Another consequence of the Doppler-broadening ef-fect is consideration of the backscatter coefficients andthe Doppler-broadened Rayleigh backscattered laserline shape in the calculation of the absorption crosssection using Eq. (4) to correct the calculation uncer-tainties due to this Doppler broadening.
Ill. Simulation
To evaluate the accuracy of ground-based water-vapor DIAL measurements using Eqs. (3) and (4) andto determine the influence of the neglected Ci terms onmeasurement accuracy, computer simulations havebeen performed. The atmospheric model used in thesesimulations is a multilayer -D model with up to 100homogeneous layers. The thickness of each layer is setto dR = 100 m. Atmospheric temperature varies lin-early with height. Pressure, H20 concentration, andaerosol scatter vary exponentially with height. Stan-
Fig. 1. Errors in the horizontal (1) and vertical (2) ground-based H2 0 DIAL measurements due to Doppler broadened Rayleigh backscattering
for clear (a) and hazy (b) atmosphere with 23- and 5-km ground level visibility, respectively. Dashed lines show measurement uncertainties
for the case where Doppler broadened Rayleigh backscattering is neglected in the calculation of the absorption cross sections. Solid lines
show measurement uncertainties for the case where errors in the cross-section calculation due to Doppler broadened Rayleigh backscatteringare corrected by using Eq. (4).
15.
10.
cc 5.0
c 0.
0-5.
LU
m -10.z
I I I I I I I I..I.... - 15 . L
2. 4. 6. 8. 10. 0.
RANGE (KM)
2. 4. 6.
RANGE (KM)
S. 10.
GLU
LUCL
LUJ
300.
290.
260.
270.
260.0.
2. a4. 6. 6. 10.
RANGE (KM)
2. 4. B. 6. 10.
RANGE (KM)
Fig. 2. Errors in vertical H2 0 DIAL measurements for clear atmosphere. Three layers with a thickness of 500 m centered at 1.5, 4, and 8
km and an aerosol concentration 50% higher than the aerosol concentration of the clear atmosphere model were stimulated in (a). Three layers
with a thickness of 500 m centered at 1.5, 4, and 8 km and characterized by dTIdR = 2 K/100 m were simulated in (b).
Table 11. H20 Absorption Line and Laser Line Parameters
Voigt-H 20-absorption lineLine center 721.1216 nmGround state energy 224.83 cm- 1
Line strength 2.95 X 10-24 cm-1/molecules/cm2Lorentz halfwidth (for STP conditions) 0.09855 cm-
Gaussian laser lineLinewidth (FWHM) 0.025 cm'
dard atmosphere conditions are assumed. Two aerosolmodels are used. They are based on similar aerosoldistributions like those given by McClatchey et al. 7 forclear and hazy atmosphere. In Table I all verticalprofile parameters are given. The model atmosphereis homogeneous along the horizontal path of measure-ment. Table II contains the assumed Voigt-H20-absorption line parameters taken from Wilkerson et al. 9and the linewidth FWHM of the Gaussian laser profileh (v). Laser line broadening due to water vapor ab-sorption is considered in the model by the variation ofFWHM with water vapor optical thickness. TheDoppler broadened laser profile g (v,Rj) is a convolutionof the incident laser profile and the intensity distribu-tion of an incident monochromatic frequency v back-scattered by air molecules, which have a Gaussian ve-locity distribution. The numerical integration over theincident and backscattered laser line shape is performedwith a spectral spacing of 0.004 cm-' over the rangefrom v to v = vi 7.5 FWHM. The Voigt profile isevaluated using numerical techniques.10 For the sakeof simplicity it will be assumed that the on-line centeris located at the H20 absorption line center. Errors inDIAL measurement due to a pressure shift of the H20absorption line of 0.015 cm-' atm-1 were estimatedto be <3% for ground level pressure, if the on-line centeris located at the H2 0 absorption line center for lowpressure.1' The off-line absorption cross section wastaken at v = v + 0.4 cm-' and set constant over thelaser profile. For calculating the lidar backscattersignals we used realistic lidar and receiver parameters(the lidar system of Max-Planck-Institut ffir Meteo-rologie, Hamburg). The calculated true lidar signals[Eq. (1)] and the absorption cross sections [Eq. (4)] wereused to determine the DIAL H20 number density pro-
files. Laser line broadening due to H2 0 absorption wasalso considered in calculating hi (v) and gi (v,k) in Eq.(4). The difference between the DIAL and input ortrue H20 number densities are shown in Figs. 1 and2.
In Fig. 1 H20 DIAL measurement uncertainties forclear and hazy atmosphere conditions are plotted. Thehorizontal measurement simulations show that the in-fluence of Doppler-broadened Rayleigh backscatteringon the measurement accuracy is negligible when theatmosphere is homogeneous along the measurementpath. In the case of vertical DIAL measurements, er-rors of more than 10% are possible for normal atmo-spheric conditions due to this line broadening effect.The sharp variations in the curves around R = 5 km arethe result of different aerosol scale heights for R 5 kmand R > 5 km assumed in the model. This indicates theimportant influence of the spatial variation of theaerosol concentration on the H20 DIAL measurementaccuracy. It should be noted that calculation of theabsorption cross section using Eq. (4) can be performedwith high accuracy when all parameters in Eq. (4) areknown. This can be seen by the solid curves for hori-zontal measurements. Consequently the solid curvesrepresenting vertical measurements are only due to theneglect of the terms C(R1) in Eq. (2), and since C2 (R1)= 0 in the simulation because of the spectral indepen-dence of the off-line absorption cross section over thespectral profile of the laser line, the errors are causedby the neglected C(RI).
C,(Rl) is mainly a function of the spatial variation ofthe Mie to Rayleigh backscatter ratio and of the Dop-pler broadened Rayleigh backscattered laser profilegi (v,R) as a function of temperature. The importantinfluence of the spatial variation of 3M/OR and T is
shown in Fig. 2. The error peaks in the curves of Fig.2(a) and the sharp variations in the curves of Fig. 2(b).occur at boundaries of aerosol and temperature inver-sion layers, respectively. The results make clear thatgreat care has to be taken in the analysis of ground-based H20 DIAL measurements when layers withstrong Mie scattering exist. The temperature effectseems to be relatively small if one considers that in-versions in the upper atmosphere are usually small.But bearing in mind that regions with temperature in-versions often cause layers with enhanced aerosol con-centration, the influence of temperature inversion onDIAL measurement accuracy can become important.
Simulations for monochromatic laser emission giveessentially the same results as for Gaussian distributedlaser emission with FWHM • 0.025 cm-1.
IV. Summary
A theoretical analysis of the influence of Dopplerbroadening by backscattering of air molecules on theaccuracy of ground-based H2 0 DIAL measurement isgiven. A derivation of the DIAL approximation start-ing from a lidar equation considering the Doppler-broadened Rayleigh backscattering is performed. Asa result, we found an equation for calculating the ab-sorption cross sections as a function of Mie to Rayleighbackscatter ratio and the Doppler-broadened Rayleighbackscattered laser profile. The derivation shows thereasons for the H20 DIAL measurement uncertaintydue to this line broadening. The influence is mainly afunction of the range dependence of the backscatter lineshape varying with Mie to Rayleigh backscatter ratioand temperature. Simulation results show that theinfluence is important in the case of vertical measure-ment and that great care has to be taken in the analysisof ground-based H20 DIAL measurements, when layerswith strong Mie scattering exist.
References
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10. B. H. Armstrong, "Spectrum Line Profiles: The Voigt Function,"
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The author would like to thank H. Hinzpeter and J.B6senberg for helpful discussions and encouragement.Financial support from the Bundesminister furForschung und Technologie under grant KF-10056 isgratefully acknowledged. This paper is based on onepresented at the OSA Topical Meeting on Optical RemoteSensing of the Atmosphere, Incline Village, Nev., 15-18Jan. 1985.
