Introducing Error Co-variances in the ARM Variational Analysis

1
Introducing Error Co-variances in the ARM
Variational Analysis
Minghua Zhang (Stony Brook University/SUNY) and
Shaocheng Xie (Lawrence Livermore National
Laboratory)
ITPA
1. Introduction Any optimization algorithm
involves the minimization of a cost function.
For multi-variable analysis, such as analysis of
ARM data with several stations, many levels and
time steps, the cost function contains a error
covariance matrix. Elements of the matrix
determines how observations are weighted to
produce the analysis. In NWP community, it is
well known that the error covariance has a major
impact on the quality of the optimal analysis.
For example, if several measurements are highly
correlated, each individual data entry should be
given small weight relative to an independent
data entry. Because of the complexities to
derive the error covariance matrix from actual
data, the current ARM variational analysis
assumes that errors are independent. Since most
of the errors are due to sampling rather than
random instrument error, this assumption needs to
be improved.
To obtain
an AR(1) model is used such that
Covariance between two levels for other variables
can be similarly calculated.
It can be also shown that
This is a symmetric polynomial matrix that can be
inverted using the Cholesky decomposition. When
the correlation length scale is short, it is a
narrow diagonal matrix.
can be similarly obtained.
The analysis is then calculated from
2. The Problem For a field experiment such as
TWP-ICE, the atmospheric state variables of winds
(u,v), temperature (?) and specific humidity (q)
at S stations, K levels, and N time steps are
written as
with constraints
The merit of the above matrix structure is that
it yields an explicit solution from the cost
function term in the E-L equation.
4. Error Structures and Correlation Matrices
Analysis increments or errors in observations
relative to the first iteration of the
variational analysis for the TWP-ICE temperature
and u wind are shown in Figure 1. The
correlation matrices in the vertical direction
for the two variables are shown in Figure 2. The
matrices derived from the AR1 model are shown in
Figure 3. The AR(1) model captures the general
features of the correlations.
Similarly
and
With truth observations as
and
We write errors
and
Maximum likelihood or minimum variance leads to
cost function
is populated by covariance among all
stations/levels/variables/ time steps.
The dimension of this matrix is
In TWP-ICE, this is (4X45X6X201)2 2170802
The minimization of I is subject to the five
constraints of column integrated conservations at
each time step
Not only this matrix is too large to invert, but
also the covariance cannot be easily obtained
from data.
3. A New Method Since the constraints are
vertically integrated, we first assume errors to
be vertically correlated. This reduces the cost
function to
Where
Figure 3
Figure 2
Figure 1
5. Summary An AR(1) model is used to
characterize the error covariance in the vertical
direction in the ARM variational analysis that
allows inversion of the covariance matrix for the
minimization of the cost function. The model
captures the de-correlation lengths and the
different matrix structures for different
variables. The numerical algorithm is being
tested and implemented into the variational
analysis of TWP-ICE data.
These matrices are KXK in dimension.