

Mean systematic error of 500 hPa
geopotential height fields
LOWRES
HIGHRES
Mean systematic error of 500 hPa
geopotential height fields
LOWRES
HIGHRES
• Reduction of z500 bias in all simulations
with model-refinement
SKEBS
Berner et al., 2012
Potential of stochastic parameterizations
to reduce model error
Ball in doublepotential well
Weak noise
Strong noise
Unimodal
Multi-modal
PDF
Potential of stochastic parameterizations
to reduce model error
Potential
Weak noise
Strong noise
Unimodal
Multi-modal
PDF
Potential of stochastic parameterizations
to reduce model error
Potential
 Stochastic
parameterizations can
change the mean and
variance of a PDF


Weak noise
Strong noise
Unimodal
Multi-modal
PDF
Impacts variability
Impacts mean bias
Key message
 Random numbers can improve
weather and climate predictions
Outline
 The stochastic parameterization schemes
 Climate application: Impact in coupled and
uncoupled simulations with the Earth System
Model CESM
 Weather application: Improving reliability and
reducing analysis error in cycled and uncycled
forecasts with the weather model WRF
Stochastically perturbed tendency
scheme (SPPT)
Rationale: Especially as resolution increases,
the equilibrium assumption is no longer valid
and fluctuations of the subgrid-scale state
should be sampled (Buizza et al. 1999, Palmer
et al. 2009, Berner et al. 2014)
¶X
= DX + (r+1)PX
¶t
Local tendency for
variable X
Dynamical tendencies
=> Resolved scales
Physical tendencies
=> Unresolved scales
 Perturbs accumulated U,V,T,Q tendencies
from physical parameterizations packages
 Same pattern for all tendencies to minimize
introduction of imbalances
Stochastic-kinetic energy backscatter
scheme (SKEBS)
Rationale: A fraction of the subgrid-scale
energy is scattered upscale and acts as random
streamfunction and temperature forcing for
the resolved-scale flow (Shutts 2005, Berner
et. al 08,09). Here simplified version with
constant dissipation rate: can be considered as
additive noise with spatial and temporal
correlations.
¶X
= DX + PX + dDX, STOCH
¶t
Local tendency for
variable X =U,V,T
Dynamical tendencies
=> Resolved scales
Physical tendencies
=> Unresolved scales
Additive stochastic
perturbation tendencies
=> Unresolved scales
Stochastic Forcing Pattern
Outline
 The stochastic parameterization schemes
 Climate application: Impact in coupled and
uncoupled simulations with the Earth System
Model CESM
 Weather application: Improving reliability and
reducing analysis error in cycled and uncycled
forecasts with the weather model WRF
Northern Annular Mode (MAM)
1st EOF of sea level pressure over Northern Hemispheric Extratropics
 CAM4 AMIP
NCEP
21%
CNTL
50%
simulations
(prescribed SSTs),
1900-2004
 Stochastic
parameterization
improves pattern
and reduces
explained variance
 Degenerate
response: SKEBS and
SPPT have same
effect
SKEBS
35%
SPPT
37%
Northern Annular Mode (MAM)
1st EOF of sea level pressure over Northern Hemispheric Extratropics
 CAM4 AMIP
NCEP
21%
CNTL
50%
simulations
(prescribed SSTs),
1900-2004
 Stochastic
parameterization
improves pattern
and reduces
explained variance
 Degenerate
response: SKEBS and
SPPT have same
effect
SKEBS
35%
SPPT
37%
Sketch: CAM4 behavior


Including a stochastic
parameterization does not
lead to large changes in the
pattern of modes of
variability, but to decreased
explained variances
This is consistent with a
flattening of a potential well
Stochastic parameterizations can
also lead to a depending of a
potential well.
Potential
without stochastic
perturbations
Potential
with stochastic
perturbations
Sketch: CAM4 behavior

Including a stochastic
parameterization does not
lead to large changes in the
pattern of modes of
variability, but to decreased
explained variances

This is consistent with a
shallowing of a potential well

Stochastic parameterizations
can also lead to a depending
of a potential well.
Potential
without stochastic
perturbations
Potential
with stochastic
perturbations
First EOF of 500hPa-field over Atlantic sector
NCEP
SKEBS
49%
46
%
CNTL
SPPT
44%
53%
First EOF of 500hPa-field over Atlantic sector
NCEP
SKEBS
49%
46
%
CNTL
SPPT
44%
53%
Impact of SPPT on sea surface
temperature (SST) variability
 Coupled simulations with CAM4, 1880-2004
 Too much variability in SSTs in Tropical Pacific
 SPPT reduces bias in SST variability in Tropical Pacific
How can a stochastic parameterization reduce variability?
Impact of SPPT on sea surface
temperature (SST) variability
 How can perturbations to the atmosphere improve the
ocean?
 SPPT reduces variability in u850 variability over the
Western Pacific
Impact of SPPT on
El Niño Southern
Oscillation
Probability density function of daily
temperatures over North America (JJA)
 AMIP simulations with
CAM4
 Too much variability in
daily temperatures in
summer compared to
reanalysis
General extreme value distributions fitted to annual
monthly temperature maxima and minima
Tagle et al. 2015
 CAM4 has to high return values for both, TMAX and TMIN
 Overestimation of extreme temperatures
 SKEBS and deteriorates 20yr return values for TMAX, but
slightly improves values for TMIN
Impact of SKEBS on precipitation bias
 Coupled control run shows significant bias due to split
inter-tropical convergence zone
 SKEBS reduced bias in precipitation
Outline
 The stochastic parameterization schemes
 Climate application: Impact in coupled and
uncoupled simulations with the Earth System
Model CESM
 Weather application: Improving reliability and
reducing analysis error in cycled and uncycled
forecasts with the weather model WRF
Representing initial uncertainty by an
ensemble of states

Forecast uncertainty in weather models:



Initial condition uncertainty
Model uncertainty
Boundary condition uncertainty

Represent initial forecast uncertainty by
ensemble of states

Reliable forecast system: Spread should
grow like ensemble mean error


\
Predictable states with small error
should have small spread
Unpredictable states with large error
should have large spread
RMS error
t0
t1
ensemble mean
analysis
t2
Resolution
g)
h)
CNTL
0.04
0.09
SKEBS
0.08
PHYS10_SKEBS
0.03
PARAM
PHYS10
PHYS3_SKEBS_PAR
Spread and error near the surface
Resolution
SKEBS
PHYS10
PHYS10_SKEBS
PHYS3_SKEBS_PARAM
Brier
Skill Score
Spread;Error
PARAM
0.06
0.02
i)a)
CNTL
0.07
0.05
0.2
2
0.1
1.5
1
0
0.5
0
0.11
12
24
36
48
Forecast Lead time
Brier Score
0.2
Dashed: spread
60
0
0
d)
12
0.12
24
36
48
Forecast Lead time
0.24
0.14
Ensemble is underdispersive
(= not enough
0.16
spread)
0.26

0.18
Unreliable and over-confident
0.28

Depending on cost-loss0.2
ratio potentially large
socio-economic impactf)(e.g. should roads be
salted)
0.22
e)
liability
Temperature at 2m
0.3
3
2.5
0.2
2
c)
Solid lines: rms
error of ensemble
mean
j)b)
Zonal Wind U at 10m
0.02
0.04
0.01
0.02
60
Resolution
Reliability
0.28
g)
0.04
e)
h)0.2
f)
0.09
0.02
0.03
0.08
0.01
0.07
CNTL
PARAM
SKEBS
PHYS10
PHYS10_SKEBS
PHYS3_SKEBS_PAR
Brier skill score near the surface
Resolution
0.06
i)a)
2.5
g)
SKEBS
PHYS10
PHYS10_SKEBS
PHYS3_SKEBS_PARAM
Resolution
Brier
Skill Score
Spread;Error
CNTL
PARAM
0.06
0.02
0.05
0.03
0.04
0.02
j)b)
0.3
h)
3
Zonal Wind U at 10m
0.2
0.04
2
0.1
1.5
0.03
1
0
0.5
0.02
0
12
24
36
48
Forecast Lead time
60
0.11
0.07
0.06
0
0.050
d)
0.12
j)
SKEBS
PHYS10
PHYS10_SKEBS
PHYS3_SKEBS_PAR
12
24
36
48
Forecast Lead time
60
0.16
0.2
0.18
0.1
0.2
0.280
liability
PARAM
0.3
0.14
0.22
0.2
0.24
0.1
0.26
e) 0
Brier skill measures
probabilistic skill in regard to a
0.02
reference (here CNTL).
Verified event: μ<x<μ+σ
0.04
CNTL
0.09
0.2
2
0.08
Brier Score
Brier Skill Score
c)
i)0.2
Temperature at 2m
12
24
36
48
Forecast Lead time
60
f) 00
0.01
0.02
12
24
36
48
Forecast Lead time
60
Berner et al., et al 2015
Reliability diagram for rain-thresholds, averaged over
forecast hours 18–36 using a 50-km neighborhood
Romine et al., 2014
WRF-DART: Verification of surface analysis
against independent observations
V-10m
CNTL
SKEBS
PHYS
T-2m
 Including a model-error representation reduces the RMS
error of the surface analysis (also prior) in 10m wind and
Temperature at 2m
Ha et al. 2015
Sketch: WRF behavior
Verifying observation
 Including a stochastic
parameterization
increased ensemble
spread
Potential
without stochastic
perturbations
 In cycled forecasts is
reduces the mean
analysis error
Potential
with stochastic
perturbations
Debate in the field: A priori vs a posteriori
Model uncertainty
added a posteriori:
Model
Process uncertainty
added a priori
during model
development:
Stochasticity
Forecast
uncertainty
Conclusions
 Random numbers can improve
weather and climate predictions
 by impacting variability and mean
 in expected (increase variability) and
unexpected (decrease variability) ways

Berner, J, K. Fossell, S.-Y. Ha, J. P. Hacker, C. Snyder 2015: “Increasing the
skill of probabilistic forecasts: Understanding performance improvements
from model-error representations, Mon. Wea. Rev., 143, 1295-1320

Berner, J., S.-Y. Ha, J. P. Hacker, A. Fournier, C. Snyder, 2011: “Model
uncertainty in a mesoscale ensemble prediction system: Stochastic versus
multi-physics representations” , Mon. Wea. Rev, 139, 1972-1995

Romine, G. S., C. S. Schwartz, J. Berner, K. R. Smith, C. Snyder, J. L.
Anderson, and M. L. Weisman, 2014: “Representing forecast error in a
convection-permitting ensemble system”, Mon. Wea. Rev, 142, 12,
4519–4541

Ha, S.-Y., J. Berner, C. Snyder, 2015: “Model-error representation in
mesoscale WRF-DART cycling”, under review at Mon. Wea. Rev.