NUMERICAL WEATHER PREDICTION
Atmospheric predictability: Ensembles
Dr Meral Demirtaş
Turkish State Meteorological Service
Weather Forecasting Department
WMO, Training Course, 26-30 September 2011
Alanya, Turkey
Outline
• Introduction
• How predictable is the atmosphere?
• Why do we need ensembles?
• What are the commonly used ensemble
techniques?
A Schematic illustration of ensemble prediction.
(Kalnay, 2003)
Predictability depends on how stable the
dynamical systems are….
– Unstable systems have finite predictability (chaos)
– Stable systems are infinitely predictable
(Kalnay, 2003)
What may lead to forecast error growth?
An NWP system may lose skill over the time because of the growth of
errors in the initial conditions (initial uncertainties) and also numerical
models describe the laws of physics only approximately -which causes
model uncertainties. Predictability is flow dependent. The Lorenz Chaos
Model (1963) illustrates this view:
(ii)
(i)
(iii)
(Palmer et al., 2007)
The prototypical Lorenz model of loworder chaos, showing that in a nonlinear system, predictability is flow
dependent. (i) A forecast with high
probability (stable), (ii) forecast with
moderate predictability (less stable),
(iii) forecast with low predictability
(unstable).
Initial conditions and the butterfly effect
In chaos theory, the butterfly
effect is the sensitive dependence
on initial conditions; where a
small change at one place in a
nonlinear system can result in
large differences to a later state.
An attractor is a set towards
which a dynamical system
evolves over time. That is,
points that get close enough to
the attractor remain close even
if slightly disturbed.
Lorenz Chaos Model (1963) in a nutshell
Lorenz (1963) introduced a 3variable model that is prototypical
example of chaos theory. The
system is non-linear (it contains
products of the dependent
variables) but autonomous (the
coefficients are time-independent).
The solution obtained by
integrating the differential
equations in time is called a flow.
A plot of the trajectory of
Lorenz system for values:
ρ=28, σ=10, β=8/3
Lorenz Chaos Model (1963) Equations:
where σ is called the Prandtl number and ρ
is called the Rayleigh number. All σ, ρ, β >
0, but usually σ=10, β=8/3 and ρ is varied.
The parameters σ, ρ, β are kept constant
within an integration, but they can be
changed to create a family of solutions of
the dynamical system defined by the
differential equations. The particular
parameter values chosen by Lorenz (1963)
were: σ=10, β=8/3 and ρ=28 -which result
in chaotic solutions (sensitively dependent
on the initial conditions).
A Schematic illustration of the evolution of a small
spherical volume in phase space in a bounded
dissipative system
(Kalnay, 2003)
The probabilistic approach to NWP:
ensemble prediction
The predictability problem
may be explained in terms
of the time evolution of an
appropriate probability
density function (PDF).
Ensemble prediction based
on a finite number of
deterministic integration
seems to be a feasible
method to predict the PDF
beyond the range of linear
growth.
Ensemble prediction basics
• An Ensemble Prediction System is a set of
integrations of one or several NWP models
that differ in their initial states and/or in their
configurations and boundary conditions.
• Ensemble prediction is an attempt to estimate
the non-linear time evolution of the forecast
error probability distribution function (PDF).
• With Ensemble forecast, it is possible
to evaluate, express and forecast uncertainty.
Some Ensemble Prediction Techniques
• Singular Vectors (SVs)
Singular vectors are the linear perturbations of a control forecast
that grow fastest within a certain time interval (Lorenz, 1965),
known as “optimization period”, using a specific norm to measure
their size.
• Bred Vectors (BVs)
Breeding is a nonlinear generalization of the method to obtain
leading Lyapunov vectors, which are the sustained fastest
growing perturbations. Bred Vectors (like leading Lyapunov
Vectors) are independent of the norm and represent the shapes of
the instabilities growing upon the evolving flow.
• Multiple data assimilation ensembles (EnKF, ETKF, LETKF)
• Accounting for model related uncertainties (see the next slide)
• Multi-model/multi-system ensembles
Main strategies practiced for sampling
• Prioritized sampling: sample leading sources of forecast
error (prioritize). Rationale: Considering complexity and
high dimensionality of a forecasting system properly
sampling the leading sources of errors is crucial. Rank
sources, prioritize, optimize sampling: growing components
will dominate forecast error growth. Following this
strategy; ECMWF have been employing “singular vectors”
and some places uses “bred vectors”.
• All-inclusive Sampling: sample all sources of forecast
errors (uncertainties). Perturb any input (observations,
boundary fields, …), any model parameter that is not
perfectly known, and etc.
Singular Vectors (1)
• Perturbations pointing along
different axes in the phase-space
of the system are characterized by
different amplification rates. As a
consequence, the initial PDF is
stretched principally along
directions of maximum growth.
• The component of an initial
perturbation pointing along a
direction of maximum growth
amplifies more than components
pointing along other directions.
Singular Vectors (2)
At ECMWF, maximum growth is
measured in terms of total energy.
A perturbation time evolution is
linearly approximated:
The adjoint of the tangent forward
propagator with respect to the totalenergy norm is defined, and the
singular vectors, i.e. the fastest
growing perturbations, are computed
by solving an eigenvalue problem:
Breeding Vectors (1)
When there is an instability, all perturbations converge towards the fastest
growing perturbation (Leading Lyapunov Vector, LLV). The LLV may be
computed by applying the linear tangent model on each perturbation of the
nonlinear trajectory random initial.
A schematic illustration of how all perturbations will converge towards
the Leading Lyapunov Vector, LLV
(Kalnay, 2003)
Breeding Vectors (2)
• Breeding: Grow naturally unstable perturbations. It is a simple
generalization of Lyapunov vectors, for finite time, finite
amplitude.
• Breeding is simply running the nonlinear model a second time,
starting from perturbed initial conditions, rescaling the
perturbation periodically.
Two tuning parameters: rescaling
amplitude and rescaling interval.
Local breeding growth rate:
Breeding Vectors (3)
The breeding cycle has been designed to mimic the analysis
cycle. Each BV is computed by:
(i) adding a random perturbation to the starting
analysis,
(ii) evolving it for 24-hours (soon to 6),
(iii) rescaling it, and then repeat steps (ii-iii).
BVs are grown non-linearly at full model resolution.
All-inclusive (Canadian approach)
• Observational errors
(random perturbations);
• Imperfect boundary
conditions (perturbed
surface fields);
• Model errors (different
parameterizations and
random error component
added to the initial
perturbations).
Ensemble spread: Good and Bad…
An ensemble forecast starts from initial perturbations to the analysis.
In a good ensemble “truth” looks like a member of the ensemble
The initial perturbations should reflect the analysis “errors of the day”.
Kalnay, 2003
Sources of Uncertainties in an NWP System
• Uncertainties in initial conditions
• Uncertainties in Boundary Conditions
• Uncertainties in the NWP models
Uncertainties in initial conditions
• Measurement errors inherent to the instruments.
• Improperly calibrated instruments.
- Systematic errors (bias)
- Random errors
• Incorrect registration of observations.
• Data assimilation errors:
- Imperfect data quality control.
- Deficiencies in trial fields –the trial fields are usually 6-h
model
forecasts.
- Unrepresentative observations and model error statistics.
- Deficiencies in the data assimilation scheme
• Data coding and transmission errors.
• Lack of coverage –incomplete information.
• Representativeness error:
- Ideally an observing system should provide information on
all model variables, at each initial time, representative at the
model scale and on the model grid.
- Model unresolved scales are sampled by observations.
Uncertainties in Boundary Conditions
Another uncertainty lies in the assignment of values for key
variables at the bottom and lateral boundaries of an NWP
model. Bottom boundary values that might be perturbed
include:
• Sea-surface temperatures (SSTs)
• Soil moisture, snow cover, roughness length, vegetation
properties
• Prescribed boundary conditions (e.g., vegetation or soil
type)
Uncertainties in Bottom Boundary Conditions
Ocean related parameters
Land related parameters
Tropical Pacific SST
Mountain /No-Mountain
Arabian Sea SST
Forest /No-Forest (Deforestation)
North Pacific SST
Surface Albedo (Desertification)
Tropical Atlantic SST
Soil Wetness
North Atlantic SST
Surface Roughness
Sea Ice
Vegetation
Global SST (MIPs)
Snow Cover
Uncertainties in the NWP models
•
•
•
•
•
•
•
•
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Space and time truncation –resolution related,
Dynamics formulation,
Approximations due to numeric,
Effects of unresolved physical processes,
Physics parameterization,
Closure assumptions,
Lack of full understanding of physics of the atmosphere,
Coding errors,
Model jumpiness –higher resolution may increase model
jumpiness,
Accounting for model uncertainty
Stochastic Physics: Perturbations (which may be formulated with
spatial/temporal structures or other dependencies) to state variables’
tendency during model integration.
Stochastic Backscatter: Return dissipated energy via scale-dependent
perturbations to wind field.
Random Parameters: Random perturbations to physics parameters (e.g.,
entrainment rate), which may be fixed prior to model integration or varied
during model integration.
Perturbed Surface Parameters: Perturbations to surface temperature,
albedo, roughness length, etc., which may be fixed before model
integration or varied during model integration
Stochastic Parameterizations: Explicit modeling of the stochastic
nature of subgrid-scale processes
Coupling to Ocean/LSM Ensemble: Explicit modeling of the
considerable uncertainty from the surface boundary
TIGGE (the THORPEX Interactive Grand
Global Ensemble eXperiment)
It is a framework for international collaboration in development
and testing of ensemble prediction systems. Key objectives are:
•
Enhance collaboration on predictability and ensemble
prediction,
•
Foster the development of new methods of combining
ensembles from different sources and correcting
systematic errors (biases, spread over-/underestimation),
•
Help understanding the feasibility of interactive
ensemble system responding dynamically to changing
uncertainty (including use of adaptive observing, variable
ensemble size, on-demand regional ensembles).
TIGGE EPS framework
• Globally; 10 operational global, medium-range ensemble systems,
BMRC, CMA, CPTEC, ECMWF, FNMOC, JMA, KMA, MSC, NCEP and
UKMO, with horizontal resolution ranging from T62 to TL639 (~32km),
and with forecast lead time ranging from 8 to 16 days.
• Over Europe; 5 operational Limited-area EPSs, SRNWP-PEPS,
COSMO-LEPS, INM, LAMEPS and PEACE, that produce daily ~50
forecasts with horizontal resolution ranging from 7 to 25 km, and with
forecast length ranging from 30 to 120 hours. Six further LEPSs:
UKMO, DMI, HMS, MeteoSwiss, SAR, PIED-SE
• Over North-America; there is 1 operational Limited-area EPSs (NCEPSREF) that produces daily 30 forecasts with horizontal resolution of 32
km, and a 63-hour forecast length. Plus Canada’s (MSC) Ensemble
Prediction System.
• Over Australia; BMRC is testing a 16-member, 0.5 degree resolution,
72-hour LEPS.
Detailed the TIGGE ensembles info
Acknowledgements:
Thanks to documents/images of ECMWF,
E. Kalnay (UMD) and various others that
provided excellent starting point
for this talk!
Thanks for attending….