Up: Parametric representation of helioseismic

Appendix B: Fixed interval smoother equations

The smoothed states for the model (13) are given by the following two recursions:

Kalman filter (forward recursion)

	$\begin{eqnarraystar} && K_k=P_{k/k-1}H^t(HP_{k/k-1}H^t+\sigma^2)\\ && \hat{\vec{... ...,k-1/k}=(I-K_kH)FP_{k-1/k-1}\\ && P_{k+1/k}=FP_{k/k}F^t+GQG^t.\end{eqnarraystar}$

The recursion is initialised with: $\hat{\vec{X}}_{0/-1}=0$ and P_0/-1=P₀ where the covariance of the state P₀ verifies P₀=FP₀F^t+GQG^t.

Fixed interval smoother (backward recursion)

	$\begin{eqnarraystar} && \Gamma_k=P_{k/k}F^tP_{k+1/k}^{-1}\\ && \hat{\vec{X}}_{k/... ...&& P_{k,k-1/N}=P_{k,k-1/k}+(P_{k/N}P_{k/k}^{-1}-I)P_{k,k-1/k}.\end{eqnarraystar}$

The recursion is initialised with: $\hat{\vec{X}}_{N/N}$ and P_N/N.

Acknowledgements

The authors are especially gratefull to John Leibacher for reading the manuscript and to Bernard Gelly for providing the Lorentzian fits. We wish to thank the GOLF team for providing us the data samples we used to test our method.

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