## 3 EMI NO\({}_{2}\) Retrieval Algorithm
### Spectral Fitting
#### 3.1.1 Wavelength Calibration  
The DOAS algorithm is very sensitive to a wavelength shift. Even a very small wavelength shift (\(\sim\)0.002 nm) will introduce large uncertainties into the SCD fitting results of NO\({}_{2}\)[46]. For EMI irradiance and radiance data, the original wavelength grid can be obtained by a third-degree polynomial:  
\[\lambda(i)=a_{0}+a_{1}i+a_{2}i^{2},i=0\,\sim N-1 \tag{2}\]  
where, \(\lambda(i)\) is the wavelength and \(i\) is the wavelength index from 0 to 1285 for VIS1 band. The coefficients \(a_{0}\), \(a_{1}\) and \(a_{2}\) are different for each CCD row and can be read from the level1 data file.  
The wavelength shift for the radiance spectrum is obtained by solving Equation (1). For irradiance spectrum, the wavelength calibration is achieved by using a highly accurate solar atlas as the reference [48], and minimizing the cost function, which describes the discrepancy between the measured solar spectrum and reference solar spectrum. Two parameters are defined during the  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameters** & **EMI** & **OMI-NASA [46]** & **TROPOMI [47]** \\ \hline
**Fitting window** & 405–465 nm & 402–465 nm & 405–465 nm \\ \hline Reference spectrum I\({}_{0}\) & Irradiance measured on 12 June, 2018 & Monthly mean solar irradiance & Annual mean (2005) solar reference \\ \hline  & & Two in seven microwalows: & \\  & & (402–410), & \\  & & (409–418), & \\ Polynomial & 5 th-order & (415–425), & 5 th-order \\  & & (424–434), & \\  & & (433–444), & \\  & & (438–453), and & \\  & & (451–465) & \\ \hline  & CHOCHO & \(\times\) & \(\surd\) (296 K) & \(\times\) \\  & O\({}_{3}\) & \(\surd\) (223 K) & \(\times\) & \(\surd\) (243 K) \\ Included cross & NO\({}_{2}\) & \(\surd\) (220 K) & \(\surd\) (220 K) & \(\surd\) (220 K) \\ Included cross & O\({}_{4}\) & \(\surd\) (293 K) & \(\times\) & \(\surd\) (293 K) \\ sections & H\({}_{2}\)O vapor & \(\surd\) (280 K) & \(\surd\) (280 K) & \(\surd\) (280 K) \\  & H\({}_{2}\)O (liquid) & \(\surd\) & \(\times\) & \(\surd\) (295 K) \\  & Ring Effect & \(\surd\) & \(\surd\)Air+water & \(\surd\) \\ \hline Offset correction & \(\surd\) & \(\times\) & \(\times\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Spectral fitting parameter settings for EMI, the Ozone Monitoring Instrument (OMI), and the TROpospheric Ozone Monitoring Instrument (TROPOMI).  
calibration: the wavelength shift that represents a wavelength increment with the unit of nm and wavelength squeeze, which is a dimensionless variable representing a correction to the width of the spectral pixel. The wavelength grid can be updated by adjusting the two parameters:  
\[\lambda^{*}(i,\alpha,\beta)=(a_{0}+\alpha)+(a_{1}\times\beta\times i)+a_{2}i^{2},i=0\sim N-1 \tag{3}\]  
where \(\alpha\) is shift, \(\beta\) is squeeze. Then, the cost function can be described as the following equation:  
\[\lambda^{2}(L,\alpha,\beta)=\frac{1}{N-2}\sum_{i=1}^{N}[G^{*}(i)-L\otimes S^{* }(i,\alpha,\beta)]^{2} \tag{4}\]  
where \(L\) is the slit function, \(G(i)\) is the measured solar spectrum, \(S^{*}(i,\alpha,\beta)\) is the reference solar spectrum that interpolated to the updated wavelength grid. The factor \(1/(N-2)\) represents the degrees of freedom and the symbol \(\otimes\) represents convolution.