Abstract

We report new modeling and error reduction methods for differential-absorption optical-depth (DAOD) measurements of atmospheric constituents using direct-detection integrated-path differential-absorption lidars. Errors from laser frequency noise are quantified in terms of the line center fluctuation and spectral line shape of the laser pulses, revealing relationships verified experimentally. A significant DAOD bias is removed by introducing a correction factor. Errors from surface height and reflectance variations can be reduced to tolerable levels by incorporating altimetry knowledge and “log after averaging”, or by pointing the laser and receiver to a fixed surface spot during each wavelength cycle to shorten the time of “averaging before log”.

© 2012 OSA

1. Introduction

1.1 Overview

Lidar remote sensing techniques are powerful tools for global measurement of atmospheric constituents and parameters [1, 2]. The required precision and accuracy of such quantitative measurements have been increased to unprecedented levels for future airborne and space missions [3, 4]. To meet such stringent requirements, nadir-viewing, direct-detection, and pulsed integrated-path differential-absorption (IPDA) lidar techniques are being developed to measure the differential absorption optical depths (DAODs) of the target species in the column to the surface [5, 6]. From the DAOD and altimetry measurements and other ancillary data of the atmosphere (temperature, pressure, water vapor, etc.), the column number densities and dry mixing ratios of the target species can be retrieved [7, 8].

This paper addresses the following common issues with such DAOD measurements that have not been well addressed in previous publications: 1) error contribution from the optical frequency noise of a pulsed laser; 2) incorporation of a bias correction factor to reduce the averaging time for “log after averaging” (as named in [9]), and thereby minimize the impact of Etalon and atmospheric turbulence effects; 3) error contribution from the surface reflectance variations [10] when beam spots shift on the surface; 4) methods to reduce errors due to variations of surface height and reflectance; 5) estimation of the measurement precision of the transmitted laser pulse energy.

We present a statistical model to address these issues, to provide insight and estimation of DAOD measurement errors and to guide lidar system design and development. The model addresses both types of DAOD measurement errors: the relative random error (RRE) and the relative systematic error (RSE) (i.e., relative bias). This paper is organized as follows. After introducing several representative IPDA lidar examples in the next subsection, we quantify the error contributions from the laser frequency noise in section 2. In section 3, additional detection noise sources and a bias correction factor are incorporated into the model. A numerical example of the model is presented in section 4. The impact of surface reflectance variation is evaluated in section 5. A new method to reduce errors due to surface height variation is described in section 6. More considerations for the lidar system are discussed in section 7. Some supporting details are provided in the Appendix.

1.2 IPDA lidar examples

The Active Sensing of CO2 Emissions over Nights, Days, and Seasons (ASCENDS) mission [3] has been planned by NASA to measure the global distribution of carbon dioxide (CO2) mixing ratios (~390 ppm in average) to ~1 ppm precision. A candidate IPDA lidar measurement approach for ASCENDS, being developed at NASA Goddard, allows simultaneous measurement of CO2 and surface height in the same path [6, 7]. As shown in Fig. 1 , a pulsed laser is wavelength-stepped across a single CO2 line at 1572.335 nm [11] to measure the DAOD. To make the measurements uniformly sensitive to concentrations in the lower troposphere, the online wavelengths (ν2 to ν7) are placed at the sides of the line where the absorption is mostly from CO2 molecules in the lower troposphere [12]. The two-way transmittance of the CO2 line from a 400-km orbit (Fig. 1, right) has a narrow linewidth (~3.5 GHz) and steep slopes on the sides, making the measurements sensitive to laser frequency uncertainties. The offline points (ν1 and ν8) are placed in the adjacent window regions where the extremely low CO2 absorption is essentially wavelength independent. To meet the mission end goal, the DAOD of the atmospheric CO2 line needs to be measured with a RRE < 0.1% at multiple wavelengths across the absorption lines, and the variations of RSE need to be kept to a small fraction of the RRE. To reduce the RSE and RRE of the DAOD measurements, the detected photon count for each wavelength is averaged across multiple laser pulses before taking the logarithm of the on/off line ratio (i.e., “log after averaging”). This lidar approach will be used as a concrete example throughout this paper.

 

Fig. 1 The laser transmitter provides the wavelength-stepped pulse train (left) to repeatedly measure at 8 points across the 1572.335 nm CO2 absorption line (right).

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The partial RRE contribution from the laser frequency noise needs to be < 0.03%, to allow more margins for other error sources. In this paper, we show that the standard deviation of the effective laser frequency noise at online points for CO2 needs to be < 0.23 MHz to meet this target. To satisfy such stringent requirements, laser pulses at each fixed wavelength are generally carved by amplitude modulating a frequency stabilized continuous-wave (CW) laser [13] (Fig. 1, left). The 1-μs wide pulses need to be at least ~100 μs apart, to completely clear the bottom ~15 km of the atmosphere, which eliminates crosstalk from cloud scattering. Chirp-free amplitude modulation [14] is used so that the laser pulses retain the instantaneous frequency of the CW laser.

A similar CO2 sounder space mission called A-SCOPE [4] has also been formulated by the European Space Agency. The A-SCOPE transmitter generates a pair of pulses, one at an online and the other at an offline wavelength, at an overall pulse pair repetition rate of 50 Hz, with a pulse energy of ~50 mJ. Comprehensive lidar sensitivity and spectroscopic analyses for the A-SCOPE instrument operating at either 1.57 μm or 2.05 μm CO2 bands have been recently published [5, 8, 15]. A similar approach has been adopted for a planned methane sounder space mission dubbed MERLIN [16].

2. Measurement errors from laser frequency noise

Throughout this paper, x¯ represents the ensemble average, σ(x)the standard deviation, and σ2(x)the variance of x.

2.1 Treatment of laser frequency noise

The instantaneous frequency ν(t) of a single frequency CW laser can be modeled as an ergodic random process and the frequency noise is defined as δν(t)ν(t)ν¯. The electric field of a pulse carved from the CW laser can be represented by the analytic signal of the CW laser field multiplied by the real single pulse amplitude a(tt0) centered at time t0, as given below

u(t)=a(tt0)exp(j[2πν¯t+ϕ(t)]).
Here the instantaneous frequency is ν(t)=v¯+(1/2π)dϕ/dt. The single-sided energy spectral density (ESD) of u(t), denoted as L(νF) as a function of the Fourier frequency νF, represents a short-term spectrum of the CW laser within a pulse duration Δt. The line-center frequency νc(t0) of L(νF)is linked to the instantaneous frequency ν(t) and a(t) [17] as
νc(t0)0νFLN(νF)dνF=ν(t)h(tt0)dt.
Here LN(νF)L(νF)/0L(νF)dνFand h(t)a2(t)/a2(t)dt are the normalized ESD of u(t) and normalized pulse intensity envelope, respectively. As to be seen in this subsection and in Appendix A, the laser frequency noise contributes to measurement errors through two complimentary factors, the line shape L(νF)and the fluctuation of the line-center frequency νc(t0). Both factors are jointly determined by the frequency noise and pulse amplitude a(t).

As shown in Eq. (2), the fluctuation of the line-center frequencyδν1(t0)νc(t0)ν¯ is an average of the CW laser frequency noise δν(t) within a pulse duration Δt weighted by h(tt0). For a top-hat pulse shape, δν1(t0) is simply a running average of δν(t) within Δt. The variance σ2(δν1) of δν1(t0) can be obtained by integrating its power spectral density (PSD), which in turn can be derived from Sδν(f), the PSD of δν(t). Since δν1(t0) is a correlation between δν(t) and h(t) as seen in Eq. (2), its PSD Sδν1(f) is related to Sδν(f) by

Sδν1(f)=Sδν(f)|H(f)|2,
where H(f) is the Fourier transform of h(t).As the ESD of h(t), |H(f)|2 serves as a window function to filter the frequency noise represented by Sδν(f). For a top-hat h(t)=rect(t/Δt)/Δt, the corresponding window function becomes |H(f)|2=sinc2(fΔt). Fast frequency noise components above its effective bandwidth 1/Δt are quickly averaged out as noise frequency f increases. As shown in subsection 2.5, this has been verified experimentally with the seed laser described in [13].

Formulation linking L(νF) to the frequency noise Sδν(f) and the pulse amplitude a(t) is further derived and discussed in Appendix A. Interestingly, the remaining frequency noise that is filtered out by |H(f)|2 contributes to the bandwidth of the line shape L(νF).

2.2 Single pulse formulation

The energy absorption of a single laser pulse through a thin layer of target molecules is given by

dL(νF,l)=L(νF,l)σ0(νF,l)N(l)dl.
Here σ0(νF,l) is the absorption cross-section of the molecules at Fourier frequency νF and path length l, and N(l) is the number density of the molecules at path length l. The laser pulse energy E(l) at a path length l can be obtained from Eq. (4) as
E(l)=E(0)exp[τ(νc,l)],τ(νc,l)0lσeff(νc,l')N(l')dl'.
Here τ(νc,l) is the effective optical depth of the species of interest and σeff(νc,l) is the effective absorption cross-section defined as
σeff(νc,l)0σ0(νF,l)LN(νF,l)dνF=0σ0(νF,l)T(νF,l)L(νF,0)dνF0T(νF,l)L(νF,0)dνF,T(νF,l)exp[τ0(νF,l)],τ0(νF,l)0lσ0(νF,l')N(l')dl'.
Here τ0(νF,l) is the monochromatic optical depth of the species of interest, and other attenuation factors are assumed to be wavelength independent.

To account for two-way absorption, the path length l is taken to be the accumulated distance that the laser pulse has traveled, running from 0 to 2rG as the laser pulse travels from the spacecraft to the surface (outgoing path) and back to the spacecraft (return path). Here rG is the distance from the spacecraft to the surface. Our model uses the effective optical depth to include the line shape factor, which causes only a bias τ(νc,l)τ0(νc,l) to the optical depth. As shown in Appendix A, this bias can be accurately corrected. When L(νF) is much narrower than the spectral width of the target absorption line, this bias even becomes negligible, as shown in subsection 2.6 for the 1572.335 nm CO2 line. When L(νF) is broad, the one-way effective optical depth may no longer be the same for the outgoing and return paths so that τ(νc,2rG)=2τ(νc,rG) may become invalid (see Appendix A for more details).

Incorporating other attenuation factors [18], the detected laser pulse energy Ws backscattered from the surface becomes

Ws=EsAsexp[τ(νc,2rG)],
where Esis the transmitted laser pulse energy, AsρTatm2D/rG2 is the lump sum of attenuation factors excluding exp[τ(νc,2rG)], D is a instrument constant, ρ is the ground surface reflectance (in sr−1), and Tatm is the one-way transmittance of the atmosphere excluding the target species.

The lidar receiver can operate in either linear mode with an internal gain or photon counting mode. To cover both cases, the analog detector signal and noise are gain normalized and expressed in detected photon count K. For any type of radiation (including thermal light), the ensemble average and the variance of K and the optical energy W entering the detector area are related by [19, 20]

K¯=αW¯,
σ2(K)=FeK¯+α2σ2(W).
Here αη/(hν¯) is proportional to the detector quantum efficiency η, and h is the Planck constant. Note that only the shot noise contribution is multiplied by the detector’s excess noise factor Fe. To estimate errors, we expand exp[τ(νc,2rG)] in Eq. (7) around ν¯ to arrive at
Ws=EsAsexp[τ(ν¯,2rG)][1(dτ/dνc)δν1(t0)+bδν12(t0)],
where b[(dτ/dνc)2d2τ/dνc2]/2.From Eqs. (8) and (10), the ensemble mean of the detected signal photon count Ks is
Ks¯=αWs¯=αEsAsexp[τ(ν¯,2rG)][1+bσ2(δν1)],
where the variance σ2(δν1) of δν1(t0) contributes a small fractional bias bσ2(δν1) to Ks¯. As illustrated in subsection 2.6, this bias is usually negligible in practice. Using Eqs. (9) and (10), the variance of Ks is related to the laser line-center frequency noise as
σ2(Ks)=FeKs¯+α2σ2(Ws)=FeKs¯+(Ks¯)2(dτ/dνc)2σ2(δν1),
where the small contribution from the quadratic term bδν12(t0) is neglected.

2.3 Averaging across multiple pulses

We now turn to estimate the two-way DAOD Δττ(ν¯on,2rG)τ(ν¯off,2rG) between online and offline laser frequencies ν¯on and ν¯off. Up to section 6, τ(νc,2rG) is assumed to be constant over the path and duration of the multiple pulses to be averaged. Although the measurement of Es is affected by the amplified spontaneous emission (ASE) in the laser amplifiers, the resulting measurement errors are negligible in practice and thus will not be included in the present model (see Appendix B for details).

For each online or offline wavelength, the laser pulses are assumed to have the same h(t) and a period of tp. From Eq. (10), the following sum of normalized pulse energies (SNE) can be introduced to include the effect of the laser frequency noise

SNE1i=0n1Ws(i)Es(i)=exp[τ(ν¯,2rG)](i=0n1As(i))[1dτdνcδνn(t)+bQn(t)],Qn(t)1i=0n1As(i)i=0n1[As(i)δν12(t+i×tp)],
where the averaged laser line-center frequency noise δνn(t) across n pulses (i=0,1,2,..,n1) is given by
δνn(t)1i=0n1As(i)i=0n1[As(i)δν1(t+i×tp)].
The mean and variance of SNE1 are found to be
SNE1¯=exp[τ(ν¯,2rG)](i=0n1As(i))(1+bσ2(δν1)),
σ2(SNE1)=(SNE1¯)2(dτ/dνc)2σ2(δνn).
Superscripts (or subscripts) on and off will be used to indicate online and offline laser frequencies, respectively. The offline σ2(SNE1off) can be neglected because of the negligible slope (dτ/dνc)off.

Corresponding to SNE1, the sum of normalized photon count Ks (SNK) is defined as SNK1i=0n1Ks(i)/Es(i) and its mean is found to be SNK1¯=αSNE1¯. Here SNK1 is named with the following scheme. S stands for sum, NK for normalized photon count, and the trailing digit 1 is used to distinguish SNK1 from SNK2 and SNK3 to be introduced in sections 3 and 6, respectively. Using Ks(i)Ks(i')¯=α2Ws(i)Ws(i')¯ [19], the variance of SNK1 is found to be

σ2(SNK1)=FeSNNK1¯+α2σ2(SNE1),SNNK1i=0n1Ks(i)Es2(i)SNK1/EsavSK/Esav2.
Here SNNK1 is named similarly, with NN indicating that Ks is normalized twice. The integrated photon count SK and average transmitted pulse energy Esav are defined as SKi=0n1Ks(i) and Esav(i=0n1Es(i))/n, respectively. SNNK1 is essentially SK normalized by Esav2. The first term in σ2(SNK1) stems from the shot noise. The second term arises solely from the laser line-center frequency noise.

From Eq. (15), Δτ is related to SNK1 by

Δτ=ln(SNK1on¯SNK1off¯)+ln(RA)+ln(Rb),
where RAi=0n1Ason(i)/i=0n1Asoff(i)1+δRA and Rb[1+bonσ2(δν1on)]/[1+boffσ2(δν1off)].For the rest of this paper, we assume the transmitted pulse energies are approximately the same for online and offline pulses so that SKon¯SKoff¯exp(Δτ). Δτ has been in effect estimated from the following “log after averaging” estimator [59]
ΔτSNK1ln(SNK1onSNK1off).
To estimate errors, approximations ln(x)¯ln(x¯)(1/2)σ2(x)/(x¯)2 (accurate to the third order) and σ2(ln(x))σ2(x)/(x¯)2 (accurate to the second order) are used in our calculations. The bias bΔτ_SNK1ΔτSNK1¯Δτ and variance of ΔτSNK1 are found to be
bΔτ_SNK1ln(RA)ln(Rb)+Fe2(SNNK1on¯SNK1on¯2SNNK1off¯SNK1off¯2)+12(dτdνc)on2σ2(δνnon),
σ2(ΔτSNK1)σ2(SNK1on)SNK1on¯2+σ2(SNK1off)SNK1off¯2FeSKoff¯[1+exp(Δτ)]+(dτdνc)on2σ2(δνnon).
As expected, the shot noise contribution to σ2(ΔτSNK1) is inversely proportional to the integrated photon count SK¯. However, it is important to use SNK1 rather than SK to estimate Δτ, to remove the effect of variation in Es(i). The laser line-center frequency noise contribution to the DAOD RRE is [(dτ/dνc)on/Δτ]σ(δνnon), where σ(δνnon) is the standard deviation of the averaged line-center frequency noise of the on-line laser (averaged across n pulses). It becomes clear that δνnon(t) represents the effective frequency noise. Obviously, an on-line laser frequency bias Δν¯on will result in a DAOD bias ~(dτ/dνc)onΔν¯on [5]. This bias can be avoided by accurate calibration and thus is neglected throughout this paper.

2.4 Frequency noise reduction from pulse averaging

We now turn to link the PSD Sδνn(f) of δνn(t) to Sδν(f). From Eq. (14) we have

Sδνn(f)=Sδν(f)|H(f)|2Wavn(f),Wavn(f)|1i=0n1As(i)i=0n1As(i)exp(j2πftp×i)|2.
Here the effect of multiple-pulse averaging on the PSD is an additional multiplicative window function Wavn(f). The overall window function |H(f)|2Wavn(f) is the ESD of the weighted pulse train function i=0n1[As(i)h(ti×tp)]/i=0n1As(i). Wavn(f)is a periodic function with a period equal to the pulse repetition rate. Using the orthogonal property of the phase shift factor exp(j2πftp×i), its averaged height is found to be
<Wavn>=i=0n1As2(i)/(i=0n1As(i))2.
When the weighting factors As(i) are the same, Wavn(f) becomes sinc2(nftp)/sinc2(ftp) with an average height of 1/n. This is consistent with n-fold variance reduction when averaging n identical but uncorrelated random variables. As shown in section 5, <Wavn> is only slightly higher than 1/n when measured surface reflectance data are used for As(i), and will be taken to be 1/n for our numerical evaluations.

The present model reveals relationships not available from previous models [5, 8]. Using Eq. (22), the effective frequency noise can be estimated in terms of Sδν(f), as illustrated in Fig. 2 for a representative frequency stabilized seeder laser. The nominal Sδν(f) can be approximated by Sδν(f)=h1/f+h0, where constant h1 and h0 quantify flicker noise and a nearly flat noise floor for f above 100 Hz, respectively. The periodical window function Wavn(f) suppresses the frequency noise except in its narrow peaks at f=mw/tp (mw=0,1,2,3,...). Frequency drifts with periods longer than the averaging time fall in the window peak at f=0 and thus are not attenuated from this averaging. Since the PSD is nearly constant over the frequency range of all other window peaks (mw=1,2,3,...), the variance of the faster frequency noise components (excluding the slow drifts) is essentially reduced by a factor of n. This frequency noise reduction has also been verified experimentally, as shown in the next subsection.

 

Fig. 2 The nominal laser frequency noise PSD Sδν(f)=h1/f+h0 (solid red) is suppressed to Sδν(f)|H(f)|2(dotted blue) by averaging within a 1-μs top-hat pulse and further reduced to Sδν(f)|H(f)|2Wavn(f) (dashed black) by averaging across 1000 identical pulses separated by a period of 1 ms.

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2.5 Experimental results

Equations (3) and (22) have been verified experimentally with our seed laser described in [13]. Figure 3 (left) shows the standard deviation of the absolute frequency of the slave laser and frequency offset between the slave and master lasers measured within a gating time from 1 μs to 1 ms. The absolute frequency noise was measured from the beatnote between two identical and independent laser units. The noise of the frequency offset was measured from the beatnote between the slave and master lasers. A fast frequency counter (HP 5371A, Agilent Technologies, Inc.) was used to measure the beatnote frequencies. The measured noise decreases as the gating time increases, matching the reducing area under Sδν(f)|H(f)|2.

 

Fig. 3 (left) The standard deviation of the absolute frequency of the slave laser and frequency offset between the slave and master lasers measured within a gating time from 1 μs to 1 ms; (right) Both frequencies were also measured periodically within a 1-µs gating time with a 1-ms period and averaged across multiple measurements. The standard deviation of each averaged frequency, computed from 100 averaged samples, is plotted as a function of the number of measurements being averaged.

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Both frequencies were also measured periodically within a 1-µs gating time with a 1-ms period and averaged across multiple measurements. The standard deviation of each averaged frequency, computed from 100 averaged samples, is plotted in Fig. 3 (right) as a function of n, the number of measurements being averaged. The variance of the averaged frequency decreases as n increases, matching the reducing area under Sδν(f)|H(f)|2Wavn(f).

2.6 Frequency stability requirements for ASCENDS

Using Eqs. (5) and (6), the relative error (ΔτΔτ0)/Δτ0 due to the finite line shape L(νF) is numerically evaluated and shown in Fig. 4 (left) as functions of the effective bandwidth 1/Δt of a top-hat laser pulse shape for the CO2 line. As detailed in Appendix A, the line shape L(νF) is simply taken to be the ESD of the top-hat pulse shape. (ΔτΔτ0)/Δτ0 is smaller than a 0.02% target [5] and thus does not need to be corrected for all three online points (with frequency offsets + 0.5 GHz, + 1.1 GHz, and + 1.7 GHz) when the pulsewidth Δt is longer than 0.5 μs. τ is taken to be the same as τ0 for the CO2 line in our numerical examples where Δt is assumed to be ~1 μs.

 

Fig. 4 Relative errors for atmospheric CO2 DAOD measurements: (left) (Δτ - Δτ0)/Δτ0 due to the finite linewidth 1/Δt of a top-hat pulse shape (calculated from Eqs. (5) and (6)); (right) laser line-center frequency noise contributions to the RRE for σ(δνnon) = 0.1 MHz (dashed red, calculated from Eq. (21)), and RSE ln(Rb)/Δτ for σ(δν1) = 10 MHz (dotted blue, from Eq. (18)) as functions of the online laser frequency offset from the absorption line center. The two-way optical depth τ (solid green) is also plotted, with blue dots marking the same online and absorption line-center frequency points used on the left.

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The partial RRE [(dτ/dνc)on/Δτ]σ(δνnon) forσ(δνnon)=0.1MHz, the partial RSE |ln(Rb)|/Δτ, and τ(νc,2rG) are plotted in Fig. 4(right) for atmospheric CO2, as functions of the online laser frequency offset from the absorption line center. Table 1 summarizes the RRE targets, relevant parameters, and resulting effective laser frequency noise requirement for atmospheric CO2 DAOD measurement. The frequency offset of the offline laser is set at 15.6 GHz. To bound the RRE to < 0.03% for an online frequency ≥ 0.5 GHz away from the absorption line center, the standard deviation σ(δνnon) of the effective frequency noise needs to be suppressed down to 0.23 MHz. This frequency stability requirement has been satisfied experimentally (see [13] and subsection 2.5).

Tables Icon

Table 1. RRE Targets, Relevant Parameters, and Resulting Effective Laser Frequency Noise Requirement for Atmospheric CO2 DAOD Measurement

The partial RSE |ln(Rb)|/Δτ for σ(δν1)=10MHz is less than 0.002% at such online frequencies, and thus negligible. However, this bias and (ΔτΔτ0)/Δτ0 (Fig. 4, left) become significant near the absorption line center, where the excessive CO2 absorption also results in higher RRE. It is thus undesirable to place an online frequency point near the absorption line center. Referring to Eq. (20), the last bias term (dτ/dνc)on2σ2(δνnon)/2 is <107when [(dτ/dνc)on/Δτ]σ(δνnon) is bound to 0.03%, and thus negligible. The third bias term can be reduced to a negligible level with photon count integration. As shown in section 5, the remaining bias term |ln(RA)| can be reduced to <1.1×104. Overall, the RSE bΔτ_SNK1/Δτ can be reduced to < 0.02%.

3. Incorporating additional detection noise sources

We now incorporate additional detection noise sources arising from the background solar radiation, detector dark count, and receiver circuitry noise for the lidar receiver. The mean and variance of the photon count Kn-n produced by the background solar radiation is given by Knn¯=2αNbgdBoΔt and σ2(Knn)=FeKnn¯, where 2Nbgd is the PSD of the background solar radiation and Bo the optical filter bandwidth. The equivalent detector dark count Kd (normalized by the mean internal gain Me of the detector) has a mean and variance given by Kd¯=λdΔt and σ2(Kd)=FdKd¯, where λd is an equivalent dark count rate and Fd an effective dark-count excess noise factor. The equivalent photon count KT arising from the receiver circuitry noise has a zero mean and a variance σ2(KT)given by

σ2(KT)=IδiΔt/(Mee)2,IδiSδi(f)sinc2(fΔt)Δtdf,
where Sδi(f)is the PSD of the equivalent input noise current of the circuit and e the electron charge. Equation (24) is derived by the same steps leading to Eq. (40) for σ2(Ksn) (see Appendix B). The integration Iδi can be approximated by Sδi(0) when the detector electrical bandwidth Be is much larger than 1/Δt.

The laser signal photon count Ks is estimated as K's=KtotK'bgd, where Ktot is the total count Ktot=Ks+Knn+Kd+KT measured within Δt, and K'bgd=K'nn+K'd+K'T is the background count within Δt scaled from a separate measurement taken between the pulse measurements. This background measurement can be taken in a longer duration βΔt, where β is a background integration time multiplier, to reduce the background variance. Obviously, K's¯=Ks¯. Referencing Eq. (9), the variance of K's is found to be

σ2(K's)FeKs¯+α2σ2(Ws)+λbgdΔt,λbgd[2αNbgdBoFe+Fdλd+Iδi/(Mee)2](1+1/β).

The transmitted pulse energy Es can be estimated as E's=E'totE'bgd where Etot is the total detected energy and the background energy E'bgd arises from the circuitry noise, dark current in the detector, and ASE in the laser amplifiers. The mean of E's equals Es and the variance of E's can be neglected (see Appendix B). The additional noise sources can be included by replacing SNK1 with SNK2i=0n1K's(i)/E's(i). Similarly, SK is replaced by SK'i=0n1K's(i). Obviously, SNK2¯=SNK1¯. The normalized variance of SNK2 is found to be

σ2(SNK2)(SNK2¯)2FeSNNK2¯(SNK2¯)2+(dτdνc)2σ2(δνn)+SNN2¯(SNK2¯)2λbgdΔt,
where SNNK2 and SNN2 are defined as SNNK2i=0n1K's(i)/E's2(i) and SNN2i=0n11/E's2(i). Here SNN2 is named similarly, with NN indicating double normalization.

As seen from Eq. (20), the “log after averaging” DAOD estimator in Eq. (19) leads to an additive bias term (1/2)Fe(SNNK1on¯/SNK1on¯2SNNK1off¯/SNK1off¯2)(1/2)Fe/SKon¯, and thus requires significant photon count integration (SKon¯~104) to bring the bias term down to a tolerable level (~10−4). When using SNK2, the additional background variance λbgdΔt in σ2(K's) results in an additional DAOD bias term (1/2)λbgdΔt(SNN2on¯/SNK2on¯2SNN2off¯/SNK2off¯2). These two bias terms can be reduced significantly by incorporating the following bias correction factor CΔτ_SNK2into the DAOD estimator given below

ΔτSNK2ln(SNK2onSNK2off)+CΔτ_SNK2,CΔτ_SNK2Fe2(SNNK2offSNK2off2SNNK2onSNK2on2)+λbgdΔt2(SNN2offSNK2off2SNN2onSNK2on2).
As shown next, the sum of the two bias terms is reduced from ~(1/2)Fe/SK'on¯ to ~(1/2)Fe2/SK'on¯2by the bias correction factor CΔτ_SNK2. The residual bias ~(1/2)Fe2/SK'on¯2 becomes tolerable (~10−4) even when the integrated photon count SK'on¯ is as low as ~100. Such a low photon count SK'on¯ can be obtained within a few pulses, or even within a single pulse without averaging.

Using the bias-corrected DAOD estimatorΔτSNK2 given by Eq. (27), the bias bΔτ_SNK2ΔτSNK2¯Δτ and variance of ΔτSNK2 are found to be

bΔτ_SNK2ln(RA)ln(Rb)+(dτ/dνc)on2σ2(δνnon)/2+bC_Δτ_SNK2,bC_Δτ_SNK212(Fe/SK'on¯)2[1exp(2Δτ)]32FenλbgdΔt[1exp(3Δτ)]/SK'on¯3,
σ2(ΔτSNK2)σΔτDET2+(dτ/dνc)on2σ2(δνnon),σΔτDET2Fe[1+exp(Δτ)]/SK'off¯+nλbgdΔt[1+exp(2Δτ)]/SK'off¯2.
It should be noted that all the parameters in the bias correction factor CΔτ_SNK2 are measureable. For example, the background variance λbgdΔt can be measured when the laser pulses are off. However, it may not be feasible to measure some of the parameters (such as Fe) constantly during the flight. Consequently, long-term variations of such parameters may degrade the bias correction.

To minimize the time-varying Etalon and atmospheric turbulence effects, it is desirable to reduce n, the number of pulses being averaged for ΔτSNK2. To do this, the overall averaging time Ttot is divided into m slots (each is Ttot/m long), and a ΔτSNK2 (denoted as ΔτSNK2(k)) is evaluated within each slot (k=0,1,2,..,m1). Δτ can then be estimated from the average of ΔτSNK2(k) given below

ΔτSNK2_av1mk=0m1ΔτSNK2(k).

The variance of ΔτSNK2_av (excluding contribution from the slow laser frequency drifts) also decreases approximately by the number of participating pulses (i.e., n×m), as in the case for ΔτSNK2. However, the averaging in Eq. (30) cannot effectively reduce the bias to Δτ.

4. Numerical estimation of measurement errors

Using realistic parameters listed in Table 2 , ΔτSNK2_av/Δτ, the RRE for atmospheric CO2 DAOD measurement, is computed and plotted as a function of τ in Fig. 5 . Partial contributions to the RRE from the signal shot noise ~Fe[1+exp(Δτ)]/SK'off¯, frequency noise, solar background, receiver circuitry noise, and detector dark count are also shown Fig. 5. When there are 3200 detected photons (in average) for each offline pulse, the RRE is ≤ 0.1% for τ between 0.66 and 3.2. This photon count can be achieved with Es=4mJ for an average ρ = 0.17. The detector specifications listed in Table 2 can be met with a state-of-the-art HgCdTe avalanche photodiode (APD) detector [21]. The detected solar radiation count Kn-n is estimated for a zenith angle of 75°. Its noise contribution is larger than that from the receiver transimpedance amplifier (TIA). The noise from the dark count Kd is negligible. It should be noted that the RRE increases sharply when the online point moves further away from the line center, confining the frequency offset of the online points to within ± 1.7 GHz from the CO2 absorption line center. The standard deviation of the effective frequency noise averaged in the 10-s measurement time is taken to be 0.23 MHz for the above calculations. When it is increased to 1 MHz, the RRE rises above the 0.1% target (also shown in Fig. 5).

Tables Icon

Table 2. Parameters Used to Evaluate the RRE of Δτ as a Function of τ

 

Fig. 5 The RRE σ(ΔτSNK2_av) /Δτ for atmospheric CO2 DAOD measurement (solid black) as a function of the two-way optical depth, computed from Eqs. (29) and (30) using parameters listed in Table 2. The blue dots mark the three online frequency points used in Fig. 4. Also plotted are partial contributions to this RRE from the shot noise (solid grey), frequency noise (dashed red), solar background (dotted brown), receiver circuitry noise (dash-dotted green), and detector dark count (long-dashed blue). The standard deviation of the effective frequency noise σ(δνnon) averaged in 10 s is 0.23 MHz for the above calculations. When it is increased to 1 MHz, the RRE (dotted black) rises above the 0.1% target (dashed black).

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It is desirable to squeeze down RRE contributions from noise sources other than the signal shot noise. This will translate into reduction in required laser power and hence the cost. The laser peak power entering the lidar receiver needs to be high enough so that the signal shot noise contribution is well above the noise contributions from the receiver circuitry and the solar background count. This is the case for Goddard’s ASCENDS lidar where online points are placed at least 0.4 GHz away from the CO2 line center with τ2.4.

5. Impact of surface reflectance variation

As discussed earlier, the surface reflectance variation adds a bias |ln(RA)|to the DAOD estimation. Since this bias varies from one path to another, it needs to be kept to a small fraction of the 0.1% RRE. Also, the PSD of the effective laser frequency noise δνn(t) is reduced by the periodical window function Wavn whose average height <Wavn> depends on the variation of As(i). Both factors are quantified here using measured surface reflectance data in [10] for realistic error estimation.

The parameters used for the evaluation are listed in Table 3 . The laser beam is assumed to travel at the same speed as the spacecraft. As shown in Fig. 1, the laser pulses are assumed to repeatedly cycle through wavelength channels 1, 2, 3, 4, 8, 7, 6, 5, 1, to minimize the separation between online (channels 2 to 6) and offline (channels 1 and 8) pulses. A set of surface reflectance data that has strong variations is used for this evaluation. The surface reflectance measurement was taken in southern Spain using ~10-m laser spot size and a step size of ~6 m [10]. To convert this data to the reflectance for our beam size, a 1-D running average is taken within our beam size (50 m) and the averaged reflectance is used as As(i) in our calculation. The raw and averaged reflectance data are plotted in Fig. 6 (left). For an averaging time of 1 s, there are n = 1000 pulses for each wavelength and the path length is 7 km. In contrast, the separation between online and offline beam spots is only 2.1 m at most. Consequently, the ratio RA is very close to 1. This point is confirmed by our calculation results shown in Fig. 6 (right). The 108-km total path length is divided into 15 sections (7 km each) and ln(RA) is evaluated for each section. For each online wavelength channel, its starting pulse for sum i=0n1Ason(i) comes in between two adjacent offline pulses, each serves as the starting pulse for its offline sum i=0n1Asoff(i). Each i=0n1Asoff(i) is used to calculate ln(RA) and the average of the two is plotted in Fig. 6 (right) for the worst case where the bias |ln(RA)|is <1.1×104. The corresponding RSE is smaller than 0.02% if the two-way DAOD is kept > 0.55.

Tables Icon

Table 3. Parameters Used for Evaluation of the Impact of Surface Reflectance Variation

 

Fig. 6 (left) Relative surface reflectance: measured data (grey) and its running average over a length of 50 m (blue), courtesy of A. Amediek of Deutsches Zentrum für Luft- und Raumfahrt (DLR); (right) ln(RA) calculated from surface reflectance data for the worst case. For each online wavelength channel, two ln(RA) are calculated from both offline sum i=0n1Asoff(i) for 1-s averaging time and the average is plotted (blue dot). An average of 10 consecutive ln(RA) values, each computed as above over 0.1 s averaging time, is also plotted (red square).

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|ln(RA)| is increased when the averaging time (hence n) is reduced. Figure 6 (right) also shows an average of 10 values of ln(RA), each computed as above over 0.1 s averaging time consecutively along the worst path. For some wavelength channels, this averaged bias |ln(RA)| becomes larger than 10−3 and thus unacceptable. To reduce the bias |ln(RA)|, it is crucial to allow a fast pulse repletion rate (≥ 1 kHz for each wavelength) and a sufficient integration time (≥ 1 s), and to properly interleave online and offline pulses to minimize their separation.

The average height <Wavn> of Wavn(f) is also evaluated for the 7-km averaging paths. The calculated <Wavn> is quite close to 1/n, varying between 1.01/n and 1.12/n.

6. Mitigating impact of surface height variation

The DAOD errors due to surface height variation can be reduced to negligible levels if pulse-by-pulse knowledge of surface height is incorporated into the formulation. Let z0(i) be the surface height at laser beam spot i relative to a certain reference ground surface, and z(i) a measured value of z0(i). rG now represents the range from the spacecraft to the reference surface. The distance from the spacecraft to the actual surface is rGz0(i). Since z0(i) could vary by tens of meters over the path of pulse averaging, τ(rGz0(i)) could fluctuate over a few tenths of a percent over the path, resulting in additional DAOD errors. To mitigate this problem, we estimate Δτ(2rG) (i.e., the two-way DAOD from the spacecraft to the reference surface and back to the spacecraft) from SNK3i=0n1K's(i)/[Az(i)E's(i)], where Az(i)exp[2σeff(ν¯,rG)N(rG)z(i)] is introduced to account for the fluctuation of the optical depth. The corresponding estimator ΔτSNK3 of Δτ(2rG) can be easily adapted from ΔτSNK2 in Eq. (27) by simply replacing E's(i) with Az(i)E's(i) in the relevant terms in ΔτSNK2 (i.e., SNK2, SNNK2, and SNN2). As shown next, the variation of the surface height z0(i) does not affect such estimation of Δτ(2rG). Nevertheless, the imperfect measurement of z0(i) still leads to additional DAOD errors.

To simplify the treatment, we assume z(i)=z0(i)+Δz+δz(i), where Δz represents a slowly varying altimetry bias, and δz(i) a zero-mean random altimetry error. Δz is treated as a constant during the n pulse averaging, and the δz(i) values among different pulses are assumed to be uncorrelated and have the same variance σz2. Expanding 1/Az(i)in SNK3 as

1/Az(i)exp(2σeffNΔz)/exp[2σeffNz0(i)](12σeffNδz(i)+2[σeffNδz(i)]2),
the bias bΔτ_SNK3ΔτSNK3¯Δτ(2rG) and variance of ΔτSNK3 are found to be
bΔτ_SNK3=bΔτ_SNK2+bΔτz,bΔτz2σeff(ν¯on,rG)N(rG)Δz2[σeff(ν¯on,rG)N(rG)σz]2(1<Wavn>),
σ2(ΔτSNK3)=σ2(ΔτSNK2)+σΔτz2,σΔτz2[2σeff(ν¯on,rG)N(rG)σz]2<Wavn>.
Here we have neglected σeff(ν¯off,rG) that is much smaller than σeff(ν¯on,rG). The additional DAOD bias bΔτz arises from the altimetry bias Δz as well as the variance σz2. The additional DAOD variance σΔτz2 scales with the altimetry variance σz2 and <Wavn>, decreasing with ~1/n. Taking atmospheric CO2 for example, an altimetry precision σz=20m results in σeff(ν¯on,rG)N(rG)(20m)<15×10-4, leading to a negligible standard deviation σΔτz<1.3×10-4 (assuming n = 500), and a negligible DAOD bias (<1.4×10-6) arising from σz2. On the other hand, the altimetry bias Δz needs to be kept below 0.66 m to limit its DAOD bias contribution to <10-4.

7. Discussions

7.1 Other error sources

Other DAOD measurement error sources have also been studied previously [58, 15, 16]. An important one is the time-varying Etalon fringes in the lidar’s frequency response. It is crucial to minimize multi-path interference (MPI) along the optical train, and to shorten the time of “averaging before log”, to minimize the impact of the Etalon effect. Using multiple wavelength channels also helps to correct for the Etalon effect. The wavelength dependency of the lidar’s spectral response in the receiving path can be partially removed by passing a small fraction of the transmitted laser trough the receiving path and measuring the transmitted pulse energy E's at the end of the path [15]. Another error in the measurement of K's arises from the broadening of the returned signal pulses mainly due to the structured ground surface. The spectral crosstalk among multiple wavelength channels must be suppressed to maintain spectral purity for the measurement [5].

The contribution of atmospheric turbulence to the bias term |ln(RA)| is negligible because Tatm essentially does not change during the time separation between an online pulse and either of its neighboring offline pulses (< 0.4 ms). The speckle noise can be also neglected in direct detection IPDA lidar instruments with a telescope diameter of ~1.5 m and laser beam spot size of ~50 m on the surface [5, 7]. The Doppler effect can be either made negligible or corrected for practical nadir viewing IPDA lidars [15, 22]. The Doppler shift due to high speed cross-wind [15] can be made negligible by minimizing the off-nadir angle of the laser beam. The 2° off-nadir angle intended for A-SCOPE lidar to avoid detector saturation from the ice cloud glint [15] appears to be unnecessary because the reflectance of the ice (and snow) is particularly small for both 1.57 μm and 2.05 μm wavelength regions [23, 24]. The Doppler shift deduced from the spacecraft velocity components can also be compensated by dynamically shifting the frequency offset of the seed laser [13]. The laser beam divergence (~ ± 62.5 μrad) in Table 2 results in a continuous spread of laser frequency of ± 278 kHz across the beam at 1570 nm [15]. Since this small spread is deterministic and symmetrical, it does not increase the variance of Δτ and its bias contribution to Δτ is negligible.

7.2 Error reduction tradeoffs

When the laser peak power entering the lidar receiver is high enough so that the signal shot noise contribution is well above the noise contributions from the receiver circuitry and the background count, the ratio η/Fe of the detector becomes a figure of merit for the DAOD measurement. For a fixed average power, further increasing the laser peak power by reducing the pulsewidth or the pulse repetition rate will no longer reduce σΔτDET2effectively. On the other hand, it is desirable to use the maximum pulse repetition rate (~10 kHz for all wavelengths combined) to reduce the bias |ln(RA)|, and to use a wider pulsewidth (along with the high pulse repetition rate) to reduce σ2(δνn) and the laser peak power. The laser peak power must be kept below the threshold of the undesirable stimulated Brillouin scattering (SBS) in the laser power amplifiers. The pulsewidth, repetition rate and laser peak power chosen for Goddard’s ASCENDS approach appear to be appropriate. Using a shorter pulse duration (< 100 ns) and a lower pulse repetition rate results in less frequency noise reduction, and may thus require a seed laser with a narrower frequency noise bandwidth and a lower noise floor.

7.3 Shortening the time of averaging before log

As shown earlier, all DAOD bias terms other than |ln(RA)| can potentially be reduced to negligible levels by averaging across just a few pulses (or even without averaging) before log. It would be great if the same can be done for the bias term |ln(RA)|, to reduce the time of averaging before log. This could be achieved by pointing the laser and receiver to a fixed surface spot during each wavelength sweep cycle (≤ 1 ms). During such a short time, the slow spacecraft pointing jitter can be ignored. As long as the online/offline laser spots coincide within each wavelength sweep, slow pointing jitters do not increase |ln(RA)|. The pointing can be achieved, for example, with piezoelectrically driven steering mirrors. While the spacecraft is traveling, the laser and receiver can be pointed to a fixed surface point for several online/offline cycles and then quickly redirected to the next fixed surface point. The mirror steering span can be reduced to only one or two beam spots to allow fast redirection. Even for a long pointing duration of 1 s, the fractional change in path length is less than 4×10-5and thus negligible.

With this pointing scheme, the overall time of averaging before log can be significantly shortened to reduce measurement errors as mentioned earlier. On the other hand, using pointing mechanism may complicate the instrument design and pose a reliability challenge.

8. Summary

New modeling and error reduction methods are presented for DAOD measurements of atmospheric constituents using direct-detection IPDA lidars. Errors from the laser frequency noise are quantified in terms of the line center fluctuation and spectral line shape of the laser pulses, revealing new relationships that have been verified experimentally. Averaging across n pulses, the effective frequency noise is quantified by the multiplicative window function that filters the PSD of the instantaneous frequency noise. This in general leads to n-fold reduction in the variance of the effective frequency noise (excluding slow drifts with periods longer than the measurement averaging time). To bound the frequency noise contribution to the DAOD RRE to 0.03%, the standard deviation of the effective frequency noise needs to be < 0.23 MHz for ASCENDS’ CO2 transmitter.

The DAOD variance decreases approximately by the number of participating pulses. A significant DAOD bias can be essentially canceled out by incorporating a correction factor. The RRE and RSE of the DAOD due to surface height and reflectance variations can be reduced to tolerable levels by incorporating pulse-by-pulse altimetry knowledge and by “log after averaging”, or by pointing the laser and receiver to a fixed surface spot during each wavelength sweep to shorten the time of averaging before log.

The predominant errors in the transmitted pulse energy measurement are found to be negligible in practice. Error reduction tradeoffs are investigated and illustrated with realistic calculations that take various detection noise sources into account. Other error sources are also summarized.

Appendix

A. Formulation linking L(νF) to Sδν(f) and a(t)

The bandwidth BLof L(νF), as defined below, is also linked to ν(t) and a(t) [17] as

BL20[νFνc(t0)]2LN(νF)dνF=BLF2+BLA2,BLF2[ν(t)νc(t0)]2h(tt0)dt,BLA2f2|A(f)|2df|A(f)|2df=14π2[da(t)/dt]2dta2(t)dt.
Here A(f) is the Fourier transform of a(t) so that |A(f)|2 is the ESD of a(t).

As seen from Eq. (34), BL2 is the sum of two components. The first component BLF2 is the time domain “variance” of δν0ν(t)νc(t0)weighted by h(tt0), and the second component BLA2 represents the bandwidth of |A(f)|2arising from the variations of the amplitude envelope a(t). From Eq. (34), the ensemble average of BLF2 is found to be

BLF2¯=[δν(t)]2¯[δν1(t0)]2¯=Sδν(f)[1|H(f)|2]df.
This shows that BLF2¯ arises from faster frequency noise components, excluding the slower components Sδν(f)|H(f)|2 that determine the line center fluctuation. Since the contributing fast frequency noise components are quite repeatable from one pulse duration to another, BLF2 has little fluctuation around its ensemble mean BLF2¯ and thus causes little variance in bτ(νc,l)τ(νc,l)τ0(νc,l). In practice, the frequency noise PSD Sδν(f) cuts off eventually, resulting in a finite BLF2¯ [17]. Since the slow frequency drifts do not contribute, BLF2¯ for a short pulse can be significantly smaller than that for the underlying CW laser.

To facilitate further analysis, we rewrite L(νF) as L0(ΔνF)L(νc(t0)+ΔνF), as a function of the Fourier frequency offset ΔνFνFνc(t0). Obviously, a broad |A(f)|2 (due to, for example, a narrow pulsewidth) can still lead to a significant bias bτ(νc,l). Fortunately, the bias bτ(νc,l) can be computed from Eqs. (5) and (6) using L0(ΔνF)=|A(ΔνF)|2 when the frequency noise contribution is negligible. Equations (5) and (6) thus allow accurate modeling even when L(νF) is deterministically broadened to suppress SBS in the laser amplifiers. Since the amplitude modulation is deterministic, the bias bτ(νc,l) is deterministic and thus can be precisely calibrated and removed. Although L(νF) for a top hat pulse shape has an infinite BLA, the relative error (ΔτΔτ0)/Δτ0 can still be negligible as shown in Fig. 4 (left). When L(νF) is broad, the one-way effective optical depth may no longer be the same for the outgoing and return paths. This is due to the progressive distortion of the laser line shape LN(νF,l) caused by the absorption of the species of interest. Since the laser line shape LN(νF,l) is no longer the same along the two paths, σeff(νc,l) may have two different values for each spatial position, one for the outgoing laser pulse and the other for the return laser pulse.

The relationship between the CW laser line shape PSD and the frequency noise PSD Sδν(f) has been established in previous studies [2527]. The present work further links L(νF), the laser pulse ESD, to Sδν(f) and the pulse shape a(t). L(νF) is the Fourier transform of the following autocorrelation of u(t)

Ru(tD)u*(t)u(t+tD)dt=a(t')a(t'+tD)exp(j2πν¯tD)exp(jΔϕ)dt',
where ttt0 and Δϕϕ(t+t0+tD)ϕ(t+t0). The ensemble average of L0(ΔνF) becomes
L0(ΔνF)¯=[a(t')a(t'+tD)exp(j[Δϕ2πδν1(t0)tD])¯dt']exp(j2πΔνFtD)dtD.
Since Δϕ2πδν1(t0)tD is a zero-mean Gaussian process, we have
exp(j[Δϕ2πδν1(t0)tD])¯=exp[12(Δϕ2πδν1(t0)tD)2¯].
Using Δϕ=t'+t0t'+t0+tDδν(t)dtand taking similar steps as in [25, 26], we finally arrive at
L0(ΔνF)¯=R0(tD)exp(j2πΔνFtD)dtD,R0(tD)a(t')a(t'+tD)exp[2π2tD2Sδν(f)|sinc(ftD)H(f)exp[j2πf(t'+tD/2)]|2df]dt'.
The phase factor exp[j2πf(t'+tD/2)] in R0(tD) stems from the time delay between Δϕ (centered at t'+tD/2+t0) and δν1(t0) (centered at t0). Following the argument in [17] leading to Eq. (34), one arrives at BL2¯=(1/2π)2(d2R0/dtD2)tD=0/R0(0) that yields BL2¯=BLA2+BLF2¯, the same result as given by Eqs. (34) and (35). The corresponding formula for CW laser [25] is a limiting case of Eq. (39) when a(t) is taken to be a constant. Similar to the CW formula, Eq. (39) in general leads to a non-analytical line shape L0(ΔνF)¯ that can only be computed numerically.

B. Impact of transmitted laser pulse energy measurement errors

We now evaluate the DAOD errors arising from the transmitted laser pulse energy measurement errors. We assume that the laser amplifier ASE received by the detector has a uniform optical power spectral density of 2NASE (NASE for each polarization) within an optical bandwidth Bo, and the laser is polarized and has a detected power of Ps during Δt. For ease of analysis, we further assume that both the laser and ASE received by the detector are uniformly distributed among NMM spatial modes and the electrical-field product of any two different spatial modes does not contribute to the detector current (due to mode orthogonality). The detected photon rate then becomes λ(t)=α[Ps+Pnn+Psn] where Pnn(t) is the ASE power, Psn(t) corresponds to the product of the laser electrical field and the ASE electrical field (referred to as s-n beat term). αPsn(t) has a zero mean and double-sided PSD Ssn(f)=2α2PsNASE/NMMwithin the detector electrical bandwidth Be [28]. The photon count variance arising from αPsn(t) is given by

σ2(Ksn)=(Δt/2Δt/2αPsn(t)dt)2¯=ΔtTri(ξΔt)Γ(ξ)dξ=IsnΔt,IsnSsn(f)sinc2(fΔt)Δtdf.
Here Γ(ξ)α2Psn(t)Psn(t+ξ)¯ is the autocorrelation of αPsn(t) and Tri(x) the triangular function. Isn can be approximated by Ssn(0)=2α2PsNASE/NMM when Be1/Δt. On the other hand, Pnn(t) produces a mean photon count Knn¯=α(2NBASEoΔt) with a variance given by [29]

σ2(Knn)=Knn¯(Fe+δc),δc=Knn¯/(2BoΔtNMM)=αN/ASENMM.

To measure Es precisely, Ps needs to be as high as ~1 mW for a high signal to noise ratio. Such a high Ps is also accompanied by a significantNASE. As a result, the variance σ2(Es) of E's arises predominately from the s-n beat term even though Pnn(t) of the ASE produces a photon count variance in great excess of the Poisson value. From Eq. (40), σ2(Es) is found to be σ2(Es)2NASEEs/NMM.

To account for the error δEsEsEs, we expand 1/ E's in SNK3 (defined in section 6) as 1/Es(1/Es)[1δEs/Es+(δEs/Es)2]. Obviously, δEs values among different pulses are uncorrelated. The additional bias and variance of ΔτSNK3 arising from δEs are found to be

bΔτENASENMM(1Esavon1Esavoff)(2<Wavn>),
σΔτE22NASENMM(1Esavon+1Esavoff)<Wavn>.

The additional variance σΔτE2 also scales with <Wavn>~ 1/n. In order to bound the corresponding RRE to < 10−4 (assuming Δτ2, n=1 and NMM=1), the ratio NASE/Esav needs to be kept below 10−8. Consider a practical case where the seed laser oscillator is polarized with a peak power Pseed and pulsewidth Δt=1μs at the input of a single mode preamplifier having a noise figure NF=4.NASE arises mainly from the ASE in the preamplifier that has an equivalent PSD of hνNF/2 (in each polarization) at the input of the amplifier [28]. Using NASE/EsavhνNF/(2PseedΔt), the minimum seed laser peak power Pseed at the input of the preamplifier is estimated to be 16dBm, which can be easily satisfied. Similarly, the additional bias bΔτE can be neglected even when the online and offline pulse energies differ significantly. Consequently, both the bias bΔτE and variance σΔτE2, arising from the laser-ASE beat noise, are neglected in the present model.

C. Noise from background solar radiation

At the lidar receiver, the ASE from the laser transmitter becomes negligible even compared to the weak background solar radiation that has a low optical power spectral density 2Nbgd. Equations (40) and (41) can still be applied to the background sun light, provided that NMM is interpreted as the number of the received speckles and NASE is replaced by Nbgd. Since αNbgd/NMM1, Knn from the background sun light becomes a Poisson process (for Fe = 1). The s-n beat term variance σ2(Ksn)=2αNbgdKs¯/NMM is much smaller than the signal shot noise FeKs¯ and thus can be neglected. For the background sun light, the shot noise variance σ2(Knn)=FeKnn¯ could be reduced to σ2(Ksn) if the optical filter bandwidth Bo could be reduced to αPs/[(1+1/β)NMMFe]. In practice, the optical filter bandwidth is much wider so that σ2(Knn) is much larger than σ2(Ksn).

Acknowledgments

The authors gratefully acknowledge Dr. J. Mao and Dr. X. Sun of NASA Goddard for fruitful discussions and help in atmospheric modeling and lidar link budget calculations. They are also indebted to Dr. A. Amediek of Deutsches Zentrum für Luft- und Raumfahrt (DLR) for sharing surface reflectance measurement data, Dr. J. Abshire and other members of the Goddard CO2 sounder team for their support. This work was supported by the NASA Goddard Internal Research and Development program and the NASA Earth Science Technology Office Instrument Incubator Program.

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14. F. Koyama and K. Oga, “Frequency chirping in external modulators,” J. Lightwave Technol. 6(1), 87–93 (1988). [CrossRef]  

15. J. Caron, Y. Durand, J. L. Bezy, and R. Meynart, “Performance modeling for A-SCOPE, a spaceborne lidar measuring atmospheric CO2,” Proc. SPIE 7479, 74790E-1 (2009). [CrossRef]  

16. C. Stephan, M. Alpers, B. Millet, G. Ehret, P. Flamant, and C. Deniel, “MERLIN: a space-based methane monitor,” Proc. SPIE 8159, 815908, 815908–815915 (2011). [CrossRef]  

17. L. Mandel, “Interpretation of instantaneous frequency,” Am. J. Phys. 42(10), 840–846 (1974). [CrossRef]  

18. W. B. Grant, “Effect of differential spectral reflectance on DIAL measurements using topographic targets,” Appl. Opt. 21(13), 2390–2394 (1982). [CrossRef]   [PubMed]  

19. J. W. Goodman, Statistical Optics (John Wiley & Sons, 1985).

20. N. Z. Hakim, B. E. A. Saleh, and M. C. Teich, “Generalized excess noise factor for avalanche photodiodes of arbitrary structure,” IEEE Trans. Electron. Dev. 37(3), 599–610 (1990). [CrossRef]  

21. J. D. Beck, R. Scritchfield, P. Mitra, W. Sullivan III, A. D. Gleckler, R. Strittmatter, and R. J. Martin, “Linear-mode photon counting with the noiseless gain HgCdTe e-APD,” Proc. SPIE 8033, 80330N, 80330N–15 (2011). [CrossRef]  

22. V. S. R. Gudimetla and M. J. Kavaya, “Special relativity corrections for space-based lidars,” Appl. Opt. 38(30), 6374–6382 (1999). [CrossRef]   [PubMed]  

23. R. N. Clark, “Water frost and ice: the near-infrared spectral reflectance 0.65–2.5 μm,” J. Geophys. Res. 86(B4), 3087–3096 (1981). [CrossRef]  

24. M. Dumont, O. Brissaud, G. Picard, B. Schmitt, J. C. Gallet, and Y. Arnaud, “High-accuracy measurements of snow bidirectional reflectance distribution function at visible and NIR wavelengths – comparison with modeling results,” Atmos. Chem. Phys. Discuss. 9(5), 19279–19311 (2009). [CrossRef]  

25. D. S. Elliott, R. Roy, and S. J. Smith, “Extracavity laser band shape and bandwidth modification,” Phys. Rev. A 26(1), 12–18 (1982). [CrossRef]  

26. G. M. Stéphan, T. T. Tam, S. Blin, P. Besnard, and M. Têtu, “Laser line shape and spectral density of frequency noise,” Phys. Rev. A 71(4), 043809 (2005). [CrossRef]  

27. G. Di Domenico, S. Schilt, and P. Thomann, “Simple approach to the relation between laser frequency noise and laser line shape,” Appl. Opt. 49(25), 4801–4807 (2010). [CrossRef]   [PubMed]  

28. N. A. Olsson, “Lightwave systems with optical amplifiers,” J. Lightwave Technol. 7(7), 1071–1082 (1989). [CrossRef]  

29. L. Mandel, “Fluctuations of photon beams: the distribution of the photo-electrons,” Proc. Phys. Soc. 74(3), 233–243 (1959). [CrossRef]  

References

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  1. R. M. Measures, Laser Remote Sensing: Fundamentals and Applications (Wiley, 1984).
  2. C. Weitkamp, Lidar: Range Resolved Optical Remote Sensing of the Atmosphere (Springer, 2005).
  3. Space Studies Board, National Research Council, Earth Science and Applications from Space: National Imperatives for the Next Decade and Beyond (National Academies Press, 2007).
  4. “A-SCOPE—advanced space carbon and climate observation of planet earth, report for assessment,” ESA-SP1313/1(European Space Agency, 2008), http://esamultimedia.esa.int/docs/SP1313-1_ASCOPE.pdf .
  5. G. Ehret, C. Kiemle, M. Wirth, A. Amediek, A. Fix, and S. Houweling, “Space-borne remote sensing of CO2, CH4, and N2O by integrated path differential absorption lidar: a sensitivity analysis,” Appl. Phys. B90(3-4), 593–608 (2008).
    [CrossRef]
  6. J. B. Abshire, H. Riris, G. Allan, X. Sun, S. R. Kawa, J. Mao, M. Stephen, E. Wilson, and M. A. Krainak, “Laser sounder for global measurement of CO2 concentrations in the troposphere from space,” in Laser Applications to Chemical, Security and Environmental Analysis, OSA Technical Digest (CD) (Optical Society of America, 2008), paper LMA4.
  7. J. B. Abshire, H. Riris, G. R. Allan, C. J. Weaver, J. Mao, X. Sun, W. E. Hasselbrack, S. R. Kawa, and S. Biraud, “Pulsed airborne lidar measurements of atmospheric CO2 column absorption,” Tellus Ser. B, Chem. Phys. Meteorol.62(5), 770–783 (2010).
    [CrossRef]
  8. J. Caron and Y. Durand, “Operating wavelengths optimization for a spaceborne lidar measuring atmospheric CO2.,” Appl. Opt.48(28), 5413–5422 (2009).
    [CrossRef] [PubMed]
  9. M. J. T. Milton and P. T. Woods, “Pulse averaging methods for a laser remote monitoring system using atmospheric backscatter,” Appl. Opt.26(13), 2598–2603 (1987).
    [CrossRef] [PubMed]
  10. A. Amediek, A. Fix, G. Ehret, J. Caron, and Y. Durand, “Airborne lidar reflectance measurements at 1.57 μm in support of the A-SCOPE mission for atmospheric CO2,” Atmos. Meas. Tech.2(2), 755–772 (2009).
    [CrossRef]
  11. J. Mao and S. R. Kawa, “Sensitivity studies for space-based measurement of atmospheric total column carbon dioxide by reflected sunlight,” Appl. Opt.43(4), 914–927 (2004).
    [CrossRef] [PubMed]
  12. S. R. Kawa, J. Mao, J. B. Abshire, G. J. Collatz, X. Sun, and C. J. Weaver, “Simulation studies for a space-based CO2 lidar mission,” Tellus Ser. B, Chem. Phys. Meteorol.62(5), 759–769 (2010).
    [CrossRef]
  13. K. Numata, J. R. Chen, S. T. Wu, J. B. Abshire, and M. A. Krainak, “Frequency stabilization of distributed-feedback laser diodes at 1572 nm for lidar measurements of atmospheric carbon dioxide,” Appl. Opt.50(7), 1047–1056 (2011).
    [CrossRef] [PubMed]
  14. F. Koyama and K. Oga, “Frequency chirping in external modulators,” J. Lightwave Technol.6(1), 87–93 (1988).
    [CrossRef]
  15. J. Caron, Y. Durand, J. L. Bezy, and R. Meynart, “Performance modeling for A-SCOPE, a spaceborne lidar measuring atmospheric CO2,” Proc. SPIE7479, 74790E-1 (2009).
    [CrossRef]
  16. C. Stephan, M. Alpers, B. Millet, G. Ehret, P. Flamant, and C. Deniel, “MERLIN: a space-based methane monitor,” Proc. SPIE8159, 815908, 815908–815915 (2011).
    [CrossRef]
  17. L. Mandel, “Interpretation of instantaneous frequency,” Am. J. Phys.42(10), 840–846 (1974).
    [CrossRef]
  18. W. B. Grant, “Effect of differential spectral reflectance on DIAL measurements using topographic targets,” Appl. Opt.21(13), 2390–2394 (1982).
    [CrossRef] [PubMed]
  19. J. W. Goodman, Statistical Optics (John Wiley & Sons, 1985).
  20. N. Z. Hakim, B. E. A. Saleh, and M. C. Teich, “Generalized excess noise factor for avalanche photodiodes of arbitrary structure,” IEEE Trans. Electron. Dev.37(3), 599–610 (1990).
    [CrossRef]
  21. J. D. Beck, R. Scritchfield, P. Mitra, W. Sullivan, A. D. Gleckler, R. Strittmatter, and R. J. Martin, “Linear-mode photon counting with the noiseless gain HgCdTe e-APD,” Proc. SPIE8033, 80330N, 80330N–15 (2011).
    [CrossRef]
  22. V. S. R. Gudimetla and M. J. Kavaya, “Special relativity corrections for space-based lidars,” Appl. Opt.38(30), 6374–6382 (1999).
    [CrossRef] [PubMed]
  23. R. N. Clark, “Water frost and ice: the near-infrared spectral reflectance 0.65–2.5 μm,” J. Geophys. Res.86(B4), 3087–3096 (1981).
    [CrossRef]
  24. M. Dumont, O. Brissaud, G. Picard, B. Schmitt, J. C. Gallet, and Y. Arnaud, “High-accuracy measurements of snow bidirectional reflectance distribution function at visible and NIR wavelengths – comparison with modeling results,” Atmos. Chem. Phys. Discuss.9(5), 19279–19311 (2009).
    [CrossRef]
  25. D. S. Elliott, R. Roy, and S. J. Smith, “Extracavity laser band shape and bandwidth modification,” Phys. Rev. A26(1), 12–18 (1982).
    [CrossRef]
  26. G. M. Stéphan, T. T. Tam, S. Blin, P. Besnard, and M. Têtu, “Laser line shape and spectral density of frequency noise,” Phys. Rev. A71(4), 043809 (2005).
    [CrossRef]
  27. G. Di Domenico, S. Schilt, and P. Thomann, “Simple approach to the relation between laser frequency noise and laser line shape,” Appl. Opt.49(25), 4801–4807 (2010).
    [CrossRef] [PubMed]
  28. N. A. Olsson, “Lightwave systems with optical amplifiers,” J. Lightwave Technol.7(7), 1071–1082 (1989).
    [CrossRef]
  29. L. Mandel, “Fluctuations of photon beams: the distribution of the photo-electrons,” Proc. Phys. Soc.74(3), 233–243 (1959).
    [CrossRef]

2011

C. Stephan, M. Alpers, B. Millet, G. Ehret, P. Flamant, and C. Deniel, “MERLIN: a space-based methane monitor,” Proc. SPIE8159, 815908, 815908–815915 (2011).
[CrossRef]

J. D. Beck, R. Scritchfield, P. Mitra, W. Sullivan, A. D. Gleckler, R. Strittmatter, and R. J. Martin, “Linear-mode photon counting with the noiseless gain HgCdTe e-APD,” Proc. SPIE8033, 80330N, 80330N–15 (2011).
[CrossRef]

K. Numata, J. R. Chen, S. T. Wu, J. B. Abshire, and M. A. Krainak, “Frequency stabilization of distributed-feedback laser diodes at 1572 nm for lidar measurements of atmospheric carbon dioxide,” Appl. Opt.50(7), 1047–1056 (2011).
[CrossRef] [PubMed]

2010

G. Di Domenico, S. Schilt, and P. Thomann, “Simple approach to the relation between laser frequency noise and laser line shape,” Appl. Opt.49(25), 4801–4807 (2010).
[CrossRef] [PubMed]

S. R. Kawa, J. Mao, J. B. Abshire, G. J. Collatz, X. Sun, and C. J. Weaver, “Simulation studies for a space-based CO2 lidar mission,” Tellus Ser. B, Chem. Phys. Meteorol.62(5), 759–769 (2010).
[CrossRef]

J. B. Abshire, H. Riris, G. R. Allan, C. J. Weaver, J. Mao, X. Sun, W. E. Hasselbrack, S. R. Kawa, and S. Biraud, “Pulsed airborne lidar measurements of atmospheric CO2 column absorption,” Tellus Ser. B, Chem. Phys. Meteorol.62(5), 770–783 (2010).
[CrossRef]

2009

A. Amediek, A. Fix, G. Ehret, J. Caron, and Y. Durand, “Airborne lidar reflectance measurements at 1.57 μm in support of the A-SCOPE mission for atmospheric CO2,” Atmos. Meas. Tech.2(2), 755–772 (2009).
[CrossRef]

J. Caron, Y. Durand, J. L. Bezy, and R. Meynart, “Performance modeling for A-SCOPE, a spaceborne lidar measuring atmospheric CO2,” Proc. SPIE7479, 74790E-1 (2009).
[CrossRef]

M. Dumont, O. Brissaud, G. Picard, B. Schmitt, J. C. Gallet, and Y. Arnaud, “High-accuracy measurements of snow bidirectional reflectance distribution function at visible and NIR wavelengths – comparison with modeling results,” Atmos. Chem. Phys. Discuss.9(5), 19279–19311 (2009).
[CrossRef]

J. Caron and Y. Durand, “Operating wavelengths optimization for a spaceborne lidar measuring atmospheric CO2.,” Appl. Opt.48(28), 5413–5422 (2009).
[CrossRef] [PubMed]

2008

G. Ehret, C. Kiemle, M. Wirth, A. Amediek, A. Fix, and S. Houweling, “Space-borne remote sensing of CO2, CH4, and N2O by integrated path differential absorption lidar: a sensitivity analysis,” Appl. Phys. B90(3-4), 593–608 (2008).
[CrossRef]

2005

G. M. Stéphan, T. T. Tam, S. Blin, P. Besnard, and M. Têtu, “Laser line shape and spectral density of frequency noise,” Phys. Rev. A71(4), 043809 (2005).
[CrossRef]

2004

1999

1990

N. Z. Hakim, B. E. A. Saleh, and M. C. Teich, “Generalized excess noise factor for avalanche photodiodes of arbitrary structure,” IEEE Trans. Electron. Dev.37(3), 599–610 (1990).
[CrossRef]

1989

N. A. Olsson, “Lightwave systems with optical amplifiers,” J. Lightwave Technol.7(7), 1071–1082 (1989).
[CrossRef]

1988

F. Koyama and K. Oga, “Frequency chirping in external modulators,” J. Lightwave Technol.6(1), 87–93 (1988).
[CrossRef]

1987

1982

W. B. Grant, “Effect of differential spectral reflectance on DIAL measurements using topographic targets,” Appl. Opt.21(13), 2390–2394 (1982).
[CrossRef] [PubMed]

D. S. Elliott, R. Roy, and S. J. Smith, “Extracavity laser band shape and bandwidth modification,” Phys. Rev. A26(1), 12–18 (1982).
[CrossRef]

1981

R. N. Clark, “Water frost and ice: the near-infrared spectral reflectance 0.65–2.5 μm,” J. Geophys. Res.86(B4), 3087–3096 (1981).
[CrossRef]

1974

L. Mandel, “Interpretation of instantaneous frequency,” Am. J. Phys.42(10), 840–846 (1974).
[CrossRef]

1959

L. Mandel, “Fluctuations of photon beams: the distribution of the photo-electrons,” Proc. Phys. Soc.74(3), 233–243 (1959).
[CrossRef]

Abshire, J. B.

K. Numata, J. R. Chen, S. T. Wu, J. B. Abshire, and M. A. Krainak, “Frequency stabilization of distributed-feedback laser diodes at 1572 nm for lidar measurements of atmospheric carbon dioxide,” Appl. Opt.50(7), 1047–1056 (2011).
[CrossRef] [PubMed]

S. R. Kawa, J. Mao, J. B. Abshire, G. J. Collatz, X. Sun, and C. J. Weaver, “Simulation studies for a space-based CO2 lidar mission,” Tellus Ser. B, Chem. Phys. Meteorol.62(5), 759–769 (2010).
[CrossRef]

J. B. Abshire, H. Riris, G. R. Allan, C. J. Weaver, J. Mao, X. Sun, W. E. Hasselbrack, S. R. Kawa, and S. Biraud, “Pulsed airborne lidar measurements of atmospheric CO2 column absorption,” Tellus Ser. B, Chem. Phys. Meteorol.62(5), 770–783 (2010).
[CrossRef]

Allan, G. R.

J. B. Abshire, H. Riris, G. R. Allan, C. J. Weaver, J. Mao, X. Sun, W. E. Hasselbrack, S. R. Kawa, and S. Biraud, “Pulsed airborne lidar measurements of atmospheric CO2 column absorption,” Tellus Ser. B, Chem. Phys. Meteorol.62(5), 770–783 (2010).
[CrossRef]

Alpers, M.

C. Stephan, M. Alpers, B. Millet, G. Ehret, P. Flamant, and C. Deniel, “MERLIN: a space-based methane monitor,” Proc. SPIE8159, 815908, 815908–815915 (2011).
[CrossRef]

Amediek, A.

A. Amediek, A. Fix, G. Ehret, J. Caron, and Y. Durand, “Airborne lidar reflectance measurements at 1.57 μm in support of the A-SCOPE mission for atmospheric CO2,” Atmos. Meas. Tech.2(2), 755–772 (2009).
[CrossRef]

G. Ehret, C. Kiemle, M. Wirth, A. Amediek, A. Fix, and S. Houweling, “Space-borne remote sensing of CO2, CH4, and N2O by integrated path differential absorption lidar: a sensitivity analysis,” Appl. Phys. B90(3-4), 593–608 (2008).
[CrossRef]

Arnaud, Y.

M. Dumont, O. Brissaud, G. Picard, B. Schmitt, J. C. Gallet, and Y. Arnaud, “High-accuracy measurements of snow bidirectional reflectance distribution function at visible and NIR wavelengths – comparison with modeling results,” Atmos. Chem. Phys. Discuss.9(5), 19279–19311 (2009).
[CrossRef]

Beck, J. D.

J. D. Beck, R. Scritchfield, P. Mitra, W. Sullivan, A. D. Gleckler, R. Strittmatter, and R. J. Martin, “Linear-mode photon counting with the noiseless gain HgCdTe e-APD,” Proc. SPIE8033, 80330N, 80330N–15 (2011).
[CrossRef]

Besnard, P.

G. M. Stéphan, T. T. Tam, S. Blin, P. Besnard, and M. Têtu, “Laser line shape and spectral density of frequency noise,” Phys. Rev. A71(4), 043809 (2005).
[CrossRef]

Bezy, J. L.

J. Caron, Y. Durand, J. L. Bezy, and R. Meynart, “Performance modeling for A-SCOPE, a spaceborne lidar measuring atmospheric CO2,” Proc. SPIE7479, 74790E-1 (2009).
[CrossRef]

Biraud, S.

J. B. Abshire, H. Riris, G. R. Allan, C. J. Weaver, J. Mao, X. Sun, W. E. Hasselbrack, S. R. Kawa, and S. Biraud, “Pulsed airborne lidar measurements of atmospheric CO2 column absorption,” Tellus Ser. B, Chem. Phys. Meteorol.62(5), 770–783 (2010).
[CrossRef]

Blin, S.

G. M. Stéphan, T. T. Tam, S. Blin, P. Besnard, and M. Têtu, “Laser line shape and spectral density of frequency noise,” Phys. Rev. A71(4), 043809 (2005).
[CrossRef]

Brissaud, O.

M. Dumont, O. Brissaud, G. Picard, B. Schmitt, J. C. Gallet, and Y. Arnaud, “High-accuracy measurements of snow bidirectional reflectance distribution function at visible and NIR wavelengths – comparison with modeling results,” Atmos. Chem. Phys. Discuss.9(5), 19279–19311 (2009).
[CrossRef]

Caron, J.

A. Amediek, A. Fix, G. Ehret, J. Caron, and Y. Durand, “Airborne lidar reflectance measurements at 1.57 μm in support of the A-SCOPE mission for atmospheric CO2,” Atmos. Meas. Tech.2(2), 755–772 (2009).
[CrossRef]

J. Caron and Y. Durand, “Operating wavelengths optimization for a spaceborne lidar measuring atmospheric CO2.,” Appl. Opt.48(28), 5413–5422 (2009).
[CrossRef] [PubMed]

J. Caron, Y. Durand, J. L. Bezy, and R. Meynart, “Performance modeling for A-SCOPE, a spaceborne lidar measuring atmospheric CO2,” Proc. SPIE7479, 74790E-1 (2009).
[CrossRef]

Chen, J. R.

Clark, R. N.

R. N. Clark, “Water frost and ice: the near-infrared spectral reflectance 0.65–2.5 μm,” J. Geophys. Res.86(B4), 3087–3096 (1981).
[CrossRef]

Collatz, G. J.

S. R. Kawa, J. Mao, J. B. Abshire, G. J. Collatz, X. Sun, and C. J. Weaver, “Simulation studies for a space-based CO2 lidar mission,” Tellus Ser. B, Chem. Phys. Meteorol.62(5), 759–769 (2010).
[CrossRef]

Deniel, C.

C. Stephan, M. Alpers, B. Millet, G. Ehret, P. Flamant, and C. Deniel, “MERLIN: a space-based methane monitor,” Proc. SPIE8159, 815908, 815908–815915 (2011).
[CrossRef]

Di Domenico, G.

Dumont, M.

M. Dumont, O. Brissaud, G. Picard, B. Schmitt, J. C. Gallet, and Y. Arnaud, “High-accuracy measurements of snow bidirectional reflectance distribution function at visible and NIR wavelengths – comparison with modeling results,” Atmos. Chem. Phys. Discuss.9(5), 19279–19311 (2009).
[CrossRef]

Durand, Y.

A. Amediek, A. Fix, G. Ehret, J. Caron, and Y. Durand, “Airborne lidar reflectance measurements at 1.57 μm in support of the A-SCOPE mission for atmospheric CO2,” Atmos. Meas. Tech.2(2), 755–772 (2009).
[CrossRef]

J. Caron and Y. Durand, “Operating wavelengths optimization for a spaceborne lidar measuring atmospheric CO2.,” Appl. Opt.48(28), 5413–5422 (2009).
[CrossRef] [PubMed]

J. Caron, Y. Durand, J. L. Bezy, and R. Meynart, “Performance modeling for A-SCOPE, a spaceborne lidar measuring atmospheric CO2,” Proc. SPIE7479, 74790E-1 (2009).
[CrossRef]

Ehret, G.

C. Stephan, M. Alpers, B. Millet, G. Ehret, P. Flamant, and C. Deniel, “MERLIN: a space-based methane monitor,” Proc. SPIE8159, 815908, 815908–815915 (2011).
[CrossRef]

A. Amediek, A. Fix, G. Ehret, J. Caron, and Y. Durand, “Airborne lidar reflectance measurements at 1.57 μm in support of the A-SCOPE mission for atmospheric CO2,” Atmos. Meas. Tech.2(2), 755–772 (2009).
[CrossRef]

G. Ehret, C. Kiemle, M. Wirth, A. Amediek, A. Fix, and S. Houweling, “Space-borne remote sensing of CO2, CH4, and N2O by integrated path differential absorption lidar: a sensitivity analysis,” Appl. Phys. B90(3-4), 593–608 (2008).
[CrossRef]

Elliott, D. S.

D. S. Elliott, R. Roy, and S. J. Smith, “Extracavity laser band shape and bandwidth modification,” Phys. Rev. A26(1), 12–18 (1982).
[CrossRef]

Fix, A.

A. Amediek, A. Fix, G. Ehret, J. Caron, and Y. Durand, “Airborne lidar reflectance measurements at 1.57 μm in support of the A-SCOPE mission for atmospheric CO2,” Atmos. Meas. Tech.2(2), 755–772 (2009).
[CrossRef]

G. Ehret, C. Kiemle, M. Wirth, A. Amediek, A. Fix, and S. Houweling, “Space-borne remote sensing of CO2, CH4, and N2O by integrated path differential absorption lidar: a sensitivity analysis,” Appl. Phys. B90(3-4), 593–608 (2008).
[CrossRef]

Flamant, P.

C. Stephan, M. Alpers, B. Millet, G. Ehret, P. Flamant, and C. Deniel, “MERLIN: a space-based methane monitor,” Proc. SPIE8159, 815908, 815908–815915 (2011).
[CrossRef]

Gallet, J. C.

M. Dumont, O. Brissaud, G. Picard, B. Schmitt, J. C. Gallet, and Y. Arnaud, “High-accuracy measurements of snow bidirectional reflectance distribution function at visible and NIR wavelengths – comparison with modeling results,” Atmos. Chem. Phys. Discuss.9(5), 19279–19311 (2009).
[CrossRef]

Gleckler, A. D.

J. D. Beck, R. Scritchfield, P. Mitra, W. Sullivan, A. D. Gleckler, R. Strittmatter, and R. J. Martin, “Linear-mode photon counting with the noiseless gain HgCdTe e-APD,” Proc. SPIE8033, 80330N, 80330N–15 (2011).
[CrossRef]

Grant, W. B.

Gudimetla, V. S. R.

Hakim, N. Z.

N. Z. Hakim, B. E. A. Saleh, and M. C. Teich, “Generalized excess noise factor for avalanche photodiodes of arbitrary structure,” IEEE Trans. Electron. Dev.37(3), 599–610 (1990).
[CrossRef]

Hasselbrack, W. E.

J. B. Abshire, H. Riris, G. R. Allan, C. J. Weaver, J. Mao, X. Sun, W. E. Hasselbrack, S. R. Kawa, and S. Biraud, “Pulsed airborne lidar measurements of atmospheric CO2 column absorption,” Tellus Ser. B, Chem. Phys. Meteorol.62(5), 770–783 (2010).
[CrossRef]

Houweling, S.

G. Ehret, C. Kiemle, M. Wirth, A. Amediek, A. Fix, and S. Houweling, “Space-borne remote sensing of CO2, CH4, and N2O by integrated path differential absorption lidar: a sensitivity analysis,” Appl. Phys. B90(3-4), 593–608 (2008).
[CrossRef]

Kavaya, M. J.

Kawa, S. R.

J. B. Abshire, H. Riris, G. R. Allan, C. J. Weaver, J. Mao, X. Sun, W. E. Hasselbrack, S. R. Kawa, and S. Biraud, “Pulsed airborne lidar measurements of atmospheric CO2 column absorption,” Tellus Ser. B, Chem. Phys. Meteorol.62(5), 770–783 (2010).
[CrossRef]

S. R. Kawa, J. Mao, J. B. Abshire, G. J. Collatz, X. Sun, and C. J. Weaver, “Simulation studies for a space-based CO2 lidar mission,” Tellus Ser. B, Chem. Phys. Meteorol.62(5), 759–769 (2010).
[CrossRef]

J. Mao and S. R. Kawa, “Sensitivity studies for space-based measurement of atmospheric total column carbon dioxide by reflected sunlight,” Appl. Opt.43(4), 914–927 (2004).
[CrossRef] [PubMed]

Kiemle, C.

G. Ehret, C. Kiemle, M. Wirth, A. Amediek, A. Fix, and S. Houweling, “Space-borne remote sensing of CO2, CH4, and N2O by integrated path differential absorption lidar: a sensitivity analysis,” Appl. Phys. B90(3-4), 593–608 (2008).
[CrossRef]

Koyama, F.

F. Koyama and K. Oga, “Frequency chirping in external modulators,” J. Lightwave Technol.6(1), 87–93 (1988).
[CrossRef]

Krainak, M. A.

Mandel, L.

L. Mandel, “Interpretation of instantaneous frequency,” Am. J. Phys.42(10), 840–846 (1974).
[CrossRef]

L. Mandel, “Fluctuations of photon beams: the distribution of the photo-electrons,” Proc. Phys. Soc.74(3), 233–243 (1959).
[CrossRef]

Mao, J.

J. B. Abshire, H. Riris, G. R. Allan, C. J. Weaver, J. Mao, X. Sun, W. E. Hasselbrack, S. R. Kawa, and S. Biraud, “Pulsed airborne lidar measurements of atmospheric CO2 column absorption,” Tellus Ser. B, Chem. Phys. Meteorol.62(5), 770–783 (2010).
[CrossRef]

S. R. Kawa, J. Mao, J. B. Abshire, G. J. Collatz, X. Sun, and C. J. Weaver, “Simulation studies for a space-based CO2 lidar mission,” Tellus Ser. B, Chem. Phys. Meteorol.62(5), 759–769 (2010).
[CrossRef]

J. Mao and S. R. Kawa, “Sensitivity studies for space-based measurement of atmospheric total column carbon dioxide by reflected sunlight,” Appl. Opt.43(4), 914–927 (2004).
[CrossRef] [PubMed]

Martin, R. J.

J. D. Beck, R. Scritchfield, P. Mitra, W. Sullivan, A. D. Gleckler, R. Strittmatter, and R. J. Martin, “Linear-mode photon counting with the noiseless gain HgCdTe e-APD,” Proc. SPIE8033, 80330N, 80330N–15 (2011).
[CrossRef]

Meynart, R.

J. Caron, Y. Durand, J. L. Bezy, and R. Meynart, “Performance modeling for A-SCOPE, a spaceborne lidar measuring atmospheric CO2,” Proc. SPIE7479, 74790E-1 (2009).
[CrossRef]

Millet, B.

C. Stephan, M. Alpers, B. Millet, G. Ehret, P. Flamant, and C. Deniel, “MERLIN: a space-based methane monitor,” Proc. SPIE8159, 815908, 815908–815915 (2011).
[CrossRef]

Milton, M. J. T.

Mitra, P.

J. D. Beck, R. Scritchfield, P. Mitra, W. Sullivan, A. D. Gleckler, R. Strittmatter, and R. J. Martin, “Linear-mode photon counting with the noiseless gain HgCdTe e-APD,” Proc. SPIE8033, 80330N, 80330N–15 (2011).
[CrossRef]

Numata, K.

Oga, K.

F. Koyama and K. Oga, “Frequency chirping in external modulators,” J. Lightwave Technol.6(1), 87–93 (1988).
[CrossRef]

Olsson, N. A.

N. A. Olsson, “Lightwave systems with optical amplifiers,” J. Lightwave Technol.7(7), 1071–1082 (1989).
[CrossRef]

Picard, G.

M. Dumont, O. Brissaud, G. Picard, B. Schmitt, J. C. Gallet, and Y. Arnaud, “High-accuracy measurements of snow bidirectional reflectance distribution function at visible and NIR wavelengths – comparison with modeling results,” Atmos. Chem. Phys. Discuss.9(5), 19279–19311 (2009).
[CrossRef]

Riris, H.

J. B. Abshire, H. Riris, G. R. Allan, C. J. Weaver, J. Mao, X. Sun, W. E. Hasselbrack, S. R. Kawa, and S. Biraud, “Pulsed airborne lidar measurements of atmospheric CO2 column absorption,” Tellus Ser. B, Chem. Phys. Meteorol.62(5), 770–783 (2010).
[CrossRef]

Roy, R.

D. S. Elliott, R. Roy, and S. J. Smith, “Extracavity laser band shape and bandwidth modification,” Phys. Rev. A26(1), 12–18 (1982).
[CrossRef]

Saleh, B. E. A.

N. Z. Hakim, B. E. A. Saleh, and M. C. Teich, “Generalized excess noise factor for avalanche photodiodes of arbitrary structure,” IEEE Trans. Electron. Dev.37(3), 599–610 (1990).
[CrossRef]

Schilt, S.

Schmitt, B.

M. Dumont, O. Brissaud, G. Picard, B. Schmitt, J. C. Gallet, and Y. Arnaud, “High-accuracy measurements of snow bidirectional reflectance distribution function at visible and NIR wavelengths – comparison with modeling results,” Atmos. Chem. Phys. Discuss.9(5), 19279–19311 (2009).
[CrossRef]

Scritchfield, R.

J. D. Beck, R. Scritchfield, P. Mitra, W. Sullivan, A. D. Gleckler, R. Strittmatter, and R. J. Martin, “Linear-mode photon counting with the noiseless gain HgCdTe e-APD,” Proc. SPIE8033, 80330N, 80330N–15 (2011).
[CrossRef]

Smith, S. J.

D. S. Elliott, R. Roy, and S. J. Smith, “Extracavity laser band shape and bandwidth modification,” Phys. Rev. A26(1), 12–18 (1982).
[CrossRef]

Stephan, C.

C. Stephan, M. Alpers, B. Millet, G. Ehret, P. Flamant, and C. Deniel, “MERLIN: a space-based methane monitor,” Proc. SPIE8159, 815908, 815908–815915 (2011).
[CrossRef]

Stéphan, G. M.

G. M. Stéphan, T. T. Tam, S. Blin, P. Besnard, and M. Têtu, “Laser line shape and spectral density of frequency noise,” Phys. Rev. A71(4), 043809 (2005).
[CrossRef]

Strittmatter, R.

J. D. Beck, R. Scritchfield, P. Mitra, W. Sullivan, A. D. Gleckler, R. Strittmatter, and R. J. Martin, “Linear-mode photon counting with the noiseless gain HgCdTe e-APD,” Proc. SPIE8033, 80330N, 80330N–15 (2011).
[CrossRef]

Sullivan, W.

J. D. Beck, R. Scritchfield, P. Mitra, W. Sullivan, A. D. Gleckler, R. Strittmatter, and R. J. Martin, “Linear-mode photon counting with the noiseless gain HgCdTe e-APD,” Proc. SPIE8033, 80330N, 80330N–15 (2011).
[CrossRef]

Sun, X.

J. B. Abshire, H. Riris, G. R. Allan, C. J. Weaver, J. Mao, X. Sun, W. E. Hasselbrack, S. R. Kawa, and S. Biraud, “Pulsed airborne lidar measurements of atmospheric CO2 column absorption,” Tellus Ser. B, Chem. Phys. Meteorol.62(5), 770–783 (2010).
[CrossRef]

S. R. Kawa, J. Mao, J. B. Abshire, G. J. Collatz, X. Sun, and C. J. Weaver, “Simulation studies for a space-based CO2 lidar mission,” Tellus Ser. B, Chem. Phys. Meteorol.62(5), 759–769 (2010).
[CrossRef]

Tam, T. T.

G. M. Stéphan, T. T. Tam, S. Blin, P. Besnard, and M. Têtu, “Laser line shape and spectral density of frequency noise,” Phys. Rev. A71(4), 043809 (2005).
[CrossRef]

Teich, M. C.

N. Z. Hakim, B. E. A. Saleh, and M. C. Teich, “Generalized excess noise factor for avalanche photodiodes of arbitrary structure,” IEEE Trans. Electron. Dev.37(3), 599–610 (1990).
[CrossRef]

Têtu, M.

G. M. Stéphan, T. T. Tam, S. Blin, P. Besnard, and M. Têtu, “Laser line shape and spectral density of frequency noise,” Phys. Rev. A71(4), 043809 (2005).
[CrossRef]

Thomann, P.

Weaver, C. J.

S. R. Kawa, J. Mao, J. B. Abshire, G. J. Collatz, X. Sun, and C. J. Weaver, “Simulation studies for a space-based CO2 lidar mission,” Tellus Ser. B, Chem. Phys. Meteorol.62(5), 759–769 (2010).
[CrossRef]

J. B. Abshire, H. Riris, G. R. Allan, C. J. Weaver, J. Mao, X. Sun, W. E. Hasselbrack, S. R. Kawa, and S. Biraud, “Pulsed airborne lidar measurements of atmospheric CO2 column absorption,” Tellus Ser. B, Chem. Phys. Meteorol.62(5), 770–783 (2010).
[CrossRef]

Wirth, M.

G. Ehret, C. Kiemle, M. Wirth, A. Amediek, A. Fix, and S. Houweling, “Space-borne remote sensing of CO2, CH4, and N2O by integrated path differential absorption lidar: a sensitivity analysis,” Appl. Phys. B90(3-4), 593–608 (2008).
[CrossRef]

Woods, P. T.

Wu, S. T.

Am. J. Phys.

L. Mandel, “Interpretation of instantaneous frequency,” Am. J. Phys.42(10), 840–846 (1974).
[CrossRef]

Appl. Opt.

Appl. Phys. B

G. Ehret, C. Kiemle, M. Wirth, A. Amediek, A. Fix, and S. Houweling, “Space-borne remote sensing of CO2, CH4, and N2O by integrated path differential absorption lidar: a sensitivity analysis,” Appl. Phys. B90(3-4), 593–608 (2008).
[CrossRef]

Atmos. Chem. Phys. Discuss.

M. Dumont, O. Brissaud, G. Picard, B. Schmitt, J. C. Gallet, and Y. Arnaud, “High-accuracy measurements of snow bidirectional reflectance distribution function at visible and NIR wavelengths – comparison with modeling results,” Atmos. Chem. Phys. Discuss.9(5), 19279–19311 (2009).
[CrossRef]

Atmos. Meas. Tech.

A. Amediek, A. Fix, G. Ehret, J. Caron, and Y. Durand, “Airborne lidar reflectance measurements at 1.57 μm in support of the A-SCOPE mission for atmospheric CO2,” Atmos. Meas. Tech.2(2), 755–772 (2009).
[CrossRef]

IEEE Trans. Electron. Dev.

N. Z. Hakim, B. E. A. Saleh, and M. C. Teich, “Generalized excess noise factor for avalanche photodiodes of arbitrary structure,” IEEE Trans. Electron. Dev.37(3), 599–610 (1990).
[CrossRef]

J. Geophys. Res.

R. N. Clark, “Water frost and ice: the near-infrared spectral reflectance 0.65–2.5 μm,” J. Geophys. Res.86(B4), 3087–3096 (1981).
[CrossRef]

J. Lightwave Technol.

F. Koyama and K. Oga, “Frequency chirping in external modulators,” J. Lightwave Technol.6(1), 87–93 (1988).
[CrossRef]

N. A. Olsson, “Lightwave systems with optical amplifiers,” J. Lightwave Technol.7(7), 1071–1082 (1989).
[CrossRef]

Phys. Rev. A

D. S. Elliott, R. Roy, and S. J. Smith, “Extracavity laser band shape and bandwidth modification,” Phys. Rev. A26(1), 12–18 (1982).
[CrossRef]

G. M. Stéphan, T. T. Tam, S. Blin, P. Besnard, and M. Têtu, “Laser line shape and spectral density of frequency noise,” Phys. Rev. A71(4), 043809 (2005).
[CrossRef]

Proc. Phys. Soc.

L. Mandel, “Fluctuations of photon beams: the distribution of the photo-electrons,” Proc. Phys. Soc.74(3), 233–243 (1959).
[CrossRef]

Proc. SPIE

J. Caron, Y. Durand, J. L. Bezy, and R. Meynart, “Performance modeling for A-SCOPE, a spaceborne lidar measuring atmospheric CO2,” Proc. SPIE7479, 74790E-1 (2009).
[CrossRef]

C. Stephan, M. Alpers, B. Millet, G. Ehret, P. Flamant, and C. Deniel, “MERLIN: a space-based methane monitor,” Proc. SPIE8159, 815908, 815908–815915 (2011).
[CrossRef]

J. D. Beck, R. Scritchfield, P. Mitra, W. Sullivan, A. D. Gleckler, R. Strittmatter, and R. J. Martin, “Linear-mode photon counting with the noiseless gain HgCdTe e-APD,” Proc. SPIE8033, 80330N, 80330N–15 (2011).
[CrossRef]

Tellus Ser. B, Chem. Phys. Meteorol.

J. B. Abshire, H. Riris, G. R. Allan, C. J. Weaver, J. Mao, X. Sun, W. E. Hasselbrack, S. R. Kawa, and S. Biraud, “Pulsed airborne lidar measurements of atmospheric CO2 column absorption,” Tellus Ser. B, Chem. Phys. Meteorol.62(5), 770–783 (2010).
[CrossRef]

S. R. Kawa, J. Mao, J. B. Abshire, G. J. Collatz, X. Sun, and C. J. Weaver, “Simulation studies for a space-based CO2 lidar mission,” Tellus Ser. B, Chem. Phys. Meteorol.62(5), 759–769 (2010).
[CrossRef]

Other

J. B. Abshire, H. Riris, G. Allan, X. Sun, S. R. Kawa, J. Mao, M. Stephen, E. Wilson, and M. A. Krainak, “Laser sounder for global measurement of CO2 concentrations in the troposphere from space,” in Laser Applications to Chemical, Security and Environmental Analysis, OSA Technical Digest (CD) (Optical Society of America, 2008), paper LMA4.

R. M. Measures, Laser Remote Sensing: Fundamentals and Applications (Wiley, 1984).

C. Weitkamp, Lidar: Range Resolved Optical Remote Sensing of the Atmosphere (Springer, 2005).

Space Studies Board, National Research Council, Earth Science and Applications from Space: National Imperatives for the Next Decade and Beyond (National Academies Press, 2007).

“A-SCOPE—advanced space carbon and climate observation of planet earth, report for assessment,” ESA-SP1313/1(European Space Agency, 2008), http://esamultimedia.esa.int/docs/SP1313-1_ASCOPE.pdf .

J. W. Goodman, Statistical Optics (John Wiley & Sons, 1985).

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Figures (6)

Fig. 1
Fig. 1

The laser transmitter provides the wavelength-stepped pulse train (left) to repeatedly measure at 8 points across the 1572.335 nm CO2 absorption line (right).

Fig. 2
Fig. 2

The nominal laser frequency noise PSD S δν (f)= h 1 /f+ h 0 (solid red) is suppressed to S δν (f)|H(f )| 2 (dotted blue) by averaging within a 1-μs top-hat pulse and further reduced to S δν (f)|H(f )| 2 W avn (f) (dashed black) by averaging across 1000 identical pulses separated by a period of 1 ms.

Fig. 3
Fig. 3

(left) The standard deviation of the absolute frequency of the slave laser and frequency offset between the slave and master lasers measured within a gating time from 1 μs to 1 ms; (right) Both frequencies were also measured periodically within a 1-µs gating time with a 1-ms period and averaged across multiple measurements. The standard deviation of each averaged frequency, computed from 100 averaged samples, is plotted as a function of the number of measurements being averaged.

Fig. 4
Fig. 4

Relative errors for atmospheric CO2 DAOD measurements: (left) (Δτ - Δτ0)/Δτ0 due to the finite linewidth 1/Δt of a top-hat pulse shape (calculated from Eqs. (5) and (6)); (right) laser line-center frequency noise contributions to the RRE for σ(δνnon) = 0.1 MHz (dashed red, calculated from Eq. (21)), and RSE ln(Rb)/Δτ for σ(δν1) = 10 MHz (dotted blue, from Eq. (18)) as functions of the online laser frequency offset from the absorption line center. The two-way optical depth τ (solid green) is also plotted, with blue dots marking the same online and absorption line-center frequency points used on the left.

Fig. 5
Fig. 5

The RRE σ(ΔτSNK2_av) /Δτ for atmospheric CO2 DAOD measurement (solid black) as a function of the two-way optical depth, computed from Eqs. (29) and (30) using parameters listed in Table 2. The blue dots mark the three online frequency points used in Fig. 4. Also plotted are partial contributions to this RRE from the shot noise (solid grey), frequency noise (dashed red), solar background (dotted brown), receiver circuitry noise (dash-dotted green), and detector dark count (long-dashed blue). The standard deviation of the effective frequency noise σ(δνnon) averaged in 10 s is 0.23 MHz for the above calculations. When it is increased to 1 MHz, the RRE (dotted black) rises above the 0.1% target (dashed black).

Fig. 6
Fig. 6

(left) Relative surface reflectance: measured data (grey) and its running average over a length of 50 m (blue), courtesy of A. Amediek of Deutsches Zentrum für Luft- und Raumfahrt (DLR); (right) ln( R A ) calculated from surface reflectance data for the worst case. For each online wavelength channel, two ln( R A ) are calculated from both offline sum i=0 n1 A s off (i) for 1-s averaging time and the average is plotted (blue dot). An average of 10 consecutive ln( R A ) values, each computed as above over 0.1 s averaging time, is also plotted (red square).

Tables (3)

Tables Icon

Table 1 RRE Targets, Relevant Parameters, and Resulting Effective Laser Frequency Noise Requirement for Atmospheric CO2 DAOD Measurement

Tables Icon

Table 2 Parameters Used to Evaluate the RRE of Δτ as a Function of τ

Tables Icon

Table 3 Parameters Used for Evaluation of the Impact of Surface Reflectance Variation

Equations (43)

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u(t)=a(t t 0 )exp( j[2π ν ¯ t+ϕ(t)] ).
ν c ( t 0 ) 0 ν F L N ( ν F )d ν F = ν(t)h(t t 0 )dt .
S δν1 (f)= S δν (f) | H(f) | 2 ,
dL( ν F ,l)=L( ν F ,l) σ 0 ( ν F ,l)N(l)dl.
E(l)=E(0)exp[ τ( ν c ,l) ], τ( ν c ,l) 0 l σ eff ( ν c ,l')N(l')dl' .
σ eff ( ν c ,l) 0 σ 0 ( ν F ,l) L N ( ν F ,l)d ν F = 0 σ 0 ( ν F ,l)T( ν F ,l)L( ν F ,0)d ν F 0 T( ν F ,l)L( ν F ,0)d ν F , T( ν F ,l)exp[ τ 0 ( ν F ,l) ], τ 0 ( ν F ,l) 0 l σ 0 ( ν F ,l')N(l')dl' .
W s = E s A s exp[ τ( ν c ,2 r G ) ],
K ¯ =α W ¯ ,
σ 2 (K)= F e K ¯ + α 2 σ 2 (W).
W s = E s A s exp[ τ( ν ¯ ,2 r G ) ][ 1(dτ/d ν c ) δ ν1 ( t 0 )+b δ ν1 2 ( t 0 ) ],
K s ¯ =α W s ¯ =α E s A s exp[ τ( ν ¯ ,2 r G ) ][ 1+b σ 2 ( δ ν1 ) ],
σ 2 ( K s )= F e K s ¯ + α 2 σ 2 ( W s )= F e K s ¯ + ( K s ¯ ) 2 ( dτ/d ν c ) 2 σ 2 ( δ ν1 ),
S NE1 i=0 n1 W s (i) E s (i) =exp[ τ( ν ¯ ,2 r G ) ]( i=0 n1 A s (i) )[ 1 dτ d ν c δ νn (t)+b Q n (t) ], Q n (t) 1 i=0 n1 A s (i) i=0 n1 [ A s (i) δ ν1 2 (t+i× t p ) ] ,
δ νn (t) 1 i=0 n1 A s (i) i=0 n1 [ A s (i) δ ν1 (t+i× t p ) ] .
S NE1 ¯ =exp[ τ( ν ¯ ,2 r G ) ]( i=0 n1 A s (i) )( 1+b σ 2 ( δ ν1 ) ),
σ 2 ( S NE1 )= ( S NE1 ¯ ) 2 ( dτ / d ν c ) 2 σ 2 ( δ νn ).
σ 2 ( S NK1 )= F e S NNK1 ¯ + α 2 σ 2 ( S NE1 ), S NNK1 i=0 n1 K s (i) E s 2 (i) S NK1 / E sav S K / E sav 2 .
Δτ=ln( S NK1 on ¯ S NK1 off ¯ )+ln( R A )+ln( R b ),
Δ τ SNK1 ln( S NK1 on S NK1 off ).
b Δτ_SNK1 ln( R A )ln( R b )+ F e 2 ( S NNK1 on ¯ S NK1 on ¯ 2 S NNK1 off ¯ S NK1 off ¯ 2 )+ 1 2 ( dτ d ν c ) on 2 σ 2 ( δ νn on ),
σ 2 (Δ τ SNK1 ) σ 2 ( S NK1 on ) S NK1 on ¯ 2 + σ 2 ( S NK1 off ) S NK1 off ¯ 2 F e S K off ¯ [1+exp(Δτ)]+ ( dτ d ν c ) on 2 σ 2 ( δ νn on ).
S δνn (f)= S δν (f)|H(f )| 2 W avn (f), W avn (f) | 1 i=0 n1 A s (i) i=0 n1 A s (i)exp(j2πf t p ×i) | 2 .
< W avn >= i=0 n1 A s 2 (i) / ( i=0 n1 A s (i) ) 2 .
σ 2 ( K T )= I δ i Δt/ ( M e e ) 2 , I δi S δ i (f) sinc 2 (fΔt)Δtdf,
σ 2 (K ' s ) F e K s ¯ + α 2 σ 2 ( W s )+ λ bgd Δt, λ bgd [ 2α N bgd B o F e + F d λ d + I δi / ( M e e ) 2 ]( 1+1/β ).
σ 2 ( S NK2 ) ( S NK2 ¯ ) 2 F e S NNK2 ¯ ( S NK2 ¯ ) 2 + ( dτ d ν c ) 2 σ 2 ( δ νn )+ S NN2 ¯ ( S NK2 ¯ ) 2 λ bgd Δt,
Δ τ SNK2 ln( S NK2 on S NK2 off )+ C Δτ_SNK2 , C Δτ_SNK2 F e 2 ( S NNK2 off S NK2 off 2 S NNK2 on S NK2 on 2 )+ λ bgd Δt 2 ( S NN2 off S NK2 off 2 S NN2 on S NK2 on 2 ).
b Δτ_SNK2 ln( R A )ln( R b )+ ( dτ/d ν c ) on 2 σ 2 ( δ νn on )/2+ b C_Δτ_SNK2 , b C_Δτ_SNK2 1 2 ( F e / S K' on ¯ ) 2 [ 1exp(2Δτ) ] 3 2 F e n λ bgd Δt[ 1exp(3Δτ) ]/ S K' on ¯ 3 ,
σ 2 (Δ τ SNK2 ) σ ΔτDET 2 + ( dτ/d ν c ) on 2 σ 2 ( δ νn on ), σ ΔτDET 2 F e [1+exp(Δτ)]/ S K' off ¯ +n λ bgd Δt[1+exp(2Δτ)]/ S K' off ¯ 2 .
Δ τ SNK2_av 1 m k=0 m1 Δ τ SNK2 (k) .
1/ A z (i)exp(2 σ eff NΔz)/exp[2 σ eff N z 0 (i)]( 12 σ eff N δ z (i)+2 [ σ eff N δ z (i)] 2 ),
b Δτ_SNK3 = b Δτ_SNK2 + b Δτz , b Δτz 2 σ eff ( ν ¯ on , r G )N( r G )Δz2 [ σ eff ( ν ¯ on , r G )N( r G ) σ z ] 2 (1< W avn >),
σ 2 (Δ τ SNK3 )= σ 2 (Δ τ SNK2 )+ σ Δτz 2 , σ Δτz 2 [ 2 σ eff ( ν ¯ on , r G )N( r G ) σ z ] 2 < W avn >.
B L 2 0 [ ν F ν c ( t 0 ) ] 2 L N ( ν F )d ν F = B LF 2 + B LA 2 , B LF 2 [ ν(t) ν c ( t 0 ) ] 2 h(t t 0 )dt , B LA 2 f 2 |A(f) | 2 df |A(f) | 2 df = 1 4 π 2 [ da(t) / dt ] 2 dt a 2 (t)dt .
B LF 2 ¯ = [ δ ν (t)] 2 ¯ [ δ ν1 ( t 0 )] 2 ¯ = S δν (f)[ 1|H(f) | 2 ]df .
R u ( t D ) u*(t)u(t+ t D ) dt= a(t')a(t'+ t D )exp(j2π ν ¯ t D )exp(jΔϕ) dt',
L 0 (Δ ν F ) ¯ = [ a(t')a(t'+ t D ) exp( j[Δϕ2π δ ν1 ( t 0 ) t D ] ) ¯ dt' ]exp(j2πΔ ν F t D )d t D .
exp( j[Δϕ2π δ ν1 ( t 0 ) t D ] ) ¯ =exp[ 1 2 ( Δϕ2π δ ν1 ( t 0 ) t D ) 2 ¯ ].
L 0 (Δ ν F ) ¯ = R 0 ( t D )exp(j2πΔ ν F t D )d t D , R 0 ( t D ) a(t')a(t'+ t D )exp[ 2 π 2 t D 2 S δν (f) | sinc(f t D )H(f)exp[ j2πf( t'+ t D /2 ) ] | 2 df ] dt'.
σ 2 ( K sn )= ( Δt/2 Δt/2 α P sn (t)dt ) 2 ¯ =Δt Tri( ξ Δt )Γ(ξ)dξ = I sn Δt, I sn S sn (f) sinc 2 (fΔt)Δtdf.
σ 2 ( K nn )= K nn ¯ ( F e + δ c ), δ c = K nn ¯ /(2 B o Δt N MM )=αN / ASE N MM .
b ΔτE N ASE N MM ( 1 E sav on 1 E sav off )( 2< W avn > ),
σ ΔτE 2 2 N ASE N MM ( 1 E sav on + 1 E sav off )< W avn >.

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