Reliably predicting uncertainty in weather and climate ... - MUCM

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Reliably predicting uncertainty in weather and climate ... - MUCM

Reliably Predicting Uncertainty in

Weather and Climate Forecasts

Hannah Arnold 1 , Irene Moroz 2 ,

Tim Palmer 1,3

1. Atmospheric, Oceanic and Planetary Physics, Univ. Oxford

2. Oxford Centre for Industrial and Applied Mathematics, Univ. Oxford

3. European Centre for Medium-Range Weather Forecasts, Shinfield Park, Reading

Uncertainty in Computer Models 2012, Sheffield, Tuesday 3 rd July


Outline

• What is the source of model uncertainty in

atmospheric models?

• Why should we represent uncertainty in

atmospheric models?

• How can we represent model uncertainty in

atmospheric models?

• Experiments in the Lorenz ‘96 simplified

atmospheric model

• Concluding remarks


Seconds

Atmospheric Scales of Motion

Meters

The COMET Program


Seconds

Atmospheric Scales of Motion

Meters

Horizontal

Resolution

of GCM

Time step in

GCM


Seconds

Atmospheric Scales of Motion

Meters

Horizontal

Resolution

of GCM

Time step in

GCM

Unresolved

by GCMs


Upscale transfer of errors

• Shallow power-law

slope in the KE

spectrum

• No observed scale

separation.

• Errors in small scale

processes infect larger

scales

-3

-5/3

Large scales Small Scales

Nastrom & Gage, 1985


Upscale transfer of errors

• Shallow power-law

slope in the KE

spectrum

• No observed scale

separation.

• Errors in small scale

processes infect larger

scales

Reverse

energy

cascade

-3

-5/3

Large scales Small Scales

Nastrom & Gage, 1985


Seconds

Atmospheric Scales of Motion

Meters

Horizontal

Resolution

of GCM

Time step in

GCM

Unresolved

by GCMs


Representing uncertainty generates

more accurate and useful forecasts

• Probabilistic forecasts

essential for risk

assessment

• Seamless prediction

paradigm: improve

climate prediction

Probabilistic

weather forecast


How can we represent model uncertainty?

e.g. clouds are unresolved ...

• Approximate the effects of

sub-gridscale clouds through

parametrisation schemes

– Effect of sub-gridscale =

function of gridscale

– Average of effects of subgridscale

• Tuneable parameters e.g.

entrainment coefficient

• Parameters constant

spatially and temporally


Deterministic representation:

Perturbed Parameters

• Parameters are poorly determined

• Perturb them within physical range

– The effect on the forecast gives indication of

uncertainty due to model approximations

• Uses a single base parametrisation


Stochastic Parametrisation

• Include random numbers in equations of

motion – probabilistic in nature

• Represent the sub-gridscale variability

• Potential realisation of sub-gridscale


dX

dt

dY

dt

Lorenz ‘96 System

kJ

k X k1

X k2

X k1

X k F Yj

b j

J ( k1)

1

j

hc

hc

cbY

J

b

X: large scale, low freq

→ cf. Global circulation

Y j2

Y

j1

cY j X int[( j1)

/ ] 1

j1


Y: small scale, high freq

→ cf. Individual convective events

Parameter Settings:

No. X variables, K = 8 Forcing term, F = 20

No. Y variables / X, J = 32 Spatial scale ratio, b = 10

Coupling constant, h = 1 Timescale ratio, c = 4 & 10

Fig. taken from Wilks, 2005


dX

dt

dY

dt

Lorenz ‘96 System

kJ

k X k1

X k2

X k1

X k F Yj

b j

J ( k1)

1

j

hc

hc

cbY

J

b

Y j2

Y

j1

cY j X int[( j1)

/ ] 1

j1


Truth Model:

> Run full model, starting at 150 initial conditions on the attractor

Forecast Model:

> Y variables unresolved

> Parametrise U as U p(X)

> N ens = 40, run from constant perfect initial conditions

Sub-grid

tendency, U


dX

dt

Lorenz ‘96 System

kJ

k X k1

X k2

X k1

X k F Yj

b j

J ( k1)

1

2

Udet b0

b1

X b2

X

Deterministic

parametrisation

b

3

X

3

U

hc

X

Sub-grid

tendency, U


dX

dt

Lorenz ‘96 System

kJ

k X k1

X k2

X k1

X k F Yj

b j

J ( k1)

1

Perturbed Parameters Stochastic Scheme

hc

Sub-grid

tendency, U

Potential

realisations


dX

dt

Lorenz ‘96 System

kJ

k X k1

X k2

X k1

X k F Yj

b j

J ( k1)

1

hc

Sub-grid

tendency, U

Perturbed Parameters Perturb 4 param.s (b 0, b 1, b 2, b 3)

defining cubic fit → 40 member

ensemble

Low:

Medium:

High:

b

b

b

p

i

p

i

p

i

i,

meas

i,

meas

Estimate s i from fitting to subsets

of U(X)

U

pp


b

p

0

b

p

1

X




( 1

b

( 1

b



p

2

s

s

i

i

X

) b

,

)

2

b

i,

meas

b

p

3

,

X

3


dX

dt

Lorenz ‘96 System

kJ

k X k1

X k2

X k1

X k F Yj

b j

J ( k1)

1

• X constrains the

probability distribution of

U, but does not determine

its exact value

• Investigated several

stochastic schemes

2

Udet

b0

b1

X b2

X

b

3

X

3

hc

Stochastic Scheme

Sub-grid

tendency, U

Potential

realisations


Stochastic Parametrisations:

• Simple Additive: U p = U det + e(t)

– e(t) is AR(1) process

• State Dependent: U p = U det + e(t)

– standard deviation of e(t) is a function of X

• Multiplicative: U p = (1 + e(t)) * U det

– e(t) is AR(1) process

• Multiplicative and Additive:

U p = (1 + e 1(t)) U det + e 2(t)

– e i(t) is same random number, appropriately scaled


Stochastic Parametrisations:

• Simple Additive: U p = U det + e(t)

– e(t) is AR(1) process

• State Dependent: U p = U det + e(t)

– standard deviation of e(t) is a function of X

• Multiplicative: U p = (1 + e(t)) * U det

– e(t) is AR(1) process

• Multiplicative and Additive:

U p = (1 + e 1(t)) U det + e 2(t)

– e i(t) is same random number, appropriately scaled


Stochastic Parametrisations:

• Simple Additive: U p = U det + e(t)

– e(t) is AR(1) process

• State Dependent: U p = U det + e(t)

– standard deviation of e(t) is a function of X

• Multiplicative: U p = (1 + e(t)) * U det

– e(t) is AR(1) process

• Multiplicative and Additive:

U p = (1 + e 1(t)) U det + e 2(t)

– e i(t) is same random number, appropriately scaled


Stochastic Parametrisations:

• Simple Additive: U p = U det + e(t)

– e(t) is AR(1) process

• State Dependent: U p = U det + e(t)

– standard deviation of e(t) is a function of X

• Multiplicative: U p = (1 + e(t)) * U det

– e(t) is AR(1) process

• Multiplicative and Additive:

U p = (1 + e 1(t)) U det + e 2(t)

– e i(t) is same random number, appropriately scaled


Test the parametrisations for:

1. Ability to predict the short term “weather” of

the model

2. How well they represent model uncertainty

3. Ability to predict the long term “climate” of

the model


Skill of parametrisations in

weather mode”

• Ranked Probability Score

RPS


J

Predicted

probability

RPSS 1







m



o j


m


y

j

m1

j 1

j1

RPS

RPS ref

2

Observed

prob.

Ref. to

clim.

Type of Parametrisation


Skill of parametrisations in

• Deterministic is worst

• Stochastic schemes:

weather mode”

– Including autocorrelation

→ significant improvement

– Simple noise sufficient

for c = 10

• Perturbed parameters

significantly better than

white stochastic schemes

Type of Parametrisation


Representation of Model Uncertainty

• Deterministic worst

• Red noise > white

noise

• PP less reliable

P(X)

– poorer representation

of model uncertainty

verification

X

Deterministic

SP: AR1 Add

SP: White Add

PP

c = 10 case


Representation of Model Uncertainty

• Summarise for all schemes

using reliability component

of Brier Score

– Smaller value = more reliable

REL


N = # bins

N 1


n i 1

N

Binned forecast

probability of

event

i

2

y o

i

i

Average

event rate

in bin


H

2

Skill of parametrisations in

1

2

climate mode”

• Define climate to be the

pdf p(X)

• Smaller the Hellinger

Distance, the closer the

forecast p(X) to the true

pdf, q(X)


2

P,

Q p(

x)

q(

x)

dx

Forecast pdf.

Truth pdf.


Skill of parametrisations in

• “Seamless Prediction”

• Palmer et al (2008) use

reliability of seasonal

predictions to verify

climate projections

climate mode”


Concluding Remarks

• It is important to represent model uncertainty

in weather and climate models.

• Stochastic Parametrisations:

– result in an improvement in weather and climate

forecasting skill

– Good representations of model uncertainty

– Spatio-Temporal autocorrelation is important

• Perturbed Parameter Schemes:

– Poorer representation of model uncertainty

• Stochastic perturbed parameter schemes...?


Thank you for listening!

Arnold et al, Stochastic Parametrisations and Model Uncertainty in the

Lorenz ‘96 System, Phil. Trans. Roy. Soc. A, 2012, in press

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