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When the measurement errors in Kalman filtering are time correlated, time-differencing approaches are conventionally applied to deal with the time-correlated errors, but they are subject to practical limitations, such as time latency and numerical issues that stem from matrix inversion. This paper proposes two new algorithms to resolve the issues by augmenting the time-correlated elements of the measurement errors into the state vector. To avoid the singularity problem of the state error covariance matrix, the gain matrix is regularized in the first algorithm and a small positive quantity is added to the diagonal elements of the state error covariance matrix in the second algorithm. The two new state-augmented algorithms are easier to keep convergent and have no time latency. Two simulations with a one-degree model and a six degree-of-freedom model demonstrate the proposed algorithms.