Statistical Downscaling

1
Statistical Downscaling

• Hans von StorchInstitute for Coastal
  ResearchGKSS Research CentreGeesthacht, Germany

18.10.2004 Central Weather Bureau, Taipei, Taiwan
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Concept of Empirical Downscaling
"Downscaling" is based on the view that regional
climate is conditioned by climate on larger, for
instance continental or even planetary, scales.
Information is cascaded "down" from larger to
smaller scales. The regional climate is the
result of interplay of the overall atmospheric,
or oceanic, circulation and of regional
specifics, such as topography, land-sea
distribution and land-use. As such,
empirical/statistical downscaling seeks to derive
the local scale information from the larger scale
through inference from the cross-scale
relationships, using a function F such
that local climate response F (external,
large scale forcing)
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Concept of Empirical Downscaling
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Jan
Example flowering date of snow drops in
Schleswig-Holstein (Germany) R spatial
distribution of the flowering date of galantis
nivalis (snow drop) in days anomalies L mean
air temperature in Jan., Feb., and March in
Europe anomalies F linear regression model
constructed through a CCA.
Feb
Mar
Maak von Storch, Intl. J. Biometeorol., 1997
Anomalies of flowering date of snow drops days
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2xCO2 climate
present climate
days (deviation from long term mean)
teacher Knuths data
fit
Maak von Storch, Intl. J. Biometeorol., 1997
days (deviation from long term mean)
Schleswig-Holstein average of flowering date
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When using downscaling for assessing regional
climate change, three implicit assumptions are
made

• (1) The predictors are variables of relevance and
  are realistically modeled by the AOGCM.
• (2) The transfer function is valid also under
  altered climatic conditions. This is an
  assumption that in principle can not be proven in
  advance. The observational record should cover a
  wide range of variations in the past ideally,
  all expected future realizations of the
  predictors should be contained in the
  observational record.
• (3) The predictors employed fully represent the
  climate change signal.

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Behind the concept local climate response F
(external, large scale forcing) stands the view
that there is a "true function F, with a unique
value F(L) for any large scale forcing L. Thus F
may be thought of a surface, with L
representing generalized coordinates thus as a
terrain F and geographical coordinates L.
For the following, see von Storch., H., 1999
Representation of conditional random
distributions as a problem of "spatial"
interpolation. In. J. Gòmez-Hernàndez, A. Soares
and R. Froidevaux (Eds.) geoENV II -
Geostatistics for Environmental Applications.
Kluwer Adacemic Publishers, Dordrecht, Boston,
London, ISBN 0-7923-5783-3, 13-23
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Example (Tim Osborn)

• R rainfall amount anomalies at some location
• L (x1, x2)with x1 flow directionand x2
  flow intensityof the regional wind
• The value S -2.5 mm/day for a flow strength of
  20 m/s and a flow direction of 100 is the
  average of rainfall amount anomalies recorded on
  many days with a flow strength of about 20 m/s
  and a flow direction of about 100.

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Of course, the surface F is unknown. It needs to
be determined by interpolating or approximating a
number of point observations Fk at some
locations Lk. This can be done by a number of
different spatial interpolation methods. The
results of the interpolation is an approximate or
estimated surface FE, which differs to some
extend from the true surface F. The purpose of
the spatial interpolation is the determination of
the continuous surface F and not the reproduction
of known points Fk.
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Skill of approximation

• The success of FE, as an estimator of F may be
  determined by comparing the estimates FE(Lk) with
  the true F(Lk) at a number of additional data
  points. "Additional" means that the data F(Lk)
  have not used for determining FE.
• Conventional measures of success are
• the bias, in the mean or in the standard
  deviation, i.e., B ltFE(L)gt ltF(L)gt, Bs sE -
  s , where the brackets ltgt represent the
  averaging operation over all considered data
  points L. sE and s are standard deviations of
  FE(L) and of F(L).
• the mean square root difference, i.e., the
  squared correlation, i.e., MSE ltFE(L) F(L)gt2 /
  ltFE(L)2gtltF(L)2gt
• the represented variance e 1 MSE / ltF(L)2gt

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Random character of the problem

• When we think of the surface no longer as a
  terrain but as the surface of the ocean, there
  is no longer a well defined height but a variable
  surface which elevation may be described in
  probabilistic terms.
• Precipitation can not completely be understood as
  a function of the variables L (i.e., as a
  function of flow direction and strength), but
  must in part be understood as being random.
• The concept of "downscaling" does not imply that
  the regional climate would be determined by the
  large-scale state for similar large-scale
  states, the associated regional states may vary
  substantially. Instead, the state is constrained
  or conditioned.
• The regional climate is a random process
  conditioned upon a driving large-scale climate
  regime. That is, precipitation may be modeled as
  a random variable R conditioned upon flow
  direction and strength.The surface F is the
  conditional expectation E of R F(L)
  E(RL).

12
Monthly mean precipitation at Orense (SW Spain)
as a function of the coefficients of the first
two EOFs of monthly mean air pressure.Winter
only.
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Skill of different techniques for interpolation
rainfall in Orense Linear regression based on
CCA, local interpolation by kriging, nearest
neighbor (analog) and neural nets (NN) The
interpolation was done with data from 1899 to
1968 the remaining independent data from the
winters 1968 to 1995 were used to determine the
skill measures.
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Application of the interpolated "surface"
The purpose of interpolation is "to guide people
in unknown terrain", i.e., to guess the state of
the system in "locations" not visited no far.
Such guesses can be of very different format,
depending on the users needs.

• In many cases, and in particular in the case of
  forecasts, the randomness is considered an
  unavoidable nuisance, since it is intrinsically
  unpredictable. Therefore, not the actual value is
  specified but the conditional expectation.
  Ideally, the specification is When the
  coordinates are close to L, on average, our
  interpolated variable will have a value of FE(L).
  The actual value will with some probability be
  within the interval FE(L) D , with some level of
  uncertainty D . Thus, for forecasting problem,
  techniques returning smooth surfaces are superior
  and the analog technique should not be used.
• On the other hand, often not "best predictions"
  are needed but "weather generators", i.e.,
  methods which generate time series with
  statistics as observed. In that case the purpose
  is "simulation", and the capability of the system
  to generate the right level of variability (and
  other aspects such as the autocorrelation
  function length of dry and wet spells) becomes
  essential. The analog method fulfills this
  request automatically, if designed properly

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Using more sophisticated predictands

• Such as high percentiles representing (possibly
  not so) extreme events see following example.
• Or characteristic statistics, such as
  probabilities of a wet day following a dry day
  (e.g., Busuioc, A., and H. von Storch, 2003
  Conditional stochastic model for generating daily
  precipitation time series, Climate Research 24,
  181-195)

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Example Percentiles as Predictands

• R percentiles of high tide water levels at a
  number of tide gaues along the North Sea coast.
  Each 3-month winter has about 180 high tides
  from the distribution of these 180 values,
  percentiles are derived and related to L
  large-scale winter mean SLP.
• Pfizenmayer, A., 1997 Zusammenhang zwischen der
  niederfrequenten Variabilität in der
  grossräumigen atmosphärischen Zirkulation und den
  Extremwasserständen an der Nordseeküste.
  Diplomarbeit Institut für Geographie Universität
  Stuttgart
• Langenberg, H. , A. Pfizenmayer, H. von Storch
  and J. Sündermann, 1999 Storm related sea level
  variations along the North Sea coast natural
  variability and anthropogenic change.- Cont.
  Shelf Res. 19821-842

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Predictand Intra-annual percentiles of water
levels
Winter mean
Winter mean
Winter percentiles after subtraction of winter
mean of high tides water levels.
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Thermal expansion not taken into account!
North Sea
Scenario 1 CO2 increase at the end of the 21st
century
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End of examples

• Summary of range of methods

18.10.2004 Central Weather Bureau, Taipei, Taiwan
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A diverse range of downscaling techniques methods
has been developed, but in principle fall into
three categories

• (a) Weather generators, which are random number
  generators of realistically looking sequences
  conditioned upon the large-scale state.
• (b) Transfer functions, where a direct
  quantitative relationship is derived through, for
  example, regression.
• (c) Weather typing schemes based on the more
  traditional synoptic climatology concept
  (including analogs and phase space partitioning)
  and which relate a particular atmospheric state
  to a set of local climate variables.

von Storch, H., B. Hewitson and L. Mearns, 2000
Review of Empirical Downscaling Techniques. T.
Iversen and B.A.K. Høiskar (Eds.) Regional
climate development under global warming. General
Technical Report No. 4. Conf. Proceedings RegClim
Spring Meeting Jevnaker, Torbjørnrud, Norway, 8.
- 9. May 2000, p. 29-46
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• Weather generators
• Weather generators are statistical models of
  observed sequences of weather variables. They can
  also be regarded as complex random number
  generators, the outputs of which resemble daily
  weather data at a particular location.
• There are two fundamental types of daily weather
  generators, based on the approach to modeling
  daily precipitation occurrence the Markov chain
  approach and the spell-length approach.
• In the Markov chain approach, a random process
  is constructed which determines a day at a
  station as rainy or dry, conditional upon the
  state of the previous day, following given
  probabilities. If a day is determined as rainy
  then the amount is drawn from a probability
  distribution.
• In the spell length approach, the time between
  two rain days is modelled.
• For statistical downscaling the parameters of the
  weather generator are conditioned upon a
  large-scale state, or relationships can be
  developed between large-scale parameters sets of
  the weather generators and local scale
  parameters.

See also Giorgi, F., B. Hewitson, J.
Christensen, M. Hulme, H. von Storch, P. Whetton,
R. Jones, L. Mearns and C. Fu, 2001 Regional
climate information - evaluation and projections.
In J.T. Houghton et al (eds.) Climate Change
2001. The Scientific Basis, Cambridge University
Press, 583-638
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Transfer functions - I The more common
approaches found in the literature are
regression-like techniques or piecewise
interpolations using a linear or nonlinear
formulation. The simplest approach is to build
multiple regression models relating free
atmosphere grid point values to surface
variables. Canonical Correlation Analysis has
found wide application. A variant of CCA is
redundancy analysis, which is theoretically
attractive as it maximizes the predictands
variance however, in practical terms it seems
similar to CCA. Also Singular Value Decomposition
has been used, which is another variant of
CCA. Most applications have dealt with
precipitation another example has successfully
specified local pressure tendencies, as a proxy
for local storminess, from large-scale monthly
mean air pressure fields. Oceanic climate and
climate impact variables have also been dealt
with salinity in the German Bight and salinity
and oxygen in the Baltic sea level and a number
of ecological variables such as abundance of
species. In addition statistics of extreme
events, expressed as percentiles within a month
or season, have been modeled storm surge levels
and ocean wave heights.
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Transfer functions - II An alternative to
linear regression is to use piecewise linear or
nonlinear interpolation geostatistics offers
elegant "kriging" tools to this end. Another
approach is to use cubic splines for specifying
precipitation. Another non-linear approach is
based on artificial neural networks (ANN), which
are generally more powerful than other
techniques, although the interpretation of the
dynamical character of the relationships is less
easy.
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• Weather typing
• The synoptic downscaling approach empirically
  defines weather classes related to local and
  regional climate variations. These weather
  classes may be defined synoptically or fitted
  specifically for downscaling purposes by
  constructing indices of airflow. The mean, or
  frequency, distributions of local or regional
  climate are then derived by weighting the local
  climate states with the relative frequencies of
  the weather classes. Climate change is then
  estimated by determining the change of the
  frequency of weather classes.
• In the "statistical/dynamical" approach,
  meso-scale atmospheric models are utilized for
  simulating a series of typical weather states.
  The advantage over the former technique is that
  in this way spatially distributed local climates
  are specified.
• The applicability of the analog method was
  demonstrated mostly for the specification of
  daily precipitation.
• Conceptually similar, but mathematically more
  demanding are techniques which partition the
  large-scale state phase space, for instance with
  Classification Tree Analysis, and use a
  randomized design for picking regional
  distributions.
• Both analog and CART approaches return the right
  level of variance and correct spatial correlation
  structures.

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Summary

• Empirical downscaling methods have matured in the
  past years.
• They are used for estimating a wide range of
  variables, both meteorological variables as well
  as statistics of not only meteorological but
  also oceanographic and ecological variables.
• In some cases, dynamical downscaling does a
  better job, but many variables (e.g. ecological)
  can not dynamically modeled or are on spatial
  scales below the resolution of any RCM.
• Thus, empirical downscaling is a methodwhich
  will remain to play an important role for
  assessing ongoing change and projecting possibel
  future changes.

28
Downloads

• The ppt-file of this talk can be downloaded
  fromhttp//w3g.gkss.de/staff/storch/PPT/statistic
  s.041018.taipeh.ppt
• Most of the literature given can be downloaded
  fromhttp//w3g.gkss.de/staff/storch/pub.htmhttp
  //w3g.gkss.de/staff/storch/recent.htmhttp//w3g.g
  kss.de/staff/storch/lit.htm

29
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30
Summary

• Empirical downscaling methods have matured in the
  past years.
• They are used for estimating a wide range of
  variables, both meteorological variables as well
  as statistics of not only meteorological but
  also oceanographic and ecological variables.
• In some cases, dynamical downscaling does a
  better job, but many variables (e.g. ecological)
  can not dynamically modeled or are on spatial
  scales below the resolution of any RCM.
• Thus, empirical downscaling is a method which
  will remain to play an important role for
  assessing ongoing change and projecting possibel
  future changes.

31
Downloads

• The ppt-file of this talk can be downloaded
  fromhttp//w3g.gkss.de/staff/storch/PPT/statistic
  s/041018.taipeh.ppt
• Most of the literature given can be downloaded
  fromhttp//w3g.gkss.de/staff/storch/pub.htmhttp
  //w3g.gkss.de/staff/storch/recent.htmhttp//w3g.g
  kss.de/staff/storch/lit.htm