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Improving Prediction Skill of Imperfect TurbulentModels through
Statistical Response and Information
Theory
Di Qi, and Andrew J. Majda
Courant Institute of Mathematical Sciences
Fall 2016 Advanced Topics in Applied Math
Di Qi, and Andrew J. Majda (CIMS) Improving Prediction Skill of
Imperfect Models Dec. 8, 2016 1 / 61
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Challenges for turbulent dynamical systems
Uncertainty quantification (UQ) deals with the probabilistic
characterization of all thepossible evolutions of a dynamical
system given an initial set of possible states as well asthe random
forcing or parameters.
Turbulent dynamical systems are characterized by a large
dimensional phasespace and high degrees of internal
instability.
Instabilities through energy-conserving nonlinear interactions
result in astatistical steady state that is usually
non-Gaussian.
Accurate quantification for the statistical variability to
general externalperturbations is important in climate change
sciences.
Major Task of this talk:Investigate a concise systematic
framework for measuring and optimizingconsistency and sensitivity
of imperfect dynamical models.
Di Qi, and Andrew J. Majda (CIMS) Improving Prediction Skill of
Imperfect Models Dec. 8, 2016 2 / 61
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General framework for statistical modelingThe system setup will
be a finite-dimensional system of, u ∈ RN , with lineardynamics and
an energy preserving quadratic part
du
dt= L [u] = (L + D) u + B (u,u) + F (t) + σk (t) Ẇk (t;ω) ,
(1)
L∗ = −L; D ≤ 0; u · B (u,u) ≡ 0.NEW STRATEGIES FOR REDUCED-ORDER
MODELS FOR PREDICTING COMPLEX TURBULENT DYNAMICAL SYSTEMS 5
Modeling Stages Math. & Computational Tools
Model SeclectionErgodic Theory
Statistical Measures in Equilibrium
Model CalibrationEmpirical Information Theory
Linear Response TheoryTotal Statistical Energy Equations
Model PredictionAccurate and Efficient Schemes
Numerical Stability Analysis
Rigorous Math Theories
Computational Methods
Figure 1.1. The general strategy for the development of
reduced-order statistical models. Threesequential stages are
required to carry out the reduced-order statistical model, and
rigorous math-ematical theories are combined with numerical
analysis to calibrate model errors and improve theimperfect model
prediction skill.
The model calibration procedure is usually carried out in a
training phase before the prediction, so that theoptimal imperfect
model parameters can be achieved through a careful calibration
about the true higher-orderstatistics. The ideal way is to find a
unified systematic strategy where various external perturbations
can be predictedfrom the same set of optimal parameters through
this training phase. To achieve this, various statistical
theoriesand numerical strategies need to be consulted. Most
importantly, we need to consider the linear response theoryto
calibrate the model responses in mean and variances; and use
empirical information theory to get a balancedmeasure for the error
in the leading order moments. In the final model prediction stage,
the optimal imperfectmodel parameters are applied for the forecast
of various model responses to perturbations. In the
constructionabout numerical models, numerical issues also need
taking into account to make sure numerical stability andaccuracy.
Especially, proper schemes with accuracy order consistent with the
reduced model approximation errorshould be proposed to ensure
optimal performance.
2. Statistical Theory Toolkits for Improving Model Prediction
Skill
In this section we introduce the general theoretical toolkits
that are useful for capturing the key statisticalfeatures in
turbulent systems like (1.1) and improving imperfect model
prediction skill. Despite the complexmodel statistical responses in
each component as the turbulent dynamical system gets perturbed,
there exists asimple and exact statistical energy conservation
principle for the total statistical energy of the system
describingthe overall (inhomogeneous) statistical structure in the
system through a simple scalar dynamical equation. Thetheory is
briefly described in Section 2.1. Then the construction about the
imperfect reduced-order models concernsabout the consistency in
equilibrium (climate fidelity) and the responses to perturbations
(model sensitivity).Equilibrium fidelity should be guaranteed in
the first place so that the reduced-order model will converge to
thetrue equilibrium statistics. To further calibrate the detailed
model sensitivity to perturbations in each statisticalcomponent,
the linear response theory can offer useful quantities to measure
for quantifying the crucial statistics inthe model structure.
Combining with the relative entropy under empirical information
theory, a general information-theoretical framework can be proposed
to tune the imperfect model parameters in a training phase, thus
optimalmodel parameters can be used for model prediction in various
dynamical regimes. We will describe the basic ideasfirst in this
section.
2.1. A statistical energy conservation principle. Despite the
fact that the exact equations for the statisticalmean (1.4) and the
covariance fluctuations (1.5) are not closed equations, there is
suitable statistical symmetry sothat the energy of the mean plus
the trace of the covariance matrix satisfies an energy conservation
principle evenwith general deterministic and random forcing. Here
we briefly introduce the theory developed in [13] about a
totalstatistical energy dynamics for the abstract system (1.1).
Di Qi, and Andrew J. Majda (CIMS) Improving Prediction Skill of
Imperfect Models Dec. 8, 2016 3 / 61
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Outline
1 Turbulent systems with quadratic nonlinearitiesGeneral setups
for turbulent systems with quadratic nonlinearities
2 Statistical dynamical models and closure methodsThe Gaussian
closure method for statistical predictionSingle point statistics
and information barrierImperfect model calibration and empirical
information theoryForecast skill of closure methods for forced
responses in L-96 model
3 Strategies for reduced order models in low dimensional
subspaceLow order models in reduced subspaceForecast skill of
reduced models for forced responses
4 Predicting statistical response in two-layer baroclinic
turbulenceTwo-layer baroclinic turbulence in ocean and atmosphere
regimesLow-dimensional reduced-order models for the two-layer
systemImperfect model prediction skill with 2× 102 resolved modes
for originalsystem with 2× 2562 dimension
Di Qi, and Andrew J. Majda (CIMS) Improving Prediction Skill of
Imperfect Models Dec. 8, 2016 4 / 61
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Outline
1 Turbulent systems with quadratic nonlinearitiesGeneral setups
for turbulent systems with quadratic nonlinearities
2 Statistical dynamical models and closure methodsThe Gaussian
closure method for statistical predictionSingle point statistics
and information barrierImperfect model calibration and empirical
information theoryForecast skill of closure methods for forced
responses in L-96 model
3 Strategies for reduced order models in low dimensional
subspaceLow order models in reduced subspaceForecast skill of
reduced models for forced responses
4 Predicting statistical response in two-layer baroclinic
turbulenceTwo-layer baroclinic turbulence in ocean and atmosphere
regimesLow-dimensional reduced-order models for the two-layer
systemImperfect model prediction skill with 2× 102 resolved modes
for originalsystem with 2× 2562 dimension
Di Qi, and Andrew J. Majda (CIMS) Improving Prediction Skill of
Imperfect Models Dec. 8, 2016 5 / 61
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General setup of turbulent systems with
quadraticnonlinearities
The system setup will be a finite-dimensional system of, u ∈ RN
, with lineardynamics and an energy preserving quadratic part
du
dt= L [u] = (L + D) u + B (u,u) + F (t) + σk (t) Ẇk (t;ω) .
(2)
L being a skew-symmetric linear operator L∗ = −L, representing
the β-effect ofEarth’s curvature, topography etc.
D being a negative definite symmetric operator D∗ = D,
representing dissipativeprocesses such as surface drag, radiative
damping, viscosity etc.
B (u, u) being the quadratic operator which conserves the energy
by itself so that itsatisfies B (u, u) · u = 0.F (t) + σk (t) Ẇk
(t;ω) being the effects of external forcing, i.e. solar forcing,
whichcan be split into a mean component F (t) and a stochastic
component with whitenoise characteristics.
Di Qi, and Andrew J. Majda (CIMS) Improving Prediction Skill of
Imperfect Models Dec. 8, 2016 6 / 61
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Exact statistical moment equationsStatistical mean and
covariance dynamics, u = ū + Zi vi , Rij =
〈Zi Z∗j
〉,
d ū
dt= (L + D) ū + B (ū, ū) + Rij B
(vi , vj
)+ F (t) ,
dR
dt= Lv R + RL
∗v + QF + Qσ .
the linear dynamics operator Lv expressing energy transfers
between the mean field and thestochastic modes (B), as well as
energy dissipation (D), and non-normal dynamics (L)
{Lv}ij =[(L + D) vj + B
(ū, vj
)+ B
(vj , ū
)]· vi .
the positive definite operator Qσ expressing energy transfer due
to external stochasticforcing
{Qσ}ij = v∗i σ∗k σk vj .
the third-order moments expressing the energy flux between
different modes due tonon-linear terms
QF =〈ZmZnZj
〉B (vm, vn) · vi + 〈ZmZnZi 〉B (vm, vn) · vj .
note that energy is still conserved in this nonlinear
interaction part
Tr [QF ] = 2 〈ZmZnZi 〉B (vm, vn) · vi= 2 〈B (Zmvm,Znvn) · Zi vi
〉 = 2
〈B(u′, u′
)· u′〉
= 0.
Di Qi, and Andrew J. Majda (CIMS) Improving Prediction Skill of
Imperfect Models Dec. 8, 2016 7 / 61
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Outline
1 Turbulent systems with quadratic nonlinearitiesGeneral setups
for turbulent systems with quadratic nonlinearities
2 Statistical dynamical models and closure methodsThe Gaussian
closure method for statistical predictionSingle point statistics
and information barrierImperfect model calibration and empirical
information theoryForecast skill of closure methods for forced
responses in L-96 model
3 Strategies for reduced order models in low dimensional
subspaceLow order models in reduced subspaceForecast skill of
reduced models for forced responses
4 Predicting statistical response in two-layer baroclinic
turbulenceTwo-layer baroclinic turbulence in ocean and atmosphere
regimesLow-dimensional reduced-order models for the two-layer
systemImperfect model prediction skill with 2× 102 resolved modes
for originalsystem with 2× 2562 dimension
Di Qi, and Andrew J. Majda (CIMS) Improving Prediction Skill of
Imperfect Models Dec. 8, 2016 8 / 61
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Statistical Accurate Closure ModelsThe true statistical
model
d ū
dt= (L + D) ū + B (ū, ū) + Rij B (vi , vj ) + F (t) , ū ∈ RN
,
dR
dt= Lv (ū) R + RL
∗v (ū) + QF + Qσ, R ∈ RN×N .
QF ,ij = 〈ZmZnZj〉B (vm, vn) · vi + 〈ZmZnZi 〉B (vm, vn) · vj ,
trQF = 0.
Basic Idea:The hierarchical moment equations will never be
closed by adding higherorder momentsEquations for the stochastic
coefficients
dZidt
= Zj [(L + D) vj + B (ū, vj ) + B (vj , ū)] · vi+ [B (u′,u′)−
Rjk B (vj , vk )] · vi + Ẇkσk · vi .
Replace nonlinear interaction term by linear damping and
Gaussian noise
Statistical closure models
B (u′,u′) · vi =∑
Bij Zi Zj −→ −dM,i (R) Zi + σM,i Ẇi .Di Qi, and Andrew J. Majda
(CIMS) Improving Prediction Skill of Imperfect Models Dec. 8, 2016
9 / 61
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Statistical Accurate Closure ModelsThe reduced-order
approximation ūM ∈ RM , M � N
d ūMdt
= (L + D) ūM + B (ūM , ūM ) + RM,ij B (vi , vj ) + F,
dRMdt
= Lv RM + RM L∗v + Q
MF + Qσ,
Basic Idea:The hierarchical moment equations will never be
closed by adding higherorder momentsEquations for the stochastic
coefficients
dZidt
= Zj [(L + D) vj + B (ū, vj ) + B (vj , ū)] · vi+ [B (u′,u′)−
Rjk B (vj , vk )] · vi + Ẇkσk · vi .
Replace nonlinear interaction term by linear damping and
Gaussian noise
Statistical closure models
B (u′,u′) · vi =∑
Bij Zi Zj −→ −dM,i (R) Zi + σM,i Ẇi .
Di Qi, and Andrew J. Majda (CIMS) Improving Prediction Skill of
Imperfect Models Dec. 8, 2016 9 / 61
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Quasilinear Gaussian (QG) closure
The simplest closure scheme for the moment problem is to
completely neglect thethird-order moments, i.e. set QF = 0.
d ū
dt= (L + D) ū + B (ū, ū) + Rij B (vi , vj ) + F (t) , (3)
dR
dt= Lv R + RL
∗v +��QF + Qσ. (4)
persistent instabilities that would cause uncontrollable growth
of the unstablemodes.
for L-96 system the QG closure scheme avoids blow-up.Consider
homogeneous statistical solutions of QG closure for the L-96
system: the meanū (t) is a time varying constant; the covariance
multiplier is diagonal in Fourier spaceR = rjδij ; the linear
operator is diagonal Fourier multiplier Lv = diag
(lj),
drj
dt= 2Relj (ū) rj , lj (ū) = (exp (2πij/J)− exp (−4πij/J)) ū −
1
dū
dt= −ū + F +
20∑j=0
rjB(vj , vj
)· v0 + B
(vj , vj
)∗ · v02
.
Di Qi, and Andrew J. Majda (CIMS) Improving Prediction Skill of
Imperfect Models Dec. 8, 2016 10 / 61
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The statistical steady state solution requires rj,∞ = 0 unless
Relj (ū) = 0, that is,
maxj
Relj (ū) = 0
⇒ ū(
cos2πj
J− cos 4πj
J
)− 1 = 0
⇒ ūcr =1
cos 2πjcrJ − cos4πjcr
J
≈ 0.8944, jcr = 8
⇒ rj 6= 0, j = 8; rj = 0, ∀j 6= 8.
Correspondingly
ūcr = F + r8B (v8, v8) · v0 + B (v8, v8)∗ · v0
2
= F + r8
(cos
4πjcrJ− cos 2πjcr
J
)< F .
Di Qi, and Andrew J. Majda (CIMS) Improving Prediction Skill of
Imperfect Models Dec. 8, 2016 11 / 61
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Number of positive eigenvalues of Lv (ū) for L-96 system with
respect to themagnitude of the mean field ū. No matter how large
the external forcing F is, theQG closure method always converges to
the unique critical value ūcr , satisfyingmarginal linear
stability.
0 0.5 1 1.5 2 2.5 3 3.5 40
1
2
3
4
5
6
7
8
9
10
Num
ber o
f pos
itive
eig
enva
lues
F=6 F=8 F=16F=5 F=32
Equilibrium point for QG models
Figure 2: Number of positive eigenvalues of Lv (ū) for L-96
with respect to the magnitude of themean field ū. The red dashed
lines indicate exact equilibrium points for di§erent value of the
forcingparameter F. The green solid lines indicate equilibrium
points for the DO UQ scheme for N = 10.
linearized dynamics - described by Lv (ū) - which will magnify
the initial uncertainty. In Figure 2we present the number of
unstable wavenumbers, i.e. the number of eigenvalue pairs with
positivereal part for the linearized matrix Lv (ū) , with respect
to the value of the steady state mean field(note that spatial
homogeneity implies spatially constant mean field). In the same
plot we showthe with dashed lines the steady state value of the
mean field for specific values of the forcingparameter F. Based on
the presence of persistent positive eigenvalues in the steady-state
we have(for su¢ciently large F ) the following energy cycle (Figure
3):
1. Energy from the external excitation F leads to the growth of
the mean field energy 12 ū.ū(equation (2)).
2. The important magnitude of ū leads to the activation of
positive eigenvalues of Lv (ū) (seeFigure 2) that essentially
absorb energy from the mean field and transform it to variance
forthe stochastic modes that are associated with this process.
3. The nonlinear conservative term B(u0,u0) absorbs part of this
energy, transferring it to thestable stochastic modes. It acts as
dissipative mechanism for the unstable modes (balancingtheir
positive eigenvalues) and external noise for the stable modes
bringing all of them into astatistical equilibrium.
6
Di Qi, and Andrew J. Majda (CIMS) Improving Prediction Skill of
Imperfect Models Dec. 8, 2016 12 / 61
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Energy flow in the quadratic systemsStatistical mean and
covariance dynamics
d ū
dt= (L + D) ū + B (ū, ū) + Rij B
(vi , vj
)+ F (t) ,
dR
dt= Lv R + RL
∗v + QF + Qσ .
In imperfect closure models, replace
QF =〈ZmZnZj
〉B (vm, vn) · vi + c.c.
−→ QMF = QMF− + Q
MF + = −2dM,k (R) rk + σ
2M,k (R)
Figure 3: Energy flow in the L-96 system. Energy flows from the
mean field to the linearly unstablemodes and then redistributed
through nonlinear, conservative terms to the stable modes.
Redarrows denotes dissipation, while the dashed box indicates terms
that conserve energy.
4. The stable modes receive energy from the unstable ones
through the nonlinear conservativeterms. A portion of this energy
is dissipated and the rest is subsequently returned to the
meanthrough the modes with negative eigenvalues. All modes
including the mean flow dissipateenergy through the negative
definite part of the linearized dynamics.
This cycle of energy in the L-96 model is representative of any
general model that contains i)unstable linearized modes whose
stability depends on the mean field energy level (i.e. that
theyabsorb energy from the mean field), ii) stable modes, and iii)
nonlinear conservative terms thattransfer energy between the modes
and through this transfer the system is reaching an
equilibrium.This structure is ubiquitous in turbulent systems in
the atmosphere and ocean with forcing anddissipation [26, 22, 1, 2]
as well as in fluid flows with lower dimensional attractors [23].
However,there are also examples of idealized truncated geophysical
flows without dissipation and forcingwith a Gaussian statistical
equilibrium where the linear operator at the climate mean state is
stablewhile the system has many positive Lyapunov exponents
[19].
7
Figure 4: Energy flow in the MQG UQ scheme. Energy flows from
the mean field to the linearlyunstable modes and then redistributed
through empirical, conservative fluxes to the stable modes.Red
arrows denotes dissipation, while the dashed box indicates terms
that conserve energy.
we substitute the nonlinear conservative mechanism by a
conservative pair of positive and negativeenergy fluxes having the
form of additional damping for the unstable modes and additive
noisefor the stable modes (Figure 4). Note that this additional
damping is chosen so that the unstableeigenvalues of the original
linearized dynamics are guaranteed to have zero real part in the
statisticalsteady state. In that sense this is the minimal amount
of additional damping and noise required toachieve marginal
stability (non-positive eigenvalues) of the correct steady state
statistics. Thus, wehave a minimally changed model compared to the
original equation that admits the correct steadystate statistics as
a stable equilibrium stage. In the next subsections we will see
that for numericalreasons it is required to add a small amount of
additional damping (and noise) so that the correctstatistical
steady state is not just marginally stable but it is associated
with eigenvalues havingpurely negative real part. Moreover, in the
transient phase of the dynamics the intensity of thenonlinear
energy fluxes should depend on the energy level of the system and
to this end we willapply a scaling factor to the additional damping
and noise (that represent the nonlinear energyfluxes) which will
take into account this dependence.Note that all of the required
fluxesQ−F1, Q
+F1 are evaluated explicitly from available information
involving the linear operator, Lv (ū1), and the covariance
matrix, R1 in a statistical steady state.In addition, since the
nonlinear flux model is kept separate from the unmodified linear
dynamics, itexpresses an inherent property of the system, a direct
link between second and third-order statisticalmoments in the same
spirit that Karman-Howarth equation [10] does for isotropic
turbulence.
11
Di Qi, and Andrew J. Majda (CIMS) Improving Prediction Skill of
Imperfect Models Dec. 8, 2016 13 / 61
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L-96 system with homogeneous assumptionThe homogeneous L-96
system
duj
dt= uj−1
(uj+1 − uj−2
)− duj + F , j = 0, 1, ..., J − 1,
satisfies 〈ui1 ui2 · · · uin
〉=〈ui1+l ui2+l · · · uin+l
〉, ∀l .
Fourier basis naturally becomes eigenfunctions for the L-96
operator, the moment equations aresimplified
Moment equations for homogeneous L-96
dū (t)
dt= −d (t) ū (t) +
1
J
J/2∑k=−J/2+1
rk (t) Γk + F (t) ,
drk (t)
dt= 2 [−Γk ū (t)− d (t)] rk (t) + QF ,kk , k = 0, 1, ...,
J/2.
Note that Γk = cos4πk
J− cos 2πk
J, and the nonlinear flux QF becomes diagonal
QF ,kk′ = 2
(1
J
)1/2∑m
Re{〈ZmZ−m−k Zk〉
(e−2πı
2m+kJ − e2πı
m+2kJ
)}δkk′ ,
with energy conservation trQF = 0.
Di Qi, and Andrew J. Majda (CIMS) Improving Prediction Skill of
Imperfect Models Dec. 8, 2016 14 / 61
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Models with consistent equilibrium single point statistics
QMF ,kk = QMF−,kk + Q
MF +,kk = −2dM,k (R) rk + σ
2M,k (R) .
Gaussian closure 1 (GC1-1pt): let
dM,k (R) = dM ≡ const., σ2M,k (R) = σ2M ≡ const.,
QGC1F = − (dM R + RdM ) + σ2M I ; (5)Gaussian closure 2
(GC2-1pt): let
dM,k (R) = �MJ
2
(trR)1/2
(trR∞)3/2≡ �M d̄ , σ2M,k (R) = �M
(trR)3/2
(trR∞)3/2
QGC2F = −�M(d̄R + Rd̄
)+ �M
(trR)3/2
(trR∞)3/2
I . (6)
Note that GC1-1pt includes parameters(dM , σ
2M
)and the nonlinear energy
trQGC1F = −2dMtrRM + Jσ2M may not be conserved, while GC2-1pt
has one parameter�M and nonlinear energy conservation is enforced
by construction trQ
GC2F = 0.
Single point statistics consistency can be fulfilled through
tuning the control parameters.But these models are calibrated by
ignoring spatial correlations and a natural informationbarrier is
present which cannot be overcome by these imperfect models.
Di Qi, and Andrew J. Majda (CIMS) Improving Prediction Skill of
Imperfect Models Dec. 8, 2016 15 / 61
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Equation for statistical energy
The statistical mean and variance at each individual grid point
focus on the single point meanand variance of the system (and
ignore the cross-correlation between different grids)
ū1pt =1
J
J−1∑j=0
uj = ū, R1pt =1
J
J/2∑k=−J/2+1
rk =1
JtrR.
By multiplying ū on both sides of the mean equation, and
summing up all the modes in thevariance equation
dū2
dt= −2dū2 + 2ūF + 2
∑k
Γk rk ū.
dtrR
dt= 2
(−∑
k
Γk rk ū
)− 2dtrR + trQF .
It is convenient to define the statistical energy as
E (t) =1
2ū2 +
1
2trR.
Thus the corresponding dynamical equation for the energy E can
be easily derived as
dE
dt= −2d (t) E + F · ū +
1
2trQF = −2E + F · ū,
with symmetry of nonlinear energy conservation assumed.
Di Qi, and Andrew J. Majda (CIMS) Improving Prediction Skill of
Imperfect Models Dec. 8, 2016 16 / 61
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Realizability of 1-point statistics
In a similar way, we can calculate EM from imperfect model. With
the same set ofassumptions, the responses between perfect and
imperfect model satisfy
d
dt(δE − δEM ) = −2d (δE − δEM ) + F · (δū − δūM ) .
Proposition
Consider a system with homogenous statistical solution and an
imperfect closuremodel with flux QMF satisfying symmetry of
nonlinear energy conservation.Statistical equilibrium fidelity for
one point statistics of the variancetrRM,∞ = trR∞ is satisfied if
the same mean state ūM,∞ = ū∞ is achieved atequilibrium.
Furthermore, with equilibrium consistency, we can successfully
predictthe response for the one point statistics using the
imperfect model‖trRδ,M − trRδ‖ = O
(δ2)
if we can predict the mean response successfully
‖ūδ,M − ūδ‖ = O(δ2).
Di Qi, and Andrew J. Majda (CIMS) Improving Prediction Skill of
Imperfect Models Dec. 8, 2016 17 / 61
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Information barrier in 1-point statistics
the information distance between the truth and imperfect model
prediction has the form
P(π, πM1pt
)= [S (πG )− S (π)] + P
πG , J−1∏j=0
πG1pt,j
+ J−1∑j=0
P(πG1pt,j , π
M1pt,j
).
The first part on the right hand side of is the inherent
information barrier in Gaussianapproximation. Using spectral random
fields
uG = ū +
J/2∑k=−J/2+1
R1/2k e
ikxj Ŵk , uG1pt = ū +
J/2∑k=−J/2+1
(∑J−1j=0 Rj
J
)1/2e ikxj Ŵk .
PropositionThe information barrier between the Gaussian random
field and the uncorrelated one pointstatistics random field is
given by
P
πG , J−1∏j=0
πG1pt
= J log det
(∑J−1j=0 Rj/J
)(∏J−1
j=0 det Rj
)1/J ∼ R−11pt (σmax − σmin)2 .
Di Qi, and Andrew J. Majda (CIMS) Improving Prediction Skill of
Imperfect Models Dec. 8, 2016 18 / 61
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Hierarchy of closure models with consistent equilibrium foreach
mode
QMF ,kk = QMF−,kk + Q
MF +,kk = −2dM,k (R) rk + σ
2M,k (R) .
Gaussian closure 1 (GC1): let
dM,k (R) = dM,k , σ2M,k (R) = σ
2M , dM,k = [−Γk ū∞ − d] + σ
2M/2rk,∞; (7)
Gaussian closure 2 (GC2): let
dM,k (R) = �1,k d̄ , σ2M,k (R) = �M
(trR)3/2
(trR∞)3/2
,
�1,k =2 [−Γk ū∞ − d] rk,∞ + �M
Jrk,∞/trR∞, d̄ =
J
2
(trR)1/2
(trR∞)3/2
. (8)
Above dM,k or �1,k is chosen so that the system has the same
equilibrium mean ū∞ andvariance rk,∞ as the true model, therefore
ensuring equilibrium consistency by finding thesteady state
solutions through simple algebraic manipulations.Modified Gaussian
closure (MQG): let
dM (R) =f (R)
f (R∞)N∞, σ
2M (R) = −
trQMQGF−
trQMQGF +,∞
[(Γk ū∞ + d) rk,∞δI+ + qs
],
with
N∞,kk = [Γk ū∞ + d] δI− −1
2qs r−1k,∞.
Above I− represents the unstable modes with Γk ū∞ + d > 0
while I+ is the stable oneswith Γk ū∞ + d ≤ 0.
Di Qi, and Andrew J. Majda (CIMS) Improving Prediction Skill of
Imperfect Models Dec. 8, 2016 19 / 61
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Model calibration blending statistical response andinformation
theory
Accurate modeling about the model sensitivity to various
external perturbationsrequires the imperfect reduced-order models
to correctly reflect the true system’s“memory” about its previous
states.
the linear response operator can characterize the model
sensitivity involvingthe nonlinear effects in the system regardless
of the specific forms of theexternal perturbations.
empirical information theory can be used as the distance between
these twooperators to calculate the unbiased and invariant measure
for modeldistributions.
Di Qi, and Andrew J. Majda (CIMS) Improving Prediction Skill of
Imperfect Models Dec. 8, 2016 20 / 61
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Empirical information theoryEmpirical information theory: A
natural way to measure the lack of informationin one probability
density from the imperfect model, πM , compared with the
trueprobability density, π, is through the relative entropy, given
by
P(π, πM
)=
∫π log
π
πM.
This functional on probability densities has two attractive
features as a metric:
P (p, q) ≥ 0 with equality if and only if p = q;P (p, q) is
invariant under general nonlinear changes of variables.
Model optimization: Consider a class of imperfect models M. The
bestimperfect model for the coarse-grained variable u is the M∗ ∈M
so that
P(π, πM
∗)
= minM∈M
P(π, πM
). (9)
Also, actual improvements in a given imperfect model with
distribution πM (u)resulting in a new πMpost (u) should result in
improved information for the perfect
model, so that P(π, πMpost
)≤ P
(π, πM
).
Di Qi, and Andrew J. Majda (CIMS) Improving Prediction Skill of
Imperfect Models Dec. 8, 2016 21 / 61
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Empirical information theory
The actual model density πL (u) only reflects the best unbiased
estimate of theperfect model given the L measurements ĒL.
P(π, πML′
)= P (π, πL) + P
(πL, π
ML′)
= [S (πL)− S (π)] + P(πL, π
ML′), for L′ ≤ L.
The entropy difference, S (πL)− S (π), precisely measures an
intrinsic error fromthe L measurements of the perfect system.
Practical setup for calibration of contemporary models involves
only mean andcovariance. If the density functions π, πM involve
only the first two moments,
P(π, πM
)=
1
2(ū− ūM )T R−1 (ū− ūM ) +
1
2
(tr(RR−1M
)− J − log det
(RR−1M
)).
First term is the signal, reflecting the model error in the mean
but weighted by theinverse of the covariance R−1, while the second
term, the dispersion, involves onlythe model error covariance
ratio, RR−1M .
Di Qi, and Andrew J. Majda (CIMS) Improving Prediction Skill of
Imperfect Models Dec. 8, 2016 22 / 61
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Theories for improving imperfect model prediction skill
Linear response theory: The linear response of a system to small
perturbationscan be predicted by observing appropriate statistics
of the system in equilibriumwithout the need of applying any
perturbations.
Assume the perfect model with equilibrium measure πeq, and
external perturbation
ut = f (u) , fδ = f + δf ′ (t) , δf ′ = δw (u) f (t) .
For any functional A (u)
EδA (u) = Eeq (A) + δE ′A + O(δ2). (10)
Eeq (A) =
∫A (u)πeq (u) du, δE
′A =
∫ t
0
RA (t − s) δf ′ (s) ds, (11)
where RA (t) is called the linear response operator.
-
Linear response operator RA (t)The linear operator RA (t) can be
calculated via the correlations in the unperturbed climate
RA (t) = 〈A (u (t + s)) B (u (s))〉 ,
with the special nonlinear functional perturbed by the change δw
(u) f (t)
B (u) ≡LFPπeq (u)
πeq= −
divu (wπeq)
πeq.
The problem in calculating the leading order response above is
that the linear response operatorRA (t) is expensive to calculate
for general systems.
kicked response1:
π |t=0= πeq (u− δu) = πeq − δu · ∇πeq + O(δ2).
For δ small enough, the linear response operator RA (t) can be
calculated bysolving the unperturbed system with a perturbed
initial distribution
δRA (t) ≡ δu · RA =∫
A (u) δπ′ + O(δ2).
1Majda, Abramov, and Grote, Information theory and stochastics
for multiscale nonlinearsystems vol. 25.
Di Qi, and Andrew J. Majda (CIMS) Improving Prediction Skill of
Imperfect Models Dec. 8, 2016 24 / 61
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Link between statistical equilibrium fidelity and
forecastingskill
Given the optimal model for the unperturbed climate πeq, how can
we assess the error in theclimate change prediction P
(πδ, π
Mδ
)based on the unperturbed climate?
Under assumptions with diagonal covariance matrices R = diag (Rk
) and equilibrium modelfidelity P
(πG , π
MG
)= 0, the relative entropy between perturbed model density πMδ
and the true
perturbed density πδ with small perturbation δ can be expanded
as
P(πδ, π
Mδ
)= S
(πG ,δ
)− S (πδ)
+1
2
∑k
(δūk − δūM,k
)R−1k
(δūk − δūM,k
)+
1
4
∑k
R−2k(δRk − δRM,k
)2+ O
(δ3).
Here in the first line S(πG ,δ
)− S (πδ) is the intrinsic error from Gaussian approximation of
the
system. Rk is the equilibrium variance in k-th component, and
δūk and δRk are the linearresponse operators for the mean and
variance in k-th component. (Majda & Gershogorin,
PNAS,2011)
Di Qi, and Andrew J. Majda (CIMS) Improving Prediction Skill of
Imperfect Models Dec. 8, 2016 25 / 61
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Optimization in the training phase
To summarize, consider a class of imperfect models, M. The
optimal modelM∗ ∈M that ensures best information consistent
responses is characterized withthe smallest additional information
in the linear response operator RA, such that
∥∥∥P(
pδ, pM∗
δ
)∥∥∥L1([0,T ])
= minM∈M
∥∥P(pδ, p
Mδ
)∥∥L1([0,T ])
,
pMδ can be achieved through a kicked response procedure in the
trainingphase compared with the actual observed data pδ in
nature;
the information distance between perturbed responses P(pδ, p
Mδ
)can be
calculated through the expansion formula;
the information distance P(pδ (t) , p
Mδ (t)
)is measured at each time instant,
so the entire error is averaged under the L1-norm inside a time
window [0,T ].
Di Qi, and Andrew J. Majda (CIMS) Improving Prediction Skill of
Imperfect Models Dec. 8, 2016 26 / 61
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Lorenz 96 system
The Lorenz 96 model was introduced to mimick the large-scale
behavior of themid-latitude atmosphere around a circle of constant
latitude. The model is adiscrete periodic system described by the
equations
dujdt
= uj−1 (uj+1 − uj−2)− duj + F , j = 0, 1, ..., J − 1. (12)
with J = 40 the number of grids and Fi the deterministic
forcing. The quadraticpart conserves energy, that is, B (u, u) · u
= 0. The L-96 model offers morecontrollable, simplified scenarios
but still mimicking the complex features of thevastly more complex
true turbulent system..
FLinear Analysis Mean Statistics
kbegin kend Reω ū E (ū) Ep
5 5 11 0.08 1.6344 53.4276 110.0132
6 4 12 0.0998 2.0127 81.0260 160.5063
8 4 12 0.1004 2.3418 109.6825 265.0064
16 4 12 0.0876 3.0869 190.5900 797.2113
Di Qi, and Andrew J. Majda (CIMS) Improving Prediction Skill of
Imperfect Models Dec. 8, 2016 27 / 61
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F = 5
space
10 20 30 40
tim
e
0
5
10
15
20
-3
-2
-1
0
1
2
3
4
5
6
7
wavenumber
0 5 10 15 20P
erc
enta
ge o
f E
nerg
y %
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4F = 5
Re(uk)
-60 -40 -20 0 20 40 60
dis
trib
utio
n function
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
pdfs for Re(uk)
mode 7mode 8
time
0 5 10 15 20 25 30
corr
ela
tion function
-0.5
0
0.5
1Autocorrelation function for F = 5
F = 8
space
10 20 30 40
tim
e
0
5
10
15
20
-6
-4
-2
0
2
4
6
8
10
12
wavenumber
0 5 10 15 20
Perc
enta
ge o
f E
nerg
y %
0.02
0.04
0.06
0.08
0.1
0.12F = 8
Re(uk)
-150 -100 -50 0 50 100 150
dis
trib
ution function
0
0.005
0.01
0.015
0.02
pdfs for Re(uk)
mode 8mode 7
time
0 5 10 15 20
corr
ela
tion function
-0.2
0
0.2
0.4
0.6
0.8
1Autocorrelation function for F = 8
Figure: Numerical solutions of L-96 model in space-time through
MC simulations for weaklychaotic (F = 5), and strongly chaotic (F =
8) regime.
Di Qi, and Andrew J. Majda (CIMS) Improving Prediction Skill of
Imperfect Models Dec. 8, 2016 28 / 61
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Tuning imperfect model in the training phase
50 100 150 200
σM
2
0.1
0.2
0.3
0.4
0.5
P(δπ
,δπ
M)
tuning model, GC1
50 100 150 200
ǫM
0.1
0.2
0.3
0.4
0.5
P(δπ
,δπ
M)
tuning model, GC2
0 0.2 0.4 0.6 0.8
qsM
0.1
0.12
0.14
0.16
0.18
0.2
P(δπ
,δπ
M)
tuning model, MQG
(a) Information distance in response operator
0 0.5 1 1.5 2 2.5 3
time
-0.5
0
0.5
1
Ru
Linear response for the mean (optimal parameter)
truthGC1GC2MQG
0 0.5 1 1.5 2 2.5 3
time
-50
0
50
100
150
200
Rσ
2
Linear response for total variance (optimal parameter)
(b) Linear response operators for the mean and total
variance
Figure: Training imperfect models in the training phase with
full dimensional models by tuningmodel parameters. Time-averaged
information distances in linear response operators for the
three closure models GC1, GC2, and MQG as the model parameter
changes are shown in the
first row (the point where the information error is minimized is
marked with a circle).
Di Qi, and Andrew J. Majda (CIMS) Improving Prediction Skill of
Imperfect Models Dec. 8, 2016 29 / 61
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Prediction skill of closure methods in L-96 system
0 5 10 15 20 25 302.3
2.4
2.5State of the Mean
truthGC1GC2MQG
0 5 10 15 20 25 30
time
500
600
700Total variance
0 5 10 15 20 25 302.3
2.4
2.5State of the Mean
truthGC1GC2MQG
0 5 10 15 20 25 30
time
500
600
700Total variance
Model with non-optimal parameterModel with optimal parameter
Figure: Upward ramp-type forcing: imperfect model predictions in
mean state and totalvariance for the closure methods GC1
(dotted-dashed blue), GC2 (dashed green), andMQG (solid red).
Di Qi, and Andrew J. Majda (CIMS) Improving Prediction Skill of
Imperfect Models Dec. 8, 2016 30 / 61
-
Prediction skill of closure methods in L-96 system
0 5 10 15 20 25 302.2
2.25
2.3
2.35State of the Mean
truthGC1GC2MQG
0 5 10 15 20 25 30
time
400
450
500
550Total variance
0 5 10 15 20 25 302.2
2.25
2.3
2.35State of the Mean
truthGC1GC2MQG
0 5 10 15 20 25 30
time
400
450
500
550Total variance
Model with non-optimal parameterModel with optimal parameter
Figure: Downward ramp-type forcing: imperfect model predictions
in mean state andtotal variance for the closure methods GC1
(dotted-dashed blue), GC2 (dashed green),and MQG (solid red).
Di Qi, and Andrew J. Majda (CIMS) Improving Prediction Skill of
Imperfect Models Dec. 8, 2016 30 / 61
-
Prediction skill of closure methods in L-96 system
0 5 10 15 202
2.2
2.4
2.6State of the Mean
truthGC1GC2MQG
0 5 10 15 20
time
400
500
600
700Total variance
0 5 10 15 202
2.2
2.4
2.6State of the Mean
truthGC1GC2MQG
0 5 10 15 20
time
400
500
600
700Total variance
Model with non-optimal parameterModel with optimal parameter
Figure: Periodic forcing: imperfect model predictions in mean
state and total variance forthe closure methods GC1 (dotted-dashed
blue), GC2 (dashed green), and MQG (solidred).
Di Qi, and Andrew J. Majda (CIMS) Improving Prediction Skill of
Imperfect Models Dec. 8, 2016 30 / 61
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Prediction skill of closure methods in L-96 system
0 5 10 15 20
2.2
2.4
2.6State of the Mean
truthGC1GC2MQG
0 5 10 15 20
time
400
500
600
Total variance
0 5 10 15 20
2.2
2.4
2.6State of the Mean
truthGC1GC2MQG
0 5 10 15 20
time
400
500
600
Total variance
Model with non-optimal parameterModel with optimal parameter
Figure: Random forcing: imperfect model predictions in mean
state and total variance forthe closure methods GC1 (dotted-dashed
blue), GC2 (dashed green), and MQG (solidred).
Di Qi, and Andrew J. Majda (CIMS) Improving Prediction Skill of
Imperfect Models Dec. 8, 2016 30 / 61
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Information errors of closure methods in L-96 system
0 10 20 308
8.2
8.4
8.6
8.8Forcing 1: upward ramp-type forcing
0 10 20 300
0.1
0.2
0.3
0.4total information error, GC1
optimal
non-optimal
0 10 20 300
0.02
0.04
0.06
0.08total information error, GC2
optimal
non-optimal
0 10 20 300
0.005
0.01
0.015
0.02total information error, MQG
optimal
non-optimal
(a) climate change forcing
0 5 10 15 207
7.5
8
8.5
9Forcing 2: periodic forcing
0 5 10 15 200
0.1
0.2
0.3total information error, GC1
optimal non-optimal
0 5 10 15 200
0.02
0.04
0.06total information error, GC2
optimal non-optimal
0 5 10 15 200
0.02
0.04
0.06total information error, MQG
optimal non-optimal
(b) periodic forcing
time
0 5 10 15 206
7
8
9Forcing 3: random forcing
time
0 5 10 15 200
0.2
0.4
0.6total information error, GC1
optimal
non-optimal
time
0 5 10 15 200
0.02
0.04
0.06
0.08total information error, GC2
optimal
non-optimal
time
0 5 10 15 200
0.02
0.04
0.06total information error, MQG
optimal
non-optimal
(c) random forcing
-
Outline
1 Turbulent systems with quadratic nonlinearitiesGeneral setups
for turbulent systems with quadratic nonlinearities
2 Statistical dynamical models and closure methodsThe Gaussian
closure method for statistical predictionSingle point statistics
and information barrierImperfect model calibration and empirical
information theoryForecast skill of closure methods for forced
responses in L-96 model
3 Strategies for reduced order models in low dimensional
subspaceLow order models in reduced subspaceForecast skill of
reduced models for forced responses
4 Predicting statistical response in two-layer baroclinic
turbulenceTwo-layer baroclinic turbulence in ocean and atmosphere
regimesLow-dimensional reduced-order models for the two-layer
systemImperfect model prediction skill with 2× 102 resolved modes
for originalsystem with 2× 2562 dimension
Di Qi, and Andrew J. Majda (CIMS) Improving Prediction Skill of
Imperfect Models Dec. 8, 2016 32 / 61
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Reduced-order models in low dimensional subspace –
Meanevolution
Use s orthonormal eigenvectors of full covariance matrix R∞ to
be {vi}si=1 (e.g.EOF modes). Form projection matrix P = [v1, · · ·
, vs ], and the reducedcovariance
Rs = P∗RP.
To compensate for the influence of quadratic terms only
partially modeled due totruncation, introduce additional forcing
G∞
d ū
dt= (L + D) ū + B (ū, ū) +
s∑
i,j=1
Rs,ij B (vi , vj ) + F + G∞.
Di Qi, and Andrew J. Majda (CIMS) Improving Prediction Skill of
Imperfect Models Dec. 8, 2016 33 / 61
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Reduced-order models in low dimensional subspace –
Meanevolution
d ū
dt= (L + D) ū + B (ū, ū) +
s∑
i,j=1
Rs,ij B (vi , vj ) + F + G∞.
Determine G∞ through statistical steady-state information for
covariance andmean. The equilibrium equation is:
G∞ = − (L + D) ū∞ − B (ū∞, ū∞)−s∑
i,j=1
Rs∞,ij B (vi , vj )− F,
where Rs∞ = P∗R∞P, thereby guaranteeing that ū∞ is the steady
state of the
truncated equation.
Di Qi, and Andrew J. Majda (CIMS) Improving Prediction Skill of
Imperfect Models Dec. 8, 2016 34 / 61
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Reduced-order models in low dimensional subspace –Covariance
evolution
The exact equation for the reduced covariance is
dRsdt
= Lv ,sRs + RsL∗v ,s + QF ,s .
QF ,s contains nonlinear dynamics between modes as well as
linear dynamicsmissing from truncation. Note in general we no
longer have conservationproperty, Tr [QF ,s ] 6= 0.Still we can use
steady state information
QF ,s∞ = −Lv ,sRs∞ − Rs∞L∗v ,s .
As done in MQG, split QF ,s∞ into positive/negative definite
parts as anoise/damping pair Q+F ,s∞, Q
−F ,s∞.
Di Qi, and Andrew J. Majda (CIMS) Improving Prediction Skill of
Imperfect Models Dec. 8, 2016 35 / 61
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Low order models in reduced subspaceSolve the statistics in a
low-order subspace spanned by leading EOFs, vk , k = 0, 1, · · · ,
s,{
dūdt
= −d (t) ū (t) + 1J
∑|k|≤s rk (t) Γk + F (t) + G ,
drkdt
= 2 [−Γk ū (t)− d (t)] rk (t) + QMF ,kk (trR) , k = 0, 1, ...,
s.
First approach:
I Get the unresolved variances in the mean dynamics from the
equilibrium statistics
G =trRs
trRs,∞G∞, G∞ = dū∞ −
1
J
∑|k|≤s
rk,∞Γk + F ;
I Scale the nonlinear flux QMF ,kk (trRs ) by only resolved
modes trRs =∑|k|≤s rk .
Further improvements:
I Get unresolved variances from linear first order correction
from FDT
rk,un ∼ rk,∞ + δr ′k = rk,∞ +∫ t
0Rrk (t − s) δF
′ (s) ds;
I Get the total variance from the scalar energy equation
dE
dt= −2d (t) E + F (t) · ū, trR = 2E (t)− ū2.
Di Qi, and Andrew J. Majda (CIMS) Improving Prediction Skill of
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Tuning parameters of reduced models in a training
regimeCOMPARISON OF PREDICTION SKILL OF REDUCED-ORDER GAUSSIAN
CLOSURE MODELS 7
0 50 100 150 2000
0.1
0.2
0.3
0.4GC1 F = 8, Information distance with reduced models
corrected model
original model
σ2
0 50 100 150 2000
0.05
0.1
0.15GC1, corrected model
signal errordispersion error
0 50 100 150 2000
0.1
0.2
0.3
0.4GC1, original model
signal errordispersion error
(a) GC1, optimal �21 = 91,�22 = 61
0 50 100 1500.02
0.04
0.06
0.08
0.1
0.12GC2 F = 8, Information distance with reduced models
corrected model
original model
ϵ
0 50 100 1500.01
0.02
0.03
0.04
0.05GC2, corrected model
signal errordispersion error
0 50 100 1500.02
0.04
0.06
0.08
0.1
0.12GC2, original model
signal errordispersion error
(b) GC2, optimal ✏1 =200, ✏2 = 20
0 0.2 0.4 0.6 0.8 10.02
0.04
0.06
0.08
0.1
0.12MQG F = 8, Information distance with reduced models
corrected model
original model
ds0 0.2 0.4 0.6 0.8 1
0
0.01
0.02
0.03
0.04
0.05
0.06ROMQG, corrected model
signal errordispersion error
0 0.2 0.4 0.6 0.8 10
0.02
0.04
0.06
0.08
0.1
0.12ROMQG, original model
signal errordispersion error
(c) ROMQG, optimalds1 = 0.455, ds2 = 0.145
Figure 2.1. Tuning parameters in the training regime for GC1,
GC2, and ROMQG. The originalmodel results are shown in red while
the corrected model results are in blue. It can be seen thatwith
the original model there is always large error in the mean (signal
error) compared with therelatively small error from the variances
in the resolved modes (dispersion error). The correctedmethods can
effectively reduce the signal error. And the fitting process is
almost to find the pointwhere the error from mean decreases to the
minimum amount. The point where the optimal valueis achieved is
marked by circles.
in Figure 3.2-3.8 are compared. It can be seen that by adding
the corrections from the global energy E, the errorin the variances
are effectively reduced while large errors appear in the
predictions for the mean. And if the linearresponse corrections for
the mean dynamics are also considered, the overall improvements for
both the mean andvariances are achieved with no further
increasement for more computational costs. Specifically, we can
summarizethe performance of these methods and the improvements
through the corrections in the following several points:
• The effects of the correction from the global energy equation
for total variance can be shown by comparingthe dispersion error
between GC1 and GC2. In GC1, we only use constant damping and noise
at eachtime instant, while in GC2 the variance equations can be
corrected by the scale factor using total variancerather than that
in the 1-dimensional subspace. Comparing the dispersion errors
between original modeland the corrected model, it can be seen that
the energy correction can effectively improve the performanceof the
prediction for variances in GC2 results (and also for the ROMQG
results when energy correction isapplied), while GC1 has little
improvement for the variance since no correction method can be
applied;
• The effects of the correction from linear response theory for
mean dynamics can be seen by checking thesignal error part for all
the three methods. When linear response corrections for the
unresolved variances areadded to the corrected models, improvements
can be seen for all three methods compared with the originalmodels
with only steady state statistics applied. Especially for the case
with also energy corrections, theaccuracy for the mean can in turn
improve the accuracy in the global energy dynamics and further
improvethe prediction for the variances. This can also be seen by
comparing GC1 and GC2 results;
Figure: Tuning parameters in the training regime for
reduced-order GC1, GC2, and ROMQGwith s = 3. The corrected methods
can effectively reduce the signal error.
Di Qi, and Andrew J. Majda (CIMS) Improving Prediction Skill of
Imperfect Models Dec. 8, 2016 37 / 61
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Check model sensitivity to perturbations
0 5 10 15 20 25 308
8.1
8.2
8.3
8.4
8.5
8.6
8.7
8.8
8.9
9upward ramp−type forcing
time
F
(a) upward ramp-type forcing
0 5 10 15 20 25 307.2
7.3
7.4
7.5
7.6
7.7
7.8
7.9
8downward ramp−type forcing
time
F
(b) downward ramp-type forcing
0 2 4 6 8 10 12 14 16 18 207.2
7.4
7.6
7.8
8
8.2
8.4
8.6
8.8
9periodic forcing
time
F
(c) periodic forcing
0 2 4 6 8 10 12 14 16 18 206.5
7
7.5
8
8.5
9random forcing
time
F
(d) random forcing
Di Qi, and Andrew J. Majda (CIMS) Improving Prediction Skill of
Imperfect Models Dec. 8, 2016 38 / 61
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Upward Ramp-type Forcing
0 5 10 15 20 25 3062
64
66
68
70
72GC2, Variance in principal direction, F = 8
0 5 10 15 20 25 302.34
2.36
2.38
2.4
2.42
2.44
2.46GC2, State of the Mean, F =8
truth
original GC2
corrected GC2
0 5 10 15 20 25 3062
64
66
68
70
72GC1, Variance in principal direction, F = 8
0 5 10 15 20 25 302.3
2.35
2.4
2.45
2.5GC1, State of the Mean, F =8
truth
original GC1
corrected GC1
time0 5 10 15 20 25 30
62
64
66
68
70
72ROMQG, Variance in principal direction, F = 8
time0 5 10 15 20 25 30
2.3
2.35
2.4
2.45
2.5ROMQG, State of the Mean, F =8
truth
original ROMQG
corrected ROMQG
time0 5 10 15 20 25 30
0
0.005
0.01
0.015
0.02
0.025total information distance, original method
GC2
GC1
ROMQG
time0 5 10 15 20 25 30
0
0.005
0.01
0.015
0.02
0.025total information distance, corrected method
-
Information Error with Upward Ramp-type Forcing— Signal part
& Dispersion part
0 5 10 15 20 25 30
×10-3
0
0.2
0.4
0.6
0.8
1GC2, signal error
original GC2
corrected GC2
0 5 10 15 20 25 30
×10-3
0
0.5
1
1.5
2
2.5
3
3.5GC2, dispersion error
0 5 10 15 20 25 30
×10-3
0
1
2
3
4
5
6
7GC1, signal error
original GC1
corrected GC1
0 5 10 15 20 25 300
0.005
0.01
0.015
0.02GC1, dispersion error
time0 5 10 15 20 25 30
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014ROMQG, dispersion error
time0 5 10 15 20 25 30
×10-3
0
2
4
6
8ROMQG, signal error
original MQG
corrected MQG
Figure: Comparison of the information errors in signal and
dispersion part separatelybetween the closure methods GC1, GC2, and
MQG.
-
Downward Ramp-type Forcing
0 5 10 15 20 25 3054
56
58
60
62
64GC2, Variance in principal direction, F = 8
0 5 10 15 20 25 302.2
2.25
2.3
2.35GC2, State of the Mean, F =8
truth
original GC2
corrected GC2
time0 5 10 15 20 25 30
54
56
58
60
62
64GC1, Variance in principal direction, F = 8
time0 5 10 15 20 25 30
2.15
2.2
2.25
2.3
2.35GC1, State of the Mean, F =8
truth
original GC1
corrected GC1
0 5 10 15 20 25 3054
56
58
60
62
64ROMQG, Variance in principal direction, F = 8
0 5 10 15 20 25 302.15
2.2
2.25
2.3
2.35ROMQG, State of the Mean F =8
truth
original ROMQG
corrected ROMQG
time0 5 10 15 20 25 30
0
0.005
0.01
0.015
0.02
0.025
0.03total information distance, original method
GC2
GC1
ROMQG
time0 5 10 15 20 25 30
0
0.005
0.01
0.015
0.02
0.025
0.03total information distance, corrected method
-
Periodic Forcing
0 2 4 6 8 10 12 14 16 18 2050
55
60
65
70
75GC2, Variance in principal direction, F = 8
0 2 4 6 8 10 12 14 16 18 202.1
2.2
2.3
2.4
2.5
2.6GC2, State of the Mean, F =8
time0 2 4 6 8 10 12 14 16 18 20
50
55
60
65
70
75GC1, Variance in principal direction, F = 8
time0 2 4 6 8 10 12 14 16 18 20
2.1
2.2
2.3
2.4
2.5
2.6GC1, State of the Mean, F =8
0 2 4 6 8 10 12 14 16 18 2050
55
60
65
70
75ROMQG, Variance in principal direction, F = 8
0 2 4 6 8 10 12 14 16 18 202.1
2.2
2.3
2.4
2.5
2.6ROMQG, State of the Mean, F =8
time0 2 4 6 8 10 12 14 16 18 20
0
0.005
0.01
0.015
0.02
0.025total information distance, original method
time0 2 4 6 8 10 12 14 16 18 20
0
0.005
0.01
0.015
0.02
0.025total information distance, corrected method
-
Random Forcing
0 2 4 6 8 10 12 14 16 18 2050
55
60
65
70
75GC2, Variance in principal direction, F = 8
0 2 4 6 8 10 12 14 16 18 202.1
2.2
2.3
2.4
2.5GC2, State of the Mean, F =8
time0 2 4 6 8 10 12 14 16 18 20
50
55
60
65
70
75GC1, Variance in principal direction, F = 8
time0 2 4 6 8 10 12 14 16 18 20
2.1
2.2
2.3
2.4
2.5
2.6GC1, State of the Mean, F =8
0 2 4 6 8 10 12 14 16 18 2050
55
60
65
70
75ROMQG, Variance in principal direction, F = 8
0 2 4 6 8 10 12 14 16 18 202.1
2.2
2.3
2.4
2.5
2.6ROMQG, State of the Mean, F =8
time0 2 4 6 8 10 12 14 16 18 20
0
0.01
0.02
0.03
0.04total information distance, original method
time0 2 4 6 8 10 12 14 16 18 20
0
0.01
0.02
0.03
0.04total information distance, corrected method
-
Check model sensitivity to perturbations in L-96 system
time0 5 10 15 20 25 30
0
0.005
0.01
0.015
0.02
0.025total information distance, original method
GC2
GC1
ROMQG
time0 5 10 15 20 25 30
0
0.005
0.01
0.015
0.02
0.025total information distance, corrected method
(g) climate change forcing
time0 2 4 6 8 10 12 14 16 18 20
0
0.005
0.01
0.015
0.02
0.025total information distance, original method
time0 2 4 6 8 10 12 14 16 18 20
0
0.005
0.01
0.015
0.02
0.025total information distance, corrected method
(h) periodic forcing
time0 2 4 6 8 10 12 14 16 18 20
0
0.01
0.02
0.03
0.04total information distance, original method
time0 2 4 6 8 10 12 14 16 18 20
0
0.01
0.02
0.03
0.04total information distance, corrected method
(i) random forcing
-
Outline
1 Turbulent systems with quadratic nonlinearitiesGeneral setups
for turbulent systems with quadratic nonlinearities
2 Statistical dynamical models and closure methodsThe Gaussian
closure method for statistical predictionSingle point statistics
and information barrierImperfect model calibration and empirical
information theoryForecast skill of closure methods for forced
responses in L-96 model
3 Strategies for reduced order models in low dimensional
subspaceLow order models in reduced subspaceForecast skill of
reduced models for forced responses
4 Predicting statistical response in two-layer baroclinic
turbulenceTwo-layer baroclinic turbulence in ocean and atmosphere
regimesLow-dimensional reduced-order models for the two-layer
systemImperfect model prediction skill with 2× 102 resolved modes
for originalsystem with 2× 2562 dimension
Di Qi, and Andrew J. Majda (CIMS) Improving Prediction Skill of
Imperfect Models Dec. 8, 2016 45 / 61
-
The two-layer flow with forcing and dissipation
The two-layer quasi-geostrophic model with baroclinic
instability is one simplebut fully nonlinear fluid model capable in
capturing the essential physics in oceanand atmosphere science.
Two-layer model
∂qψ
∂t+ J
(ψ, qψ
)+ J (τ, qτ ) + β
∂ψ
∂x+ U
∂∆τ
∂x= −
κ
2∆ (ψ − τ)− ν∆s qψ + Fψ ,
∂qτ
∂t+ J (ψ, qτ ) + J
(τ, qψ
)+ β
∂τ
∂x+ U
∂
∂x
(∆ψ + k2dψ
)= −
κ
2∆ (τ − ψ)− ν∆s qτ + Fτ .
Barotropic and baroclinic modes:
qψ = ∆ψ, ψ =1
2(ψ1 + ψ2) ,
qτ = ∆τ − k2dτ, τ =1
2(ψ1 − ψ2) .
Normalized energy-consistent modes:
pψ,k =qψ,k|k| = − |k|ψk,
pτ,k =qτ,k√|k|2 + k2d
= −√|k|2 + k2dτk.
Di Qi, and Andrew J. Majda (CIMS) Improving Prediction Skill of
Imperfect Models Dec. 8, 2016 46 / 61
-
Flow in high-latitude homogeneous regimes
regime N β kd U κ (kmin, kmax) σmax (kx , ky )max
ocean, high lat. 256 10 10 1 9 (2.25, 14.61) 0.411 (4, 0)
atmosphere, high lat. 256 1 4 0.2 0.2 (1.58, 6.78) 0.099 (2, 0)4
Reduced-order models with homogeneous mean flow 14
linear growth rate, ocean regime
-15 -10 -5 0 5 10 15zonal wavenumber
-15
-10
-5
0
5
10
15
me
rid
ion
al w
ave
nu
mb
er
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4nonlinear flux, sum of eigenvalues
-10 -5 0 5 10zonal wavenumber
-10
-5
0
5
10
me
rid
ion
al w
ave
nu
mb
er
-1.5
-1
-0.5
0
0.5
0 5 10 15-8
-6
-4
-2
0
2linear growth rate
0 5 10 15wavenumber
-8
-4
0
4
8nonlinear flux eigenvalues
(a) high-latitude ocean regime
linear growth rate, atmosphere regime
-6 -4 -2 0 2 4 6zonal wavenumber
-6
-4
-2
0
2
4
6
me
rid
ion
al w
ave
nu
mb
er
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09nonlinear flux, sum of eigenvalues
-4 -2 0 2 4zonal wavenumber
-4
-3
-2
-1
0
1
2
3
4
me
rid
ion
al w
ave
nu
mb
er
-0.1
-0.05
0
0.05
0.1
0 1 2 3 4 5 6-0.5
0
0.5linear growth rate
0 1 2 3 4 5 6wavenumber
-0.2
0
0.2nonlinear flux eigenvalues
(b) high-latitude atmosphere regime
Fig. 4.4: Stability from linear analysis and nonlinear flux in
ocean (upper) and atmosphere (lower) regime using param-eters in
Table 1. The growth rate from linear analysis including Ekman
damping effect, and the eigenvaluesof the nonlinear flux trQF,k in
each wavenumber combining barotropic and baroclinic mode are
displayed inthe two-dimensional spectral domain. The last column
shows the radial averaged growth rate and nonlinearflux eigenvalues
in positive and negative components.
and validates the feasibility of using quasi-Gaussian
approximation in calculating the linear response operators as a2⇥2
blocked system.
4.2 Testing reduced-order models in homogeneous regime
In the previous section we displayed the unperturbed statistical
structures of the two-layer QG system in high-latituderegime with
important nonlinear non-Gaussian features. The major task now is to
test the reduced-order model skillsin predicting statistical
responses to both stochastic and deterministic forcing
perturbations as prescribed in (4.1) and(4.2) using only low-order
closures. Only the large-scale modes |k| < 10 are calculated
here, which cover the regime ofmost energetic directions. And to
investigate the model sensitivity in each component, the
perturbations in barotropicmode and baroclinic mode are applied
individually in the tests. Three statistical quantities are of
special importance
in characterizing the two-layer system, that is, the barotropic
energy,��py,k
��2, baroclinic energy��pt,k
��2, and the heatflux ikxy⇤k tk. Due to the homogeneous
statistics as we have shown before, the mean states become zero and
thus wecan focus on the second-order variances in this situation.
Therefore we will majorly check the reduced-order method’sability
in capturing the responses in these key quantities. Like the
Algorithm summarized in Section 3.3, the modelingprocess are
decomposed into a training phase for finding optimal model
parameters and a prediction phase for gettingresponses to various
perturbations.
4.2.1 Training phase with linear response operator
Equilibrium consistency for the reduced-order methods
In testing the reduced-order models, we need to first guarantee
climate consistency with the true unperturbed equilib-rium in
statistical steady state. In the construction of low-order
correction in Section 3.2, higher-order statistics fromequilibrium
are combined with additional damping and noise corrections. It
needs to be emphasized that neither theadditional damping and noise
(3.7) nor the equilibrium high-order correction (3.6) is stable on
its own even with the
Di Qi, and Andrew J. Majda (CIMS) Improving Prediction Skill of
Imperfect Models Dec. 8, 2016 47 / 61
-
Flow in low/mid-latitude regimes with zonal jets
5 Reduced-order models with inhomogeneous jet flow 25
0 20 40 60 80 100 120 1400
50
100
150mean in relative vorticity
barotropicbaroclinic
100 101 102
100
variance in relative vorticity
barotropicbaroclinic
100 101 102
100
statistical energy
barotropicbaroclinicpotential
k-1.6
k-3.6
(a) ocean regime
0 20 40 60 80 100 120 1400
0.5
1
1.5
2mean in relative vorticity
barotropicbaroclinic
100 101 102
100
variance in relative vorticity
barotropicbaroclinic
100 101 10210-10
100
statistical energy
barotropicbaroclinicpotential
k-3.5
k-1.5
(b) atmosphere regime
Fig. 5.4: Time-averaged statistics (in radial average) in mean
and second-order moments in low/mid-latitude regime.The first row
compares the statistical mean states. The following two rows show
the variances, and statisticalenergy, in barotropic and baroclinic
modes, as well as the potential energy.
0 5 10 15 20 25 30-1
0
1mode (6,3)
0 5 10 15 20 25 30-1
0
1mode (6,2)
0 5 10 15 20 25 30-1
0
1mode (6,1)
-10 -5 0 5 100
0.2
0.4mode (6,3)
-10 -5 0 5 100
0.2
0.4mode (6,2)
-10 -5 0 5 100
0.2
0.4mode (6,1)
(a) ocean
0 10 20 30 40 50-1
0
1mode (1,0)
0 50 100 150 200-1
0
1mode (0,1)
0 50 100 150 200-1
0
1mode (0,2)
-3 -2 -1 0 1 2 30
0.5
1mode (1,0)
-3 -2 -1 0 1 2 30
0.5
1
1.5mode (0,1)
-3 -2 -1 0 1 2 30
1
2mode (0,2)
(b) atmosphere
Fig. 5.5: Autocorrelation functions and the probability
distribution functions in low/mid-latitude ocean and atmo-sphere
regime. The first three most energetic baroclinic modes are
displayed. In the autocorrelations, thesolid lines show the real
part while the dashed lines are the imaginary part of the
functions. In the pdfs, thecorresponding Gaussian distributions
with the same variance are also plotted in dashed black lines.
Di Qi, and Andrew J. Majda (CIMS) Improving Prediction Skill of
Imperfect Models Dec. 8, 2016 48 / 61
-
Exact statistical moment equations
The rescaled set of equations of (1) can be summarized in the
abstract form
dpkdt
= Bk (pk, pk) + (Lk −Dk) pk + Fk, pk = (pψ,k, pτ,k)T ,∑
k
pk · Bk (pk, pk) ≡ 0,
where the normalized state variable pk =(pψ,k, pτ,k
)Tis in barotropic and baroclinic mode, the
linear operator is decomposed into non-symmetric part Lk
involving β-effect and shear flow Uand dissipation part Dk,
together with the forcing Fk combining deterministic component
andstochastic component.
Lk =
ikxβ|k|2 − ikx U√1+(kd/|k|)2−ikx U 1−(kd/|k|)
2√
1+(kd/|k|)2ikxβ
|k|2+k2d
, Dk = κ2
−1 1√1+(kd/|k|)21√
1+(kd/|k|)2− 1
1+(kd/|k|)2
,
Fk =
fψ,k|k| + σψ,kẆψ,k|k|fτ,k√|k|2+k2
d
+στ,kẆτ,k√|k|2+k2
d
.
Di Qi, and Andrew J. Majda (CIMS) Improving Prediction Skill of
Imperfect Models Dec. 8, 2016 49 / 61
-
Exact statistical moment equations
The rescaled set of equations of (1) can be summarized in the
abstract form
dpkdt
= Bk (pk, pk) + (Lk −Dk) pk + Fk, pk = (pψ,k, pτ,k)T ,∑
k
pk · Bk (pk, pk) ≡ 0,
where the normalized state variable pk =(pψ,k, pτ,k
)Tis in barotropic and baroclinic mode, the
linear operator is decomposed into non-symmetric part Lk
involving β-effect and shear flow Uand dissipation part Dk,
together with the forcing Fk combining deterministic component
andstochastic component.
Bk (pk,pk) =
[Bψ,kBτ,k
]
=
∑m+n=k
m⊥·n|k|
(|n||m|pψ,mpψ,n +
√|n|2+k2d|m|2+k2d
pτ,mpτ,n
)
∑m+n=k
m⊥·n√|k|2+k2d
(√|n|2+k2d|m| pψ,mpτ,n +
|n|√|m|2+k2d
pτ,mpψ,n
)
.
Di Qi, and Andrew J. Majda (CIMS) Improving Prediction Skill of
Imperfect Models Dec. 8, 2016 49 / 61
-
Exact statistical moment equations
Statistical energy in each spectral mode
Rk = pk∗pk =
[|pψ,k|2 p∗ψ,kpτ,k
pψ,kp∗τ,k |pτ,k|
2
], p∗1,kp2,k = p̄
∗1,kp̄2,k + p
′∗1,kp′2,k.
Rk combines the variability in both mean and variance. The true
statisticaldynamical equations form a 2× 2 system about Rk ∈
C2×2�
�dRkdt = (Lk −Dk) Rk + QF ,k + Qσ,k + c.c., |k| ≤ N,
QF ,k = pk∗Bk (pk,pk) =
[p∗ψ,kBψ,k p
∗ψ,kBτ,k
p∗τ,kBψ,k p∗τ,kBτ,k
],∑
k
trQF ,k ≡ 0.
Di Qi, and Andrew J. Majda (CIMS) Improving Prediction Skill of
Imperfect Models Dec. 8, 2016 50 / 61
-
Exact statistical moment equations
Statistical energy in each spectral mode
Rk = pk∗pk =
[|pψ,k|2 p∗ψ,kpτ,k
pψ,kp∗τ,k |pτ,k|
2
], p∗1,kp2,k = p̄
∗1,kp̄2,k + p
′∗1,kp′2,k.
Rk combines the variability in both mean and variance. The true
statisticaldynamical equations form a 2× 2 system about Rk ∈
C2×2�
�dRkdt = (Lk −Dk) Rk + QF ,k + Qσ,k + c.c., |k| ≤ N,
QF ,k = pk∗Bk (pk,pk) =
[p∗ψ,kBψ,k p
∗ψ,kBτ,k
p∗τ,kBψ,k p∗τ,kBτ,k
],∑
k
trQF ,k ≡ 0.
Di Qi, and Andrew J. Majda (CIMS) Improving Prediction Skill of
Imperfect Models Dec. 8, 2016 50 / 61
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Statistical energy conservation principle
The total statistical energy dynamical equation2 concerns the
evolution of thetotal variability in mean and variance in response
to external perturbations
E =1
2
∑1≤|k|≤N
|k|2 |ψk|2 +(|k|2 + k2d
)|τk|2 =
1
2
∑1≤|k|≤N
|pψ,k|2 + |pτ,k|2.
The exact dynamics for the statistical energy can be derived
as
dE
dt+ Hf = −κE +
κ
2F − νH + Qσ.
Hf is the meridional heat flux due to baroclinic instability, F
is the additional damping effectsdue to the non-symmetry in Ekman
drag only applied on the bottom layer
Hf = k2d U
∫ψxτ = k
2d U∑
ikxψ∗k τk, F =∑
k2d |τk|2 + 2 |k|2 Reψ∗k τk.
2Majda, Statistical energy conservation principle for
inhomogeneous turbulent dynamicalsystems, PNAS, 2016
Di Qi, and Andrew J. Majda (CIMS) Improving Prediction Skill of
Imperfect Models Dec. 8, 2016 51 / 61
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Reduced-Order Statistical Energy ClosureThe true statistical
model
dRkdt
= (Lk −Dk) Rk + QF ,k + Qσ,k + c.c., |k| ≤ N.
Qψ,k =∑
m+n=k
m⊥ · n|k|
|n||m|
p∗ψ,kpψ,mpψ,n +
√√√√ |n|2 + k2d|m|2 + k2d
p∗ψ,kpτ,mpτ,n
,Qτ,k =
∑m+n=k
m⊥ · n√|k|2 + k2d
√|n|2 + k2d|m|
p∗τ,kpψ,mpτ,n +|n|√|m|2 + k2d
p∗τ,kpτ,mpψ,n
.A preferred approach for the nonlinear flux QM,k combining both
the detailedmodel energy mechanism and control over model
sensitivity is proposed��
�
QM,k = Q−M,k+Q+M,k = f1 (E ) [− (NM,k,eq + dM ) RM,k]+f2 (E
)
[Q+F ,k,eq + σ
2M,k
].
Higher-order corrections from equilibrium statistics:
QF ,k,eq = Q−F ,k,eq + Q
+F ,k,eq = − (Lk −Dk) Rk,eq + c.c., NM,k,eq =
1
2Q−F ,k,eqR
−1k,eq.
Additional damping and noise to model nonlinear flux:
QaddM,k = −dM RM,k + σ2M,k, dM =[
dM,ψ iωM−iωM dM,τ
].
Statistical energy-consistent scaling to improve model
sensitivity:
f1 (E) =
(E
Eeq
)1/2, f2 (E) =
(E
Eeq
)3/2.
Di Qi, and Andrew J. Majda (CIMS) Improving Prediction Skill of
Imperfect Models Dec. 8, 2016 52 / 61
-
Reduced-Order Statistical Energy ClosureThe reduced-order
approximation
dRM,kdt
= (Lk −Dk) RM,k + QM,k + Qσ,k + c.c., |k| ≤ M � N.
A preferred approach for the nonlinear flux QM,k combining both
the detailedmodel energy mechanism and control over model
sensitivity is proposed��
�
QM,k = Q−M,k+Q+M,k = f1 (E ) [− (NM,k,eq + dM ) RM,k]+f2 (E
)
[Q+F ,k,eq + σ
2M,k
].
Higher-order corrections from equilibrium statistics:
QF ,k,eq = Q−F ,k,eq + Q
+F ,k,eq = − (Lk −Dk) Rk,eq + c.c., NM,k,eq =
1
2Q−F ,k,eqR
−1k,eq.
Additional damping and noise to model nonlinear flux:
QaddM,k = −dM RM,k + σ2M,k, dM =[
dM,ψ iωM−iωM dM,τ
].
Statistical energy-consistent scaling to improve model
sensitivity:
f1 (E) =
(E
Eeq
)1/2, f2 (E) =
(E
Eeq
)3/2.
Di Qi, and Andrew J. Majda (CIMS) Improving Prediction Skill of
Imperfect Models Dec. 8, 2016 52 / 61
-
Reduced-Order Statistical Energy Closure
A preferred approach for the nonlinear flux QM,k combining both
the detailedmodel energy mechanism and control over model
sensitivity is proposed��
�
QM,k = Q−M,k+Q+M,k = f1 (E ) [− (NM,k,eq + dM ) RM,k]+f2 (E
)
[Q+F ,k,eq + σ
2M,k
].
Higher-order corrections from equilibrium statistics:
QF ,k,eq = Q−F ,k,eq + Q
+F ,k,eq = − (Lk −Dk) Rk,eq + c.c., NM,k,eq =
1
2Q−F ,k,eqR
−1k,eq.
Additional damping and noise to model nonlinear flux:
QaddM,k = −dM RM,k + σ2M,k, dM =[
dM,ψ iωM−iωM dM,τ
].
Statistical energy-consistent scaling to improve model
sensitivity:
f1 (E) =
(E
Eeq
)1/2, f2 (E) =
(E
Eeq
)3/2.
Di Qi, and Andrew J. Majda (CIMS) Improving Prediction Skill of
Imperfect Models Dec. 8, 2016 52 / 61
-
Climate fidelity for equilibrium
Equilibrium fidelity refers to the convergence to the same final
unperturbedstatistical equilibrium Req in the reduced-order models
RM in each resolvedcomponent.
Specifically, it requires that the model nonlinear flux
correction term QMconverges to the truth, QM → QF ,eq, when no
external perturbation is added
dRM,eqdt
= 0 = Lv (ūeq) RM,eq + RM,eqL∗v (ūeq) + Q
MF ,eq + Qσ → RM,eq = Req.
the first component(
NM,eq,Q+F ,eq
)comes from the true equilibrium
statistics.
climate consistency requires the second component correction
makes nocontribution in the unperturbed case
ΣM =1
2dM Req, f1 (Eeq) = 1, f2 (Eeq) = 1.
Di Qi, and Andrew J. Majda (CIMS) Improving Prediction Skill of
Imperfect Models Dec. 8, 2016 53 / 61
-
Set-up for the numerical problem
The true statistics is calculated by a pseudo-spectra code with
128 spectralmodes zonally and meridionally, corresponding to 256×
256× 2 grid points intotal.
In the reduced-order methods, only the large-scale modes |k| ≤
10 areresolved, which is about 0.15% of the full model
resolution.
External forcing in stochastic and deterministic component:
The amplitude of the stochastic forcing σk Ẇk is introduced
according to theequilibrium energy so that
σ2ψ,k = δσ20 |qψ,k|2eq, σ2τ,k = δσ20 |qτ,k|
2eq.
The deterministic forcing is introduced through a perturbation
in thebackground shear Uδ = U + δU
δfψ,k = δUikx(− |k|2
)τk, δfτ,k = δUikx
(− |k|2 + k2d
)ψk.
Di Qi, and Andrew J. Majda (CIMS) Improving Prediction Skill of
Imperfect Models Dec. 8, 2016 54 / 61
-
Tuning parameters in the training phase4 Reduced-order models
with homogeneous mean flow 18
tuning parameters with energy correction
0.50.50.5
0.5
0 0.05 0.1 0.15 0.2
dψ
0
0.05
0.1
0.15
0.2
dτ
0
0.5
1
1.5
(a) tuning with energy scaling
tuning parameters without energy correction
2.5
2.5
2.5
0 0.05 0.1 0.15 0.2
dψ
0
0.05
0.1
0.15
0.2
dτ
1.5
2
2.5
3
3.5
(b) tuning without energy scaling
0 1 2 3 4 5 6 7 8 9
wavenumber
0
0.2
0.4
0.6
0.8
1information error
w/ energy correctionw/o energy correction
0 1 2 3 4 5 6 7 8 9
wavenumber
0
2
4
6
8baroclinic energy
unperturbed spectrumperturbed responsew/ energy correctionw/o
energy correction
(c) prediction and info. errors
Fig. 4.10: Tuning imperfect model parameters in the training
phase. The information errors with varying model pa-rameters, dM
=
�dy ,dt
�, are plotted for stochastic barotropic perturbation case. The
errors using total
energy as scalar factor from the statistical equation and method
without the scaling factor are compared.The prediction skill and
information error with and without using the total energy
correction are comparedin the last row for a typical test case of
perturbing the barotropic mode.
corrections from low-order moments (that is, mean and variance)
are used.The high-latitude ocean regime responses in barotropic and
baroclinic energy and heat flux in large-scale wavenum-
bers to stochastic perturbations are first shown in Figure 4.11.
The perturbation amplitude is chosen as ds20 = 0.5 of
theequilibrium energy in the stochastic forcing (4.1) so that the
response is large and nonlinear. We compare the responsesin
perturbing only the barotropic mode and baroclinic mode. The most
energetic and most sensitive scales take place atwavenumbers |k| =
4,5,6. Both barotropic and baroclinic perturbations can lead to
large changes in a wide spectrumin both barotropic and baroclinic
component due to the strong coupling between the modes. In the
reduced-order meth-ods, only the first large-scale modes |k| <
10 are resolved, while the responses in these dominant modes are
all capturedwith accuracy in both perturbation cases though the
complicated higher-order interactions with small-scale modes arenot
computed explicitly. Further the time-series with the total
statistical energy from the equation (3.11) are compared.The dashed
black lines mark the level of energy in unperturbed and perturbed
case. In this regime, the total statisticalenergy can also be
recovered exactly with little error. This in turn explains the high
skill of the reduced-order modelsin predicting this regime.
Instead, if we only consider the energy in the resolved subspace
shown by blue lines, a largegap can be observed compared with the
total energy. Figure 4.12 shows the results in the high-latitude
atmosphereregime. Alternating blocked and unblocked structures
appear in this regime and generate quite complicated
statisticalfeatures. The leading mode |k| = 1 contains most of the
energy and becomes highly sensitive to perturbations.
Thereduced-order method keeps the skill in capturing the responses
in the most sensitive directions in this difficult regime.Also it
is observed that the baroclinic perturbation case becomes a little
less accurate in both spectra and total statisticalenergy. This
might be due to the stronger nonlinear energy interactions from
baroclinic to barotropic mode.
A further test requires to check the model’s robustness in
predicting perturbations with different amplitudes. Figure4.13
displays the prediction results with changing stochastic forcing
amplitude ds20 in the barotropic modes. Thereduced-order model
maintains the skill in predicting responses with various forcing
strength, and the nonlinear trendsin the total resolved barotropic
and baroclinic energy as well as the heat flux are captured
compared with the linearprediction in the FDT shown by dashed
lines.
Figure: Tuning imperfect model parameters in the training phase.
The information errorswith varying model parameters, dM = (dψ, dτ
), are plotted for stochastic barotropicperturbation case.
Di Qi, and Andrew J. Majda (CIMS) Improving Prediction Skill of
Imperfect Models Dec. 8, 2016 55 / 61
-
High-latitude: Stochastic perturbation
The model is perturbed with stochastic forcing with variance
amplitudeδσ20 = 0.5;
Either the barotropic or the baroclinic modes in large scales
|k| ≤ 10 areperturbed.
4 Reduced-order models with homogeneous mean flow 19
0 1 2 3 4 5 6 7 8 90
5
10barotropic energy
unperturbedperturbed truthred. model
0 1 2 3 4 5 6 7 8 90
5
10baroclinic energy
0 1 2 3 4 5 6 7 8 9
wavenumber
-0.2
-0.1
0heat flux
0 1 2 3 4 5 6 7 8 90
5
10barotropic energy
unperturbedperturbed truthred. model
0 1 2 3 4 5 6 7 8 90
5
10baroclinic energy
0 1 2 3 4 5 6 7 8 9
wavenumber
-0.2
-0.1
0heat flux
0 5 10 15 20 25 30
time
30
35
40
45time-series of total energy, barotropic perturbation
resolved
total
truth
(a) barotropic perturbation
0 5 10 15 20 25 30
time
30
35
40
45time-series of total energy, baroclinic perturbation
resolved
total
truth
(b) baroclinic perturbation
Fig. 4.11: Reduced-order model predictions to stochastic
perturbations with amplitude ds20 = 0.5 in barotropic (left)and
baroclinic (right) mode in high-latitude ocean regime. The spectra
for the resolved modes 1 |k| < 10are compared. Black lines with
circles show the perturbed model responses in the normalized
variables,��py,k
��2 (barotropic energy),��pt,k
��2 (baroclinic energy), and ikxy⇤k tk (heat flux). The dashed
black lines arethe unperturbed statistics. And the reduced order
model predictions are in red lines. The last row shows themodel
prediction of the energy equation in red lines and the energy in
the resolved subspace shown in bluelines. For comparison, the
unperturbed and perturbed total energy from the true system is
shown in dashedblack lines.
(a) ocean regime
4 Reduced-order models with homogeneous mean flow 20
0 1 2 3 4 5 6 7 8 90
5
10
barotropic energy
unperturbedperturbed truthred. model
0 1 2 3 4 5 6 7 8 90
2
4
baroclinic energy
0 1 2 3 4 5 6 7 8 9
wavenumber
-0.2
-0.1
0heat flux
0 1 2 3 4 5 6 7 8 90
5
10barotropic energy
unperturbedperturbed truthred. model
0 1 2 3 4 5 6 7 8 90
2
4baroclinic energy
0 1 2 3 4 5 6 7 8 9
wavenumber
-0.2
-0.1
0heat flux
0 50 100 150 200
time
2
4
6
8
time-series of total energy, barotropic perturbation
resolved
total
truth
(a) barotropic perturbation
0 50 100 150 200
time
3
4
5
6
7time-series of total energy, baroclinic perturbation
resolved
total
truth
(b) baroclinic perturbation
Fig. 4.12: Reduced-order model predictions to stochastic
perturbations with amplitude ds20 = 0.5 in barotropic (left)and
baroclinic (right) mode in high-latitude atmosphere regime. The
spectra for the resolved modes 1 |k| < 10 are compared. The last
row shows the model prediction of the energy equation in red lines
and theenergy in the resolved subspace shown in blue lines. For
comparison, the unperturbed and perturbed totalenergy from the true
system is shown in dashed black lines.
0 1 2 3 4 5 6 7 8 90
2
4
6
8barotropic energy
0 1 2 3 4 5 6 7 8 90
2
4
6
8baroclinic energy
0 1 2 3 4 5 6 7 8 9
wavenumber
-0.2
-0.15
-0.1
-0.05
0heat flux
(a) response in spectra
0 0.2 0.4 0.6 0.8
25
30
35
response in barotropic energy
0 0.2 0.4 0.6 0.836
38
40
42
44
46
response in baroclinic energy
0 0.2 0.4 0.6 0.8
δσ2
-0.9
-0.8
-0.7
-0.6response in heat flux
(b) total responses
Fig. 4.13: Imperfect model predictions to responses with
changing perturbation amplitude ds20 in the high-latitudeocean
regime (with barotropic perturbation). In the first part on the
left we the predicted spectra with threedifferent perturbation
amplitude, ds20 = 0.1,0.5,0.8, are shown. On the right the
responses in total energyand heat flux with changing amplitudes
ds20 2 [0,0.8] are plotted. For clarification in display, we only
plotreduced model predictions by red markers and the truth is in
black lines.
(b) atmosphere regime
Di Qi, and Andrew J. Majda (CIMS) Improving Prediction Skill of
Imperfect Models Dec. 8, 2016 56 / 61
-
High-latitude: Mean shear flow perturbation
The model is perturbed by changing the background zonal flow
strength U;
The entire spectral is perturbed due to the mean flow advection
in eachspectral mode.
5 Reduced-order models with inhomogeneous jet flow 21
0 1 2 3 4 5 6 7 8 90
5
10barotropic energy
0 1 2 3 4 5 6 7 8 90
5
10baroclinic energy
0 1 2 3 4 5 6 7 8 9
wavenumber
-0.2
-0.1
0heat flux
(a) U = 1.05
0 1 2 3 4 5 6 7 8 90
2
4
6barotropic energy
0 1 2 3 4 5 6 7 8 90
5
10baroclinic energy
0 1 2 3 4 5 6 7 8 9
wavenumber
-0.2
-0.1
0heat flux
(b) U = 0.95
Fig. 4.14: Reduced-order model predictions to mean shear flow
perturbation dU = ±0.05 (that is, 5% of the originalvalue U0) in
the ocean regime. The spectra for the resolved modes 1 |k| < 10
are compared. Black lineswith circles show the perturbed model
responses in the normalized barotropic energy, baroclinic energy,
andheat flux. The dashed black lines are the unperturbed
statistics. And the reduced order model predictionsare in red
lines.
Model responses to the perturbed mean shear dU
In checking the model responses to deterministic forcing, we
introduce the forcing perturbation by changing the back-ground jet
strength U as in (4.2). The same perturbation is tested in [24] for
a more complicated reduced-order modifiedquasi-Gaussian closure
(RoMQG), and we test the same perturbation form here under our
systematic reduced-ordermodeling framework. Note that the
deterministic perturbation in (4.2) forms a more difficult test
case compared withthe stochastic forcing (4.1) because the forcing
is applied along all wavenumbers with stronger mean-fluctuation
in-teractions involved. On the other hand, for the reduced order
methods, only the perturbations at the limited resolvedmodes are
quantified. This gives the inherent difficulty for applying the
reduced order models to this kind of perturba-tions since we have
no knowledge of the unresolved modes where large amount of energy
is contained. Therefore thestatistical energy equation (3.11) plays
a crucial role.
The results with mean flow perturbations dU = ±0.05 in the ocean
regime and perturbations dU = 0.02,�0.01 inthe atmosphere regime
are shown in Figure 4.14 and 4.15 separately. The perturbation
accounts for about 5%-10% ofthe original shear strength U , and the
correspon