Ensemble-based Data
Assimilation and Forecasting
The amplification of small errors in the specification of
initial conditions and model inadequacies place limits on the
ability to accurately forecast any system, including the
atmosphere and oceans. Imperfect observations will frustrate
any attempt to estimate a system’s exact state. In fact,
even given a perfect model and a time-series of noisy
observations into the infinite past, it is impossible to
uniquely identify a system’s true state. The correct
expression of the system state is therefore a probability
density function (PDF) that defines the probability of the
state lying in a region of state space.
Because the correct description of initial conditions is
probabilistic, it follows that the correct forecast is also
probabilistic. A popular approach for estimating the evolution
of the forecast PDF is the so-called ensemble approach. In
ensemble forecasting, random sampling is used to approximate
the initial PDF, and each sample is propagated forward under
the forecast model. The resulting collection of forecasts is
interpreted as a random sample from the forecast PDF. There is
no single forecast; the correct expression of the future
system state is the PDF.
Probabilistic, or ensemble, forecasting requires random
draws from a PDF of initial states. Where does this initial
PDF come from? One typically has incomplete initial
observations of the system, as well as an estimate of the
system state from a short-term forecast. Data assimilation is
the process of blending these two independent pieces of
information (and their associated uncertainty) to produce an
estimate of the initial state that is superior to either in
isolation. Just as one can take a probabilistic approach to
forecasting, it is possible to take a probabilistic approach
to data assimilation using ensembles. This integrated approach
is capable of producing correct probabilistic initial state
estimates and forecasts for a number of different data
assimilation techniques when a perfect model is in hand.
When only imperfect models are available (i.e. always),
it is impossible to produce correct probabilistic forecasts.
Finding methods to account for model error in probabilistic
data assimilation and forecasting is an active area of
research. The aim is not simply to produce good forecasts, but
to use data assimilation and forecasting as tools to better
understand the physical system and, hopefully, improve models.
Contact Prof.
Jim Hansen for more information.
