Tuesday, 15 January 2002: 4:59 PM

The Kalman-LÉvy filtering: Sequential assimilation methodology for power law and LÉvy law noises

We extend the sequential data assimilation
methodology to the case where the probability
of errors and noises are distributed according
to not only Gaussian but power and Lévy laws with
heavy tails. The main tool required to solve this
``Kalman-Lévy'' filter is the ``tail-covariance''
matrix. It is a natural generalization of
the covariance matrix which is mathematically
ill-defined when the exponents of the power law
correspodns to the heavy tail regime. We present
the full solution for the Kalman-Lévy filter and
discuss its properties on pedagogical examples.
The standard Kalman-Gaussian filter is recovered
as a special case when the exponent equals 2
The optimal Kalman-Lévy filter is found to differ
substantially from the standard Kalman-Gaussian
filter as the exponent deviates from $2$. For
smaller exponents, error forecast by the Kalman-Lévy
filter in terms of the tail-covariance matrix result
in less growth for divergent (unstable) dynamics
and less decay for convergent (stable) dynamics.
Furthermore the Kalman-Lévy gain for optimization
using the forecast and observed variables favors
more strongly the best of the two as estimated
from the tail-covariance matrix. Our theory also
estimates the error caused by the standard
Kalman-Gaussian filter when it is applied to the
system with heavy tail probability distribution.
Direct numerical experiments confirms our
theoretical predictions. We also discuss
implication of heavy-tail stochastic noises to
forecast skill for atmospheric low-frequency
variability.

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