Covariance of derived quantities Propagation of covariances can be used to determine the covariance of derived quantities. Example latitude de and radius. e is co-lat R is radius. AN. ae and a are north east and radial changes(all in distance units Geocentric Case cos(O)cos(n)-cos(O)sin(a) sin(O)IAX sin(λ) CoS 0|/F X/R YIR ZIR AZ A matrix for use in propagation from Vax 03/1703 12540Lec11 Estimation in parts/Sequential estimation A very powerful method for handling large data sets, takes advantage of the structure of the data covariance matrix if parts of it are uncorrelated (or assumed to be uncorrelated) 0 0V,0 00 00† † 03/17/03 12.540 Lec 11 15 • Propagation of covariances can be used to determine longitude and radius. q is co-latitude, l is longitude, R is radius. DN, DE and DU are north, east and radial changes (all in distance units). DN DE DU È Î Í Í Í ˘ ˚ ˙ ˙ ˙ = -cos(q)cos(l) -cos(q) l) q) - l) cos(l) 0 X / R Y / R Z / R È Î Í Í Í ˘ ˚ ˙ ˙ ˙ A 1 2 3 DX DY DZ È Î Í Í Í ˘ ˚ ˙ ˙ ˙ Covariance of derived quantities the covariance of derived quantities. Example latitude, Geocentric Case : sin( sin( sin( matrix for use in propagation from Vxx 4444444 4444444 8 03/17/03 12.540 Lec 11 16 V1 0 0 0 V2 0 0 0 V3 È Î Í Í Í ˘ ˚ ˙ ˙ ˙ -1 = V1 -1 0 0 0 V2 -1 0 0 0 V3 -1 È Î Í Í Í ˘ ˚ ˙ ˙ ˙ Estimation in parts/Sequential estimation • A very powerful method for handling large data sets, takes advantage of the structure of the data covariance matrix if parts of it are uncorrelated (or assumed to be uncorrelated)