Minimum-entropy velocity estimation from GPS position time series

Abstract

We propose a nonparametric minimum entropy method for estimating an optimal velocity from position time series, which may contain unknown noise, data gaps, loading effects, transients, outliers and step discontinuities. Although nonparametric, the proposed method is based on elementary statistical concepts familiar to least-squares and maximum-likelihood users. It seeks a constant velocity with a best possible (realistic) variance rather than a best variable velocity fit to the closest position data. We show, based on information theory, synthetic and real data, that minimum-entropy velocity estimation: (1) accounts for colored noise without assumptions about its distribution or the extent of its temporal correlations; (2) is unaffected by the series deterministic content such as an initial position and the heights of step discontinuities and insensitive to small-amplitude periodic variations and transients; (3) is robust against outliers and, for long time series, against step discontinuities and even slight non-stationarity of the noise; (4) does not involve covariance matrices or eigen/singular value analysis, thus can be implemented by a short and efficient software; (5) under no circumstances results in a velocity variance that decays as \(1/N\), where \(N\) is the number of observations. The proposed method is verified based on synthetic data and then applied to a few hundred NGL (Nevada Geodetic Lab) position time series of different characteristics, and the results are compared to those of the Median Interannual Difference Adjusted for Skewness (MIDAS) algorithm. The compared time series include continuous and linear ones used to test the agreement between the two methods in the presence of unknown noise, data gaps and loading effects, discontinuous but linear series selected to include the effect of a few (1–4) discontinuities, and nonlinear but continuous time series selected for including the effects of transients. Both the minimum-entropy and MIDAS methods are nonparametric in the sense that they only extract the velocity from a position time series with hardly any explicit assumptions about its noise distribution or correlation structure. Otherwise, the two methods differ in every single possible technical sense. Other than pointing to a close agreement between the derived velocities, the comparisons consistently revealed that minimum-entropy velocity uncertainties suggest a smaller degree of temporal correlations in the NGL time series than the MIDAS does.