L1-norm pre-analysis measures for geodetic networks

Abstract. Several pre-analysis measures which help to expose the behavior of L1 -norm minimization solutions are described. The pre-analysis measures are primarily based on familiar elements of the linear programming solution to L1-norm minimization, such as slack variables and the reduced-cost vector. By examining certain elements of the linear programming solution in a probabilistic light, it is possible to derive the cumulative distribution function (CDF) associated with univariate L1-norm residuals. Unlike traditional least squares (LS) residual CDFs, it is found that L1-norm residual CDFs fail to follow the normal distribution in general, and instead are characterized by both discrete and continuous (i.e. piecewise) segments. It is also found that an L1 equivalent to LS redundancy numbers exists and that these L1 equivalents are a byproduct of the univariate L1 univariate residual CDF. Probing deeper into the linear programming solution, it is found that certain combinations of observations which are capable of tolerating large-magnitude gross errors can be predicted by comprehensively tabulating the signs of slack variables associated with the L1 residuals. The developed techniques are illustrated on a two-dimensional trilateration network.