NSF Award Abstract - #0426845
CMG COLLABORATIVE RESEARCH: Improved Bayesian Estimators for Uncertainty in Climate System Properties
Abstract
The investigators are developing Bayesian statistical models for
the study of the distribution of climate system properties. The
study is based on output from the MIT 2DLO climate model as well
as an estimation of the natural climate variability from
atmosphere-ocean general circulation models (AOGCM). The
statistical models account for all uncertainties by focusing on
the estimation of the main patterns of natural variability. This
is achieved by building prior distributions for the covariance
matrix from ensemble runs of AOGCMs. Particular attention is paid
to the spectral decomposition of the covariance matrix. In
addition the statistical models are hierarchical in order to
consider all sources of errors in a comprehensive way. These
errors include the interpolation error due to the impossibility of
evaluating climate models in a time short enough to embed it
within a Monte Carlo iterative estimation method. The proposed
research falls clearly into the ``Representing uncertainty in
geosystems'' theme of the NSF Program for Collaborations in
Mathematical Geosciences. The main focus is to improve the
estimates of parameters that govern the large-scale behavior of
the climate system. The resulting analysis will include an
assessment of the uncertainty of those estimates. The research is
a collaborative effort between climate scientists and
statisticians as it requires the use of climate system models as
well as analyzing climate observational datasets. The broader
aspect of this project is that the estimated uncertainties in
climate system behavior can be used for uncertainty analysis of
climate change projections. By enhancing the ability to analyze
the risks of climate change on society, this research will provide
valuable input to policymakers.
Technical issues: We are addressing two issues to improve our
previous method. First, we are implementing a Markov Chain Monte Carlo
approach and thereby, implementing a more complete Bayesian
framework. Second, we are exploring methods for estimating the inverse
of the noise covariance matrices as used in the goodness-of-fit
statistics required for the Likelihood function in Bayes' Theorem. The
noise covariance matrices are estimated from unforced control
simulations in AOGCMs and are typically rank deficient (i.e., not
enough degrees of freedom in the data to estimate all
covariances). This rank deficiency results both from the high number
of spatial-temporal averages required to resolve climate changes in
the 20th century and from the length of the control runs only allowing
order 10-20 realizations of the noise. For example, if we include 10
decadal means and 4 zonal averages for the surface temperature changes
in the 20th century (1901-2000), we obtain a 40 dimensional space-time
pattern of climate change for the 20th century. This then requires
inverting a 40x40 covariance matrix to be used in the likelihood
function. This matrix has 820 unique elements that must be estimated
from 10 realizations of the noise in a 1000-year control simulation
(or 400 data points). Thus, 400 < 820 implies that the problem is
rank deficient. One choice is to get more data, a second is to try
and estimate a noise model from the data and impose a criteria that
the degrees of freedom of the noise model are much less than the dof
for the available data. Both are options but getting more data
requires further runs of the AOGCM in "spin-up" mode and is
undesirable from the standpoint of the AOGCM modelling centers.
Chez CEF