Functional data analysis for glaciated valley profile analysis

Over the last 135 years, a research question in glacial geomorphology has been to understand the observed shape of glaciated valleys to better understand the relationship between the erosive process and the observed form. The shape of a valley is typically assessed by selecting a single elevation cross-profile from a valley and describing its shape using either a polynomial, power law or generalized power law (GPL) model. It has been observed that glaciated valleys tend to be U-shaped in contrast to incised river valleys that tend to be V-shaped, but the techniques used to classify observed profiles according to these shapes have been relatively limited. New techniques are applied to the problem of describing and comparing valley profiles.; Random sampling in ArcView from a digital elevation model (DEM) is discussed along with improvements to conventional techniques using nonlinear regression methods. Model selection criteria for nonlinear regression are also discussed. Functional data analysis (FDA) techniques are also applied to describe and compare profiles. Derivative estimates are compared to results from simulated U and V-shaped profiles using model selection criteria on different linear and nonlinear regression models. Curve registration (Ramsay and Silverman 1997) is used to align important features of curves to allow easier comparison between the curves. Cluster analysis is used to investigate different groupings of profile shapes. Different types of cluster analysis are compared in this application, comparing functional and discrete clustering methods as well as clustering using derivatives. Functional linear modeling is also used to assess the explanatory power of a couple of different potential covariates for explaining the variability in the shapes of profiles.