Propagation of covariances Given a data noise covariance matrix. the characteristics of expected values can be used to determine the covariance matrix of any linear combination of the measurements Given linear operation: y=Ax with Vas covariance matrix of x yy =< yy >=< AXx'A>=A<XXAT V=AⅴAT Propagation of covariance Propagation of covariance can be used for any linear operator applied to random variables whose covariance matrix is already Specific examples: Covariance matrix of parameter estimates from least squa Covariance matrix for post-fit residuals from least Covariance matrix of derived quant such latitude, longitude and height from coordinate estimates 03/703 12540Lec11† 03/17/03 12.540 Lec 11 7 Propagation of covariances y = Ax with Vxx x Vyy =< yyT >=< AxxT A T >= A < xxT > A T Vyy = AVxxA T • Given a data noise covariance matrix, the characteristics of expected values can be used to determine the covariance matrix of any linear combination of the measurements. Given linear operation : as covariance matrix of 03/17/03 12.540 Lec 11 8 Propagation of covariance known. – least squares – Covariance matrix for post-fit residuals from least squares – Covariance matrix of derived quantities such as estimates. • Propagation of covariance can be used for any linear operator applied to random variables whose covariance matrix is already • Specific examples: Covariance matrix of parameter estimates from latitude, longitude and height from XYZ coordinate 4