2,989 research outputs found
Linear theory for filtering nonlinear multiscale systems with model error
We study filtering of multiscale dynamical systems with model error arising
from unresolved smaller scale processes. The analysis assumes continuous-time
noisy observations of all components of the slow variables alone. For a linear
model with Gaussian noise, we prove existence of a unique choice of parameters
in a linear reduced model for the slow variables. The linear theory extends to
to a non-Gaussian, nonlinear test problem, where we assume we know the optimal
stochastic parameterization and the correct observation model. We show that
when the parameterization is inappropriate, parameters chosen for good filter
performance may give poor equilibrium statistical estimates and vice versa.
Given the correct parameterization, it is imperative to estimate the parameters
simultaneously and to account for the nonlinear feedback of the stochastic
parameters into the reduced filter estimates. In numerical experiments on the
two-layer Lorenz-96 model, we find that parameters estimated online, as part of
a filtering procedure, produce accurate filtering and equilibrium statistical
prediction. In contrast, a linear regression based offline method, which fits
the parameters to a given training data set independently from the filter,
yields filter estimates which are worse than the observations or even divergent
when the slow variables are not fully observed
Combining Stochastic Parameterized Reduced-Order Models with Machine Learning for Data Assimilation and Uncertainty Quantification with Partial Observations
A hybrid data assimilation algorithm is developed for complex dynamical
systems with partial observations. The method starts with applying a spectral
decomposition to the entire spatiotemporal fields, followed by creating a
machine learning model that builds a nonlinear map between the coefficients of
observed and unobserved state variables for each spectral mode. A cheap
low-order nonlinear stochastic parameterized extended Kalman filter (SPEKF)
model is employed as the forecast model in the ensemble Kalman filter to deal
with each mode associated with the observed variables. The resulting ensemble
members are then fed into the machine learning model to create an ensemble of
the corresponding unobserved variables. In addition to the ensemble spread, the
training residual in the machine learning-induced nonlinear map is further
incorporated into the state estimation that advances the quantification of the
posterior uncertainty. The hybrid data assimilation algorithm is applied to a
precipitating quasi-geostrophic (PQG) model, which includes the effects of
water vapor, clouds, and rainfall beyond the classical two-level QG model. The
complicated nonlinearities in the PQG equations prevent traditional methods
from building simple and accurate reduced-order forecast models. In contrast,
the SPEKF model is skillful in recovering the intermittent observed states, and
the machine learning model effectively estimates the chaotic unobserved
signals. Utilizing the calibrated SPEKF and machine learning models under a
moderate cloud fraction, the resulting hybrid data assimilation remains
reasonably accurate when applied to other geophysical scenarios with nearly
clear skies or relatively heavy rainfall, implying the robustness of the
algorithm for extrapolation
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