Tropical Biases in the COLA Coupled Model

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(No Transcript)
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CHAOTIC SYSTEMS CAN SYNCHRONIZE DESPITE
SENSITIVITY

• two coupled chaotic systems can fall into
  synchronized motion
• along their strange attractors when linked
  through only one variable

z (t)
x ?(y-x) y ?x-y-xz z -?zxy
y1 ?x-y1-x(z1) z1 -??z1)x(y1)
(also works for y-coupling, but not for
z-coupling)

(Pecora and Carroll 90)
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SUPPOSE THE WORLD IS A LORENZ SYSTEM AND ONLY X
IS OBSERVED

• two coupled chaotic systems can fall into
  synchronized motion
• along their strange attractors when linked
  through only one variable

z (t)
x ?(y-x) y ?x-y-xz z -?zxy
y1 ?x-y1-x(z1) z1 -??z1)x(y1)
(also works for y-coupling, but not for
z-coupling)

(Pecora and Carroll 90)
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TWO CHANNEL MODELS SYNCHRONIZE WHEN DISCRETELY
COUPLED - makes weather prediction possible
Truth Model
(Duane and Tribbia, PRL 01, JAS 04)
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Part I Treatment of Nonlinearities in the
Synchronization Approach Part II
Synchronization for Parameter Estimation, Model
Learning and Fusion of Climate Models
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Analysis Synchronization with Noisy Coupling

SDEs dxA/dt f (xA)
dxB/dt f (xB) C (xA- xB ?)
? is white noise lt ?(t)
?T(t) gt R d(t- t) linearize de/dt Fe
Ce C ????????e ? xA- xB F ? Df(xA) ?
Df(xB) Fokker-Planck eqn for PDF p(e) ?p/ ??t
? e ? p (F-C) e ½ t??(CTRC?p) Gaussian
ansatz p N exp(-eTKe) ?pdne 1
?p/ ??t 0 Choose C to minimize the spread
B ? (2K)-1 of the distribution.
Fluctuation-Dissipation Relation B (C-F)T
(C-F) B ?CRCT for C ? C dC (dC
arbitrary), let dB be such that B ? B dB
dB0 if C Copt
(1/ t) B R-1
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Standard Data Assimilation As a Continuous
Process (as in Einsteins treatment of
Brownian motion)
Standard methods xAxbkd B(BR)-1(xT
- xbkd noise)

(perfect model)
dxT/dt f(xT)
dxbkd/dt f(xbkd) (1/??B(BR)-1(xT - xbkd
?)
O (B(BR)-1)2
f(xbkd) (1/??
BR-1 (xT - xbkd ?) ??is the time between
analyses in incremental data assimilation The
coupling C (1/ t) B R-1 Copt So, the standard
methods of data assimilation (3DVar, Kalman
Filtering) are also optimal for synchronization
under local linearity assumption! (Exact
treatment of discrete analysis cycle as a map
gives Copt (1/ t)
B (BR)-1. )
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OPTIMAL COUPLING IN FULLY NONLINEAR CASE
de/dt (F-C)e C? Ge2 He3 ?M ansatz
pN exp(Ke2Le3Me4) Model error covariance Qlt
?M ?MTgt ?p/ ??t ? e ? (F-C) e Ge2 He3 p
½ t??(CTRC?p)
In one dimension, Fokker-Planck eqn?
(F-C)e Ge2 He3 ½ ? C2R (-2Ke 3Le2
4Me3) ?
F-C ½ ? C2R (-2K)
G ½ ? C2R (-3L)
H ½ ? C2R
(-4M) background error B B(K,L,M)
?e2p(e)de B(K(C),L(C),M(C)) optimize B as a
function of C ? general correction to KF If we
restrict form of C, e.g. CF BR-1
?
cov. inflation factor F
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Choose G and H so that the dynamics are those of
motion in a two-well potential dx/dt f
(x) e.g. for d1,d2 matching the distances
between the fixed points in the Lorenz 84
system with F1, one finds G .15 H -.75
Minimize background error B as a function of
coupling C Find C 1.51, B0.145
B
C
If C F B/(dR), then we have a covariance
inflation factor F 1.04 (where
R1, d 0.1)
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d1
No model error (Q0)
d2
d1
Model error equal to 50 of the resolved tendency
d2
The need for inflation is shaped by the
nonlinearities, regardless of the amount of
model error.
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WHAT ABOUT SAMPLING ERROR?

• Suppose undersampling ? uncertainty in
  estimate of B
• multiplicative noise in assimilation
• dxT/dt f(xT)
• dxbkd/dt f(xbkd) (1/??B(BR)-1 ?S(xT
  - xbkd ?)
• Fokker Planck equation
• 
  ?S2 lt ?S
  ?STgt
• ?p/ ??t ? e ? p (F-C ½?S2) e
• ½
  t?2(CTRC ?S2 e2 - 2 ?Se/? (CTRC))p
• Use change of variables p p(CTRC ?S2 e2 - 2
  ?Se? (CTRC))
• Arguably, effect is small if ?S lt BR-1
• g

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Multidimensional Case e.g. D2
Consider two wells separated in one
dimension. Assume R (can
arrange by rescaling)
Choose a basis such that the dynamical
equations are given by a direct product of
motion in a two-well potential and simple linear
dynamics. R is still diagonal. The FP
equation ?p/ ??t ? e ? p (F-C) e ½
t??(CTRC?p) separates.

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Summary Covariance Inflation in the
Synchronization Approach

• In the synchronization approach, the rough
  magnitudes
• of covariance inflation factors used in
  practice might be explained
• from first principles
• Model error due to unresolved physics makes
  little difference the
• requirement for inflation is shaped by
  nonlinearities in the dynamics
• Refinements may yield treatments of
  nonlinearities that improve on
• covariance inflation

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Part I Treatment of Nonlinearities in the
Synchronization Approach Part II
Synchronization for Parameter Estimation, Model
Learning and Fusion of Climate Models
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WHAT IF THE MODEL IS IMPERFECT?

• can synchronize parameters as well as states
• Lorenz system example
• add parameter adaptation laws r1 (y-y1)
  x1
• 
  n (y1-y) y1
• 
  m y-y1

x1 ?(y-x1) y1 ?1x1- ny1-x1(z1)m z1
-??z1)x1(y1)
x ?(y-x) y ?x-y-xz z -?zxy

• these augmented equations minimize a Lyapunov
  function
• V ex2 ey2 ez2 rr2rn2rm2
• where ex
  x-x1, ey.. rr r-r1, rn.
• since it can be shown that dV/dt lt 0, and V is
  bounded below
• So as t?8, (x1,y1,z1) ?(x,y,z) and also r1? r,
  n?1, m? 0
  
• 
  i.e. the model learns

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General Rule for Parameter Estimation, If
Systems Synchronize with Identical Parameters
dx/dt f(x,p) dp/dt0 dy/dtf(y,q) u(y,s)
ss(x) (30) dq/dtN(y,x-y)
(31) e?y-x
r?q-p h? f(y,q)- f(y,p)
Truth Model
(Duane , Yu, and Kocarev, Phys. Lett. A 06)
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Example A Column Model With an Unknown Surface
Moisture Availability Parameter
Column model summary
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Parameter Adaptation Rule
Prognostic equation for humidity
nudging term
soil moisture
moisture availability parameter
Adapt M according to

• interpretation decrease or increase M in
  proportion to the
• covariance between the synchronization
  (forecast) error and the
• factor multiplied by M in the dynamical equations

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RESULTS
observations at 7 points in column nudging at 1
point nudging coeffiicient .01
M-MT
time
-alternating periods of slow convergence to
synchronization and rapid bursts
away -apparently can always identify the true
value of M
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.other configurations show same pattern
observations and nudging at 7 points, coefficient
.0025
as previous, but with nudging
coefficient .015
observations and nudging at 4 points, coefficient
.015
Actual details of model as implemented in
software were unknown!
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..because the state variables also do not
converge completely in the time interval shown
qT
qm
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Single-Realization vs Ensembles

• in principle, should be able to replace ensemble
  averages with
• time averages to estimate relatively constant
  quantities
• (cf. ergodicity)
• learn on the fly ? AI view of data
  assimilation
• compare to Lagged Average Forecasting
• (Hoffman and Kalnay 83 ) use a single
  realization with
• different initialization times to create an
  artificial ensemble

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Which parameters should we adapt?
TAKE A COLLECTION OF THE BEST MODELS,
COUPLE THEM TO ONE ANOTHER, AND ADAPT THE
COUPLING COEFFICIENTS
Ki constant data assimilation
adapt Clij learning
CONSENSUS
-couple corresponding model elements l
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Test Case Fusing 3 Lorenz Systems With
Different Parameters
Average Output of Models (Unfused)
z from Model With Best z Eqn
Fused Models
zMavg-zT
zMbest-zT
zMavg-zT
not adapting Clij0
adapting
time
time
time
dCxij/dt a(xj-xi)(x ??xk)
dCyij/dt. dCzij/dt.
- Model fusion is superior to any weighted
averaging of outputs
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Parameter Adaptation in the QG Channel Model
What if foB ? foB ?
model B
truth A
n0
Add terms to FB to assimilate medium scales of A.
Then adapt foB
foB?(q-qB)(qA-qB)d2x
foB?foA
timestep n
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Proposed Adaptive Fusion of Different Channel
Models
forcing in Atlantic
forcing in Pacific
(k-dependence suppressed)
Fo fo(q-q) Fofo(q-q)

• If the parallel channels
• synchronize, their common
• solution also solves the
• single-channel model with
• the average forcing

To find
c adaptively dc/dt
?d2x J(y,q-q)(q-qobs) ?d2x
J(y,q-q)(q-qobs)
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FUSION OF REAL CLIMATE MODELS
typical scenario
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SST
CAM
CAM_MOM3
Heat Flux
Momentum Flux
MOM
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SST
COLA
COLA_MOM3
Heat Flux
Momentum Flux
MOM
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SST
CAM
COLA
Heat Flux
Momentum Flux
Interactive Ensemble
MOM
CAM_COLA_MOM3
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SST
CAM
COLA
Heat Flux
Interactive Ensemble
Momentum Flux
MOM
COLA_CAM_MOM3
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Heat Flux COLA Momentum Flux CAM
Heat Flux CAM Momentum Flux COLA
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Observations
COLA_MOM3
CAM_MOM3
Heat Flux COLA Momentum Flux CAM
Heat Flux CAM Momentum Flux COLA
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COLA_MOM3
CAM_MOM3
All Model Error
Heat Flux COLA Momentum Flux CAM
Heat Flux CAM Momentum Flux COLA
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COLA_MOM3
CAM_MOM3
COLA Heat Flux Errors
Heat Flux COLA Momentum Flux CAM
Heat Flux CAM Momentum Flux COLA
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COLA_MOM3
CAM_MOM3
Error Amplified by CAM Momentum Flux
Heat Flux COLA Momentum Flux CAM
Heat Flux CAM Momentum Flux COLA
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COLA_MOM3
CAM_MOM3
CAM Heat Flux Error
Heat Flux COLA Momentum Flux CAM
Heat Flux CAM Momentum Flux COLA
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INFERENCES ABOUT SOURCES OF ERROR WERE USED TO
FORM A FUSED CAM-COLA MODEL
Guiding principle For each model element, make
the choice of
model that reduces truth-model
synchronization error
-simplified form of the automated adjustment of
coupling coefficients (which need not be
binary) proposed here
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Adaptive Consensus Formation Approach is
Empirical

• -reminiscent of learning in neural networks
• Hebbs rule Cells that fire together,
  wire together
• here Model elements wire
  together directionally,
• until they
  collectively fire in sync with reality
• Can the role of synchronization in the consensus
  formation scheme
• be compared to its proposed role in
  consciousness, via the highly
• intermittent synchronization of the 40 Hz
  oscillation in widely
• separated regions of the brain?

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Conclusion Adaptive consensus formation among
models can likely reduce error in long-range
climate forecasts
But what if the dynamical parameters change
drastically in the 21st century as compared to
the training period?
Lorenz test case
Attractors
Average of outputs (unfused)
Fusion
adaptation
r28
r50
r100
Other possible issues -local vs. global
optima in coupling coefficients -climate vs.
weather prediction
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Suggestive of Measure Synchronization

• in jointly Hamiltonian systems, trajectories can
  become the same,
• while states differ at any instant of time
  (Hampton Zanette PRL 99)
• Afraimovich et al. 97 nonisochronic
  synchronization of
• dissipatively coupled systems