Expected and Actual Forecast Errors by a Non-normal Model of El Nio - - PowerPoint PPT Presentation

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Expected and Actual Forecast Errors by a Non-normal Model of El Nio - - PowerPoint PPT Presentation

Jan 05, 2023 181 likes •535 views

Expected and Actual Forecast Errors by a Non-normal Model of El Nio Ccile Penland NOAA/ESRL/Physical Sciences Division Outline Some phenomenology: What is El Nio The Annual Cycle Deviations from it, especially El Nio An

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Expected and Actual Forecast Errors by a Non-normal Model of El Niño

Cécile Penland NOAA/ESRL/Physical Sciences Division

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  • Some phenomenology: What is El Niño

–The Annual Cycle –Deviations from it, especially El Niño

  • An empirical-dynamical model
  • Uncertainty and errors

Outline

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Size of the annual cycle (oC). Typical size of El Niño ~ 1-3 oC

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Linear Inverse Modeling

Assume linear dynamics (dropping the primes): dT/dt = BT + ξ, with < ξ (t+τ) ξT (t) > = Q(t)δ(τ) For now, we’ll assume additive noise, although that assumption is false. Q(t) is periodic. Corresponding FPE:

[ ] [ ]

) , ( ) ( 2 1 ) , ( ) , (

2

t p t Q T T t p T B T t t p

ij ij j i ij j ij i

T T T

  • +
  • =
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From the FPE. p(T,t +τ|Το,t) is Gaussian, centered on G(τ) Το where G(τ) = exp(Bτ) = <T(t+τ)TT(t) >< T(t)TT(t) >-1 . The covariance matrix of the predictions: Σ (t,τ) = <T(t+τ)TT(t +τ) > − G(τ) < T(t)TT(t) > GT (τ) . Further,

t

  • < T(t)TT(t) > = B < T(t)TT(t) > + < T(t)TT(t) > BT + Q( t)
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Digression : The disturbing assumption of additive noise. Instead of dT/dt = BT + ξ , the system is actually of the form dT/dt = BT + (AT +C)ξ1 +Dξ2. All of the LIM formalism follows through, with the identification B → B + A2/2; Q → <(AT +C) (AT +C)T > + DDT G(τ) → exp { (B +A2/2) τ } Note: p(T,t +τ|Το,t) is no longer Gaussian, but G(τ) Το is still the best prediction in the mean square sense.

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If LIM’s assumptions are valid, the prediction error ε = = Τ(t+τ) − G(τ) Το does not depend on the lag at which the covariance matrices are evaluated. This is true for El Niño; it is not true for the chaotic Lorenz system. Eigenvectors of G(τ) are the “normal” modes {ui}. Eigenvectors of GT(τ) are the “adjoints” {vi}, (Recall: G(τ) = <T(t+τ)TT(t) >< T(t)TT(t) >-1) and uvT = uTv = 1 . Most probable prediction: T(t+τ) = G(τ) Tο (t) The neat thing: G(τ) ={G(το) } τ/ το .

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Several sources of expected error and uncertainty:

  • Stochastic forcing:

Σ (t,τ) = <T(t+τ)TT(t +τ) > − G(τ) < T(t)TT(t) > GT (τ)

  • Uncertain initial conditions:

<δT(t+τ)δTT(t +τ) >i.c. = G(τ) <δT(t)δTT(t) > GT (τ)

  • Sampling errors when estimating G(τ) :

<δT(t+τ)δTT(t +τ)|T(t) >ij,Samp=

< Gik

km

  • Gjm > Tk(t)Tm(t)
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Expected error in Niño 3.4 anomaly forecast

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Conclusions

  • Expected and actual errors can be a useful

diagnostic tool

  • El Niño is mainly a linear process

maintained by additive and multiplicative cyclostationary stochastic forcing

  • Initial condition errors grow and then decay