Sensitivity analysis and correction algorithms for atmospheric CO<sub>2</sub> measurements with 1.57-&#x00B5;m airborne double-pulse IPDA LIDAR

Yadan Zhu; Yadan Zhu; Yadan Zhu; Jiqiao Liu; Jiqiao Liu; Jiqiao Liu; Xiao Chen; Xiao Chen; Xiaopeng Zhu; Decang Bi; Weibiao Chen; Weibiao Chen; Weibiao Chen

doi:10.1364/OE.27.032679

1. Introduction

Long-term observations of the sources and sinks of greenhouse gases (GHGs) are essential for predicting global climate changes. Several ground-based light detection and ranging (LIDAR) systems have been developed to measure GHGs. However, these systems have limited detection ranges. In contrast, space-borne integrated-path differential absorption (IPDA) LIDAR systems can operate globally during both day and night and also have the advantage of a wide detection range. They are thus considered a more effective tool for measuring the global GHGs (carbon dioxide and methane, among others) levels and are critical for studying the global carbon cycle [1–4]. Several IPDA LIDAR instruments for the active detection of CO₂ were developed for NASA’s Active Sensing of CO₂ Emission over Nights, Days, and Seasons (ASCENDS) and Advanced Space Carbon and Climate Observation of Planet Earth (A-SCOPE) missions [5,6], including 1.57 and 2.05 µm wavelengths using pulsed and continuous-wave (CW) techniques.

Airborne IPDA LIDAR systems provide a means of verifying the performance of space-borne LIDAR systems as well as data inversion methods used by them. Owing to current detection techniques and spectral absorption characteristics of CO₂ molecule, wavelengths of 1.57 and 2.05 µm have been used of the CO₂ column-averaged dry-air mixing ratio (XCO₂) measurements in many airborne campaigns. A pulsed multi-wavelength IPDA approach was selected as a candidate for the ASCENDS mission. The pulsed IPDA LIDAR incorporates a rapidly wavelength-tunable, step-locked seed laser in the transmitter and frequency was modulated into a 10 kHz pulse train by an electro-optic (EO) modulator on the CO₂ line with a center wavelength of 1572.335 nm. Abshire et al. showed the latest results of an airborne experiment with a pulsed IPDA LIDAR system using a multiple-wavelength-locked laser and HgCdTe APD detector. And the mean values of XCO₂ as determined by the IPDA LIDAR system was consistently within 1 ppm of that determined using an in-situ sensor [7]. Intensity-modulated continuous-wave (IM-CW) IPDA LIDAR is another critical instrument for high-precision, XCO₂ measurements, which has online wavelength positioned at the center of the CO₂ absorption line at 1571.112 nm [8]. Stable, accurate XCO₂ measurements with systematic variations of <0.3 ppm compared to in-situ measurements for extensive flight periods over generally-uniform environments were observed by Lin et al. by using the airborne IM-CW IPDA LIDAR [9,10]. The 2-µm airborne IPDA LIDAR system, whose online frequencies could be tuned with an offset of 1 to 6 GHz was implemented in a B-200 aircraft. As compared to XCO₂ of 404.08 ppm, derived from air sampling, IPDA measurement resulted in a value of 405.22 ± 4.15 ppm with 1.02% uncertainty and 0.28% bias [11]. The IPDA LIDAR system CHARM-F onboard the German research aircraft HALO was able to measure both CO₂ and CH₄ simultaneously. The technique of Nd: YAG-pumped OPOs was used in the laser transmitter, which generated the double pulses with an online wavelength of 1572.02 nm and an offline wavelength of 1572.12 nm for XCO₂ measurement. The measurement precision of below 0.5% was found [12].

Simulations of IPDA LIDAR systems are usually performed based on their parameters in order to reduce errors and analyze their sources. In 2008, Ehret et al. optimized the online wavelength and simulated the random and systematic errors during space-borne remote sensing of CO₂, CH₄, and N₂O by an IPDA LIDAR system [1]. In 2011, Kiemle et al. simulated the random errors based on the parameters of the MERLIN space-borne IPDA LIDAR system and analyzed the effects of the atmospheric parameters [3]. In 2012, Chen et al. proposed an error reduction method for IPDA LIDAR measurements [13]. In 2015, Refaat et al. analyzed the influence of laser energy fluctuations on the inversion results and designed a set of energy self-calibration and detection devices to improve the system performance [14]. In 2018, Hu et al. investigated energy monitoring methods for IPDA LIDAR systems and used ground glass diffusers to reduce the speckle effect during energy monitoring [15].

In this study, we assumed that the detector noise, background noise, and signal noise obey the Gaussian distribution and that the quantization noise exhibits a uniform distribution. After the averaging of 148 shots, the relative random error (RRE) caused by the above noise was 0.057% for a typical target reflectivity of 0.1 sr⁻¹. Space-borne IPDA LIDAR with 20 Hz laser repetition frequency will be developed, and the expected horizontal resolution over land is 50 km. When the velocity of the satellite is approximately 6.75 km/s, the simulation considered a typical number of 148 shots per averaging window, approximately corresponding to 50 km along the satellite ground track. Therefore, 148 shots averaging used in the simulation is consistent with the planned space-borne system. We develop parametric optimization at differential target reflectivity to derive an unsaturated signal and a sufficient signal-to-noise ratio (SNR). In addition, the systematic errors caused by the laser pulse energy, linewidth, spectral purity, and frequency drift, as well as the atmospheric parameters related to flight experiments are investigated. Finally, the relative error (RE) caused by an uncorrected integral path and the Doppler shift is analyzed using the aircraft attitude angle and velocity data. Correction algorithms of the integral path and Doppler shift are proposed to eliminate the RE by accurately calculating the IWF. Hence, this study analyzes the random errors and system errors based on the actual system parameters of the airborne IPDA LIDAR system. It is verified that the system parameters meet the requirement that the measurement bias of XCO₂ is less than 0.25% (1ppm). According to the error analysis, the influence of each error source can be obtained, which is helpful to improve the system and further reduce the errors. Furthermore, it is helpful to verify the detection performance of future space-borne CO₂ measurement IPDA LIDAR and provides an important reference for parameter optimization of the space-borne system. Moreover, the integral path and Doppler shift correction algorithms contribute to more accurate inversion results.

2. Airborne LIDAR system for CO₂ measurements

2.1 Principle of IPDA LIDAR

The airborne IPDA LIDAR system developed in this study is based on the integrated path differential absorption at two different wavelengths, which are hereafter denoted as online and offline wavelengths. A laser pulse emitted at the online wavelength is attenuated because of absorption by the trace gas molecules during propagation through the atmosphere. In contrast, the offline pulse is only weakly attenuated by molecular absorption [12]. In this study, the wavelengths of online and offline laser have been selected by avoiding interference from other molecules except for carbon dioxide, and their wavelengths are very close that the difference caused by aerosol and atmosphere molecules scattering and absorption between them can be neglected. Therefore, the study mainly focuses on CO₂ molecular absorption. The observed difference in the echo intensities of the two laser beams with the different wavelengths is thus primarily caused by the absorption by the trace gas molecules. The echo power is given by the simplified hard target LIDAR equation [1,16], which can be written as

(1)$${P_\textrm{e}}(\lambda ,{R_A}) = {\eta _r} \cdot {O_r} \cdot \frac{A}{{{{({R_A} - {R_G})}^2}}} \cdot \frac{{E(\lambda )}}{{\Delta t(\lambda )}} \cdot {\rho ^ \ast } \cdot {T_m} \cdot \exp [ - {\tau _{\textrm{C}{\textrm{O}_2}}}(\lambda ,{R_A})], $$

where ${P_\textrm{e}}$ is the echo power, ${\eta _r}$ is the optical efficiency of the receiver, ${O_r}$ is the overlap function, A is the area of the telescope, ${R_A}$ is the altitude of airborne platform, ${R_G}$ is the altitude of the hard target above sea level, $E$ is the transmitted laser energy, $\Delta t$ is the effective pulse width of the return pulse, ${\rho ^ \ast }$ is the target reflectivity defined as the reflected power per steradian towards the receiver divided by the incident power [17], ${\tau _{\textrm{C}{\textrm{O}_2}}}$ is the total integrated double-path optical depth caused by atmospheric CO₂ molecule, and ${T_m}$ is the atmospheric transmission efficiency including scattering and absorption of other atmosphere molecules and aerosols. The IPDA double-path differential absorption optical depth (DAOD) for CO₂, denoted as $\Delta {\tau _{C{O_2}}}$, is expressed as

(2)$$\Delta {\tau _{C{O_2}}} = 2\int_{{R_G}}^{{R_A}} {\Delta {\sigma _{C{O_2}}}(P(r),T(r))} {N_{C{O_2}}}(r)dr = \ln \frac{{{P_{off}}{E_{o{n_0}}}}}{{{P_{on}}{E_{of{f_0}}}}}, $$

where $\Delta {\sigma _{C{O_2}}}$ is the differential cross-section between the online and offline wavelengths; ${N_{C{O_2}}}$ is the number density of the CO₂ molecules; ${E_{o{n_\textrm{0}}}}$ and ${E_{of{f_\textrm{0}}}}$ are energies of the online and offline monitoring pulses, respectively; and ${P_{on}}$ ${P_{off}}$ are the powers of online and offline echo pulses, respectively. During the airborne experiments, the vertical-path ${X_{C{O_2}}}$ (in ppm) is given by [16]

(3)$${X_{C{O_2}}} = \frac{{\Delta {\tau _{C{O_2}}}}}{{2 \times {{10}^{ - 6}} \cdot IWF}}, $$

(4)$$IWF = \int_{{R_G}}^{{R_A}} {\frac{{{N_A} \cdot P(r) \cdot \Delta {\sigma _{C{O_2}}}(P(r),T(r))}}{{RT(r)(1 + {X_{{H_2}o}}(r))}}dr}, $$

where ${N_A}$ is Avogadro’s number, $R$ is the gas constant, $P$ is the pressure, $T$ is the temperature, ${X_{{H_2}O}}$ is the dry-air mixing ratio of water vapor, and $IWF$ represents the integrated weight function.

2.2 Instrument configuration

A schematic diagram of the airborne IPDA LIDAR system is shown in Fig. 1. It includes four modules: a laser transmitter subsystem, a receiver subsystem, a timing controller, and a data acquisition system. The primary performance parameters of the system are listed in Table 1.

Fig. 1. Schematic diagram of the airborne IPDA LIDAR system. OS, optical switch; FOS, fiber optical splitter; AOM, acoustic-optic modulator; FSS, frequency stabilization system; BSM, beam splitting mirror; BF1, band-pass filter 1; BF2, band-pass filter 2; IS, integrating sphere; CL, collimating lens; BA, beam attenuator; RM, reflecting mirror; ID, iris diaphragm; FL, focusing lens; APD, avalanche photodiode.

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Table 1. Primary parameters of airborne double-pulse 1.57 µm IPDA LIDAR system

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The laser transmitter subsystem includes a seeder laser, a frequency stabilization system, a pulsed laser, and a few optical components. The seed laser can produce radiation at two stabilized wavelengths (1572.024 nm and 1572.085 nm) and hence acts both as the online seeder and the offline one. The root mean square (RMS) of the frequency drifts in the online and offline seeders is less than 0.3 MHz. Thus, they can be considered reliable Refs. [18]. The seeder wavelength can be switched from the online one to the offline one by using a timing-controlled optical switch (OS) with an interval of 200 µs and a repetition frequency of 30 Hz. The fiber optical splitter (FOS) divides the seeder laser into two parts. One part is injected into an optical parametric oscillator (OPO) cavity in the pulsed laser, and the other part is frequency shifted by 400 MHz from the acoustic-optic modulator (AOM). Two beams from the OPO cavity and AOM output are combined by FOS and mixed. The mixed signals are detected by the photo detector in a frequency stabilization system (FSS) [19] to obtain frequency stabilized 1572 nm pulse laser. The final transmitted pulsed laser beam is amplified by the optical parametric amplifier (OPA) in the pulsed laser transmitter. The timing controller provides timing triggers for OS and the pulsed laser to produce switched wavelengths from the online one to the offline one of the final transmitted pulsed laser beam with an interval of 200 µs and a repetition frequency of 30 Hz.

The laser receiver subsystem consists of integrating sphere (IS), a telescope, iris diaphragm (ID) and some optical elements. The beam splitting mirror (BSM) with a ratio of 1:9 divides the transmitted laser beam into two parts. One part is used for energy monitoring and passes through the optical filter (OF) and band-pass filter 1 (BF1) into IS. Then it passes through a multi-mode fiber with a core diameter of 300 µm, collimating lens (CL), beam attenuator (BA), reflecting mirror (RM), BSM, band-pass filter 2 (BF2), focusing lens (FL) and finally is focused on 1572 nm InGaAs APD. Data acquisition is implemented with a 125 MS/s digitizer (NI PXIe-1071). As stated previously, the energies of the online and offline monitoring pulses are denoted ${E_{o{n_0}}}$ and ${E_{of{f_0}}}$, respectively. The other part is transmitted into the atmosphere where it is reflected by a hard target. The echo signal is collected by the telescope with a diameter of 150 mm and is reflected onto the ID by the RM. It then passes through the CL, BSM, BF2, and FL and is received by the same detector. As stated previously, the powers of the online and offline echo pulses are denoted ${P_{on}}$ and ${P_{off}}$, respectively. IS can reduce the impact of the pointing jitter on monitoring error. BF1 and BF2 with a bandwidth of 0.45nm can reduce the influence of background light on IS and APD detector and improve the SNR of the IPDA LIDAR system.

3. Random error analysis

When simulating the random error, the power of the monitoring signal and echo signal, which are the key factors with respect to XCO₂ inversion, should be simulated simultaneously. After CO₂ absorption, water vapor absorption, and aerosol extinction attenuation, the echo signals are reflected by the hard target and received by the telescope. The received power can be calculated using Eq. (1) and is converted into an electrical signal by the detector as per Eq. (5):

(5)$${V_{_{on,\;off}}} = \frac{{M \cdot {P_{on,\;off}} \cdot {\Re _i} \cdot {R_L} \cdot \Delta t}}{{\Delta {t_{eff}}}}, $$

(6)$$\Delta {t_{eff}} = \sqrt {\Delta {t^2} + {{(\frac{1}{{3\Delta f}})}^2} + {{(\frac{{2\Delta h}}{c})}^2}}, $$

where ${\Re _i}$ is the detector responsivity, ${R_L}$ is the feedback resistor (1 MΩ), $M$ is the internal gain factor of the detector, $\Delta f$ is the electrical bandwidth of the detector, $\varDelta h$ is the effective altitude of the target within the footprint of the laser pulse, and c is the speed of light. The laser power of the monitoring pulses is attenuated by the integrating sphere, beam attenuator, and band-pass filters and is subsequently converted into an electrical signal by the detector.

3.1 Noise analysis

The expressions of detector noise, background noise, and signal noise have been shown by Ehret et al. [1] and Kiemle et al. [3]. Figure 2(a) is the superposition noise of detector noise and background noise measured by the IPDA LIDAR system in the absence of echo signal. Figure 2(b) shows that the superposition noise obeys the Gaussian distribution. The overall bias of noise does not affect the distribution of noise and can be corrected in data inversion processing.

Fig. 2. (a) Superposition noise of detector noise and background noise measured by the IPDA LIDAR system in the absence of echo signal. (b) The distribution of the superposition noise and the Gaussian fitting.

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However, signal noise is not easy to measure because it is accompanied by optical signals, and superimposed with optical signals. When a photoelectric material interacts with a radiation field, the number of exciting photoelectrons fluctuates around an average, obeying a Poisson distribution [20,21]. According to the central limit theorem, the Poisson distribution can be approximated as a Gaussian distribution, when the number of photons within the signal is large [22]. Therefore, it is assumed that the signal noise obeys the Gauss distribution. As is known to all, the quantization noise in the analog signal converted by Analog-to-Digital Converter (ADC) is uniformly distributed [23]. The noise model used to generate the simulated signals is based on the parameters of the IPDA LIDAR system, which is presented in Table 1.

3.2 Simulation

The simulation condition is assumed that the altitude of the airborne platform is approximately 8 km, the vertical-path XCO2 is 400 ppm. The temperature, pressure, and humidity profiles used for calculation of echo power were measured by an in-situ sensor during the spiral-down stage of the airborne IPDA LIDAR measurements (cf. Section 4.1). Therefore, the echo power can be calculated using Eq. (1) and the spectroscopy database HITRAN 2012 [24]. The discussion on random noise in section 3 mainly affects signals received by the IPDA LIDAR system and does not affect the calculation of IWF. All the simulations were performed under the above-stated conditions unless stated otherwise. The hard target reflectivity ranges from 0.01 sr-1 to 0.3 sr-1 for different targets and seasons, while the optical depth (OD) of aerosols ranges from 0.01 to 0.3 for different regions and atmospheric conditions [4,25,26]. The absolute value of the echo voltage varies with the hard target reflectivity and aerosol OD (AOD), as shown in Fig. 3. Here, the online and offline echo voltages were simulated under the assumption that the internal gain factor of the detector is equal to 1.

Fig. 3. (a) Variations in absolute values (V) of (a) online and (b) offline echo voltages with hard target reflectivity and AOD.

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Since the echo voltage is very low for low hard target reflectivity and high AOD values, it must be amplified by adjusting the internal gain factor of the detector in order to improve the SNR. In the meantime, the monitoring voltage must also be amplified, as it is also received by the same detector. To ensure unsaturation and a high SNR while adjusting the internal gain factor, the monitoring voltage is optimized to −0.3 V under the condition that the detector internal gain factor is equal to 1.

The random error was simulated while assuming that the target reflectivity is 0.1 sr⁻¹ and AOD is 0.25. The four simulated signals received by the detector are shown in Fig. 4. The simulation results for the double-path DAOD for a 9000-shot sample is shown in Fig. 5(a), while the average for 148 shots is shown in Fig. 5(b). The vertical-path XCO₂ is shown in Fig. 5(c); its standard deviation is 2.799 ppm. The XCO₂ corresponding to the average of 148 shots is shown in Fig. 5(d); its standard deviation is 0.230 ppm. The relative and absolute errors in XCO₂ for an AOD of 0.25 and typical target reflectivity of 0.3, 0.1, 0.05 and 0.02 sr⁻¹ are listed in Table 2. As can be seen from the table, lower target reflectivity resulted in larger errors, with a standard deviation in the case of the average of 148 shots being the highest at 0.841 ppm. This can be reduced by increasing the detector’s internal gain.

Fig. 4. Four signals received by the APD detector.

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Fig. 5. (a) Single-shot double-path DAOD. (b) Averaged double-path DAOD for 148 shots. (c) Single-shot XCO₂. (d) Averaged XCO₂ for 148 shots in case of a 9000-shot sample taken in 5 min.

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Table 2. Relative and absolute errors in CO₂ concentration as measured at typical target reflectivity

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4. Systematic error analysis

4.1 Effects of atmospheric parameters

Since the temperature, pressure, and humidity affect the absorption of the online and offline signals by the atmospheric CO₂, the measurement errors in their profiles will cause calculation errors in the IWF. The temperature, pressure, and humidity profiles measured by the in-situ sensor during the spiral-down stage of the airborne IPDA LIDAR measurements during a single-flight experiment are shown in Figs. 6(a), 7(a), and 8(a), respectively.

When the error source only causes an error in the DAOD, the relative system error (RSE) in XCO₂ can be approximated as

(7)$$\frac{{\delta XC{O_2}}}{{XC{O_2}}} \approx \frac{{\delta \Delta \tau }}{{\Delta \tau }}, $$

here, the effect of the IWF is ignored, and it is assumed that there is no correlation between the different error sources.

On the other hand, when the error source only causes an error during the calculation of the IWF, the RSE of XCO₂ can be approximated as

(8)$$\frac{{\delta XC{O_2}}}{{XC{O_2}}} \approx \frac{{\delta IWF}}{{IWF}}, $$

here, the effect of the DAOD is ignored, and it is assumed that there is no correlation between the different error sources.

The temperature measurement error is assumed to obey the Gaussian distribution, and the standard deviation is taken as its uncertainty. The RSE of XCO₂ varies with the uncertainty in the temperature as calculated using Eq. (8) (see Fig. 6(b)). The RSE of XCO₂ for a temperature uncertainty of 1 K was 0.0123%, while the absolute error was 0.0492 ppm.

Fig. 6. (a) Temperature profile measured by the in-situ sensor during a spiral-down stage of airborne IPDA LIDAR measurements. (b) Changes in RSE of XCO₂ with uncertainty in temperature.

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As can be seen from Fig. 7(a), the atmospheric pressure decreases exponentially with the increase of altitude. The pressure deviation ratio is defined as the deviation ratio at each measured pressure. Hence, a deviation ratio of 0.1% at one atmosphere is 100 Pa. The RSE of XCO₂ as calculated by Eq. (8) varies with the pressure deviation ratio, as shown in Fig. 7(b). The RSE of XCO₂ for a pressure deviation ratio of 0.1% was 0.034%, while the absolute error was 0.137 ppm.

Fig. 7. (a) Pressure profile measured by the in-situ sensor during a spiral-down stage of airborne IPDA LIDAR measurements. (b) Change in RSE of XCO₂ with the pressure deviation ratio.

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The same method was used to simulate the error caused by the humidity. It was found that the RSE of XCO₂ varies with the humidity deviation ratio, which was calculated using Eq. (8) (see Fig. 8(b)). The RSE of XCO₂ for a humidity deviation ratio of 10% was 0.033%, while the absolute error was 0.134 ppm.

Fig. 8. (a) Humidity profile measured by the in-situ sensor during a spiral-down stage of airborne IPDA LIDAR measurements. (b) Changes in RSE of XCO₂ with uncertainty in humidity.

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4.2 Effects of laser parameters

4.2.1 Frequency drift

During the simulations, we ignored the effects of deviations in the offline laser frequency, as this signal is only marginally attenuated by the trace molecules. However, in case of a deviation in the frequency of the online laser, the IPDA double-path DAOD can be expressed as

(9)$$\Delta {\tau _{\Delta \nu }} = 2 \times {10^{ - 6}} \times {X_{C{O_2}}} \times \int_{{R_G}}^{{R_A}} {\frac{{{N_A} \cdot P(r) \cdot (\sigma ({\nu _{on}} + \Delta \nu ) - \sigma ({\nu _{off}}))}}{{RT(r)(1 + {X_{{H_2}o}}(r))}}dr}, $$

where $\Delta \nu $ is the deviation in the central frequency of the online laser. As shown in Fig. 9(a), the IPDA double-path DAOD varies sharply when the frequency deviates significantly from the center frequency.

Fig. 9. (a) Double-path DAOD varies greatly with deviation from the central frequency of the online laser. (b) Schematic of the frequency stabilization system. DAQ, digital-to-analog converter; PD, photodiode; IPC, industrial personal computer.

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To prevent frequency deviations, a frequency stabilization method based on the optical heterodyne technique was used with the injection-seeded OPO and M. Wirth et al. showed first results by the OPO heterodyne frequency stabilization technique in 2009 [19,27,28]. A schematic diagram of the frequency stabilization system is shown in Fig. 9(b). The AOM is used to shift the frequency of the seeder laser by 400 MHz in order to reduce the overlap between the optical pulse intensity envelope and the optical heterodyne beat frequency signal. Figure 10(a) shows the drift between the OPO output laser frequency and the seeder laser frequency (DDOSF), which is changed by AOM, as recorded in real-time during flight. The difference between the frequency drift and 400 MHz is used as the error signal to determine the voltage to be applied to the PZT actuator so that the OPO cavity length and the seeder laser frequency can be matched in a stable manner. The frequency deviation is also the difference between the frequency of the transmitted laser and the desired wavelength (1572.024 nm). During airborne experiments, the frequency deviation can be recorded in real-time, and the GPS time can be tagged for data correction. The frequency deviation distribution for 9000 shots recorded during one experiment is shown in Fig. 10(b); the standard deviation is 4.862 MHz.

Fig. 10. (a) Drift between the frequency of OPO output laser and seeder laser frequency shifted by AOM. (b) Frequency deviation distribution of 9000 shots. (c) Corresponding double-path DAOD. (d) Double-path DAOD for averaged 148 shots.

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The variation in the double-path DAOD is shown in Fig. 10(c). The double-path DAOD calculated by Eq. (9) for 9000 shots has a standard deviation of 0.0019 and causes a standard deviation of 0.765 ppm in XCO₂. The standard deviation has a significant impact on the performance of IPDA LIDAR, and there are two methods for limiting its effect. The first method is averaging the result for 148 shots, as shown in Fig. 10(d). After the averaging of 148 shots, the standard deviation for the 9000 shot frequencies was 0.108 MHz; in this case, the standard deviation of the double-path DAOD reduced to 0.004% and the standard deviation of the XCO₂ was 0.018 ppm. The second method involves correcting the IWF, which is given by Eq. (4), using the measured frequency instead of the designed online frequency. However, calculating the IWF for different temperature, humidity, and pressure profiles as well as different frequencies is time-consuming and computationally intensive. To solve this problem, look-up tables of the absorption cross-section were established for deviations of ± 40 MHz in the central frequency of the online laser. Each look-up table had dimensions of 1054×241 and was calculated for a pressure range of 105325 Pa to 25 Pa, at intervals of 100 Pa, and for a temperature range of 320.15 K to 200.15 K, at intervals of 0.5 K. This method resulted in an error of 0.005% compared to that when the IWF was calculated directly.

4.2.2 Linewidth

Because of the effects of natural broadening, collision broadening, and Doppler broadening, the actual spectral line type is not a geometric line; but a narrow linewidth spectrum. To elucidate the effects of the spectral linewidth, the spectral energy distribution for nanosecond laser pulses is modeled based on the following laser line shape [1]:

(10)$$L(v) = \frac{{{\pi ^2}}}{b}\frac{1}{{2\cosh (\frac{{2{\pi ^2}v}}{b}) + 2\cos (\pi \frac{a}{b})}}, $$

where $a = 1/4(1/{\tau _r} - 1/{\tau _f})$ and $b = 1/4(1/{\tau _r} + 1/{\tau _f})$ ${\tau _r}$ ${\tau _f}$ are the rise and fall times, respectively. The linewidth $L(v)$ is given by $FWHM = b/{\pi ^2}\ln ((R + \sqrt {{R^2} - 4} )/2)$ with $R = 2(2 + \cos (\pi a/b))$.

The spectrum of the OPO output laser can be obtained from the Fourier transform of the beat-frequency signal from the frequency stabilization system. The laser linewidth of the OPO can be recorded in real-time and is approximately 40 MHz. The linewidth of the OPA as measured in the laboratory is approximately 60 MHz. However, it cannot be recorded in real-time during the experiments. Figure 11(a) shows the actual line shape of the laser output from the OPO cavity and the 40 MHz (FWHM) model line shape calculated using Eq. (10) and a ratio of ${\tau _f}/{\tau _r} = 3$ [29]. With the exception of the measurement noise at the bottom of the actual line type, the line types are very similar for corresponding bandwidths. Figure 11(b) shows the modeled line shapes for 40 and 60 MHz.

(11)$$\Delta {\tau _{eff}} = \log \left. {\left\{ {\frac{{\int_0^\infty {L(v)dv} }}{{\int_0^\infty {L(v)\exp \left[ { - 2\int_{{R_G}}^{{R_A}} {({\sigma (r,\;v) - \sigma (r,{v_{off}})} )\ast {N_{c{o_2}}}(r)dr} } \right]dv} }}} \right.} \right\}$$

Equation (2) for the double-path DAOD is valid only for monochromatic light. The effective double-path DAOD can be determined by substituting the spectral energy distribution model into Eq. (2), as shown in Eq. (11). The systematic error caused by variations in the laser linewidth is shown in Fig. 12(a). The relative and absolute errors of XCO₂ for an OPA output laser spectral linewidth of 60 MHz are 0.027% and 0.109 ppm, respectively. The variations in the RSE of XCO₂ with the OPA output laser linewidth drift at 60 MHz is shown in Fig. 12(b). As can be seen from the figure, the RSE of the linewidth drift is very small and can thus be ignored.

Fig. 11. (a) Actual line shape of OPO output laser and model line shape for full width at half maximum (FWHM) of 40 MHz. (b) Modeled line shapes for FWHM values of 40 and 60 MHz.

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Fig. 12. (a) RSE of XCO₂ at different laser linewidths. (b) RSE of XCO₂ caused by variations in laser linewidth when FWHM is 60 MHz.

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4.2.3 Spectral purity

The laser transmitter of the airborne IPDA LIDAR system uses an injection-seed OPO to ensure a single longitudinal-mode output. However, amplified spontaneous emissions and other effects result in impurities in the spectrum. Since the offline wavelength is at the bottom of the CO₂ absorption line, as shown in Fig. 13(b), we primarily considered the effects of the online spectral purity. A long-path gas absorption cell was used as an optical filter to measure the spectral purity. The underlying principle of the method and the experimental setup used have been described elsewhere [30]. Figure 13(a) shows the spectral purity of the online laser as measured in the laboratory. The average spectral purity is 99.996%, while the standard deviation is 1.600×10⁻⁵. Without seed injection, the output spectral range of the free-running OPO cavity, measured with a spectrograph, is approximately 20 GHz. If the remaining 0.004% impure partial spectrum is a Lorentz distribution over a spectral width of 20 GHz, and the laser line profile is given by

(12)$$\tilde{L}(v) = (1 - {P_{spec}}){L_b}(v) + {P_{spec}}{L_n}(v), $$

where ${P_{spec}}$ is the spectral purity, then ${L_b}$ is the impure partial spectral line type, and ${L_n}(v)$ is the narrow-band pure spectral line type of the output laser of the OPA. If the remaining 0.004% impure partial spectrum is a uniform distribution, the laser line profile is given by

(13)$$\tilde{L}(v) = (1 - {P_{spec}}){L_B}(v) + {P_{spec}}{L_n}(v), $$

where ${L_B}(\nu )$ is the impure partial spectral line type (denoted ${L_B}(v){\sim }U( - \frac{B}{2},\frac{B}{2})$; where $B$ is the spectral distribution range). Figure 13(b) shows the energy distributions of the two line profiles and the variation in the double OD over the spectral width of 20 GHz.

Fig. 13. (a) Spectral purity of airborne IPDA LIDAR system. (b) Energy distributions of two line profiles, offline position, and the variations in double OD.

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Considering the effects of the spectral purity, the effective double-path DAOD can be expressed as

(14)$$\Delta \tau _{eff}^s = \log \left\{ {\frac{{\int_{ - \frac{{\Delta {B_f}}}{2}}^{\frac{{\Delta {B_f}}}{2}} {\tilde{L}(v)dv} }}{{\int_{ - \frac{{\Delta {B_f}}}{2}}^{\frac{{\Delta {B_f}}}{2}} {\tilde{L}(v) \cdot \exp \left( { - 2\int_{{R_G}}^{{R_A}} {[{\sigma (r,\;v) - \sigma (r,{v_{off}})} ]\times {N_{C{O_2}}}(r)dr} } \right)} dv}}} \right\}, $$

where $\Delta {B_f}$ is the bandwidth of the narrow-band filter added in front of the detector. Since the online and offline lasers are incident on the same detector and because their frequency difference is approximately 7.32 GHz, the bandwidth of the narrowband filter cannot be lower than 7.32 GHz. The effective ODs for the filter bandwidths used with this IPDA LIDAR system (10 GHz, 15 GHz, and 54 GHz) can be calculated using Eq. (14). The relative and absolute errors of XCO₂ for different filter bandwidths are shown in Table 3.

Table 3. RSE and absolute errors in XCO₂ for different filter bandwidths

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4.3 Correction algorithms for calculating IWF

During the airborne flight experiments, the electronic control system of the IPDA LIDAR system was installed within the aircraft, while the transceiver system was mounted in a pod outside the aircraft. To accurately measure the attitude angles (pitch angle $\theta $; roll angle $\gamma $; yaw angle $\varphi $), velocity (east speed ${\upsilon _x}$; north speed ${\upsilon _y}$; upward speed ${\upsilon _z}$) and position parameters of the LIDAR system during the experiment, an INS was installed on the same base as the LIDAR transceiver system.

A reference Cartesian coordinate system for the emitted laser was established, as shown in Fig. 14. The y-axis of the system is parallel to the base and points in the flying direction, while the z-axis points in the direction opposite to that of the laser. The x-axis is perpendicular to the plane composed of the y-axis and z-axis and forms a right-handed coordinate system with the y-axis and z-axis. The origin of the coordinate is the laser emission point. The coordinates of the laser footprint in this reference coordinate system are assumed to be ${({x_{SL}},{y_{SL}},{z_{SL}})^T}$. The direction of laser emission is along z-axis and perpendicular to the plane composed of the x-axis and y-axis, so its component on the x-axis and y-axis is zero. Therefore, the coordinates can be expressed as

(15)$$\left[ \begin{array}{l} {x_{SL}}\\ {y_{SL}}\\ {z_{SL}} \end{array} \right] = \left[ \begin{array}{c} 0\\ 0\\ - \rho \end{array} \right], $$

where $\rho = \frac{1}{2}ct$ is the distance between the laser emission point and the laser footprint, c is the velocity of light, and $t$ is the time delay of the laser pulse from the time of emission to the receiving of the echoes. As shown in Fig. 14, since the INS and the laser are mounted closely to each other on the same base, the issues arising from the misalignment of the system center can be ignored. Here, we mainly study the errors caused by attitude angles and velocity. The local geographic coordinate system is defined as one, in which the y_t-axis points in the North direction, the z_t-axis points to the sky zenith direction, and the x_t-axis points in the East direction. In order to facilitate calculation, the origin of local geographic coordinates is set at the laser emission point. Assuming that the coordinates of the laser footprint in the local geographic coordinate system are ${({x_{LH}},{y_{LH}},{z_{LH}})^T}$, we can say that

(16)$$\left[ \begin{array}{l} {x_{LH}}\\ {y_{LH}}\\ {z_{LH}} \end{array} \right] = {R_N}\left[ \begin{array}{l} {x_{SL}}\\ {y_{SL}}\\ {z_{SL}} \end{array} \right], $$

where ${R_N} = \left[\begin{array}{{ccc}} {\cos \varphi \cos \gamma + \sin \varphi \sin \theta \sin \gamma }&{\sin \varphi \cos \theta }&{ - \sin \varphi \sin \theta \cos \gamma + \cos \varphi \sin \gamma }\\ {\cos \varphi \sin \theta \sin \gamma - \sin \varphi \cos \gamma }&{\cos \varphi \cos \theta }&{ - \sin \varphi \sin \gamma - \cos \varphi \sin \theta \cos \gamma } \\ {\cos \theta \sin \gamma }& - \sin \theta & \cos \theta \cos \gamma \end{array}\right]$ is the matrix for transforming the INS reference coordinate system into the local geographic coordinate system. After the transformation, the coordinates of the laser footprint in the local geographic coordinate system are

(17)$$\left[ \begin{array}{l} {x_{LH}}\\ {y_{LH}}\\ {z_{LH}} \end{array} \right] ={-} \rho \left[ \begin{array}{c} - \sin \varphi \sin \theta \cos \gamma + \cos \varphi \sin \gamma \\ - \sin \varphi \sin \gamma - \cos \varphi \sin \theta \cos \gamma \\ \cos \theta \cos \gamma \end{array} \right]. $$

Fig. 14. LIDAR transceiver system and INS/GPS system installation diagram.

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4.3.1 Correction algorithm of integral path

Because of the nonzero pitch and roll angles during the flight (see Fig. 15), the direction of laser propagation is not always vertical to the ground. In this case, the corrected IWF can be expressed as

(18)$$IW{F_d} = \int_{R_G^d}^{{R_A}} {\frac{{P(r) \cdot {N_A} \cdot ({\sigma ({v_{on}},\;r) - \sigma ({v_{off}},\;r)} )}}{{R \cdot T(r) \cdot (1 + {X_{{H_2}O}}(r)) \cdot \cos \theta \cdot \cos \gamma }}} dr, $$

where $R_G^d = {R_A} - \rho \cdot \cos \theta \cdot \cos \gamma$ is ground elevation, which can be calculated by the altitude of aircraft minus the absolute value of the component of laser path in sky zenith direction.

Fig. 15. Height and attitude angles measured during an airborne experiment.

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The RE of XCO₂ caused by an uncorrected integral path was analyzed based on the altitude and attitude angles, as shown in Fig. 15(a). It varied with the aircraft attitude angles, and the maximum RE caused by the uncorrected integral path was 0.273%, with the corresponding absolute error being 1.090 ppm. The IWF can be corrected using Eq. (18) while the attitude can be measured with the INS. However, the INS exhibits installation errors, which can be determined by placing a laser electron theodolite on the base of the LIDAR transceiver system (see Table 4). As can be seen from Fig. 16(b), the maximum RE caused by the installation error of the INS was 4.10×10⁻⁴ and thus could be ignored.

Fig. 16. (a) RE caused by the uncorrected integral path. (b) RE caused by the installation error of INS owing to the integral path.

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Table 4. Comparison of results obtained using the INS and laser electronic theodolite

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4.3.2 Correction algorithms of Doppler shift

The head of the aircraft is lifted slightly because of the pitch during flight. This can result in a laser transmission direction that differs from the direction of flight. Hence, the flying speed will have a component in the direction of laser transmission, resulting in a Doppler shift. If the flying speed vector as measured by the INS is assumed to be ${({\upsilon _x},{\upsilon _y},{\upsilon _z})^T}$, the component of the flying speed in the direction of laser transmission can be expressed as

(19)$${\upsilon _L} = \frac{{({x_{LH}},{y_{LH}},{z_{LH}}) \cdot ({\upsilon _x},{\upsilon _y},{\upsilon _z})}}{{\sqrt {x_{LH}^2 + y_{LH}^2 + z_{LH}^2} }}. $$

The frequency offset caused by the Doppler shift is given by

(20)$$\Delta v = \frac{{{\upsilon _L}}}{\lambda } = \frac{{{\upsilon _L}{v_0}}}{c}. $$

The IWF with a correction for the Doppler shift (in MHz) is expressed as shown below:

(21)$$IW{F_f} = \int_{R_G^d}^{{R_A}} {\frac{{P(r) \cdot {N_A} \cdot ({\sigma ({v_{on}} + \Delta v,\;r) - \sigma ({v_{off}},\;r)} )}}{{R \cdot T(r) \cdot (1 + {X_{{H_2}O}}(r)) \cdot \cos \theta \cdot \cos \gamma }}} dr. $$

The RE of XCO₂ caused by an uncorrected Doppler shift was analyzed based on the three angles in Fig. 15 and along with the speed in the three directions in Fig. 17 using Eqs. (20) and (21), which is shown in Fig. 18(a). The maximum RE was 0.479%, and the corresponding absolute error was 1.916 ppm. The RE of XCO₂ because of the INS installation error after the Doppler shift correction is shown in Fig. 18(b). The maximum RE was 9.70×10⁻³, and the corresponding absolute error was 0.039 ppm, which is 2.025% of the RE caused by the Doppler shift. However, that caused by the INS installation error owing to the Doppler shift can be ignored. Thus, Doppler shift correction should be performed during the inversion process.

Fig. 17. (a) East speed ${\upsilon _x}$. (b) North speed ${\upsilon _y}$. (c) Upward speed ${\upsilon _z}$. (d) Frequency offset caused by the Doppler shift.

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Fig. 18. (a) RE caused by Doppler shift. (b) RE caused by the installation error of INS because of the Doppler shift.

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4.4 Results of simulation of systematic error

The systematic error in XCO₂ was simulated by taking the atmospheric (pressure, temperature, and humidity) and laser (frequency drift, linewidth, and spectral purity) parameters into account. The experiment of using ground glass diffusers to reduce the speckle effect during energy monitoring with this IPDA LIDAR has been thoroughly studied elsewhere [15]. The experiment results in the literature [15] showed that the standard deviation of the normalized energy ratio was 7.57×10⁻⁴. Therefore, the RSE of XCO₂ caused by energy monitoring was reduced to 0.077%.

The contributions of the various error sources, as well as the RSE and absolute error, are shown in Table 5. The total systematic error when added geometrically using Eq. (22) was 0.855 ppm, and became 0.381 ppm after the averaging of 148 shots based on frequency.

(22)$$\frac{{\delta XC{O_2}}}{{XC{O_2}}} = \sqrt {\sum\limits_i {{{\left( {\frac{{\delta \Delta {\tau_i}}}{{\Delta {\tau_i}}}} \right)}^2}} + \frac{1}{{IW{F^2}}}\sum\limits_j {{{\left( {\frac{{\partial IWF}}{{\partial {\upsilon_j}}} \cdot \delta {\upsilon_j}} \right)}^2}} }, $$

where subscript j represents the pressure, temperature, and humidity, and subscript i represents the frequency drift, linewidth, and spectral purity.

Table 5. Contributions of various error sources of airborne IPDA LIDAR instrument

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5. Conclusions

In this study, sensitivity analysis of the airborne IPDA LIDAR was completed with actual LIDAR and platform parameters. During the modeling of the monitoring and echo voltages, it was assumed that the detector speckle noise, background noise, and signal noise exhibit a Gaussian distribution while the quantization noise shows a uniform distribution. The online and offline echo signal voltages were simulated for different target reflectivity and AOD values. The double-path DAOD was calculated from the simulated signals, and the XCO₂ was inverted using the DAOD and IWF. The results indicate that low target reflectivity results in a larger random error. After the averaging of 148 shots, the standard deviation of XCO₂ for a typical target reflectivity of 0.1 sr⁻¹ was 0.230 ppm, with the highest standard deviation being 0.841 ppm under target reflectivity of 0.02 sr⁻¹.

The systematic errors in XCO₂ were simulated using actual atmospheric (pressure, temperature, and humidity) and laser (frequency drift, linewidth, and spectral purity) parameters. When the errors corresponding to these error sources were added geometrically, the RSE was 0.214% while the absolute error was 0.855 ppm; this was for a temperature uncertainty of 1 K, bias in the pressure profile of 0.1%, and bias in the humidity profile of 10%. Meanwhile, the measurement uncertainties in the energy monitoring, linewidth, and spectral purity as determined in the laboratory were 7.57×10⁻⁴ (normalized energy ratio), 60 MHz, and 99.996%, respectively. The STANDARD DEVIATION of the frequency drift as measured during the flight was 4.862 MHz. As can be seen from the results, the RSE was primarily caused by the frequency drift. Two methods were proposed for reducing the RSE caused by the frequency drift. The first method involves averaging 148 shots. After the averaging of 148 shots, the RSE was reduced to 0.096% while the absolute error was reduced to 0.381 ppm. The second method involves calculating the IWF using the measured real-time frequency instead of the fixed online wavelength of 1572.024 nm. However, this is time-consuming and computationally intensive. Thus, look-up tables for absorption cross-section were established to solve this problem. The proposed method exhibited an error of 0.005% with respect to that in the case of directly calculating the IWF. Further, the RSE could also be reduced to 0.096% using this method.

Based on the actual aircraft attitude angles, velocity and position data during a flight experiment, it was found that the RE caused by using the uncorrected integral path for calculating the IWF was 0.273%, while that caused by the Doppler shift was 0.479%. Two correction algorithms were used for calculating the IWF. In the case of the correction algorithm of the integral path, the IWF was corrected by a factor of $\cos \theta \cdot \cos \gamma$. For the correction algorithm of the Doppler shift, the IWF was corrected by the real-time laser frequency plus the Doppler shift, where the Doppler shift is $\Delta v = \frac{{{\upsilon _L}{v_0}}}{c}$ and ${\upsilon _L}$ is the component of the flying speed in the direction of the laser transmission. Thus, by applying correction algorithms for the integral path and Doppler shift, the RE can be eliminated by accurately calculating the IWF.

Field experiments of this airborne IPDA LIDAR system were performed at sea level, mountainous regions and on flat land subsequently by our group. This study confirmed the validity of system parameters and, as such, it can serve as a reference for other researchers who study similar IPDA LIDAR systems. In particular, the sensitivity analysis of the airborne IPDA LIDAR system can provide a reference to future data inversions. Moreover, the proposed correction algorithms will contribute to more accurate inversion results. However, the factors of H₂O line interference and the percentage of the footprint overlap between online and offline laser pulses were excluded from this study, owing to their negligible effects on this particular IPDA LIDAR system. Nevertheless, these factors should be investigated in future studies with other IPDA LIDAR systems.

Funding

National Key R&D Program of China (2017YFF0104600); Pre-research Project of Civilian Space (D040103); ACDL LIDAR project.

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Transmitter	Parameters	Receiver	Parameters
Online wavelength	1572.024 nm	Receiver optical efficiency	0.3797
Offline wavelength	1572.085 nm	Estimated flight altitude	8 km
Pulse energy(on/off)	6/6 mJ	Telescope diameter	150 mm
Pulse width(on/off)	17 ns	Field-of-View	1 mrad
Frequency drift(RMS)	2.7 MHz	Detector type	IAG350H1D
Repetition rate	30 Hz	Responsivity	0.94 A/W
Beam divergence	0.62 mrad	NEP	64 fw/rtHz
Pulse spectral width(OPA)	60 MHz	Excess noise factor	3.2(@M = 10)
Emission optical efficiency	0.8955	Data acquisition	125 MS/s

Target relativity	Mean number of pulse (shot)	Relative Error	Absolute error
0.3 sr⁻¹	1	0.524%	2.096 ppm
0.3 sr⁻¹	148	0.042%	0.167 ppm
0.1 sr⁻¹	1	0.699%	2.799 ppm
0.1 sr⁻¹	148	0.057%	0.230 ppm
0.05 sr⁻¹	1	1.096%	4.391 ppm
0.05 sr⁻¹	148	0.954%	0.382 ppm
0.02 sr⁻¹	1	2.430%	9.727 ppm
0.02 sr⁻¹	148	2.102%	0.841 ppm

Background emission distribution	Narrow-band filter bandwidth	RSE	Absolute error (ppm)
	10 GHz	0.0052%/0.0053%	0.0210/0.0212
Lorenz/Uniform	15 GHz	0.0053%/0.0063%	0.0212/0.0251
	54 GHz	0.0053%/0.0073%	0.0212/0.0291

Angles	Theodolite/°	Inertial navigation system/°	Angle deviation/°
Pitch angle	0.6065	0.6030	0.0035
Roll angle	1.5490	1.6250	−0.0760
Yaw angle	121.6803	124.8190	−3.1387

Error source	RSE (%)	Absolute error (ppm) (@ $Δ X C O_{2} = 400 p p m$ )	Uncertainty for CO₂
Atmosphere
Temperature	0.012	0.049	1 K (standard deviation)
Pressure	0.034	0.137	0.1% (bias)
Humidity	0.033	0.134	10% (bias)
Laser
Energy monitoring	0.077	0.306	7.57×10⁻⁴ (standard deviation)
Frequency drift	0.191	0.765	4.8624 MHz (standard deviation)
Frequency drift	0.004	0.018	0.1083 MHz (148 shots averaging)
Linewidth	0.027	0.109	60MHz
Spectral purity	0.005	0.021	99.996% with 0.45 nm filter (Lorenz distribution)
integral path	4.10×10⁻⁴	0.002	Angle deviation in Table 4
Doppler shift	9.70×10⁻³	0.039	Angle deviation in Table 4
Error budget	0.214	0.856	Frequency without averaging
(Geometrically added)	0.096	0.384	Frequency with 148 shots averaging

Sensitivity analysis and correction algorithms for atmospheric CO₂ measurements with 1.57-µm airborne double-pulse IPDA LIDAR

Abstract

1. Introduction

2. Airborne LIDAR system for CO₂ measurements

2.1 Principle of IPDA LIDAR

2.2 Instrument configuration

3. Random error analysis

3.1 Noise analysis

3.2 Simulation

4. Systematic error analysis

4.1 Effects of atmospheric parameters

4.2 Effects of laser parameters

4.2.1 Frequency drift

4.2.2 Linewidth

4.2.3 Spectral purity

4.3 Correction algorithms for calculating IWF

4.3.1 Correction algorithm of integral path

4.3.2 Correction algorithms of Doppler shift

4.4 Results of simulation of systematic error

5. Conclusions

Funding

References

Cited By

Figures (18)

Tables (5)

Equations (22)

Optics Express

Abstract

1. Introduction

2. Airborne LIDAR system for CO2 measurements

2.1 Principle of IPDA LIDAR

2.2 Instrument configuration

3. Random error analysis

3.1 Noise analysis

3.2 Simulation

4. Systematic error analysis

4.1 Effects of atmospheric parameters

4.2 Effects of laser parameters

4.2.1 Frequency drift

4.2.2 Linewidth

4.2.3 Spectral purity

4.3 Correction algorithms for calculating IWF

4.3.1 Correction algorithm of integral path

4.3.2 Correction algorithms of Doppler shift

4.4 Results of simulation of systematic error

5. Conclusions

Funding

References

Cited By

Figures (18)

Tables (5)

Equations (22)

Optics Express

2. Airborne LIDAR system for CO₂ measurements