complementary study of differential absorption lidar optimization in direct and heterodyne...

Complementary study of differential absorption lidaroptimization in direct and heterodyne detections

Didier Bruneau, Fabien Gibert, Pierre H. Flamant, and Jacques Pelon

A detailed study using both analytical and numerical calculations of direct and heterodyne differentialabsorption lidar (DIAL) techniques is conducted to complement previous studies. The DIAL measurementerrors depend on key experimental parameters, some of which can be adjusted to minimize the statisticalerror. Accordingly, the pertinent criteria on optical thickness, the number of photons emitted at the on andoff wavelengths, are discussed to reduce the relative error on the total column content or range-resolvedmeasurements that rely on either hard target or atmospheric backscatter returns. In direct detection, theoptimal optical thickness decreases from 1.3 to 0.8 when the background increases while the on-line-to-off-line optimal energy ratio decreases from 3.6 to 2.7. In heterodyne detection, the minimum error is obtainedfor an optical thickness of 1.2 and an energy ratio of 4.3. © 2006 Optical Society of America

OCIS codes: 010.3640, 030.5290.

1. Introduction

The differential absorption lidar (DIAL) technique isan efficient means of measuring trace gases in theatmosphere. It relies on the comparison of the returnsignals at two frequencies, one on a spectral absorptionline of the targeted species (on-line frequency), theother at a minimum of absorption (off-line frequency).The signal can be backscattered by clear air, clouds, orsolid targets (such as the ground in the case of airborneor spaceborne instruments). The measurement can ei-ther be performed on an atmospheric column (inte-grated measurements) or range resolved.

The following study aims to minimize the statisti-cal error on the trace gas concentration for a giventotal emitted energy (the sum of the on-line andoff-line transmitted energies on a shot series) orequivalently to minimize this error for a given totalemitted energy by optimizing the energy per shot, theon-line to off-line emitted energy ratio, and the ab-sorption optical thickness. Such an optimization hasalready been investigated but only under restricted

conditions. Remsberg and Gordley1 first establishedthe optimal optical thickness in shot-noise-limiteddirect detection for an equal on-line and off-line emit-ted energy and number of shots. Mégie and Menzies2

then studied the DIAL optimization with equal emit-ted energies but a different number of shots for bothdirect and heterodyne detections. In direct detection,they obtained an optimal optical thickness and anon-line-to-off-line averaging ratio for the shot-noise-and background-limited cases. In heterodyne detec-tion, they limited their study to the case of lowcarrier-to-noise ratio (CNR). Both studies neglectedthe optical or speckle noise.

Here some considerations of signal processing en-able us to go further in the optimization of the DIALtechnique. In this study we include the speckle noisefor both direct and heterodyne detections and coverall the background (in direct detection) or CNR (inheterodyne detection) conditions. We first establishfor both detections the condition on the signal levelfor the measurement to approach the closest to theshot-noise-limited case. We consider different on-lineand off-line emitted energies and optimize the energyratio in addition to the optical thickness. We alsoconsider different on-line and off-line averaging andcalculate the error when the system deviates from theoptimum.

Section 2 summarizes the basic DIAL measure-ment theory and presents the optimization problem.Then, the optimization of integrated DIAL mea-surements is discussed in Sections 3 and 4 fordirect and heterodyne detection, respectively. Range-

D. Bruneau ([email protected]) and J. Pelon arewith the Institut Pierre-Simon Laplace, Service d’Aéronomie, Uni-versité Pierre et Marie Curie, 4 Place Jussieu, 75252 Paris, Cedex05, France. F. Gibert and P. H. Flamant are with the InstitutPierre-Simon Laplace, Laboratoire de Météorologie Dynamique,Ecole Polytechnique, 91120 Palaiseau Cedex, France.

Received 20 December 2005; accepted 1 February 2006; posted 7February 2006 (Doc. ID 66804).

0003-6935/06/204898-11$15.00/0© 2006 Optical Society of America

4898 APPLIED OPTICS � Vol. 45, No. 20 � 10 July 2006

resolved measurements for both detection types arediscussed in Section 5. Finally, the main results of thestudy are summarized in Section 6. The acronymsand main parameters used in this paper are summa-rized in Table 1.

2. Differential Absorption Lidar Basic Equationsand Dilemma

The lidar equation that gives the return power can bewritten as2

Pi�R� � �iEi

bi�R�Ac

2R2 exp��2�i�R� � 2�i0�R��, (1)

where R is the measurement range; Pi�R� is the i-line(on- and off-line) measured return signal power; Ei isthe transmitted i-line energy in J�pulse; �i�R� is theintegrated i-line optical thickness due to absorptionby the constituent of interest, with

�i�R� ��0

R

n�r��i�r�dr, (2)

where �i�r� is the i-line absorption cross section andn�r� is the concentration of the species of interest;�i

0�R� is the i-line optical thickness including extinc-tion and absorption but excluding the absorbing con-stituent of interest; �i is the optical efficiency of thereceiver; bi�R� is the distributed backscatter coeffi-cient at the i wavelength; A is the receiver area; andc is the light velocity.

Throughout this paper, we will use the reducedform of Eq. (1),

Pi�R� � Ki�R�Ei exp��2�i�R��, (3)

where

Ki�R� � �i

bi�R�Ac

2R2 exp��2�i0�R��. (4)

Furthermore, we shall consider that the on-line andoff-line frequencies are sufficiently close and that theinstrument is properly adjusted so that for any range

Kon�R� � Koff�R� � K�R�.

By use of Eq. (3), DIAL measurements are providedby the ratio

Pon�R�Poff�R�

�Kon�R�Koff�R�

Eon

Eoffexp��2��on�0, R� � �off�0, R��.

(5)

Then, the integrated optical thickness � � �on � �offbetween the instrument and a target (either solid ordistributed) at range R is given by

��R� �12 lnPoff�R�Eon

Pon�R�Eoff. (6)

Range-resolved measurements can be seen as mea-surements of the local optical thickness between tworanges R1 and R2 by the difference of the integratedoptical thickness of these two ranges. This case willbe discussed in Section 5.

Using Eq. (2), we also have ��R� � �0R n�r��on�r�

� �off�r��dr. The integrated optical thicknessmeasurement leads to the measurement of an aver-aged species concentration n� �0, R� � ��R��0

R ��on�r�� off�r��dr. The relative errors on optical thicknessand concentration are then equal. Consequently,minimizing the relative error on the species concen-tration estimate is tantamount to minimizing the rel-ative error on the optical thickness.

If we assume, as a first approximation, that theuncertainties on the emitted energies can be neglectedwith respect to the errors on the returned powers, thevariance on optical thickness is given from Eq. (6) by

var�� 14 var�Pon�

Pon2 �

var�Poff�Poff

2 � 2cov�Pon, Poff�

PonPoff,

(7)

where we take into account all the observed returnpower fluctuations, which is either coming from thedetection statistics or from the target reflectivity andatmospheric fluctuations (especially those induced byturbulence in the case of heterodyne detection).

Table 1. Acronyms and Main Parameters

BSR Background-to-signal ratioCNR Carrier-to-noise ratio (ratio of the signal to

detection noise average powers in heterodynedetection)

DCP Detected coherent photons (heterodyne detection)DDP Directly detected photons (direct detection)DELB DIAL error lower boundDOF Degree of freedomLOT Local optical thicknessSNL Shot-noise limitSNR Signal-to-noise ratio (ratio of the mean signal to

the detection, including speckle, noise)NT Number of detected photons in the absence of

absorptionNon, Noff Number of actually detected photons, on-line and

off-line, respectivelyNBon, NBoff Number of detected background photons, on-line

and off-line, respectivelyMd, Mt Number of DOF (speckles) in direct and

heterodyne detection, respectively� Ratio of on-line to off-line emitted pulse energies� Ratio of the on-line to off-line averaging shot

numbers�(X) Standard deviation of X� Integrated optical thickness�� Local optical thickness (LOT)

10 July 2006 � Vol. 45, No. 20 � APPLIED OPTICS 4899

If we assume that the on-line and off-line signalsare captured simultaneously, the probed volumes areidentical, so the fluctuations of the target reflectivityand atmospheric transmission are completely corre-lated and their related variances in Eq. (7) are can-celled by the covariance term. We then come to arelative error on optical thickness given by

��

�12� ��Pon�

Pon2

� ��Poff�Poff

2�1�2

, (8)

where ��Pi� is the standard deviation on Pi account-ing only for detection statistics. The relative error onreturn power we consider in this paper refers thenonly to the detection process and is the inverse of thesignal-to-noise ratio (SNR). This equation is tanta-mount to the DIAL error expression initially estab-lished by Schotland.3

On the one hand, from the lidar equation, we cansee that as the optical thickness is increased the on-line signal strongly decreases. Consequently, therelative error on the return power measurement in-creases dramatically (the SNR drops down). On theother hand, Eq. (8) shows that the relative error onthe optical thickness contains a factor of 1�� thataccounts for the method sensitivity. We then come tothe DIAL dilemma: An optical thickness that is toosmall decreases the sensitivity of the measurement,and an optical thickness that is too large decreasesthe on-line SNR and the power measurement accu-racy. It is then clear that an optimization of the op-tical thickness is possible, as already addressed inRefs. 1 and 2 for some specific cases. Practically, theoptical thickness can be matched by choosing theappropriate absorption line or by tuning the on-lineemission on the line edge. In addition, the emittedenergies at both wavelengths do not need to be equal.For a given total emitted energy, the ratio of on-lineto off-line emitted energies can also be optimized.These optimizations are presented in Sections 3 and4 for heterodyne and direct detections. Consideringdifferent on-line and off-line averaging times is notan optimal solution since the on-line and off-lineprobe volumes will be necessarily different (espe-cially for airborne and spaceborne applications), andthe uncorrelated fluctuations of atmospheric returnwill increase the optical thickness measurement er-ror [and invalidate Eq. (8)]. However, under somefavorable conditions (ground-based systems, strato-spheric observations, weak atmospheric heterogene-ities, etc.), these additional fluctuations can beneglected or considered correlated for sufficientlyshort on-line and off-line shot series. For practicalreasons (the same laser sequentially switched be-tween on-line and off-line emissions), it can then bevaluable to consider the measurement optimizationplaying on the averaging on a different number ofon-line and off-line shots with equal emitted energy.This optimization will be presented in Appendix A. InSections 3, 4, and 5 we shall first establish the opti-mal conditions for a single shot pair (with unbalanced

emitted energies) and then discuss how to achieve thebest measurement for a series of shot pairs.

3. Integrated Differential Absorption LidarMeasurements in Direct Detection

In direct detection, the detector output current isproportional to the optical return power. The relativestandard deviation on the directly detected photoelec-tron (DDP) number, or power estimation P, of a ther-mal partially coherent light is given by4

��P��P � 1 � N�Md

N �1�2

, (9)

where N is the average DDP number in the rangegate and Md � MsMt is the total number of degrees offreedom (DOFs), a product of the number of spatialcoherence cells (speckles) Ms and the number of tem-poral speckles Mt.

Ms is a function of the beam divergence d�, thecollecting area A, and the operating wave-length � that can be approximated by4 Ms ��1 � �Ad�2�4�2��. The number of temporal specklesin the range gate of duration T can be approximatedby Mt � 1 � T�Tc, where Tc is the coherence time ofthe backscattered signal.

Additionally, in the case of lidar detection we have toconsider background radiation coming from either theobserved scene or the detector. The relative error onthe power estimate for the emitted line i on a singleshot is then

��Pi�Pi

� 1 � �NBi�Ni� � �Ni�Md�Ni

1�2

, (10)

where Ni � �Pi�2hBd is the total number of thesignal DDP in the range gate, NBi is the total numberof background DDPs in the range gate, and Bd is thedetection bandwidth. We also assume here that thenumber of DOFs for the background light is nearlyinfinity.

The power relative error decreases when Ni and Md

increase and NBi decreases. If the emitted beam di-vergence is large compared to the diffraction limit ofthe collecting telescope, the optical (speckle) noisecan be neglected with regard to the detection noise�Ni�Md �� 1�. If, additionally, the background issmall with regard to the signal �NBi�Ni �� 1�, wereach the shot-noise limit (SNL) and obtain��Pi��Pi�SNL � Ni

�1�2, which represents the ultimatelower limit in power measurement accuracy. Notethat for the SNL case only the total energy of a shotseries counts; it is indifferent to increasing the emit-ted energy per shot or the number of shots.

If we look now for the signal level that brings animperfect detection closest to the SNL, we have tominimize the quantity ��Pi��Pi�2 � ��Pi��Pi�SNL

2��Pi��Pi�SNL

2 with regard to Ni. In that case we obtainNi � �MdNBi�1�2, which is an equal number of DDPsper DOF and background-to-signal ratio (BSR):


Ni�Md � NBi�Ni. Once the system has reached thisoperation point, any increase in emitted energy by afactor of Mp will produce an error decrease slowerthan the SNL; that is, the relative error is divided byless than �Mp�1�2. It is therefore more beneficial to useMp shots with the same emitted energy. The condi-tion of an equal number of DDPs per DOF and BSRis then an optimum operation point for power mea-surement on a shot series.

If we now consider a DIAL system with a totalnumber of emitted photons, NT

e � None � Noff

e

� �Eon � Eoff��h and unbalanced on- and off-lineemitted energies such as Non

e � �Noffe, the number of

on-line and off-line DDPs can be written as

Noff � NT

11 � �

, (11)

Non � NT

�

1 � �exp��2��, (12)

with NT � KNTe. Note that NT is not actually the total

(on-line plus off-line) number of DDPs but is the totalnumber of DDPs in the absence of absorption (i.e., for� � 0). This notation has the advantage that NT isindependent of � and is directly proportional to thetotal emitted energy.

With the notations of Eqs. (11) and (12), and as-suming that the number of DOFs is the same foron-line and off-line signals, the relative error on theoptical thickness [Eq. (8)] can be written as

��

�1

2�NT1�2 �1 � �1 � ��

NBoff

NT�1 � ��

� 1 ��1 � ��

�

NBon

NTexp�2��

� 1 � �

� � exp�2�� 2NT

Md�1�2

. (13)

The best accuracy of optical thickness measurementis obtained for both on-line and off-line SNL signals�NBon � NBoff � 0, Md infinite). The relative error forthe SNL system becomes

��

SNL�

1

2�NT1�2 ��1 � ��1 � exp�2��1�2.

(14)

By minimizing this expression with regard to � and �,we find �0 � 1.28 [solution of � � 1 � exp��] and�0 � 3.60 ��0 � exp��0��. With this optimization, theSNL system presents the DIAL error lowest bound(DELB):

��

0� 1.8NT

�1�2 � 3.86Noff�1�2 � 7.32Non

�1�2.

(15)

As for the single line power measurement, an imper-fect DIAL system �NBon 0, NBoff 0, Md finite� willoperate under optimal conditions when the quantity

F�NT, Md, �, �� 2

� ��

SNL

2 ��

SNL

2

is minimized as a function of NT. From Eqs. (13) and(14) we have

F �1 � �

�

NBoff

NT�2 �

NBon

NTexp�4�� 2

NT

Md �

1 � ��2

� � exp�2��.

(16)

By nullifying, dF�dNT, we obtain the condition

NT

Md�

opt�

12 1 � �

� �2NBoff

NT�2 �

NBon

NTexp�4��,

(17)

or, using the more experimentally accessible quanti-ties,

Non

Md�

opt�

12 exp�2�� NBoff

Noff� �

NBon

Nonexp�2��,

(18)

Noff

Md�

opt�

12� NBoff

Noff� �

NBon

Nonexp�2��. (19)

Equation (17) defines the optimum emission level. Ifthe single-shot total emitted energy is multiplied byMp, the error will decrease by less than Mp

�1�2, that is,less than for the Mp shot pairs of the optimum energy.

We can numerically calculate the optical thicknessrelative error given by Eq. (13) as a function of opticalthickness � and energy ratio �. The results are pre-sented on Fig. 1 for the SNL case and for a high BSRNBon�NT � NBoff�NT � 0.2, keeping the optimum en-ergy level defined by Eq. (17). As compared to theSNL case, when the on-line and off-line BSR increase,the optimum � and � decrease and the minimumrelative error increases quickly. Note also that theoptimum in � and � becomes narrower. Besides, wecan determine the optimum on-line to off-line emittedenergy ratio for a given optical thickness by minimiz-ing Eq. (13) with respect to �. The optimum energyratio is given by the real positive solution of

2�4NBoff�NT � �3�1 � 2NBoff�NT� � ��exp�2�� 2NBon�NT exp�4�� 2NBon�NT exp�4�� 0. (20)

The optimum optical thickness is then determined byminimizing numerically Eq. (13) with respect to �,using the optimal NT�Md and � obtained previously.


The optimum optical thickness and energy ratioare presented in Figs. 2 and 3, respectively, as afunction of the on-line and off-line BSRs. For theseresults, the optimum energy level per shot pair[Eq. (17)] is maintained. Figure 4 presents the corre-sponding off-line number of DDPs per DOF as a func-tion of the BSRs. The relative optical thickness errorat optimum is presented in Fig. 5 as a function of theBSRs; the result is divided by the DELB given byEq. (15). In these four figures, a main diagonalstraight line would correspond to equal off-line andon-line background levels, NBon � NBoff. We can notethat the optimal optical thickness and the relativeerror are much more dependent on the on-line thanon the off-line BSR. This is a consequence of the factthat the optical thickness, in addition to impinging onthe measurement sensitivity, dramatically affectsthe on-line signal level.

4. Integrated Differential Absorption LidarMeasurements in Heterodyne Detection

In a heterodyne system, the detector output currentat the beat-wave frequency is proportional to the am-plitude of the received optical field. In contrast todirect detection, only one coherent cell (or spatialspeckle) can be detected at a time. A simple andefficient technique for determining the return opticalpower Pi consists of squaring the beat-wave currentand averaging it over a range gate. The relative errorof the squarer power estimator for a single shot in arange gate of duration T is5

��Pi�Pi

�1

�Mt 1 �

1CNR�, (21)

where CNR is the carrier-to-noise ratio (that is, theratio of the average signal power to the averagenoise power, not accounting for speckle noise) andMt � 1 � T�TC is the number of coherence cells in therange gate duration or number of DOFs of the power

Fig. 1. Relative optical thickness error in direct detection as afunction of optical thickness � and on-line-to-off-line emittedenergy ratio �. Left plot, for SNL conditions. Right plot, forNBon�NT � NBoff�NT � 0.2 with optimized NT�Md. The error isdivided by the DELB.

Fig. 2. Optimum optical thickness � for direct detectionintegrated measurements as a function of NBon�Non and NBoff�Noff.

Fig. 3. Optimum on-line-to-off-line emitted energy ratio � fordirect detection integrated measurements as a function ofNBon�Non and NBoff�Noff.

Fig. 4. Optimum number of off-line DDPs per DOF, Noff�Md, fordirect detection integrated measurements as a function ofNBon�Non and NBoff�Noff.


measurement (assumed to be the same for on-lineand off-line frequencies). We consider here only thecase of a single detector; the use of multiple detectorswould multiply the number of DOFs. Other powerestimators can be applied (such as the Levin estima-tor), but the error is not significantly different.5 If weassume that the local oscillator shot noise is muchlarger than the other noise sources, the CNR is de-fined by

CNRi ��H�Pi

hBh, (22)

where �H is the heterodyne mixing efficiency, � is thedetector quantum efficiency, h is the photon energy,and Bh is the detection bandwidth, all assumed equalfor on-line and off-line signals.

The lowest bandwidth required for analyzing thesignal without losing information is the inverse of thecoherence time TC. This condition can be obtained ifthe Doppler shift is known (hard target) or if thebandwidth is reduced after estimation of the meansignal frequency. The CNR is then simply expressedby

CNRi �Ni

Mt, (23)

where Ni � �H��Pi�h�T is the number of detectedcoherent photons (DCPs) in a range gate for a singleshot. The relative error on the power estimation isthen

��Pi�Pi

�1

�Mt 1 �

Mt

Ni�. (24)

We can easily obtain from Eq. (24), as it has alreadybeen shown,5 that the optimal power measurementon a single frequency occurs when Ni�Mt � 1, that isfor a single DCP per DOF (CNR � 1).

However, in the case of DIAL measurements, theoptimum DCP per DOF is slightly different. By use ofEq. (24) and the same notations for the emitted num-ber of photons as for direct detection [Eqs. (11) and(12)], the relative error on the optical thickness[Eq. (8)] can be written

��

�1

2��Mt�1 �

Mt

NT�1 � ��2

� 1 �Mt

NT 1 � �

� �exp�2��2�1�2

. (25)

By nullifying the derivative of �� with respect toMt we obtain the optimum number of DOF, for agiven Nt, such as

NT

Mt�

1 � �

� �2 � exp�4��2 1�2

. (26)

Consequently, the optimal number of DCPs per DOF(i.e., CNR) at each wavelength is given by

Noff

Mt� CNRoff �

a�

, (27)

Non

Mt� CNRon �

aexp�2��

, (28)

with

a � �2 � exp�4��2 1�2

.

Figure 6 presents the optimum CNR as a function of� and �. We note that the optimal CNR is different forthe two wavelengths and also different from unity,which optimizes single wavelength measurements.5

By replacing the number of DCPs per DOF by their

Fig. 5. Optical thickness relative error for direct detection inte-grated measurements in optimal conditions as a function ofNBon�Non and NBoff�Noff, divided by the DELB.

Fig. 6. Optimum CNR for integrated heterodyne detection mea-surements as a function of optical thickness � and on-line-to-off-line emitted energy ratio �. Left, CNRon; right, CNRoff.


optimum values, we can write Eq. (25) as

��

�1

2�NT1�2 1 � �

� �a1�2� 1 ��

a�2

� 1 �exp�2��

a 2�1�2

. (29)

A numerical solution of Eq. (29) is then used to cal-culate the relative error of the optical thickness withrespect to � and � for a given total emitted energy.The minimum value of �� 3.71��NT is reachedfor � � 4.28 and � � 1.23, which correspond toNT�Mt � 10.9, CNRon � 0.75, and CNRoff � 2.06.Figure 7 presents the relative error in heterodynedetection for the optimum CNRs, divided by theDELB (obtained for SNL direct detection). The min-imum relative error is 2.06 times the DELB. A quitebroad, nearly optimal area exists. Relative errors be-low 2.1 times the DELB are achieved for 1.2 � �� 1.5 and 2.5 � � � 7. For equal energies �� 1�, theminimum relative error is 2.4 times the DELB for anoptical thickness of 1.05.

Assuming low CNR conditions enables us to findpreviously published results.2 For NT�Mt �� 1and � � 1, Eq. (25) becomes �� Mt�1�2�NT��1� exp�4��1�2��, and the minimum of the opticalthickness error is obtained for � � 0.55. This result,which leads to a measurement error of three timesthe DELB, is significantly higher than the optimum.

As the range gate duration is usually of the sameorder of magnitude as the pulse coherence time, Mt

cannot be much larger than a few units (Mt � 1 for ahard target and a transform-limited pulse). Since theoptimum CNRs are moderate, the relative error ofintegrated DIAL measurements for a single shot pairwill be large. To reduce this error, the collection of alarge number Mp of shot pairs is then usually neces-sary in heterodyne detection. The error is then di-vided by �Mp. If we consider now a measurement of aseries of shots with a given total emitted energy, the

optimal conditions are obtained when the singlepulse-pair measurement is optimized, that is, for� � 4.28 and � � 1.23 and when the energy of eachpulse is adjusted so that CNRon � 0.75 and CNRoff� 2.06. For the same total emitted energy, higher orlower CNRs will yield higher optical thickness rela-tive errors.

Table 2 compares the relative optical thickness er-rors for heterodyne and direct detection. We can seethat, unless the background reaches levels largerthan the signal, direct detection yields better resultsthan heterodyne detection. It is well known that het-erodyne receivers offer poor performance in terms ofpower measurement accuracy because of the highspeckle noise. Furthermore, this comparison refers todetected photons. The detection efficiency of a hetero-dyne system is limited by the mixing efficiency, whichitself imposes a limitation on the telescope diameter(owing to the loss of coherence caused by index tur-bulence), as compared to direct detection. Neverthe-less, heterodyne power measurements remaincompetitive in spectral domains where detectorsbring a large background in direct detection (mostlyin the infrared).

5. Range-Resolved Measurements

Range-resolved concentration measurements areobtained from the measurement of a local opticalthickness (LOT) �� between two ranges R1 andR2 (not necessarily adjacent) using n�R1, R2� ��R1

R2 ��on�r� � �off�r��dr. The LOT is the difference ofintegrated optical thicknesses �1 and �2 at range R1and R2, respectively: �� 2 � �1. The relative errorof the LOT, equal to the relative local concentrationerror, is then

��

�1�� 2��1� � �2��2��1�2. (30)

The task is now to establish the optimal conditionsfrom a concentration measurement in a defined at-

Fig. 7. Relative optical thickness error for heterodyne detectionintegrated measurements as a function of optical thickness � andon-line-to-off-line emitted energy ratio � with optimized CNR.

Table 2. Optimal Conditions and Relative Error of Optical Thicknessfor Integrated Measurements for Direct Detection and

Heterodyne Detection

Conditions Direct DetectionHeterodyneDetection

Background SNL Low High Local Oscillator

NBon�NT � NBoff�NT 0 0.01 0.2 Shot-noiselimited

NBon�Non 0 0.2 2.74 N.A.a

NBoff�Noff 0 0.08 1.14 N.A.(NT�M)opt 0 1.34 22.1 10.90(Non�M)opt 0 0.89 9.3 0.75(Noff�M)opt 0 0.27 3.9 2.6�opt 1.28 1.11 0.8 1.23�opt 3.6 3.2 2.7 4.28(�(�)��)opt�DELB 1 1.11 2.04 2.06

aN.A., not applicable.


mospheric layer measurement characterized by theparameter ��2. The limit case ��2 � 1 correspondsto a first measurement range before the atmosphericlayer where the absorbing species are present. Thiscase is nevertheless different from an integratedmeasurement since two optical thickness measure-ments are performed. For ��2 �� 1 we have �1

� �2 and Eq. (30) becomes �� 2��2��2��2�, which leads to the same optimal �2 and � asfor the integrated measurements.

The signal level at range R1�NT1� and range R2�NT2�present a ratio of k � NT1�NT2 that can be differentfrom unity, but we assume that the number of DOFsis the same for both ranges. A large variety of condi-tions can be met for range-resolved measurement de-pending on the range difference and the nature of thetargets. For measurements in clear air at sufficientlyremote ranges we have k � 1, but, if the second rangegate includes the echo of a significantly reflectivetarget (ground or cloud, for instance) k can be large.Conversely, if the first range is close to the instru-ment and the second one much more distant, k can besignificantly smaller than 1, owing to the decrease inthe 1�R2 signal.

We shall not try here to discuss all the aspects ofthe problem. In Subsection 5.A we establish the equa-tions that give the LOT error and optimize the opticalthickness and energy ratio with an adjustable param-eter k. Nevertheless, we shall illustrate only the caseof k � 1, which is close to the most-frequent measure-ment conditions. More-specific measurement condi-tions need to be modeled with a dedicated numericalsimulation.

A. Direct Detection

With the notations and assumptions previously in-troduced and by use of Eq. (13), Eq. (30) can be writ-ten as

��

�1

2��NT21�2 ��1 � ��1 � k� �

1 � �

�

� exp�2�2��1 � k exp��2��

� �1 � ��2�1 � k2�NBoff

NT2� 1 � �

� �2

� exp�4�2��1 � k2 exp�4��

�NBon

NT2� 4

NT2

Md�1�2

. (31)

In the same way as for the integrated measurements,we can look for the number of DDPs per DOF thatbrings the error closest to the SNL case and find

NT2

Md�

14 1 � �

� �2�NBoff

NT2�2�1 � k2� �

NBon

NT2

� exp�4�2��1 � k2 exp��4��. (32)

The LOT relative error [Eq. (31)] can then be numer-ically minimized as a function of �2 and � for given��2 and k.

Figure 8 presents the optimum �2 and � as func-tions of ��2 for k � 1 and for different BSRs,NBon�NT � NBoff�NT � 0, 0.01, and 0.2. The optimal �2do not differ significantly from the results obtainedfor integrated measurements, regardless of ��2. Incontrast, the optimal energy ratio decreases signifi-cantly when ��2 increases, that is, when the firstrange on-line signal is less attenuated by the speciesabsorption. This effect reduces the need for a highenergy ratio because at least the first on-line SNR ismore favorable. The reduction is less pronounced fora high BSR than for SNL conditions because a largerpart of the noise does not depend on the signal level.

B. Heterodyne Detection

With the notations and assumptions previously in-troduced and by use of Eq. (25), Eq. (30) can be writ-ten as

��

�1

2��Mt�1 �

kMt

NT2�1 � ��2

� 1 �kMt

NT2 1 � �

� �exp�2�1�2

� 1 �Mt

NT2�1 � ��2

� 1 �Mt

NT2 1 � �

� �exp�2�2�2�1�2

. (33)

Similar to the integrated measurement optimization,we can find an optimum number of DCPs per DOF:

NT2

Mt�

a2

1 � �

�,

Fig. 8. Optimum optical thickness at the most distant range �2

and on-line-to-off-line emitted energy ratio � as a function of therelative LOT ��2 for range-resolved direct detection measure-ments with different BSRs. NBoff�NT � NBon�NT � 0 (solid curve),NBoff�NT � NBon�NT � 0.01 (dashed curve), NBoff�NT � NBon�NT �0.2 (dotted–dashed curve), and for a signal ratio of k � 1.


with

a � ��2�1 � k2� � exp�4�2��1 � k2 exp��4��1�2.(34)

Equation (33) becomes

��

�1

2��NT21�2 1 � �

�a1�2[ 1 �

�

a�2

� 1 � k�

a�2

� 1 �exp�2�2�

a 2

��1 � kexp�2��2 � ��

a �2]1�2

. (35)

The LOT relative error [Eq. (35)] can then be numer-ically minimized as a function of �2 and � for given��2 and k. Figure 9 presents the optimum �2 and �as a function of ��2 for k � 1. The results are rela-tively similar to those for SNL direct detection withnevertheless lower �2 and higher � values. Only val-ues of ��2 above 0.2 make the optima differ signif-icantly from the integrated measurement ones. Therelative error in heterodyne detection remains twiceas high as the DELB.

6. Conclusion

A general optimization of direct and heterodyne DIALstatistical error is presented for integrated and range-resolved DIAL measurements under different operat-ing conditions. It is worth noting that the study ofheterodyne detection is not restricted to low CNR, thatis, when the speckle noise is neglected. For direct de-tection our study accounts for signal, background, andspeckle noises. For integrated heterodyne DIAL mea-surements, we first established the conditions of theoptimal signal level on single shots. In direct detectionthe optimum number of DDPs per DOF that makes thereception closest to the SNL is a function of the BSR.

For a given BSR and DOF, this determines the opti-mum total emitted energy per shot pair. The lowestpossible DIAL error (DELB) is obtained for a SNLdirect detection with an optical thickness of 1.28 andan to on-line-to-off-line energy ratio of 3.6. When thedirect detection background increases, the optimal op-tical thickness and energy ratio decrease and the errorincreases. The direct detection relative error reachestwice the DELB for an off-line BSR just above 1. Inheterodyne detection we directly found an optimum forthe on-line and off-line CNR (0.75 and 2.6, respec-tively). The minimum optical thickness relative errorin heterodyne detection is a little more than twice theDELB and is obtained for an optical thickness of 1.23and an on-line-to-off-line emitted energy ratio of 4.28.

The relatively low optimal CNR in heterodyne de-tection can usually be reached with a moderate energyper shot pair. A large number of shots is then requiredto reduce the error. Direct detection with a substantialbackground typically necessitates a high energy pershot pair and, consequently, an emitter divergencethat is much higher than the diffraction limit of thecollecting aperture in order to match the spatialspeckle number to the DDP. Consequently, fewer shotpairs are required to obtain a given relative error indirect detection than in heterodyne detection. For atotal emitted energy on a shot series and comparabledetection efficiencies, the error will be inferior in directdetection as compared to heterodyne detection, whenthe background remains lower than the off-line signal.The optimization of range-resolved measurements isstudied in the case when the off-line signal of twoadjacent ranges is of comparable amplitude. The opti-mal conditions are similar to those obtained for inte-grated measurements. It is only when the total opticalthickness is concentrated in the measurement cell�� 2� and for conditions closed to the SNL that theoptimal on-line-to-off-line emitted energy ratio de-creases significantly (down to approximately 1.5).

Conveniently, all these optima are quite broad, anderrors not exceeding the minimum by more than 5%can be obtained for significant variations in opticalthickness and energy ratio. This allows efficientDIAL measurements over extended ranges and for asufficiently large domain of optical thickness account-ing for the natural variation in the species concen-tration.

We also considered the case of equal on-line andoff-line transmitted energies but with an averagingperformed on a different number of shots (see Appen-dix A). In direct detection, the on-line-to-off-line av-eraging ratio is very close to the energy ratio foundpreviously, and the optimal optical thickness is onlyslightly lower. In heterodyne detection, the resultsare more significantly different. The optimal condi-tions are reached for an optical thickness of 1.1 and aaveraging ratio of 1.83. The minimum error is 1.13times the error obtained with unbalanced energies.

Appendix A

We consider in this appendix the case of measure-ments performed with equal on-line and off-line emit-

Fig. 9. Optimum optical thickness at the most-distant range of �2

and on-line-to-off-line emitted energy ratio � as a function of therelative LOT ��2 for range-resolved heterodyne detection mea-surements and for a signal ratio of k � 1.


ted energies �� 1 in Eqs. (11) and (12) so thatNoff � NT�2 and Non � NT�2 exp��2��] but averagedover a different number of shots Mon and Moff with

Mon � �Moff. (A1)

The equivalent total number of shot pairs, Mp, is thenMp � �Mon � Moff��2. We briefly present here theintegrated DIAL optimization in direct and hetero-dyne detection. The logic of the optimization issimilar to those presented in Sections 3 and 4. Therange-resolved case shall not be presented since theresults are similar to those obtained withthe energy-ratio optimization.

A. Direct Detection

With the notations presented above, the relative er-ror of the integrated optical thickness [Eq. (8)] can bewritten as

��

�1

2�MP1�2NT

1�2 �1 � ��1 �exp�2��

�

� 2NBon

NT

exp�4��

� 2NBoff

NT�

1 � �

2�

NT

Md1�2

.

(A2)

Under SNL conditions �NBon � NBoff � 0, Md � �� theoptimal optical thickness is equal to that obtainedwith equal averaged shots of unbalanced energy�0 � 1.28 and the optimal shot number ratio � isequal to the optimal energy ratio � � 3.6. This resultis identical to that obtained by Mégie and Menzies2

for nighttime conditions. The fact that the optima areexactly the same as those obtained in Section 3 is nota surprise since under SNL conditions the error isindependent of the way the photoelectrons are

summed (large energy of a few shots or large numberof shots of small energy). Both optimized methodsreach the DELB under SNL conditions.

With a background and a finite number of DOFs,the optimal measurement conditions (closest to theSNL performance) are obtained for

NT

Md�

41 � � NBon

NTexp�4��

NBoff

NT�. (A3)

The optical thickness relative error then becomes

��

�1

2�MP1�2NT

1�2 �1 � ��1 �exp�2��

�

� 4NBon

NT

exp�4��

� 4NBoff

NT1�2

. (A4)

Figure 10 presents the relative integrated opticalthickness error divided by the DELB for the sameconditions as in Fig. 1 (SNL conditions and NBon�NT

� NBoff�NT � 0.2) as a function of � and �. As re-marked above, the SNL case is strictly identical tothat presented in Fig. 1. In the presence of thebackground, the results are also quite similar. Nev-ertheless, the error is slightly higher than for theenergy-ratio optimization and is obtained with aslightly lower optical thickness and a shot numberratio that is slightly larger than the optimal energyratio (which can be explained by the fact that sum-ming the signals—instead of increasing the energy—also sums the backgrounds). For example, to becompared with Table 2, in the case NBon � NBoff� 0.2NT, the minimal error is 2.18 times the DELBfor an optimal � of 0.73 and � of 3.3, with NT�Md

� 4.1.

B. Heterodyne Detection

In heterodyne detection, with the notations pre-sented above, the optical thickness error is

Fig. 10. Relative optical thickness error in direct detection as afunction of optical thickness � and on-line-to-off-line averagingratio �. Left plot, for SNL conditions. Right plot, for NBon�NT �NBoff�NT � 0.2 with optimized NT�Md. The error is divided by theDELB.

Fig. 11. Relative optical thickness error for heterodyne detectionintegrated measurements as a function of optical thickness � andon-line-to-off-line averaging ratio �, with optimized CNR.


��

�1

2��MtMp

1 � �

2 �1� 1 � 2

Mt

NTexp�2��2

� 1 � 2Mt

NT�2�1�2

. (B1)

As in Section 4, we can determine an optimal numberof DCPs per DOF corresponding to

NT

Mt� 2� �

1 � � �1 � exp�4��. (B2)

Assuming this condition is respected, the relative in-tegrated optical thickness error, divided by theDELB, is presented in Fig. 11 as a function of � and�. The results are significantly different from the un-balanced energy case, which can be explained by thefact that the optimal CNR cannot be obtained on both

lines with equal emitted energies. The optimal erroris 2.32 times the DELB and is reached for � � 1.1 and� � 1.83. The CNRs are then 0.81 and 7.2 for theon-line and off-line signals, respectively.

References1. E. E. Remsberg and L. L. Gordley, “Analysis of differential

absorption lidar from the Space Shuttle,” Appl. Opt. 17, 624–630 (1978).

2. G. Mégie and R. T. Menzies, “Complementarity of UV and IRdifferential absorption lidar for global measurements of atmo-spheric species,” Appl. Opt. 19, 1173–1183 (1980).

3. R. M. Schotland, “Errors in the lidar measurements of atmo-spheric gases by differential absorption,” J. Appl. Meteorol. 13,71–77 (1974).

4. B. Saleh, Photoelectron Statistics (Springer, 1978), pp. 195–196.

5. B. J. Rye and R. M. Hardesty, “Estimate optimization param-eters for incoherent backscatter heterodyne lidar,” Appl. Opt.36, 9425–9436 (1997).


complementary study of differential absorption lidar optimization in direct and heterodyne...

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