NPGNonlinear Processes in GeophysicsNPGNonlin. Processes Geophys.1607-7946Copernicus GmbHGöttingen, Germany10.5194/npg-22-601-2015A framework for variational data assimilation with superparameterizationGroomsI.ian.grooms@colorado.eduhttps://orcid.org/0000-0002-4678-7203LeeY.Center for Atmosphere Ocean Science, Courant Institute of Mathematical Sciences, New York University, New York, USADepartment of Applied Mathematics, University of Colorado, Boulder, Colorado, USAI. Grooms (ian.grooms@colorado.edu)9October20152256016119March201520March201525August201528September2015This work is licensed under a Creative Commons Attribution 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0/This article is available from https://npg.copernicus.org/articles/22/601/2015/npg-22-601-2015.htmlThe full text article is available as a PDF file from https://npg.copernicus.org/articles/22/601/2015/npg-22-601-2015.pdf
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 {Yi}, with large and small
scale parts, and the SP approximation to the system is straightforward. With
the new assimilation framework the SP model approximates the large scale
dynamics of the true system accurately.
Introduction
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 , and has been
implemented in global atmosphere and climate models
. SP couples a large scale, low resolution
model to an array of local small scale, high resolution simulations embedded
within the computational grid of the large scale model. The computational
cost is kept down through a variety of methods, most prominently by reducing
the dimensionality of the small scale simulations, e.g., using one vertical
and one horizontal coordinate in the aforementioned atmospheric applications.
Although atmosphere and climate models with SP are particularly successful at
producing a realistic Madden-Julian oscillation and diurnal cycle of
convection over land , there are as yet no data assimilation
systems designed for use with these models. Instead, the large scale
variables are initialized from state estimates generated with non-SP models
and the small scale variables are initialized with small-amplitude noise
. Once the SP model has been initialized, there is no practical
framework for combining observational data with the multiscale model forecast
to produce a new initial condition.
The authors recently developed an ensemble Kalman filter framework for data
assimilation with SP hereafter GLM14. This framework was
developed in the context of stochastic SP, a variant of SP that reduces
computational cost by replacing the small scale simulations of SP with
quasilinear stochastic models . Stochastic SP has only been
developed for idealized turbulence models , and
is not yet implemented in global atmosphere, ocean, or climate models. The
relatively high cost and computational complexity of global atmosphere and
climate models with SP and the extra cost associated with an ensemble-based
data assimilation system makes it unlikely that it will be possible to use
these models with the framework of GLM14 in the near future. Here we develop
a 3D-Var variational data assimilation framework for SP that builds on and
modifies the framework of GLM14.
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. we develop a new system of
ordinary differential equations based on the two-scale
Lorenz-'96 (L96) model , and an SP approximation to that
system. This new model serves as a test bed in which to demonstrate our new
SP 3D-Var framework. The 3D-Var framework with SP is presented in
Sect. , and assimilation experiments using the new framework
and the new system are described in Sect. , followed by
conclusions.
A multiscale Lorenz-'96 model with superparameterization
In this section we develop a new simple model for SP in which to demonstrate
our data assimilation framework. developed an idealized model of
SP, but the system suffers from one major drawback: it does not consist of an
SP approximation to an idealized system, but rather consists only of an
idealized SP model. used the model of to develop a
data assimilation strategy for SP, but with the assumption that direct
observations of the large scale variables were available, rather than having
both large and small scale contributions to the observations.
have recently investigated a range of multiscale assimilation methods in a
highly condensed model where the “large scale” consists of a single scalar
with no spatial extent.
developed an SP approximation for the two-scale Lorenz-'96
system, which has the following form :
X˙k=-Xk-1Xk-2-Xk+1-Xk-hcb∑j=1JYj,k+FY˙j,k=c-bYj+1,kYj+2,k-Yj-1,k-Yj,k+hbXk.
The Xk variables have periodicity Xk=Xk+K, and the
Yj,k variables have periodicity Yj+J,k=Yj,k+1 and
Yj,k+K=Yj,k, where j= 1, …, J and
k= 1, …, K. The combined index j+J(k- 1) is
naturally associated with spatial location along a latitude circle, and the
local average J-1∑j=1J serves to separate large and
small spatial scales. This system is primarily useful as a two-timescale
model, since for large c the Yj,k variables are faster than the
Xk variables. Wilks's SP approximation to this system reflects this fact
by treating the Yj,k variables as purely small-scale; also, in his SP
approximation the periodicity of the Yj,k variables is replaced by
defining Y0,k=YJ+1,k=YJ+2,k, which are set to
a constant value. Considering that the multiscale nature of SP is primarily
based on spatial scale separation rather than timescale separation, a more
natural SP approximation to the two-scale Lorenz-'96 system might make the
Yj,k variables locally periodic: Yj+J,k=Yj,k.
Nevertheless, there would still be two sets of large scale variables
(Xk, and the j-average of Yj,k) but only one set of small scale
variables (Yj,k minus its j average). Rather than bend the
two-timescale Lorenz-'96 model to our two-space-scale purpose, we develop a
new two-space-scale version of the Lorenz-'96 model that is more naturally
suited to an SP approximation.
The new model is defined by the following equation:
Y˙=hNY(Y)+JTTNX(TY)-Y+F1JK,
where Y={Yi}i=1JK, where 1JK is a vector of
length JK with all elements equal to 1, T is a matrix in
RK×JK, and the index i, which is periodic
Yi+JK=Yi, is analogous to spatial location on a latitude circle,
similar to the original L96 model . The nonlinear functions
NY and NX are defined as
NY(Y)i=-Yi+1Yi+2-Yi-1,NX(X)k=-Xk-1Xk-2-Xk+1,
where Eqs. () and () are evaluated assuming
periodicity for the vectors X={Xk}k=1K and
Y: Xk+K=Xk and Yi+JK=Yi. The
matrix T extracts the large scale part of Y; we choose to
let T be defined as the projection onto the first K discrete
Fourier modes, followed by evaluation on an equispaced grid of K points.
The large scale dynamics are obtained by applying T to
Eq. () from the left:
X˙=hTNY(Y)+NX(X)-X+F1K,
where we define the large scale component X=TY, and
use that JTTT is the identity matrix and that
T 1JK=1K (these are true for our choice of a
Fourier projection, but other choices of T are possible). Note
that when h= 0 the dynamics are those of the single-scale Lorenz-'96 model
with K modes, and when h≠0 the nonlinearity NY(Y)
couples large and small scales. Energy conservation for the nonlinear terms
in Eq. () is obtained by noting that Eq. ()
implies YTNY(Y)= 0, and that Eq. () implies
YTTTNX(TY)=XTNX(X)= 0.
The matrix JTT interpolates from RK to
RJK, and it is convenient to define notation for the small
scale part of Y:
y=yii=1JK=Y-JTTTY.
The superparameterization approximation is governed by
Y˙j,k=-hYj+1,kYj+2,k-Yj-1,k-Xk-1Xk-2-Xk+1-Yj,k+F,
where Xk=J-1∑j=1JYj,k, and there is local as well as global
periodicity: Yj+J,k=Yj,k and Xk+K=Xk. The large scale dynamics
in the SP approximation are obtained by j-averaging Eq. (), which gives
X˙k=-hJ∑j=1JYj+1,kYj+2,k-Yj-1,k-Xk-1Xk-2-Xk+1-Xk+F.
When h= 0 the large scale dynamics of the SP approximation and the
true system are equivalent. As in more complex SP applications, the small
scale variables (here Yj,k-Xk) are locally periodic, and are
coupled to the large scale using a local average over a periodic domain in a
manner analogous to the coupling in more complex SP models
e.g.,. The Xk variables in the SP model attempt to
accurately model the dynamics of X in the true system, but the small
scale variables of the SP approximation are only statistically related to the
small scale variables of the true system; i.e., one does not expect an SP
variable Yj,k to be a direct approximation of any of the true system
variables Yi.
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 J= 128 so that there is
a good scale separation (the SP approximation should break down for
small J). The number of large scale modes is set to K= 41; we
choose 41 rather than the usual 40 so that the discrete Fourier modes
associated with the large scale variables are 0, ±1, …, ±20, and
the twentieth mode is not split between large and small scales. It remains to
choose F and h. In general, for fixed nonzero h the small scale
variables become more chaotic and larger amplitude as F increases, and
similarly for fixed F as h increases. As the small scales become more
chaotic and larger amplitude, the large scale variables become less chaotic.
This behavior is perhaps counterintuitive, but similar behavior has been
observed in the two-scale Lorenz-'96 system by . Balancing
the desire for complex large scale dynamics and turbulent small scale
dynamics, we choose to focus on two parameter regimes:
F= 30, h= 0.4;
F= 21, h= 0.35.
Climatological statistics in regime I. (a) Time series of
the Xk variables. (b) Time series of the Xk variables from
the SP approximation. (c) A snapshot showing Yi (blue), the
large scale part of Yi defined by projection onto the first 41 discrete
Fourier modes (red), and the Xk variables (yellow circles).
(d) Time-averaged energy spectrum |Y^κ|2 where
Y^κ is the discrete Fourier coefficient of Yi with wave
number κ. (e) Time-lagged autocorrelation functions for
Xk (blue) and the small scale part of Yi (red), defined by
projecting out the first 41 Fourier modes. (f) Space-lagged
autocorrelation functions for Xk from the true dynamics (blue) and the
SP approximation (red).
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. and , respectively. In regime I the large scale
dynamics consist of a train of eight propagating and nonlinearly interacting
“waves”, as seen in the time series of the X variables in
Fig. a. The large scale dynamics of the SP approximation are
qualitatively similar, as shown in Fig. b. The time-lagged
autocorrelation function of the Xk variables (averaged over k) is
shown in Fig. e, and displays an oscillatory structure associated
with the wave train. The initial decay of the time-lagged autocorrelation is
approximated by an exponential of the form
exp{-(λ+iω)t} with decorrelation time
λ-1= 0.84 and oscillation period 2π/ω= 0.71;
the resurgence of correlation between 6 and 8 time units is associated with
the time it takes a single wave to propagate once around the domain. The
regularity of the wave train is also reflected in the space-lagged
autocorrelation function for the Xk variables shown in
Fig. f, which is well approximated by the SP dynamics.
Figure c shows the Yi variables at an instant of time (blue),
along with the large scale part (red; the projection onto the first
41 Fourier modes) and the Xk variables (yellow circles). There is
clearly strong small scale variability, but not so strong that it completely
obscures the large scale pattern, and the amplitude of the small scale
variability varies over the domain. Figure d shows the
time-averaged energy spectrum |Y^κ|2, where
Y^κ is the discrete Fourier coefficient of Yi with wave
number κ. There is a clear separation in amplitude between the large
scale Fourier modes (κ≤ 20) and the small scale modes, showing
that the large scale energy is concentrated near wave numbers
κ= 7 and 8, while the small scale energy is more broadly
distributed among Fourier modes. The broad distribution of small scale energy
among Fourier modes is indicative of the strongly chaotic small scale
dynamics, as is the rapid temporal decorrelation of the small scale variables
yi shown in Fig. e. The decorrelation time of the small scale
variables yi is estimated as 0.2 using the integral of the time-lagged
autocorrelation function.
In regime II the large scale dynamics are more chaotic, though wave trains
are still evident in the time series of X in Fig. a. The
large scale dynamics of the SP approximation are again qualitatively similar,
as shown in Fig. b. The time-lagged autocorrelation function of
the Xk variables in Fig. e decays much more rapidly than in
regime I. The initial decay of the time-lagged autocorrelation is
approximated by an exponential of the form
exp{-(λ+iω)t} with decorrelation time
λ-1= 0.38 and oscillation period 2π/ω= 0.95,
and there is no resurgence of correlations at long lag times. The decreased
regularity of the wave train is reflected in the space-lagged autocorrelation
function for the Xk variables shown in Fig. f, which is
again well approximated by the SP dynamics. The snapshot of the Yi
variables in Fig. c shows a diminished level of small scale
variability overall, with some regions having almost no small scale activity
and others having strong small scale variability. The energy spectrum in
Fig. d shows that the energy is more broadly distributed among
large scale Fourier modes, though there is still a peak at wave number
κ= 8. The broad distribution of small scale energy among Fourier
modes is again indicative of the strongly chaotic small scale dynamics, as is
the rapid temporal decorrelation of the small scale variables yi shown
in Fig. e. The decorrelation time of the small scale variables
yi is estimated as 0.23 using the integral of the time-lagged
autocorrelation function.
The Yi variables have a uniform time mean of 3.8 and 3.6 in regimes I
and II, respectively, which is accurately reproduced by
the SP approximation. The Xk variables have variance 31 and 32 in regimes I
and II, respectively, and their SP counterparts have
slightly higher variances of 33 and 34. The small scale variables yi have
climatological variance of 70 in regime I and 29 in regime II, though
Figs. c and c show that this
variability is unevenly distributed over the physical domain at any given instant.
Variational data assimilation with superparameterization
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
u‾ (the overbar does not denote a statistical mean), and
let the small scale variables be denoted ũ. In the context of
the new Lorenz-'96 model, u‾=X and
ũ={Yj,k-Xk}j,k. In most SP applications there is a
set of small scale variables at every point of the large scale computational
grid. The small scale variables exist on local periodic domains so that the
small scale variables at each coarse grid point are disconnected from those
at surrounding coarse grid points, and the small scale variables have zero
average across the periodic directions. Each location in the small scale
periodic domains does not correspond to a different location in the real
physical domain. Instead, all points in a given periodic domain are best
thought of as existing at one physical location: the associated coarse grid point.
In GLM14, observations are related to the SP model variables using the
following observation model
v=H(L(u‾+u′))+ε,
where H is the observation operator and ε is a
vector of zero-mean normal random variables associated with observation
error. The vector u′ has the same size as u‾, and
models the small scale contribution to physical variables at the coarse grid
points; i.e., u=u‾+u′ is the
vector of real physical variables at the coarse model grid points. The
physical variables u are interpolated to the location of the
observations by L. The vector u′ is not the same as the
small scale SP variables ũ. Instead, the mean and covariance
of u′ are computed from the statistics of the small scale SP
variables ũ. Although the true small scale variables
u′ can in principle have nonzero statistical mean, the small scale SP
variables ũ always have zero mean because their average over
the local periodic domains is always zero by definition. For example, in the
context of the new Lorenz-'96 model the GLM14 version of the vector
u′ has length K, has zero mean, and has a diagonal covariance with
entries
Varuk′=1J-1∑j=1JYj,k-Xk2.
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
u′ to include small scale variables only at the points where
observations are taken. We also assume that the statistics of the small scale
variables vary on large scales and can therefore be smoothly interpolated
from the coarse grid points, where small scale SP statistics are available,
to the locations of the observations.
Let P denote the number of different physical locations where observations
are available (for simplicity of exposition, we assume that there is only one
observation per location, i.e., v∈RP). The
updated observation model for the pth location is
vp=HpLp(u‾)+up′+εp,
where Lp interpolates the large scale model variables
u‾ to the observation location and εp is a
zero-mean Gaussian random variable. There is thus one vector up′ of
small scale variables per observation location. The updated observation model
for all P observations can be written in vector form as
v=H(L(u‾)+u′)+ε,
where u′ is no longer defined as in GLM14, but according to the
discussion above.
The covariance of the small scale variables P′ is computed from
the small scale variables of the SP model, and thus changes from one
assimilation cycle to the next. Specifically, it is first assumed that the
small scale variables at different observation locations are uncorrelated
from each other so that one needs only compute the covariance matrices
Pp′ of the up′ variables. This assumption is reasonable
as long as the observations are taken at locations reasonably well separated
compared to the correlation length of the small scale variables. (The
framework could be updated for situations where the observations are closer
than this, e.g., by using spatial correlation information for the small scale
variables computed from the SP simulation, but this is beyond the scope of
the present investigation.) To compute Pp′ we begin by computing
auxiliary small scale sample covariance matrices P̃k using
the small scale SP variables ũ at each coarse grid point. Let
{ũk,j}j=1J be the small scale SP variables located
in a periodic domain at the kth coarse grid point, where there are J grid
points in the periodic embedded domain. Then, recalling that their
average over J is zero, the auxiliary small scale sample covariance matrix is
P̃k=1J-1∑j=1Jũj,kũj,kT,
where the superscript T denotes a vector transpose. It is typically the
case that J is large enough that P̃k is full rank, and
we do not consider exceptions here. In the context of the new Lorenz-'96
model the auxiliary small scale sample covariances are given by
Eq. (). Finally, the small scale covariance matrices at the
observation locations Pp′ are obtained by interpolating the
elements of the matrices P̃k from the coarse grid to the
locations of the observations, which assumes that the small scale statistics
vary smoothly on the large scale. The interpolation method used to
interpolate the small scale covariance matrices need not be the same as
L, and should have positive coefficients in order to ensure that
the small scale covariance matrices remain positive definite. (It may not be
necessary to compute sample covariance matrices P̃k at
every coarse grid point; one only needs to compute them at points needed in
the interpolation.) For comparison, in GLM14 the covariance of the small
scale variables P′ is the same size as the large scale background
covariance B‾, and consists of the auxiliary small scale
sample covariance matrices P̃k arranged in block-diagonal
form. When observations are taken at every coarse grid location the GLM14
formulation is equivalent to the new one.
To complete the specification of the 3D-Var framework we specify a prior
joint distribution for u‾ and u′ with mean
E[u‾]=μ‾,E[u′]=0
and covariance
B‾00P′.
As typical in a 3D-Var setting, the background covariance matrix
B‾ for the large scale variables is independent of time,
and the prior mean for the large scales is given by a single forecast of the
large scale part of the SP model. The small scale variable u′ is
assumed to be uncorrelated with the large scale variable. In practice, the
large and small scale variables are certainly not independent, but as shown
in GLM14 the assumption that they are uncorrelated is reasonable within the
context of an SP model where the small scale variables have zero mean. To
wit, the joint probability distribution of large and small scale variables
can be factored into the large scale marginal and the small scale conditional
distributions p(u‾, u′)=pM(u‾)pC(u′|u‾).
The cross-covariance between large and small scale variables is
∬(u‾-μ‾)u′TpM(u‾)pC(u′|u‾)du‾du′=∫(u‾-μ‾)pM(u‾)∫u′TpC(u′|u‾)du′du‾=0,
where the term in square brackets is zero because the small scale variables
are assumed to have zero mean regardless of the state of the large scale variables.
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 :
Υ(u‾,u′)=(u‾-μ‾)TB‾-1(u‾-μ‾)+u′TP′-1u′+(v-H(L(u‾)+u′))TR-1(v-H(L(u‾)+u′)),
where R is the covariance matrix of the observation error vector ε.
When the observation operator is linear, H=H, the analysis
can be computed from the Kalman filter formulas , which in
this case gives
u‾a=μ‾+K‾(v-HLμ‾),K‾=B‾(HL)THLB‾(HL)T+HP′HT+R-1,u′a=K′(v-HLμ‾),K′=P′HTHLB‾(HL)T+HP′HT+R-1,
where the superscript “a” denotes the analysis estimate. A key feature of
these formulas is that the large scale and small scale estimates can be
computed independently. In particular, the large scale estimate can also be
computed as the minimizer of the following objective function:
Υ‾(u‾)=(u‾-μ‾)TB‾-1(u‾-μ‾)+(v-HLu‾)THP′HT+R-1(v-HLu‾).
In cases where the small scale estimate is not used and the observation
operator is linear, the small scale estimate does not need to be computed. It
can be seen from Eqs. () and () that the
observed small scale covariance matrix HP′HT
acts as a time-varying estimate of the representation error since it inflates
the measurement error covariance R.
In GLM14 the small scale covariance matrix P′ is defined
differently (as described above) and the small scale vector u′ is the
same size as the large scale vector u‾. In the GLM14
formulation the final term in the objective function
Eq. () is replaced by
(v-H(L(u‾+u′)))TR-1(v-H(L(u‾+u′))).
For linear observations the GLM14 versions of the Kalman filter formulas are
u‾a=μ‾+K‾(v-HLμ‾)K‾=B‾(HL)THL(B‾+P′)(HL)T+R-1u′a=K′(v-HLμ‾)K′=P′(HL)THL(B‾+P′)(HL)T+R-1.
In the new approach there is one set of small scale variables for each
location where observations are available, whereas in GLM14 there are small
scale variables at each coarse grid point. In global atmosphere and climate
models there are typically fewer observations than coarse grid points; when
the observation operator is nonlinear the new formulation is more efficient
because the objective function has fewer degrees of freedom. Another key
difference is in the assumptions that go into the specification of the small
scale background covariance: in GLM14 the small scale variables are tacitly
assumed to vary smoothly over the physical domain, since they are smoothly
interpolated between coarse grid points, whereas in the present approach only
the small scale covariance is assumed to vary smoothly over the domain.
Assimilation experiments
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 P=MK equispaced points with M= 1, 2, and 4;
specifically, observations are taken at ip= 1 +pJ/M for p= 1, …, P.
Observations are either linear, with vp=Yip+εp, or
nonlinear, with vp= (Yip+ 30)2/50+εp. In both cases the
observation errors εp are iid Gaussians with zero mean and
variance 0.1. Observations are assimilated every Δt time units. In
regime I we test Δt= 0.2 and 0.6; for comparison the
decorrelation times of the small scale and large scale variables in this
regime are 0.2 and 0.84. In regime II we test Δt= 0.2 and 0.4,
which are close to the decorrelation times of the small scale and
large scale variables, respectively.
Specification of the background covariance matrix is a crucial aspect of any
3D-Var assimilation system. We consider the simplest possible estimate
B‾=σ2IK where
IK is the K×K identity matrix
and σ2 is a tunable parameter. Assimilation experiments are run
over a range of σ2 and the optimal value is chosen based on rms
(root mean square) errors; the results are very weakly sensitive to σ2 as
long as it is within a factor of 2 of the diagnosed forecast error variance.
Since our observing system includes at least one observation for every
Xk variable, it is less important to build a background covariance
matrix with correlations between the Xk variables.
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
Y of the true system. We view the 3D-Var assimilation as primarily
aimed at estimating the large scale model variables Xk, and error
statistics are tracked only for the large scale variables. We track two
performance metrics for the large scale variables, the time averaged rms
error
rmserror=|X-XSP|2
and the time averaged pattern correlation
Patterncorrelation=XTXSP|X|2|XSP|2
both for the forecast and for the analysis.
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
Xk as a prediction: Xk= 3.8 in regime I and Xk= 3.6
in regime II. The climatological rms error is simply the square root of the
climatological variance: 5.6 in regime I and 5.7 in regime II. The
climatological pattern correlation is the time averaged pattern correlation
between Xk and its uniform climatological mean value: the climatological
pattern correlation is 0.57 in regime I and 0.53 in regime II. If the
forecast has a larger rms error or smaller pattern correlation than the
climatological values, then the forecast is of very limited utility.
Results of the assimilation experiments for regime I. There are
P=MK equispaced observations, assimilated at time intervals of
Δt, and σ2 is the amplitude of the background covariance
matrix. For comparison, the climatological rms error and pattern correlation
are 5.6 and 0.57.
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 K Fourier modes.
For example, when M= 1 there are K observations and the “smoothed
observation” estimate of the Xk variables is simply
Xk≈vp for the linear case and
Xk≈50vp- 30 for the nonlinear case. The rms
errors in the smoothed observation estimate are tracked over the course of
each assimilation experiment, rather than computing climatological values.
The 3D-Var should at a minimum perform better than the smoothed observations.
The results for both regimes are presented in Tables
and in the format Forecast→Analysis. In all cases the errors
decrease as M increases, and the analysis significantly improves over the
forecast.
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
Δt= 0.2, the forecast and analysis for linear and nonlinear
observations both have rms errors smaller than both the climatological error
of 5.6 and the error in the smoothed observation estimate. The nonlinear
observations generate slightly more accurate results than the linear
observations when M= 1, and the linear observations generate slightly
more accurate results for M= 4, but overall the results are similar.
With a longer observation time Δt= 0.6 the results are,
naturally, less accurate. In every case the analysis is more accurate than
both the climatological error and the smoothed observations, but the
forecasts are more accurate than the climatological mean only with
M= 4. With M= 1 and 2, the rms forecast errors are worse than
the climatological error, but the forecast pattern correlations are still a
bit better than the climatological pattern correlation. As with the shorter
observation time, the results are more accurate with the nonlinear
observations.
In regime II the results with the linear and nonlinear observations
are very similar in all cases. With a short observation time Δt= 0.2,
the forecast is always more accurate than the climatological mean, and the
analysis is always more accurate than the smoothed observations. With a
longer observation time Δt= 0.4 the forecasts are no more accurate
than climatology, but the analysis is still more accurate than the smoothed
observations, though at M= 4 the analysis is only slightly more accurate.
Results of the assimilation experiments for regime II. There are
P=MK equispaced observations, assimilated at time intervals of
Δt, and σ2 is the amplitude of the background covariance
matrix. For comparison, the climatological rms error and pattern
correlation are 5.7 and 0.53.
To put the foregoing results into perspective we compare to the results of an
ensemble adjustment Kalman filter EAKF; using an
ensemble of 100 simulations of the true, non-SP model dynamics. The
experiments were run with relatively frequent (Δt= 0.2),
relatively plentiful (M= 4), linear observations in an effort to
obtain the best possible results. Experiments were run with multiplicative
covariance inflation factors from 0 to 20 % and covariance localization
radii of two, four, and six grid points , and optimal results
were obtained with 5 % inflation and a localization radius of four grid
points.
In regime I the rms forecast errors of the Xk variables were 5.1,
decreasing to 4.6 after the analysis; the rms forecast pattern correlation
was 0.61, improving to 0.69 after the analysis. In regime II the rms forecast
errors of the Xk variables were 5.9, decreasing to 5.6 after the
analysis; the rms forecast pattern correlation was 0.53, and remained
essentially unchanged at 0.52 after the analysis.
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 Y, whereas the SP 3D-Var is only estimating the large
scale part. From the point of view of the EAKF the observations are very
sparse, since there is only one observation for every 32 variables, whereas
from the point of view of the SP 3D-Var there are four observations for every
large scale Xk variable. The significant improvement in both cost and
accuracy of using the SP 3D-Var instead of a perfect-model ensemble Kalman
filter underscores the utility of the present approach, though it bears
noting that one should be very hesitant to extrapolate results such as these
to the far more complex setting of SP atmosphere models. Furthermore, whether
or not SP 3D-Var will be more accurate than a state of the art ensemble
Kalman filter in an atmospheric model context is somewhat beside the point
since the goal here is to provide a practical framework for data assimilation
with SP models where no such framework currently exists.
Conclusions
Superparameterization (SP) is a multiscale computational approach that has
been successfully applied to modeling atmospheric dynamics, and that shows
promise for more general applications .
have developed an ensemble Kalman filter framework for use with
SP. However, the standard approach to SP in global atmosphere and climate
models, where small scale nonlinear dynamics are simulated on an array of
periodic domains embedded in the computational grid of a large scale model,
is too computationally demanding for use in an ensemble framework. As a
result, there is at present no practical framework for data assimilation with
SP models. We here develop a 3D-Var variational data assimilation framework
for SP that builds on and modifies the framework of GLM14. The main update to
the GLM14 framework, in addition to using a variational as opposed to
ensemble Kalman filter setting, is that small scale estimates are computed at
locations where observations are taken, rather than at every point of the
large scale model's computational grid. The computational costs of the new
framework are such that it could be used with computationally demanding
global atmosphere and climate SP models.
The data assimilation framework is demonstrated in a new system of ordinary
differential equations based on the two-scale Lorenz-'96 model
. Unlike the two-scale Lorenz-'96 model the new model has only
one set of variables, Yi, and these variables have large and small scale
parts. An SP approximation to the new system is developed, which is perhaps
the simplest idealized model of SP. The new data assimilation framework is
tested in two regimes of the new model, with both linear and nonlinear
observation operators. In regime I the large scale dynamics consist
of a weakly chaotic wave train, with relatively strong small scale
variability superposed. In regime II the large scale dynamics are
more strongly chaotic, and there is less small scale variability. In both
regimes the data assimilation performs as expected (and better than an
ensemble Kalman filter using 100 simulations of the true dynamics), with
increased accuracy as the number of observations increases.
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 e.g.,. In addition, the new
framework removes one of the difficulties associated with development of a
3D-Var framework for large scale models: the small scale simulations in the
multiscale SP computation provide direct information on the small scale
statistics, obviating, or at least simplifying, the need to develop models of
representation error.
I. Grooms designed the research, I. Grooms and Y. Lee carried out the
experiments, and I. Grooms prepared the manuscript.
Acknowledgements
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
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