estimating random errors due to shot noise in backscatter lidar ... · we discuss the estimation of...

Estimating random errors due to shot noise inbackscatter lidar observations

Zhaoyan Liu, William Hunt, Mark Vaughan, Chris Hostetler, Matthew McGill,Kathleen Powell, David Winker, and Yongxiang Hu

We discuss the estimation of random errors due to shot noise in backscatter lidar observations that useeither photomultiplier tube (PMT) or avalanche photodiode (APD) detectors. The statistical character-istics of photodetection are reviewed, and photon count distributions of solar background signals andlaser backscatter signals are examined using airborne lidar observations at 532 nm using a photon-counting mode APD. Both distributions appear to be Poisson, indicating that the arrival at the photo-detector of photons for these signals is a Poisson stochastic process. For Poisson-distributed signals, aproportional, one-to-one relationship is known to exist between the mean of a distribution and itsvariance. Although the multiplied photocurrent no longer follows a strict Poisson distribution in analog-mode APD and PMT detectors, the proportionality still exists between the mean and the variance of themultiplied photocurrent. We make use of this relationship by introducing the noise scale factor (NSF),which quantifies the constant of proportionality that exists between the root mean square of the randomnoise in a measurement and the square root of the mean signal. Using the NSF to estimate random errorsin lidar measurements due to shot noise provides a significant advantage over the conventional errorestimation techniques, in that with the NSF, uncertainties can be reliably calculated from or for a singledata sample. Methods for evaluating the NSF are presented. Algorithms to compute the NSF aredeveloped for the Cloud-Aerosol Lidar and Infrared Pathfinder Satellite Observations lidar and testedusing data from the Lidar In-space Technology Experiment. © 2006 Optical Society of America

OCIS codes: 280.3640, 040.5160, 270.5290.

1. Introduction

Lidar or laser radar has been used for atmosphericremote sensing since the early 1960s to measure im-portant atmospheric parameters such as wind andtemperature, and constituents such as aerosols,clouds, trace gases, etc. Accurately estimating andaccounting for the measurement errors (or uncertain-ties) introduced by various lidar system componentsis an important issue that must be addressed in orderto ensure the reliable application of lidar data prod-

ucts to atmospheric studies. Well-established error-propagation theory1 is usually used in the erroranalysis of backscatter lidar observations. Based onthis theory, an algebraic expression2 can be derivedthat computes the total uncertainty as a function ofthe various error sources. However, application ofthis expression requires estimates of the uncertain-ties attributable to each significant source.

There are two major types of uncertainty in lidarobservations: random errors and bias (systematic) er-rors. Random errors are generally caused by randomfluctuations (or noise) inherent in the measurement.For backscatter lidar measurements, these randomfluctuations result primarily from the following: (a)quantum noise (also known as shot noise) due to thediscrete nature of the incident light, charge carriers,and the interaction of light with the photodetector(i.e., photoemission); (b) thermal noise due to the ran-dom motion of electrons arising within the photode-tector, load resistor, and amplifier, and other noisesources (e.g., 1�f noise, etc.3); and (c) excess noiseintroduced in the multiplication process when a pho-tomultiplier tube (PMT) or an avalanche photodiode(APD) is operated in the analog detection mode. Ran-

Z. Liu ([email protected]) is with the National Institute of Aero-space, 100 Exploration Way, Hampton, Virginia 23666. W. Hunt,M. Vaughan, and K. Powell are with Science Applications Inter-national Corporation, Mail Stop 475, NASA Langley ResearchCenter, Hampton, Virginia 23681. C. Hostetler, D. Winkler, and Y.Hu are with NASA Langley Research Center, Mail Stop 475,Hampton, Virginia 23681. M. McGill is with NASA Goddard SpaceFlight Center, Laboratory for Atmospheres, Code 912, Building 33,Room B424, Greenbelt, Maryland 20771.

Received 9 September 2005; revised 6 December 2005; accepted9 December 2005; posted 19 December 2005 (Doc. ID 64669).

0003-6935/06/184437-11$15.00/0© 2006 Optical Society of America

20 June 2006 � Vol. 45, No. 18 � APPLIED OPTICS 4437

dom errors can be reduced by averaging or by repeat-ing the measurement. Systematic errors, on the otherhand, generally arise from sources such as inaccuratecalibration, nonlinearities in the photodetector re-sponse, defects in optical components, and�or a sys-tematic electronic noise. This type of error canproduce a fixed amount of bias that cannot be reducedby averaging. In contrast to the random error, how-ever, it is sometimes possible to reduce the effects ofsystematic errors when their sources are known. Asthe focus of this paper is random error, we will not bediscussing systematic errors in any further detail.

In lidar observations, the noise arising from back-ground radiation and detector dark current, but ex-cluding those fluctuations due to the scatteringsignal, is generally referred to as the backgroundnoise. Background noise is easily measured and isindependent of range from the laser transmitter. Thestandard deviation of the background signal can bedetermined, for example, from the samples acquiredbefore firing the laser (i.e., when there is no backscat-tered signal), or from the samples corresponding tovery high altitudes (e.g., �40 km) where the laserbackscatter is negligibly small when compared to themagnitude of the background signal. In contrast tothe background noise, the magnitude of the noiseassociated with the scattering signal depends on therange-resolved intensity of the backscattered light,and thus needs to be estimated separately for eachdata sample. In this paper we will focus our discus-sion on this latter type of error, and on methods forestimating its magnitude.

For lidar measurements, the conventional methodwidely used to estimate the random error is to com-pute the standard deviation of a series of consecutivesamples. These samples can be obtained either ver-tically, from a sequence of consecutive range binswithin a single lidar profile, or horizontally, fromsamples at the same range bin obtained over somenumber of consecutive profiles. When using these sta-tistical techniques, however, the natural variabilityof the atmosphere can cause significant overesti-mates of the random component of the measurementerror. This effect is especially severe in those areaswhere the atmospheric composition changes rapidly(e.g., within clouds). Given that measuring the vari-ability of the atmosphere is one of the fundamentalobjectives to be realized by the use of backscatterlidar observations, it is thus highly desirable to havean error estimate that can be generated in a mannerwholly independent of the ambient atmospheric con-tent.

In this paper we introduce the noise scale factor(NSF) to estimate the random error due to signal shotnoise. The derivation of the NSF is based on the factthat when the intensity of an incident light field doesnot fluctuate during the time of observation (i.e.,when it remains in a statistically stationary state),photons sampled during this time will follow a Pois-son stochastic process.4,5 In Section 2 of this paper wereview the statistical basis of photodetection. Themathematical derivation of the NSF is presented in

Section 3. Practical techniques for ascertaining thecorrect value for the NSF are developed in Section 4.This development is illustrated via application to thelidar that is flying aboard the Cloud-Aerosol Lidarand Infrared Pathfinder Satellite Observations(CALIPSO) satellite,6 and tested using data acquiredduring the Lidar In-Space Technology Experiment(LITE) mission.7 Issues of transferring the NSF fromone signal domain to another, and concerns, arisingfrom averaging partially correlated samples, are dis-cussed in Section 5. Concluding remarks and a sum-mary are given in Section 6.

2. Statistics of Photodetection

A. Shot Noise

PMTs and APDs, operating either in a photon-counting mode or in an analog mode, are the standardphotodetectors used for backscatter lidar observa-tions. We will therefore focus our discussions on thestatistics of photodetection using PMTs and APDs.

Even if the radiation field is of constant intensity,the number of photons arriving at the photodetectorduring any time increment is inherently uncertaindue to the quantum nature of light. Straightforward,statistical proofs exist showing that if photon arrivalrates are time independent (i.e., they can be de-scribed as being a statistically stationary process),the total number of photons arriving during any timeinterval � is Poisson distributed.4 Theoretical studieshave established the correspondence between thenumber of photons incident on the detector and thenumber of photoelectrons emitted, and thus the pho-toelectrons also have a Poisson distribution.5 Theprobability of emitting np photoelectrons during time� is given by

p�np� ��n̄p�np

np! e�n̄p. (1)

In this expression, n̄p � ��P�h� represents the meannumber of photons emitted. P is the power of theincident field, � is the quantum efficiency of detector,h represents Planck’s constant, � describes the fre-quency of the field, and h� is the energy of the photon.Both here and afterwards, an overbar (e.g., n̄p) is usedto indicate that a quantity represents a mean or av-erage value. For a Poisson distribution, the varianceis equal to the mean, so that

�np2 � n̄p, (2)

where �np represents the standard deviation. Thevariance quantifies the uncertainty in the measure-ment due to shot noise. The Poisson distribution ap-plies to light emitted from an ideal laser havingdeterministic intensity, or from a thermal radiationsource such as the sun that has a coherence time �cmuch smaller than the sampling time �.5 Examples ofthe photon-count distributions of solar backgroundsignals and laser scattering signals are given in

4438 APPLIED OPTICS � Vol. 45, No. 18 � 20 June 2006

Fig. 1. These data were acquired by the Cloud Phys-ics Lidar (CPL),8 which is an airborne, down-looking system that uses photon-counting detectionand can thus provide direct photon-count measure-ments. The data shown in Fig. 1 was obtained fromdaytime measurements, where detector (APD) darkcounts are negligibly small when compared to thebackground light signal. The background signal dis-tribution in Fig. 1(a) was compiled using 100 sub-surface samples (i.e., signals containing no laserbackscatter) from 1000 profiles (a total of 100 000samples). Each CPL raw sample is acquired in acounting time period of 0.2 �s (corresponding to a30 m vertical resolution) and accumulated over 500shots. This results in an effective counting time of0.1 ms for each raw sample. The composite atmo-spheric scattering distribution (laser backscatter �background signal) shown in Fig. 1(b) was derivedusing six samples (range bins) from �10 km in thesame 1000 profiles (a total of 6000 samples). For com-parison, Poisson distributions having the samemeans as the measured data are also shown in eachpanel, and it is clearly seen that both the laser scat-

tering signal and the solar background signal sharethe same type of distribution—Poisson.

In general, if the radiation field intensity varieswith time, the photodetection statistics are governedby a compound Poisson distribution (also known asMandel’s formula),5,9 whose rate density is propor-tional to the instantaneous electromagnetic energycollected by the detector. In this case, the variance isgiven by

��np�2 � n̄p ��h��2�W2, (3)

where W is the integrated optical intensity overtime interval �, and �W is the standard deviation ofW. The additional term in the expression for the vari-ance, ��h��2�W2, results from the field fluctuationsof the incident radiation. This term, which in photo-detection of thermal light is sometimes called“photon-bunching noise,” is a consequence of the cor-relation of fluctuations in the thermal light inten-sity.5,9 In backscatter lidar observations, this excessnoise may arise from fluctuations in the laser sourceand�or the natural variation of the atmospheric scat-tering media. Fluctuations of laser output are usuallysmall, and are monitored during the observations inorder to energy normalize the lidar data prior to sub-sequent analyses. As a result, the effect of laser fluc-tuations is ignored in our analysis. However, thevariability of the atmosphere and its components,especially clouds, can be very large. As mentionedpreviously, characterizing this atmospheric variabil-ity is one of the primary objectives of backscatterlidar measurements. We therefore do not include anatmospheric variability term in our random uncer-tainty estimates.

B. Excess Noise (Multiplication Noise)

For a PMT or an APD operated in the analog detec-tion mode, the output electrons (multiplied photoelec-trons) at the anode do not obey Poisson statistics,even if the incident photons (or emitted photoelec-trons) do.10–12 This is because the photoelectron mul-tiplication in these detectors is also a stochasticprocess, which can introduce an excess noise. In atypical PMT, the photoelectrons emitted from thephotocathode are multiplied by a set of dynodes viathe secondary emission of electrons. The probabilitydistributions of the multiplication gains of thesePMTs can be described by a multiple stochastic (com-pound) Poisson distribution.10,12 In APDs, on theother hand, photoelectrons can initiate impact ion-ization to produce extra hole-electron pairs, which inturn result in more hole-electron pairs as they movethrough the space-charge region (avalanche region ormultiplying region). The photocurrent is thus multi-plied. For a uniform APD having a thick multiplyingregion, the probability distribution of gains can becharacterized analytically by a local-field theory.11

Fig. 1. Examples of photon count distributions derived from CPLmeasurements at 532 nm for (a) solar background signals and (b)laser scattering signals mixed with solar background signals. Inboth examples, the photon counts that comprise the input datawere accumulated over an interval of 0.1 ms.


The variance of multiplied electrons can be ex-pressed as10–12

��nm�2 � FmGmn̄m, (4)

where nm is the number of multiplied electrons andn̄m is the mean number of such electrons. n̄m is deter-mined by

n̄m � Gmn̄p, (5)

where Gm is the average gain of the multiplicationand n̄p is the mean number of photons incident on thedetector. The Fm term in Eq. (4) represents the excessnoise factor that is used to quantify the extra noisecaused by the variability of the multiplication gainin a PMT or APD. The excess noise factor is a func-tion of the average gain, Gm, for both PMTs andAPDs.10–12 For PMTs, Fm normally ranges from 1 to2, and decreases as Gm increases. For APDs, Fm in-creases with increasing Gm and is normally largerthan 2. The larger excess noise introduced in the APDis due to the greater uncertainty of the APD multi-plication gain. The APD gain variation arises fromtwo sources: (i) the randomness in the locations atwhich ionizations may occur, and (ii) the feedbackprocess associated with the fact that both electronsand holes can produce impact ionizations as theymove in opposite directions. In contrast, in standardPMTs only one carrier—electrons—causes second-ary emissions (or multiplication), and this occurs onlyat fixed locations (dynodes). For PMTs having iden-tical gain factors m for each dynode, the excess noisefactor is given by10,12

Fm �m

m � 1. (6)

In this case, Gm � mN, where N is the number ofdynodes. For uniformly multiplying APDs,11 the ex-cess noise factor is

Fm � kGm �1 � k��2 �1

Gm�, (7)

where k is the ratio of ionization coefficients due toholes and electrons. As an example, Fm � 1.5 forPMTs when m � 3, and Fm � 5 for APDs whenk � 0.03 and Gm � 100.

3. Noise Scale Factor

As shown by the discussion in Subsection 2.B., thereexists a proportional relation between the varianceand the mean of the shot noise for both PMTs andAPDs operated in either a photon-counting detectionmode or an analog mode. Based on this proportion-ality, we introduce the noise scale factor to estimate

the standard deviation, �x, of the shot noise in ameasurement x from its mean, x, using

�x � NSF · x1�2. (8)

NSF has units of the square root of the units for x. Forlidar observations using photon counting (e.g., Refs. 8and 13), the random error due to shot noise can beestimated from the number of photon counts based onEq. (2). In this case, NSF � 1�counts1�2� in the photon-counts domain. For the analog detection, the NSF inthe multiplied-photoelectron domain is given by

NSF � �FmGm�1�2. (9)

For lidar observations, the data is normally sampledusing a digitizer. In the digitizer-readings domain,NSF can be derived from the signal-to-noise ratioanalysis for the lidar measurements (see, e.g., Ref.14), and is computed using

NSF � �2eBFmGmGA�1�2. (10)

Here e is the electron charge, B � 1�2�T0 is thespectral bandwidth of the lidar receiver, and �T0 isthe integration time. GA is a gain factor that convertsthe anode current of the detector to digitizer counts,with the assumption that linear amplifiers are used.GA is a product of a number of converting or scalingfactors and gains.

In practice, some amount of background signal,arising from the background radiation, detector darkcurrent, etc., is unavoidably included in the lidarmeasurements. Thus each digitized sample, V, can bewritten as V � Vs Vb, where Vs represents the laserbackscatter signal and Vb represents the backgroundcontribution. The overall random uncertainty foreach sample is therefore the sum of the uncertaintiesin each of these quantities:

�V � �NSF2Vs ��Vb�21�2. (11)

In this expression, Vs is the mean of the scatteringsignal, and �Vb is the background noise; i.e., the stan-dard deviation of the background signal. �Vb can bemeasured directly from the samples where there isno laser scattering signal (e.g., subsurface samplesor very high-altitude samples). Generally Vs is un-known. However, if the measurement is not verynoisy, the uncertainty can be estimated from a singlesample using

�V � �NSF2Vs ��Vb�21�2. (12)

Note that, in practice, Vs is typically derived by sub-tracting the measured mean value of the backgroundsignal, Vb, from the raw digitizer reading V; i.e., Vs

� V � Vb. When computed in this manner, Vs is alsoa random variable, and thus an additional uncer-tainty, �Vb, which represents the uncertainty in the


measured Vb, must be introduced into the calculation;that is,

�V � �NSF2Vs ��Vb�2 ��Vb�21�2. (13)

�Vb is usually determined by computing the standarddeviation of a number of samples where there is noscattering signal and V̄b is the mean of these samples.Therefore, the error in the estimate of the mean is

�Vb �1

Nb

�Vb, (14)

where Nb is the number of the samples from which Vb

is computed. This number is usually quite large, sothat �Vb is typically much smaller than �Vb.

The advantages of using the NSF to estimate theuncertainties inherent in lidar backscatter measure-ments are illustrated by the CPL profile measure-ments shown in Fig. 2. To derive the conventionalerror estimates, standard deviations (with respect tothe mean signals) have been computed for each alti-tude bin between 0 and 16 km for a sequence of 100

consecutive profiles. These values are plotted using adashed curve. For comparison, standard deviationsestimated from a single profile using the NSF tech-nique are plotted using a solid curve. The uncertain-ties computed using the two methods are generallyconsistent in the aerosol-free region above �1.5 km,where only molecular scattering exists. However, inthe aerosol layer between 0 and 1.5 km, significantoverestimates appear in the uncertainties computedusing the conventional method. This behavior is dueto the horizontal variation of the particle concentra-tion within the aerosol layer [i.e., the implicit inclu-sion of the W2 term in Eq. (3)]. This comparisonclearly shows that the conventional method can over-estimate the random error. More importantly, a num-ber of horizontally homogeneous profiles are requiredin order to derive accurate results using the conven-tional method. On the other hand, the NSF methodcan estimate the random error using only a singlesample.

4. Noise Scale Factor Measurement

When the parameters in Eq. (10) are all known, thecalculation of NSF is straightforward. GA and B canbe determined accurately based on laboratory ex-periments, and they generally do not vary duringthe observation period. Gm and Fm, however, mayvary during the observation period, in concert withchanges in the lidar operating environment. An ex-ample where this situation can be expected to occur isprovided by the Cloud Aerosol Lidar with OrthogonalPolarization6 (CALIOP) that is flying aboard theCALIPSO satellite. CALIOP (pronounced “calliope”)is a satellite-borne, two-wavelength (532 and 1064nm), polarization-sensitive (at 532 nm) lidar that, fol-lowing its launch in April 2006, will conduct contin-uous observations of the atmosphere from space forthree years. Two PMTs and one APD operated inanalog mode are used to detect the two 532 nm po-larization signals and the single 1064 nm total sig-nal. The gains (and consequently excess noise factors)of these detectors (especially the PMTs) may changesignificantly during the course of the three-year mis-sion. They may also change considerably during thelaunch phase from the ground to space, owing to thesevere vibration and huge change in temperature.Consequently, the NSF must be monitored con-stantly during on-orbit operations in order to makeuse of this factor in estimating contributions to thesignals from random noise. This section discussestechniques for measuring NSF using the solar back-ground signals, and develops operational algorithmsfor use by the CALIPSO lidar. Because the CALIOPdetectors are operated in analog mode, and because,in general, NSF � 1�counts1�2� for both PMTs andAPDs operated in the photon-counting mode, we fo-cus our discussion on the measurement of NSF forthe analog detection mode of the two detector types.

The CALIPSO satellite was launched successfullyon 28 April 2006. However, backscatter data have yetto be acquired, and hence we illustrate the algorithm

Fig. 2. Examples of uncertainty estimates in attenuated back-scatter (m�1 sr�1) derived from airborne lidar measurements usingphoton counting detection: standard deviations computed for eachaltitude bin using 100 consecutive profiles (conventional method)and using the NSF. The uncertainties computed using the conven-tional method are generally consistent with those derived usingthe NSF in the aerosol-free region (above �1.5 km) where theatmosphere is relatively stable. However, due to the horizontalvariability of the aerosol layer, the conventional method is seen tosignificantly overestimate the uncertainties below �1.5 km in theprofile.


development discussion using data acquired by LI-TE.7 LITE was the world’s first space-borne lidar, athree-wavelength backscatter system that flewaboard NASA space shuttle flight STS-64 in Septem-ber 1994. Figure 3 presents an example of a single-shot lidar profile measured at 532 nm during thenighttime portion of LITE orbit 117. Like CALIPSO,LITE used a PMT for the 532 nm measurements andan APD for 1064 nm channel.7 The LITE data systemacquired 5500 range-resolved samples per profile at a10 MHz sampling rate (i.e., 15 m per range bin). Thebackground signal (dc component) was measured andrecorded onboard by a background monitor, and au-tomatically removed from each profile prior to digiti-zation. In the figure, the return signal below �40 kmis seen to increase with decreasing height. This in-crease is due to the increasing atmospheric molecularnumber density and the greater incidence of sus-pended particles (aerosols). The scattering signalfrom the upper atmosphere ��40 km� is very smallcompared with the background signal (backgroundradiation and dark current, etc.).

For a PMT in the analog mode, the dark noise isgenerally negligibly small when compared with thesolar background noise during daytime measure-ments. For an APD in the analog mode, the darknoise (which is predominantly amplifier noise7) isdominant during nighttime measurements and iscomparable to the solar radiation noise during day-time measurements. The NSF can then be derived,

based on Eq. (8), using

NSF ��Vb

Vb

(15)

when the solar radiation noise is dominant (i.e., day-time measurements), and using

NSF ��Vb�2 � ��Vd�21�2

�Vb � Vd�1�2 (16)

when the dark noise cannot be ignored (nighttimemeasurements). In these equations, �Vb and �Vd rep-resent the rms noise of the total background signaland the component due to detector dark current, andVb and Vd are their means, respectively. We note thatEq. (15) is an approximation of Eq. (16), valid onlywhen the dark noise is negligibly small. In the fol-lowing subsections, methods for computing each ofthese quantities are described. In addition, test re-sults derived using LITE measurements at both 532and 1064 nm are presented for both PMTs and APDs,and are discussed in detail.

A. Noise Scale Factor Estimation forPhotomultiplier Tubes

The rms noise and the mean of the solar backgroundsignal must be derived in order to compute NSF usingEq. (15). The rms background noise is estimated bycalculating the standard deviation over a large num-ber of samples in each profile, selected from a regionwhere the laser scattering signal is negligiblysmall (i.e., above �40 km; refer to Fig. 3). For LITEand CALIPSO, the background signal is derived byconverting the background monitor reading from itsnative units into equivalent science digitizer counts.In Fig. 4(a) we present the square root of the back-ground signal and the rms noise derived from thehigh altitude region for LITE measurements at532 nm (i.e., PMT detection) acquired during orbit117. It is shown that the solar radiation backgrounddominates the background signal for the daytime por-tion of the orbit. The NSF values derived using Eq.(15) are shown in Fig. 4(b). It is seen that the NSF isgenerally constant for the daytime portion of the or-bit. However, at profile number 2200 there is a stepchange of �10%, which is most likely due to an un-documented change in the PMT gain. The suddenspike in NSF values (to �10) for the profiles from�2000 to 2200 is due to the saturation of the back-ground monitor digitizer, which can be seen in theflat-line segments of the square-root curve in Fig.4(a). The NSF values calculated for the nighttimeportion are generally smaller than that for the day-time portion, where the background signal is at avery low level, and is dominated by the detector darkcurrent. This difference is most likely due to the lim-ited resolution of the acquisition system (especiallythe background monitors, which used separate digi-tizers having fewer bits than the science data digitiz-

Fig. 3. Single-shot lidar return profile at 532 nm acquired usingPMT from the LITE orbit 117 measurement.


ers), which could yield errors and�or biases that maybe significant when recording very weak signals.However, for those nighttime regions where lunarlight (backscattered from dense clouds, etc.) domi-nates the background signal and has a higher signallevel, the NSFs have values similar to those com-puted during the daytime portion. The nighttimeNSF values are also substantially noisier, due to thevery low levels of background illumination combinedwith the limited resolution of the LITE backgroundmonitors.

B. Noise Scale Factor Estimation forAvalance Photodiodes

Due to the presence of large amounts of dark noise,NSF measurement for an APD in analog mode is

relatively complicated. The APD dark current repre-sents the dominant noise source in the nighttimeportion of the data, and is comparable to the solarbackground signal for the daytime portion. The com-putational difficulties arising from this situation areillustrated in the sequence of plots shown in Fig. 5.The upper panel [Fig. 5(a)] shows the square root ofthe 1064 nm background monitor reading and therms noise of the background signal at 1064 nm com-puted over the same data segment shown in Fig. 4.The rms noise is once again estimated using the sam-ples above �40 km, where the laser backscatter isnegligible. Figure 5(b) shows the NSF estimates thatwould be computed without first correcting the mea-surements for dark components; i.e., by using Eq. (15)rather than Eq. (16). The large NSF oscillations seen

Fig. 4. NSF calculations using LITE orbit 117data acquired at 532 nm: (a) standard deviationand square root of the background signals, com-puted using the uppermost 2500 samples of eachsingle-shot profile; and (b) NSF computed using Eq.(15). The arrows indicate daytime and nighttimeportions of the orbit. All calculations are derivedfrom data acquired using a photomultiplier (PMT).

Fig. 5. (Color online) NSF calculations using theorbit 117 data acquired at 1064 nm: (a) the squareroot and rms noise of the background signal, com-puted over the same altitude regime used in Fig. 4;(b) NSF computed using Eq. (15); and (c) NSF com-puted using Eq. (16) (pale gray curve) and Eq. (17)with c � 12 490 (black curve). All calculations arederived from data acquired using an avalanche pho-todiode (APD). The data segment displayed is iden-tical to that shown in Fig. 4.


in the daytime segment of the APD data comparepoorly with the consistent results obtained using thePMT measurements, and are a direct consequence ofthe dark noise contributions from the APD and theamplifier.

The APD NSF computed for the daytime portionusing Eq. (16) with the dark components removed ispresented in Fig. 5(c). The mean value, Vd, and rmsnoise, �Vd, of the dark current were determined fromthe nighttime portion of the data, under the assump-tion that these quantities do not change significantlyin the transition from nighttime to daytime observa-tions. The computed NSF using Eq. (16) is generallyconstant. However, very large variations appear inregions where the solar background signal is quitesmall compared with the dark current. This is be-cause, when such conditions occur, the magnitude ofVb becomes very similar to that of Vd in the denomi-nator of Eq. (16), and the uncertainties in their de-termination become bigger than the difference oftheir average values. These near-zero values in thedenominator give rise to the very noisy behavior ofthe NSF estimate computed via Eq. (16) and seen onthe left-hand side of Fig. 5(c). To stabilize the calcu-lation of the NSF, a modified form of the equation isderived, such that

NSF ��Vb

Vb c, (17)

where c is a constant that satisfies

c � Vd��Vd2�Vd

NSF2 � 1� (18)

under the assumption that the NSF and �Vd2�Vd do

not change for the chosen data segment. c is chosenby trial so that the NSF curve is flattest over theentire data segment. The NSF determined accordingto Eq. (17) is also presented in Fig. 5(c). It is seen tobe constant over the entire data segment, with amean of 1.39, and is generally consistent with theNSF computed using Eq. (16). This modified ap-proach appears to be much less sensitive to noisewhen the background levels are low.

We note that the modified method described byEqs. (17) and (18) could also be applied to the night-time 532 nm data shown in Fig. 4. Although we didnot test this approach, its application could some-what improve the estimate derived for the 532 nmNSF.

5. Noise Scale Factor Application Issues

A. Transferring Noise Scale Factor

The value of NSF is signal domain dependent. Theformula for a linear transform of NSF from a domainV to another domain V� � KV is given by

NSFV� � K1�2NSFV. (19)

K is a conversion factor independent of V or V�. Thederivation of this formula is straightforward. As anexample, the application of this formula to the lidarmeasured attenuated backscatter coefficients, whichare a fundamental lidar product, is discussed below.

Raw lidar measurements are usually further pro-cessed in order to produce additional meaningful dataproducts. The attenuated backscatter coefficients,��r�, are derived by range-correcting and scaling thebackground-subtracted samples, Vs � V � Vb, as fol-lows:

��r� � �r�T2�r� �r2

C Vs�r�. (20)

Here �r� is the atmospheric backscatter coefficient(including both molecular and particulate contribu-tions) at range r; T is the atmospheric transmittance,which accounts for signal attenuation between thelidar and the volume of atmosphere at range r; and Cis the lidar calibration constant. Vb is the measuredbackground signal, which is usually determined fromthe mean of the subsurface samples where there is nolaser scattering signal (e.g., as for CPL and otherdown-looking lidars) or from the samples acquired athigh altitudes (e.g., 65–80 km for the CALIPSO lidar)where the laser scattering signal due to the atmo-spheric molecules and particles is negligibly small.The uncertainty in � due to shot noise can be esti-mated using

�� r2

C ��NSFV�2Vs ��Vb�2 ��Vb�21�2

��r2

C�NSFV�2� �r2

C�2

��Vb�2 ��Vb�2�1�2

(21)

or

�� NSF��2� �r2

C�2

��Vb�2 ��Vb�2�1�2

,

(22)

where �Vb � �Vb�Nb, and Nb is the number of thesamples used to compute Vb and �Vb. NSFV is thenoise scale factor in the V domain and NSF�

� �r2�C�1�2NSFV [refer to Eq. (19)] is the noise scalefactor in the � domain.

Note however, that NSF is not constant in somedomains. For example, in the ��r� domain, NSF is afunction of r. In practice, it is usually more conve-nient (and less error prone) to derive and apply theNSF in a domain in which its value is constant.

B. Sample Average

To produce high quality lidar data products, signalaveraging over a number of range bins or over anumber profiles (laser shots) is usually required.


(Note, however, that while averaging is an effectiveway to reduce noise, as a trade-off it also degrades theresolution of the data.) When the samples are totallyuncorrelated and N samples are averaged, the rmsnoise (standard deviation) can be reduced by a factorequal to the square root of N.1,10,11 Therefore ifthe samples used in averaging are totally indepen-dent (uncorrelated), the random error due to noise inan averaged measurement, Vavg � j�1

Nshot i�1Nbin Vj,i�

�NshotNbin�, is estimated by

�Vavg �1

Nshot� 1

Nbin�NSF2Vavg ��Vb,avg�2

��Vb, avg�2�1�2

, (23)

where Nbin and Nshot are the number of range binsand laser shots, respectively, used to compute theaverage; i.e., ��Vb,avg�2 � j�1

Nshot ��Vb, j�2�Nshot and��Vb,avg�2 � j�1

Nshot ��Vb, j�2�Nshot.

C. Correlation Correction

Lidar design considerations (e.g., bandwidth andsampling frequency) may lead to the acquisition ofsamples that are partially correlated with neighbor-ing samples. For example, the sampling interval ofthe LITE data is 15 m while the fundamental rangeresolution of the system is limited by the bandwidthof the lidar receiver (amplifier) to a resolution slightlygreater than 30 m (i.e., more than two sample inter-vals). As a result, neighboring samples (2–3 bins) in aLITE backscatter profile are partially correlated.Fig. 6(a) shows the autocorrelation function derivedfrom the LITE orbit 117 measurements. The calcula-tion was restricted to the uppermost 2500 samples(i.e., data from above 40 km, where atmosphericbackscatter is negligible), and averaged over 6000profiles, so that the backscattered solar signal is es-sentially constant. The plot clearly shows that eachLITE sample is at least partially correlated with thetwo samples before or after it.

Though the rms noise is expected to reduce by afactor of N�1�2 when N independent samples areaveraged, if the samples are partially correlated thecorrect expression for the relationship described byEq. (23) becomes more complicated. To illustratethis, Fig. 6(b) presents the standard deviation as afunction of number of range bins used to computethe average. All values were computed from thesame data segment of the LITE orbit 117 measure-ments. For comparison, the standard deviation pre-dicted by the N�1�2 relation is also presented. Figure6(b) clearly shows that the actual reduction of noiseis not as large as would be predicted by the N�1�2

relation. This is due to the partial correlation be-tween the neighboring samples, as demonstrated inFig. 6(a). The ratio of the measured standard devi-ation curve to the N�1�2 curve is presented in Fig.6(c) (dashed curve). This ratio is larger than 1.5

when the number of samples averaged is largerthan 10.

When using correlated data, the difference be-tween the measured and predicted values of �Vavg canbe significant. Therefore, when using the NSF to es-timate random error in averaged measurements, acorrection is required to compensate for effects ofsample-to-sample correlation. Introducing the corre-lation correction function f, Eq. (23) can be modifiedas

�Vavg �1

�Nshot�1�2�f�Nbin�Nbin

�NSF2Vavg ��Vb,avg�2

��Vb, avg�2�1�2

. (24)

Note that signal averaging does not reduce �Vb,avg,and that the samples acquired from different lasershots are uncorrelated, so that a correction for aver-aging over multiple profiles is not necessary.

The f function can be either measured directly [i.e.,

Fig. 6. (a) Autocorrelation function derived from uppermost 2500samples and averaged over 6000 profiles from the LITE orbit 117measurement. (b) Standard deviations as a function of average binnumber Nbin from the measurement and predicted using(Nbin)�1�2. (c) Correlation correction function.


the dashed curve in Fig. 6(c)] or computed from theautocorrelation function using

f�Nbin� � �1 2 m�1

Nbin�1 �Nbin � mNbin

�R�m��1�2

. (25)

Here R is the autocorrelation function, as shown inFig. 6(a). Values of f computed using Eq. (25) are alsoplotted [solid curve in Fig. 6(c)], and are generallyconsistent with the measurements (dashed curve)when small numbers of samples are averaged. Ana-lytically derived values of f are smaller than themeasurements for large averages, due probably tosystematic errors such as the baseline ripple and�orother electronic oscillations imposed in the measure-ment.7 The photon-bunching noise5 [i.e., the secondterm on the right-hand side of Eq. (3)] arising fromfluctuations of the emission rate of dark counts (athermal emission process) in the PMT and of thebackscattered solar light intensity due to the vari-ability of the underlying atmosphere (though we havechosen a horizontally relatively homogeneous clear-air region profile) may also contribute to thisdiscrepancy. The difference, however, is acceptablysmall ��3%�.

6. Summary

In the analysis of lidar data, there are two types oferrors (uncertainties) that must be considered: ran-dom and systematic. This paper focuses on the esti-mation of random errors in the received signal due tonoise inherent in the backscatter lidar measurement.The statistical characteristics of photodetection usingboth photomultipliers and avalanche photodiodeshave been reviewed. In general, the distribution ofsampled photons (photon counts) is a doubly stochas-tic (compound) Poisson distribution. The multipli-cation process in a PMT or an APD is a stochasticprocess, and hence generates excess noise. Conse-quently, the multiplied carriers (electrons for PMTand electron-hole pairs for APD) no longer follow thePoisson statistics even if the incident photons arePoisson distributed. For both PMT and APD, how-ever, there still exists a proportional relation betweenthe standard deviation (rms noise) and square rootof the mean of multiplied carriers. Based on this fact,the noise scale factor (NSF) has been introduced toestimate the random error due to the shot noise. Theuse of the NSF greatly facilitates the random errorestimation; it allows an estimate of random error foreach individual sample in the lidar backscatter pro-file. The traditional method widely used for estimat-ing random error computes statistics from anensemble of lidar measurements, and its applicationthus requires a large number of samples. Further-more, as shown in this work, when applying the con-ventional technique, an overestimation of the randomerror frequently results from the natural atmosphericvariability. This bias error is especially acute in themeasurement targets of greatest interest, such asboundary layer aerosols and clouds.

Background noise is another important error source.This error, however, can be measured directly; it canbe determined from subsurface samples where no scat-tering signals exist, or from samples acquired at veryhigh altitudes (e.g., 40–80 km) where the scatteringsignal is negligibly small. Two major components—background radiation signal and detector darkcurrent—are included in the background signal. Thedistributions of these signals have also been investi-gated in this paper. The analysis using the CPL mea-surements at 532 nm, which used photon-countingdetection, showed that the photon counts due to thesolar radiation follow the same statistics as the pho-ton counts due to the laser scattering, i.e., Poissonstatistics. Based on the statistical characteristics ofthe Poisson distribution, algorithms have been devel-oped for the CALIPSO lidar that use the solar radi-ation background signal to determine the NSF of theanalog modes for PMTs and APDs. The algorithms tocompute NSF from the solar background signal havebeen tested with the LITE data. It was shown thatthe NSF measurement for the PMT is largely unaf-fected by the dark current, because the dark currentis very small when compared with the solar back-ground signal. The NSF measurement for the APD,however, is significantly affected by the presence ofdark current, because the dark current is large andmay, in the presence of significant amplifier noise,behave statistically different from the optical signal.When computing the NSF for the APD, either thedark current must be subtracted from the solar signalor the modified algorithm must be used.

The authors thank the two anonymous reviewersfor their constructive and insightful comments.

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