non-equilibrium statistical mechanics and the … · non-equilibrium statistical mechanics and the...

!!!! Non-equilibrium Statistical Mechanics and the Theory of Extreme Events in Earth Science

! ! ! ! ! ! ! ! ! ! !! 30 October 2013!

!Data-driven reduction and climate prediction: nonlinear

stochastic, energy-conserving models with memory effects !!

Dmitri Kondrashov !!

University of California, Los Angeles!Joint work with M.D. Chekroun (UCLA), M. Ghil (UCLA, ENS), !

S. Kravtsov (UWM), A.W. Robertson (IRI/CU)!!

http://www.atmos.ucla.edu/tcd/!!!

Motivation!

• Observed spatio-temporal climate variability can be fairly well represented by a few number of "resolved" (predictable, quasi-oscillatory) modes of “slow” (time) scales • Low-order models (LOM): – linear and nonlinear interactions between the resolved modes – estimation of “fast&small” scales - unresolved modes (“noise”) – proper description of interactions between the few resolved and the large number of unresolved modes • Empirical Model Reduction (EMR) - data-driven LOM methodology - no prior assumptions on scale separation between resolved and unresolved modes. - multi-level nonlinear equations driven by stochastic forcing

- estimates dynamical operator and driving noise from the data

Outline! • Under certain conditions, EMR can be understood as a data-driven discrete formulation of nonlinear Multi-Layer Stochastic Model (MSM) in the context of Mori-Zwanzig formalism that estimates unresolved dynamics as (time-domain) orthogonal, and represents resolved-unresolved interaction as non-Markovian and stochastic. • Orthogonal property of unresolved dynamics facilitates energy-conserving EMR formulation: illustrative toy model examples

• Pathwise prediction context: “Past-Noise” Forecasting (PNF) • Climate applications: ENSO, Madden-Julian Oscillation Low-order stochastic model and “past-noise forecasting” of the Madden-Julian oscillation, (2013) D. Kondrashov, M. Chekroun, A. Robertson, M. Ghil, GRL, 40, doi:10.1002/grl.50991 .

!

!   x(t) is given time series (observations, or high-end, complex model simulation) of resolved modes– typically leading subset of EOFs

!   Unresolved modes r(m) are estimated and modeled by “matrioshka” of addt’l levels. Model coefficients F, A, B and L(m) are computed by multiple linear regression recursively from main level down to the last, using estimated tendencies Δx, Δr(m) as predictants.

!   r(m) are estimated as regression residuals at each level and are orthogonal in time domain to the variables from the previous levels [x, r(0), ..., r(m−1)], i.e

!   The data series x(t) is reconstructed exactly when LOM is driven by last level regression residual r(M)

!  

!   Figure 2: Prediction skill for the (RMM1, RMM2) pair of EOF-filtered indices3 of the Madden-Julian

!   Oscillation (MJO): EMR (red) and PNF (blue) prediction, while the black curve shows damped persistence for a base comparison.

!   yr of the Real-time Multivariate MJO Index (RMM) record; the respective forecasts were validated on the record’s last 3.5 yr.

EMR Basic Facts !

�x, r(m�1)� = 0;�r(j), r(m)� = 0, � j, m � {1, ...,M � 1}, with j � m� 1

dx =�� Ax + �B(x,x) + �F

�dt + r(0)dt,

dr(m�1) = L(m)�(x)T , (r(0))T , ..., (r(m�1))T

�Tdt + r(m)dt, � m � {1, ...,M}

r(M) = �(t) � �dW

EMR as Multi-layered Stochastic Model!

!  For a special case of resolved-unresolved interactions when

additional levels can be integrated and expressed formally as

It allows to formally integrate matrioshkas from the last level to the top and obtain

Interpretation in the context of Mori-Zwanzig formalism: The evolution of x(t) relies on stochastic integro-differential system of equations with the deterministic terms in (a) that rule the nonlinear self-interactions of x; it is the Markovian contribution. The terms in (b) convey memory effects through repeated convolutions involving the past history of x and constitute thus the non-Markovian contribution; (c) represents last level random forcing orthogonal to x .

L(m)�(x)T , (r(0))T , ..., (r(m�1))T

�T = Cmx�Dmr(m�1)

dxdt

= �

(a)� �� Ax + B(x,x) + F+

(b)� �� M�1�

l=0

l�

m=0

(µ0 � · · · � µm � �l+1 � x)(t)+

(c)� �� (µ1 � · · · � µM � r(M))(t) .

r(m�1)(t) =� t

0e�(t�s)DmCmx(s)ds +

� t

0e�(t�s)Dmr(m)(s)ds

�m(t) = etDmCm;µm(t) = etDm

r(m�1)(t) = (�m � x)(t) + (µm � r(m))(t)

Change of Basis in a Partially Observed Case!

!   Apply EMR to x(t=1,...N) time series from linear model integration

!   a=-2, A = -1, q=1

!   In the limit (dt->0, N->∞) EMR imposes similarity transformation S (x, y) → (x, r) that preserves eigenvalues of original model. EMR model has different model coefficients since x(t) and r(t) are orthogonal, but x(t) and y(t) are not.



! yr of the Real-time Multivariate MJO Index (RMM) record; the respective forecasts were validated on the record’s last 3.5 yr.

!   Due to orthogonality of unresolved dynamics, on average there is zero net energy flux between resolved and unresolved modes

The following conditions then should hold for EMR to be considered formally as stable forced-dissipative system:




Energy-Conserving EMR formulation -I!!  “The RHS of scalar EMR model (for univariate time series) has to be a gradient

of the potential function, and the lowest possible value of nonlinearity K that results in a stable model is equal to K=3”! [Kravtsov, Kondrashov and Ghil, 2005]

!  The multivariate case is more interesting one to consider: with enough data record and proper regularization even non-constrained EMR models are well behaving [Lorenz 63,Kravtsov, Kondrashov and Ghil, 2005]

�x� =M�

i=1

x2i < �;

x · |x = F�Ax + B(x,x) + r0t

�x · r0

t dt � 0

Bijkxixjxk � 0;Aijxixj > 0,x �= 0

Less is More!!

!  “Toy” climate model [Majda et al. 2005, Franzke et al. 2007] with four modes: “slow” (resolved) x1-x2; “fast” (unresolved) y1-y2 modes that carry most of the variance.

!  Energy-conserving form for nonlinear and linear part.

!  We will apply EMR to slow modes only (x1-x2) with a goal to reproduce their statistics (PDF, ACF) by reduced model.

!  Multiplicative and additive noise forcing (bias in reduced model) is appropriately parametrized and modeled by multi-level unresolved EMR variables.

!  Figure 2: Prediction skill for the (RMM1, RMM2) pair of EOF-filtered indices3 of the Madden-JulianOscillation (MJO): EMR (red) and PNF (blue) prediction, while the black curve shows damped persistence for a base comparison.yr of the Real-time Multivariate MJO Index (RMM) record; the respective forecasts were validated on the record’s last 3.5 yr.

Less is More: Numerical Results - ACFs!

!  3-level quadratic EMR captures reasonably well ACFs of x1,x2 even for cases without scale separation.

!  Figure 2: Prediction skill for the (RMM1, RMM2) pair of EOF-filtered indices3 of the Madden-JulianOscillation (MJO): EMR (red) and PNF (blue) prediction, while the black curve shows damped persistence for a base comparison.yr of the Real-time Multivariate MJO Index (RMM) record; the respective forecasts were validated on the record’s last 3.5 yr.

Less is More: Numerical Results - PDFs!

EMR-ENSO Prediction model!

Long-term (>8mo) operational ENSO forecasts need improvement- PNF!

Barnston et al., 2012: Skill of Real-Time Seasonal ENSO Model Predictions during 2002–11: Is Our Capability Increasing?. Bull. Amer. Meteor. Soc., 93, 631–651, 2012. !

EMR-ENSO prediction (ensemble mean) “has the highest seasonally combined correlation skill among the statistical models exceeded by only a few dynamical models… as well as one of the smallest RMSE”. !

!

Predictions by EMR-ENSO model are submitted monthly into IRI forecast plume of dynamical and statistical models. !

Chekroun, M. D., D. Kondrashov and M. Ghil, 2011: Predicting stochastic systems by noise sampling, and application to the El Niño-Southern Oscillation, Proc. Nat. Acad. Sci. USA, 18, 4425–4444.

Kondrashov, D., S. Kravtsov, A. W. Robertson, and M. Ghil, 2005: A hierarchy of data-based ENSO models, J. Climate, 18, 4425–4444

Past-Noise Forecasting (PNF)!• !• Given that weather (small and fast scales) does affect climate (large and slow scales) it might be useful to adopt it for climate prediction. ! • Similar to Constructed Analogues (“future” ≈ ”past”) but in the context of LOM-estimated noise and pathwise prediction. !• Chekroun et al. (2013): two circumstances when it can be done:

– presence of oscillatory climate modes -- low-frequency variability (LFV). – linear response of the LOMs w.r.t. noise perturbation.

• The crucial idea is to composite ensemble of noise “snippets” according to the current phase of the LFV, and use this ensemble to drive the LOM into the future - hence the acronym Past-Noise Forecasting (PNF).

• Singular Spectrum Analysis (SSA) and Hilbert Transform to estimate current phase of LFV. !

• Predictability study for MJO!

Chekroun, M. D., D. Kondrashov and M. Ghil, 2011: Predicting stochastic systems by noise sampling, and application to the El Niño-Southern Oscillation, Proc. Nat. Acad. Sci. USA, 18, 4425–4444.!

PNF and EMR!

MJO Basic Facts!!   MJO is the dominant intraseasonal

(30–90 days) variability in the tropical atmosphere and represents large-scale coupling between atmospheric circulation and tropical deep convection.

!   Unlike ENSO which is mostly a standing pattern, MJO is a traveling eastward pattern.

!   Real-time Multivariate MJO (RMM1,2) represent EOF-based prefiltering of 850 hPa zonal wind, 200 hPa zonal wind, and outgoing long-wave radiation (OLR) data.

PNF for MJO!!   A quadratic, energy-conserving three-level EMR model with seasonal

cycle was developed to model and predict the leading pair (RMM1, RMM2) of real-time, daily MJO indices.

!   Shorter time-scale of MJO LFV (30-50) day compared to ENSO (2-4yr) allows robust testing of EMR/PNF in ~35yr of data.

!   The EMR model was fitted, and the PNF method was trained, on the first 30 yr of data (1974-2004) and validated on the last 4yr (2005-2008).

!   Apply the Hilbert Transform to the SSA reconstruction (Feliks et al., 2010) over the whole time series of the RMM1,2 to find instantaneous LFV phase.

!   Refine subset of noise samples by choosing only those that correspond to initial states in the raw (RMM1, RMM2) data closest to those at the start time of the forecast.

!  

!   EMR

!   PNF Perstistence

!   EMR

!   PNF Perstistence




Energy-conserving EMR-MJO Model!

�x1,n+1 � x1,n

x2,n+1 � x2,n

�= F�A

�x1,n

x2,n

�+ B

�

�x2

1,n

x1,nx2,n

x22,n

�

� + S

�

��

sin(�n)x1,n sin(�n)x2,n sin(�n)

cos(�n)x1,n cos(�n)x2,n cos(�n)

�

��+

�r(0)1,n

r(0)2,n

�,

�r(0)1,n+1 � r(0)

1,n

r(0)2,n+1 � r(0)

2,n

�= L(1)

�

��

x1,n

x2,n

r(0)1,n

r(0)2,n

�

�� +

�r(1)1,n

r(1)2,n

�,

�r(1)1,n+1 � r(1)

1,n

r(1)2,n+1 � r(1)

2,n

�= L(2)

�

��

x1,n

x2,n

r(0)1,n

r(0)2,n

r(1)1,n

r(1)2,n

�

��

+ Q�

�1,n

�2,n

�.

A =�

0.0268 0.1150�0.1150 0.0278

�,

B =�

0 �0.0064 �0.00020.0064 0.0002 0

�

Bijkxixjxk � 0;Aijxixj > 0,x �= 0

Q =�

1.0 0.00.0412 0.9992

�;

�1 � N (0, 0.1704), �2 � N (0, 0.1688)

Low-order stochastic model and “past-noise forecasting” of the Madden-Julian oscillation, (2013) D. Kondrashov, M. Chekroun, A. Robertson, M. Ghil, GRL, 40, doi:10.1002/grl.50991 .

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PNF and EMR for MJO - I!

A quadratic, three-level EMR model with seasonal cycle for daily (RMM1, RMM2)

PNF improvement is best during

energetic MJO events

PNF and EMR for MJO - II!

PNF improvement is best during

energetic MJO events

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Prediction by PNF ensemble members (green) tracks observed (black); (blue) – PNF ensemble mean; (red) – EMR ensemble mean

PNF and EMR for MJO - III!

(a) Bivariate correlation (corr) and (b) RMSE showing increase in predictability of RMM1-2 beyond 15 days by PNF (blue) vs. EMR (red); the black curve shows damped persistence as a basis for comparison. (c–f) Prediction skill as a function of the MJO strength at the start of the forecast: PNF improvement is most pronounced for weak MJO, especially in correlation. The PNF skill is useful up to 30 days and comparable with the skill demonstrated by a state-of-the art dynamical multi-model ensemble [Zhang, 2013]. Similar skill for non-energy conserving formulation (no constraints)

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non-equilibrium statistical mechanics and the … · non-equilibrium statistical mechanics and the...

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