## Abstract

In this study, a 1.57-µm airborne double-pulse integrated-path differential absorption (IPDA) light detection and ranging (LIDAR) system was developed for CO_{2} measurements. This airborne IPDA LIDAR is integrated with a real-time frequency monitoring system, an integrated sensor for temperature, pressure, and humidity, an inertial navigation system, and a global positioning system. The random errors of the LIDAR system, which are caused by the signal noise, background noise, and detector noise, among other factors, are analyzed for different target reflectivities at a flight altitude of 8 km. After parametric optimization, the signal is unsaturated at high target reflectivity. Further, it can be detected at low target reflectivity by adjusting the detector gain. After the averaging of 148 shots, the relative random error (RRE) was 0.057% for a typical target reflectivity of 0.1 sr^{−1}. Moreover, the systematic errors caused by the laser pulse energy, linewidth, spectral purity, and frequency drift, as well as the atmospheric parameters related to the flight experiments are also investigated. The relative system error (RSE) was 0.214% as determined based on an analysis of the systematic errors, which are primarily caused by the frequency drift. Two methods are proposed to reduce the RSE caused by the frequency drift. The first is the averaging of 148 shots, which can reduce the RSE to 0.096%. The other method involves calculating the integral weight function (IWF) using real-time frequency. However, this is a time-consuming and computationally intensive process. Hence, look-up tables for the absorption cross-section were created to overcome this issue, resulting in a decrease in the RSE to 0.096%. Using actual aircraft attitude angles, velocity, and position data from flight experiments, the relative errors (REs) in the IWF caused by the uncorrected integral path and Doppler shift were determined to be 0.273% and 0.479%, respectively. However, it was found that corrections to the integral path and Doppler shift based on accurate calculations of the IWF cause the airborne platform to turn in such a way that the REs are eliminated. Hence, this study confirms the validity of system parameters and provides a reference for other researchers who study similar IPDA LIDAR systems. Further, the sensitivity analysis of the airborne IPDA LIDAR system can provide a reference to future data inversions. Moreover, the proposed correction algorithms for the integral path and Doppler shift contribute to more accurate inversion results.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 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 CO_{2} were developed for NASA’s Active Sensing of CO_{2} 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 CO_{2} molecule, wavelengths of 1.57 and 2.05 µm have been used of the CO_{2} column-averaged dry-air mixing ratio (XCO_{2}) 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 CO_{2} 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 XCO_{2} 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, XCO_{2} measurements, which has online wavelength positioned at the center of the CO_{2} absorption line at 1571.112 nm [8]. Stable, accurate XCO_{2} 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 XCO_{2} 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 CO_{2} and CH_{4} 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 XCO_{2} 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 CO_{2}, CH_{4}, and N_{2}O 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 XCO_{2} 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 CO_{2} 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 CO_{2} 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 CO_{2} 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

*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 CO

_{2}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 CO

_{2}, denoted as $\Delta {\tau _{C{O_2}}}$, is expressed as

_{2}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]

#### 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.

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 XCO_{2} inversion, should be simulated simultaneously. After CO_{2} 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):

*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.

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.

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 XCO_{2} is shown in Fig. 5(c); its standard deviation is 2.799 ppm. The XCO_{2} 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 XCO_{2} 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.

## 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 CO_{2}, 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 XCO_{2} can be approximated as

On the other hand, when the error source only causes an error during the calculation of the IWF, the RSE of XCO_{2} can be approximated as

The temperature measurement error is assumed to obey the Gaussian distribution, and the standard deviation is taken as its uncertainty. The RSE of XCO_{2} varies with the uncertainty in the temperature as calculated using Eq. (8) (see Fig. 6(b)). The RSE of XCO_{2} for a temperature uncertainty of 1 K was 0.0123%, while the absolute error was 0.0492 ppm.

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 XCO_{2} as calculated by Eq. (8) varies with the pressure deviation ratio, as shown in Fig. 7(b). The RSE of XCO_{2} for a pressure deviation ratio of 0.1% was 0.034%, while the absolute error was 0.137 ppm.

The same method was used to simulate the error caused by the humidity. It was found that the RSE of XCO_{2} varies with the humidity deviation ratio, which was calculated using Eq. (8) (see Fig. 8(b)). The RSE of XCO_{2} for a humidity deviation ratio of 10% was 0.033%, while the absolute error was 0.134 ppm.

#### 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

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.

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 XCO_{2}. 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 XCO_{2} 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]:

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.

_{2}for an OPA output laser spectral linewidth of 60 MHz are 0.027% and 0.109 ppm, respectively. The variations in the RSE of XCO

_{2}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.

### 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 CO_{2} 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

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

_{2}for different filter bandwidths are shown in Table 3.

#### 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

*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

### 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

The RE of XCO_{2} 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.

### 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

_{2}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 XCO

_{2}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.

#### 4.4 Results of simulation of systematic error

The systematic error in XCO_{2} 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 XCO_{2} 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.

*j*represents the pressure, temperature, and humidity, and subscript

*i*represents the frequency drift, linewidth, and spectral purity.

## 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 XCO_{2} 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 XCO_{2} 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 XCO_{2} 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 H_{2}O 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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