Empirical-Statistical Downscaling & Record-Statistics R.E. Benestad Rasmus.benestad@met.no
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Principles of Downscaling Why downscaling? Interpolated Temperatures v.s. station Observations Skillful spatial scale : ~ 8 grid-pts. Grotch & McCracken (1991), J. Clim, 4, p. 286 temperature 0 2 4 6 8 Annual oslo oksoy nesbyen Local climatic differences are not resolved in GCMs. ECHAM4 GSDIO 1850 1900 1950 2000 2050 Time Interpolated NCEP & AOGCM and station observations
Principles of Downscaling Large-scale (GCMs, re-analysis) Geographical influence (physiography) slp: 1st EOF: var=25.7968% Latitude 90 80 70 60 50 40 X region : large-scale regional condition Empirical-statistical downscaling: 6000 4000 2000 Incorporates influence of regional 0 conditions and geographical influences using information from the past. 2000 30 4000 20 eof an ea slp atl.nc J u m n 150 100 50 0 50 100 150 Longitude 6000 X local = ψ(x regnion, physiography) Small-scale (Direct measurements)
Methods Regression example for Innsbruck Empirical Downscaling ( ncep_t2m [ 10W50E 30N60N ] > Tmpr ) Tmpr ( deg C ) 20 30 40 50 60 70 Tmpr ( deg C ) 6 4 2 0 2 Empirical Downscaling ( ncep_t2m [ 10W50E 30N60N ] > Tmpr ) Obs. Fit GCM Trends Jan: Trend fit: P value=92%; Projected trend= 0.02+ 0.25 deg C/decade 80 60 40 20 0 20 40 Time Calibration: Jan Tmpr at INNSBRUCK using ncep_t2m: R2=81%, p value=0%. 1960 1970 1980 1990 Time Calibration: Jan Tmpr at INNSBRUCK using ncep_t2m: R2=81%, p value=0%.
Methods Empirical-statistical & Dynamical downscaling: 2 completely different approaches - independent modelling strategies. 2.5 2.0 Projected change in annual mean temperature DD ED Deg C 1.5 1.0 0.5 Dynamical DS not necessarily better than empirical-statistical. 0.0 Stationarity-problems associated with parameterisation (statistical) and Hanssen-Bauer not et more al. (2003) physically Clim. Res., consistent 25, 15 (systematic biases also see figure!). R1 R2 R3 R4 R5 R6
Methods 2-dimensional data matrix converted to a 1D vector: Y ij R n m Example of data grid n X14 X24 X34 X44 X54 X64 X74 X84 X94 m Latitude (deg N) X13 X12 X23 X22 X33 X32 X43 X42 X53 X52 X63 X62 X73 X72 X83 X82 X93 X92 X11 X21 X31 X41 X51 X61 X71 X81 X91 Longitude (deg E) X11 X12... Time Observations Model A question of of how to to organize the the data
Methods common EOF :: combine two different data sets Gridded Observations/ re-analysis Time axis GCM results space PCA: Singular Vector Decomposition (SVD): X = U Σ V T U Σ V U: spatial pattern common Σ: Eigenvalues (variance) V: time series describing the loadings (principal components) Mathematically identical to Empirical orthogonal functions (EOF).
Methods U V Σ U PCA: observations Σ V U PCA: GCM 1950 1960 1970 1980 1990 2000 Time met.no Klima DataVareHus Regression ψ OSLO BLINDERN ψ scenario 30 20 Precipitation (mm) 40 50 60 ϕψ scenario 10 60 50 40 30 10 20 Match Patterns ϕ VT VT U 0 Precipitation (mm) Common EOF based downscaling 0 U OSLO BLINDERN Σ Common PCA Prefect prognosis Traditional downscaling VT V 1950 1960 1970 1980 1990 2000 Time met.no Klima DataVareHus Regression ψ
Methods Example of downscaling: Perfect prog & common EOF Common EOF Perfect prog Benestad (2001) Int. J. Clim. 21 1645
Methods & Uncertainties Choice of domain can affect your results Annual mean temperature No No at inflation bjoernoeya 99710 has been used here. Less need for for inflation Bjoernoeya: 19.02 E,74.52 N than in in Perfect Prog approach Temperature (deg C) 8 6 4 2 0 2 4 6 (von (von Storch, 1999, J. J. Clim, 12, 12, 3505) obs 10E50E52N75N 40E20E67N85N 40E40E52N80N 60E40E42N70N 1900 1950 2000 2050 Time (year) Downcaled from ECHAM4/OPYC3 GSDIO Benestad (2002) Clim. Res.,21 (2), p. 105-125: warning about domain choice.
Methods Experiment: downscaling using a set of different predictor domains. Check robustness (flat structure). Common EOF Perfect prog Benestad (2001) Int. J. Clim. 21 1645
Methods Empirical Orthogonal Functions (EOFs) and Principal Component Analysis (PCA). Eigenvectors of the co-variance matrix: S-mode and T-mode Variance-covariance matrix S of X is 1/(n-1) X T X where X = X - X S e = λ e (Eigenfunctions) x =E u u m = e T m x Singular Vector Decomposition (SVD) X = U Σ V T [x 1, x 2,..x m ] X e T i e j=δ ij [e 1, e 2,..e n ] E Literature: Wilks, D.S. (1995) Statistical Methods in the Atmospheric Sciences. Academic Press Press W.H., Flannery B.P., Teukolsky S.A, & Vetterling W.T..(1989) Numerical Recipes, Cambridge Preisendorfer R.W. (1989) Principal Component Analysis in Meteorology and Oceanology, Elsevier Science Press
Methods Regression & other Statistical models on Relationships Regression: Example: lm versus projection single & multiple regression. Least Squares fit Projection of y= a 0 + a 1 x 1 + a 2 x 2 + a 3 x 3 + lm data multivariate regression & matrix projection. Projection & least squares: A x = y y= A(A T A) -1 A T x y = a x a = (x T y)/(x T x) Strang, G. (1988) Linear Algebra and its Applications, Harcourt Brace & Company Benestad (1999) MVR applied to Statistical Downscaling for Prediction of Monthly Mean Land Surface Temperatures: Model Documentation, DNMI Klima, 2/99. pp.35 Oslo Temperature (deg C) 10 5 0 5 linear models, generalised linear models, non-linear models. d2 d1 6 4 2 0 2 4 Ualand Temperature (deg C)
Methods Canonical Correlation Analysis (CCA) classical & Barnet-Preisendorfer CCA. ECHAM4 Bretherton et al. (1992) An Intercomparison of Methods for finding Coupled Patterns in Climate Data, J. Clim., 5, 541. T(2m). Coupled fields (from CCA). Benestad (1998) CCA applied to Statistical Downscaling for Prediction of Obs. Monthly Mean Land Surface Temperatures: Model Documentation, DNMI Klima, 28/98, pp.96 SLP. T(2m). SLP. Find patterns with the maximum correlation. X 1 = G U T, X 2 =H V T U T V = L M R T = C Downscaling: X 1 = G M (H T H) -1 X 2
Methods Singular Vector Decomposition (SVD) not to be confused with Singular Vector Decomposition (SVD) Benestad (1998) SVD applied to Statistical Downscaling for Prediction of Monthly Mean Land Surface Temperatures: Model Documentation, DNMI Klima, 30/98, pp. 38 X 1 = G svd S 12 S 22-1 H T svd X 2 Maximize co-variance (CCA maximizes correlation) Other types of models Neural nets and Self-Organising Maps
Methods & Uncertainties Which parameters as predictors? Strong & well-understood relationship (reflecting a physical mechanism) Field that GCMs can skilfully reproduce Parameters that carry the essential signal (e.g. a gradual global warming is not well-represented in SLP) Temperature trend (deg C per decade) 2 1 0 1 2.slp..slp.temp..temp.
Methods & Uncertainties Choice of method: e.g. linear v.s. analog Other methods may also be be incorporated into clim.pact in in the the future, such as as neural nets. Various regression models are are available (lm, glm, etc.), and CCA/SVDbased models may also be be included in in the the future. In In the the common EOF framework it it is is possible to to add corrections to to the the model results: by by setting PC PC loadings for for present-day climate to to have same spread and location (µ (µ& σ) σ) as as in in the the observations and use the the same adjustments for for the the future.
Methods & Uncertainties Probing uncertainties through multi-model ensembles: Spread caused by different model shortcomings, natural variability & different initialisation processes.
Methods & Uncertainties Downscaling, ensembles & geographical distribution
Methods & Uncertainties Precipitation Slope estimates Linear trend rate (mm/month per decade) 0 5 10 15 20 3 2.5 2 1.5 1 0.5 0 0.5 1 3 2.5 2 1.5 1 0.5 0 0.5 1 3 2.5 2 1.5 1 0.5 0 0.5 1 3 2.5 2 1.5 1 0.5 0 0.5 1 OSLO BLINDERN BERGEN FLORIDA TORSHAVN KOEBENHAVN TROMSOE 3 2.5 2 1.5 1 0.5 0 0.5 1 3 2.5 2 1.5 1 0.5 0 0.5 1 3 2.5 2 1.5 1 0.5 0 0.5 1 3 2.5 2 1.5 1 0.5 0 0.5 1 HELSINKI STOCKHOLM SKJAAK split.merge even odd ERA40 NCEP 2 4 6 8 10 Predictors= prec, slp & mix
Methods & Uncertainties Validation of models Bergen: reconstruction from gridded SLP Seasonal accum. precipitation (mm/month) 400 350 1500 300 250 200 150 100 50 350 0 300 250 1000 200 150 100 50 350 0 300 250 200 150 100 500 50 350 0 300 250 200 150 100 50 0 Reconstr. Predict Station ERA40 DNMI: SON: R2= 59 71 % DNMI: JJA: R2= 30 45 % DNMI: MAM: R2= 60 70 % DNMI: DJF: R2= 49 74 % Calibration interval ERA40: SON: R2= 78 90 % ERA40: JJA: R2= 68 75 % ERA40: MAM: R2= 46 82 % ERA40: DJF: R2= 61 87 % 1850 1900 Time clim.pact: DNMI.slp (Benestad & Melsom, 2002, Clim. Res.,Vol 23, 67 79) 1950 2000
Methods & Uncertainties When a fraction of the variance can be accounted for
The upper tail Daily rainfall Linear (regression) models fail to give a good representation of the tails of the disribution. Other approach: the analog model.
The upper tail Analog model not representative for a situation where the upper tail of the distribution (pdf) is being stretched. Original pdf Distorted pdf from the analog True pdf in changed climate Observed range
The upper tail Record-events= values outside historical sample range
The upper tail Caveat: the traditional analog model cannot predict values outside the observed range. A random variable of rational numbers with independent and identical distribution (iid) has following property: Pr(n=record) = 1/n (assuming no ties)
The upper tail Record-event statistics
The upper tail Test of number of record-events: iid-test.
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The upper tail Dependency of pdf on local conditions For many situations, the linear-log relationships are approximately linear, indicating that an exponential law provides a reasonable fit. But, still some discrepancies in the upper tail.
The upper tail Exponential law: simpler than the gamma distribution pdf: p(x)= -m exp{-mx} m varies with local mean temperature and precipitation
The upper tail Ways of modelling the tails Gamma functions OSLO BLINDERN DJF f(x) 0.0 0.1 0.2 0.3 0.4 0.5 scale=0.8, shape=1 scale=5, shape=1 scale=11, shape=1 scale=0.8, shape=0.1 scale=11, shape=0.1 scale=5, shape=5 scale=11, shape=5 shape scale 0.0 0.5 1.0 1.5 2.0 0 5 10 15 20 moments method maximum likelihood 1950 1960 1970 1980 1990 2000 moments method maximum likelihood Time OSLO BLINDERN DJF 1950 1960 1970 1980 1990 2000 0 20 40 60 80 100 Time x
The upper tail Scenario for the future: derived from downscaled changes in the mean temperature and precipitation.
Exponential law: simple expression for upper percentiles Q p =log(1-p)/m M = -1/m S=-2/m 2
Other extremes severe events: Severe events cyclones (usually, we are not particularly interested inextremes because they are rare, but because they are severe)
Severe events Cyclone count 1955 1994 Downscaling cyclones Validation: analysis of observations Cyclone count per year count/month 0 10 20 30 40 1960 1970 1980 1990 Latitude (deg N) 20 30 40 50 60 70 80 60 40 20 0 20 40 Longitude (deg E) Period: 1955 1994 psl0= 1000 Mean storm count/month 80 0 20 40 60 80 60 40 20 0 time region: 5E...35E / 55N...72N. Threshold= 1000 seasonal cyclone variability Jan Feb Mar Apr 5 10 15 20 May Jun Jul Aug Sep Oct Nov Dec Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Month region: 5E...35E / 55N...72N. Threshold= 1000
Severe events Testing the models & toolscyclone count per year LAtitude (deg N) 45 50 55 60 65 70 The Great 1987 Storm 17 Oct 1987 PSL=978 16 Oct 1987 PSL=971 16 Oct 1987 PSL=973 Latitude (deg N) 30 40 50 60 70 60 40 20 0 20 20 10 0 10 20 Longitude (deg E) October 15 16 Reproduces known storms Longitude (deg E) Period: 2081 2100 psl0= 1000 GCMs may not have sufficient spatial resolution for proper representation of cyclones.
Severe events Downscaled storm frequency over Fennoscandia Use the observed time series of cyclone counts as predictand treating it like a station series and applying an ordinary downscaling analysis to this, based on monthly SLP.
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