## 5 A Simple Radiance Assimilation Algorithm
The departure is weighted by a \(n\times n\) 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 the error covariance among the state variables, for example between the temperature and water vapor profiles. The second term represents the penalty for \(m\) simulated radiances \(F(x)\) (or brightness temperatures) departing from \(m\) observed radiances (or brightness temperatures) \(Y^{M}\), weighted by the sum of the instrument and forward \(m\times m\) modeling error covariance matrix \(E^{-1}\). \(F\) 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 \(x\):  
\[\
abla_{x}J(x)=B^{-1}(x-x_{b})-K^{T}(x)E^{-1}(Y^{M}-F(x)) \tag{2}\]  
where \(K^{T}(x)\) is the adjoint model, which is equivalent to the transpose of Jacobian radiance with respect to the state vector. Mathematically speaking, the local minimum may be obtained by setting (gradients) Equation (2) to zero. However, we can only have an explicit solution in the 1 DVAR case. For the 3 DVAR case, the solution may be approached within the inner loop of the minimization. For the 1 DVAR case, using a first order Taylor series expansion:  
\[F(x)=F(x_{b})+K(x)(x-x_{b}) \tag{3}\]  
It is obvious that Equation (3) is only valid when the Jacobian operator \(K(x)\) is linear or when the increment \((x-x_{b})\) is small enough. Inserting Equation (3) into Equation (2) and letting Equation (2) equal zero (where the cost function \(J(x)\) will have a local minimum), we obtain  
\[(x-x_{b})=\Big{[}B^{-1}+K^{T}(x)E^{-1}K(x)\Big{]}^{-1}K^{T}(x)E^{-1}\Big{(}Y^{ M}-H(x_{b})\Big{)} \tag{4}\]  
Taking an iterative approach, we may rewrite Equation (4) as  
\[\begin{split}&(x_{k+1  
The increment at each step can be large, so Equation (3) may not be accurate for a non-linear problem since the increment is the departure directly from the background instead of the small change with respect to the previous iteration value at each step. It is known that the inversion is an ill-posed problem (i.e., there are multiple possible solutions that are consistent with the satellite observations). In MiRS, the values of the diagonal elements in the observation error covariance matrix are increased if the differences between the observations and the simulations are large. The adjustment seems reasonable and mitigates the ill-posed problem. In radiance assimilation, the minimization of the cost function is composed of the inner loop and outer loop iterations. The background and observational error covariance matrices are fixed in both the inner and outer loops. For every outer loop step, there are many dozens of inner loop iterations. Within inner loop iterations, the CRTM is only called one time, and the background remains the same. The minimization is conducted for the fixed Jacobian and fixed background. For the next outer loop step, the CRTM is called again, and the background is updated as well. The process ensures small increments at each outer loop step since the background is updated toward the final state. It may seem that the observation information content is utilized more than once. However, this is just an internal process. For the entire radiance assimilation process, the information used is the 6-h forecast, satellite radiances, and other observations. In this study, we follow similar steps as the operational 3 DVAR FGAT radiance assimilation. The time interpolation utilizes the observations with the background at the observation time. Our simple radiance assimilation scheme is similar to the operational 3 DVAR FGAT scheme but with a one-dimensional variational procedure and error covariance matrix in the cost function. In 1 DVAR, the cost function minimization problem can be explicitly solved, while the solution in 3 DVAR minimization can only be approached in the inner loop iteration.