## 6 Experimental OMPS Radiance Assimilation
Figure 5: Total transmittance from surface to satellite (black line). The red line is the accumulated CRTM Jacobian radiance to ozone profile. Symbol “c” is the position at 331 nm used to estimate surface reflectance. The symbol “o” are the two channels that we propose to estimate the surface reflectance. The surface reflectance for other channels is either interpolated or extrapolated from the two reflectance points at 347.6 nm and 371.8 nm. As shown in Figure 4, the spectral variation of the normalized radiance can be 100 times larger between the low and high normalized radiances. In the ozone retrieval, the logarithm of the normalized radiance (so called \(N_{value}\)) was used instead of the normalized radiance itself. The \(N_{value}\) is defined as [45]  
\[N_{value}=-100\times log_{10}\frac{Radiance}{solar\_flux} \tag{8}\]  
We replotted Figure 4 using the \(N_{value}\). One can see from Figure 6 that the variation of \(N_{value}\) is much smaller, less than a factor of 4. The CRTM simulation and the OMPS NM measurements generally agree well. However, we see a noticeable difference for large wavelengths between observations (red line) and CRTM simulations using a constant surface reflectance (black line). Again, the CRTM simulation using two surface reflectance values is in better agreement with the observations. Large differences in the \(N_{value}\) are also found for wavelengths between 300 nm and 305 nm, where the dichroic filter transitions from reflection at 300 nm to transmission at 305 nm. In this OMPS NM radiance assimilation, we use the \(N_{value}\) in the cost function equation, Equation (1). The Jacobian equation will be the derivative of the \(N_{value}\) with respect to geophysical parameters. That is  
\[\frac{\partial N_{value}}{\partial x}=\ \frac{-100}{\ln(10)\times Radiance} \times\frac{\partial Radiance}{\partial x} \tag{9}\]  
To run the assimilation experiments, we used both analysis and forecast data from the ECMWF modelling system. ECMWF analysis data are available at 00Z, 06Z, 12Z, and 18Z. ECMWF forecast data are available at 03Z and at 09Z from an initialization time at 00Z and at 15Z and at 21Z from an initialization time at 12Z. We conducted the first experiment at 06Z on 2 March 2021. The background was the mean of the 3 h and 9 h forecasts from 00Z. The satellite data within the time window between 03Z and 09Z were assimilated. We compared the total column ozone in Dobson Units between our assimilation and the ECMWF analysis at 06Z. OMPS NM channels 61, 75, 77, 82, 94, 103, 117, 127, and 175 at the wavelengths 324.52, 330.36, 331.20, 333.28, 338.29, 342.05, 347.89, 352.06, and 372.09 in nm were assimilated. The nine channels were similar to the channels that are used in ozone retrievals. As shown in Figure 4, channels 117 and 175 are suitable for estimating the surface reflectance, and Channels 61, 75, 77, 82, 94 and 103 have good sensitivities to ozone. Figure 7 compares the difference of the normalized radiance for those channels between \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Channel** & **Wavelength** & **OMPS** & **Stokes Model** & **Scalar Model** & **Modified Scalar** \\
**Number** & **(nm)** & **NM** & **Error (\%)** & **Error (\%)** & **Model Error (\%)** \\ \hline
61 & 324.53 & 0.06646 & \(-\)0.030 & 4.228 & \(-\)0.015 \\
75 & 330.38 & 0.07006 & \(-\)0.029 & 4.168 & \(-\)0.014 \\
77 & 331.20 & 0.06837 & \(-\)0.015 & 4.183 & \(-\)0.015 \\
82 & 333.28 & 0.07068 & \(-\)0.014 & 4.188 & \(-\)0.014 \\
94 & 338.30 & 0.06935 & \(-\)0.029 & 4.167 & \(-\)0.014 \\
103 & 342.06 & 0.68780 & \(-\)0.015 & 4.115 & \(-\)0.015 \\
117 & 347.90 & 0.06615 & \(-\)0.015 & 4.006 & \(-\)0.015 \\
127 & 352.07 & 0.06422 & \(-\)0.016 & 3.955 & \(-\)0.016 \\
175 & 372.09 & 0.05607 & \(-\)0.018 & 3.674 & \(-\)0.018 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparison of OMPS normalized radiance (OMPS NM) by solar irradiance with the CRTM Stokes, scalar, and modified scalar calculations.