Automatic segmentation and parallel phased least squares for highly efficient geodetic network parameter estimation

Abstract

This paper describes a novel automatic segmentation algorithm and parallel phased least squares strategy. These developments have been motivated by a requirement to repeatedly estimate extremely large parameter sets from massive and constantly changing geodetic networks in a highly efficient manner. The challenge of solving parameter sets efficiently, together with the inverse of the normal equations to obtain the full matrix of parameter estimate precisions, may be addressed by using highly optimised parallel matrix factorisation libraries. While offering notable reductions in time on multi-core CPU architectures, the simultaneous solution of extremely large measurement sets still requires excessive amounts of time and computational resources (if attainable). Historically, the challenge has been addressed by geodesists through a range of Helmert blocking or matrix partitioning techniques and sequential least squares models. While capable of producing rigorous parameter estimates, certain methods suffer from computational inefficiency; implementation complexity; inflexibility in re-forming the parameterised expressions upon the introduction of new measurements or non-rigorous variance matrices. Tienstra's phased least squares method is able to overcome all of the aforementioned problems, provided an efficient and flexible process is available for rigorously subdividing the parameters and measurements. In this contribution, we present an automatic segmentation algorithm and a strategy for parallelising Tienstra's phased least squares method using a dynamic scheduling queue. These developments offer a breakthrough in geodetic methodology, in that for the first time they allow for the rigorous solution of both parameters and variances from massive systems of observation equations subject to continual change in an adaptive and highly efficient manner.