## 9 Discussion and Conclusions  
This paper examines the problem of state estimation in linear state space models when non-zero cross-correlations exist between the dynamical noise and observation noise terms. Lag-zero and lag-one cross-correlations are distinguished, and it is highlighted that it depends on which form is chosen for the state space model (the contemporaneous form or the future form) whether a given cross-correlation matrix refers to lag-zero or lag-one cross-correlations. As previously noted by [PERSON] (2007), the existence of these two forms leads to two distinct, non-equivalent Kalman filtering algorithms for the case of correlated noise. The main contributions of this paper are as follows: First, the weighted least-squares framework, as developed by [PERSON] and [PERSON] (2007) for the problem of state estimation in linear state space models, has been generalised to the case of correlated noise terms, thereby obtaining Kalman filter and smoother algorithms, both for the contemporaneous form and the future form. While the resulting generalised Kalman filter algorithms coincide with already known information filter algorithms, the resulting generalised smoother algorithms may be of wider interest, as in the available literature, smoothing in the case of correlated noise terms is rarely discussed. In order to obtain a recursion for the covariance of the  
Figure 2: Left panel: state estimates vs. time for the first 100 samples of the hydroacoustic time series: smoothed state estimates obtained by the non-recursive weighted least-squares approach in Equation (65) (lines and symbols in red colour) and filtered state estimates obtained by the generalised Kalman filter in Equations (97)–(104) (lines and symbols in green colour). Right panel: perpendicular scatter plot of the smoothed state estimates vs. the filtered state estimates for all 2048 samples of the time series (symbols in blue colour). smoothed state estimates, the original recursive inversion algorithm by [PERSON] was augmented by two additional expressions, given by Equations (56) and (57). Second, it has been demonstrated that [PERSON]'s recursive algorithm for inverting one-block-banded matrices via blockwise LDL decomposition, as reviewed in Section 4, provides a general canonical structure for information filters and RTS smoothers. This structure differs considerably from the structure that is usually chosen; see, e.g., the difference between Equations (25) and (26) and Equations (88) and (89). As shown in the Appendix A, by suitable algebraic transformations Equation (88) can be transformed into Equation (25), and without doubt, the same holds true for the other pairs of corresponding equations for filtered state estimates and their covariance matrices. Actually, in order to prove the equivalence of these pairs of equations, such explicit algebraic transformations are not necessary since the equivalence is already proven by the fact that each pair of equations has been derived from the same underlying state space model. Therefore there is no need to develop a formal proof. It is possible that the equivalence of Equation (26) and Equation (89) has already been established in the theory of Riccati equations, but the author has not verified this. Third, a generalisation to the case where both lag-zero and lag-one cross-correlations are present simultaneously has been proposed, building upon the previously developed theory for lag-zero and lag-one cross-correlations of the noise terms in state space models. This generalisation offers the potential to merge the filter algorithms for the contemporaneous form and the future form. Strictly speaking, as it has been shown by using the weighted least-squares approach, in this case, no recursive algorithm for optimally solving the corresponding state estimation problem can be derived. Nevertheless, an optimal solution to the state estimation problem can still be obtained through the non-recursive approach of directly solving the weighted least-squares problem.