There have been many papers written about tuning Kalman filters. Such tuning usually consists of adjustments to values used for the covariances of the model and observation noise. In this paper we described a procedure using the observations with a linear process model to develop estimates for the effective values of the covariance matrices of both the process noise (Q) and the measurement noise (R). These are needed for maximum likelihood state estimation in control and optimization. A horizon state estimator is derived that is linearly unbiased and has a constant state estimation error covariance. The process and measurement models are combined with the state estimates and a constant state estimation error covariance to generate cumulative error covariances that are also constant. A maximum likelihood function and a linear regression technique are then utilized to obtain the diagonal elements of covariance matrices of the process noise and the measurement noise. A simulation example for two chemical reactors in series is presented.