TR2021‑1 develops two new algorithms that use a measurement vector to update the estimate for a state vector at each discrete time step: a batch estimator and a sequential stepwise estimator. Both algorithms can minimize variances of random estimation errors. Both can impose inequality constraints on Kaman estimates for state variables. Both can mitigate pathological divergence caused by suspected outliers. Both consider nonzero covariances between random estimation errors for the state vector and random measurement errors, which is essential for reverse smoothing of a multivariate time‑series with the Kalman filter. The batch algorithm is feasible if the covariance matrix for the innovation residuals is full‑rank and well‑conditioned, which is often true if there are few measurement variables. Otherwise, the sequential stepwise algorithm is feasible, regardless of the number of measurement variables, and the rank and numerical condition of the covariance matrix.