# Evaluating Model Output

## Overview

The Proper Orthogonal Decomposition (POD) method was implemented to facilitate data analysis and as a tool to pattern recognition in the ice sheet dynamics modeled by GLIMMER using data collected from the model state evolution under various experimental scenarios.

**Mathematical Formulation**

The Proper Orthogonal Decomposition (POD) method^{[1]} (Karhunen-Loève decomposition)
provides an optimal representation of an ensemble data set collected from observations and/or snapshots of the model state

into a low-dimensional state subspace by performing the singular value decomposition (svd) of the snapshot matrix

The POD modes are the left singular vectors in the svd decomposition

and are eigenvectors to the -dimensional spatial correlation matrix

Since in practice , a computationally efficient procedure for computing the reduced-order space is to find the eigenvectors to the -dimensional temporal correlation matrix

then obtain the POD modes as

The fraction of total information (``energy") in the data captured by the leading POD modes

is used to determine the dimension of the reduced-space by requiring that , where is an user-defined threshold factor in the vicinity of the unity. Data variability around the ensemble average is analyzed through the svd of the modified snapshot matrix

that defines the covariance matrix of the ensemble data set

Outcomes of the reduced-order data analysis and pattern identification using the POD methodology are illustrated for model configurations in the GLIMMER EISMINT-2 test cases. Applications to reduced order modeling for parameter estimation and data assimilation^{[2]} will be further investigated.