The use of copulas for the description of the spatial variability of environmental variables András Bárdossy Universität Stuttgart

Spatial patterns are not necessarily symmetrical

Paul Klee: Copula

Dependence Goal: dependence between two variables: To recognize To quantify Correlation (Signifikance Normal distribution) Linear Regression

Dependence changes through a Transformation of the Marginal distribution Correlations between 0.4 and 0.8 Idea: take the same marginal distribution Uniform in [0,1] Transformation using the pdf

Dependence  Marginal distributions

Bivariate Distributions and Copulas Bivariate Copula = bivariate pdf with uniform marginals

Copulas are a new way of modelling the correlation structure between variables. They dissociate the correlation structure from the marginal distributions of the individual variables.

Entropy = measure of Information (Shannon) Differential entropy: Conditional Differential entropy Interesting for „extreme“ (v large)

Multivariate Distributions and Copulas Sklar (1959) all F pdf can be written in this form and C is unique if F is continuous Measures of the dependence Differential Entropy Rank correlation (Spearman) Kendalls tau

Copula density

Multivariate normal copula Copula density:

Normal copula

Can copulas be used for the description of spatial variability ? Do we need this ?

Geostatistics Z(x) Random function – Realisation z(xi) Assumption – „uniform continuity“ No differences are known a-priori Independent of the location – depends only on h (Semi)Variogramm  Covariance function

Experimental Variogramm EC

Spatial copulas Assumption: Multivariate copula exists for any number of points The bi-variate marginal copulas corresponding to pairs separated by a vector h are translation invariant How to find such copulas ?

Empirical copulas Set of pdf pairs corresponding to points separated by the vector h Generalization of the variogram Empirical density using kernel smoothing

Nitrat und Phosphat

Copula Nitrate GW 5 km

Entropy: Nitrat 3000 m und 30000m

Cl Variogramm

Copula pH groundwater

pH Variogramm

n-dimensional Chi-square copula

Chi-Quadrat Copulas

Testing the multivariate copulas Analytically difficult due to the dependence of the pairs Simulation and Bootstrap to compare bi-variate marginals to theoretiacal

Summary and conclusions Copulas offer an interesting alternative Many natural variables show a non-Gaussian spatial dependence

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Indikator Variablen Indikator variablen Indikator variogram

Spatial dependence – 5 km Event 70

Spatial dependence – 5 km Event 347

Spatial dependence – 5 km Event 159

Radarniederschlag 29. Dezember 2001 11:20-13:20

Copula Radarniederschlag 29. Dezember 2001 11:20-13:20

Gauss – Chi-square

Statistische Tests Problem: Paare sind nicht unabhängig Deshalb klassische Tests nicht verwendbar Lösung: Bootstrap

Zusammenfassung Zusammenhänge können mit Copulas „einheitlich“ quantifiziert werden Viele natürliche Parameter zeigen assymetrische Zusammenhänge Diese Eigenschaft kann bei der räumlichen Betrachtung berücksichtigt werden

Simulation results

Räumlicher Zusammenhang – 5 km Ereignis 71

Räumlicher Zusammenhang – 5 km Ereignis 117

Räumlicher Zusammenhang – 5 km Ereignis 348

Asymmetrie des Zusammenhanges Hohe Werte zeigen einen anderen Zusammenhang wie die niedrigen

Empirische Copuladichten Nitrat pH

Radarniederschlag 29. Dezember 2001 8:20-9:20

Niederschlag Geeignete Copulas finden Bessere Interpolation (HW relevant) Theoretische Gebietsabminderungsfunktionen Simulationsmodelle mit beliebigen Randverteilungen Realistische Extreme über räumliche und zeitliche Skalen ( Fraktale Modelle)

Neckar Einzugsgebiet 13 sc! HBV-Model 2.1 Study area and database The upper Neckar catchment is situated in the SW part of Germany, between the Black Forest to the west and the Schwäbische Alb to the south-east. The southern border of the catchment is the European Watershed, which separates the two big catchments of Danube and Rhine. The river Neckar has its origin at an altitude of 706 m and reaches the outlet of the upper part after 163 km at an altitude of 245 m at Plochingen. With an area of 3995 km2 the catchment represents approximately 28% of the whole Neckar catchment. The highest points lie in the Black Forest (1030 m) and on the Westalb (1014 m), the lowest (245 m) at the outlet in Plochingen. For the Neckar catchment time series of observed daily data is available at a great number of locations. Precipitation data are available at 44 stations inside the catchment and additionally 244 stations in the vicinity. Temperature, snow and wind data are available at 43 stations in and around the catchment. Runoff data are available at 22 gauges in the Upper Neckar catchment. The observation time period for all parameters is from 1961 to 1990.

Korrelation – Zusammenhang der Extreme (99,5%)

Zusammenfassung Zusammenhaänge können mit Copulas beschrieben werden Viele der Zusammenhänge in der Hydrologie sind asymmetrisch Extreme sind of stärker abhängig als mittlere Korrelation ist hierfür kein gutes Maß

Introduction - Modelling Hydrological modeling is necessary Design Changes Climate change Land use change Unobserved catchments (PUB) Forecasts In combination with Quality Ecology For understanding

Introduction - Variability Variability due to natural conditions Weather Annual cycle Random variability Catchment reaction State Output - discharge

The Upper Neckar Catchment 13 sc! HBV-Model 2.1 Study area and database The upper Neckar catchment is situated in the SW part of Germany, between the Black Forest to the west and the Schwäbische Alb to the south-east. The southern border of the catchment is the European Watershed, which separates the two big catchments of Danube and Rhine. The river Neckar has its origin at an altitude of 706 m and reaches the outlet of the upper part after 163 km at an altitude of 245 m at Plochingen. With an area of 3995 km2 the catchment represents approximately 28% of the whole Neckar catchment. The highest points lie in the Black Forest (1030 m) and on the Westalb (1014 m), the lowest (245 m) at the outlet in Plochingen. For the Neckar catchment time series of observed daily data is available at a great number of locations. Precipitation data are available at 44 stations inside the catchment and additionally 244 stations in the vicinity. Temperature, snow and wind data are available at 43 stations in and around the catchment. Runoff data are available at 22 gauges in the Upper Neckar catchment. The observation time period for all parameters is from 1961 to 1990.

Dependence between discharge series Cross correlations (Pearson) Cross rank correlations (Spearman) Copulas

Dependence results Cross correlations 0.66 – 0.95 for all pairs >0.89 for the best pair for each site Cross rank correlations 0.65 – 0.98 for all pairs >0.88 for the best pair for each site

Dependence structure Dependence between Quantiles instead of Variable values Copula – Dependence separated from the Marginal distributions

Copula density of the pair C8 and C9

Spatial dependence – 5 km Event 347

Spatial dependence – 5 km Event 159

Simulation models Multivariate normal copulas Non Gaussian copulas (non-central chi-square copulas) Correct marginal distribution

Copula density of a bivariate non-central chi-square distribution

n-dimensional Chi-square copula

Summary Hydrological modeling is necessary and difficult Discharge series are similar Events are different Hydrological models  small differences Spatial resolution is not the answer As mentioned, we are connected globally. The Institute brings learning opportunities to you no matter where you are, to make it convenient for you, in many instances you don’t even need to leave your office… Webinars: These are usually one-hour, online learning opportunities where you can participate free of charge as long as you have Internet access and a phone. Topics focus on products, emerging risks, risk analytics, claims and legal. Webcasting: This expands the concept of webinars to integrate video of the speaker(s) to create a more robust presentation experience. You only need a computer, no phone. Again, accessed at your convenience. Classroom: We offer Practical Thought Leadership; Classes focus on technical training, GE leadership training and customized needs. Most classes can be done in a location convenient for you. Symposiums and Summits: Our Symposiums focus on emerging risks and are held in different areas of the world. We also conduct Global Summits; they have been held in the US, Europe and Asia.  Spatial variability is partly responsible Rainfall variability is “asymmetrical”

Interpolation

Radar Measurement

Future work Compare different sets of downscaled discharge extremes (3 versions) Calculate spatial indices Find appropriate copulas Assess scenario dependent extremes Develop new methodology for spatial extremes

Variability of discharge Distribution of discharge for a single site Depends on the aggregation Skewed distribution Few maxima – due to precipitation (snow melt)

Natural Variability Discharge series are similar because of the spatial extent of rainfall events.