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First results and methodological approach to parameter perturbations

in GEM-LAM simulationsPART I

Leo Separovic, Ramon de Elia and Rene Laprise

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MOTIVATION• Sub-grid parameterization schemes are source of “parametric uncertainty”:

- well-known processes that can be exactly represented (e.g. radiation transfer) but need to be approximated so that they do not take excessive computational time;- less-well understood processes (e.g. turbulent energy transfer) that are situation dependent; parameters rely on mixture of theoretical understanding and empirical fitting;- measurable parameters - measurement error, - non-measurable parameters uncertainty associated with representativity.

• Tuning can eliminate only reducible component of the model error.

• Parametric uncertainty can be (at least theoretically) quantified by perturbing parameters and measuring the impact on model output.

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CONTENTS• Detection of the model response to perturbations of parameters in a large

domain

- noise: brief analysis of internal variability

- signal: sensitivity of seasonal climate to selected perturbations of a trigger parameter of KF convection- statistical significance (signal-to-noise ratio)

- trade-off between statistical significance and computational cost

• Detections of model response in a small domain- effects of reduction of domain size on magnitude of the signal and noise

• Future work- intermediate domain size, next parameter, multiple parameter perturbations

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EXPERIMENTAL CONFIGURATION• GEM-LAM 140x140• DX=0.5 deg (max 55.5 km at JREF=65), NLEV=52

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EXPERIMENTAL CONFIGURATION

• Five start dates: November 1-5 1992 00GMT• End date: November 30 1993 00GMT

4 seasons DEC01-NOV30

• Time step: 30 min

• Nesting data: ERA 40

• PTOPO: npex=4 npey=4• Estimated time: 12hrs/year

• Output frequency: once per 6 hours

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Physics packageVersion: RPN-CMC4.5

• RADIA: CCCMARAD• SCHMSOL: ISBA• GWDRAG: GWD86• LONGMEL: BOUJO• FLUVERT: CLEF• SHLCVT: CONRES, KTRSNT_MG• CONVEC: KFC• KFCPCP: CONSPCPN• STCOND: CONSUN

• Stomate: .false.• Typsol: .true.• Snowmelt: .false.

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Trigger vertical velocity in KFC scheme• The KFCTRIG values that are deemed to be appropriate at the limits of the resolution

interval in which the KFC scheme is to be used (B. Dugas, 2005):KFCTRIG (170 km) = 0.01KFCTRIG (10 km) = 0.17

• It is assumed that: KFCTRIG (RES) * RES = 1.7 = C (#)

• KFCTRIG is a function of the grid-tile area:KFCTRIG = KFCTRIG0 * RES0 / sqrt (DXDY)

• At the nominal resolution of 50km (#) gives KFCTRIG=0.034 (REFERENCE)

• We performed 2 perturbations (ONE PER TIME):KFCTRIG1=0.020 and KFCTRIG2=0.048

These values would be deemed appropriate at resolution of85km and 35km.

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THREE ENSEMBLES

• WKLCL =0.034 (REFERENCE) 5 members

• WKLCL=0.029 (-) single 5 members

• WKLCL=0.048 (+) single 5 members

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Internal variability in the reference 5-member ensembleTA-ESTD-PCP

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Internal variability in the reference 5-member ensembleTA-ESTD-PCP normalized

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ESTD-TA-PCP

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(ESTD-TA-PCP)/(EA-TA-PCP)

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ESTD-TA-Tscn

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Detection of the model response to parameter perturbations

follows the Student’s distribution with (nR+nP-2) degrees of freedom.

• Null hypothesis: The two means are computed from two samples drawn from a unique distribution.

€

t ≡XP

E− XR

E

1

nR+

1

nP

⎛

⎝ ⎜

⎞

⎠ ⎟

(nR −1) σ E2 (XR ) + (nP −1) σ E

2 (XP )

nR + nP − 2

⎛

⎝ ⎜

⎞

⎠ ⎟

⎡

⎣ ⎢

⎤

⎦ ⎥

1/ 2

• Signal: difference between the ensemble averages of - reference ensemble XR: nR=5 members

- perturbed-parameter ensemble XP: nP= members

• Error: sample STD of the ensemble averages: E2(XR)/nR and E

2(XP)/nP.

• If the true variances of XR and XP are equal then the quantity

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PCPKFCTRIG=0.020 (-)

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PCP – level of rejection

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Signal (TTscn)

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TTscn: rejection level

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Trade-off between number of parameter perturbations and significance

• We need to find a trade-off between P and t

• Let’s relate the two ensemble sizes: then

€

nR = bnP

€

t ≈ΔE

σ E× np ×

b

b+1

€

P ≡$

np−b

€

t ≈ΔE

σ E×

nR nPnR +nP

€

$ ≡ nR + P nPSignificance t

Internal variability σ

Computational resources $

No of parameter perturbations P

signal

and

i) One should invest in nR because of its low cost but not more than b=5 (diminishing returns)

ii) np=1 & b>>0 minimizes the cost but also minimizes the signal-to-noise ratio