## 5 A Simple Radiance Assimilation Algorithm  
Satellite radiance data have been used in many applications: radiance assimilation in supporting of weather forecasts; retrievals of environmental data records; and long-term radiance records for climate studies. Microwave and infrared observations are unique to studying atmospheric temperature changes in both the troposphere and stratosphere. Direct satellite radiance assimilation has greatly improved weather forecasting. Passive microwave observations alone contribute to around 30\(-\)40% of the overall improvement of short-range forecast performance and provide around 15\(-\)30% of the total number assimilated observations [34]. UV radiances contain the most unique and the most important information about ozone amount in the atmosphere. UV radiances, which are measured from the sensor on the satellite platform, contain sufficient sensitivity to stratospheric ozone, which plays an essential role in absorbing harmful UV radiation. Therefore, direct UV radiance assimilation is highly desirable. However, four-dimensional radiance assimilation at NWP centers demands very large computational resources and significant manpower. In the context of this study, these resources are beyond our capability. For demonstration purposes, we adopted a one-dimensional variational (1 DVAR) scheme to assimilate OMPS NM radiance. Similar to an operational radiance assimilation system, a 6-h forecast is used as the background. The radiance assimilation in our 1 DVAR was first conducted at the time and location of the satellite observations, and then the ozone increments from the 1 DVAR were propagated to NWP model grids and at the time of analysis through an adjoint technique. The observation data are four-dimensional. The treatment of the observation time is called the first guess at appropriate time (FGAT). The 3 DVAR FGAT has been used in radiance assimilations at NWP centers [35]. FGAT compares the observations with the background at the observation time. The 1 DVAR technique had been very successful and has demonstrated clear and consistent benefits for forecast skill and has been implemented operationally at the European Centre for Medium-Range Weather Forecasts (ECMWF) in June 1992 [36]. The 1 DVAR technique is very common in generating environmental data records. The 1 DVAR algorithm is the core of the NOAA Microwave Integrated Retrieval System (MiRS) system [37] that has been implemented operationally at the U.S. National Oceanic and Atmospheric Administration since 2007 [38; 39]. The MiRS has generated many environmental data records (i.e., satellite retrieval products) for NOAA-18, NOAA-19, Metop-B and C AMSUA-MHS, SNPP ATMS, NOAA-20 ATMS, DMSP F17 and F18 SSMIS, GCOM-W1 AMSR-2, and GPM GMI. These satellite products are atmospheric profiles of temperature and water vapor, cloud liquid water, ice water content, rainfall rate, snow cover and snow water equivalent, snow fall rate, surface temperature and microwave emissivity, and sea ice concentration ([[https://www.star.nesdis.noaa.gov/mirs/highresolution.php](https://www.star.nesdis.noaa.gov/mirs/highresolution.php)]([https://www.star.nesdis.noaa.gov/mirs/highresolution.php](https://www.star.nesdis.noaa.gov/mirs/highresolution.php)), accessed on 25 August 2021). Radiance assimilation and 1 DVAR retrieval all aim to minimize a cost function that weighs the relative contribution of background (a priori) information and satellite observations. Assuming that the errors in the observations and in the prior information are neither biased nor correlated and have Gaussian distributions, the best estimate will minimize the cost function [40]:  
\[J(x)=\frac{1}{2}(x-x_{b})^{T}B^{-1}(x-x_{b})+\frac{1}{2}(Y^{M}-F(x))^{T}E^{-1}(Y ^{M}-F(x)) \tag{1}\]  
where \(x\) is a retrieved state vector of \(n\) elements. The state vector may include temperature, water vapor, and cloud water content at each atmospheric layer. The first item on the right represents the penalty for departing from the background \(x_{b}\) (a priori).