Predicting the cloud patterns of the Madden-JulianOscillation through a low-order nonlinear stochastic

model(Auxiliary Material)

N. Chen, A. J. Majda and D. Giannakis

Department of Mathematics and Center for Atmosphere Ocean Science,Courant Institute of Mathematical Sciences, New York University, New

York, NY 10012, USA.

1 Calibration of the Nonlinear Stochastic Models

Recall the low-order nonlinear stochastic model

du1dt

= (−du u1 + γ (v + vf (t))u1 − (a+ ωu)u2) + σu Wu1 , (1.1a)

du2dt

= (−du u2 + γ (v + vf (t))u2 + (a+ ωu)u1) + σu Wu2 , (1.1b)

dv

dt= (−dv v − γ (u21 + u22)) + σv Wv, (1.1c)

dωu

dt= (−dωωu + ωu) + σω Wω, (1.1d)

wherevf (t) = f0 + ft sin(ωf t+ φ). (1.2)

The optimal parameters in the stochastic model is determined by systematically minimiz-ing the information distance (model error) of the signal equilibrium PDF of the stochasticmodel q compared with that of the actual data p [1, 2],

P(p, q) =

∫p ln

(p

q

). (1.3)

The sensitivity analysis is shown in Figure 1.1, which indicates the robustness of thelow-order nonlinear stochastic model with respect to the parameters.

Prediction with random suboptimal parameters are shown in Figure 1.2 and 1.3, wherethe random suboptimal parameter are given by

σu = σ∗u + U(−0.05, 0.05), du = d∗u + U(−0.35, 0.25), ft = f ∗

t + U(0, 1)

γ = γ∗ + U(0, 0.2), σv = σ∗v + U(−0.2, 0.2), dv = d∗v + U(−0.35, 0.25),

(1.4)

where the variables with stars are the optimal parameters and U(a, b) is the uniformdistribution in the interval [a, b].

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Model error as a function of ft

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Model error as a function of σu

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du

Model error as a function of du

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dv

Model error as a function of dv

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σv

Model error as a function of σv

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γ

Model error as a function of γ

(a) (b) (c)

(d) (e) (f)

Figure 1.1: Sensitivity analysis of parameters ft, σu, du, dv, σv and γ in the low-ordernonlinear stochastic model (1.1).

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True MJO1Prediction

15 days lead prediction (MJO 1)

25 days lead prediction (MJO 1)

Months Months Months

Figure 1.2: 15 and 25 days lead prediction skill with random suboptimal parameters.

The prediction with random suboptimal parameters has comparable prediction skillas that with optimal parameters, implying the robustness of the low-order nonlinearstochastic model in prediction.

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Figure 1.3: Prediction skill starting from November 1, January 10 and March 1 withrandom suboptimal parameters.

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2 Details of data assimilation algorithm

In the prediction stage, the initial values of the hidden variables v and ωu are determinedvia data assimilation algorithm.

One feature of the low-order nonlinear stochastic system (1.1) is that it is a conditionalGaussian system with respect to the observations u1 and u2, meaning that once u1 andu2 are given, the system is a linear equation with Gaussian statistics provided that theinitial condition is Gaussian. For such type of system, the conditional distribution of vand ωu given the observations u1 and u2, which is Gaussian, can be explicitly writtendown, which is contributed by Lipster and Shiryaev [3].

To run this data assimilation algorithm, we rewrite the low-order nonlinear stochasticmodel (1.1) in the following abstract form,

dUt = [A0(t,U) + A1(t,U)Γt]dt+ B1(t,U)dW1(t), (2.1a)

dΓt = [a0(t,U) + a1(t,U)Γt]dt+ b2(t,U)dW2(t), (2.1b)

where Ut = (u1, u2)T is a vector containing the observed variables while Γt = (v, ωu)T

represents for the unobserved processes. The matrices and vectors A0, A1, a0, a1, B1 andb2 for the low-order nonlinear stochastic are given as follows

A0(t,U) =

(−duu1 − au2 + γvf (t)−duu2 + au1 + γvf (t)

), A1(t,U) =

(γu1 −u2γu2 u1

),

a0(t,U) =

(−γ(u21 + u22)

ωu

), a1(t,U) =

(−dv

−dω

),

B1(t,U) =

(σu

σu

), b2(t,U) =

(σv

σω

).

(2.2)

The Theorem in [3] states that assuming the conditional initial conditions being Gaus-sian and some moments bound for the corresponding processes, the conditional mean µt

and conditional covariance Rt of Γ based on the observed process U are expressed asfollows

dµt =[a0(t,U) + a1(t,U)µt]dt+ (RtA∗1(t,U))(B1B

∗1)

−1(t,U)[dUt − (A0(t,U) + A1(t,U)µt)dt],

dRt ={a1(t,U)Rt +Rta

∗1(t,U) + (b2b

∗2)(t,U)− (RtA

∗1(t,U))(B1B

∗1)

−1(t,U)(RtA∗1(t,U))∗

}dt.

(2.3)The initial value of the hidden processes for this filtering process is taken as the Gaus-sian fit of the model equilibrium state. Therefore, with (2.3) the conditional Gaussiandistribution of the hidden processes v and ωu at each fixed time is determined.

As a remark, we point out that the filter (2.1) and (2.3) is an optimal filter if andonly if the signal is generated from system (2.1). Yet, our observed signal is MJO indiceswhile we utilize the low-order nonlinear stochastic model as the filter, and therefore ourfilter is merely a suboptimal filter.

In Figure 2.1 we show the posterior mean and variance of the recovery of the hiddenvariable v and ωu in the historic period from 1983/09/03 to 1999/12/31. The cross-covariance of v and ωu is negligible of order O(10−18) and is not shown here. Note thatwhen intermittency occurs, the posterior covariance is small, meaning that the recoveredstate has small uncertainty.

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f

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Posteriorvariance of ω

u

Posteriorvariance of v

Posteriormean of ω

u

Posteriormean of v

MJO 1

Recovery of v and ωu from 1983/09/03 to 1999/12/31

Figure 2.1: Recovery of posterior mean and variance of stochastic damping v and stochas-tic phase ωu in (1.1) from 1983/09/03 to 1999/12/31 as a function of time compared withthe observations of MJO indices. The cross-covariance of v and ωu is negligible of orderO(10−18) and is not shown here.

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3 Twin experiment

In this final section, we consider a twin experiment. That is, we assume the true signalis generated from the low-order nonlinear stochastic model (1.1). Thus, the data assim-ilation algorithm is based on a perfect filter. The signal generated by (1.1) is shown inFigure 3.1. In particular, the signal has the same length as MJO indices and containsweak and strong events in the prediction period. Figure 3.2 and 3.3 show the RMS errorand bivariate correlation for 1-60 days lead prediction skill and the 15 and 25 days leadprediction curves. The MJO prediction skill in the paper is comparable to this internalprediction skill, suggesting the low-order nonlinear stochastic model has significant skillfor determining the predicability limits of the large scale cloud patterns of the borealwinter MJO.

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u1

u2

Months

Figure 3.1: Twin experiment. The signal is generated from nonlinear model (1.1) thathas the same length as the MJO data with the training period (1983/09/03–1999/12/31)and prediction period (2000/01/01–2006/06/30).

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Figure 3.2: Twin experiment. Skill scores with RMSE (top) and bivariate correlation(bottom) for prediction in different years.

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Figure 3.3: Twin experiment. Prediction skill of u1 at a 15 (top) and 25 (bottom) dayslead. The blue line shows the true signal and the red line shows the ensemble average ofthe predicted signal with 50 ensemble members.

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References

[1] Majda, A. J., and B. Gershgorin (2010), Quantifying uncertainty in climate changescience through empirical information theory, Proceedings of the National Academyof Sciences, 107 (34), 14,958–14,963.

[2] Majda, A. J., and B. Gershgorin (2011), Improving model fidelity and sensitivity forcomplex systems through empirical information theory, Proceedings of the NationalAcademy of Sciences, 108 (25), 10,044–10,049.

[3] Liptser, R. S., and A. N. Shiryaev (2001), Statistics of Random Processes II: II.Applications, vol. 2, Springer.

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