Error reduction methods for integrated-path differential-absorption lidar measurements

doi:10.1364/OE.20.015589

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Error reduction methods for integrated-path differential-absorption lidar measurements

Jeffrey R. Chen, Kenji Numata, and Stewart T. Wu »View Author Affiliations

Optics Express, Vol. 20, Issue 14, pp. 15589-15609 (2012)
http://dx.doi.org/10.1364/OE.20.015589

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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”.

1. Introduction

1.1 Overview

Lidar remote sensing techniques are powerful tools for global measurement of atmospheric constituents and parameters [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).

]. The required precision and accuracy of such quantitative measurements have been increased to unprecedented levels for future airborne and space missions [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.

]. 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

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. B 90(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.

]. 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

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 CO_2.,” Appl. Opt. 48(28), 5413–5422 (2009). [CrossRef] [PubMed]

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

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]

]), and thereby minimize the impact of Etalon and atmospheric turbulence effects; 3) error contribution from the surface reflectance variations [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]

] 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

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

] 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

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]

] 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

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]

]. 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).

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

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]

] (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

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

] 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

“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.

] 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

J. Caron and Y. Durand, “Operating wavelengths optimization for a spaceborne lidar measuring atmospheric CO_2.,” Appl. Opt. 48(28), 5413–5422 (2009). [CrossRef] [PubMed]

, 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]

]. A similar approach has been adopted for a planned methane sounder space mission dubbed MERLIN [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]

2. Measurement errors from laser frequency noise

Throughout this paper,

\bar{x}

represents the ensemble average,

σ (x)

the standard deviation, and

σ^{2} (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

δ_{ν} (t) \equiv ν (t) - \bar{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 (t - t_{0})

centered at time

t_{0},

as given below

u (t) = a (t - t_{0}) \exp (j [2 π \bar{ν} t + ϕ (t)]) .

(1)

Here the instantaneous frequency is

ν (t) = \bar{v} + (1 / 2 π) d ϕ / d t .

The single-sided energy spectral density (ESD) of

u (t),

denoted as

L (ν_{F})

as a function of the Fourier frequency

ν_{F},

represents a short-term spectrum of the CW laser within a pulse duration Δt. The line-center frequency

ν_{c} (t_{0})

L (ν_{F})

is linked to the instantaneous frequency

ν (t)

and

a (t)

[17

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

] as

ν_{c} (t_{0}) \equiv \int_{0}^{\infty} ν_{F} L_{N} (ν_{F}) d ν_{F} = \int_{- \infty}^{\infty} ν (t) h (t - t_{0}) d t .

(2)

Here

L_{N} (ν_{F}) \equiv L (ν_{F}) / \int_{0}^{\infty} L (ν_{F}) d ν_{F}

and

h (t) \equiv a^{2} (t) / \int_{- \infty}^{\infty} a^{2} (t) d t

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 (ν_{F})

and the fluctuation of the line-center frequency

ν_{c} (t_{0}) .

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 frequency

δ_{ν 1} (t_{0}) \equiv ν_{c} (t_{0}) - \bar{ν}

is an average of the CW laser frequency noise

δ_{ν} (t)

within a pulse duration Δt weighted by

h (t - t_{0}) .

For a top-hat pulse shape,

δ_{ν 1} (t_{0})

is simply a running average of

δ_{ν} (t)

within Δt. The variance

σ^{2} (δ_{ν 1})

δ_{ν 1} (t_{0})

can be obtained by integrating its power spectral density (PSD), which in turn can be derived from

S_{δ ν} (f),

the PSD of

δ_{ν} (t) .

Since

δ_{ν 1} (t_{0})

is a correlation between

δ_{ν} (t)

and

h (t)

as seen in Eq. (2), its PSD

S_{δ ν 1} (f)

is related to

S_{δ ν} (f)

S_{δ ν 1} (f) = S_{δ ν} (f) {| H (f) |}^{2},

(3)

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

S_{δ ν} (f) .

For a top-hat

h (t) = r e c t (t / Δ t) / Δ t,

the corresponding window function becomes

| H (f) |^{2} = {sinc}^{2} (f Δ t) .

Fast frequency noise components above its effective bandwidth

1 / Δ t

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 (ν_{F})

to the frequency noise

S_{δ ν} (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 (ν_{F}) .

2.2 Single pulse formulation

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

d L (ν_{F}, l) = - L (ν_{F}, l) σ_{0} (ν_{F}, l) N (l) d l .

(4)

Here

σ_{0} (ν_{F}, l)

is the absorption cross-section of the molecules at Fourier frequency

ν_{F}

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

\begin{array}{l} E (l) = E (0) \exp [- τ (ν_{c}, l)], \\ τ (ν_{c}, l) \equiv \int_{0}^{l} σ_{e f f} (ν_{c}, l') N (l') d l' . \end{array}

(5)

Here

τ (ν_{c}, l)

is the effective optical depth of the species of interest and

σ_{e f f} (ν_{c}, l)

is the effective absorption cross-section defined as

\begin{array}{l} σ_{e f f} (ν_{c}, l) \equiv \int_{0}^{\infty} σ_{0} (ν_{F}, l) L_{N} (ν_{F}, l) d ν_{F} = \frac{\int_{0}^{\infty} σ_{0} (ν_{F}, l) T (ν_{F}, l) L (ν_{F}, 0) d ν_{F}}{\int_{0}^{\infty} T (ν_{F}, l) L (ν_{F}, 0) d ν_{F}}, \\ T (ν_{F}, l) \equiv \exp [- τ_{0} (ν_{F}, l)], \\ τ_{0} (ν_{F}, l) \equiv \int_{0}^{l} σ_{0} (ν_{F}, l') N (l') d l' . \end{array}

(6)

Here

τ_{0} (ν_{F}, 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

2 r_{G}

as the laser pulse travels from the spacecraft to the surface (outgoing path) and back to the spacecraft (return path). Here

r_{G}

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 (ν_{c}, l) - τ_{0} (ν_{c}, l)

to the optical depth. As shown in Appendix A, this bias can be accurately corrected. When

L (ν_{F})

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 (ν_{F})

is broad, the one-way effective optical depth may no longer be the same for the outgoing and return paths so that

τ (ν_{c}, 2 r_{G}) = 2 τ (ν_{c}, r_{G})

may become invalid (see Appendix A for more details).

Incorporating other attenuation factors [18

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

], the detected laser pulse energy W_s backscattered from the surface becomes

W_{s} = E_{s} A_{s} \exp [- τ (ν_{c}, 2 r_{G})],

(7)

where

E_{s}

is the transmitted laser pulse energy,

A_{s} \equiv ρ T_{a t m}^{2} D / r_{G}^{2}

is the lump sum of attenuation factors excluding

\exp [- τ (ν_{c}, 2 r_{G})],

D is a instrument constant, ρ is the ground surface reflectance (in sr⁻¹), and T_atm 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

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]

]

\bar{K} = α \bar{W},

(8)

σ^{2} (K) = F_{e} \bar{K} + α^{2} σ^{2} (W) .

(9)

Here

α \equiv η / (h \bar{ν})

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

F_{e} .

To estimate errors, we expand

\exp [- τ (ν_{c}, 2 r_{G})]

in Eq. (7) around

\bar{ν}

to arrive at

W_{s} = E_{s} A_{s} \exp [- τ (\bar{ν}, 2 r_{G})] [1 - (d τ / d ν_{c}) δ_{ν 1} (t_{0}) + b δ_{ν 1}^{2} (t_{0})],

(10)

where

b \equiv [{(d τ / d ν_{c})}^{2} - d^{2} τ / d ν_{c}^{2}] / 2.

From Eqs. (8) and (10), the ensemble mean of the detected signal photon count K_s is

\bar{K_{s}} = α \bar{W_{s}} = α E_{s} A_{s} \exp [- τ (\bar{ν}, 2 r_{G})] [1 + b σ^{2} (δ_{ν 1})],

(11)

where the variance

σ^{2} (δ_{ν 1})

δ_{ν 1} (t_{0})

contributes a small fractional bias

b σ^{2} (δ_{ν 1})

\bar{K_{s}} .

As illustrated in subsection 2.6, this bias is usually negligible in practice. Using Eqs. (9) and (10), the variance of K_s is related to the laser line-center frequency noise as

σ^{2} (K_{s}) = F_{e} \bar{K_{s}} + α^{2} σ^{2} (W_{s}) = F_{e} \bar{K_{s}} + {(\bar{K_{s}})}^{2} {(d τ / d ν_{c})}^{2} σ^{2} (δ_{ν 1}),

(12)

where the small contribution from the quadratic term

b δ_{ν 1}^{2} (t_{0})

is neglected.

2.3 Averaging across multiple pulses

We now turn to estimate the two-way DAOD

Δ τ \equiv τ ({\bar{ν}}_{o n}, 2 r_{G}) - τ ({\bar{ν}}_{o f f}, 2 r_{G})

between online and offline laser frequencies

{\bar{ν}}_{o n}

and

{\bar{ν}}_{o f f} .

Up to section 6,

τ (ν_{c}, 2 r_{G})

is assumed to be constant over the path and duration of the multiple pulses to be averaged. Although the measurement of E_s 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 t_p. From Eq. (10), the following sum of normalized pulse energies (SNE) can be introduced to include the effect of the laser frequency noise

\begin{array}{l} S_{N E 1} \equiv \sum_{i = 0}^{n - 1} \frac{W_{s} (i)}{E_{s} (i)} = \exp [- τ (\bar{ν}, 2 r_{G})] (\sum_{i = 0}^{n - 1} A_{s} (i)) [1 - \frac{d τ}{d ν_{c}} δ_{ν n} (t) + b Q_{n} (t)], \\ Q_{n} (t) \equiv \frac{1}{\sum_{i = 0}^{n - 1} A_{s} (i)} \sum_{i = 0}^{n - 1} [A_{s} (i) δ_{ν 1}^{2} (t + i \times t_{p})], \end{array}

(13)

where the averaged laser line-center frequency noise

δ_{ν n} (t)

across n pulses

(i = 0, 1, 2, .., n - 1)

is given by

δ_{ν n} (t) \equiv \frac{1}{\sum_{i = 0}^{n - 1} A_{s} (i)} \sum_{i = 0}^{n - 1} [A_{s} (i) δ_{ν 1} (t + i \times t_{p})] .

(14)

The mean and variance of

S_{N E 1}

are found to be

\bar{S_{N E 1}} = \exp [- τ (\bar{ν}, 2 r_{G})] (\sum_{i = 0}^{n - 1} A_{s} (i)) (1 + b σ^{2} (δ_{ν 1})),

(15)

σ^{2} (S_{N E 1}) = {(\bar{S_{N E 1}})}^{2} {(d τ / d ν_{c})}^{2} σ^{2} (δ_{ν n}) .

(16)

Superscripts (or subscripts) on and off will be used to indicate online and offline laser frequencies, respectively. The offline

σ^{2} (S_{N E 1}^{o f f})

can be neglected because of the negligible slope

(d τ / d ν_{c})_{o f f} .

Corresponding to S_NE_1, the sum of normalized photon count K_s (SNK) is defined as

S_{N K 1} \equiv \sum_{i = 0}^{n - 1} K_{s} (i) / E_{s} (i)

and its mean is found to be

\bar{S_{N K 1}} = α \bar{S_{N E 1}} .

Here S_NK₁ is named with the following scheme. S stands for sum, NK for normalized photon count, and the trailing digit 1 is used to distinguish S_NK₁ from S_NK₂ and S_NK₃ to be introduced in sections 3 and 6, respectively. Using

\bar{K_{s} (i) K_{s} (i')} = α^{2} \bar{W_{s} (i) W_{s} (i')}

[19

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

], the variance of S_NK₁ is found to be

\begin{array}{l} σ^{2} (S_{N K 1}) = F_{e} \bar{S_{N N K 1}} + α^{2} σ^{2} (S_{N E 1}), \\ S_{N N K 1} \equiv \sum_{i = 0}^{n - 1} \frac{K_{s} (i)}{E_{s}^{2} (i)} ≅ S_{N K 1} / E_{s a v} ≅ S_{K} / E_{s a v}^{2} . \end{array}

(17)

Here S_NNK₁ is named similarly, with NN indicating that K_s is normalized twice. The integrated photon count S_K and average transmitted pulse energy E_sav are defined as

S_{K} \equiv \sum_{i = 0}^{n - 1} K_{s} (i)

and

E_{s a v} \equiv (\sum_{i = 0}^{n - 1} E_{s} (i)) / n,

respectively. S_NNK₁ is essentially S_K normalized by

E_{s a v}^{2} .

The first term in

σ^{2} (S_{N K 1})

stems from the shot noise. The second term arises solely from the laser line-center frequency noise.

From Eq. (15), Δτ is related to S_NK₁ by

Δ τ = - \ln (\frac{\bar{S_{N K 1}^{o n}}}{\bar{S_{N K 1}^{o f f}}}) + \ln (R_{A}) + \ln (R_{b}),

(18)

where

R_{A} \equiv \sum_{i = 0}^{n - 1} A_{s}^{o n} (i) / \sum_{i = 0}^{n - 1} A_{s}^{o f f} (i) \equiv 1 + δ_{R A}

and

R_{b} \equiv [1 + b^{o n} σ^{2} (δ_{ν 1}^{o n})] / [1 + b^{o f f} σ^{2} (δ_{ν 1}^{o f f})] .

For the rest of this paper, we assume the transmitted pulse energies are approximately the same for online and offline pulses so that

\bar{S_{K}^{o n}} ≅ \bar{S_{K}^{o f f}} \exp (- Δ τ) .

Δτ has been in effect estimated from the following “log after averaging” estimator [5

–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]

]

Δ τ_{S N K 1} \equiv - \ln (\frac{S_{N K 1}^{o n}}{S_{N K 1}^{o f f}}) .

(19)

To estimate errors, approximations

\bar{\ln (x)} ≅ ln (\bar{x}) - (1 / 2) σ^{2} (x) / {(\bar{x})}^{2}

(accurate to the third order) and

σ^{2} (\ln (x)) ≅ σ^{2} (x) / {(\bar{x})}^{2}

(accurate to the second order) are used in our calculations. The bias

b_{Δ τ_S N K 1} \equiv \bar{Δ τ_{S N K 1}} - Δ τ

and variance of Δτ_SNK₁ are found to be

b_{Δ τ_S N K 1} ≅ - \ln (R_{A}) - \ln (R_{b}) + \frac{F_{e}}{2} (\frac{\bar{S_{N N K 1}^{o n}}}{{\bar{S_{N K 1}^{o n}}}^{2}} - \frac{\bar{S_{N N K 1}^{o f f}}}{{\bar{S_{N K 1}^{o f f}}}^{2}}) + \frac{1}{2} {(\frac{d τ}{d ν_{c}})}_{o n}^{2} σ^{2} (δ_{ν n}^{o n}),

(20)

σ^{2} (Δ τ_{S N K 1}) ≅ \frac{σ^{2} (S_{N K 1}^{o n})}{{\bar{S_{N K 1}^{o n}}}^{2}} + \frac{σ^{2} (S_{N K 1}^{o f f})}{{\bar{S_{N K 1}^{o f f}}}^{2}} ≅ \frac{F_{e}}{\bar{S_{K}^{o f f}}} [1 + \exp (Δ τ)] + {(\frac{d τ}{d ν_{c}})}_{o n}^{2} σ^{2} (δ_{ν n}^{o n}) .

(21)

As expected, the shot noise contribution to

σ^{2} (Δ τ_{S N K 1})

is inversely proportional to the integrated photon count

\bar{S_{K}} .

However, it is important to use S_NK₁ rather than S_K to estimate Δτ, to remove the effect of variation in E_s(i). The laser line-center frequency noise contribution to the DAOD RRE is

[(d τ / d ν_{c})_{o n} / Δ τ] σ (δ_{ν n}^{o n}),

where

σ (δ_{ν n}^{o n})

is the standard deviation of the averaged line-center frequency noise of the on-line laser (averaged across n pulses). It becomes clear that

δ_{ν n}^{o n} (t)

represents the effective frequency noise. Obviously, an on-line laser frequency bias

Δ {\bar{ν}}_{o n}

will result in a DAOD bias ~

{(d τ / d ν_{c})}_{o n} Δ {\bar{ν}}_{o n}

]. 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

S_{δ ν n} (f)

δ_{ν n} (t)

S_{δ ν} (f) .

From Eq. (14) we have

\begin{array}{l} S_{δ ν n} (f) = S_{δ ν} (f)| H (f {)|}^{2} W_{a v n} (f), \\ W_{a v n} (f) \equiv {| \frac{1}{\sum_{i = 0}^{n - 1} A_{s} (i)} \sum_{i = 0}^{n - 1} A_{s} (i) \exp (j 2 π f t_{p} \times i) |}^{2} . \end{array}

(22)

Here the effect of multiple-pulse averaging on the PSD is an additional multiplicative window function

W_{a v n} (f) .

The overall window function

| H (f {)|}^{2} W_{a v n} (f)

is the ESD of the weighted pulse train function

\sum_{i = 0}^{n - 1} [A_{s} (i) h (t - i \times t_{p})] / \sum_{i = 0}^{n - 1} A_{s} (i) .

W_{a v n} (f)

is a periodic function with a period equal to the pulse repetition rate. Using the orthogonal property of the phase shift factor

\exp (j 2 π f t_{p} \times i),

its averaged height is found to be

< W_{a v n} > = \sum_{i = 0}^{n - 1} A_{s}^{2} (i) / {(\sum_{i = 0}^{n - 1} A_{s} (i))}^{2} .

(23)

When the weighting factors A_s(i) are the same,

W_{a v n} (f)

becomes

{sinc}^{2} (n f t_{p}) / {sinc}^{2} (f t_{p})

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,

< W_{a v n} >

is only slightly higher than 1/n when measured surface reflectance data are used for A_s(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

J. Caron and Y. Durand, “Operating wavelengths optimization for a spaceborne lidar measuring atmospheric CO_2.,” Appl. Opt. 48(28), 5413–5422 (2009). [CrossRef] [PubMed]

]. Using Eq. (22), the effective frequency noise can be estimated in terms of

S_{δ ν} (f),

as illustrated in Fig. 2 for a representative frequency stabilized seeder laser. The nominal

S_{δ ν} (f)

can be approximated by

S_{δ ν} (f) = h_{- 1} / f + h_{0},

where constant

h_{- 1}

and

h_{0}

quantify flicker noise and a nearly flat noise floor for f above 100 Hz, respectively. The periodical window function

W_{a v n} (f)

suppresses the frequency noise except in its narrow peaks at

f = m_{w} / t_{p}

(m_{w} = 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

(m_{w} = 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

S_{δ ν} (f) = h_{- 1} / f + h_{0}

(solid red) is suppressed to

S_{δ ν} (f)| H (f {)|}^{2}

(dotted blue) by averaging within a 1-μs top-hat pulse and further reduced to

S_{δ ν} (f)| H (f {)|}^{2} W_{a v n} (f)

(dashed black) by averaging across 1000 identical pulses separated by a period of 1 ms.

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

S_{δ ν} (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.

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

S_{δ ν} (f)| H (f {)|}^{2} W_{a v n} (f) .

2.6 Frequency stability requirements for ASCENDS

Using Eqs. (5) and (6), the relative error

(Δ τ - Δ τ_{0}) / Δ τ_{0}

due to the finite line shape

L (ν_{F})

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 (ν_{F})

is simply taken to be the ESD of the top-hat pulse shape.

(Δ τ - Δ τ_{0}) / Δ τ_{0}

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(δ_νn^on) = 0.1 MHz (dashed red, calculated from Eq. (21)), and RSE ln(R_b)/Δτ for σ(δ_ν₁) = 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.

The partial RRE

[(d τ / d ν_{c})_{o n} / Δ τ] σ (δ_{ν n}^{o n})

for

σ (δ_{ν n}^{o n}) = 0.1 MHz,

the partial RSE

| \ln (R_{b}) | / Δ τ,

and

τ (ν_{c}, 2 r_{G})

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 (δ_{ν n}^{o n})

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).

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

Requirement	Value	Parameter	Value
Budget for total RRE σ(Δτ) /Δτ	0.1%	τ(ν_c, 2r_G) online (typical)	~2.0
Budget for partial RRE from laser frequency noise	0.03%	τ(ν_c, 2r_G) offline (15.6 GHz laser frequency offset)	0.013
Upper bound of σ(δ_νn^on) (0.5 GHz laser frequency offset)	0.23 MHz	Sensitivity of RRE to σ(δ_νn^on) (0.5 GHz frequency offset)	0.13% /MHz

The partial RSE

| \ln (R_{b}) | / Δ τ

for

σ (δ_{ν 1}) = 10 MHz

is less than 0.002% at such online frequencies, and thus negligible. However, this bias and

(Δ τ - Δ τ_{0}) / Δ τ_{0}

(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

(d τ / d ν_{c})_{o n}^{2} σ^{2} (δ_{ν n}^{o n}) / 2

< 10^{- 7}

when

[(d τ / d ν_{c})_{o n} / Δ τ] σ (δ_{ν n}^{o n})

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 (R_{A}) |

can be reduced to

< 1.1 \times 10^{- 4} .

Overall, the RSE

b_{Δ τ_S N K 1} / Δ τ

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 K_n-n produced by the background solar radiation is given by

\bar{K_{n - n}} = 2 α N_{b g d} B_{o} Δ t

and

σ^{2} (K_{n - n}) = F_{e} \bar{K_{n - n}}

, where

2 N_{b g d}

is the PSD of the background solar radiation and B_o the optical filter bandwidth. The equivalent detector dark count K_d (normalized by the mean internal gain M_e of the detector) has a mean and variance given by

\bar{K_{d}} = λ_{d} Δ t

and

σ^{2} (K_{d}) = F_{d} \bar{K_{d}}

, where λ_d is an equivalent dark count rate and F_d an effective dark-count excess noise factor. The equivalent photon count K_T arising from the receiver circuitry noise has a zero mean and a variance

σ^{2} (K_{T})

given by

\begin{array}{l} σ^{2} (K_{T}) = I_{δ}_{i} Δ t / {(M_{e} e)}^{2}, \\ I_{δ i} \equiv \int_{- \infty}^{\infty} S_{δ}_{i} (f) {sinc}^{2} (f Δ t) Δ t d f, \end{array}

(24)

where

S_{δ i} (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

σ^{2} (K_{s - n})

(see Appendix B). The integration

I_{δ i}

can be approximated by

S_{δ}_{i} (0)

when the detector electrical bandwidth B_e is much larger than 1/Δt.

The laser signal photon count K_s is estimated as

K'_{s} = K_{t o t} - K'_{b g d},

where K_tot is the total count

K_{t o t} = K_{s} + K_{n - n} + K_{d} + K_{T}

measured within Δt, and

K'_{b g d} = K'_{n - n} + 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,

\bar{K'_{s}} = \bar{K_{s}}

. Referencing Eq. (9), the variance of K'_s is found to be

\begin{array}{l} σ^{2} (K'_{s}) \equiv F_{e} \bar{K_{s}} + α^{2} σ^{2} (W_{s}) + λ_{b g d} Δ t, \\ λ_{b g d} \equiv [2 α N_{b g d} B_{o} F_{e} + F_{d} λ_{d} + I_{δ i} / {(M_{e} e)}^{2}] (1 + 1 / β) . \end{array}

(25)

The transmitted pulse energy E_s can be estimated as

E'_{s} = E'_{t o t} - E'_{b g d}

where E_tot 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 E_s and the variance of E'_s can be neglected (see Appendix B). The additional noise sources can be included by replacing S_NK₁ with

S_{N K 2} \equiv \sum_{i = 0}^{n - 1} K'_{s} (i) / E'_{s} (i) .

Similarly, S_K is replaced by

S_{K'} \equiv \sum_{i = 0}^{n - 1} K'_{s} (i)

. Obviously,

\bar{S_{N K 2}} = \bar{S_{N K 1}}

. The normalized variance of S_NK₂ is found to be

\frac{σ^{2} (S_{N K 2})}{{(\bar{S_{N K 2}})}^{2}} ≅ F_{e} \frac{\bar{S_{N N K 2}}}{{(\bar{S_{N K 2}})}^{2}} + {(\frac{d τ}{d ν_{c}})}^{2} σ^{2} (δ_{ν n}) + \frac{\bar{S_{N N 2}}}{{(\bar{S_{N K 2}})}^{2}} λ_{b g d} Δ t,

(26)

where S_NNK₂ and S_NN₂ are defined as

S_{N N K 2} \equiv \sum_{i = 0}^{n - 1} K'_{s} (i) / E'_{s}^{2} (i)

and

S_{N N 2} \equiv \sum_{i = 0}^{n - 1} 1 / E'_{s}^{2} (i) .

Here S_NN₂ 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) F_{e} (\bar{S_{N N K 1}^{o n}} / {\bar{S_{N K 1}^{o n}}}^{2} - \bar{S_{N N K 1}^{o f f}} / {\bar{S_{N K 1}^{o f f}}}^{2}) ≅ (1 / 2) F_{e} / \bar{S_{K}^{o n}}

, and thus requires significant photon count integration (

\bar{S_{K}^{o n}}

~10⁴) to bring the bias term down to a tolerable level (~10⁻⁴). When using S_NK₂, the additional background variance

λ_{b g d} Δ t

σ^{2} (K'_{s})

results in an additional DAOD bias term

(1 / 2) λ_{b g d} Δ t (\bar{S_{N N 2}^{o n}} / {\bar{S_{N K 2}^{o n}}}^{2} - \bar{S_{N N 2}^{o f f}} / {\bar{S_{N K 2}^{o f f}}}^{2})

. These two bias terms can be reduced significantly by incorporating the following bias correction factor

C_{Δ τ_S N K 2}

into the DAOD estimator given below

\begin{array}{l} Δ τ_{S N K 2} \equiv - \ln (\frac{S_{N K 2}^{o n}}{S_{N K 2}^{o f f}}) + C_{Δ τ_S N K 2}, \\ C_{Δ τ_S N K 2} \equiv \frac{F_{e}}{2} (\frac{S_{N N K 2}^{o f f}}{S_{N K 2}^{o f f}^{2}} - \frac{S_{N N K 2}^{o n}}{S_{N K 2}^{o n}^{2}}) + \frac{λ_{b g d} Δ t}{2} (\frac{S_{N N 2}^{o f f}}{S_{N K 2}^{o f f}^{2}} - \frac{S_{N N 2}^{o n}}{S_{N K 2}^{o n}^{2}}) . \end{array}

(27)

As shown next, the sum of the two bias terms is reduced from

~ (1 / 2) F_{e} / \bar{S_{K'}^{o n}}

~ (1 / 2) F_{e}^{2} / {\bar{S_{K'}^{o n}}}^{2}

by the bias correction factor

C_{Δ τ_S N K 2} .

The residual bias

~ (1 / 2) F_{e}^{2} / {\bar{S_{K'}^{o n}}}^{2}

becomes tolerable (~10⁻⁴) even when the integrated photon count

\bar{S_{K'}^{o n}}

is as low as ~100. Such a low photon count

\bar{S_{K'}^{o n}}

can be obtained within a few pulses, or even within a single pulse without averaging.

Using the bias-corrected DAOD estimator

Δ τ_{S N K 2}

given by Eq. (27), the bias

b_{Δ τ_S N K 2} \equiv \bar{Δ τ_{S N K 2}} - Δ τ

and variance of

Δ τ_{S N K 2}

are found to be

\begin{array}{l} b_{Δ τ_S N K 2} ≅ - \ln (R_{A}) - \ln (R_{b}) + {(d τ / d ν_{c})}_{o n}^{2} σ^{2} (δ_{ν n}^{o n}) / 2 + b_{C_Δ τ_S N K 2}, \\ b_{C_Δ τ_S N K 2} \equiv - \frac{1}{2} {(F_{e} / \bar{S_{K'}^{o n}})}^{2} [1 - \exp (- 2 Δ τ)] - \frac{3}{2} F_{e} n λ_{b g d} Δ t [1 - \exp (- 3 Δ τ)] / {\bar{S_{K'}^{o n}}}^{3}, \end{array}

(28)

\begin{array}{l} σ^{2} (Δ τ_{S N K 2}) ≅ σ_{Δ τ D E T}^{2} + {(d τ / d ν_{c})}_{o n}^{2} σ^{2} (δ_{ν n}^{o n}), \\ σ_{Δ τ D E T}^{2} \equiv F_{e} [1 + \exp (Δ τ)] / \bar{S_{K'}^{o f f}} + n λ_{b g d} Δ t [1 + \exp (2 Δ τ)] / {\bar{S_{K'}^{o f f}}}^{2} . \end{array}

(29)

It should be noted that all the parameters in the bias correction factor

C_{Δ τ_S N K 2}

are measureable. For example, the background variance

λ_{b g d} Δ t

can be measured when the laser pulses are off. However, it may not be feasible to measure some of the parameters (such as F_e) 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

Δ τ_{S N K 2} .

To do this, the overall averaging time

T_{t o t}

is divided into m slots (each is

T_{t o t} / m

long), and a

Δ τ_{S N K 2}

(denoted as

Δ τ_{S N K 2} (k)

) is evaluated within each slot

(k = 0, 1, 2, .., m - 1) .

Δτ can then be estimated from the average of

Δ τ_{S N K 2} (k)

given below

Δ τ_{S N K 2_a v} \equiv \frac{1}{m} \sum_{k = 0}^{m - 1} Δ τ_{S N K 2} (k) .

(30)

The variance of

Δ τ_{S N K 2_a v}

(excluding contribution from the slow laser frequency drifts) also decreases approximately by the number of participating pulses (i.e.,

n \times m

), as in the case for

Δ τ_{S N K 2} .

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 ,

Δ τ_{S N K 2_a v} / Δ τ,

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

~ F_{e} [1 + \exp (Δ τ)] / \bar{S_{K'}^{o f f}}

, 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

E_{s} = 4 mJ

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

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]

]. The detected solar radiation count K_n-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 K_d 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).

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

Parameter	Value	Parameter	Value
Operating wavelength	1572 nm	Detector output dark current i_d	1 nA
Orbit altitude	400 km	Single-sided TIA equivalent input noise current	1 pA/Hz^1/2
Telescope diameter	1.5 m	B_e (electrical bandwidth)	1.6 MHz
Receiver field of view (full angle)	150 μrad	B_o (optical bandwidth)	48 GHz
Beam spot diameter on surface	50 m	Background solar photon count rate K_n-n/Δt	129 MHz
T_atm (one-way transmittance of atmosphere excluding CO₂)	0.7	β (background integration time multiplier)	10
ρ (surface reflectance)	0.17 sr⁻¹	Δt (pulse duration)	1 μs
Receiver path transmittance	0.51	E_s (transmitted pulse energy)	4 mJ
A_s (see Eq. (7))	1.48x10⁻¹³	K_s (signal photon count)	3200
η (detector quantum efficiency)	70%	T_tot (overall averaging time)	10 s
M_e (detector internal gain)	400	m (see Eq. (30))	10
F_e (excess noise factor)	2	effective n in T_tot/m	500
F_d (dark-count excess noise factor)	2	σ(δ_νn^on) averaged in T_tot	0.23 MHz

Fig. 5 The RRE σ(Δτ_{SNK2_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(δ_νn^on) 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).

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

τ \leq 2.4

5. Impact of surface reflectance variation

As discussed earlier, the surface reflectance variation adds a bias

| \ln (R_{A}) |

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

δ_{ν n} (t)

is reduced by the periodical window function

W_{a v n}

whose average height

< W_{a v n} >

depends on the variation of A_s(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 A_s(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 R_A 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 (R_{A})

is evaluated for each section. For each online wavelength channel, its starting pulse for sum

\sum_{i = 0}^{n - 1} A_{s}^{o n} (i)

comes in between two adjacent offline pulses, each serves as the starting pulse for its offline sum

\sum_{i = 0}^{n - 1} A_{s}^{o f f} (i)

. Each

\sum_{i = 0}^{n - 1} A_{s}^{o f f} (i)

is used to calculate

\ln (R_{A})

and the average of the two is plotted in Fig. 6 (right) for the worst case where the bias

| \ln (R_{A}) |

< 1.1 \times 10^{- 4} .

The corresponding RSE is smaller than 0.02% if the two-way DAOD is kept > 0.55.

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

Parameter	Value	Parameter	Value
Speed of spacecraft	7 km/s	Pulse rate for each channel	1 kHz
Beam spot diameter on surface	50 m	Pulse duration	1 μs
Number of wavelength channels	8	Duration of averaging	1 s
Combined pulse rate	8 kHz	Path length within averaging time	7 km

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

\sum_{i = 0}^{n - 1} A_{s}^{o f f} (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).

| \ln (R_{A}) |

is increased when the averaging time (hence n) is reduced. Figure 6 (right) also shows an average of 10 values of

\ln (R_{A}),

each computed as above over 0.1 s averaging time consecutively along the worst path. For some wavelength channels, this averaged bias

| \ln (R_{A}) |

becomes larger than 10⁻³ and thus unacceptable. To reduce the bias

| \ln (R_{A}) |,

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

< W_{a v n} >

W_{a v n} (f)

is also evaluated for the 7-km averaging paths. The calculated

< W_{a v n} >

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

z_{0} (i)

be the surface height at laser beam spot i relative to a certain reference ground surface, and

z (i)

a measured value of

z_{0} (i) .

r_{G}

now represents the range from the spacecraft to the reference surface. The distance from the spacecraft to the actual surface is

r_{G} - z_{0} (i) .

Since

z_{0} (i)

could vary by tens of meters over the path of pulse averaging,

τ (r_{G} - z_{0} (i))

could fluctuate over a few tenths of a percent over the path, resulting in additional DAOD errors. To mitigate this problem, we estimate

Δ τ (2 r_{G})

(i.e., the two-way DAOD from the spacecraft to the reference surface and back to the spacecraft) from

S_{N K 3} \equiv \sum_{i = 0}^{n - 1} K'_{s} (i) / [A_{z} (i) E'_{s} (i)],

where

A_{z} (i) \equiv \exp [2 σ_{e f f} (\bar{ν}, r_{G}) N (r_{G}) z (i)]

is introduced to account for the fluctuation of the optical depth. The corresponding estimator

Δ τ_{S N K 3}

Δ τ (2 r_{G})

can be easily adapted from

Δ τ_{S N K 2}

in Eq. (27) by simply replacing

E'_{s} (i)

with

A_{z} (i) E'_{s} (i)

in the relevant terms in

Δ τ_{S N K 2}

(i.e., S_NK₂, S_NNK₂, and S_NN₂). As shown next, the variation of the surface height

z_{0} (i)

does not affect such estimation of

Δ τ (2 r_{G}) .

Nevertheless, the imperfect measurement of

z_{0} (i)

still leads to additional DAOD errors.

To simplify the treatment, we assume

z (i) = z_{0} (i) + Δ z + δ_{z} (i),

where

Δ z

represents a slowly varying altimetry bias, and

δ_{z} (i)

a zero-mean random altimetry error.

Δ z

is treated as a constant during the n pulse averaging, and the

δ_{z} (i)

values among different pulses are assumed to be uncorrelated and have the same variance

σ_{z}^{2} .

Expanding

1 / A_{z} (i)

in S_NK₃ as

1 / A_{z} (i) \equiv \exp (- 2 σ_{e f f} N Δ z) / \exp [2 σ_{e f f} N z_{0} (i)] (1 - 2 σ_{e f f} N δ_{z} (i) + 2 {[σ_{e f f} N δ_{z} (i)]}^{2}),

(31)

the bias

b_{Δ τ_S N K 3} \equiv \bar{Δ τ_{S N K 3}} - Δ τ (2 r_{G})

and variance of

Δ τ_{S N K 3}

are found to be

\begin{array}{l} b_{Δ τ_S N K 3} = b_{Δ τ_S N K 2} + b_{Δ τ z}, \\ b_{Δ τ z} \equiv 2 σ_{e f f} ({\bar{ν}}_{o n}, r_{G}) N (r_{G}) Δ z - 2 {[σ_{e f f} ({\bar{ν}}_{o n}, r_{G}) N (r_{G}) σ_{z}]}^{2} (1 - < W_{a v n} >), \end{array}

(32)

\begin{array}{l} σ^{2} (Δ τ_{S N K 3}) = σ^{2} (Δ τ_{S N K 2}) + σ_{Δ τ z}^{2}, \\ σ_{Δ τ z}^{2} \equiv {[2 σ_{e f f} ({\bar{ν}}_{o n}, r_{G}) N (r_{G}) σ_{z}]}^{2} < W_{a v n} > . \end{array}

(33)

Here we have neglected

σ_{e f f} ({\bar{ν}}_{o f f}, r_{G})

that is much smaller than

σ_{e f f} ({\bar{ν}}_{o n}, r_{G}) .

The additional DAOD bias

b_{Δ τ z}

arises from the altimetry bias

Δ z

as well as the variance

σ_{z}^{2} .

The additional DAOD variance

σ_{Δ τ z}^{2}

scales with the altimetry variance

σ_{z}^{2}

and

< W_{a v n} >,

decreasing with ~1/n. Taking atmospheric CO₂ for example, an altimetry precision

σ_{z} = 20 m

results in

σ_{e f f} ({\bar{ν}}_{o n}, r_{G}) N (r_{G}) (20m) < 15 \times 10^{-4},

leading to a negligible standard deviation

σ_{Δ τ z} < 1 .3 \times 10^{-4}

(assuming n = 500), and a negligible DAOD bias (

< 1 .4 \times 10^{-6}

) arising from

σ_{z}^{2} .

On the other hand, the altimetry bias

Δ z

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 [5

–8

J. Caron and Y. Durand, “Operating wavelengths optimization for a spaceborne lidar measuring atmospheric CO_2.,” Appl. Opt. 48(28), 5413–5422 (2009). [CrossRef] [PubMed]

, 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]

]. 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

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]

]. 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 (R_{A}) |

is negligible because T_atm 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

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]

, 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]

]. The Doppler shift due to high speed cross-wind [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]

] 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

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]

] 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

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]

]. 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

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]

]. 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 / F_{e}

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

σ_{Δ τ D E T}^{2}

effectively. 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 (R_{A}) |,

and to use a wider pulsewidth (along with the high pulse repetition rate) to reduce

σ^{2} (δ_{ν n})

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 (R_{A}) |

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 (R_{A}) |,

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 (R_{A}) | .

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}^{-5}

and 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.

Appendices

Appendix

A. Formulation linking L(ν_F) to S_δν(f) and a(t)

The bandwidth

B_{L}

L (ν_{F}),

as defined below, is also linked to ν(t) and a(t) [17

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

] as

\begin{array}{l} B_{L}^{2} \equiv \int_{0}^{\infty} {[ν_{F} - ν_{c} (t_{0})]}^{2} L_{N} (ν_{F}) d ν_{F} = B_{L F}^{2} + B_{L A}^{2}, \\ B_{L F}^{2} \equiv \int_{- \infty}^{\infty} {[ν (t) - ν_{c} (t_{0})]}^{2} h (t - t_{0}) d t, \\ B_{L A}^{2} \equiv \frac{\int_{- \infty}^{\infty} f^{2} | A (f) |^{2} d f}{\int_{- \infty}^{\infty} | A (f) |^{2} d f} = \frac{1}{4 π^{2}} \frac{\int_{- \infty}^{\infty} {[d a (t) / d t]}^{2} d t}{\int_{- \infty}^{\infty} a^{2} (t) d t} . \end{array}

(34)

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),

B_{L}^{2}

is the sum of two components. The first component

B_{L F}^{2}

is the time domain “variance” of

δ_{ν 0} \equiv ν (t) - ν_{c} (t_{0})

weighted by

h (t - t_{0}),

and the second component

B_{L A}^{2}

represents the bandwidth of

| A (f) |^{2}

arising from the variations of the amplitude envelope a(t). From Eq. (34), the ensemble average of

B_{L F}^{2}

is found to be

\bar{B_{L F}^{2}} = \bar{{[δ_{ν} (t)]}^{2}} - \bar{{[δ_{ν 1} (t_{0})]}^{2}} = \int_{- \infty}^{\infty} S_{δ ν} (f) [1 - | H (f) |^{2}] d f .

(35)

This shows that

\bar{B_{L F}^{2}}

arises from faster frequency noise components, excluding the slower components

S_{δ ν} (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,

B_{L F}^{2}

has little fluctuation around its ensemble mean

\bar{B_{L F}^{2}}

and thus causes little variance in

b_{τ} (ν_{c}, l) \equiv τ (ν_{c}, l) - τ_{0} (ν_{c}, l) .

In practice, the frequency noise PSD

S_{δ ν} (f)

cuts off eventually, resulting in a finite

\bar{B_{L F}^{2}}

[17

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

]. Since the slow frequency drifts do not contribute,

\bar{B_{L F}^{2}}

for a short pulse can be significantly smaller than that for the underlying CW laser.

To facilitate further analysis, we rewrite

L (ν_{F})

L_{0} (Δ ν_{F}) \equiv L (ν_{c} (t_{0}) + Δ ν_{F}),

as a function of the Fourier frequency offset

Δ ν_{F} \equiv ν_{F} - ν_{c} (t_{0}) .

Obviously, a broad |A(f)|² (due to, for example, a narrow pulsewidth) can still lead to a significant bias

b_{τ} (ν_{c}, l) .

Fortunately, the bias

b_{τ} (ν_{c}, l)

can be computed from Eqs. (5) and (6) using

L_{0} (Δ ν_{F}) = | A (Δ ν_{F}) |^{2}

when the frequency noise contribution is negligible. Equations (5) and (6) thus allow accurate modeling even when

L (ν_{F})

is deterministically broadened to suppress SBS in the laser amplifiers. Since the amplitude modulation is deterministic, the bias

b_{τ} (ν_{c}, l)

is deterministic and thus can be precisely calibrated and removed. Although

L (ν_{F})

for a top hat pulse shape has an infinite

B_{L A}

, the relative error

(Δ τ - Δ τ_{0}) / Δ τ_{0}

can still be negligible as shown in Fig. 4 (left). When

L (ν_{F})

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

L_{N} (ν_{F}, l)

caused by the absorption of the species of interest. Since the laser line shape

L_{N} (ν_{F}, l)

is no longer the same along the two paths,

σ_{e f f} (ν_{c}, 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

S_{δ ν} (f)

has been established in previous studies [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]

–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]

]. The present work further links

L (ν_{F}),

the laser pulse ESD, to

S_{δ ν} (f)

and the pulse shape a(t).

L (ν_{F})

is the Fourier transform of the following autocorrelation of u(t)

R_{u} (t_{D}) \equiv \int_{- \infty}^{\infty} u * (t) u (t + t_{D}) d t = \int_{- \infty}^{\infty} a (t') a (t' + t_{D}) \exp (j 2 π \bar{ν} t_{D}) \exp (j Δ ϕ) d t',

(36)

where

t^{'} \equiv t - t_{0}

and

Δ ϕ \equiv ϕ (t^{'} + t_{0} + t_{D}) - ϕ (t^{'} + t_{0}) .

The ensemble average of

L_{0} (Δ ν_{F})

becomes

\bar{L_{0} (Δ ν_{F})} = \int_{- \infty}^{\infty} [\int_{- \infty}^{\infty} a (t') a (t' + t_{D}) \bar{\exp (j [Δ ϕ - 2 π δ_{ν 1} (t_{0}) t_{D}])} d t'] \exp (- j 2 π Δ ν_{F} t_{D}) d t_{D} .

(37)

Since

Δ ϕ - 2 π δ_{ν 1} (t_{0}) t_{D}

is a zero-mean Gaussian process, we have

\bar{\exp (j [Δ ϕ - 2 π δ_{ν 1} (t_{0}) t_{D}])} = \exp [- \frac{1}{2} \bar{{(Δ ϕ - 2 π δ_{ν 1} (t_{0}) t_{D})}^{2}}] .

(38)

Using

Δ ϕ = \int_{t' + t_{0}}^{t' + t_{0} + t_{D}} δ ν (t) d t

and taking similar steps as in [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]

], we finally arrive at

\begin{array}{l} \bar{L_{0} (Δ ν_{F})} = \int_{- \infty}^{\infty} R_{0} (t_{D}) \exp (- j 2 π Δ ν_{F} t_{D}) d t_{D}, \\ R_{0} (t_{D}) \equiv \int_{- \infty}^{\infty} a (t') a (t' + t_{D}) \exp [- 2 π^{2} t_{D}^{2} {\int_{- \infty}^{\infty} S}_{δ ν} (f) {| sinc (f t_{D}) - H (f) \exp [j 2 π f (t' + t_{D} / 2)] |}^{2} d f] d t' . \end{array}

(39)

The phase factor

\exp [j 2 π f (t' + t_{D} / 2)]

R_{0} (t_{D})

stems from the time delay between Δϕ (centered at

t' + t_{D} / 2 + t_{0}

) and

δ_{ν 1} (t_{0})

(centered at

t_{0}

). Following the argument in [17

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

] leading to Eq. (34), one arrives at

\bar{B_{L}^{2}} = - {(1 / 2 π)}^{2} {(d^{2} R_{0} / d t_{D}^{2})}_{t_{D} = 0} / R_{0} (0)

that yields

\bar{B_{L}^{2}} = B_{L A}^{2} + \bar{B_{L F}^{2}}

, the same result as given by Eqs. (34) and (35). The corresponding formula for CW laser [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]

] 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

\bar{L_{0} (Δ ν_{F})}

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

2 N_{A S E}

(

N_{A S E}

for each polarization) within an optical bandwidth

B_{o},

and the laser is polarized and has a detected power of P_s during Δt. For ease of analysis, we further assume that both the laser and ASE received by the detector are uniformly distributed among

N_{M M}

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 [P_{s} + P_{n - n} + P_{s - n}]

where

P_{n - n} (t)

is the ASE power,

P_{s - n} (t)

corresponds to the product of the laser electrical field and the ASE electrical field (referred to as s-n beat term).

α P_{s - n} (t)

has a zero mean and double-sided PSD

S_{s - n} (f) = 2 α^{2} P_{s} N_{A S E} / N_{M M}

within the detector electrical bandwidth

B_{e}

[28

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

]. The photon count variance arising from

α P_{s - n} (t)

is given by

\begin{array}{l} σ^{2} (K_{s - n}) = \bar{{(\int_{- Δ t / 2}^{Δ t / 2} α P_{s - n} (t) d t)}^{2}} = Δ t \int_{- \infty}^{\infty} Tri (\frac{ξ}{Δ t}) Γ (ξ) d ξ = I_{s - n} Δ t, \\ I_{s - n} \equiv \int_{- \infty}^{\infty} S_{s - n} (f) {sinc}^{2} (f Δ t) Δ t d f . \end{array}

(40)

Here

Γ (ξ) \equiv α^{2} \bar{P_{s - n} (t) P_{s - n} (t + ξ)}

is the autocorrelation of

α P_{s - n} (t)

and

Tri (x)

the triangular function.

I_{s - n}

can be approximated by

S_{s - n} (0) = 2 α^{2} P_{s} N_{A S E} / N_{M M}

when

B_{e} ≫ 1 / Δ t .

On the other hand,

P_{n - n} (t)

produces a mean photon count

\bar{K_{n - n}} = α (2 N {}_{ASE}B_{o} Δ t)

with a variance given by [29

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

]

\begin{array}{l} σ^{2} (K_{n - n}) = \bar{K_{n - n}} (F_{e} + δ_{c}), \\ δ_{c} = \bar{K_{n - n}} / (2 B_{o} Δ t N_{M M}) = α N {}_{ASE}/ N_{M M} . \end{array}

(41)

To measure E_s precisely, P_s needs to be as high as ~1 mW for a high signal to noise ratio. Such a high P_s is also accompanied by a significant

N_{A S E} .

As a result, the variance

σ^{2} ({E^{'}}_{s})

of E'_s arises predominately from the s-n beat term even though

P_{n - n} (t)

of the ASE produces a photon count variance in great excess of the Poisson value. From Eq. (40),

σ^{2} ({E^{'}}_{s})

is found to be

σ^{2} ({E^{'}}_{s}) ≅ 2 N_{A S E} E_{s} / N_{M M} .

To account for the error

δ {E^{'}}_{s} \equiv {E^{'}}_{s} - E_{s},

we expand 1/ E'_s in S_NK₃ (defined in section 6) as

1 / {E^{'}}_{s} \equiv (1 / E_{s}) [1 - δ {E^{'}}_{s} / E_{s} + {(δ {E^{'}}_{s} / E_{s})}^{2}] .

Obviously,

δ {E^{'}}_{s}

values among different pulses are uncorrelated. The additional bias and variance of Δτ_SNK₃ arising from

δ {E^{'}}_{s}

are found to be

b_{Δ τ E} \equiv - \frac{N_{A S E}}{N_{M M}} (\frac{1}{E_{s a v}^{o n}} - \frac{1}{E_{s a v}^{o f f}}) (2 - < W_{a v n} >),

(42)

σ_{Δ τ E}^{2} \equiv \frac{2 N_{A S E}}{N_{M M}} (\frac{1}{E_{s a v}^{o n}} + \frac{1}{E_{s a v}^{o f f}}) < W_{a v n} > .

(43)

The additional variance

σ_{Δ τ E}^{2}

also scales with

< W_{a v n} >

~ 1/n. In order to bound the corresponding RRE to < 10⁻⁴ (assuming

Δ τ \sim 2

n = 1

and

N_{M M} = 1

), the ratio

N_{A S E} / E_{s a v}

needs to be kept below 10⁻⁸. Consider a practical case where the seed laser oscillator is polarized with a peak power

P_{s e e d}

and pulsewidth

Δ t = 1 μs

at the input of a single mode preamplifier having a noise figure

N_{F} = 4.

N_{A S E}

arises mainly from the ASE in the preamplifier that has an equivalent PSD of

h ν N_{F} / 2

(in each polarization) at the input of the amplifier [28

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

]. Using

N_{A S E} / E_{s a v} ≅ h ν N_{F} / (2 P_{s e e d} Δ t),

the minimum seed laser peak power

P_{s e e d}

at the input of the preamplifier is estimated to be

- 16 dBm,

which can be easily satisfied. Similarly, the additional bias

b_{Δ τ E}

can be neglected even when the online and offline pulse energies differ significantly. Consequently, both the bias

b_{Δ τ E}

and variance

σ_{Δ τ E}^{2},

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

2 N_{b g d} .

Equations (40) and (41) can still be applied to the background sun light, provided that

N_{M M}

is interpreted as the number of the received speckles and

N_{A S E}

is replaced by

N_{b g d} .

Since

α N_{b g d} / N_{M M} ≪ 1,

K_{n - n}

from the background sun light becomes a Poisson process (for F_e = 1). The s-n beat term variance

σ^{2} (K_{s - n}) = 2 α N_{b g d} \bar{K_{s}} / N_{M M}

is much smaller than the signal shot noise

F_{e} \bar{K_{s}}

and thus can be neglected. For the background sun light, the shot noise variance

σ^{2} (K_{n - n}) = F_{e} \bar{K_{n - n}}

could be reduced to

σ^{2} (K_{s - n})

if the optical filter bandwidth B_o could be reduced to

α P_{s} / [(1 + 1 / β) N_{M M} F_{e}] .

In practice, the optical filter bandwidth is much wider so that

σ^{2} (K_{n - n})

is much larger than

σ^{2} (K_{s - n})

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.

References and links

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. B 90(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 CO_2.,” 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. 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]

OCIS Codes
(030.6600) Coherence and statistical optics : Statistical optics
(120.0280) Instrumentation, measurement, and metrology : Remote sensing and sensors
(280.1910) Remote sensing and sensors : DIAL, differential absorption lidar

ToC Category:
Remote Sensing

History
Original Manuscript: March 14, 2012
Revised Manuscript: May 25, 2012
Manuscript Accepted: June 12, 2012
Published: June 26, 2012

Citation
Jeffrey R. Chen, Kenji Numata, and Stewart T. Wu, "Error reduction methods for integrated-path differential-absorption lidar measurements," Opt. Express 20, 15589-15609 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-14-15589

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References

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 .
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]
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.
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]
J. Caron and Y. Durand, “Operating wavelengths optimization for a spaceborne lidar measuring atmospheric CO2.,” Appl. Opt.48(28), 5413–5422 (2009). [CrossRef] [PubMed]
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]
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. 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]
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]
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]
F. Koyama and K. Oga, “Frequency chirping in external modulators,” J. Lightwave Technol.6(1), 87–93 (1988). [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]
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]
L. Mandel, “Interpretation of instantaneous frequency,” Am. J. Phys.42(10), 840–846 (1974). [CrossRef]
W. B. Grant, “Effect of differential spectral reflectance on DIAL measurements using topographic targets,” Appl. Opt.21(13), 2390–2394 (1982). [CrossRef] [PubMed]
J. W. Goodman, Statistical Optics (John Wiley & Sons, 1985).
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. 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]
V. S. R. Gudimetla and M. J. Kavaya, “Special relativity corrections for space-based lidars,” Appl. Opt.38(30), 6374–6382 (1999). [CrossRef] [PubMed]
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]
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]
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]
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]
N. A. Olsson, “Lightwave systems with optical amplifiers,” J. Lightwave Technol.7(7), 1071–1082 (1989). [CrossRef]
L. Mandel, “Fluctuations of photon beams: the distribution of the photo-electrons,” Proc. Phys. Soc.74(3), 233–243 (1959). [CrossRef]

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Error reduction methods for integrated-path differential-absorption lidar measurements

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Abstract

1. Introduction

1.1 Overview

1.2 IPDA lidar examples

2. Measurement errors from laser frequency noise

2.1 Treatment of laser frequency noise

2.2 Single pulse formulation

2.3 Averaging across multiple pulses

2.4 Frequency noise reduction from pulse averaging

2.5 Experimental results

2.6 Frequency stability requirements for ASCENDS

3. Incorporating additional detection noise sources

4. Numerical estimation of measurement errors

5. Impact of surface reflectance variation

6. Mitigating impact of surface height variation

7. Discussions

7.1 Other error sources

7.2 Error reduction tradeoffs

7.3 Shortening the time of averaging before log

8. Summary

Appendices

Appendix

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

B. Impact of transmitted laser pulse energy measurement errors

C. Noise from background solar radiation

Acknowledgments

References and links

References

Am. J. Phys.

Appl. Opt.

Appl. Phys. B

Atmos. Chem. Phys. Discuss.

Atmos. Meas. Tech.

IEEE Trans. Electron. Dev.

J. Geophys. Res.

J. Lightwave Technol.

Phys. Rev. A

Proc. Phys. Soc.

Proc. SPIE

Tellus Ser. B, Chem. Phys. Meteorol.

Other

Cited By

Figures

A. Formulation linking L(ν_F) to S_δν(f) and a(t)