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

**Reliably** Predict**in**g Uncerta**in**ty **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, Sh**in**field Park, Read**in**g

Uncerta**in**ty **in** Computer Models 2012, Sheffield, Tuesday 3 rd July

Outl**in**e

• What is the source of model **uncerta inty**

atmospheric models?

• Why should we represent **uncerta inty**

atmospheric models?

• How can we represent model **uncerta inty**

atmospheric models?

• Experiments **in** the Lorenz ‘96 simplified

atmospheric model

• Conclud**in**g 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 **in**fect 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 **in**fect 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

Represent**in**g **uncerta inty** 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 **uncerta inty**?

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.

entra**in**ment coefficient

• Parameters constant

spatially **and** temporally

Determ**in**istic representation:

Perturbed Parameters

• Parameters are poorly determ**in**ed

• Perturb them with**in** physical range

– The effect on the forecast gives **in**dication of

**uncerta inty** due to model approximations

• Uses a s**in**gle base parametrisation

Stochastic Parametrisation

• Include r**and**om 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 **in**t[( j1)

/ ] 1

j1

Y: small scale, high freq

→ cf. Individual convective events

Parameter Sett**in**gs:

No. X variables, K = 8 Forc**in**g term, F = 20

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

Coupl**in**g 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 **in**t[( j1)

/ ] 1

j1

Truth Model:

> Run full model, start**in**g at 150 **in**itial conditions on the attractor

Forecast Model:

> Y variables unresolved

> Parametrise U as U p(X)

> N ens = 40, run from constant perfect **in**itial 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

Determ**in**istic

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)

def**in****in**g 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 fitt**in**g 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 constra**in**s the

probability distribution of

U, but does not determ**in**e

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)

– st**and**ard 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 r**and**om 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)

– st**and**ard 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 r**and**om 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)

– st**and**ard 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 r**and**om 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)

– st**and**ard 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 r**and**om number, appropriately scaled

Test the parametrisations for:

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

the model

2. How well they represent model **uncerta inty**

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**

• Determ**in**istic is worst

• Stochastic schemes:

“**weather** mode”

– Includ**in**g autocorrelation

→ significant improvement

– Simple noise sufficient

for c = 10

• Perturbed parameters

significantly better than

white stochastic schemes

Type of Parametrisation

Representation of Model Uncerta**in**ty

• Determ**in**istic worst

• Red noise > white

noise

• PP less reliable

P(X)

– poorer representation

of model **uncerta inty**

verification

X

Determ**in**istic

SP: AR1 Add

SP: White Add

PP

c = 10 case

Representation of Model Uncerta**in**ty

• Summarise for all schemes

us**in**g reliability component

of Brier Score

– Smaller value = more reliable

REL

N = # b**in**s

N 1

n i 1

N

B**in**ned forecast

probability of

event

i

2

y o

i

i

Average

event rate

**in** b**in**

H

2

Skill of parametrisations **in**

1

2

“**climate** mode”

• Def**in**e **climate** to be the

pdf p(X)

• Smaller the Hell**in**ger

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”

Conclud**in**g Remarks

• It is important to represent model **uncerta inty**

**in** **weather** **and** **climate** models.

• Stochastic Parametrisations:

– result **in** an improvement **in** **weather** **and** **climate**

forecast**in**g skill

– Good representations of model **uncerta inty**

– Spatio-Temporal autocorrelation is important

• Perturbed Parameter Schemes:

– Poorer representation of model **uncerta inty**

• Stochastic perturbed parameter schemes...?

Thank you for listen**in**g!

Arnold et al, Stochastic Parametrisations **and** Model Uncerta**in**ty **in** the

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