## 3 NOAA Microwave Integrated Retrieval System (MiRS)  
The NOAA Microwave Integrated Retrieval System (MiRS) system has been implemented operationally at the U.S. National Oceanic and Atmospheric Administration since 2007 and has been generating many environmental data records (i.e., satellite products) for NOAA-18, NOAA-19, SNPP ATMS, NOAA-20 ATMS, Metop-B and C, SSMIS, AMSR-2, and GMI. The 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))). The MiRS is based on the 1D-VAR retrieval algorithm [1; 2], which includes the community radiative transfer model (CRTM) for computing radiances and the gradient of radiance (or Jacobian) [13], and a scheme for minimizing the 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 a priori information are neither biased nor correlated, and have Gaussian distributions, the best estimate will minimize the cost function:  
\[J(x)\ =\ \left[\frac{1}{2}(x-x_{0})\times B^{-1}(x-x_{0})\right]+\left[\frac{1} {2}(Y_{m}-H(x))\times E^{-1}(Y_{m}-H(x))\right] \tag{1}\]  
where \(x\) is a retrieved state vector. The state vector may include the temperature, water vapor, and cloud water contents at each atmospheric layer. The first item on the right represents the penalty for departing from the background \(x_{0}\) (a priori). The departure is weighted by the error covariance matrix \(B^{-1}\). The diagonal elements of \(B^{-1}\) are the standard deviations of the state variables while the off-diagonal elements are error correlations among state variables, for example, between the temperature and water vapor profiles. The second term represents the penalty for the simulated radiances \(H(x)\) (or brightness temperatures) departing from the observed radiances (or brightness temperatures) \(Y_{m}\), weighted by the sum of the instrument and forward modeling errors \(E^{-1}\). \(H\) is a forward operator. The minimum of the cost function may be found from an iterative process. It computes descent directions that are gradients of the cost function (Equation 1) to the state vector \(x\):  
\[\
abla_{x}J(x)\ =\ B^{-1}(x-x_{0})-K^{T}(x)E^{-1}(Y_{m}-H(x)) \tag{2}\]  
where \(K^{T}(x)\) the adjoint model, which is equivalent to the transpose of radiance Jacobian with respect to the state vector. Using the first Taylor expansion:  
\[H(x)\ =\ H(x_{0})+K(x)(x-x_{0}) \tag{3}\]  
By inserting Equation (3) into Equation (2) and letting Equation (2) equal to zero (where the cost function \(J(x)\) will have a local minimum), we obtain:  
\[(x-x_{0})\ =\ \left[B^{-1}+K^{T}(x)E^{-1}K(x)\right]^{-1}\!\!K^{T}(x)E^{-1}(Y_{m} -H(x_{0})) \tag{4}\]  
Equation (4) may be solved in an iterative process. It is an iterative physical inversion system that finds a consistent geophysical solution to fit all radiometric measurements simultaneously. One of the particularities of the system is its applicability in cloudy and precipitating conditions. In order to be generic, the MiRS system adds a state vector for surface emissivity to automatically consider various surface properties. The inclusion of the emissivity spectrum in the state vector makes the system applicable globally, with the only differences between land, ocean, sea ice, and snow backgrounds residing in the covariance matrix chosen to spectrally constrain the emissivity. Similarly, the inclusion of the cloud and hydrometeor parameters within the inverted state vector makes the assimilation/inversion of cloudy and rainy radiances possible, and therefore, it provides an all-weather capability to the system.