# 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.

Data analysis using the singular value decomposition for the EISMINT-2 test H using a uniform basal heat flux (left column) and a non-uniform basal heat flux (right column). Displayed are the time-averaged basal temperature ($^\circ$C) for a 200,000-years time integration, $\Delta t = 1000$-years, the leading left SV of the data correlation matrix, the leading left SV of the data covariance matrix, and the percentage of information (energy) captured by the leading modes. In each case ~90% of the energy is captured by the 1st mode.

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

${\mathbf x}^{(i)}\in \mathbb R^n, i=1,2,\dots , m$

into a low-dimensional state subspace by performing the singular value decomposition (svd) of the snapshot matrix ${\mathbf X} =\left[{\mathbf x}^{(1)}, {\mathbf x}^{(2)}, \dots , {\mathbf x}^{(m)}\right] \in {\mathbb R}^{n\times m}$

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

$\frac{1}{\sqrt m}{\mathbf X} = {\mathbf U}{\mathbf \Sigma} {\mathbf V}^T$

and are eigenvectors to the $n\times n$-dimensional spatial correlation matrix

$\frac{1}{m}{\mathbf X}{\mathbf X}^T {\mathbf u}_j = \sigma^2_j{\mathbf u}_j$

Since in practice $m\ll n$, a computationally efficient procedure for computing the reduced-order space is to find the eigenvectors to the $m\times m$-dimensional temporal correlation matrix

$\frac{1}{m}{\mathbf X}^T{\mathbf X}{\mathbf v}_j = \sigma_j^2{\mathbf v}_j$ then obtain the POD modes as

${\mathbf u}_j = \frac{1}{\sqrt m \sigma_j}{\mathbf X}{\mathbf v}_j$

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

$I(k) = \left(\sum_{j=1}^k\sigma_j^2\right)/\left(\sum_{j=1}^m\sigma_j^2\right)$

is used to determine the dimension $k$ of the reduced-space by requiring that $I(k) \ge \gamma$, where $0<\gamma <1$ is an user-defined threshold factor in the vicinity of the unity. Data variability around the ensemble average $\bar{\mathbf x} = (1/m)\Sigma_{i=1}^m{\mathbf x}^{(i)}$ is analyzed through the svd of the modified snapshot matrix

$\bar {\mathbf X} =\left[{\mathbf x}^{(1)} - \bar{\mathbf x}, {\mathbf x}^{(2)} - \bar{\mathbf x}, \dots , {\mathbf x}^{(m)} - \bar{\mathbf x}\right]$

that defines the covariance matrix of the ensemble data set

${\mathbf C} = \frac{1}{m}\bar {\mathbf X} \bar {\mathbf X}^T$

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.

## References

1. Antoulas, A.C. 2005: Approximation of Large-Scale Dynamical Systems. Advances in Design and Control Series, No. 6, SIAM, 481pp.
2. Daescu, D.N. and I.M. Navon, 2008: A dual-weighted approach to order reduction in 4DVAR data assimilation. Monthly Weather Review, 136, 1026-1041.