IGS Analysis Center Workshop - IGS - NASA

..standard deviations. This is convenient for looking at an unfamiliar solution beforedeciding what to do with it (e.g., to perform some acceptance test). The disadvantageis that loose a priori constraints must be applied for every fiducial-free solution in orderto remove the rank deficiency problem when computing the covariance matrix. Anotherdisadvantage is that M must be inverted to form the weight matrix every time solutionsare combined. This is not as serious a disadvantage as it might seem at first, because inpractice, methods (b) and (c) below also require a matrix inversion if the solutions areto be checked prior to combination.Examples of systems that currently use representation (a) include STAMRG (and someother GIPSY modules) [~PL, 1993], and GLOBK (MIT program for combiningloosely constrained solutions from GPS and VLBI experiments [Herring, 1993; Dong,1993; Feigl et al., 1993].(b) A traditional representation in geodesy is the system of normal equations, includingthe coefficient matrix N = A~CA and the vector of normalized estimates u = ATCZ.This is more computationally efficient for combining solutions, since no inverses needto be calculated until the final solution is desired, Technically, no a priori constraintsare necessary until the last step; however, as pointed out above, it is likely that, inroutine operation, an inverse will be taken for purposes of acceptance testing.An example of systems that currently use representation (b) is the GFZ software [Zhuet al., 1993].Going from system (b) to system (a): M = N-l and x = N-lUGoing from system (a) to system (b): N = M-l and u = M -l X(c) A traditional representation in space navigation, closely related to (b), is the squareroot information formalism (SRIF) [Biennan, 1977], including information matrix R =HA, where H is a householder transformation designed to make R upper triangular,and the vector of normalized estimates y = Hz. This is the most numerically stablerepresentation. When using normal equations, a computational precision is required thatis equal to the square of that needed when using SRIF [Vanicek and Krakiws@,1986]. Like (b), no inversion is required until the last step, with the additionaladvantage that inversion can be very rapidly achieved using back substitution. Like(b), no a priori constraints need be imposed until the last step, but unlike (b), validsolutions can be computed for observable parameters even if rank deficiencies exist.Partial inversion for a subset of parameters is easy (by inverting only the lower righthand corner). It is less convenient than (a) for preliminruy acceptance testing, butconversion to (a) is computationally quick (see below).Examples of systems that currently use representation (c) include AMBIGON (andsome other GIPSY modules) [Blewitf, 1989]. The SRIF method was used to combineglobal and regional GPS solutions by Lindqwister et al. [1991]. Although the GIPSYmodule for network combination (STAMRG) currently exchanges files inrepresentation (a), all internal computations are done after conversion to (c).9Going from system (c) to system (a): M = R -l (R - l)T and x = R-l yGoing from system (c) to system (b): N = R T R and u = RTY9JpL “se. Bjem~ns~ )Z~timation subr~”ti~~ ~lb~~~ (“EsL”’, public dorn~n, FORW) to ~flollll thesquare-root factorization in going from system (a) to system (c), and the back substitution for R] [Biennun,1977].75