STATISTICAL ORBIT DETERMINATION Smoothing Monte Carlo Simulation

1. STATISTICALORBIT DETERMINATIONSmoothingMonte Carlo Simulation ASEN 5070 LECTURE 33 11/18/09

2. Smoothing

3. Smoothing After someafter some algebra algebra After some algebra it can be shown that

4. Smoothing • Finally

5. Smoothing The equation for the smoothed covariance is given by The equation for the smoothed covariance is given by the

6. Smoothingcomputational algorithm

7. Smoothingcomputational algorithm

10. where

13. Solve for the smoothed history of using the computational algorithm for smoother. Plot the true values of η, the filter values (determined by the algorithm below), and the smoothed values. Compute the RMS of the smoothed fit (data is noised single cycle of a sine wave). The sequential algorithm is given by where

15. The smoother covariance at stage k is given by where is the filter covariance at stage k and is the filter covariance at stage k + 1 based on k observations

16. Calculate the n x n difference matrix difference between the filter and smoother covariance at stage k for k {0, 1, 2, …, }. should be non negative definite, i.e. have no negative eigenvalues Denote the square root of the ith main diagonal element of as Calculate the n x 1 difference vector between the filtered and smoothed state estimate

17. Calculate the ratio for each i {0, 1, 2, …, n} and k {0, 1, 2, …, } If for each i and k, we have Then the filter-smoother solutions are said to be globally consistent. If for each i and k, we have The filter-smoother solutions have globally failed the consistency test.

18. Monte Carlo Simulation George H. Born ASEN 5070 Statistical Orbit Determination The University of Colorado, Boulder