On dynamics underlying variance of mass balance estimation in Chilean glaciers

Abstract

Mass balance of a glacier is an accepted measure of how much mass a glacier gains or loses. In theory, it is typically computed by integral functional and empirically, it is approximated by arithmetic mean. However, the variability of such an approach was not studied satisfactory yet. In this paper we provide a dynamical system of mass balance measurements under the constrains of 2nd order model with exponentially decreasing covariance. We also provide locations of optimal measurements, so called designs. We study Ornstein-Uhlenbeck (OU) processes and sheets with linear drifts and introduce K optimal designs in the correlated processes setup. We provide a thorough comparison of equidistant, Latin Hypercube Samples (LHS), and factorial designs for D- and K-optimality as well as the variance. We show differences between these criteria and discuss the role of equidistant designs for the correlated process. In particular, applications to estimation of mass balance of Olivares Alfa and Beta glaciers in Chile is investigated showing that simple application of full raster design and kriging based on inter- and extrapolation of points can lead to increased variance. We also show how the removal of certain measurement points may increase the quality of the melting assessment while decreasing costs. Blow-ups of solutions of dynamical systems underline the empirically observed fact that in a homogenous glaciers around 11 well-positioned stakes suffices for mass balance measurement.