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Nonlinear Processes in Geophysics An interactive open-access journal of the European Geosciences Union
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Discussion papers
https://doi.org/10.5194/npg-2019-16
© Author(s) 2019. This work is distributed under
the Creative Commons Attribution 4.0 License.
https://doi.org/10.5194/npg-2019-16
© Author(s) 2019. This work is distributed under
the Creative Commons Attribution 4.0 License.

Research article 23 Apr 2019

Research article | 23 Apr 2019

Review status
This discussion paper is a preprint. A revision of this manuscript was accepted for the journal Nonlinear Processes in Geophysics (NPG) and is expected to appear here in due course.

Joint state-parameter estimation of a nonlinear stochastic energy balance model from sparse noisy data

Fei Lu1, Nils Weitzel2,3, and Adam H. Monahan4 Fei Lu et al.
  • 1Department of Mathematics, Johns Hopkins University, Baltimore, Marlyand, USA
  • 2Institut für Umweltphysik, Ruprecht-Karls-Universität Heidelberg, Heidelberg, Germany
  • 3Institut für Geowissenschaften und Meteorologie, Rheinische Friedrich-Wilhelms-Universität Bonn, Bonn, Germany
  • 4School of Earth and Ocean Sciences, University of Victoria, Victoria, British Columbia, Canada

Abstract. While nonlinear stochastic partial differential equations arise naturally in spatiotemporal modeling, inference for such systems often faces two major challenges: sparse noisy data and ill-posedness of the inverse problem of parameter estimation. To overcome the challenges, we introduce a strongly regularized posterior by normalizing the likelihood and by imposing physical constraints through priors of the parameters and states. We investigate joint parameter-state estimation by the regularized posterior in a physically motivated nonlinear stochastic energy balance model (SEBM) for paleoclimate reconstruction. The high-dimensional posterior is sampled by a particle Gibbs sampler that combines MCMC with an optimal particle filter exploiting the structure of the SEBM. In tests using either Gaussian or uniform priors based on the physical range of parameters, the regularized posteriors overcome the ill-posedness and lead to samples within physical ranges, quantifying the uncertainty in estimation. Due to the ill-posedness and the regularization, the posterior of parameters presents a relatively large uncertainty, and consequently, the maximum of the posterior, which is the minimizer in a variational approach, can have a large variation. In contrast, the posterior of states generally concentrates near the truth, substantially filtering out observation noise and reducing uncertainty in the unconstrained SEBM.

Fei Lu et al.
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Fei Lu et al.
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Short summary
ll-posedness of the inverse problem and sparse noisy data are two major challenges in the modeling of high-dimensional spatiotemporal processes. We present a Bayesian inference method with a strongly regularized posterior to overcome these challenges, enabling joint state-parameter estimation and quantifying uncertainty in the estimation. We demonstrate the method on a physically motivated nonlinear stochastic partial differential equation arising from paleoclimate construction.
ll-posedness of the inverse problem and sparse noisy data are two major challenges in the...
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