3.1
Systematic Error Detection in Dynamical Models: A Framework
S. Lakshmivarahan, University of Oklahoma, Norman, OK; and J. Lewis
It is the rule rather than the exception that operational weather prediction models exhibit systematic error. Over the Gulf of Mexico, the NCEP/EMC operational model has been plagued with systematic error in the marine-layer moisture for several decades — a negative bias from the late-1980s to the mid-1990s followed by a positive bias from the mid-1990s to the present. Despite a sustained effort by the research and operational community at NCEP/NHC, the biased forecast remains. The symptoms of the bias forecast can be ameliorated to a certain extent, i. e., from knowledge of the bias, the data assimilation scheme can empirically alter the analysis. Clearly, it is preferable to identify and remove the source of the error. We have built a framework that searches for the source(s) of systematic error. We apply our methodology to the cool season weather regime that is linked to the systematic errors mentioned above — return flow over the Gulf where the consensus of opinion indicates that erroneous turbulent moisture/heat flux is the principal culprit. Rather than using the general operational model, a primitive equation model with a variety of parameterizations, we will use the mixed layer model that has been found to be appropriate for the generation of the convective boundary layer that forms during the outbound phase of return flow. The data we use were collected during the Gulf of Mexico Experiment (GUFMEX) in 1988. These data provide an exceptional depiction of the air mass transformation that took place as the air moved over the coastal waters toward the Loop Current in mid-Gulf. Our framework follows: Let denote the state of a dynamical system at time
. Let
and
denote a set of (
) parameters of which the first
components denote the initial conditions, i. e.,
for
and the last
components denote the physical parameters. Let
and
(1)
denote the model dynamics. To emphasize the dependence of the solution on the parameters , we explicitly denote the state system state as
. Given the current operating point
, the solution
denotes the model forecast. Let
be a mapping such that
(2)
denotes the model predicted observations. It is tacitly assumed that the model and the map
are exact, i. e., there are no errors in the model or in the relation that links observations to the model state. Assume observations
are available at a specific time
where these observations are denoted by
. The difference between observations and model counterparts to the observations is given by
. (3)
This constitutes a measure of the forecast error. Since and
are assumed to be exact, a little reflection reveals that the forecast error is only due to errors or biases in the parameter vector
. Given the forecast error in (3), our goal is to find a correction
to the parameter
such that
closely resembles
. We start by quantifying the first variation in
resulting from the perturbation of
by
. To this end, first define the Jacobian,
, of
with respect to
, given by
(4)
where . Then
(5)
The matrix
is also known as the first-order sensitivity matrix. The variation
in
induces a variation
in
that is given by
(6)
where the Jacobian is an
matrix given by
(7)
and . Combining (5) and (6), we obtain
. (8)
Given the operating point and
in (3), we search for
such that
. (9)
Combining (8) and (9), given , we seek
such that
. (10)
Defining . (11)
Then (10) becomes . (12) Since
in general, this equation is solved as a linear least squares problem. From Lewis et al. (2006), we get the solution as
if
(13)
or if
. (14)
A number of observations are in order.
1. The matrix defined in (11) is a product of two Jacobian matrices
and
. Given
, the matrix
can be explicitly computed. However, to compute the second factor
, we need to know the dependence of
on
. That is, we need to know the solution
of (1). Since (1) is in general non-linear, except for special cases, we cannot obtain the solution
. Hence clever numerical schemes must be designed to compute the evolution of
with
.
2. There is a vast literature dealing with the computation of first-order sensitivity matrices. The anthology edited by J. B. Cruz (1979) contains many of the classic results developed in the context of control theory. The review paper by Rabitz et al. (1983) contains a very readable account of the key results developed in the context of chemical kinetics. The recent monographs by Cacuci (2003) and Cacuci et al. (2005) contain an up to date coverage of topics related to sensitivity analysis.
3. Our goal is to demonstrate the utility of this framework. Accordingly, we have chosen the 3-variable nonlinear mixed layer model mentioned above. In a companion paper, we plan to illustrate the numerical process of computing using well-known methods discussed in the literature.
4. Since we have the luxury of an exact solution in the mixed-layer model case, we also illustrate the computation of second-order sensitivity.
We have generated systematic error in the mixed layer model that mimics the results found in operational practice. The methodology outlined above has succeeded in identifying the sources of error in a variety of cases.
Session 3, Advances in Modeling and Forecasting-I
Monday, 12 January 2009, 2:00 PM-2:30 PM, Room 126A
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