Quarterly Journal of the Royal Meteorological Society Q. J. R. Meteorol. Soc. 00: 1–27 (2017)
Stochastic representations of model uncertainties at ECMWF:
State of the art and future vision
Martin Leutbecher
a∗
, Sarah-Jane Lock
a
, Pirkka Ollinaho
b
, Simon T. K. Lang
a
,
Gianpaolo Balsamo
a
, Peter Bechtold
a
, Massimo Bonavita
a
, H. M. Christensen
c
,
Michail Diamantakis
a
, Emanuel Dutra
a
, Stephen English
a
, Michael Fisher
a
, Richard M. Forbes
a
,
Jacqueline Goddard
a
, Thomas Haiden
a
, Robin J. Hogan
a
, Stephan Juricke
c
, Heather Lawrence
a
,
Dave MacLeod
c
, Linus Magnusson
a
, Sylvie Malardel
a
, Sebastien Massart
a
,Irina Sandu
a
,
Piotr K. Smolarkiewicz
a
, Aneesh Subramanian
c
, Fr
´
ed
´
eric Vitart
a
, Nils Wedi
a
, Antje Weisheimer
a,c,d
a
European Centre for Medium-Range Weather Forecasts, Reading, UK
b
Finnish Meteorological Institute, Helsinki, Finland
c
Atmospheric, Oceanic and Planetary Physics, University of Oxford, UK
d
Department of Physics, National Centre for Atmospheric Science, University of Oxford, UK
∗
Correspondence to: M. Leutbecher, ECMWF, Shinfield Park, Reading, RG2 9AX, UK; E-mail: M.Leutbecher@ecmwf.int
Members in ensemble forecasts differ due to the representations of initial uncertainties
and model uncertainties. The inclusion of stochastic schemes to represent model
uncertainties has improved the probabilistic skill of the ECMWF ensemble by
increasing reliability and reducing the error of the ensemble mean. Recent progress,
challenges and future directions regarding stochastic representations of model
uncertainties at ECMWF are described in this paper. The coming years are likely to
see a further increase in the use of ensemble methods in forecasts and assimilation. This
will put increasing demands on the methods used to perturb the forecast model. An area
that is receiving a greater attention than 5 to 10 years ago is the physical consistency of
the perturbations. Other areas where future efforts will be directed are the expansion of
uncertainty representations to the dynamical core and to other components of the Earth
system as well as the overall computational efficiency of representing model uncertainty.
Key Words: ensemble forecasts, ensemble data assimilation, weak-constraint 4D-Var, numerical weather prediction,
dynamical core, Earth system model, model uncertainty, stochastic parametrization
Received . . .
1. Introduction
Weather forecasting is an initial value problem solved with
numerical models that describe the evolution of the state of the
atmosphere and other Earth system components interacting with
it. Since the seminal work of Edward Lorenz in the 1960s, it is
understood that the equations describing the atmospheric flow
exhibit sensitive dependence on initial conditions that leads to
forecast error growth and eventual loss of predictability (Lorenz
1969; Buizza and Leutbecher 2015).
Some forecast error will be due to model imperfections even
if the model had been initialised with an initial state that
corresponded exactly to the true state of the Earth system. We
propose to define model error as the error in forecasts and model
climate that would be observed had the model been initialised with
the initial state corresponding exactly to the true state. This is not
without subtlety as the map from the physical system to the state
space of the model involves some filter operation on the scales that
are absent in the model. This will render the state of the model
that corresponds exactly to the true state ambigous. By assuming
there is a model state corresponding to the true state of the system,
we include the filtering operation of the initial conditions in our
definition of model error.
Due to the chaotic nature of geophysical fluid dynamics, model
error will limit predictability in addition to initial error. We
expect that for systems exhibiting sensitive dependence on initial
conditions, solutions will also exhibit sensitive dependence on
the models used in the numerical integrations. This becomes
obvious when considering an n-day forecast as an (n − `)-day
forecast initialised from an `-day forecast (` < n). The (n − `)-
day forecast depends sensitively on its initial condition, which is
the `-day forecast, which in turn is model-dependent.
The process of translating the laws of physics governing
the evolution of the Earth system to a numerical code,
i.e. the “model”, inevitably requires many simplifications and
approximations. These arise from incomplete knowledge of a
physical process and its ancilliary data, reduced complexity
to limit computational costs, omission or mis-representation of
processes and system components, from uncertain parameters in
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2 M. Leutbecher et al.
parametrizations that do not have a directly observable equivalent,
and last but not least from discretisation. Related to the latter are
errors due to the omission of fluctuations on the unresolved scales
as discussed for instance by Palmer (2001).
Quantification of forecast uncertainties became established
in operational numerical weather prediction (NWP) in the
1990s using ensemble forecasting techniques (Lewis 2005). The
physical equations describing the evolution of the atmosphere are
nonlinear and therefore ensemble-based Monte-Carlo approaches
appear as the only feasible way of estimating the future probability
distribution of the state of the atmosphere. Building an ensemble
forecast system requires the specification of the sources of
uncertainties. It seems useful to distinguish between uncertainties
in the initial conditions and uncertainties in the forecast model
both in terms of the methods as well as the implications for
predictability. From now on, for brevity, we will refer to them as
initial uncertainties and model uncertainties, respectively. In this
paper, we distinguish between actual model errors, where there is
only one realisation per model and forecast, and model uncertainty
representations, which sample model perturbations from some
distribution to make a prediction of the forecast uncertainty. The
perturbations should ideally have the same statistics as the error
but are likely to have different statistics in practical applications.
An ensemble that represents only initial uncertainties consistent
with the true distribution of initial condition errors is known to
be underdispersive and therefore will lack reliability (e.g. Wilks
2005; Palmer et al. 2005). This motivates the development of
methodologies to represent model uncertainties.
In the following, we will focus on stochastic representations
of model uncertainties (e.g. Palmer 2012). These are schemes
that sample perturbations from some underlying distribution that
defines the scheme that represents model uncertainties. A key
advantage is that these schemes involve a distribution that can
be adjusted to control the characteristics of the model error
representation. Generally, the aim of stochastic representations
of model uncertainty is to simulate the effect of the random
component of the errors of the model tendencies. However, this
may include flow-dependent systematic errors that appear like
random errors.
ECMWF first introduced a stochastic representation of model
uncertainties in the medium-range ensemble in October 1998
using a scheme that multiplies the total parametrized physics
tendencies with a random number (Buizza et al. 1999). The
scheme was originally referred to as “stochastic physics”. It
is now commonly referred to as the Stochastically Perturbed
Parametrization Tendency scheme (SPPT). Major revisions of
the stochastic representation of model uncertainties took place in
September 2009 and November 2010 (Palmer et al. 2009; Shutts
et al. 2011). These changed aspects of the probability distribution
sampled by the SPPT scheme. In addition, a Stochastic Kinetic
Energy Backscatter scheme (SKEB) was activated in ECMWF
ensemble forecasts in the November 2010 upgrade.
As ensembles are becoming more widely used in forecasting
and assimilation, the need for representations of model
uncertainties is increasingly recognised and this fuels an
expansion of research on this topic. In April 2016, ECMWF
and the World Weather Research Programme (WWRP) jointly
organised a workshop on model uncertainty with over 80
participants. The workshop proceedings (ECMWF/WWRP 2016)
contain summaries of the talks as well as the recommendations
from the three working groups. Looking at the research presented
at the workshop and reviewing the literature reveals that a range
of alternative approaches to represent model uncertainties have
been explored and are currently developed across the wider
community (see references in Secs. 2, 3 and 7.2). The main
reasons for the multitude of approaches are that: (i) it is difficult
to accurately characterise model error and (ii) there are many
different sources of model error. Thus, it is not straighforward
to decide whether one way of representing model error is better
than another. A related challenge is the difficulty in disentangling
initial uncertainties from model uncertainties as the estimation of
the initial state involves the use of a forecast model.
This paper is based on a special topic paper prepared for
the ECMWF Scientific Advisory Committee held at ECMWF
in October 2016. Its purpose is to (i) report on progress that
has been made since the last special topic paper on stochastic
parametrization prepared for the ECMWF Scientific Advisory
Committee in 2009 (Palmer et al. 2009) and to (ii) discuss ideas
that influence plans for future work on the representation of model
uncertainties in ECMWF’s prediction system. Forecast errors also
arise from systematic errors that lead to biases. Such errors and
work on identifying their root causes are beyond the scope of this
paper. However, the impact of stochastic representations of model
uncertainties on the model climate will be covered. This paper will
also look at the use of model uncertainty representations in data
assimilation.
The outline of the paper is as follows. The operational
stochastic representations of model uncertainties used at ECMWF
are described in Section 2. Work on developing a process-
oriented representation of model uncertainties is summarized in
Section 3. The impact of model uncertainty representations on
a range of applications at ECMWF is documented in Section 4.
Unrepresented sources of model uncertainty in the Earth system
are discussed in Section 5. Future directions for work on model
uncertainty are presented in Section 6. We conclude with a brief
discussion and summary in Sections 7 and 8.
2. Operational methods in the IFS
The Integrated Forecasting System (IFS) is ECMWF’s operational
NWP model. This section is dedicated to the operational
stochastic parametrizations used in the IFS, i.e. SPPT and SKEB.
2.1. The Stochastically Perturbed Parametrization Tendency
scheme
The Stochastically Perturbed Parametrization Tendency scheme
(SPPT) makes the assumption that the dominant error of
the parametrized physics tendency is proportional to the net
physics tendency (Buizza et al. 1999). SPPT generates perturbed
parametrization tendencies p stochastically by multiplying the net
physics tendency p
D
provided by the physics package with a 2D
random field r
p = (1 + µr)p
D
. (1)
Here, p denotes the vector of perturbed tendencies of temperature,
specific humidity, and wind components in a model column. The
factor µ is an optional tapering function that depends on the model
level only and has values in the range [0, 1].
The number of NWP models in which a SPPT scheme
has been implemented keeps rising. It is used operationally
in global ensembles by Environment Canada (Charron et al.
2010, Separovic 2016, ECMWF/WWRP workshop) and by Japan
Meteorological Agency. M
´
et
´
eo-France uses SPPT in the regional
ensemble based on the Application of Research to Operations
at Mesoscale convection-permitting model (AROME, Bouttier
et al. 2012). It has been tested as well in the Weather Research
and Forecasting model (WRF) in ensembles with parametrized
convection and in convection-permitting ensembles (Berner et al.
2015; Romine et al. 2014). Pegion (2016, ECMWF/WWRP
workshop) reported recent tests of SPPT in the United States
National Centers for Environmental Prediction’s Global Forecast
System model (GFS) and Sanchez et al. (2016) describe tests
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Stochastic representations of model uncertainties 3
in the Met Office Unified Model. In summary, SPPT has been
found to be effective in generating additional ensemble spread and
improving probabilistic skill in a range of NWP ensembles.
Shutts and Pallar
`
es (2014) diagnosed coarse-grained differ-
ences of tendencies from integrations with different horizontal
resolution and examined the relationship between the width of
the probability distribution and the mean value of the tendency of
the coarser resolution model. They find different relationships for
different physical processes, e.g. convection and radiation. This
finding motivated work by Arnold (2013), who explored the effect
of applying noise independently to the tendencies from radiation,
vertical mixing and orographic drag, convection, cloud processes,
non-orographic drag, and methane oxidation. Christensen et al.
(2016) summarize more recent tests with this so-called indepen-
dent SPPT (iSPPT), which results in an increase in ensemble
spread in regions with significant convective activity. The iSPPT
approach also offers a larger degree of flexibility, which could
facilitate sensitivity studies.
Recently, studies by Davies et al. (2013) and Peters et al.
(2013) raised doubts whether the multiplicative ansatz in SPPT
was consistent with observations of the relationship between
deep convection and the large-scale state. Watson et al. (2015)
examined this relationship for the IFS and found that it can
reproduce the observed relationships in a deterministic integration
as well as in an integration perturbed with SPPT.
2.1.1. The ECMWF methodology
The random field r in (1) is obtained through first order auto-
regressive processes in spectral space. A multi-scale approach is
implemented in IFS with
r =
J
X
j=1
r
j
(2)
where the component random fields r
j
are independent and
represent different scales. Table 1 lists the standard deviation in
grid point space, the spatial auto-correlation scale L and the time
decorrelation scale τ of the components of the three-scale pattern
(J = 3) used in the medium-range, extended-range and seasonal
(System 4) ensemble forecasts since 2010. Shutts et al. (2011)
show an example of a realisation of the three components. The
ensemble of 4D-Var data assimilations (EDA) uses the single-
scale pattern with the fast small-scale pattern only (J = 1) mainly
for reasons of technical simplicity and lack of investigations
assessing the impact of the slower and larger-scale correlations
on the background error structures. A first assessment of using the
three-scale pattern in the EDA is summarized in Section 4.4.
The implementation of SPPT in the Integrated Forecast System
(IFS) uses a tapering function µ which is 1 in the free troposphere
and reduces the amplitude gradually to 0 close to the surface and
in the stratosphere. The latter is done to avoid large amplitude
perturbations of the radiative tendencies in the stratosphere, which
are presumed to be more accurate, and the former to avoid
numerical instabilities in the boundary layer.
Apart from the limiters mentioned below, r samples a
Gaussian distribution with a grid-point space variance of
σ
2
=
P
J
j=1
σ
2
j
, with σ
2
j
being the grid-point variance of the
component pattern r
j
. When r falls outside of the range [−1, 1],
the magnitude of r is reduced to 1. This is required for
numerical stability and avoids sign reversal of the tendencies.
In addition, a supersaturation limiter reduces the magnitude of
r to avoid excessive supersaturation due to the perturbation.
The supersaturation limiter determines a vertically consistent
reduction of the amplitude of the T and q perturbations based on
a mean over the 200 hPa layer requiring the largest reductions.
scale j σ L (km) τ (d)
1 0.52 500 0.25
2 0.18 1000 3
3 0.06 2000 30
Table 1. Characteristics of the three-scale random field used in SPPT: Standard
deviation σ, horizontal decorrelation length scale L, time decorrelation scale
τ. The operational EDA uses only the first scale j = 1.
2.1.2. Global fix for tendency perturbations
In long runs with the EC-Earth climate model version 3.1, SPPT
caused substantial imbalances in the radiative fluxes, the surface
fluxes of precipitation (P) and evaporation (E) and for the radiation
budgets (Davini et al. 2017). The SPPT scheme systematically
reduced humidity in the atmosphere which was compensated
by overly strong evaporation. The global P−E imbalance in the
stochastically perturbed simulations was increased from about
−0.016 mm d
−1
in the runs without SPPT to about −0.16
mm d
−1
. This is based on a 10-year mean of simulations with
and without SPPT at TL255L91 resolution
∗
. In 30-day forecasts
with IFS CY41R1, a qualitatively similar signal is diagnosed with
an increase of the P−E imbalance from a value of 0.03 mm d
−1
to a value of −0.15 mm d
−1
when SPPT is activated. It is worth
noting that P and E are the raw model output as provided to users
and consistent with the practices in the forecast evaluation without
accounting explicitly for imbalances introduced by SPPT (see also
Sec. 6.2.2).
SPPT also had a significant impact on the energy fluxes
at the surface and the top of the atmosphere. The 10-year
mean of global top-of-the-atmosphere net flux changed from
−1.71 to −2.77 W m
−2
through the activation of SPPT while
the surface net flux changed from −0.46 to −3.66 W m
−2
in the
aforementioned EC-Earth runs. Imbalances exceeding ∼ 1 W m
−2
are considered unacceptable for climate models (Mauritsen et al.
2012). For comparison, the present-day radiative forcings due to
anthropogenic well-mixed greenhouse gases is about 3 W m
−2
(Myhre et al. 2013).
To address these imbalances, a modification of the SPPT
scheme was developed. A correction is added to the perturbed
tendency, which results in the global integral of the perturbed
tendency being equal to that of the unperturbed tendency. Details
are described in Appendix A. In terms of medium-range and
extended-range ensemble forecast scores, it was found that this
SPPT modification has a neutral to slightly positive impact. In
view of this impact and the fact that it makes the climate of
the forecasts perturbed with SPPT more similar to the climate of
unperturbed forecasts, this modification of SPPT was activated in
the model upgrade (CY43R1) in November 2016.
2.1.3. Discussion
SPPT can be viewed as a “holistic approach” that maintains
the overall balance between the tendencies due to different
physical processes (Tim Palmer, pers. comm.). In the presence of
compensating biases of the tendencies from different processes,
the bias of the total tendency can be smaller than the biases of
individual tendencies. A potential strength of SPPT is that its
formulation results in unbiased perturbations of the physics when
supersaturation limiter, interactions and subsequent nonlinearities
are neglected. The problems with the systematic impact on the
humidity tendencies discussed above indicate that in practice it is
difficult to obtain unbiased perturbations with SPPT.
∗
Henceforth, we will refer to triangular truncation at wavenumber NNN using a
linear grid by TLNNN; the L91 refers to the 91 level vertical discretisation with a
model top at 1 Pa.
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The mean and the perturbation structure of the tendencies
are controlled by the IFS physics. In the limit of vanishing
perturbation variance, the scheme converges to the deterministic
IFS physics. The scheme is relatively simple and it is
computationally cheap: less than 2% of total runtime, or less than
4% with the fix described in Sec. 2.1.2. Furthermore, the scheme
requires only modest resources for maintenance. As will be seen
later, it is efficient in generating ensemble spread and contributes
positively to the probabilistic skill of the ECMWF ensemble
forecasts. There are also noticeable impacts on the model climate
(cf. Sec. 4.3) and on the EDA and thus data assimilation in general
(cf. Sec. 4.4).
However, there are also several limitations of SPPT. It assumes
that the error of the physics tendency is always proportional to
the deterministic tendency p
D
while the true uncertainty is likely
to have also variance in many directions not parallel to p
D
.
Considering a single time step, the perturbations are confined
to a one-dimensional subspace in a space with a dimension of
a few hundred (4 variables times the number of model levels).
For example, uncertainty in the shape of a heating profile cannot
be captured with SPPT. The validity of the simple multiplicative
ansatz has previously been questioned by Shutts and Palmer
(2007) as well as Shutts and Pallar
`
es (2014). Their coarse-grained
tendencies show evidence of non-vanishing uncertainty when the
parameterised physics tendency vanishes.
The modulation of the amplitude of the perturbation with
the total tendency implies that the same level of uncertainty is
assigned to all processes and all atmospheric situations. This
may not reflect actual variations in model uncertainty well. For
instance, one would expect that on average longwave radiative
cooling in a clear-sky situation is more certain than the heating
associated with parametrized deep convection. Moreover, the
amplitude modulation implies that the error vanishes where the
total tendency is zero while the true error distribution will
also be influenced by non-compensating errors of the individual
processes.
Another drawback of the current formulation of SPPT is
that it does not respect conservation laws. Fluxes at the top of
the atmosphere and the surface are not perturbed. This implies
that local budgets of energy and moisture that are satisfied
by the deterministic parametrizations are violated by the SPPT
perturbations.
All implementations of SPPT share the basic principle that the
parametrized total physics tendency is multiplied by a random
number. However, there are potentially important differences in
detail. These differences include the variance of the perturbations,
the space and time auto-correlation of the random pattern, the
shape of the distribution that is sampled and whether perturbations
are suppressed in some regions of the atmosphere or for some
processes. For example, the implementations of SPPT in the
model of the COnsortium for Small-scale MOdelling (COSMO)
and the WRF model do not use the tapering to zero of the
perturbations in the boundary layer. While these differences
will limit the ability to transfer conclusions obtained with one
model to other models, they may motivate a range of sensitivity
experiments in the future.
2.2. The Stochastic Kinetic Energy Backscatter scheme
Stochastic Kinetic Energy Backscatter scheme (SKEB) aims to
represent model uncertainties associated with scale interactions
that take place in the real atmosphere but are absent in a truncated
numerical model. Motions on scales that would be fully resolved
in the model interact with motions on scales that would be near
grid-scale or subgrid-scale in the model.
Following ideas in Large Eddy Simulation (LES), Shutts (2005)
and Berner et al. (2009) developed a stochastic forcing for the IFS
model targeting this uncertainty. The SKEB scheme is also used
in the Environment Canada and in the United Kingdom MetOffice
(UKMO) global ensembles (Charron et al. 2010; Tennant et al.
2011). Sanchez et al. (2016) propose improvements to the SKEB
scheme used by UKMO. Berner et al. (2011) and Berner et al.
(2015) study the impact of SKEB on WRF ensembles. Pegion
(2016, ECMWF/WWRP workshop) summarizes initial tests of
SKEB in global GFS ensembles. We note in passing that there
are potentially important differences in the detail of the various
implementations of SKEB in different models. For example, the
versions in the Canadian model and in the Met Office model
perturb different ranges of wavenumbers and the implementation
in WRF is purely additive, i.e. state-independent.
Recently, Shutts (2015) introduced a stochastic convective
backscatter scheme, which focusses entirely on the random model
error arising from the interaction of parametrized deep convection
with the model dynamics near the grid scale.
2.2.1. The ECMWF methodology
In the IFS, SKEB introduces a stochastic streamfunction forcing
∂Ψ/∂t given by an evolving 3-dimensional pattern F with an
amplitude modulation determined by a horizontally smoothed
local estimate of kinetic energy sources at the subgrid-scale,
referred to as the dissipation rate D
∂Ψ
∂t
SKEB
= [bD]
1/2
F (3)
Here b is the backscatter ratio that controls the global amplitude
of the perturbations. The horizontal structure of the pattern is
determined by a power-law variance distribution in spectral space
while the vertical structure is obtained from random-phase shifts
that decorrelate the structure in the vertical. The dissipation rate
is a 3D field diagnosed from the model state. In the original
version of the scheme, the dissipation rate estimate is obtained
as the sum of three terms describing subgrid-scale energy sources
due to orographic gravity wave drag, numerical dissipation and
parametrized deep convection. The streamfunction perturbations
are tapered to zero in the boundary layer similar to the approach
used by ECMWF for SPPT.
The scheme evolved recently to a version with only
the contribution from deep convection. The dissipation rate
contribution from orographic gravity-wave drag was abandoned as
it generated excessive ensemble spread in lower tropospheric wind
near steep orography, i.e. the Andes. The term linked to numerical
dissipation estimated from the horizontal diffusion resulted in
spurious kinetic energy spectra with the new cubic octahedral
grid (Malardel et al. 2016) implemented in March 2016 due to
the inconsistency in representing horizontal mixing in the model
and in SKEB. Therefore, numerical dissipation was deactivated in
SKEB.
The computational cost of SKEB is higher than that of SPPT
due to requiring spectral transforms of two 3D fields: the pattern
F and the dissipation rate estimate D. The operational version
does not update these fields every model time step to save
computational time.
2.2.2. Discussion
As described above, the IFS implementation of SKEB has
evolved to a scheme that is conceptually similar to the Stochastic
Convective Backscatter algorithm (SCB) proposed by Shutts
(2015). The latter scheme is effective in generating ensemble
spread and introduces strong wind perturbations in the boundary
layer. However, at present it is unknown whether these low-level
perturbations are a good or a bad characteristic and whether they
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Stochastic representations of model uncertainties 5
target the main model uncertainties associated with deep moist
convection. Another difference between the operational version
of SKEB and Shutts’ SCB is that the latter perturbs the velocity
potential and the streamfunction on synoptic scales in the vicinity
of parametrized deep convection while the former is limited to
streamfunction perturbations.
With increasing spatial resolution, the model is able to represent
more of the scale interactions explicitly and the need for SKEB
should be reduced. This, could be at least partly a justification
for deactivating the numerical dissipation rate contribution with
the higher resolution TCo639 ensemble
†
. Nevertheless, until
deep convection is explicitly resolved, one may see a need
for including a stochastic convective backscatter. However, the
stochastic convective momentum transport perturbations can also
be introduced in an alternative manner (cf. Section 3). Recent
work by Malardel and Wedi (2016) looking at the energy
transfer in high-resolution, convection-permitting IFS integrations
indicates that backscatter happens through available potential
energy.
There is a generic challenge in keeping the actual perturbations
used by SKEB in line with the spatial scales that can be justified
theoretically — while large-scale perturbation structures appear
to be most effective in generating ensemble spread, the dominant
uncertainty due to the interaction with the unresolved scales
should be at scales close to the truncation scale.
3. Towards process-level representation of model
uncertainties
Parametrizations of physical processes include a number of
tunable parameters, which quantify efficiencies, rates of change,
etc. of phenomena that the parametrizations seek to represent.
Uncertainty in the parameter values leads to a source of model
uncertainty. Sampling values of uncertain key parameters provides
a way to represent uncertainties at their sources and thus links to
individual physical processes.
A scheme with stochastically perturbed parameters has
been developed in the UKMO global ensemble (Bowler
et al. 2008b) and later, applied in the UKMO convection-
permitting ensemble (Baker et al. 2014). The original Random
Parameters (RP) scheme was based on parameters that vary
stochastically, but discontinuously, in time. McCabe et al.
(2016) report improvements of ensemble fog forecasts in the
2.2 km regional UKMO ensemble from implementing the RP
scheme. In their experiments, the parameters vary more gradually
in time (”RP2”) than in previous implementations of the
RP scheme. Highlighting future developments, Tennant (2016,
ECMWF/WWRP workshop) demonstrated further improvement
in global ensemble forecasts from enabling the stochastic
parameters to vary in time and space (”RP3”).
Stochastic parameter perturbations are beginning to be
considered beyond the atmosphere in other Earth system
components. Juricke et al. (2014) have explored the impact
of stochastic perturbations of sea ice strength on sea ice
predictability. Brankart et al. (2015) introduce generic uncertainty
representations in the Nucleus for European Modelling of the
Ocean model (NEMO) that include a form of SPPT as well
as stochastic parameter perturbations. They propose exploring
stochastic parameter perturbations to represent uncertainty in
marine ecosystem modelling that arises from restricting the
diversity of species. In addition, they also propose perturbing
the sea ice strength parameter. Stochastic and fixed parameter
perturbations also start to be explored in the H-TESSEL
‡
land
†
We refer to triangular truncation at wavenumber NNN using a cubic octahedral
grid by TCoNNN.
‡
Revised Hydrology for the Tiled ECMWF Scheme for Surface Exchanges over
Land
Table 2. The parameters and variables perturbed by SPP in the physical
process parametrization schemes.
TURBULENT DIFFUSION & SUBGRID OROGRAPHY
transfer coefficient for momentum (ocean/land)
coefficient in turbulent orographic form drag scheme
stdev. of subgrid orography
length scale for vertical mixing in stable boundary layer
CONVECTION
entrainment rate
shallow entrainment rate
detrainment rate for penetrative convection
conversion coefficient cloud to rain
zonal convective momentum transport
meridional convective momentum transport
adjustment time scale in CAPE closure
CLOUD & LARGE-SCALE PRECIPITATION
RH threshold for onset of stratiform cond.
diffusion coefficient for evaporation by turbulent mixing
critical cloud water content for autoconversion
threshold for snow autoconversion
RADIATION
cloud vertical decorrelation height in McICA
fractional stdev. of horizontal distribution of water content
effective radius of cloud water and ice
scale height of aerosol normalised vertical distribution
optical thickness of aerosol
surface model with the aim of representing model uncertainties
(MacLeod et al. 2016; Orth et al. 2016).
3.1. ECMWF’s Stochastically Perturbed Parametrization
methodology
At ECMWF, work has started on stochastic parameter per-
turbations in the framework of the Stochastically Perturbed
Parametrization scheme (SPP, Ollinaho et al. 2017). SPP provides
a framework in the IFS code to represent some of the key random
errors of the parametrized tendencies close to their sources within
the physical processes. Like SPPT, SPP is strongly guided by
the existing deterministic parametrizations. The proximity of the
uncertainty model to the processes permits the exploitation of
physically consistent relationships between different variables.
For instance, SPP can produce perturbations to the fluxes at
the top of the atmosphere and the surface that are consistent
with the tendency perturbation in the model column. Thus local
budgets of moisture, momentum and energy, which are respected
by the deterministic parametrizations, remain closed. SPP can also
represent uncertainty beyond a simple amplitude error, e.g. the
uncertainty in the shape of a heating profile.
SPP can be seen as a generalisation of the concept of perturbed
parameters as it introduces local stochastic perturbations to
parameters and variables in the parametrizations. Ollinaho et al.
(2017) describe the methodology in detail and give a justification
for the selection of parameters and variables. During the
initial development phase of SPP, experts working on the IFS
parametrization of individual processes identified 20 parameters
and variables that are considered uncertain and when changed
introduce significant changes in the forecast (Tab. 2).
Figure 1 displays the distributions sampled by SPP in the
parametrizations of (a) turbulent diffusion and subgrid orographic
drag, (b) convection, (c) cloud and large-scale precipitation
and (d) radiation. SPP samples the distributions independently
for each parameter and variable. Thus, the perturbations are
uncorrelated. The perturbations evolve using the same type of
auto-regressive AR(1) pattern generator in spectral space as
c
2017 Royal Meteorological Society
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