Performance Evaluation of an A Band Differential Absorption LIDAR Model and Inversion for the Ocean Surface Pressure from Low-Earth Orbit

Abstract

Remote sensing of ocean surface pressure from space is very important, and differential absorption LIDAR and differential absorption radar are only two kinds of remote sensing instruments with this potential. The differential absorption LIDAR works with the integral path mode from the spacecraft in the 400 km low-Earth orbit. The differential optical depth of the oxygen A-band is measured, and then the ocean surface pressure is obtained using a circle-iterative calculation. Performance evaluation of the differential absorption LIDAR model was based on feasibility to the advanced system parameters of the space instrument, whilst weak echo pulse energy at ocean surface yielded random errors in the surface pressure measurement. On the other hand, uncertain atmospheric temperature profiles and water vapor mixture profiles resulted in a primary systematic error in the surface pressure. The error of the surface pressure is sensitive to the jitter of the central frequency of laser emission. Under a strict implementation of the error budget, the time resolution is 6.25 s and the along-orbit distance resolution is 44 km, 625 echoes from ocean surface was cumulatively averaged. Consequently, if the jitter of the central frequency of laser emission exceeded 10 MHz, controlling the error of the surface pressure below 0.1% proved almost hopeless; further, the error could be expected to within 0.1–0.2%; however, the error limited within 0.2–0.3% is an achievable indicator.

1. Introduction

Atmospheric pressure plays a vital role in several atmospheric processes related to atmospheric dynamics. Low pressure, high pressure, low-pressure troughs, high-pressure ridges, anticyclones, and other related information have been introduced into the number weather model. Hurricanes are profound low-pressure systems that originate from low-pressure cyclones in the tropical or subtropical ocean regions. The accurate prediction of their formation, landing direction, and movement trajectory requires atmospheric pressure gradient data. Meanwhile, the release of radio sounding balloons is restricted two times per day; thus, continuous detection during the entire day is not possible. Brown, R.A. and Levy, G. announced that the accuracy of the weather models is primarily limited by the regional sparsity of the input data [1]. Specifically, atmospheric pressure data is very sparse in large areas of the ocean. Consequently, the International Meteorological Organization aims to achieve remote sensing of the surface pressure at an accuracy of 0.1–0.3% [2] (WMO-ICSU, 1973).

In 1983, the Laboratory for Atmospheres, NASA Goddard Space Flight Center, a method of detecting atmospheric pressure using differential absorption LIDAR with the trough between oxygen absorption lines was proposed by Korb, C. L. et al. [3] In 1987, Schwemmer et al. built a novel differential absorption LIDAR system, which employs a flash-pump alexandrite laser to emit a pulse beam for two wavelengths at approximately 13,160 cm⁻¹,coupled with an oxygen photo-acoustic absorption cell and a high-precision wavelength meter to stabilize the emission wavelength. Moreover, the seed source is a continuous wave from either a Ti:sapphire single longitudinal mode laser or a diode laser [4]. The relationship between the differential optical depth obtained using the LIDAR and square difference of atmospheric pressure were expressed as:

$$\underset{z_0}{\overset{z}{{\displaystyle \int }}}\overline{K\left(Z\right)}dZ=C^{*}\left|p^2\left(z\right)-p_0^2\right|$$

(1)

$\overline{K\left(Z\right)}$ is the differential absorption coefficient, and the constant C* needs to be calibrated via radiosonde.

In June and July 1989, a series of flight measurement tests were conducted on the east coast of the United States [5]. In 1999, Flamant, C. N. et al. published their report “Pressure measurements Using and Airborne Differential absorption LIDAR. Part I: Analysis of the systematic error sources,” where the instrumental and systematic error sources of atmospheric pressure profile for differential absorption LIDAR were Analyzed [6].

In the ASCENDS (Active Sensing of CO₂ Emission over Nights, Days, and Seasons) mission, the surface pressure was required in order to accurately measure the CO₂ dry mixing ratio [7,8]. 2007 and 2013 period, Stephen, M.Krainak, et al. of the NASA Goddard Space Flight Center and Allan, G. R. of Sigma Space Corporation reported on the use of an aircraft as a platform to continuously transmit pulse trains of multiple wavelengths of around 764.7 nm and receive echo [9,10,11,12,13,14]. Thus, pulse train returns were accumulated, and the oxygen absorption spectrum curve of the 764.5–764.9 nm trough segment was plotted. Subsequently, the differential optical depth of oxygen was calculated from the transmittance curve.

24 September 2021, B. Lin and Z. Liu published a paper on <Earth and Space Science>, a new concept of Martian differential absorption Lidar operating at the 2.0 μm absorption band for atmospheric CO₂, and pressure observation for Martian were introduced. The paper shows that the relative errors of the CO₂ differential absorption optical depth measured are equivalent to corresponding relative errors of atmospheric CO₂ and pressure profile in observation [15].

Laser pulses with dual-wavelength (detection wavelength/reference wavelength) are transmitted downwards from the space platform; consequently, the return pulses energy from the ocean surface or the top of a cloud are received. Subsequently, the atmospheric optical depth and flight time of the laser pulses passing through the air column are measured. Thus, the atmospheric pressure and altitude of the surface/cloud top can be simultaneously obtained, and the top of the cloud and ocean surface can be distinguished. Such data is meaningful for various meteorological applications. By obtaining the pressure values on the ocean surface and cloud tops and combining the results with a vertical temperature profile obtained from other sensors or weather models and utilizing quasi-statistic equations, the vertical profile of the atmospheric pressure can be obtained. The differential absorption LIDAR is installed on a sun-synchronous orbit, and it makes a polar orbit around the earth from south to north. Atmospheric pressure data for surface/cloud top than ground meteorological stations could be much dense [16].

This paper is structured as follows. Section 1 presents the introduction, and the mechanism of differential absorption LIDAR for inversion ocean surface pressure is introduced in Section 2. Section 3 evaluates the performance of a differential absorption LIDAR model. Finally, Section 4 presents the conclusions.

2. Mechanism of Inversion Ocean Surface Pressure with Differential Absorption LIDAR

The differential absorption LIDAR selects two wavelengths in the A absorption band of oxygen (759–770 nm). The laser beam with one wavelength among the A absorption band passes through the atmosphere twice; its absorption coefficient is insensitive to changes in atmospheric temperature, is sensitive to variations in atmospheric pressure. This wavelength is this wavelength is called the detection wavelength (online). Further, the absorption coefficient of another wavelength for the laser beam is relatively smaller, and it is called the reference wavelength (offline), the reference wavelength value is close to the detection wavelength value.

Let the atmospheric pressure at altitude R₀ where the LIDAR located be p₀; the atmospheric pressure at altitude R be p(R); and g(z) be the gravitational acceleration at altitude z. The difference of the atmospheric pressure between altitude R₀ and R is equal to the weight of the air column between R₀ and R per unit area, where the dry air molecular mass m_dry = 28.9644 g/mol and water vapor molecular mass m_wv = 20 g/mol.

Atmospheric quasi-static Equation:

$$dp=-n_{dry}\left(z\right)\cdot \left(m_{dry}+m_{wv}{\chi }_{wv}\left(z\right)\right)\cdot gdz$$

(2)

Gas state Equation:

$$p\left(z\right)=n_{dry}\left(z\right)\cdot \left(1+{\chi }_{wv}\left(z\right)\right)\cdot kT\left(z\right)$$

(3)

$$p\left(R\right)=p_{surface}\cdot ex{p}^{-{\int }_{R_0}^R\frac{\left(m_{dry}+m_{wv}{\chi }_{wv}\left(z\right)\right)g\left(z\right)}{kT\left(z\right)\left(1+{\chi }_{wv}\left(z\right)\right)}dz}$$

(4)

This integration is performed between altitude surface and altitude z; n_dry(z) is the density of dry air molecules; χ_wv(z) is the water vapor volume mixing ratio; p_surface represent ocean surface pressure; and k is the Boltzmann constant. Thus, via remote sensing the weight or mass per unit area of a vertical air column between two altitudes, the difference in atmospheric pressure between these two altitudes can be obtained.

Oxygen is the most stable components in the atmosphere in terms of space and time. $n_{O_2}\left(z\right)$ is the number density of oxygen molecules at altitude z. The number of oxygen molecules accounts for a fixed proportion of 20.948% of the number of dry air molecules. Further, the optical depth of the atmosphere between R₀ and R is the integral of its extinction coefficient respect to the beam path, which can be expressed as:

$$OD\left(R_0,R\right)={{\displaystyle \int }}_{R_0}^R\left[{\alpha }_a\left(\lambda ,z\right)+{\alpha }_m\left(\lambda ,z\right)+n_{O_2}\left(z\right)\sigma \left(\lambda ,p\left(z\right),T\left(z\right)\right)\right]dz$$

(5)

where OD is the optical depth in Beer’s theorem; σ is the absorption cross-section of the oxygen molecule to the A-band λ wavelength; and ∆σ is the difference for σ (λ_on, p(z), T(z)) subtracting σ (λ_off, p(z), T(z)). Further, α_a (λ, z) and α_m (λ, z) are the aerosol extinction coefficient and the extinction coefficient of atmospheric molecules excepting oxygen absorption, respectively, and $n_{O_2}\left(z\right)\sigma \left(\lambda ,p\left(z\right),T\left(z\right)\right)$ is the oxygen absorption coefficient of the corresponding wavelength. The difference of the single-pass optical depth compared to the dual-wavelength between R₀ and R is called as the differential optical depth dOD(R₀, R).

$$dOD\left(R_0,R\right)={{\displaystyle \int }}_{R_0}^R{n}_{O_2}\left(z\right)\left(\sigma \left({\lambda }_{on},z\right)-\sigma \left({\lambda }_{off},z\right)\right)dz$$

(6)

$$n_{O_2}\left(z\right)=\frac{0.20948p\left(z\right)}{kT\left(z\right)\cdot \left(1+\chi \left(z\right)\right)}$$

(7)

where N_s,on(R)/N_s,off(R) represents the online/offline dual-wavelength echo pulse energy (number of photons) received by the LIDAR, which is expressed with the LIDAR Equations as follows:

$$N_{s,on}\left(R_0,R\right)=\frac{{\lambda }_{on}\cdot E_{on}}{h\cdot c}\cdot A_r\cdot {\eta }_d\cdot {\eta }_r\cdot \left(\frac{{\rho }_{eff}}{\pi }\right).\frac{\mathit{exp}\left[-2OD\left(R_0,R\right)\right]}{{\left(R_0-R\right)}^2}$$

(8)

$$N_{s,off}\left(R_0,R\right)=\frac{{\lambda }_{off}\cdot E_{off}}{h\cdot c}\cdot A_r\cdot {\eta }_d\cdot {\eta }_r\cdot \left(\frac{{\rho }_{eff}}{\pi }\right).\frac{\mathit{exp}\left[-2OD\left(R_0,R\right)\right]}{{\left(R_0-R\right)}^2}$$

(9)

where E_on/E_off is the energy of a single shot laser for both online/offline. Further, η_r is the receiving efficiency of the light beam; η_d is the quantum efficiency of the detector; A_r is the effective receiving area of the telescope; the detection wavelength and reference wavelength are λ_on and λ_off; the Planck constant is h; ρ_eff is the surface reflectivity; and c is the speed of light. In the space-to-earth observation, the integrate path differential absorption Lidar (IPDA) receives the return echo from ocean wave, the beam of dual-wavelength has the same path, receiving/sending time, footprint, and random process in the atmosphere. Further, except for σ(λ, p(z),T(z)), all other parameters are considered to be similar (but not equal).

Equation (8) divided by (9), where the differential optical depth can be calculated by measuring the energy of the pulse emitted and the energy of the received return echo using the LIDAR as following Equation (10):

$$dOD\left(R_0,R\right)=-\frac{1}{2}\mathit{ln}\left\{\left[\frac{N_{on}\left(R\right)}{N_{off}\left(R\right)}\right]\left(\frac{E_{off}}{E_{on}}\right)\right\}+C$$

(10)

$$\begin{gathered} C={{\displaystyle \int }}_{R_0}^R\left[{\alpha }_a\left({\lambda }_{on},z\right)-{\alpha }_a\left({\lambda }_{off},z\right)\right]dz \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{{\displaystyle \int }}_{R_0}^R\left[{\alpha }_m\left({\lambda }_{on},z\right)-{\alpha }_m\left({\lambda }_{off},z\right)\right]dz \end{gathered}$$

(11)

Although the weight of the atmospheric column between R₀ and R per unit area is unknown, but the differential optical depth can be measured with Equation (10). Where C in Equation (11) represents a wavelength dependent systematic error between the value $\left(\frac{1}{2}\mathit{ln}\left(\frac{N_{off}\left(R\right)}{N_{on}\left(R\right)}\frac{E_{on}}{E_{off}}\right)\right)$ and the differential optical depth. Thus, laser shots of two wavelengths are simultaneously emitted and the returns from the surface are received. However, owing to the difference in oxygen absorption, the atmospheric transmittance of the two wavelengths is different. The logarithm of the ratio $\left(\frac{N_{off}\left(surface\right)}{N_{on}\left(surface\right)}\frac{E_{on}}{E_{off}}\right)$ can be used to obtain the differential optical depth from the spacecraft to the surface. The IPDA transmit several laser pulses from the space platform to the ocean surface, and it detects the surface pressure, with point R₀ representing the spacecraft location. Further, there is almost no air pressure, p(R₀) = 0.0, and p(R) represents the surface pressure p_surface.

In the path of the laser beam, only the section from the altitude of 71 km to the ground has a significant effect on the optical depth, whereas the effect of the atmosphere above an altitude of 71 km can be ignored. Further, the gravitational acceleration g(z) can be regarded as a constant 9.80665 N/m² at an atmospheric altitude below 71 km.

On transforming the elevation z coordinates in Equation (2) into atmospheric pressure p coordinates, the following Equation (12) is obtained:

$$n_{dry}\left(z\right)dz=\frac{dp}{\left(m_{dry}+m_{wv}{\chi }_{wv}\left(p\right)\right)g\left(p\right)}$$

(12)

Thus, the absorption cross-section σ (λ, p(z), T(z)) is related to atmospheric temperature and pressure, and thus it can be rewritten as σ(λ, p, T(p)) in pressure p coordinates. Combining Equation (12), we can transform Equation (6) in the elevation z coordinate to Equation (13) in the pressure p coordinate. The differential optical depth dOD associated with pressure p coordinates is expressed using Equation (13) as follow:

$$dOD\left(p_{surface},p_{top}\right)=0.20948{{\displaystyle \int }}_{p_{top}}^{p_{surface}}\frac{\sigma \left({\lambda }_{on},p,T\left(p\right)\right)-\sigma \left({\lambda }_{off},p,T\left(p\right)\right)}{\left(m_{dry}+m_{wv}{\chi }_{wv}\left(p\right)\right)g\left(p\right)}dp$$

(13)

Here because of the pressure at the top of the atmosphere is p_top = 0.0 and the atmospheric pressure at the surface (or cloud top) is p_surface. In the pressure p coordinate, the differential optical depth dOD(p_surface)is expressed through the integral Equation (14) as follows:

$$dOD\left(p_{surface}\right)={{\displaystyle \int }}_0^{p_{surface}}\frac{\sigma \left({\lambda }_{on},p,T\left(p\right)\right)-\sigma \left({\lambda }_{off},p,T\left(p\right)\right)}{2.251667\times {10}^{-24}\times \left(1+0.69{\chi }_{wv}\left(p\right)\right)}dp$$

(14)

Equation (14) establishes the implicit expression of the differential optical depth of the entire atmosphere with respect to the surface pressure p_surface. Theoretically, the true value of the differential optical depth is a function of the atmosphere state and is independent of LIDAR parameters, further, it is independent of the measurement method. However, the measurement error in the differential optical depth is nearly related to the LIDAR parameters and inversion method.

On differentiating both sides of Equation (14) with respect to p_surface and considering the derivative function of dOD (p_surface) with respect to p_surface, we obtain Equation (15) as follow:

$$\begin{array}{ll}\frac{\partial \left(dOD\left(p_{surface}\right)\right)}{\partial p_{surface}}& \\ & =\frac{\sigma \left({\lambda }_{on},p_{surface},T\left(p_{surface}\right)\right)-\sigma \left({\lambda }_{off},p_{surface},T\left(p_{surface}\right)\right)}{2.251667\times {10}^{-24}\times \left(1+0.69{\chi }_{wv}\left(p_{surface}\right)\right)}\end{array}$$

(15)

Subsequently, the relationship between the errors of the surface pressure and the errors of differential optical depth of the entire atmosphere is obtained as follow Equation (16):

$$\begin{array}{l}\delta p_{surface}\\ =\frac{2.251667\times {10}^{-24}\times \left(1+0.69{\chi }_{wv}\left(p_{surface}\right)\right)}{\sigma \left({\lambda }_{on},p_{surface},T\left(p_{surface}\right)\right)-\sigma \left({\lambda }_{off},p_{surface},T\left(p_{surface}\right)\right)}\delta \left[dOD\left(p_{surface}\right)\right]\end{array}$$

(16)

Assuming that the vertical profile of atmospheric temperature T(R) and the vertical profile of water vapor mixing ratio χ_wv(R) are known from other sensors or weather models, the surface pressure can be inversed from the differential optical depth of the entire atmosphere measurement. The steps are shown in Figure 1.

a.

Measurement value of the differential optical depth (dOD)_m from the return signal N_s and emission energy E of the differential absorption LIDAR is calculated with Formula (10).

b.

Utilizing the atmospheric temperature profile T(R) coupled with the pressure profile and the surface pressure in the standard atmosphere mode as the initial value of the atmospheric pressure profile p₁(R) and the initial value of the surface pressure p_surface,1, respectively, and using the oxygen HITRAN database, the initial value of the absorption coefficient profile of the entire atmosphere is calculated. Thereafter, the initial value of the differential optical depth (dOD)_c,1 of the entire atmosphere is calculated from Formula (6).

c.

If the differential optical depth (dOD)_c,i of the entire atmosphere is numerically calculated in the i-th cycle and (dOD)_c,i is not equal to the differential optical depth (dOD)_m measured using the LIDAR, then the surface pressure (p_surface,i) calculated using the numerical value is not equal to the true value p_surface of the pressure at the footprint, and thus (dOD)_m is subtracted from (dOD)_c,i.

d.

The surface pressure varies with the differential optical depth. In the i-th cycle, the difference between (dOD)_c,i and (dOD)_m is multiplied by a coefficient $\frac{2.251667\times {10}^{-24}\times \left(1+0.69{\chi }_{wv}\left(p_{surface,i}\right)\right)}{\sigma \left({\lambda }_{on},p_{surface,i},T_{surface}\right)-\sigma \left({\lambda }_{off},p_{surface,i},T_{surface}\right)}$ in Equation (16) as the compensation amount and added to the calculated value of the surface pressure (p_surface,_i). Consequently, the resulting sum shown Formula (17) as follow is used as the new surface pressure (p_surface,i+1).

$$p_{surface,i+1}=p_{surface,i}+\frac{2.251667\times {10}^{-24}\times \left(1+0.69{\chi }_{wv}\left(p_{surface}\right)\right)}{\sigma \left({\lambda }_{on},p_{surface,i},T_{surface}\right)-\sigma \left({\lambda }_{off},p_{surface,i},T_{surface}\right)}\left(dO{D}_{c,i}-dO{D}_m\right)$$

(17)

e.

Subsequently, with the atmospheric temperature profile T(R) and water vapor mixing ratio χ_wv(R) provided via other sensors or numerical weather models, coupled with the surface pressure result p_surface,i obtained in the i + 1-th cycle, the new atmospheric pressure profile p_i+1(R) is calculated from Formula (4).

f.

Further, with the atmospheric temperature profile T(R), the profile for the water vapor mixing ratio χ_wv(R) and the atmospheric pressure profile p_i+1(R), based on the HITRAN database, differential optical depth (dOD)_c,i+1 are calculated repeated with Formula (6) repetitively.

g.

Repeat steps c–f. In the case of the above-mentioned iterative process, with the increasing in i, the difference between (dOD)_c,i+1 and (dOD)_m decreases till i = M, whilst the (p_surface,M+1-p_surface,M) is stable. If that happens, the iterative loop stops. Herein, the output surface pressure (p_surface,M) calculation result is considered to be sufficiently close to the true value (p_surface).

The above calculation steps also suggest that the parameters related to the temperature, pressure, and humidity of the atmosphere should be remote sense synchronously in the future, as input conditions for each other, and simultaneously iterated.

3. Performance Evaluation of an Integrated Path Differential Absorption LIDAR Model

3.1. A-Band Absorption Spectrum of Oxygen

Within the A absorption band of oxygen (759–770 nm) [17], the spectral transmittances in the vicinity of 760 and 765 nm were relatively insensitive to temperature, with this band with low interference from water vapor and carbon dioxide molecules.

The wavelength in the trough between the oxygen absorption line P13Q12 and P13P13, between the oxygen absorption line P15P15 and P15Q14, and even between absorption lines P17Q16 and P17P17, or the trough between the absorption line RR13 and R13Q14, RR12 and RQ13, RR11 and RQ12 can be selected as the detection wavelength.

Table 1. Differential optical depth and differential absorption cross section of 4 pairs of wavelengths in standard atmospheric mode.

Wavelength (nm)		${{\displaystyle \int }}_0^{71}\left({\mathit{\alpha }}_{\mathit{m}}+{\mathit{\alpha }}_{\mathit{a}}\right)\mathit{d}\mathit{z}$	${{\displaystyle \int }}_0^{71}\left({\mathit{n}}_{{\mathit{O}}_2}\mathit{\sigma }\right)\mathit{d}\mathit{z}$	OD(0,71)	dOD(0,71)	Δσ(psurface)
on	764.6840	0.1852	0.493	0.678	0.428	1.80 × 10−25
off	764.9097	0.1851	0.0653	0.251
on	765.1600	0.1851	0.389	0.574	0.325	1.46 × 10−25
off	764.9707	0.1851	0.0638	0.249
on	765.1736	0.1849	0.373	0.558	0.328	1.47 × 10−25
off	765.3883	0.1849	0.0448	0.230
on	765.6735	0.1848	0.231	0.416	0.192	0.925 × 10−25
off	765.4637	0.1849	0.0391	0.224

lists certain parameters of the six absorption lines: for example, the linear function σ_q($\lambda$ − λ₀₁) of the absorption cross-section of the oxygen absorption line P13Q12 with respect to the wavelength λ₀₁, and the line function σ_p(λ − λ₀₂) of the absorption cross-section of the P13P13 with respect to the wavelength λ₀₂. Further, the wavelength(λ_on/off) selected is located at the minimum of the absorption cross-section between the two spectral lines, its absorption cross-section is the superposition of the values of the extension lines of two adjacent Voigt linear functions at λ_on. Moreover, its absorption cross-section σ_on(λ) = σ_p($\lambda$_on) + σ_q($\lambda$_on) is the addition of the wings of the two Voigt line-shape functions with their pressure expansion. Differential optical depth and differential absorption cross section of 4 pairs of wavelengths in standard atmospheric mode is shown in Table 1.

In addition, the plots of absorption spectra in Figure 2 show that the oxygen absorption features a number of smaller and sharper absorption spikes, and that it contains isotope of oxygen molecules, showing some subtle differences. Near ocean surface absorption cross section near 765 nm is shown in Figure 2.

Equation (16) clearly indicates various factors that influence δ[dOD(p_surface)], the error of the differential optical depth conditionally causes δp_surface the error of the surface pressure. Additionally, the surface pressure error is inversely proportional to the paired wavelength absorption cross-section difference Δσ(p_surface) near the ocean surface. Evidently, a key factor affecting the DIAL sensitivity is the online and offline wavelength positions in A band. However, for the candidate wavelengths marked in Figure 2 used as detection wavelengths, each would offer its own advantages and disadvantages, and consequently, a comprehensive evaluation is required. When the jitter of the central frequency laser exceeded 10 MHz, the wavelength in the trough (example 765.6735 nm) between the oxygen absorption line should be selected as online. If the jitter of the central frequency laser was within 1 MHz order, online wavelength could be selected from the trough nearby (example 765.6968 nm).

3.2. Differential Absorption LIDAR System Model

The research results on the Alexandrite laser were reported by Coney et al. [18,19,20,21,22,23,24]. The DIAL transmitter uses an injection seeded alexandrite laser pumped diode. To achieve the narrow linewidth operation out of the alexandrite laser cavity, injection seeding is employed via two CW fiber coupled tunable distributed feedback (DFB) lasers [25], a EDFA (Erbium-Doped Fiber Amplifier) and a periodically poled SHD (Second Harmonic Generation).

The CW lasers are stabilized and operated at twice of the online and offline wavelengths. First, a distributed-feedback laser diode (DFB-LD) was frequency-locked to the Hydrogen Cyanide (or acetylene) line center by “Pound-Drever-Hall” technique, limiting its peak-to-peak frequency drift to 0.3 MHz at 0.8 s averaging time over 72 hours. Second, wavelengths of another two DFB-LDs were then offset locked to twice of the online and offline wavelengths using phase-locked loops, retaining virtually the same absolute frequency stability [25].

Stabilization of the transmitter wavelength is implemented by locking the laser cavity length referenced to the seed wavelength using the “Ramp-Delay-Fire” technique [26]. When the ramp voltage is applied to the PZT, the PZT moves and periodically scans the length of the oscillation cavity. If the ramp voltage on the PZT is rising, when the seed laser resonates with the oscillation cavity, the photodiode will monitor a peak signal, and the system will record the voltage value of the PZT at this time; in the falling phase of the scanning voltage of the PZT, once the photodiode monitors the peak signal again, the system will apply a trigger signal to the Pockels cell and turn on the Q switch. At this time, the oscillation cavity will quickly send out a single-frequency laser pulse based on the seed laser.

Along with those reported by Wulfmeyer and ¨osenberg, et al. [27], referring to the ADM-Aeolus in orbit ALADIN system parameters of the Aeolus mission [28], the receiver is based on the GLAS-Mission-1064 nm receiver, the orbit altitude is 400 km, and diameter of the telescope is 1.5 m. Consequently, the model parameters of the differential absorption LIDAR have been proposed, as shown in

Table 2. System parameters of differential absorption LIDAR.

Transmitter	Alexandrite Laser
Laser pulse energy	100 mJ
Laser pulse Width	<100 ns
Pulse repetition rate	100 Hz
Laser divergence angle	90 μrad for ±3σ
Spectral purity	99.99%
Fluctuations of central optical frequency	≤10 MHz(rms)
Receiver
Telescope diameter (Ar)	1.5 m(SiC) Cassegrain
Receiver field-of-view (full)	100 μrad
Optical filter bandwidth (FW)	0.8 nm (FWHM)
Fabry-Perot Elton bandwidth (depth = 2 mm)	25 pm (free spectral range ≈ 0.1 nm)
Receiver Efficiency	50%
Combined filter width	0.025 nm
Detection
Detector (Laser Components DG, Inc.)	Si-APD(SAR1500/C30956/S3884-04)
APD quantum efficiency (ηd)	75%
Detector diameter	Φ1.5 mm
Electronic system bandwidth (BW)	3 MHz
APD dark current (Id)	1 nA type
APD gain(M)	100
APD excess noise factor(F)	2.4
APD capacitance (Cd)	4 pF
Trans-impedance amplifier gain (Rf)	20 kV/A
Trans-impedance amplifier input current noise (InA)	2.5 pA/Hz1/2
Trans-impedance amplifier input voltage noise (VnA)	20 nV/Hz1/2
Operate temperature	293 K
Platform and Environment
Orbit altitude and velocity	400 km, 7 km/s
Orbit type	Polar, sun synchronous, dawn/dusk
Along-track resolution	44 km
Simulation top altitude	71 km
Viewing geometry	Nadir
Pointing stability	<50 μrad
(765 nm) surface albedo over ocean	0.1575
Atmosphere model	US standard atmosphere
Aerosol model	Median aerosol profile
Spectroscopic data base	HITRAN 2012

. The transmitter model parameters, with the exception of the pulse energy of 100 mJ, have been respectively reported in different documents [20]. However, these indicators have been achieved in the same laser, and thus, more research is required.

As reported in the reference [29], the equivalent Lambertian reflection coefficient of the sub-spacecraft point laser on the ocean surface has an empirical relationship:

$${\rho }_{eff}=\frac{\rho }{4S^2}$$

(18)

where the Fresnel reflection coefficient is ρ = 0.02, and <S²> is the variance of the wave steepness distribution.

Further, Bufton et al. [30] and Menzies et al. [31] individually adopted the relationship as follows:

$$〈S^2〉=\left\{\begin{array}{c}\left(ln{U}_{10}+1.2\right)\times {10}^{-2}{U}_{10}\le 7.0\mathrm{m}/\mathrm{s}\\ \left(0.85ln{U}_{10}-1.45\right)\times {10}^{-1}{U}_{10}>7.0\mathrm{m}/\mathrm{s} \end{array}\right.$$

(19)

where U₁₀ is the wind speed of segment 10 m above the ocean surface. The ocean surface wind speed with the larger probability of occurrence is between level 4 and level 5, the ocean surface wind speed is calculated as 8 m/s and the wave altitude is 2 m.

3.3. Performance Evaluation of A-band DIAL System

3.3.1. Random Error of Differential Optical Depth Caused by Noise

The number of received return echo photons N_s,on and N_s,off is obtained using the LIDAR Equation (8) and Equation (9). Equations (20)–(25) are commonly used for the on channel and off channel.

R − R₀ = 400 km, OD(R₀, R) = OD(0,71 km). Further, the effective pulse width τ_w of the echo signal is a combination of the emitted laser pulse width τ_L, and the detection electronic system bandwidth BW (Hz), and effective the wave altitude within the laser footprint ΔH, can be expressed as [32]:

$${\tau }_w=\sqrt{{\tau }_L^2+{\left(\frac{1}{3}\cdot BW\right)}^2+{\left(\frac{2\cdot \Delta H}{c}\right)}^2}$$

(20)

ΔH = 2 m, the background signal N_BG (photoelectrons), assuming a Lambertian surface and zenith sun, is calculated as:

$$N_{BG}\left(\lambda \right)=\frac{\lambda \cdot S_{BG}}{h\cdot c}{\tau }_w\cdot A_r\cdot {\eta }_d\cdot {\eta }_r\cdot \left(\frac{{\rho }_{eff}}{\pi }\right)\cdot {\left(\frac{FOV}{2}\right)}^2\pi \cdot FW·\mathit{exp}[-2OD\left(R_0,R\right)]$$

(21)

where N_BG is the exo-atmospheric solar irradiance value (1.221 W·m⁻²·nm⁻¹) [33] at 765 nm, FW is the bandwidth of the optical filter (0.025 nm × 4) and the field of view (FOV) (unit rad) of the receiving telescope. The bandwidth of the Fabry–Perot etalon, free spectral range, and width of the narrowband filter was 25 pm, 0.1 nm, and 0.8 nm, respectively. There are eight longitudinal modes of the Fabry–Perot etalon that can pass through. However, the transmittance of each longitudinal mode is different, and thus, the equivalent solar window width is 25 pm × 4 = 0.1 nm. Further, ρ_eff/π is the backscattering coefficient on the surface of ocean during the daytime [32] and ρ_eff was calculated as 0.1575.

Here the q electrons charge is 1.6 × 10⁻¹⁹ C, and M is the gain of silicon avalanche diode (APD). The total noise associated with the detection signal is divided into fixed circuit noise and signal-dependent shot noise. The total circuit noise current spectral density (unit A/Hz^1/2) I_n can be expressed as [34]:

$$I_n=\sqrt{2\cdot q\cdot I_d\cdot M\cdot F+I_{nA}^2+\frac{V_{nA}^2}{R_f^2}+\frac{4\cdot k\cdot T}{R_f+\frac{{\left(2\cdot \pi \cdot V_{nA}\cdot C_d\cdot BW\right)}^2}{3}}}$$

(22)

where I_d and F are the dark current and excess noise factors of the detector, respectively; I_nA and V_nA are the preamplifier integrated input current and input voltage noise spectral density, respectively; R_f is the feedback resistance of the preamplifier; and C_d is the equivalent input capacitance of the amplifier and the detector.

The circuit noise is often limited by the shot noise of the dark current of the detector or the noise of the preamplifier. In this analysis, all circuit noises refer to the detector input and the equivalent circuit noise-generated photoelectrons N_n,C is calculated as [34]:

$$N_{n,C}=\frac{I_n\cdot {\tau }_w\cdot \sqrt{BW}}{q\cdot M}$$

(23)

Similarly, the equivalent shot noise-generated photoelectrons, N_n,S, are calculated as [34]:

$$N_{n,S}=\sqrt{2\cdot N_S\cdot F\cdot {\tau }_w\cdot BW}$$

(24)

Further, the photoelectron N_n,BG, equivalent to the equivalent shot noise associated with the background radiation can be calculated as [34]:

$$N_{n,BG}=\sqrt{2\cdot N_{BG}\cdot F\cdot {\tau }_w\cdot BW}$$

(25)

These noises are regarded as the equivalent photoelectron number generated in the detector (before the multiplication process) and are proportional to the actual detected photoelectron number. The total signal-to-noise ratio is expressed as follows [32]:

$$SN{R}_{on/off}=\frac{N_{s,on/off}}{\sqrt{N_{n,C}^2+N_{n,S,on/off}^2+N_{n,BG,on/off}^2}}$$

(26)

where s is the number of echo signal pulses accumulated and averaged via the LIDAR. Error ε_R, caused by the noise of the LIDAR receiving a single echo, it is a random error. Moreover, it is necessary to calibrate the LIDAR echo signal detection channel and the laser emission pulse energy monitoring channel to remove nonlinear and nonzero biased background voltage.

In addition, error ε_A caused by the uncertainty of the atmospheric environment (atmospheric temperature profile, atmospheric water vapor mixing ratio profile), and the associated error ε_T of the laser emission characteristics (jitter of the center wavelength of the emitted light, the emission spectrum width, and the purity of the emission spectrum) are all systematic errors. The total error in the differential optical depth can be expressed as [34]:

$$\delta \left[dOD\right]=\frac{{\epsilon }_R}{\sqrt{s}}+\sqrt{{\epsilon }_A{}^2+{\epsilon }_T{}^2}$$

(27)

Equation (10) indicates that the random noise of the echo signals N_s,on(R) and N_s,off(R) and the random measurement error in the pulse energies E_on and E_off result in the random error in the differential optical depth ε_R = δ[dOD(p_surface)]_R, as follows:

$$\begin{gathered} {\epsilon }_R=\delta {\left[dOD\left(p_{surface}\right)\right]}_R \\ =\frac{1}{2}\sqrt{{\left(\frac{\delta N_{s,on}\left(R\right)}{N_{s,on}\left(R\right)}\right)}^2+{\left(\frac{\delta N_{s,off}\left(R\right)}{N_{s,off}\left(R\right)}\right)}^2+{\left(\frac{\Delta E_{on}}{E_{on}}\right)}^2+{\left(\frac{\Delta E_{off}}{E_{off}}\right)}^2} \end{gathered}$$

(28)

The signal-to-noise ratio of LIDAR can be calculated using Equations (20)–(26). Simultaneously, it is considered that the measurement error in the pulse energy ΔE_on/E_on ≈ ΔE_off/E_off is very small and can thus be ignored. The random error of the LIDAR echo (noise) measurement is calculated as:

$${\epsilon }_R=\frac{1}{2}\sqrt{SN{R}_{on}^{-2}+SN{R}_{off}^{-2}}$$

(29)

When the laser irradiates the ocean surface (for example, the average wind speed is 8 m/s), 0.1575 represents the median for the ocean surface reflectivity, and the random error ε_R of the differential optical depth is calculated considering the single pulse echo; with a time resolution of at least 6.25 s, the distance resolution along the track of 44 km and s = 625 laser pulse echoes are taken as a group for a cumulative average; the random error in the atmospheric differential optical depth above the ocean surface is $\frac{{\epsilon }_R}{\sqrt{625}}$.

The ocean surface possesses low laser reflectivity and weak echoes. The averaging method employed is as follows: first, add up 625 echoes (Equation (30)); thereafter, subtract B (Equation (31)) and normalize, where B is the level background baseline of the LIDAR output; and finally Equation (32) provides the differential optical depth. The error from signal-to-noise ratio were listed in

Table 3. Error from signal-to-noise ratio.

Wavelength (nm)	SNR	${\mathit{\sigma }}_{{\mathit{O}}_2}$ (cm2)	Single Shot Noise Error	Average Noise Error/25
764.6840	97.74	2.08 × 10−25	0.0061	2.44 × 10−4
764.9097	152.1	2.80 × 10−26
765.1600	109.0	1.74 × 10−25	0.0056	2.24 × 10−4
764.9707	152.3	2.77 × 10−26
765.1736	110.9	1.67 × 10−25	0.0055	2.20 × 10−4
765.3883	155.4	2.01 × 10−26
765.6735	128.4	1.10 × 10−25	0.0050	2.00 × 10−4
765.4637	156.3	1.78 × 10−26

.

$$E_{on/off}={\displaystyle \sum }_{i=1}^{625}{E}_{i,on/off}$$

(30)

$$N_{s,on/off}={\displaystyle \sum }_{i=1}^{625}{N}_{i,s,on/off}-B$$

(31)

$$dOD=O{D}_{on}-O{D}_{off}=-\frac{1}{2}\mathit{ln}\left(\frac{N_{s,on}-B}{E_{on}}\right)+\frac{1}{2}\mathit{ln}\left(\frac{N_{s,off}-B}{E_{off}}\right)$$

(32)

3.3.2. Uncertainty of Vertical Distribution Profile of Atmospheric Temperature

Within a certain time resolution (distance resolution), the uncertainty of the vertical profile of the atmospheric temperature results in an absolute systematic error with respect to the differential optical depth [34].

It can be considered that within the range of 0–71 km, the probability of temperature +1 K error and −1 K error is equivalent.

Altitude resolution of temperature profile is 250 m in the boundary layer, 500 m in the troposphere, 1000 m in the upper stratosphere, and 2000 m in the 30–71 km altitude. In the interval [z_i-1, z_i], the temperature may show a +1 K error, and in the next interval [z_i, z_i+1] the temperature might have a −1 K error; the optical depth is integrated over the interval [0–71 km], and the accumulation of +1 K and −1 K errors makes the net optical depth error smaller than the accumulated optical depth error in the case of pure +1 K or pure −1 K temperature error; data of

Table 4. Temperature sensitivity of differential optical depth in case of ±1 K uncertainty.

Regimentation	$\mathbf{\Delta }\left|\mathit{d}\mathit{O}\mathit{D}\left( \mathit{T}\left(\mathit{z}\right)-1\right)\right|$	$\mathbf{\Delta }\left|\mathit{d}\mathit{O}\mathit{D}\left( \mathit{T}\left(\mathit{z}\right)+1\right)\right|$	Differential Optical Depth Error
764.6840/764.9097	0.00216	0.00215	21.55× 10−4
765.1600/764.9707	0.000929	0.000933	9.31 × 10−4
765.1736/765.3883	0.00102	0.00102	10.2× 10−4
765.6735/765.4637	0.000139	0.000146	1.425 × 10−4

is the average of the two cases (+1 K or −1 K), the error of the optical depth caused by the ±1 K temperature uncertainty may be:

$$\mathsf{\Delta }{\left[dOD\left(surface\right)\right]}_{\pm 1K}=\underset{0}{\overset{71\mathrm{km}}{{\displaystyle \int }}}\left\{\frac{0.20948p\left(z\right)\left[\Delta \sigma \left(p\left(z\right),T\left(z\right)\pm 1\right)\right]}{k\left[T\left(z\right)\pm 1\right]\cdot \left[1+\chi \left(z\right)\right]}-\frac{0.20948p\left(z\right)\left[\Delta \sigma \left(p\left(z\right),T\left(z\right)\right)\right]}{kT\left(z\right)\cdot \left[1+\chi \left(z\right)\right]}\right\}dz$$

(33)

3.3.3. Error of Differential Optical Depth Caused Uncertainty of the Mixture Ratio of Water Vapor

The mixture ratio of near-ocean surface water vapor χ_wv(p_surface) in the standard atmospheric mode is higher than1.247%. The ±20% uncertainty of profile for the water vapor mixture ratio introduces uncertainty in the differential optical depth [dOD(0,p_surface)]_wv, it was listed in

Table 5. Error differential optical depth caused uncertainty of the mixture of water vapor.

Wavelength (nm)	Differential Optical Depth Error (±20% Water Vapor)
764.6840	5.30 × 10−4
764.9097
765.1600	4.16× 10−4
764.9707
765.1736	4.18 × 10−4
765.3883
765.6735	2.52× 10−4
765.4637

.

The error in the optical depth caused by the ±20% water vapor mixture ratio uncertainty may be:

$$\Delta {\left[dOD\left(P_{surface}\right)\right]}_{\pm 20\%wv}\approx \underset{0}{\overset{71km}{{\displaystyle \int }}}\frac{0.20948p\left(z\right)\times 20\%\chi \left(z\right)}{kT\left(z\right)\left(1+\chi \left(z\right)\right)}\left(\sigma \left({\lambda }_{on},z\right)-\sigma \left({\lambda }_{off},z\right)\right)$$

(34)

We believe that within the range of 0–15 km, the probability of a +20% error and −20% error in the mixing ratio of water vapor is equivalent, both of which are 50%, so the net error of the differential optical depth is smaller than the accumulated optical depth error in the case of a pure +20% or pure −20% error of the vapor mixing ratio.

3.3.4. Error of the Differential Optical Depth Caused the Uncertainty of the Altitude of the Ocean Wave

The largest oxygen density is near the sea level, and thus, the differential optical depth is sensitive to the uncertainty of the ocean wave altitude, ΔH = 2 m. Differential optical depth error caused by the 2 m altitude of the ocean wave as shown in

Table 6. Differential optical depth error caused by the 2 m altitude of the ocean wave.

Wavelength	Error [dOD]
764.6840	0.4578 × 10−4
764.9097
765.1600	0.3723 × 10−4
764.9707
765.1736	0.3737 × 10−4
765.3883
765.6735	0.2356 × 10−4
765.4637

.

$$\begin{array}{l}\Delta {\left[dOD\left(P_{surface}\right)\right]}_{\Delta H}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\approx {{\displaystyle \int }}_0^{71000}{n}_{O_2}\left(z\right)\left(\sigma \left({\lambda }_{on},z\right)-\sigma \left({\lambda }_{off},z\right)\right)dz\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-{{\displaystyle \int }}_{\pm 2}^{71000}{n}_{O_2}\left(z\right)\left(\sigma \left({\lambda }_{on},z\right)-\sigma \left({\lambda }_{off},z\right)\right)dz\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\approx {{\displaystyle \int }}_0^{\pm 2}{n}_{O_2}\left(z\right)\left(\sigma \left({\lambda }_{on},z\right)-\sigma \left({\lambda }_{off},z\right)\right)dz\end{array}$$

(35)

3.3.5. Relative Error in Calibration Channels for the Echo and Energy Monitoring

Further, the systematic error comprises calibration channels for the echo and energy monitoring and AD conversion error. The absolute error in the differential optical depth is dOD × 0.025% [32], which also belongs to the systematic error, which were listed in

Table 7. Calibration error for echo detection channels and energy monitoring channels.

Wavelength (nm)	dOD(0,71 km)	Error in dOD
764.6840	0.428	1.07 × 10−4
764.9097
765.1600	0.325	0.81 × 10−4
764.9707
765.1736	0.328	0.82 × 10−4
765.3883
765.6735	0.192	0.48 × 10−4
765.4637

.

3.3.6. Error of Optical Depth Due to the Wavelength Dependence of Aerosol Backscattering

In the standard atmosphere mode, Mie and Rayleigh backscattering of the 765 nm dual wavelength (online/offline) are similar but not equal. The coefficient C in Formula (11) expresses the systematic error caused by the difference between online wavelength and offline wavelength. This value is different under different weather conditions. We take the differential extinction coefficient of dual-wavelength under the standard atmospheric model.

The errors in molecule Rayleigh backscattering with the wavelength dependence can be eliminated using a correction. The wavelength dependence of the extinction coefficient of aerosol is uncertain and varies with the particle size, shape, and concentration of the aerosol, so it will cause a systematic error in the differential optical depth, as shown in

Table 8. Optical depth error caused by the wavelength dependence of the atmospheric backscattering.

Wavelength (nm)		C
on	764.6840	0.90 × 10−4
off	764.9097
on	765.1600	0.76 × 10−4
off	764.9707
on	765.1736	0.86 × 10−4
off	765.3883
on	765.6735	0.84 × 10−4
off	765.4637

.

3.3.7. Error in Differential Optical Depth from the Spectral Purity of the Online/Offline Laser

The spectral purity ξ of the spaceborne IPDA LIDAR is 99.99%, which results in an increase in the on-channel echo and the absolute error in the optical depth. If a spectral purity is 100%, the relationship between the two on/off channel signals is considered to be $N_{s,on}{}^{\text{'}}=N_{s,off}{}^{\text{'}}{e}^{-2dOD}$ and $dOD=-\frac{1}{2}\mathit{ln}\left(\frac{N_{s,on}{}^{\text{'}}}{N_{s,off}{}^{\text{'}}}\right)$. However, because the spectral purity is not 100%, it only is ξ, the relationship between the two channel signals is approximately $N_{s,on}=N_{s,off}\left[\left(1-\xi \right)+\xi e^{-2dOD}\right]$, $ln\left(\frac{N_{s,on}}{N_{s,off}}\right)=\mathit{ln}\left[\left(1-\xi \right)+\xi e^{-2dOD}\right]$, the spectral purity yields the following error in the optical depth [35]:

$${\epsilon }_{\xi }=\frac{1}{2}\mathit{ln}\left(\frac{N_{s,on}}{N_{s,off}}\right)-\frac{1}{2}\mathit{ln}\left(\frac{N_{s,on}{}^{\prime }}{N_{s,off}{}^{\prime }}\right)\approx dOD+\frac{1}{2}ln\left[1-\xi \left(1-e^{\left(-2dOD\right)}\right)\right]$$

(36)

The errors in the molecule Rayleigh backscattering with the spectral purity can be eliminated using a correction, so it was not in

Table 9. Error in differential optical depth from spectral purity of 99.99%.

Wavelength	dOD(0,71 km)	Error (Spectral Purity of 99.99%)
764.6840	0.428	6.79 × 10−5
764.9097
765.1600	0.325	4.60 × 10−5
764.9707
765.1736	0.328	4.66 × 10−5
765.3883
765.6735	0.192	2.36 × 10−5
765.4637

.

3.3.8. Error of Differential Optical Depth Caused Jitter of Central Optical Frequency in the Transmitter

The laser source of the spaceborne IPDA has a jitter of ν central optical frequency, whereas the online/offline emission laser spectrum has a central frequency jitter of 10 MHz [21]. Δν_on = ±10 MHz introduces uncertainty in the oxygen molecule absorption cross-section σ(ν_on), resulting in a systematic error in the optical depth. However, for solid-state lasers, stabilizing the center wavelength within ±10 MHz compared to within ±1 MHz is easier. For example, if the center wavelength is offset by ±10 MHz, the jitter of the emitted laser frequency that affects the jitter of the optical depth could be expressed. The absorption of the Mie backscattered echo is relatively significantly affected by the uncertainty of the laser center wavelength, while the width of the Rayleigh backscatter is in the order of 1 GHz, so the errors in the molecule Rayleigh backscattering with the jitter of the central optical frequency can be decreased using a correction. It is assumed that Mie scattering and Rayleigh scattering have the same intensity in the backscattering, so the value in

Table 10. A 10 MHz jitter in the emission laser frequency ν causes change in the differential optical depth.

λ (nm)	$\left[\mathit{d}\mathit{O}\mathit{D}\right]-\left[\mathit{d}\mathit{O}\mathit{D}\left(10 \mathbf{M}\mathbf{H}\mathbf{z}\right)\right]$		$0.5\mathit{m}\mathit{a}\mathit{x}\left\{\begin{array}{c}│\mathbf{\Delta }\left(+10 \mathbf{M}\mathbf{H}\mathbf{z}\right)│\\ │\mathbf{\Delta }\left(-10 \mathbf{M}\mathbf{H}\mathbf{z}\right)│\end{array}\right\}$
	│Δ(+10 MHz)│	│Δ(−10 MHz)│
764.6840	6.18 × 10−7	9.62 × 10−7	4.81 × 10−7
764.9097
765.1600	2.23 × 10−5	2.38 × 10−5	1.19 × 10−5
764.9707
765.1736	2.00 × 10−6	2.81× 10−6	1.41× 10−6
765.3883
765.6735	1.04 × 10−6	8.53 × 10−7	5.2 × 10−7
765.4637

is taken as half of the larger value (±10 MHz). Linewidth of the emitted laser is generally approximate to 20 MHz, which is larger than the 10 MHz jitter of the central optical frequency. The optical depth error in Table 10 is expressed by 50%.

$$\begin{array}{c}{\left[dOD\left(0,surface\right)\right]}_{10\mathrm{MHz}}\\ ={{\displaystyle \int }}_0^{71\mathrm{km}}\frac{0.20948p\left(z\right)}{kT\left(z\right)\left(1+\chi \left(z\right)\right)}\left(\sigma \left({\upsilon }_{on}\pm 10\mathrm{MHz},z\right)-\sigma \left({\upsilon }_{off}\pm 10\mathrm{MHz},z\right)\right)dz\hfill \\ -{{\displaystyle \int }}_0^{71\mathrm{km}}\frac{0.20948p\left(z\right)}{kT\left(z\right)\left(1+\chi \left(z\right)\right)}\left(\sigma \left({\upsilon }_{on},z\right)-\sigma \left({\upsilon }_{off},z\right)\right)\hfill \end{array}$$

(37)

According to the comprehensive analysis of the above differential optical depth errors induced in many factors, the comprehensive evaluation index items of the wavelength to be selected are shown in

Table 11. Comprehensive of various errors at near 765 nm.

Wavelength (nm)	764.6840/764.9097	765.1600/764.9707	765.1736/765.3883	765.6735/765.4637
(SNR) Random error	2.44 × 10−4	2.24 × 10−4	2.20 × 10−4	2.00 × 10−4
(±1 K) Temperature	21.55 × 10−4	9.31 × 10−4	10.2 × 10−4	1.425× 10−4
(20%) Vapor mixing ratio	5.30×10−4	4.16×10−4	4.18 × 10−4	2.52 × 10−4
Energy monitor channel calibration	1.07 × 10−4	0.813 × 10−4	0.820 × 10−4	0.480 × 10−4
Echo channel calibration	1.07 × 10−4	0.813 × 10−4	0.820 × 10−4	0.480 × 10−4
Elevation 2 m error	0.458× 10−4	0.372 × 10−4	0.374 × 10−4	0.236 × 10−4
Aerosol Mie scattering	0.903 × 10−4	0.756 × 10−4	0.857 × 10−4	0.837 × 10−4
(99.99%) spectral purity	0.679 × 10−4	0.460 × 10−4	0.466 × 10−4	0.236 × 10−4
(10 MHz) Frequency jitter	4.81 × 10−7	1.19 × 10−5	1.41 × 10−6	5.2 × 10−7
Geometrically added	24.717× 10−4	12.55× 10−4	13.33× 10−4	5.123× 10−4
Differential absorption cross section (m2)	1.80 × 10−29	1.46 × 10−29	1.47 × 10−29	0.925 × 10−29
Absolute error (Pa)	213.593	133.707	141.05	86.148
Relative error (%)	0.211	0.132	0.139	0.085

.

Consequently, according to Table 11, from the first row to the ninth row, the absolute errors of the differential optical depth induced by various single factors in the standard atmospheric model are displayed respectively; The tenth line shows the combined result of the absolute error of the optical depth caused by all above factors; The eleventh line shows the differential absorption cross section of each wavelength pairs at sea level under the standard atmospheric model; The twelfth line shows the absolute error of the ocean surface pressure calculated by Equation (16) based on above values; The last row shows the relative error of ocean surface pressure under the standard atmospheric model.

4. Conclusions

The calculation process of retrieving the ocean surface pressure from the differential optical depth was introduced. The performance of the differential absorption LIDAR model was also evaluated. The pulse energy, pulse repetition rate, time resolution, and distance resolution along the track are 100 mJ, 100 Hz, 6.25 s, and 44 km, respectively. Owing to the uncertainty of profile for the atmospheric temperature (1 K) and the uncertainty of the atmospheric mixing ratio (20%) and the jitter of the central frequency of laser emission spectrum (10 MHz), maintaining the relative error of the ocean surface pressure below 0.1% is a near-impossible task. The main factor affecting the random error of the ocean surface pressure is the weak sea reflectivity, while the relative error in the ocean surface pressure could be expected to within range of 0.1–0.2%; a relative error of 0.2%–0.3% is a desirable goal.