1. Statistical tools in Climatology René Garreaud www.dgf.uchile.cl/rene

2. Fundamental problem in (paleo) Climatology: How to link local-scale phenomena with large scale atmospheric patterns (if there such a link)? Local signal Large scale climate Internal dynamics Example of local scale signals: Rainfall, wind, temperature, tree-ring growth, lake level, wet deposition, charcoal accumulation, etc. Some times, we have local events: heavy rainfall, extended drought, lake desiccation, etc. Example of large scale climate: wind patterns, SST, precipitation elsewhere, etc….

3. Longitude Latitude time 1 time 2 time 3 time n time 4 Time series at a fixed grid point  lat  lon Map Gridded Analysis: set of maps with data on a regular grid (e.g., lat-lon, lon-lev) Spatial structure at a given time (displayed with isolines or colors) Time series at each grid point

4. time 1 time 2 time 3 time n time 4  lat Standard deviation Mean field Basic Operation with gridded data Mean and Std Deviation Average, standard deviation and other statistical parameters (max, min, skewness, etc.) are calculated using the time series at each grid point

5. Example of mean fields…

6. Use compositing analysis to describe the spatial structure of selected climate variables during the occurrence of events at your site. Sometime, the events are associated to extreme values of a continuous time series. In this case, you can also use regression analysis (1Point correlation map), but the later method will fail if there is no a linear relationship… Your site specific variable SLP elsewhere Your events

7. time 1 time 2 time 3 time n time 4  lat Basic Operation with gridded data Compositing Analysis (composite map) All mean Event! Composite mean Composite anomalies _ = Here one can use a univariate method (e.g., t-test) at each point to test for statistical significance of the anomalies

8. Example of compositing analysis…time series

9. Example of compositing analysis…spatial fields

10. Cold minus Warm composites to enhance signal

11. Composite maps of (a,b) rainfall and (c,d) SST anomaly overlaid on vectors of wind stress anomaly during dry and wet years. ® Mathew England 2005

12. time 1 time 2 time 3 time n time 4  lat Correlation map Basic Operation with gridded data 1Point Correlation Map time 1 time 2 time 3 time n time 4 Maps of Variable Y Time Series of variable X Here one can use a univariate method at each point to test for statistical significance of correlation

13. r : Measure the covariability of two time series (-1 to +1) r 2 : Fraction of variance in series Y explained by linear fit of series X

14. Annual mean Precip/SAT regressed upon index of large-scale modes (50 years of data)

15. How is the spatial pattern of SST when there is above/below normal Rainfall over the Altiplano?

16. Correlation map Basic Operation with gridded data Point-Point Correlation Map Maps of Variable Y time 1 time 2 time 3 time n time 4 time 2 time 3 time n time 4 Maps of Variable X Here one can use a univariate method at each point to test for statistical significance of correlation

17. Precipitación en latitudes medias Garreaud 2007

18. Large annual cycle Small interannual variability Small annual cycle Large interannual variability Intraseasonal noise?

19. Nice trend Moderate annual cycle Some noise… Abrupt climate change? Strong annual cycle?

20. Full time series Mix regular cycles and interannual variability Stgo. Rainfall 1-Point Correlation map Stgo. Rainfall – 1000 hPa air temperature

21. Stgo. Winter Rainfall 1-Point Correlation map Stgo. Winter Rainfall – 1000 hPa air temperature

22. Full time series Annual cycle Anomaly time series Mean annual cycle Anomalies Alternatively, use the anomaly filter, i.e. substract annual mean cycle Caution with seasonally dependent signal

23. 1 Point Correlation Map between U300’ over the Andes and Precipitation DJFJJA All months

24. Most geophysical (met) variables exhibit some spatial coherence Adjacent grid points covary with reference point…but how far? U300 winter Stgo Winter Precip

25. EOF / PC Analysis Powerful method to synthesize (large) data sets Allows to obtain the mode(s) that explain most of the variance in the data Each mode composed by a spatial pattern (EOF), time pattern (PC, loading factors) and fraction of variance accounted by It uses linear algebra: EOF/PC are linear combination of original data EOF = Empirical orthogonal functions (instead of analytical orthogonal functions)

26. EOF / PC Analysis User has to decide spatial domain (large-small) Pre-filter data (at least remove mean field) How many modes are retained (North’s rule) Each mode is orthogonal with the rest…not always true in nature Rotated EOFs / Extended EOFs

27. X(space) variable time 1 time 2 time 3 time 4 time 5 time N 1 2 3 4 5 ….. N time variable Leading EOF (dominant mode) loading Principal Component EOF Analysis Original dataSynthesized information Let the data speak by itself : fraction of variance accounted by a mode X(space)