Superparameterization (SP) is a multiscale computational approach wherein a
large scale atmosphere or ocean model is coupled to an array of simulations
of small scale dynamics on periodic domains embedded into the computational
grid of the large scale model. SP has been successfully developed in global
atmosphere and climate models, and is a promising approach for new
applications, but there is currently no practical data assimilation framework
that can be used with these models. The authors develop a 3D-Var variational
data assimilation framework for use with SP; the relatively low cost and
simplicity of 3D-Var in comparison with ensemble approaches makes it a
natural fit for relatively expensive multiscale SP models. To demonstrate the
assimilation framework in a simple model, the authors develop a new system of
ordinary differential equations similar to the two-scale Lorenz-'96 model.
The system has one set of variables denoted

Superparameterization (SP) is a multiscale computational method for
parameterizing small scale effects in large scale atmosphere and ocean
models. It was originally developed and has been particularly effective as a
cloud parameterization in atmosphere models

The authors recently developed an ensemble Kalman filter framework for data
assimilation with SP

Observations of physical variables have large scale and small scale parts, the former of which is equated with the large scale model variables, and the latter with the variables of the small scale embedded simulations. A key feature of SP is that the small scale simulations are periodic, so a location on the small scale computational grid does not correspond precisely to any location in the real physical domain; as a result, the small scale simulations provide only statistical information about the small scales, and this information can be used as a prior in the data assimilation context. In GLM14 an ensemble of SP simulations provides prior information on the large scale variables, but in the present approach the prior information on the large scales comes from a single SP simulation and a time-independent “background” covariance matrix for the large scale variables. When the observation operator is linear the analysis estimates of the large and small scale variables can be computed independently of each other, and the small scale covariance information effectively provides a time- and state-dependent estimate of representation error. When the observation operator is nonlinear the large and small scale analysis must be computed simultaneously by minimizing an objective function. Although analysis estimates of the small scale variables can be computed with linear observations, and must be computed with nonlinear observations, our framework does not at this time use the small scale analysis estimate to update any of the small scale SP variables because the latter cannot be unambiguously associated with any real physical location. A key update of the GLM14 framework is that we here compute a small scale analysis estimate at locations where observations are available, rather than at every coarse grid point. This can result in significant computational savings in the case of a nonlinear observation operator. We also update the GLM14 framework to better handle observations at locations off the coarse grid.

The complex superparameterized atmosphere and climate models mentioned above
are not particularly convenient for the development of a new data
assimilation framework, and existing toy models of SP are of limited utility
for this purpose. In Sect.

In this section we develop a new simple model for SP in which to demonstrate
our data assimilation framework.

The new model is defined by the following equation:

The superparameterization approximation is governed by

The purpose of this research is not to study the SP approximation in this
system, but rather to use the system as a test bed for our data assimilation
framework. We therefore choose to focus on parameter regimes where the SP
approximation is reasonably accurate, setting

Climatological statistics in regime I.

Climatological statistics in regime II. Panels are the same as
Fig.

Some characteristics of the dynamics in regimes I and II are presented in
Figs.

In regime II the large scale dynamics are more chaotic, though wave trains
are still evident in the time series of

The

The primary difficulty in developing a data assimilation framework for an SP model is that observations of the true system include contributions from large and small scales, and it is necessary to relate the observations to the large and small scale variables of the SP model. GLM14 provided a framework for relating observations to SP model variables, and we improve on this framework below.

Let the large scale variables of the SP simulation be denoted

In GLM14, observations are related to the SP model variables using the
following observation model

As noted in GLM14, it is unrealistic to use the same interpolation operator
for both the large and small scale variables because it assumes that the
small scale variables vary smoothly between the coarse grid points, whereas
the small scale variables should by definition vary over shorter distances.
(Observations in GLM14 were taken only on the coarse grid points, avoiding
the issue.) Instead of specifying an alternative interpolation operator for
the small scales, we update the framework by altering the definition of

Let

The covariance of the small scale variables

To complete the specification of the 3D-Var framework we specify a prior
joint distribution for

Having thus specified the observation model and prior mean and covariance,
the 3D-Var analysis estimate of the system state minimizes the following
objective function

When the observation operator is linear,

In GLM14 the small scale covariance matrix

In this section we describe data assimilation experiments in both regimes of
the test model using the 3D-Var framework from Sect.

Observations are taken at

Specification of the background covariance matrix is a crucial aspect of any
3D-Var assimilation system. We consider the simplest possible estimate

A single assimilation experiment consists of 1000 cycles, where the SP
variables for the first forecast are initialized directly from the true model
variables. Although the assimilation system provides estimates of the small
scale part of the true system at the location of the observations, this
information is far from sufficient to provide an estimate of the full state

As a point of comparison for the performance of the forecast in the
assimilation experiments, we consider climatological values of rms error and
pattern correlation defined using the uniform climatological mean value of

Results of the assimilation experiments for regime I. There are

As a point of comparison for the performance of the analysis estimate in the
assimilation experiments we take a “smoothed observation” estimate that is
obtained by projecting the observations onto the largest

We also compare to the performance of an ensemble adjustment Kalman filter using the true system dynamics. These experiments and their results are described in Section “Comparison to EAKF”.

The large scale dynamics are more predictable in regime I than in regime II,
but the small scale variance is larger as well, making it harder to obtain an
accurate estimate of the large scales. With a short observation time

In regime II the results with the linear and nonlinear observations
are very similar in all cases. With a short observation time

Results of the assimilation experiments for regime II. There are

To put the foregoing results into perspective we compare to the results of an
ensemble adjustment Kalman filter

In regime I the rms forecast errors of the

In both regimes the EAKF estimates the large scale part of the solution very
poorly, much worse than the SP 3D-Var. This poor performance is presumably
associated with the fact that the EAKF is attempting to estimate the full
system state

Superparameterization (SP) is a multiscale computational approach that has
been successfully applied to modeling atmospheric dynamics, and that shows
promise for more general applications

The data assimilation framework is demonstrated in a new system of ordinary
differential equations based on the two-scale Lorenz-'96 model

Our work lays a foundation for 3D-Var data assimilation with existing SP
models. In order to implement our framework with an SP atmosphere or climate
model, it would be necessary to specify an appropriate background covariance
matrix for the large scale model, but this should be straightforward given
the extensive use of the 3D-Var approach in atmosphere and ocean data
assimilation

I. Grooms designed the research, I. Grooms and Y. Lee carried out the experiments, and I. Grooms prepared the manuscript.

The authors acknowledge funding from the United States Office of Naval Research MURI grant N00014-12-1-0912. The authors thank A. J. Majda for suggesting the addition of a second regime (regime II), and two anonymous reviewers. Edited by: Z. Toth Reviewed by: two anonymous referees