A Forecast Test for Reducing Dynamical Dimensionality of Model Emulators

The climate system can be numerically represented by a set of physically based dynamical equations whose solution requires substantial computational resources. This makes computationally efficient, low dimensional emulators that simulate trajectories of the underlying dynamical system an attractive alternative for model evaluation and diagnosis. We suggest that since such an emulator must adequately capture anomaly evolution, its construction should employ a grid search technique where maximum forecast skill determines the best reference model. In this study, we demonstrate this approach by testing different bases used to construct a Linear Inverse Model (LIM), a stochastically forced multivariate linear model that has often been used to represent the evolution of coarse-grained climate anomalies in both models and observations. LIM state vectors are typically represented in a basis of the leading Empirical Orthogonal Functions (EOFs), but while dominant large-scale climate variations often are captured by a subset of these statistical patterns, key precursor dynamics involving relatively small scales are not. An alternative approach is balanced truncation, where the dynamical system is transformed into its Hankel space, whose modes span both precursors and their subsequent responses. Constructing EOF- and Hankel-based LIMs from monthly observed anomalous Pacific sea surface temperatures, both for the 150-year observational record and a perfect model study using 600 years of LIM output, we find that no balanced truncation model of any dimension can outperform an EOF-based LIM whose dimension is chosen to maximize independent skill. However, the dynamics of a high-dimensional EOF-based LIM can be efficiently reproduced by far fewer Hankel modes. The climate system may be physically and dynamically represented by a set of partial differential equations, from which one can obtain various numerical models and typically require substantial computational resources to solve. It is therefore attractive to find a low order representation of the dynamical system that captures key climate phenomena. One such approach is the Linear Inverse Model (LIM), a multivariate linear empirical model that builds on the large scale climate variability, identified from principal component analysis, and infers the lagged response relationship between two temporally varying climate states. This approach, however, may not adequately identify the small scale precursor dynamics, the occurrences of which often leads to large scale phenomena. Alternatively, the balanced truncation approach is capable of capturing the precursor dynamics and the large-scale response, by which we find the balanced truncation is more effective in approximating the dynamical system using a lower dimension than LIM. Our finding implies potential use of the balanced truncation approach in many numerical dynamical systems that may benefit from reducing the dimensions and improving the efficiency. Low dimensional emulators of the climate system should capture predictable dynamics, which may be evaluated using forecast skill metricsObserved predictable dynamics of high-dimensional linear inverse model (LIM) can be reproduced by lower dimensional balanced truncationHowever, no balanced truncation model of any dimension can outperform a LIM whose dimension is chosen to maximize independent skill