Development of a SNR parameterization scheme for general lidar assessment

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Appl. Phys. B 00, 1–12 (2005)

Applied Physics B

DOI: 10.1007/s00340-005-1783-8

Lasers and Optics


Development of a SNR parameterization scheme for general lidar assessment 1 City

College of the City University of New York, 140 St. & Convent Ave., New York, NY 10031, USA State Technical University, 10 K. Marx St., Kazan, Tatarstan 420111, Russia

2 Kazan

Received: 30 June 2004 / Final version: 7 February 2005 Published online: 30 March 2005 • © Springer-Verlag 2005

We present a general methodology for evaluating the capabilities of a general lidar system encompassing both backscatter (elastic and Raman lidar) and topographic targets. By introducing a well-defined atmospheric reference medium and by individually examining and decomposing the contributions of lidar system parameters including lidar transmitter power, field of view, receiver noise, atmospheric conditions, and sky background on the signal-to-noise ratio, we obtain a simple dimensionless parameterization of the lidar system. Using this parameterization, numerical simulations are carried out to determine achievable lidar performance including operation range, minimum detectable gas concentration, and so on.




42.68.Wt; 42.79.Qx; 92.60.Sz; 42.62.Fi


Over the last 30 years, many practical researchlevel systems for remote atmospheric monitoring have emerged. However, as lidar (Light Detection and Ranging) technology matures, many lidar developers are shifting focus toward the development of lidar systems with dramatic reductions in size, weight, energy consumption, and operator involvement. Lidar designs vary widely [1–8] depending on the specific application, available hardware components, and experience of the developers. While simple qualitative design issues such as the use of a more powerful laser transmitter, a largeraperture receiving telescope, and/or more sensitive photodetectors will obviously achieve a greater operation range, better retrieval accuracy, etc., cost constraints often limit such designs. To conduct quantitative tradeoff studies, a significant number of instrumental parameters and external environmental factors must be taken into account, and it is often not clear from this representation how each system and/or environmental parameter can quantitatively effect the ultimate performance. While it is straightforward to assess a given lidar system, an assessment of the tradeoffs necessary to develop and

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assess an effective design is not straightforward due to the sheer number of parameters used to describe the lidar system and the environment under investigation. In particular, the identification of universal design parameters central to lidar performance has not been sufficiently investigated. Analysis of lidar performance is traditionally based on examination of the signal-to-noise ratio (SNR) at the photodetector output [1, 2, 4–7, 9–12]. SNR is a frequently used comprehensive criterion for lidar instrument efficiency and is presented in Fig. 1. While the comprehensive nature of the SNR criterion makes it a very useful tool for assessing a given lidar system [10, 11], it is also a weak point [12, 13] because it obscures the impact of the different components. For example, an increase or reduction of SNR can be caused not only by the scattering efficiency of the target under study but also by changes in ‘weather’ conditions for signal propagation [1, 6, 7, 11], by change of background conditions [12, 13], or by other factors. If the influence of the different factors cannot be evaluated individually, it is difficult to evaluate subsystem or overall system measurement capabilities [13]. From these considerations, it is clear that a universal parameterization over many lidar technologies can be very useful as a design and assessment tool. In the absence of such an approach, system developers use rather complex analytical expressions and empirical formulas, which are often applicable


Conventional approach to analysis of lidar capabilities


Applied Physics B – Lasers and Optics

only for a very narrow range of parameters and specific experimental setups [1, 6, 15]. The purpose of the present work is to develop criteria that can be widely used to evaluate a broad range of lidar system capabilities for a variety of lidar remote sensing applications, and based on these criteria develop a methodology for selection of appropriate lidar system parameters for a specific application. To do this, it is necessary to choose a reference atmospheric state which will serve as a basis for comparison and evolve an appropriate parameterization scheme for expressing lidar system parameters in a generalized manner that can be applied to differing systems. Preliminary work in this direction has recently been performed [13, 14]. In [13], we introduced a dimensionless spatial filtering efficiency criterion to compare different lidar instruments under intense external background conditions and their stability against background radiation. This formalism allowed us to give recommendations on how to improve existing subsystems and to choose the best approach. In [14], we obtained a SNR decomposition and parameterization which was primarily used to study the performance and design of DIAL (Differential Absorption Lidar) and lidars with remote topographic targets. As the present paper is a logical continuation of our paper [14] mentioned earlier, it would be useful to underline significant and new scientific results obtained in the present paper. Based on examination and illustration of the conventional approach to analysis of lidar capabilities, a reference test object as a key element for dimensionless parameterization of lidar is introduced and explained in the present paper. A decomposition of the total SNR into five dimensionless parameters representing the transmitter and receiver conditions, background noise, target efficiency, atmospheric operation conditions, and a scale length is carried out. The nature of a novel approach to general lidar assessment and interrelations between introduced lidar parameters are explained in detail and schematically illustrated. A newly introduced Qx-parameter of lidar including backscatter magnitudes and simply describing the backscatter efficiency of an arbitrary species to the molecular reference is considered and discussed in detail. The modified lidar system parameter (Vequ ), which now includes the background influence, is introduced and its generalized role is analyzed. The influence of the generalized dimensionless system parameters (responsible for impact of the backscatter efficiency, lidar instrument performance, sky background illumination, and the optical ‘weather’ along the path) on the lidar operation range is studied and illustrated. Finally, estimations of a minimum detectable value of the Qx-parameter and a minimum detectable gas concentration for a general backscatter lidar scheme are carried out. Therefore, in the present paper, we extend the formalisms developed in the previous paper [14] to include all standard lidar configurations and develop the necessary tools to study their capabilities. The main advantage of our new approach is that it provides generalized, uniform, and objective criteria for the evaluation and comparison over a broad range of lidar types (aerosol, Raman, DIAL), operating on different targets (backscatter or topographic), and it can be used within the lidar community to compare different lidars.


Illustration of the reference test-object introduction


Parameterization of the lidar signal-to-noise ratio


Molecular atmosphere as measurement reference

We propose that the most useful reference atmosphere is the standard molecular atmosphere (see Fig. 2), whose parameters are well characterized. In particular, at λref ≡ λr0 = 0.55 µm, the extinction coefficient of the molecular atmosphere under standard conditions and at sea level is α0 = 0.0116 km−1 and the backscattering indicatrix of the molecular atmosphere is i ϕ0 (γ = π ) = 3/8π . Using this reference has several advantages. 1. Comparison of lidar returns under a common reference in the absence of atmospheric variability provides the most direct comparison of lidar capability. 2. Comparison of lidar returns from an arbitrary atmospheric target normalized to the reference molecular atmosphere gives the most direct estimate of the measurement sensitivity. 3. Comparison of the reference signal to those signals generated under different atmospheric states (high aerosol loading, etc.) along the lidar path provides a direct measure of the effects of atmospheric transmission on the measurement capabilities. As a reference signal we choose an echo signal Ps0 obtained from an elastic lidar received from a reference range R0 (this range can be thought of as a natural length scale, which may or may not have any particular significance in the system in question) at λ0 = 550 nm under standard molecular atmosphere conditions. In this case, Ps0 (λ0 , R0 ) =

1 cτp βπ0 (λ0 , α0 )Ar R0−2 P0 (λ0 )ξ (λ0 )T02 (λ0 , R0 ), 2



et al. Development of a SNR parameterization scheme for general lidar assessment

where c is the light velocity, τp the laser pulse duration, βπ0 the backscattering coefficient for the molecular atmosphere (βπ0 = (3/8π )α0 ), R0 the reference range, P0 the transmitted pulse power, ξ the optical efficiency of the lidar, and the path transmittance T0 (λ0 , R0 ) = exp[−α0 (λ0 )R0 ]. 2.2

Decomposition of lidar signal-to-noise ratio

Since the total SNR is not useful in comparing different lidar classes, we focus on decomposing the noise into several factors, each factor being connected to a particular aspect of the lidar system and/or the measurement medium. To begin, the power of the received echo signal has the following form: PsX (λL , λX , R) = K X (λL , λX , R)P0 (λL )ξ (λX )Tfw (λL , R)Tbw (λX , R),


where the subscript symbol X defines the following lidar types. 1. X = BS for backscattering lidar. 2. X = Ram for Raman lidar. 3. X = Top for lidar with a topographic target. Here, the factor KX describes the cumulative spatial efficiency of the system, PsX is the echo-signal power received at wavelength λX for range R, Tfw is the atmospheric transmission in the forward direction at the laser wavelength λL , and Tbw is the atmospheric transmission of the atmosphere in the backward direction at the echo-signal wavelength λX . (Only for Raman systems is λL = λX .) To develop the parameterization we write the input signalto-noise ratio ψX in the form PsX PsX Ps0 −1 X ≡ = U , Pt Ps0 Pt0


where Pt is the photodetector threshold power (which is the power of the minimal detectable signal Psmin ), Pt0 the threshold power in the absence of sky background, and U the background factor describing the increase of the photodetector threshold due to the sky background. The photodetector threshold power is given as a product Pt = Pt0 U.


An introduction and description of the background factor U is given in Sect. 3.4 (see Appendix A3). 2.3

Echo-signal power normalized by reference signal for different lidars

To evaluate Eq. (2), we need to determine the effective ratio KEfX of an echo signal of any type of lidar PsX to the reference echo signal Ps0 . It is easy to see from Eq. (2) (see Appendix A1) that, in general,    Tfw (λ, R)Tbw (λX , R) KX PsX (λ, λX , R) = K EfX = . Ps0 (λ, λ0 , R0 ) K BS0 T02 (λ0 , R0 )


Taking into account Eqs. (2) and (5), the specific values of the PsX /Ps0 ratios for different lidars are as follows: X = BS molecular :    PsBSm T (λ, R) 2 −2 αm (λ) = r Ps0 α0 (λ0 ) T0 (λ0 , R0 )   −4  T (λ, R) 2 −2 λ = r . λ0 T0 (λ0 , R0 ) X = BS aerosol :    PsBSa T (λ, R) 2 −2 βπa (λ) + βπm (λ) = r . Ps0 βπ0 (λ0 ) T0 (λ0 , R0 ) X = Raman :    Tfw (λ, R)Tbw (λR , R) −2 PsRam βπR (λR ) = r . Ps0 βπ0 (λ0 ) T02 (λ0 , R0 )




X = Topographic : PsTop ρa Ar /π R 2 = Ps0 (1/2)cτp βπ0 Ar R0−2    2ρa /π cτp T (λ, R) 2 −2 = r . βπ0 (λ0 ) T0 (λ0 , R0 )


Here, r = R/R0 is the normalized lidar range and ρa the backscatter albedo of the topographic target. From analysis of the above expressions, each case can be factored in the form PsX = QxW 2r −2 , Ps0 where the explicit representations of Qx and W appear in Sects. 3.2 and 3.3. 3

Lidar parameter interpretation

Combining the explicit signal ratios obtained in Eq. (6) with the general decomposition of X in Eq. (3) shows that the signal-to-noise ratio can be decomposed into the following dimensionless parameterization : ψX = V QxW 2 U −1r −2 .


Thus, the S/N ratio at the lidar photodetector is given as a product of five independent dimensionless parameters, each of which follows from a different source. The block diagram in Fig. 3 shows the relationships between the normalized parameters and the components of the measuring system including the atmospheric state, the lidar transmitter/receiver, the background noise, and the reference medium. 3.1


In Eq. (7), V is defined as the ratio of the echosignal power Ps0 received from the reference range R0 for the reference atmosphere to the threshold power Pt0 in the absence of background noise:

Ps0 (5) V = Pt0 .



Applied Physics B – Lasers and Optics

is only necessary to know the receiving and transmitting subsystems parameters and the optical parameters of a standard molecular atmosphere. According to Eq. (9), due to wide variations of lidar instrument parameters for different lidar types (P0 , τp , Ar , ξ , R0 , noise-equivalent power (NEP), f), the value of the Vparameter can change dramatically from 10−10 to 107 . It is easy to see that large values of V lead to better operation performance of the lidar, but such values correspond to more expensive electro-optical components. 3.2


Block diagram of interrelations between the introduced lidar



While the V-parameter probes the effect of transmitter and receiver operation on a reference atmosphere, backscatter magnitudes are included in the Qx-parameter, which simply describes the backscatter efficiency of an arbitrary species relative to the molecular reference. For different types of lidar, the Qx-parameter can be written as follows: For molecular backscattering lidar:  −4 βπm (λ) αm (λ) λ Q BSm (λ, λ0 ) = . = = βπm (λ0 ) α0 (λ0 ) λ0

For aerosol backscattering lidar:    ba (λ)αa (λ) + bm (λ)αm (λ) βa (λ) + βm (λ)   1 2 2 2 = Pt0 = PtB=0 = ρout Pn2 /Pq2 ∼ Pq 1 + 1 + 4/ρout Pq Q BS (λ, λ0 ) = = ρout βm (λ0 ) bm (λ)α0 (λ0 ) 2 ba (λ)αa (λ) + (3/8π )αm (λ) for Pn  Pq ; . = (3/8π )α0 (λ0 ) Pq = 2hc f F/ηλ is the quantum noise power and ρ out is the signal-to-noise ratio at the photodetector output taken from For Raman lidar: Ng σR (λR ) βπR (λR ) Eqs. (A3) and (A6). Q Ram (λR , λ0 ) = = . βπ0 (λ0 ) Nm σm (λ0 ) When comparing different types of lidars it is useful to identify the V-parameter as a universal parameter describing For lidar with a topographic target: the energetic potential of the lidar instrument: 2ρa /π cτp ρa /π R Q Top (λ, λ0 ) = = . Ps0 K EfX (λ, R)P0 (λ)ξ (λ)Tfw (λ, R)Tbw (λ, R) β (λ ) βπ0 (λ0 ) π0 0 V = = . (9) 2 Pt0 ρout Pq Here ba = βπa /αa and bm = βπm /αm are the lidar ratios for To illustrate this, the V-parameters and the subsystem parame- aerosol and molecular atmospheres and R = cτp /2 is the ters for lidars at CUNY (City College of New York) are given potentially achievable range resolution. Sample calculation results of the values of Qx for difin Table 1. Here, η is the quantum efficiency of the photodetector, ferent types of lidars and for different intervening atmo f the transmission band of the receiving subsystem, F the spheres for a range of Qx-parameter magnitudes are given in excess noise factor of the photodetector, and Psmin the power Table 2. When estimating the Qx-parameter for a lidar with a toof the minimum detectable signal. As is seen from Eqs. (2), (5), and (9), to calculate the pographic target, we took the surface albedo to be ρa = 0.1 V-parameter for existing or proposed lidar instruments it and a laser pulse duration of τp = 10−8 s is assumed.


Lidar CUNY-Lab


Ar (m2 )


R0 (km)

λ0 (nm)

E0 (J)

τp (ns)


f (MHz)

Sa (A/W)


NEP (W/Hz1/2 )

Ps0 (α0 , R0 ) (W)

Psmin (W)


0.2 0.2 0.2 0.1 0.1 0.1

0.3 0.3 0.3 0.4 0.4 0.4

1 1 1 1 1 1

1064 532 355 1064 532 355

0.50 0.25 0.20 0.65 0.30 0.10

3.5 3.5 3.0 7.0 7.0 7.0

0.41 0.18 0.19 0.31 0.18 0.19

250 250 250 250 250 250

34.0 3.8 × 104 3.7 × 105 25.0 3.8 × 104 3.7 × 105

5.0 1.3 1.3 5.0 1.3 1.3

1.5 × 10−14 3.7 × 10−17 3.7 × 10−17 1.5 × 10−14 3.7 × 10−17 3.7 × 10−17

2.5 × 10−5 1.2 × 10−5 1.0 × 10−5 1.6 × 10−5 7.4 × 10−6 2.5 × 10−6

1.3 × 10−9 1.3 × 10−9 1.8 × 10−9 1.3 × 10−9 1.3 × 10−9 1.8 × 10−9

1.8 × 104 9.1 × 103 5.4 × 103 1.3 × 104 5.5 × 103 1.4 × 103

TABLE 1 Main performance of CUNY lidar instruments and their V-parameters

et al. Development of a SNR parameterization scheme for general lidar assessment


Elastic lidar

5 Raman lidar

Lidar type

Topographic lidar

Aerosol atmosphere

Molecular atmosphere


H2 O

Range of Qx-parameter

102 to 104

100 to 102

10−1 to 101

10−5 to 10−3

10−7 to 10−5

TABLE 2 Range of Qx-parameter



The third factor in Eq. (7) is a normalized atmospheric component W2 that is determined by the transparency ratio of the atmosphere state and the standard molecular atmosphere. The normalized transparency along the sounding path is defined differently for elastic and Raman lidars:  R T (λL , R) WBS = [αa (R  , λL ) = exp − T0 (λL , R0 ) 0

+ αm (R  , λL )]R  dR  − α0 (λL )R0 , √

Tfw (λ, R)Tbw (λR , R) T0 (λ0 , R0 )  1 R = exp − [αa (R  , λ) + αm (R  , λ) + αa (R  , λR ) 2 0


αm (R  , λ0 ) dR .




WRam =

+ αm (R  , λR )] dR +

W is a direct measure of the degradation of the lidar performance due to the attenuation along the beam path, which increases both as a function of the extinction coefficient of the aerosol and the range parameter r, as illustrated in Fig. 4. When sounding scattering media, the W-parameter characterizes the optical ‘weather’ along the path. For absorbing media, the W-parameter includes the absorber concentrations of the trace constituents under investigation, which were considered in [14].



For homogeneous atmospheres, the W-parameter can be expressed as     αa + αm r −1 WBS = exp −α0 R0 α0     αa + αm r −1 . (12) = exp −τ0 α0

FIGURE 4 Estimations of W-parameter values for different optical ‘weather’ conditions

Frequently for aerosol lidars with moderate pulse energies, or in Raman lidar applications, the most important factor that limits the detection of weak signals in daytime is the background sky radiation. The fourth factor of Eq. (7) describes the influence of the background clutter on the sensitivity threshold of the lidar photodetector. The U-parameter is defined as the ratio of the photodetector threshold powers Pt and Pt0 defined in the presence and absence of the background noise, respectively (see Appendix A3):      2 (Pb /Pq ) + Pn2 /Pq2 1 + 4/ρout 1 + Pt  U≡ , (13) =    Pt0 1 + 1 + 4/ρ 2 P 2 /P 2 out



where Pb is the background power, Pq = 2hc f F/ηλ characterizes the quantum limit of the detector sensitivity or the threshold power due to the shot (quantum) noise, Pn = √ NEP f is the power of the internal noise of the photodetector referred to the photodetector input, and ρ out is the output signal-to-noise ratio. The U-parameter is the excess noise factor quantifying the influence of the background noise. For lidar applications, the receiving subsystem bandwidth that determines Pq is very broad, so as a rule Pn  Pq , and Eq. (13) can be written as  

4 Pb 1 U≈ . 1+ 1+ 2 2 ρout Pq From Fig. 5, which illustrates the background factor (U) behavior, the smaller the internal noise of the photodetector, the greater the influence of the external background. When the background power increases, the threshold sensitivity increases resulting in a decreased operation range, increased minimum detectable concentration, etc. While the general expression for obtaining the Uparameter, which measures the decrease in S/N due to increased noise components, is derived in the appendix A3, under conditions most relevant to us (Pb ≈ Pq  Pn ) the Uparameter becomes a universal function of background to shot noise. To estimate values of both the threshold power Pt and


Applied Physics B – Lasers and Optics

we can further generalize the V-parameter to absorb the Uparameter, resulting in a final equivalent parameter √ V N . (14) Vequ = U 4

FIGURE 5 Illustration of the photodetector threshold sensitivity worsened by background clutter

the U-parameter, we assumed the following range of sky background brightness and lidar receiver parameters (Table 3). It should be pointed out that this treatment is identical for all lidar types. 3.5

Finally, the fifth term in Eq. (7) is the normalized range factor r = R/R0 , which compares the current range R to the reference range R0 . This parameter should be interpreted only as a scale parameter, which must be known when intercomparing different lidars and should be fixed to a universal value for intercomparisons. 3.6

Equivalent V-parameter

The V-parameter was only defined for single-shot operation. However, it is essential to account for shot averaging. When accumulating N lidar signals with repetition frequency fmod over a time of observation Tobs , an in√ crease in the signal-to-noise ratio equal to N occurs, where N = f mod Tobs . In this case, instead of the V-parameter defined ∗ in Eq. (9), the equivalent system parameter Vequ can be defined as ∗ Vequ =V


f mod Tobs .

In this way, potential capabilities of different lidars (including, for example, micro-lidars [16], which usually have a very small energy per pulse and a high repetition frequency) can be correctly compared. Finally, since increased background noise is equivalent to decreasing the V-parameter,

Spectral brightness, Bλ (W/m m2 sterad) 106 to 3 × 108

Telescope area, Ar (m2 )

The determination of lidar criteria for the detection of atmospheric components is based on the condition that the received echo signals exceed the internal and external noise (i.e. SNR > 1). In Sects. 2 and 3, we used the molecular atmosphere as a reference medium and decomposed the resultant SNR into a five-parameter expression. In this section, the parameterization formalism will be applied to estimate lidar operation range independent of the particular design principles used. This formalism can then be used to predict the potential performance of lidar for remote sensing of different atmospheric objects using different lidar technologies under very different conditions of optical ‘weather’ and sky background. 4.1


Optical ξ

Field of view,  (rad)

Maximum operation range of lidar for horizontal sounding

Let us first consider horizontal sounding in the lower layers of the atmosphere. From Eq. (11), it is easy to determine the normalized operation range assuming that ψ = 1. Using the S/N parameterization in Eq. (11), the maximum operating range reduces to the solution of a nonlinear equation:  (15) rmax = V QxW (rmax )U −1 = f (rmax ), which can be solved by iteration. For the simplest case of sounding in a molecular atmosphere, where Qx = 1 and the W-parameter is determined from Eqs. (10) and (15), this becomes much simpler:     α rmax = V exp −2α0 R0 rmax − 1 U −1 . (16) α0 Finally, if the background radiation is low (U = 1) and atmospheric optical densities α0 R < 1, the maximum operation range reduces to √ (17) rmax = V . From here, a physical interpretation of the system parameter V becomes clear: its numerical value defines a square of a normalized operation range of a lidar at horizontal sounding at standard molecular atmosphere conditions (with α0 R < 1) in the absence of background noise.

Interference filter bandwidth λ (nm)

0.1 0.3 10−3 1 From here: the background power Pb = 3 × 10−11 to 10−8 W

TABLE 3 Sky background brightness and lidar receiver parameters

Application of the generalized system parameters for estimations of potential capabilities of lidars

Quantum efficiency, efficiency, η 0.22

Frequency bandwidth, f (Hz)

2.5 × 108 Shot-noise power Pq = 10−9 W


et al. Development of a SNR parameterization scheme for general lidar assessment

Normalized operation range rmax = Rmax /R0 of a backscatter lidar for horizontal sounding of a molecular atmosphere as a function of system V-parameter for different reference ranges R0


In the general case, when sky background effects are significant and signal averaging is used to improve the SNR, one should use the equivalent V-parameter from Eqs. (14) and (17). Results for the normalized operation range of a backscattering lidar at horizontal sounding (height h ∼ 0) of a molecular atmosphere when neglecting the background noise (U = 1) are given in Fig. 6. It is important to note that both r and the V-parameter depend on the choice of R0 , so that R0 must be included when specifying a particular lidar system. 4.2

Operational range of a real-world horizontal lidar

For horizontal sounding, allowing for a particularly easy representation of the intervening weather, the scattering sensitivity of a particular target (aerosol particulates, molecular, etc.) at range r is totally specified by its Qx-parameter, while the intervening medium (weather) is quantified by the W-parameter. The dependence of the maximum operation range rmax on the Qx-parameter for various optical conditions of the atmosphere obtained by numerical solution of


Eq. (15) is shown in Fig. 7. The range of the Qx-parameter used in our calculations corresponds to the reflected echo signals from a topographic target, backscatter from dense haze, backscatter from weak haze, molecular scattering, and finally Raman scattering (nitrogen at a concentration of 78% and water vapor at concentrations of 1% and 1 ppm (Qx = 104 , 102 , 101 , 100 , 10−4 , 10−6 , and 10−10 , accordingly). If the target is an extended body such as haze, it should be emphasized that the haze below the range r is contained in the W-parameter. Curves of the maximum operation range versus optical weather conditions for different Qx-parameters are illustrated in Fig. 8. Using the curves in Figs. 7 and 8 it is possible to estimate the operating range of lidars over many different target and weather scenarios. In the above calculations, background noise was ignored, but it is clear that increased background noise will degrade lidar performance. Simulations of the background influence carried out on the basis of Eq. (15) determine the operation range of a lidar as a function of the normalized background power Pb /Pq (i.e. for various values of the Qx-parameter) and are presented in Fig. 9. From Fig. 9, the background noise can considerably decrease the operation range of any lidar. Furthermore, the effect of background noise is enhanced when the target parameter Qx is small. In particular, for Raman lidar (Qx = 10−6 and 10−10 ) the operation range increased by a factor of five from daylight to night-time conditions. These curves illustrate that daylight is a severe restriction for Raman lidar performance. In the previous discussion, background noise was treated separately. However, since the case of most interest in assessing lidar performance is to consider the highest background noise and a fixed data-processing (shot-averaging) scheme, it is often more useful to incorporate the background noise and signal processing into the equivalent V-parameter according to Eq. (14). In this reduced representation, the dependence of the operating range resulting from the equivalent system parameter Vequ = V N 1/2 U −1 is shown in Fig. 10. Note that the influence of the background factor is to reduce the V-parameter, while averaging of echo signals will increase the V-parameter.

FIGURE 7 Maximum operation range of lidar as a function of Qx-parameter when sounding under conditions of a molecular atmosphere, light and dense haze, and a fog ((αa + αm )/α0 = 100 , 101 , 102 , and 103 at λ = 550 nm, accordingly) for the V-parameter values V = 104 (left) and 107 (right) in absence of background clutter. R0 = 1 km


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FIGURE 8 Maximum operation range of lidar versus optical weather conditions for various values of the Qx-parameter in the absence of background clutter. R0 = 1 km

FIGURE 9 Lidar operation range versus normalized background power Pb /Pq for different Qx- and V-parameters: optical weather condition is set at (αa + αm )/α0 = 10. R0 = 1 km

Operation range versus equivalent Vparameter for different values of the Qx-parameter at (αa + αm )/α0 = 10 (a) and 100 (b). R0 = 1 km


Therefore, the results of calculations presented in Fig. 10 allow quantitative estimation of the influence of background and signal averaging tradeoffs. It is also important to note that the equivalent parameter Vequ can be used directly instead of the V-parameter when analyzing the achievable operation range rmax . For example, using Fig. 10 it is possible to determine the number of averaged shots necessary to achieve the desired operational range. 4.3

Vertical sounding

Up to this point, only horizontal lidar sensing was considered, allowing for a particularly easy representation of the intervening weather. In the case of vertical sounding,

the reference medium is no longer constant and the range dependence of the reference medium needs to be included. For vertical sounding, the height-dependent changes of the molecular scattering coefficient α0 will significantly impact the maximum range Zmax due to the decrease in molecular (aerosol) concentration. For a molecular reference, the gas concentration can be estimated from the equilibrium model as N0 (Z ) = N0 (0)exp(−Z /Z a ) [11], where Z is the vertical height, Z a ≈ 8 km is the atmospheric scale height, and N0 (0) is the standard atmospheric air concentration at the ground level. Then, Eq. (15) can be written as  z max =

V Q BSmol W02 (Z max ), U



et al. Development of a SNR parameterization scheme for general lidar assessment

FIGURE 11 W0 -parameter for the backscatter lidar for vertical sounding versus height for different values of the initial height Z0

where the term corresponding to atmospheric transmittance is  Z  2 α(z) dz W0 (Z ) = exp −2 Z0

   z dz exp − Za Z0    Z0 = exp −2Z a α0 (0)exp − Za    Z0 , × 1 − exp (1 − z) Za

 = exp −2α0 (0)



where Z0 is the initial height (similar to R0 ). The illustration of the dependence of W02 on height is given in Fig. 11. From Fig. 11, it is clear that due to the decreasing gas concentration with height, the attenuation effects are much less for a homogeneous layer and will therefore lead to higher operating ranges for the same lidar V-parameter. Explicitly, the maximum relative operation range is derived as before from the transcendental equation z max = z0

     V Z0 Z0 1 − exp (1 − z max ) . exp −2Z a α0 (0)exp − U Za Za

(20) Some values of the relative height Z max = Z max /Z 0 versus the V-parameter (supposing U = 1) corresponding to the case of the vertical sounding at molecular atmosphere conditions at λ = 0.55 µm are presented in Table 4. While we have used the condition S/N = 1 to determine the maximum threshold range, it is quite clear that, in general, it is important to have an echo -signal with S/N  1 to obtain


Minimum achievable value of the normalized parameter Q xnorm = QxU/V N 1/2 as a function of operation range for different optical weather conditions. R0 = 1 km FIGURE 12

both an accurate concentration measurement as well as to calibrate lidars to a reference molecular signal through the aerosol boundary layer. The addition of these effects is accomplished by simply redefining the V-parameter to account for these 2 effects so that Veff = ψdes V /Taer , where des represents the desired input S/N ratio and Taer represents the transmission of the aerosol component (treated as a constant) and can be determined from sun-photometry measurements. Of course, the use of sun-photometry optical thickness is overly conservative and overestimates the attenuation, thus underestimating the operating range. This can in principle be improved if a vertical model of aerosol loading is used but, for long-range lidars, the improvement is negligible. 5

Estimation of minimum detectable value of Qx-parameter and minimum detectable gas concentration

The target backscatter efficiency is directly related to the value of Qx, so determination of the minimum value of Qx for a given range r determines the minimum detectable target. To determine the lower limit of lidar sensitivity, we equate the signal-to-noise ratio in Eq. (7) to unity, which results in V QxW 2 U −1r −2 = Qxnorm W 2r −2 = 1, where, for generality, a normalized Q xnorm ≡ Qx/Vequ = QxU/V N 1/2 is used. The dependences of the normalized parameter Qxnorm as a function of the operation range for different optical weather conditions are shown in Fig. 12. From these curves, we can determine the minimum measurable value of the equivalent parameter Qxnorm and hence Qx. Given the relevant backscatter cross section of the target and the molecular reference, the lower limit of Qx allows us

λ = 550 nm, α 0 = 0.0116 km−1 , Qx = 1, U = 1 System parameter, V, relative units Maximum relative operation range, zmax , relative units

102 9.8

103 29.9

104 92.1

105 290.6

106 918.9

107 2906.0

TABLE 4 Maximum relative operation range of a backscatter lidar for vertical sounding of the molecular atmosphere


Applied Physics B – Lasers and Optics Sky brightness, Bλ = 3 × 10−1 W/nm m2 sterad

Sky brightness, Bλ = 0 Receiver bandwidth, f (MHz)

CUNY-Lab lidar’s V-parameter

Qxmin (R = 1 km)

NH2 O min (R = 1 km)

Vequ min = V/Umax

Qxmin (R = 1 km)

NH2 O min (R = 1 km)

20 250

6.8 × 104 5.4 × 103

5 × 10−5 6 × 10−4

0.037 0.46

4.3 × 103 3.6 × 102

7 × 10−4 9 × 10−3

0.55 7.0

TABLE 5 Estimations of CUNY-Lab lidar water vapor mixing ratio NHrel2 O min capabilities

Minimum achievable value of parameter Qx and minimum detected gas concenrel tration Ngas min as functions of V-parameter for different values of the normalized operation range rmax . Optical weather condition is set at (αa + αm )/α0 = 10. R0 = 1 km FIGURE 13

Minimum achievable value of parameter Qxmin and minimum detected gas concentration Ngasmin as functions of operation range for different values of V-parameter. Optical weather condition is set at (αa + αm )/α0 = 10. R0 = 1 km


to calculate the minimum retrieval volume concentration as Ngas min σm (λ0 ) rel . Ngas = Q R min min ≡ Nm σR (λR ) Using Raman lidar as an example, we estimate the minimum detected concentration of water vapor at 1 km from the CUNY-Lab Raman lidar for a wavelength λL = 355 nm under conditions of a light haze. Given that σR = Raman scattering cross section at λR (cm2 sterad−1 ), σ0 = molecular scattering cross section at λ0 = 550 nm, cm2 , estimates of spectral brightness of the sky Bλ , the threshold power of the receiver, etc., we can determine the relevant parameters and capabilities of the CUNY system as follows (Table 5). As illustrated in Table 5, under intense background clutter conditions, the minimum detected concentration of water vapor increases from the mixing ratio M = 0.037 to M = 0.55 for the lidar receiver bandwidth f = 20 MHz. Using these values, we arrive at the design curves Qxmin (r) rel and Ngas min (r ) in Fig. 13. As an example, to calculate the number of shots necessary to detect 1% watervapor concentration in the lower atmosphere at R = rel 1 km, we intersect the Ngas min = 0.01 level with a curve

1 km), giving corresponding to r = 1 (r = R/R √ 0 , R0 = shots Vequ = 3 × 105 . Since Vequ = V N /U , Nmin = [Vequ × Umax /V ]2 = [3 × 105 /4.3 × 103 ]2 = 4867. For a pulse train with frequency f rep = 50 Hz, the accumulation time necessary is tacc = N shots / f rep = 97 s. Illustrations of the minimum achievable values of Qxmin rel and Ngas min versus V- and r-parameters are shown in Figs. 13 and 14. They can be used for the determination of the minimum necessary value of the V-parameter for a given range r (Fig. 13), or for the determination of the maximum operating range to detect the desired species concentration for a given V- (or Vequ -) parameter (Fig. 14).



In this paper, a general method for the validation and performance comparison of various lidar systems is proposed. The method is based on introducing a universal atmospheric reference medium and a decomposition of the total SNR into five dimensionless parameters representing the transmitter and receiver conditions, background noise,


et al. Development of a SNR parameterization scheme for general lidar assessment


target efficiency, atmospheric operation conditions, and a scale length. The main advantage of this approach is that it provides generalized, uniform, and objective criteria for the evaluation of a broad range of lidar types and systems (aerosol, Raman, DIAL), operating on different targets (backscatter or topographic) and it can be used within the lidar community to compare different lidars.

For a lidar with a topographic target:


A3 U-parameter of lidar

Q Top (λ, λ0 ) =

2ρa /π cτp ρa /π R = . βπ0 (λ0 ) βπ0 (λ0 )


Here, ba = βπa /αa and bm = βπm /αm are the lidar ratios for aerosol and molecular atmospheres and R = cτp /2 is the range resolution.

A1 Spatial efficiency of different types of lidar

According to Eq. (2), the spatial efficiency factors KX (R) for different lidars can be described as follows: For a reference lidar signal: 1 3 cτp α0 (λ0 )Ar R0−2 . K 0 = cτp βπ0 (λ0 )Ar R0−2 = 2 16π


For a molecular elastic lidar: K BSm

1 3 cτp αm (λ)Ar R −2 . (A1b) = cτp βπm (λ)Ar R −2 = 2 16π

For an aerosol elastic lidar: K BSa =

1 cτp [βπa (λ) + βπm (λ)]Ar R −2 . 2


For a Raman lidar: K Ram =

1 cτp βπR (λR )Ar R −2 . 2


For a lidar with a topographic target: K Top =

ρa Ar ρa = r . 2 πR π


Here, the product 1/2cτp β πa is the backscattered power over the resolution interval, c the speed of light, and Ar /R 2 = r the receiving solid angle of the receiving aperture at range R; βπa and βπm are respectively the aerosol and molecular backscattering coefficients. A2 Qx-parameter of lidar

For different types of lidar, the Qx-parameter can be written as follows: For a molecular backscattering lidar:  −4 αm (λ) λ βπm (λ) = = . (A2a) Q BSm (λ, λ0 ) = βπm (λ0 ) α0 (λ0 ) λ0 For an aerosol backscattering lidar: ba (λ)αa (λ) + bm (λ)αm (λ) βa (λ) + βm (λ) = Q BS (λ, λ0 ) = βm (λ0 ) bm (λ)α0 (λ0 ) =

ba (λ)αa (λ) + (3/8π )αm (λ) . (3/8π )α0 (λ0 )


The SNR at the output of the generalized subsystem of direct photodetection with internal gain and external and internal noise is represented as follows [1, 2, 10, 11]: ρout = 

Ng σR (λR ) βπR (λR ) = . βπ0 (λ0 ) Nm σm (λ0 )


(2hc f F/ηλ)(Ps + Pb ) + NEP2 f



where h is Planck’s constant, Ps the echo-signal power reaching the photodetector, Pb the power of the background radiation, η the photodetector quantum efficiency, F the excessnoise factor of the photodetector, f the receiver electrical bandwidth, and NEP the noise-equivalent power not depending on signal and background radiation and corresponding to the bandwidth of 1 Hz. The background radiation power can be written as [1, 11] Pb = Bλ Ar  λξ0 , where Bλ is the spectral radiance of the zone of the sky falling in the receiver field of view, Ar the effective area of the optical receiving system,  the field-of-view solid angle, λ the spectral width of the optical band-pass filter in the receiving system, and ξ 0 the optical receiver transmittivity. For a given signal-to-noise ratio ρout at the photodetector output, from Eq. (A3) we can obtain the relative echo-signal power ψq = Ps /Pq normalized to the quantum noise power Pq = 2hc f F/ηλ that characterize the quantum limit of the detector sensitivity or the threshold power at the input determined by the shot (quantum) noise: 2 = ρout


Ps2 (2hc f F/ηλ)(Ps + Pb ) + NEP2 f (Ps /Pq )2 Ps2 . = (Ps + Pb )/Pq + (Pn /Pq )2 Pq (Ps + Pb ) + NEP2 f

Then, we have 2 2 ψ2q − ρout ψq − ρout [Pb /Pq + (Pn /Pq )2 ] = 0,

where Pn = NEP f 1/2 is the power of the internal noise of the photodetector referred to the photodetector input. Therefore, the normalized limit of the sensitivity is

For a Raman lidar: Q Ram (λR , λ0 ) =


ψmin q

    Pb 1 2 4 Pn2  = ρout 1 + 1 + 2 + 2 2 Pq ρout Pq



Applied Physics B – Lasers and Optics

and the absolute threshold signal power Pt (or the power of the minimal detectable signal, Psmin ) is

     Pb 1 4 Pn2 min 2  Ps ≡ Pt = ρout Pq 1 + 1 + 2 + 2 . (A5) 2 Pq ρout Pq

ACKNOWLEDGEMENTS The authors acknowledge partial support of this work by grants from NOAA (No. NA17AE1625) and NASA (No. NCC-1-03009). We thank Prof. Adolfo Comeron for fruitful discussions of some parameterization scheme features.

Equation (A5) determines the threshold signal power at the input of the direct photodetection subsystem. From here, one can easily obtain particular cases corresponding to modes limited by background, internal, and quantum noise. For example, in the absence of background noise, we have     1 4 2 Pt0 = PtB=0 = ρout Pq 1 + 1 + 2 Pn2 Pq2 2 ρout

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2 Pq ≈ ρout

for Pn  Pq .



In other words, Pt is the noise power due to both the internal photodetector and the shot noise. During daytime, however, the threshold power is basically determined by background noise, which can be written as    1 2 4 Pb Pt ≈ ρout Pq 1 + 1 + 2 = Pt0 U, 2 ρout Pq 2 where Pt0 = ρout Pq and the parameter U is a measure of the increase in threshold power due to increased background noise signal,    1 4 Pb U≈ . (A7) 1+ 1+ 2 2 ρout Pq

Finally, we mention that in the case of heterodyne detection the 2 Pq /2 [1, 2, 6, 10]. threshold power is determined as Pt = ρout


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