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 Optical Society of America
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 ), and thereby minimize the impact of Etalon and atmospheric turbulence effects; 3) error contribution from the surface reflectance variations  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  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  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 . 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.
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  (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  is used so that the laser pulses retain the instantaneous frequency of the CW laser.
A similar CO2 sounder space mission called A-SCOPE  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 .
2. Measurement errors from laser frequency noise
Throughout this paper, represents the ensemble average, the standard deviation, and 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 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 centered at time as given below17] as
As shown in Eq. (2), the fluctuation of the line-center frequency is an average of the CW laser frequency noise within a pulse duration Δt weighted by For a top-hat pulse shape, is simply a running average of within Δt. The variance of can be obtained by integrating its power spectral density (PSD), which in turn can be derived from the PSD of Since is a correlation between and as seen in Eq. (2), its PSD is related to by13].
Formulation linking to the frequency noise and the pulse amplitude is further derived and discussed in Appendix A. Interestingly, the remaining frequency noise that is filtered out by contributes to the bandwidth of the line shape
2.2 Single pulse formulation
The energy absorption of a single laser pulse through a thin layer of target molecules is given byEq. (4) as
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 as the laser pulse travels from the spacecraft to the surface (outgoing path) and back to the spacecraft (return path). Here 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 to the optical depth. As shown in Appendix A, this bias can be accurately corrected. When 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 is broad, the one-way effective optical depth may no longer be the same for the outgoing and return paths so that may become invalid (see Appendix A for more details).
Incorporating other attenuation factors , the detected laser pulse energy Ws backscattered from the surface becomes
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]Eq. (7) around to arrive atEqs. (8) and (10), the ensemble mean of the detected signal photon count Ks isEqs. (9) and (10), the variance of Ks is related to the laser line-center frequency noise as
2.3 Averaging across multiple pulses
We now turn to estimate the two-way DAOD between online and offline laser frequencies and Up to section 6, 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
Corresponding to SNE1, the sum of normalized photon count Ks (SNK) is defined as and its mean is found to be 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 , the variance of SNK1 is found to be
From Eq. (15), Δτ is related to SNK1 by5–9]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 of to From Eq. (14) we have
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 as illustrated in Fig. 2 for a representative frequency stabilized seeder laser. The nominal can be approximated by where constant and quantify flicker noise and a nearly flat noise floor for f above 100 Hz, respectively. The periodical window function suppresses the frequency noise except in its narrow peaks at Frequency drifts with periods longer than the averaging time fall in the window peak at and thus are not attenuated from this averaging. Since the PSD is nearly constant over the frequency range of all other window peaks 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.
2.5 Experimental results
Equations (3) and (22) have been verified experimentally with our seed laser described in . 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
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
2.6 Frequency stability requirements for ASCENDS
Using Eqs. (5) and (6), the relative error due to the finite line shape 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 is simply taken to be the ESD of the top-hat pulse shape. is smaller than a 0.02% target  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.
The partial RRE for the partial RSE and 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 of the effective frequency noise needs to be suppressed down to 0.23 MHz. This frequency stability requirement has been satisfied experimentally (see  and subsection 2.5).
The partial RSE for is less than 0.002% at such online frequencies, and thus negligible. However, this bias and (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 is when 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 can be reduced to Overall, the RSE 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 and , where 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 and , 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 given byEquation (24) is derived by the same steps leading to Eq. (40) for (see Appendix B). The integration can be approximated by when the detector electrical bandwidth Be is much larger than 1/Δt.
The laser signal photon count Ks is estimated as where Ktot is the total count measured within Δt, and 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, . Referencing Eq. (9), the variance of K's is found to be
The transmitted pulse energy Es can be estimated as 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 Similarly, SK is replaced by . Obviously, . The normalized variance of SNK2 is found to be
As seen from Eq. (20), the “log after averaging” DAOD estimator in Eq. (19) leads to an additive bias term , and thus requires significant photon count integration (~104) to bring the bias term down to a tolerable level (~10−4). When using SNK2, the additional background variance in results in an additional DAOD bias term . These two bias terms can be reduced significantly by incorporating the following bias correction factor into the DAOD estimator given below
Using the bias-corrected DAOD estimator given by Eq. (27), the bias and variance of are found to be
To minimize the time-varying Etalon and atmospheric turbulence effects, it is desirable to reduce n, the number of pulses being averaged for To do this, the overall averaging time is divided into m slots (each is long), and a (denoted as ) is evaluated within each slot Δτ can then be estimated from the average of given below
The variance of (excluding contribution from the slow laser frequency drifts) also decreases approximately by the number of participating pulses (i.e., ), as in the case for 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 , 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 , 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 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 . 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).
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 .
5. Impact of surface reflectance variation
As discussed earlier, the surface reflectance variation adds a bias 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 is reduced by the periodical window function whose average height depends on the variation of As(i). Both factors are quantified here using measured surface reflectance data in  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 . 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 is evaluated for each section. For each online wavelength channel, its starting pulse for sum comes in between two adjacent offline pulses, each serves as the starting pulse for its offline sum . Each is used to calculate and the average of the two is plotted in Fig. 6 (right) for the worst case where the bias is The corresponding RSE is smaller than 0.02% if the two-way DAOD is kept > 0.55.
is increased when the averaging time (hence n) is reduced. Figure 6 (right) also shows an average of 10 values of each computed as above over 0.1 s averaging time consecutively along the worst path. For some wavelength channels, this averaged bias becomes larger than 10−3 and thus unacceptable. To reduce the bias 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 of is also evaluated for the 7-km averaging paths. The calculated 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 be the surface height at laser beam spot i relative to a certain reference ground surface, and a measured value of now represents the range from the spacecraft to the reference surface. The distance from the spacecraft to the actual surface is Since could vary by tens of meters over the path of pulse averaging, could fluctuate over a few tenths of a percent over the path, resulting in additional DAOD errors. To mitigate this problem, we estimate (i.e., the two-way DAOD from the spacecraft to the reference surface and back to the spacecraft) from where is introduced to account for the fluctuation of the optical depth. The corresponding estimator of can be easily adapted from in Eq. (27) by simply replacing with in the relevant terms in (i.e., SNK2, SNNK2, and SNN2). As shown next, the variation of the surface height does not affect such estimation of Nevertheless, the imperfect measurement of still leads to additional DAOD errors.
To simplify the treatment, we assume where represents a slowly varying altimetry bias, and a zero-mean random altimetry error. is treated as a constant during the n pulse averaging, and the values among different pulses are assumed to be uncorrelated and have the same variance Expanding in SNK3 as
7.1 Other error sources
Other DAOD measurement error sources have also been studied previously [5–8, 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 . 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 .
The contribution of atmospheric turbulence to the bias term 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  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  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 . 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 . 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 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 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 and to use a wider pulsewidth (along with the high pulse repetition rate) to reduce 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 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 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 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 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.
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.
A. Formulation linking L(νF) to Sδν(f) and a(t)
The bandwidth of as defined below, is also linked to ν(t) and a(t)  as
As seen from Eq. (34), is the sum of two components. The first component is the time domain “variance” of weighted by and the second component represents the bandwidth of arising from the variations of the amplitude envelope a(t). From Eq. (34), the ensemble average of is found to be17]. Since the slow frequency drifts do not contribute, for a short pulse can be significantly smaller than that for the underlying CW laser.
To facilitate further analysis, we rewrite as as a function of the Fourier frequency offset Obviously, a broad |A(f)|2 (due to, for example, a narrow pulsewidth) can still lead to a significant bias Fortunately, the bias can be computed from Eqs. (5) and (6) using when the frequency noise contribution is negligible. Equations (5) and (6) thus allow accurate modeling even when is deterministically broadened to suppress SBS in the laser amplifiers. Since the amplitude modulation is deterministic, the bias is deterministic and thus can be precisely calibrated and removed. Although for a top hat pulse shape has an infinite , the relative error can still be negligible as shown in Fig. 4 (left). When 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 caused by the absorption of the species of interest. Since the laser line shape is no longer the same along the two paths, 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 has been established in previous studies [25–27]. The present work further links the laser pulse ESD, to and the pulse shape a(t). is the Fourier transform of the following autocorrelation of u(t)25, 26], we finally arrive at17] leading to Eq. (34), one arrives at that yields , the same result as given by Eqs. (34) and (35). The corresponding formula for CW laser  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 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 ( for each polarization) within an optical bandwidth 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 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 where is the ASE power, corresponds to the product of the laser electrical field and the ASE electrical field (referred to as s-n beat term). has a zero mean and double-sided PSD within the detector electrical bandwidth . The photon count variance arising from is given by29]
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 significant As a result, the variance of E's arises predominately from the s-n beat term even though of the ASE produces a photon count variance in great excess of the Poisson value. From Eq. (40), is found to be
To account for the error we expand 1/ E's in SNK3 (defined in section 6) as Obviously, values among different pulses are uncorrelated. The additional bias and variance of ΔτSNK3 arising from are found to be
The additional variance also scales with ~ 1/n. In order to bound the corresponding RRE to < 10−4 (assuming , and ), the ratio needs to be kept below 10−8. Consider a practical case where the seed laser oscillator is polarized with a peak power and pulsewidth at the input of a single mode preamplifier having a noise figure arises mainly from the ASE in the preamplifier that has an equivalent PSD of (in each polarization) at the input of the amplifier . Using the minimum seed laser peak power at the input of the preamplifier is estimated to be which can be easily satisfied. Similarly, the additional bias can be neglected even when the online and offline pulse energies differ significantly. Consequently, both the bias and variance 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 Equations (40) and (41) can still be applied to the background sun light, provided that is interpreted as the number of the received speckles and is replaced by Since from the background sun light becomes a Poisson process (for Fe = 1). The s-n beat term variance is much smaller than the signal shot noise and thus can be neglected. For the background sun light, the shot noise variance could be reduced to if the optical filter bandwidth Bo could be reduced to In practice, the optical filter bandwidth is much wider so that is much larger than .
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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