# Performance Evaluation for China's Planned CO\({}_{2}\)-Ipda
## 2 Materials and Methods
### Random Errors  
The total error consists of systematic and random errors. Systematic errors have several sources, but a major proportion is caused by hardware defects, a global and constant bias that has little impact  
Figure 1: Overall schematic diagram for the work carried out in this study. on retrieved CO\({}_{2}\) sources and sinks. The remaining proportion fluctuates over various temporal and spatial scales, but is relatively small compared to the constant bias and the random error. Reduction of systematic errors relies on rigid instrument calibration and accurate meteorological parameters, but not on the subsequent processing methods. Random error refers to errors that are highly localized in space and time; these can yield false CO\({}_{2}\) sources and sinks. Spatial and temporal averaging can help reduce random errors, meaning that there is a trade-off between resolution and accuracy. Consequently, we will focus on evaluations of the random error in this work. The random error of XCO\({}_{2}\) measurements is equivalent to that of optical depth, owing to the differential absorption effect of CO\({}_{2}\), which can be expressed as shown in Equation (1). \[\frac{\Delta\delta_{\text{CO}_{2}}}{\delta_{\text{CO}_{2}}}=\frac{1}{2\delta _{\text{CO}_{2}}}\sqrt{\frac{1}{n_{shots}}\left(\frac{1}{SNR_{0n}^{2}}+\frac{1 }{SNR_{off}^{2}}+\frac{1}{\left(SNR_{on}^{L}\right)^{2}}+\frac{1}{\left(SNR_{ off}^{L}\right)^{2}}\right)} \tag{1}\]  
where \(\delta_{\text{CO}_{2}}\) is the optical depth due to absorption of CO\({}_{2}\), n\({}_{shots}\) denotes the number of statistically independent pulse pairs, \(SNR_{on,off}^{L}\) represents the statistical fluctuation of laser pulse energy measurements, and \(SNR_{on,off}\) is the mean signal-to-noise ratio (SNR) of the lidar returns. A typical lidar setup that uses direct detection allows the averaging of a large number of statistically independent speckle cells, satisfying the condition that the mean carrier-to-noise ratio is considerably greater than the speckle-related SNR. Additionally, only two dominant error sources, dark current noise and the solar background noise, are considered herein for simplification. \(SNR_{on,off}\) can then be expressed as in Equation (2)  
\[SNR_{on,off}=\frac{P_{on,off}MR}{B\left(2 eM^{2}FR\left(P_{on,off}+P_{back} \right)+i_{D}^{2}\right)} \tag{2}\]  
where \(M\) is the internal gain factor of the detector, \(R\) denotes the responsivity of the detector, \(B\) is the electrical bandwidth, \(e\) is the elementary charge, \(F\) is the excess noise factor of the detector (which accounts for additional noise due to the internal amplification statistics), and \(i_{D}^{2}\) is the dark current noise density. \(P_{back}\) is the solar background radiance, which is expressed in Equation (3). \[P_{back}=\frac{\text{FOV}^{2}A^{2}LQ}{4} \tag{3}\]  
where FOV is the field of view, \(L\) is the solar radiance at the top of the atmosphere, \(A\) is the effective detection area of the telescope, and \(Q\) is the surface reflectance. \(P_{on,off}\) is expressed in Equation (4). \[P_{on,off}(r_{G})=D_{on,off}\frac{A}{r_{G}^{2}}O_{on,off}\frac{Q_{on,off}}{ \pi}\tau_{on,off}^{2}\frac{E_{on,off}}{\Delta t_{eff}} \tag{4}\]  
\[\Delta t_{eff}=\sqrt{\Delta\tau_{\text{L}}{}^{2}+\left(\frac{1}{3B}\right)^{2 }+\left(\frac{2\Delta\text{h}}{c}\right)^{2}} \tag{5}\]  
where \(D\) is the total optical efficiency of the CO\({}_{2}\)-IPDA, \(O\) is the overlap function, \(E\) is the pulse energy of the transmitted laser radiation, \(\Delta t_{eff}\) is introduced to account for the effective pulse length of the lidar returns and is calculated using Equation (5), \(\Delta\tau_{\text{L}}\) is the pulse length, \(B\) is the electronic bandwidth, \(\Delta\)h is the effective target altitude within the footprint of the laser pulse, \(c\) is the speed of light, \(\tau\) is the one-way atmospheric transmission, and \(r_{G}\) is the distance between the target and the laser.