Weather Generator. Downscaling Summer School in Lodz, 21 June Deliang Chen

Transcription

1 Weather Generator Downscaling Summer School in Lodz, 21 June 2007 Deliang Chen Professor of Physical Meteorology and August Röhss Chair in Physical Geography (Geoinformatics) Department of Earth Sciences Göteborg University, Sweden 1

2 Topics covered Statistical downscaling: Techniques and methodologies What is a Weather Generator (WG)? Two types of WG How WG is used in downscaling? Suggested further readings 2

3 Acknowledgement Some of the OHs appeared in my lectures are adopted from CCIS, Dr. Pierluigi Calanca and Dr. Rob Wilby. 3

4 Temporal and spatial scales of climate (Global climate model) GCM RCM (Regional climate model) 4

5 How to build the statistical model? Transfer functions: quantify relationship between large-scale climate (predictor) and local variables (predictand) by e.g. regression. Most often used for monthly temperature and precipitation. Related to Perfect Prognosis (PP) and Model Output Statistic (MOS) in numerical weather prediction. Weather typing: relate a particular atmospheric state to a set of local climate variables (Lamb weather typing, Grosswetterlagen, others..) Weather generators: generate realistic looking sequences of e.g. daily precipitation based on the large-scale atmospheric state 5

6 Statistical downscaling: methods linear in large majority of studies regression canonical correlation / covariance (SVD) analysis nonlinear neural networks analog 6

7 Statistical downscaling: Reproduction of variance underestimated variance in downscaled data two ways of reproduction of variance inflation (enhancement of anomalies by a constant factor) physically questionable all forcing comes from large-scales adding noise white noise regression residuals noise with pre-specified statistical characteristics 7

8 Statistical downscaling: reproduction of variance observed downscaled: stddev = 58% obs downscaled inflated 8

9 Statistical downscaling: Reproduction of variance observed downscaled: stddev = 58% obs white noise downscaled with added white noise 9

10 Statististical downscaling: Reproduction of variance obs down infl added white n. correlation with OBS autocorrel

11 A successful statistical downscaling requires A strong statistical relationship between local and large scale climate variables be established; The statistical relationship should be physically interpretable and temporally stationary; The large scale variables must be those that can be reliably produced by GCM; Attempts should be made to understand the unexplained part by the statistical model and to assess/compensate its effect if possible. 11

12 Added value of downscalings (SD,DD) Hellstroem et al., 2001 Vänersborg, One station in southern Sweden obs SDE DDE ECHA Precipitation (mm/month) obs SDH DDH HadC Month b 12

13 Differences between real world and GCM-world 13

14 Two extremes to categories of downscaling: Transfer functions relating atmospheric forcing to target variable Stochastic functions and pure weather generators For both: variance explained as a function of the large scale flow, residual variance can only be stochastically generated. For future climate, only change due to the signal contained in the GCM scale forcing can be accounted for.. Variance explained Synoptic Forcing Sub-grid scale forcings Synoptic scale Local scale 14

15 Transfer Functions Area Grid Box Select predictor variables Calibrate and verify model Predictor variables e.g., MSLP, 500, 700 hpa geopotential heights, zonal/meridional components of flow, areal T&P Transfer function e.g., Multiple linear regression, principal components analysis, canonical correlation analysis, artificial neural networks Extract predictor variables from GCM output Drive model Observed station data for predictand Site variables for future, e.g.,

16 Weather Typing Identify weather types Derive Observed weather variables Select classification scheme Relationships between weather type and local weather variables Pressure fields from GCM Calculate weather types Drive model Local weather variables for, say, 2050 Statistically relate observed station or area-average meteorological data to a weather classification scheme. Weather classes may be defined objectively (e.g. by PCA, neural networks) or subjectively derived (e.g., Lamb weather types [UK], European Grosswetterlagen) 16

17 Weather Typing Fundamental Assumption the relationships between weather type and local climate variables will continue to be valid under future radiative forcing ADVANTAGES founded on sensible physical linkages between climate on the large scale and weather on the local scale 17

18 Example: Investigate the link between circulation and winter temp. (1) (Chen, 2000: A monthly circulation climatotolgy for Sweden and its application to a winter temperature case study. Int. journal of Climatology 20, Approach: transfer function (multiple regression) Predictant=f(predictors) winter temp. circulation (strength of geostrophic flow and vorticity) T a = v+0.32V+0.17ξ circulation indices 18

19 Example 1: cont How to quantify of atmospheric circulation? Circulation indices and weather typing based on the system by Lamb ξ 6 circulation indices: U V u: west-east geostr. wind v: north-south geostr. wind W: total geostr. wind ξ u : westerly shear vorticity ξ v : southerly shear vorticity ξ: total vorticity (after Chen and Li, 2003) 19

20 Example 1: cont Temperature anomaly ( o C) January temperature in SW Sweden R=0.84 N=122 observation reconstruction Year Chen,

21 General categories of methodologies Transfer Weather Stochastic conditioned Stochastic Function Typing on weather type Stochastic / weather generators calibrated on observed data, conditioned on atmospheric state. Very effective at capturing high frequency variance, peaks and extremes Requires long term data sets to effectively define stochastic characteristics Question of stationarity 21

22 What is a WG? WG is a and stochastic (based on observed probability) model of meteorological variables. It can generate unlimited length of data. 22

23 The key problem->why WG? Output of GCM ~ 300 km ~ 1 mo Ecosystem and other Impact models ~ 1-10 km ~ 1 h-1 day Disparity of spatial (and temporal) scales 23

24 Stochastic weather generators: basic idea given slow set of statistics (monthly means and standard deviations, Y, from downscaling), generate the high frequency variability of the weather (y) assuming auto- and cross correlation: y(t) = O T [Y, y(t-1)] where O T is the time operator. 24

25 Matching the scales: two common approaches (i) nesting Regional Climate Models (RCMs, horizontal resolution Δx < 10 km) into GCMs. Advantage: mathematical description of system is basically preserved! However: CPU ~ 1/(Δx) 2 or even 1/(Δx) 3 Running a RCM at Δx ~ 1 km requires 10 4 to 10 6 as much CPU as at Δx ~ 100 km! 25

26 (ii) Statistical downscaling (space dimension) at low temporal resolution (say Δt ~ 1 mo), followed by stochastic generation of weather sequences (time dimension) at a fixed location. Advantage: computationally efficient large number of simulations feasible! 26

27 Key publications in the development of daily WG Site(s) Observation Source Brussels Wet and dry days tend to cluster Quetelet (1852) Kew, Aberdeen, Greenwich, Valencia Probability of a rain day is greater if the previous day was wet Rothamstead, UK; Wet and dry spell lengths have a geometric five Canadian cities distribution Tel Aviv Use of Markov chain to reproduce geometric distribution of wet and dry spell lengths? Combined Markov occurrence model with exponential distribution for rainfall amounts USA Generation of max/min temperature, and solar radiation conditional on rain occurrence USA Multi-site generalization of daily stochastic precipitation model Newnham (1916); Besson (1924); Gold (1929); Cochran (1938) Williams (1952); Longley (1953) Gabriel and Neumann (1962) Todorovic and Woolhiser (1975) Richardson (1981) Bras and Rodriguez- Iturbe (1976) Source: Wilby 27

28 Precipitation occurrence process Most weather generators contain separate treatments of the precipitation occurrence and intensity processes. A first-order Markov chain for precipitation occurrence is fully defined by two conditional probabilities and p 01 = Pr{precipitation on day t no precipitation on day t-1} p 11 = Pr{precipitation on day t precipitation on day t-1} which are called transition probabilities. 28

29 Precipitation occurrence processes (cont.) The transition probabilities for Cambridge, UK are as follows dry-to-wet (p 01 ) = wet-to-wet (p 11 ) = Therefore it follows (for a two state model) that dry-to-dry (p 00 ) = 1 - p 01 = wet-to-dry (p 10 ) = 1 - p 11 = This approach may be extended from a first-order to n th -order model by considering transitions that depend on states on days t-1, t-2...t-n (as in Gregory et al., 1993). 29

30 Precipitation amount processes Daily precipitation amounts are typically strongly skewed to the right. The simplest reasonable model is the exponential distribution, as it requires specification of only one parameter, μ, and whose probability density function is: f(x) = 1 μ x exp μ The two-parameter gamma distribution is another popular choice, defined by the shape α and scale parameter β: f(x) = α-1 ( ) [ ] x β exp - βγ( α) x β Most weather generators make the assumption that precipitation amounts on successive wet days are independent. 30

31 Precipitation amount processes (cont.) January precipitation at Ithaca, New York represented by three pdfs: exponential gamma mixed exponential Source: Wilks and Wilby (1999) 31

32 Daily total (mm) Precipitation and its PDF [1] raw data [2] empirical pdf Daily to tal (mm) 32

33 Other meteorological variables Condition the statistics of the daily variables (typically maximum/ minimum temperatures and solar radiation) on occurrence of precipitation (a proxy for other processes such as cloud cover). In the classic WGEN model, multiple variables are modelled simultaneously with auto-regression: z () [ ] ( ) [ ] () t = A z t Where z(t) are normally distributed values for today s nonprecipitation variables, z(t-1) are corresponding values for the previous day, and [A] and [B] are K K matrices of parameters, and ε(t) is white-noise forcing. B ε t 33

34 Other meteorological variables (cont.) The z(t) are transformed to weather variables dependent on rainfall occurrence: T k () t = { μ μ k,0 k,1 + σ + σ k,0 k,1 () t z () t k () t z () t k if day t is dry if day t is wet where each T k is any of the nonprecipitation variables, μ k,0 and σ k,0 are its mean and standard deviation for dry days, and μ k,1 and σ k,1 are its mean and standard deviation for wet days. Seasonal dependence of the means and standard deviations is usually achieved through Fourier harmonics (i.e., sine and cosines). 34

35 WG type I: Markov chain (transition probabilities) Source: Wilks and Wilby (1999) 35

36 WG type II: spell-lengths Source: Wilks and Wilby (1999) 36

37 The Richardson type WG Precipitation Process Occurrence Amount Non-precipitation variables Maximum temperature Minimum temperature Solar radiation Model calibration Synthetic data generation Climate scenarios Climate scenarios 37

38 Weather Generators Precipitation Process Occurrence Amount Non-precipitation variables Maximum temperature Minimum temperature Solar radiation LARS-WG: wet and dry spell length Model calibration Synthetic data generation Climate scenarios 38

39 One application of WG: Create data for unsampled sites Assumption: compared to a meteorological variable it self, parameters of a WG for that variable can be more accurately interpolated from the sampled sites. 39

40 Spatial Downscaling with WG Calibrate weather generator using area-average weather Calibrate weather generator for each individual station within area Calculate changes in parameters from grid box data Area Area parameter set Station parameter set Grid Box Apply changes in parameters derived from difference between area and grid box parameter sets to individual station parameter files; generate synthetic data for scenario 40

41 Statistical spatial downscaling Source: Wilks (1999) Changes in station-series means and variances will be proportional to changes in the respective areaaverage (GCM grid) moments: μ down E S( T) E[ S( T) ] station E S( T) = Tπ [ ] [ ] down GCMfuture GCMpresent where S(T) is the sum of T daily precipitation amounts,π is the unconditional probability of precipitation, and μ is the mean wet-day amount. 41

42 Temporal Downscaling->high temporal resolution! Use of monthly scenarios Observed station data WG Parameter file containing statistical characteristics of observed station data Monthly scenario information from GCM, RCM or SD Generate daily weather data corresponding to the monthly scenario 42

43 Weather Generators Fundamental Assumption The statistical correlations between climatic variables derived from observed data are assumed to be valid under a changed climate. Advantages the ability to generate time series of unlimited length opportunity to obtain representative weather time series in regions of data sparsity, by interpolating observed data ability to alter the WG s parameters in accordance with scenarios of future climate change - changes in variability as well mean changes 43

44 Weather Generators Disadvantages seldom able to describe all aspects of climate accurately, especially persistent events, rare events and decadal- or centuryscale variations designed for use, independently, at individual locations and few account for the spatial correlation of climate 44

45 NCC/RCG-WG Version 1: released in April 2004 with parameters determined for 672 Chinese and 300 Swedish rainfall stations. Precipitation only. Version 2.0: released with parameters determined for 671 Chinese (temperature and precipitation) and 300 Swedish rainfall stations. bcc.cma.gov.cn 45

46 Application of NCC/RCG-WG for China 0.0~0.2 Annual mean P(WD) 0.3~0.5 Annual mean P(WW) 0.7~0.9 46

47 Application of NCC/RCG-WG for China 0.60 转移概率 P(WD) 转移概率 (PWW) 齐齐哈尔乌鲁木齐北京常德海口 月份 月份 齐齐哈尔乌鲁木齐北京常德海口 Five stations are chosen to represent regional differences For most months it is true that P(WW)> P(WD) The seasonal cycle of P(WD) is relatively similar among different regions, while the seasonal cycle of P(WW)show greater difference. 47

48 GAMMA distribution of daily precipitation The probability of daily precipitation may be described by a Gamma function: where 48

49 Two examples Beijing Haike 49

50 Two parameters of the GAMMA function ALPHA BETA 月份 齐齐哈尔 乌鲁木齐 北京 常德 海口 月份 The two parameters do not very much with the regions. This is especially true for ALPHA. -> great petential for accuarte spatial interpolations!!! 齐齐哈尔乌鲁木齐北京常德海口 50

51 Regional distribution of GAMMA parameters 1~3 >18 51

52 Simulated and observed monthly and annual mean rain rates y = x R 2 = Annual precipitation 模拟 y = x R 2 = 实测 模拟值 Monthly precipitation 实测值 52

53 Simulated and observed monthly mean rain days and mean annual number of days with precipitation rate greater than 20 mm/day 30 y = x 25 R 2 = 模拟值 monthly mean rain days y = x R 2 = 实测值 模拟 Day number with P>20 mm/day 实测 53

54 Simulated and observed daily mean and standard deviation of precipitation 模拟湿日平均降水量 (mm) y = x R 2 = Mean daily Precipitation y = x R 2 = 实测湿日平均降水量 (mm) Standard deviation 模拟湿日降水量标准差 实测湿日降水量标准差 54

55 Suggested Further Readings IPCC TAR - Chapter 10 ( Richardson, C.W. (1981): Stochastic simulation of daily precipitation, temperature and solar radiation. Water Resources Research 17, Wilks, D.S. (1992): Adapting stochastic weather generation algorithms for climate change studies. Climatic Change 22, Wilks, D.S. and Wilby, R.L. (1999): The weather generation game: a review of stochastic weather models. Progress in Physical Geography 23,