Parameter tuning for a multi-fidelity dynamical model of the magnetosphere

Abstract

Geomagnetic storms play a critical role in space weather physics with the potential for far reaching economic impacts including power grid outages, air traffic rerouting, satellite damage and GPS disruption. The LFM-MIX is a state-of-the-art coupled magnetosphericionospheric model capable of simulating geomagnetic storms. Imbedded in this model are physical equations for turning the magnetohydrodynamic state parameters into energy and flux of electrons entering the ionosphere, involving a set of input parameters. The exact values of these input parameters in the model are unknown, and we seek to quantify the uncertainty about these parameters when model output is compared to observations. The model is available at different fidelities: a lower fidelity which is faster to run, and a higher fidelity but more computationally intense version. Model output and observational data are large spatiotemporal systems; the traditional design and analysis of computer experiments is unable to cope with such large data sets that involve multiple fidelities of model output. We develop an approach to this inverse problem for large spatiotemporal data sets that incorporates two different versions of the physical model. After an initial design, we propose a sequential design based on expected improvement. For the LFM-MIX, the additional run suggested by expected improvement diminishes posterior uncertainty by ruling out a posterior mode and shrinking the width of the posterior distribution. We also illustrate our approach using the Lorenz '96 system of equations for a simplified atmosphere, using known input parameters. For the Lorenz '96 system, after performing sequential runs based on expected improvement, the posterior mode converges to the true value and the posterior variability is reduced.

Model Calibration via Deformation

“…Our approach herein does not use the higher resolution model and is able to identify a single region of parameter space that optimally aligns warped low resolution LFM-MIX with observational data. Additionally, our posterior densities approximately overlap with those derived by [27].…”

“…With our initial design this results in 554040 total points of model output, and hence a dimension reduction approach is desired. We follow the emulator of [27], which was also developed for the LFM-MIX, an approach we briefly describe here. The basic idea is to decompose the LFM-MIX output as weighted sums of spatial empirical orthogonal functions (EOFs).…”

“…Briefly, mean function parameters are estimated by ordinary least squares, and conditional on these the spatial parameters of variance and ranges are estimated by maximum likelihood. The interested reader can find a detailed description of how the model physics and EOF structure correspond to likelihood assumptions in [27]. Note that similar modeling strategies have been employed in [23,24].…”

“…We experimented with allowing β 1 to vary and found posterior calibration distributions that were multimodal and favored no particular region of parameter space. This suggests there is a confounding between the input parameters and potential multiplicative bias, which is expected given that α and R contribute directly and nonlinearly to the magnitude of the storm's energy (see [27] for discussion). We additionally experimented with inflating λ to 10, and the posterior calibration densities updated slightly from those shown in Figure 6, with the posterior mode generally centering around (0.22, 0.07, 0.027).…”

“…[27] proceeded under the assumption that a higher resolution version of the LFM-MIX is unbiased at an optimal input setting. By linking the lower resolution version (that which is used in this paper) to a higher resolution version (that is computationally more expensive), they were able to identify three regions of parameter space which may contain optima and developed a sequential design based on expected improvement to further hone in on the optimum.…”