1. Singular value decomposition (SVD) – a tool for VLBI simulations Markus Vennebusch VLBI – group of the Geodetic Institute of the University of Bonn M. Vennebusch – singular value decomposition – a tool for VLBI simulationsJanuary 9, 2006slide 1 IVS-General Meeting 2006

2. Idea Idea: - Use of an advanced (not very popular) method for parameter estimation (without normal equations) - So far: Investigations of parameters (cofactors, EVD, correlations,....) - Now: Investigation of parameters AND observations (entire schedule) - Goals: - Improvement of Schedules by (direct) investigations of design matrix - Gain insight into adjustment process: „Which observation affects which parameter? Is a particular observation more important than another observation?“ - This talk: No explanations of theoretical background, just applications M. Vennebusch – singular value decomposition – a tool for VLBI simulationsslide 2January 9, 2006

3. Least-squares adjustment Least-squares adjustment: Overdetermined system of linear equations: Normal equation method: Solution of normal equations: + simple + fast + yields accuracy of parameters - inversion necessary - numerical problems (condition becomes worse) direct methods (design matrix used) : + numerically more stable (even with bad condition) + no inversion (S = diagonal) + yields geometrical insight into adjustment process - complex, slowly QR-decomposition Singular value decomposition M. Vennebusch – singular value decomposition – a tool for VLBI simulationsslide 3January 9, 2006

4. Linear Algebra: - provides different methods for solution of (overdetermined) systems of linear equations: Matrix decompositions: Cholesky, LU (Gauß-Algorithm), QR, EVD, SVD,... - Singular value decomposition (Extended EVD for rectangular matrices): Least-squares adjustment Jacobi-Matrix / Designmatrix Left singular vectors Singular values Right singular vectors (= eigenvectors of NEQ) = M. Vennebusch – singular value decomposition – a tool for VLBI simulationsslide 4January 9, 2006

5. Singular value decomposition Least-Squares-solution by SVD: =+... +++ r = Rank elements = scaling factors M. Vennebusch – singular value decomposition – a tool for VLBI simulationsslide 5 Left and right singular vectors (u i and v i ): - u i shows impact (or weight) of observations - v i shows parameters affected - s i shows whether v i can be determined or not SVD yields geometrical insight into LSM-adjustment: - Vector spaces, bases of vector spaces, projections - column space = Data space, row space = Model space => further investigations January 9, 2006

6. Data resolution matrix (= U r U r T, U r = U(:,1:r)): - shows „importance“ of observations in general (information of observation already contained?) - shows observations which might be neglected - system / solution is very sensitive to errors in observations with large importance (!) Model resolution matrix (= V r V r T, V r = V(:,1:r)): - shows dependencies (correlations, separability) between parameters (even in case of rank deficiency) Q xl -Matrix (= V r S r -1 U r T ): - shows „correlations“/dependencies between observations and unknowns (more precise: common stochastic behaviour) - shows impact of variations in observations on parameters (measure of sensitivity to errors) M. Vennebusch – singular value decomposition – a tool for VLBI simulationsslide 6 Singular value decomposition January 9, 2006

7. Example Example I04267 (04SEP23XU): M. Vennebusch – singular value decomposition – a tool for VLBI simulationsslide 7 1 hour duration: 16 observations, 6 pseudo-observations (not necessary) SVD of (22 x 5)-design matrix (from OCCAM) yields: - Matrix U: Dimension 22 x 22 - Matrix S: Dimension 22 x 5 (Singular values on main diagonal) - Matrix V: Dimension 5 x 5 => Rank r = 5 January 9, 2006

8. Example M. Vennebusch – singular value decomposition – a tool for VLBI simulationsslide 8 ZD KOKEE ZD WETTZELL CL0 KOKEE XPOL DUT1 Number of observation v1v1 u1u1 v2v2 v3v3 u2u2 u3u3 Singular vectors 1 – 3 (out of 5): January 9, 2006

9. Data resolution matrix M. Vennebusch – singular value decomposition – a tool for VLBI simulationsslide 9 Source statistics: Largest importance January 9, 2006

10. Outlook Problems: - Parameters must have the same unit (=> parameter rates are difficult to investigate) - SVD heavily depends on parametrisation Further investigations: - Piecewise linear parameters: useful parametrisation? - Comparison of different schedules - Identification of important observations - Identification of undeterminable parameters - Impact of datum constraints -... Conclusion: - SVD is a very useful tool to understand adjustment processes - Contribution to improve VLBI schedules and data analysis => Suitable method for VLBI simulations M. Vennebusch – singular value decomposition – a tool for VLBI simulationsslide 10January 9, 2006

11. Singular value decomposition – a tool for VLBI simulations Markus Vennebusch VLBI – group of the Geodetic Institute of the University of Bonn M. Vennebusch – singular value decomposition – a tool for VLBI simulationsslide 11 IVS-General Meeting 2006 January 9, 2006

12. Q xl -Matrix M. Vennebusch – singular value decomposition – a tool for VLBI simulationsslide 12 Q xl -Matrix: - impact of variations in observations on parameters (sensitivity on errors) January 9, 2006