Copula-Based Stat. Refinement of Precipitation in Rcms Simulations

icsinPpr|Dsuso ae icsinPpr|Dsuso ae | Paper Discussion | Paper Discussion | Paper Discussion | Paper Discussion Hydrol. Earth Syst. Sci. Discuss., 8, 3001–3045, 2011 Hydrology and www.hydrol-earth-syst-sci-discuss.net/8/3001/2011/ Earth System HESSD doi:10.5194/hessd-8-3001-2011 Sciences 8, 3001–3045, 2011 © Author(s) 2011. CC Attribution 3.0 License. Discussions

Copula-based This discussion paper is/has been under review for the journal Hydrology and Earth stat. reﬁnement of System Sciences (HESS). Please refer to the corresponding ﬁnal paper in HESS precipitation in RCMs if available. simulations

P. Laux et al. Copula-based statistical reﬁnement of precipitation in RCM simulations over Title Page Abstract Introduction complex terrain Conclusions References

P. Laux1, S. Vogl2, W. Qiu1, H. R. Knoche1, and H. Kunstmann1,2 Tables Figures

1 Karlsruhe Institute of Technology (KIT), Institute for Meteorology and Climate Research J I (IMK-IFU), Kreuzeckbahnstrasse 19, 82467 Garmisch-Partenkirchen, Germany 2 University of Augsburg, Institute for Geography, Regional Climate and Hydrology, J I 86135 Augsburg, Germany Back Close Received: 17 March 2011 – Accepted: 21 March 2011 – Published: 29 March 2011 Full Screen / Esc Correspondence to: P. Laux ([email protected])

Published by Copernicus Publications on behalf of the European Geosciences Union. Printer-friendly Version

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3001 icsinPpr|Dsuso ae icsinPpr|Dsuso ae | Paper Discussion | Paper Discussion | Paper Discussion | Paper Discussion Abstract HESSD This paper presents a new Copula-based method for further downscaling regional climate simulations. It is developed, applied and evaluated for selected stations in the 8, 3001–3045, 2011 alpine region of Germany. Apart from the common way to use Copulas to model the 5 extreme values, a strategy is proposed which allows to model continous time series. In Copula-based this paper, we focus on the positive pairs of observed and modelled (RCM) precipita- stat. refinement of tion. As the concept of Copulas requires independent and identically distributed (iid) precipitation in RCMs random variables, meteorological fields are transformed using an ARMA-GARCH time simulations series model. The dependence structures between modelled and observed precipita- 10 tion are conditioned on the prevailing large-scale weather situation. The impact of the P. Laux et al. altitude of the stations and their distance to the surrounding modelled grid cells is analyzed. Based on the derived theoretical Copula models, stochastic rainfall simulations are performed, finally allowing for bias corrected and locally refined RCM simulations. Title Page

Abstract Introduction

1 Introduction Conclusions References

15 GCMs and RCMs are a central prerequisite for the conduction of climate change impact Tables Figures studies that require time series of climatic variables. The projections of future climate

usually follow the so-called delta-change approach, considering the diﬀerences be- J I tween present and future climate. If time series of RCMs are used directly as input for external impact models such as hydrological or agricultural models with nonlinear re- J I

20 sponses to the climate signal, the delta-change approach may fail (e.g., Graham et al., Back Close 2007; Sennikovs and Bethers, 2009). One reason is the spatial resolution which does not allow for local scale climate differences, particularly in complex terrain. Usually Full Screen / Esc there exist tremendous biases between modelled and observed climate statistics (e.g. Schmidli et al., 2007; Smiatek et al., 2009). Therefore, further statistical refinement Printer-friendly Version 25 and bias correction methods are required to obtain reliable meteorological information at local scale (Wilby and Wigley, 1997). Precipitation is one of the most important Interactive Discussion 3002 icsinPpr|Dsuso ae icsinPpr|Dsuso ae | Paper Discussion | Paper Discussion | Paper Discussion | Paper Discussion variables for climate change impact studies (Schmidli et al., 2006). At the same time it is the most difficult to model. HESSD Statistical downscaling and bias correction methods can be divided into (i) direct 8, 3001–3045, 2011 point-wise techniques which relate specific (mostly adjacent RCM gridd cells) to a sta- 5 tion of interest such as mean value adaptation (e.g. Kunstmann et al., 2004; Jung and Kunstmann, 2007), quantile mapping/histogram equalization methods (e.g. Leung et Copula-based al., 1999; Wood et al., 2002; Themeßl et al., 2010; Senatore et al., 2011) or local inten- stat. refinement of sity scaling (e.g. Schmidli et al., 2006; Yang et al., 2010), and (ii) indirect methods which precipitation in RCMs relate meteorological fields to the station such as the analogue method (e.g. Bliefer- simulations 10 nicht and Bardossy´ , 2007), weather or circulation pattern classification techniques (e.g. Bardossy´ et al., 2002), or empirical orthogonal functions (e.g. von Storch and Zwiers, P. Laux et al. 1999). The dependence structure of hydrometeorological data such as between modelled Title Page and observed rainfall is usually very complex, both in time and space. Using sim- 15 ple correlation of the multivariate normal is often not appropriate (Bardossy´ and Pe- Abstract Introduction gram, 2009). It depends on the rainfall generating process, i.e. stratiform or convective events, and thus, on the season. For the mid-latitudes, large-scale stratiform events Conclusions References

can be represented well by climate models resulting in a relatively good agreement Tables Figures between modelled (grid cell) and measured rainfall amounts (point scale). The models 20 generally perform worse for convective events, which are highly variable in time and J I space. As a result, the discrepancies between modelled and observed rainfalls can be very large especially during the summer season with prevailing convective rainfall J I processes (e.g. Schmidli et al., 2007). Potential reasons for this are e.g. (i) the variabil- Back Close ity of observed rainfall within one corresponding modelled grid cell could be very high; 25 rain gauges in the near surrounding can measure large diﬀerences in rainfall amounts; Full Screen / Esc (ii) diﬃculties to capture the location of convective rainfall events by the climate models; this is mostly due to coarse resolution of the land surface model (LSM) and the partly Printer-friendly Version chaotic nature of convection, and (iii) wrong or inadequate model parametrisations for convection. Interactive Discussion

3003 icsinPpr|Dsuso ae icsinPpr|Dsuso ae | Paper Discussion | Paper Discussion | Paper Discussion | Paper Discussion The dependence structures of multivariate distributions can be modelled using classical distributions such as a multivariate normal. Alternatively, a Copula approach HESSD can be used to describe the dependence structure independently from the marginal 8, 3001–3045, 2011 distributions (e.g. Genest and Favre, 2007; Dupois, 2007), and thus, to use diﬀerent 5 marginal distributions at the same time without any transformations. There is an increase in applications of Copulas in hydrometeorology over the past Copula-based years. Copula-based models have been introduced for multivariate frequency analysis, stat. reﬁnement of risk assessment, geostatistical interpolation and multivariate extreme value analyses precipitation in RCMs (e.g. De Michele and Salvadori, 2003; Bardossy´ , 2006; Genest and Favre, 2007; Re- simulations 10 nard and Lang, 2007; Scholzel¨ and Friederichs, 2008; Bardossy´ and Li, 2008; Zhang and Singh, 2008). For rainfall modelling, De Michele and Salvadori (2003) used Cop- P. Laux et al. ulas to model intensity-duration of rainfall events. Favre et al. (2004) utilized Copulas for multivariate hydrological frequency analysis. Zhang and Singh (2008) carried out Title Page a bivariate rainfall frequency analysis using Archimedean Copulas. Renard and Lang 15 (2007) investigated the usefulness of the Gaussian Copula in extreme value analysis. Abstract Introduction Kuhn et al. (2007) employed Copulas to describe spatial and temporal dependence of weekly precipitation extremes. Serinaldi (2008) studied the dependence of rain gauge Conclusions References

data using the non parametric Kendall’s rank correlation and the upper tail depen- Tables Figures dence coefficient (TDC). Based on the properties of the Kendall correlation and TDC, 20 a Copula-based mixed model for modelling the dependence structure and marginals is J I suggested. Recently, Copula-based models for estimating error fields of radar information are developed (Villarini et al., 2008; AghaKouchak et al., 2010a,b) J I The intermittend nature of daily and sub-daily rainfall time series (zero-inflated data) Back Close can be modelled by mixed distributions, which are distributions with a continuous part 25 describing data larger than zero and a discrete part accounting for probabilities to ob- Full Screen / Esc serve zero values (Serinaldi, 2009). While the univariate form of such distributions was studies by many authors, the bi- or multivariate case received less attention (Serinaldi, Printer-friendly Version 2009). Interactive Discussion

3004 icsinPpr|Dsuso ae icsinPpr|Dsuso ae | Paper Discussion | Paper Discussion | Paper Discussion | Paper Discussion Conventional rainfall models operate under the assumption of either constant variance or season-dependent variances using an AutoRegressive Moving Average HESSD (ARMA) model. However, it could be shown that daily rainfall data are affected by non- 8, 3001–3045, 2011 linear characteristics of the variance often referred to as variance clustering or volatility, 5 in which large changes tend to follow large changes, and small changes tend to follow small changes. This nonlinear phenomenon of the variance behaviour can be found Copula-based e.g. in monthly and daily streamflow data (Wang et al., 2005), but also in meteorological stat. refinement of time series such as temperature (Romilly, 2005). It is analyzed in this paper if volatility precipitation in RCMs in daily precipitation series can be modelled using Generalized AutoRegressive Condi- simulations 10 tional Heteroskedasticity (GARCH) models. This paper addresses the questions of (i) how to model the temporal characteristics, P. Laux et al. i.e. serial dependence and time varying variance (volatility) of daily rainfall series, and (ii) how to describe the complex joint dependence structure of measured daily rainfall Title Page series and corresponding simulated rainfall series obtained from a RCM model. The 15 method of choice in this paper is the bivariate Copula. The dependence structure is Abstract Introduction investigated for each observation station separately. The innovation of this paper mainly is: Conclusions References – Application of an ARMA-GARCH algorithm to analyze daily precipitation time se- Tables Figures ries for seasonal variation and volatility, and to generate independent and identi- 20 cally distributed (hereinafter iid) residuals for the Copula approach. J I

– Description and modelling of the joint dependence structure between RCM mod- J I elled and observed precipitation, accounting for the prevailing ﬂow situations Back Close caused by large-scale circulation patterns. Full Screen / Esc

2 Regional climate model simulations and observed data Printer-friendly Version

25 Regional climate simulations used in this study are based on the Penn State/NCAR Interactive Discussion Mesoscale Model (MM5) and ECMWF/ERA15 reanalysis data for 1979–1993 at 3005 icsinPpr|Dsuso ae icsinPpr|Dsuso ae | Paper Discussion | Paper Discussion | Paper Discussion | Paper Discussion 19.2 km spatial resolution. The climate simulations have been carried out within the framework of the QUIRCS-DEKLIM project (Kotlarski et al., 2005). A comparison of HESSD the MM5 simulations with gridded observation data for Germany obtained from the 8, 3001–3045, 2011 German weather service (DWD) reveal that rainfall is overestimated by MM5 for the 5 eastern part of Germany, and strongly underestimated for the Rhine valley and the alpine region of Germany (Fig. 1, bottom). The underestimation in the alpine region is Copula-based possibly due to the complex terrain with strong gradients of altitude (Fig. 1, top). stat. refinement of In order to statistically refine and correct precipitation obtained by RCM (MM5) cli- precipitation in RCMs mate simulations, daily rainfall data of 132 observation stations within the alpine region simulations 10 of Germany are retrieved from the webwerdis data portal of the DWD. Our study focusses on the alpine subregion round Garmisch-Partenkirchen. For the analysis of P. Laux et al. the dependence structure between modelled and observed precipitation, a subset of 14 observation stations with large altitudinal differences is selected (see Fig. 2 and Ta- Title Page ble 1). These stations correspond to three different grid cells of the RCM output where 15 the model bias is comparatively large. Abstract Introduction

Conclusions References 3 Modelling the dependence structure between modelled and observed rainfall Tables Figures The procedure followed in this paper to model the dependence structure between RCM modelled and observed rainfall, and to finally generate random samples of locally re- J I fined and bias corrected pseudo-observations, requires multiple steps (Fig. 4) that can J I 20 be comprised as follows: Back Close 1. A suitable ARMA-GARCH model is fitted to the modelled and observed rainfall series (positive values only) to capture the seasonal variation of variance (see Full Screen / Esc Sect. 3.1.1) of both RCM simulated and station observed precipitation. 2. The marginals are fitted to semi-parametric Generalized Pareto Distributions Printer-friendly Version 25 (GPD) for an improved representation of the tails (see Sect. 3.1.2). Interactive Discussion

3006 icsinPpr|Dsuso ae icsinPpr|Dsuso ae | Paper Discussion | Paper Discussion | Paper Discussion | Paper Discussion 3. The bivariate empirical Copula (bivariate probability density plot), which is inde- pendend from their corresponding marginal distributions, is derived from the resid- HESSD uals of the ARMA-GARCH model. 8, 3001–3045, 2011 4. Using the marginal distributions of the original data (positive pairs only) and the 5 iid residuals obtained from the ARMA-GARCH model, a theoretical Copula model Copula-based is estimated (see Sect. 3.2.2). stat. reﬁnement of 5. Stochastic simulations are performed using the conditional CDF of the theoretical precipitation in RCMs Copula (see Sect. 3.2.3). simulations

6. Stochastic simulations are performed using conditional CDFs of the theoretical P. Laux et al. 10 Copula of diﬀerent large-scale weather patterns (see Sect. 3.3).

3.1 Modelling the marginals Title Page

Modelling the single marginal distributions requires the observations to be iid. Most Abstract Introduction climatological time series, however exhibit some degree of autocorrelation and heteroskedasticity. In the sequel the ARMA-GARCH composite model to generate iid Conclusions References 15 variables is introduced, followed by the description of how to ﬁt a GPD to the marginals, Tables Figures and to derive a joint distribution function (Copula) to model the dependence between modelled and observed rainfall time series. J I

3.1.1 ARMA-GARCH ﬁlter J I

This section describes brieﬂy the theory of the ARMA/GARCH composite model and Back Close 20 how it is used to simulate the univariate time series in the presence of conditional Full Screen / Esc mean as well as conditional time-varying variance, i.e. heteroskedasticity or volatility, to produce iid residuals. An ARMA model is used to compensate for autocorrelation, Printer-friendly Version and a GARCH model to compensate for the heteroskedasticity. The term conditional in GARCH – Generalized Autoregressive Conditional Het- Interactive Discussion 25 eroskedasticity – implies explicitely the dependence on a past sequence of 3007 icsinPpr|Dsuso ae icsinPpr|Dsuso ae | Paper Discussion | Paper Discussion | Paper Discussion | Paper Discussion observations, and autoregressive describes a feedback mechanism that incorporates past observations into the present. GARCH is a time series modelling technique that HESSD includes past variances for predicting present or future variances. 8, 3001–3045, 2011 A univariate model of an observed time series yt can be written as

5 y f t ,X ε t = ( − 1 ) + t (1) Copula-based In this equation, the term f (t − 1,X ) represents the deterministic component of the stat. reﬁnement of current value as a function of the information known at time t−1, including past innova- precipitation in RCMs simulations tions {εt−1, εt−2, ...}, past observations {yt−1, yt−2, ...} and other relevant time series data X . Bollerlslev (1986) developed GARCH as a generalization of the ARCH volatility P. Laux et al. 10 modelling technique (Engle, 1982). The distribution of the residuals, conditional on the time t, is given by

2 2 Title Page Vart−1 (yt) = Et−1 εt = σt (2) Abstract Introduction where P Q Conclusions References X X σ2 κ G σ2 A ε2 t = + i t−i + j t−j (3) Tables Figures i=1 j=1

2 J I 15 One can see that σt is the prediction of the variance, given the past sequence of variance predictions, σ2 , and past realizations of the variance itself, ε2 . When P = 0, t−i t−j J I the GARCH(0,Q) model becomes the original ARCH(Q) model introduced by Engle (1982). This equation mimics the variance clustering of the variable (i.e. precipitation Back Close and temperature). The lag lengths P and Q and the coefficients G and A determine i j Full Screen / Esc 20 the degree of persistence. A common assumption is that the innovations are serially independent, however, Printer-friendly Version GARCH(P,Q) innovations, {εt}, are modelled as Interactive Discussion εt = σt zt. (4) 3008 icsinPpr|Dsuso ae icsinPpr|Dsuso ae | Paper Discussion | Paper Discussion | Paper Discussion | Paper Discussion σt is the conditional standard deviation given by the square root of Eq. (3), and zt is the standardized iid random draw from some specified probability distribution. Usually, HESSD a Gaussian distribution is assumed such that ε∼N(0, σ2). Reflecting this, Eq. (5) illus- t 8, 3001–3045, 2011 trates that a GARCH innovations process {εt} simply rescales an iid process {zt} such 5 that the conditional standard deviation incorporates the serial dependence of Eq. (3). Copula-based 3.1.2 Generalized pareto distribution stat. refinement of precipitation in RCMs This subsection describes the fitting of semi-parametric cumulative distribution func- simulations tions (CDFs). First, the empirical CDF of each parameter is estimated using a Gaussian kernel function (using a kernel width of 50 points) to eliminate the staircase pattern. P. Laux et al. 10 This provides a reasonably good fit to the interior of the distribution of the residuals. This procedure, however, tends to perform poorly when applied to upper and lower tails. Title Page The upper and lower tails therefore are fitted separately from the interior of the distri- Abstract Introduction bution. For this reason, the peaks over threshold (POT) method is applied: A threshold 15 value of 0.1 is chosen, i.e. the upper and lower 10% of the residuals are reserved for Conclusions References each tail. The extreme residuals (beyond the threshold) are fitted to a parametric GPD, which can be described as Tables Figures −1− 1 1 (x − θ) k y = f (x|k, σ, θ) = 1 + k (5) J I σ σ J I using a maximum likelihood approach. Given the exceedances in each tail, the neg- Back Close 20 ative log-likelihood function is optimized to estimate the tail index/shape parameter k

and the scale parameter σ of the GPD. The composite GPD function allows for inter- Full Screen / Esc polation in the interior of the CDF but also for extrapolation in the lower and upper tails. Printer-friendly Version

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3009 icsinPpr|Dsuso ae icsinPpr|Dsuso ae | Paper Discussion | Paper Discussion | Paper Discussion | Paper Discussion 3.2 Copula based joint distribution functions of modelled and observed rainfall HESSD Copulas are functions that link multivariate distribution F (x1, ... xn) to their univariate marginals F (x ). Sklar (1959) proved that every multivariate distribution F (x , ... x ) 8, 3001–3045, 2011 Xi i 1 n can be expressed in terms of a Copula C and its marginals F (x ): Xi i Copula-based 5 F (x , ... x ) = C F (x ), ..., F (x ) (6) 1 n X1 1 Xn n stat. refinement of precipitation in RCMs C : [0, 1]n → [0, 1]. (7) simulations Copulas allow to merge the dependence structure and the marginal distributions to P. Laux et al. form a joint multivariate distribution. The Copula function is unique when the marginals are steady functions. As the Copula is a reflection of the dependency structure itself, 10 its construction is reduced to the study of the relationship of the correlated iid variables, Title Page giving freedom for the choice of the univariate marginal distributions. Further information about Copulas can be found e.g. in Joe (1997); Frees and Valdez (1998); Nelson Abstract Introduction (1999); Salvadori et al. (2007). The Copula approach allows to account for the fact that the dependence structure Conclusions References 15 between regional and local meteorological fields and between simulated and observed Tables Figures fields is more complex than it can be modelled by the multivariate normal distribution or ordinary dependency measures such as e.g. the Pearson correlation coefficient. J I The complex multivariate dependence structure is analyzed between RCM modelled precipitation (MM5 output) and station observed precipitation. J I 20 As there exist no unique characterization of the Copula for dry days, our work fo- Back Close cusses on the positive pairs (RCM precipitation > 0, observed precipitation > 0). For dry days, only the conditional marginals can be identified (Yang, 2008). Full Screen / Esc

3.2.1 Empirical Copula Printer-friendly Version

The dependence structure of daily measured precipitation and simulated precipitation Interactive Discussion 25 is studied. Since the underlying (theoretical) Copula is not known in advance, it is 3010 icsinPpr|Dsuso ae icsinPpr|Dsuso ae | Paper Discussion | Paper Discussion | Paper Discussion | Paper Discussion necessary to analyze the empirical Copula, which is purely based on the data (De- heuvels, 1979). The ranks of the residuals of modelled and observed rainfall from HESSD day 1 to day n, obtained from the original data as well as the ARMA-GARCH time 8, 3001–3045, 2011 series model, are {r1(1), ..., r1(n)} and {r2(1), ..., r2(n)}, respectively. The empirical 5 Copula is deﬁned as: n Copula-based X r1(t) r2(t) C (u,v) = 1/n 1 u, v (8) stat. reﬁnement of n n 6 n 6 t=1 precipitation in RCMs where u indicates the percentile of the modelled rainfall residuals, v indicates the per- simulations centile of the measured rainfall residuals and 1(...) is denoting the indicator function. P. Laux et al. 3.2.2 Estimation of the theoretical Copula

Title Page 10 A goodness-of-ﬁt test for Copulas is applied comparing the empirical Copula Cn (Eq. 8) with the parametric estimate of a theoretical Copula model Cθ derived under the null Abstract Introduction hypothesis. The test is based on the Cramer-von´ Mises statistic (Genest and Favre, 2007): Conclusions References n Tables Figures X 2 Sn = n {Cθ (ut, vt) − Cn (ut, vt)} . (9) t=1 J I 15 As the deﬁnition of Sn involves the theoretical Copula function, the distribution of this statistic depends on the unknown value of θ under the null hypothesis that C is J I from the class C (Gregoire´ et al., 2008). Therefore, the approximate p-values for the θ Back Close test statistic are obtained using a parametric bootstrap (Genest and Remillard, 2008; Genest et al., 2009) as well as a fast multiplier approach (Kojadinovic and Yan, 2011,?). Full Screen / Esc

20 3.2.3 Copula-based rainfall simulations Printer-friendly Version

After the estimation of the Copula-based joint distribution – that is FX (x), FY (y) and Interactive Discussion Cθ(u,v) are obtained – conditional random samples from this distribution are generated 3011 icsinPpr|Dsuso ae icsinPpr|Dsuso ae | Paper Discussion | Paper Discussion | Paper Discussion | Paper Discussion through Monte Carlo simulations. We follow the procedure of Salvadori for the conditional simulation using Copulas Salvadori et al. (2007). The simulation is based on HESSD conditional probabilities of the form: 8, 3001–3045, 2011 ∂ P (V ≤ v|U = u) = C (u, v); (10) ∂u Copula-based ∂ 5 P (U ≤ u|V = v) = C (u, v). (11) stat. reﬁnement of ∂v precipitation in RCMs For the Gumbel-Hougaard Copula e.g. it is: simulations ∂ θ θ 1/θ C(u,v) = u−1e−(log(u) )+(log(v) ) (−log(u))−1+θ((−log(u)θ)+(−log(v))θ)−1+1/θ. (12) P. Laux et al. ∂u The concept pseudo-observation simulation from model data is as follows: a pair of

variates (u, v) with Copula C(u, v) needs to be generated which ﬁnally can be trans- Title Page 10 formed into (x, y), using the probability integral transformation −1 Abstract Introduction U = FX (x) ⇔ X = FX (U) (13) Conclusions References −1 V = FY (y) ⇔ Y = FY (V ). (14) Tables Figures The complete algorithm is divided into three steps:

1. Computation u = FX (x), where x denotes one value of the modelled rainfall and J I 15 FX (x) is the marginal distribution of the variate X . J I 2. Generation of random samples for the variate v ∗ from the conditional CDF −1 ∗ −1 Back Close CV |U (v|u) = cu(v) and calculation of v = cu (v ), where cu denotes the generalized inverse of cu (Nelsen, 1999). Full Screen / Esc 3. Calculation of the corresponding y-values using the probability integral transfor- −1 Printer-friendly Version 20 mation FY (v) = y.

The final result for y is a sample of pseudo-observations which lies in the original Interactive Discussion data space and can be compared with the observed data series. 3012 icsinPpr|Dsuso ae icsinPpr|Dsuso ae | Paper Discussion | Paper Discussion | Paper Discussion | Paper Discussion 3.3 Usability of weather patterns for conditional simulations HESSD Especially for complex terrain, it is assumed that the direction of advection is of crucial importance for the observed precipitation amounts. The combinations of terrain 8, 3001–3045, 2011 exposition and advection direction leads to luv and lee side effects, i.e. the stations can 5 lie in the rainshadow or can be exposed to intense rainfall. As independent from the Copula-based RCM simulations, large-scale weather patterns are used to further improve the results stat. refinement of of the bias correction. Besides the advection direction, large scale information about precipitation in RCMs cyclonality and tropospheric humidity is evaluated. The objective weather pattern clas- simulations sification method of the German Weather Service is used (Bissolli and Dittmann, 2001). 10 The classification domain is Germany, and the meteorological criteria for the classifi- P. Laux et al. cation are (i) the direction of advection of air masses, (ii) the cyclonality, and (iii) the humidity of the troposphere. This leads to numerical indices from which the weather types are derived (Bissolli and Dittmann, 2001). There exist 40 predefined types, which Title Page

can be used. Due to the limited occurrence frequencies of single weather types, their Abstract Introduction 15 usability for conditional simulations is restricted. However, the usage of the numerical indices provides the possibility to group the types to different classes. Conclusions References For this study, the following grouping strategies are evaluated: Tables Figures 1. Grouping types due to the direction of the advection of air masses at 700 hPa: the weather types (WTs) are grouped into northeasterly, southeasterly, southwesterly, J I 20 and northwesterly flow. J I 2. Grouping types due to the cyclonality at 950 hPa and 500 hPa: this leads to four classes, namely anticyclonal – anticyclonal (AA), anticyclonal – cyclonal (AC), Back Close cyclonal – anticyclonal (CA), and cyclonal – cyclonal (CC). Full Screen / Esc 3. Grouping types due to the humidity of the troposphere: this leads to the discrim- Printer-friendly Version 25 ination of dry (D) and wet (W). Therefore, a humidity index is calculated as the weighted areal mean of the precipitable water integrated over the 950, 850, 700, Interactive Discussion 500, and 300 hPa levels. 3013 icsinPpr|Dsuso ae icsinPpr|Dsuso ae | Paper Discussion | Paper Discussion | Paper Discussion | Paper Discussion For each group of weather types, a theoretical Copula model is estimated separately. For sake of simplicity, the Gumbel-Hougaard Copula model is used. HESSD 8, 3001–3045, 2011 4 Simulation results Copula-based In this section simulation results of both, the obtained RCM and corresponding ob- stat. refinement of 5 served precipitation time series are exemplarily presented. Based on iid residuals precipitation in RCMs obtained by ARMA-GARCH models the empirical and theoretical Copulas, and the simulations marginal distributions are estimated and analyzed and locally refined and bias cor- P. Laux et al. rected pseudo-observations are generated.

4.1 Analysis of ARMA-GARCH time series models Title Page

10 The autocorrelation function and Ljung-Box Q-test is applied to the original time se- Abstract Introduction ries, the squared original time series as well as the resulting standardized residuals and standardized squared residuals of the ARMA-GARCH model. According to the Conclusions References autocorrelation function plots the original time series show serial dependence and het- Tables Figures eroskedasticity, i.e. non-iid behaviour (Fig. 3). After application of the ARMA-GARCH 15 model, the time series of the residuals can be seen as serially independent (Fig. 5). The Ljung-Box Q-test conﬁrms the results of the autocorrelation function (not shown J I

here). J I The K-plots indicate that there remains a positive dependence in the upper tails of the residuals (Fig. 6). Further information about how to calculate and to interprete Back Close

20 the K-plots can be obtained e.g. by Genest and Favre (2007). Therefore, a sensitivity Full Screen / Esc analysis is conducted accounting for the order of ARMA-GARCH models (using orders

for AR, MA, P, and Q between 1 and 3, respectively), the threshold value for a wet day Printer-friendly Version (0.01 mm, 0.1 mm, and 1 mm), and the peak-over-threshold (POT) value for lower and upper tail (10% and 20%). It is found that the larger the wet day threshold, the higher is Interactive Discussion

3014 icsinPpr|Dsuso ae icsinPpr|Dsuso ae | Paper Discussion | Paper Discussion | Paper Discussion | Paper Discussion the distortion of the upper tail, which can be partly explained by the fat tail behaviour. The POT and the order of the ARMA-GARCH are less sensitive. HESSD Figure 7 (top) shows the empirical and fitted exceedance probability for the upper tail 8, 3001–3045, 2011 of the observed rainfall residuals at station Garmisch-Partenkirchen. Both, for observed 5 and modelled rainfall, the Generalized Pareto Distribution seems to be a good choice to fit the upper tails of the data. Figure 7 (bottom) illustrates the composite of the three Copula-based piecewise CDFs for modelled and observed rainfall residuals. stat. refinement of precipitation in RCMs 4.2 Analysis of empirical and theoretical Copula models simulations P. Laux et al. Figure 8 (top) shows the empirical Copula density between modelled and measured 10 rainfall for station Garmisch-Partenkirchen. Only the positive pairs of modelled and measured rainfall are shown, using a threshold of 0.01 mm to define a wet day. It Title Page can be seen from the figure that the distribution is strongly asymmetrical for the minor diagonal, and that the density in the upper corner is highest. This implies that modelled Abstract Introduction

and observed rainfall are strongly concordant in the higher ranks of the distribution, Conclusions References 15 whereas the concordance is weaker in the lower ranks. This empirical density structure may be remarkably different compared to the ARMA-GARCH transformed residuals Tables Figures (Fig. 8, bottom). Table 2 shows the results for the goodness-of-fit (GOF) test statistics using the para- J I metric bootstrap procedure. In order to chose between the three different Copula fami- J I 20 lies, namely Normal, Gumbel-Hougaard, and Clayton Copula, the parametric bootstrap algorithm of Genest and Remillard (2008) is applied to observed and modelled precipi- Back Close tation. 1000 bootstrap values of the Cramer-von-Mises´ test statistic are produced, and Full Screen / Esc the proportion of those values that are larger than Sn (p-values) is estimated. From the p-values obtained the usability of the Gumbel-Hougaard Copula is concluded. The Printer-friendly Version 25 Copula parameters which are used for Copula-based stochastic simulations are also

given in Table 2. Interactive Discussion

3015 icsinPpr|Dsuso ae icsinPpr|Dsuso ae | Paper Discussion | Paper Discussion | Paper Discussion | Paper Discussion 4.3 Dependence on altitude and distance HESSD The dependence of the altitude is shown in Table 1. The variability of the simulated results (shown as empirical CDFs) are obviously not related to differences in the Copula 8, 3001–3045, 2011 parameters, just weak differences of θ can be found (see Table 2). No clear functional 5 dependence between the altitude of the stations and the Copula parameter θ exists. Copula-based Additionally, neither shape parameter nor scale parameter of the marginal distributions stat. refinement of are systematically differing with the station altitude. At least one could suspect that precipitation in RCMs the scale parameter for the observed precipitation is increasing with height, but there simulations are just three grid cells available for this inspection which is clearly not significant. 10 The dependence of the distance between observed rainfall and modelled rainfall of P. Laux et al. the corresponding and surrounding RCM grid cells is analyzed using the Kendall’s τ (Fig. 17). In general, higher coefficients are obtained for nearest grid cells, and the correlations decrease anisotrophically around the stations. Moreover, the pattern Title Page

shown in this ﬁgure mimics well the regional topography given in the RCM model. In Abstract Introduction 15 some cases, the corresponding grid cell does not show the highest correlation. There is a functional relationship between the classical dependence parameters Conclusions References such as Kendall’s τ and Spearman’s ρ namely ZZ ZZ Tables Figures ρ = 12 u v dCθ (u, v) − 3 = 12 Cθ (u, v) du dv − 3 (15) 2 2 [0, 1] [0, 1] J I and ZZ J I 20 τ = 4 Cθ (u, v) dCθ (u, v) − 1 (16) [0, 1]2 Back Close

or for Archimeadean Copulas with generator ϕ Full Screen / Esc Z ϕ(t) τ = 1 + 4 . (17) 0 [0, 1] ϕ (t) Printer-friendly Version θ For the Gumbel-Hougaard Copula with its generator ϕ(t) = (−ln(t)) it is found that Interactive Discussion 1 θ = 1−τ , so θ is a increasing function of τ. Thus the “Copula map” shown in Fig. 17 3016 icsinPpr|Dsuso ae icsinPpr|Dsuso ae | Paper Discussion | Paper Discussion | Paper Discussion | Paper Discussion may be interpreted in the light of dependence. Higher Copula parameters θ reveal a stronger dependence. HESSD

4.4 Dependence of large-scale weather situation 8, 3001–3045, 2011

The dependence structure between modelled and observed rainfall, given the large- Copula-based 5 scale weather situation, is analysed. The method used for classifying large-scale stat. refinement of weather types is described in Sect. 3.3. The empirical Copulas are calculated us- precipitation in RCMs ing different grouping strategies for the WTs. Based on the empirical Copulas, as well simulations as the conditional CDFs, the usability for conditional simulations is investigated. Using four different weather types and one indefinite type for advection (Fig. 10) P. Laux et al. 10 can have additional value, and thus be used for conditional stochastic simulations. Figure 11 illustrates the empirical Copula density for modelled and observed precipitation for Garmisch-Partenkirchen grouping the weather types due to the cyclonality in Title Page 950 hPa and 500 hPa into four classes. One can observe that for the four classes sig- Abstract Introduction nificant differences within the dependence structure between modelled and observed 15 rainfall exist. The classification due to the humidity of the troposphere (Fig. 12) does not Conclusions References lead to a clear discrimination between the empirical Copulas. Both, the wet and the dry Copula density is similar to the unconditional Copula densities (compare with Fig. 8). Tables Figures The empirical CDFs of observed precipitation in Garmisch-Partenkirchen based on a given WT and certain groups of WTs are illustrated in Fig. 13. J I

J I 20 4.5 Conditional stochastic simulations of pseudo-observations Back Close Figure 9 shows the results of Copula-based stochastic simulations (100 realizations) of pseudo-observations asssuming that the modelled RCM precipitation is given. A split- Full Screen / Esc sampling approach is used to subdivide the data into calibration and validation period. It can be seen from the figure that the observations are usually underpredicted by the Printer-friendly Version 25 model, and that the Copula-based technique can partly correct for that effect. For the very high RCM rainfall amounts, the Copula-based approach tends to overestimate Interactive Discussion 3017 icsinPpr|Dsuso ae icsinPpr|Dsuso ae | Paper Discussion | Paper Discussion | Paper Discussion | Paper Discussion the observations. After this first graphical comparison the improvements attained using the Copula approach are analyzed further with selected performance measures. A HESSD first hint is given by the correlations between the observations, RCM and the pseudo- 8, 3001–3045, 2011 observations i.e. the bias corrected prediction (see Table 3). The Pearson correlation 5 coefficient between observations and RCM is calculated as 0.3 for the iid transformed data of Garmisch-Partenkirchen, and between observations and the mean value of the Copula-based random sample, generated through the Copula approach, it is 0.36. This corroborates stat. refinement of the usability of the Copula based bias correction of precipitation. Including large-scale precipitation in RCMs information about advection, cyclonality, and humidity is increasing the correlation co- simulations 10 efficient between observations and pseudo-observations. The correlation between the RCM and the pseudo-observations is higher. However, the Pearson correlation is just P. Laux et al. a general measure, operating on the complete time series, and does not mirror the quality of the new method for specific subsets of the rank space. In turn, the probability Title Page plot (Fig. 15) provides a performance measure for the quantiles of the distribution. 15 It can be seen from Fig. 15 that the RCM underestimates the observations over the Abstract Introduction whole range of the distribution. Taking this as a reference, the Copula-based stochastic simulations of the pseudo-observations lead to significant improvements. However, Conclusions References

the simulations still suﬀer from fat tail characteristics. Including large-scale conditional Tables Figures information contributes moderately to a reduction of the bias. Including information 20 about humidity of the troposphere, the performance skill is comparable to the Copula- J I based stochastic simulations without any large-scale information. J I

Back Close 5 Discussion Full Screen / Esc It is shown that ARMA-GARCH models are able to model serial dependence and volatility in the precipitation time series. Consequently, they are generally useful to generate Printer-friendly Version 25 iid random variables. However, even high order models are not able to fully capture the Interactive Discussion fat tail behaviour. Filtering the time series before fitting to a theoretical Copula model 3018 icsinPpr|Dsuso ae icsinPpr|Dsuso ae | Paper Discussion | Paper Discussion | Paper Discussion | Paper Discussion is reducing the correlation between RCM (here: MM5) and observed precipitation and thus the estimated Copula parameter θ, which is directly related to Kendall’s τ. HESSD From the theoretical Copula models analyzed, the Gumbel-Hougaard Copula is 8, 3001–3045, 2011 found to be a suited choice to model the joint distribution of modelled (gridded) and 5 observed precipitation. While the Copula parameter is relatively stable for the joint distribution functions between different locations within the same grid cell, the shape Copula-based and scale parameters of the fitted marginal distributions of the observation stations can stat. refinement of differ significantly. It is also found that the correlation between station and grid cell is precipitation in RCMs not necessarily highest for the corresponding grid cell. Assessing the dependence of simulations 10 the distance between observed rainfall and rainfall of the surrounding RCM grid cells potentially allows for spatial interpolation in ungauged regions. However, further inves- P. Laux et al. tigations will be necessary. The empirical Copula density plot is used to analyze the dependence structure Title Page between modelled and observed precipitation. As computational inexpensive they 15 are suitable to (i) find a theoretical Copula model, and (ii) screen variables such as Abstract Introduction e.g. large-scale weather types which could additionally improve the performance of the bias correction. Conclusions References

The objective weather pattern classification method of the German Weather Ser- Tables Figures vice (Bissolli and Dittmann, 2001) shows only moderate potential to further constraint 20 the model. Including information about the humidity of the troposphere can slightly J I increase the skill for bias correction compared to the Copula-based stochastic simulations without using large-scale information. This can be seen from the conditional CDFs J I and the corresponding probability plots for the different groups, which are not very dis- Back Close criminative (compare Sects. 3.3 and 4.4). Using the single 40 weather types (without 25 grouping) could potentially increase the discriminative power (see e.g. Fig. 16), but de- Full Screen / Esc creases the sample size for certain WTs far too much to reliably estimate the Copula parameter(s) and the marginal distributions. Another limitation of the approach shown Printer-friendly Version in this paper is that the same theoretical Copula model, i.e. the Gumbel-Hougaard Cop- ula, is fitted to each WT class. It is obvious from the empirical Copula density plots, Interactive Discussion

3019 icsinPpr|Dsuso ae icsinPpr|Dsuso ae | Paper Discussion | Paper Discussion | Paper Discussion | Paper Discussion that this does not necessarily provide an adequate fit for all groups of weather types. It is also well-known from other studies that the domain size (here: whole Central Europe HESSD domain) can strongly impact the classification results (e.g. Laux, 2009), and thus the 8, 3001–3045, 2011 subsequent conditional modelling. 5 In this study a stationary approach for the Copula parameter θ is chosen. Further improvements are expected by accounting for the temporal variability of θ. Figure 17 Copula-based illustrates the temporal variability of τ, which is empirically linked with the Copula pa- stat. refinement of rameter. As seen in Sects. 4.2 and 4.4, all the empirical Copulas derived in this study precipitation in RCMs show a strong asymmetry with respect to the minor axis u = 1 − v of [0, 1]2. This simulations 10 asymmetry can not be depicted by the common Copula families such as the Clayton, Normal or Gumbel-Hougaard Copulas, acting as basic set of possible candidates for P. Laux et al. the performed GOF tests. Nonmonotonic transformation to construct asymmetric multivariate Copulas from the Gaussian (Bardossy´ , 2006) could reflect the asymmetries in Title Page the empirical Copulas and thus also improve the bias correction. Abstract Introduction

15 6 Conclusions Conclusions References

The presented Copula-based approach is potentially useful for statistical downscaling, Tables Figures bias correction, and local refinement of RCMs. The performance will be evaluated and compared to established methods for bias correction. J I Asymmetries are found in the empirical Copula densities which cannot be repro- J I 20 duced by the theoretical Copulas used in this study. Therefore, it is generally difficult to find a theoretical Copula model which is not rejected by the applied GOF test. Back Close Fitting the marginal distributions is of crucial importance as it strongly impacts the simulation results (more than the Copula parameter θ). Full Screen / Esc Large-scale weather patterns could be used to further constrain the model, and thus, 25 increase the performance of the simulation results. Printer-friendly Version

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3020 icsinPpr|Dsuso ae icsinPpr|Dsuso ae | Paper Discussion | Paper Discussion | Paper Discussion | Paper Discussion Acknowledgements. This research is funded by the Bavarian State Ministry of the Environ- ment and Public Health (reference number VH-ID:32722/TUF01UF-32722). Additional funding HESSD by the Federal Ministry of Education and Research (research project: Land Use and Climate Change Interactions in Central Vietnam LUCCi, reference number 01LL0908C), by the HGF 8, 3001–3045, 2011 5 Impuls- und Vernetzungsfond (research project: Regional Precipitation Observation by Cellular Network Microwave Attenuation and Application to Water Resources Management PROCEMA, Virtual Institute, reference number VH-VI-314), and by HGF (research project: Terrestrial Envi- Copula-based ronmental Observatories TERENO – http://www.tereno.net) is highly acknowledged. The help stat. reﬁnement of of Thomas Rummler, who created Fig. 2 is appreciated. precipitation in RCMs simulations

10 References P. Laux et al.

AghaKouchak, A., Bardossy,´ A., and Habib, H.: Copula-based uncertainty modeling: Ap- plication to multi-sensor precipitation estimates, Hydrol. Process., 24(15), 2111–2124, Title Page doi:10.1002/hyp.7632, 2010a. 3004 AghaKouchak, A., Bardossy,´ A., and Habib, H.: Conditional simulation of remotely sensed Abstract Introduction 15 rainfall data using a non-gaussian v-transformed copula, Adv. Water Resour., 33(6), 624– 634, doi:10.1016/j.advwatres.2010.02.010, 2010b. 3004 Conclusions References Bardossy,´ A.: Copula-based geostatistical models for groundwater quality parameters, Water Resour. Res., 42(11), W11416.1–W11416.12, doi:10.1029/2005WR004754, 2006. 3004, Tables Figures 3020 20 Bardossy,´ A. and Li, J.: Geostatistical interpolation using copulas. Water Resour. Res., 44(7), J I W07412.1–W07412.15, doi:10.1029/2007WR006115, 2008. 3004 Bardossy,´ A. and Pegram, G. G. S.: Copula based multisite model for daily precipitation simula- J I tion, Hydrol. Earth Syst. Sci., 13, 2299–2314, doi:10.5194/hess-13-2299-2009, 2009. 3003 Bardossy,´ A., Stehl´ık, J., and Caspary, J.: Automated objective classiﬁcation of daily circula- Back Close 25 tion patterns for precipitation and temperature downscaling based on optimized fuzzy rules, Full Screen / Esc Climate Res., 23, 11–22, 2002. 3003 Bissolli, P. and Dittmann, E.: The objective weather type classiﬁcation of the German Weather Service and its possibilities of application to environmental and meteorological investigations, Printer-friendly Version Meteorol. Z., 10(4), 253–260, 2001. 3013, 3019 Interactive Discussion

3021 icsinPpr|Dsuso ae icsinPpr|Dsuso ae | Paper Discussion | Paper Discussion | Paper Discussion | Paper Discussion Bliefernicht, J. and Bardossy,´ A.: Probabilistic forecast of daily areal precipitation focusing on extreme events, Nat. Hazards Earth Syst. Sci., 7, 263–269, doi:10.5194/nhess-7-263-2007, HESSD 2007. 3003 Bollerslev, T.: Generalized autoregressive conditional heteroskedasticity, J. Econometrics, 8, 3001–3045, 2011 5 31(3), 307–327, doi:10.1016/0304-4076(86)90063-1, 1986. 3008 De Michele, C. and Salvadori, J.: A generalized pareto intensity-duration model of storm rainfall exploiting 2-copulas, J. Geophys. Res.-Atmos., 108, D2111, doi:10.1029/2002JD002534, Copula-based 2003. 3004 stat. refinement of Dupuis, D.: Using copulas in hydrology: Benets, cautions, and issues, J. Hydrol. Eng., 12(4), precipitation in RCMs 10 381–393, 2007. 3004 simulations Engle, R.: Autoregressive Conditional Heteroskedasticity with Estimates of United Kingdom Inflation, Econometrica, 50, 987–1008, 1982. 3008 P. Laux et al. Favre, A.-C., El Adlouni, S., Perreault, L., Thiemonge,´ N., and Bobee, B.: Multivariate hydrological frequency analysis using copulas, Water Resour. Res., 40, 112, 2004. 3004 15 Frees, E. W. and Valdez, E. A.: Understanding relationships using copulas, N. Am. Actuarial J., Title Page 2(1), 1–25, 1998. 3010 GARCH toolbox: GARCH toolbox User’s Guide, The MathWorks, Inc., Natick, Massachusetts, Abstract Introduction USA, 1999. Genest, C. and Favre, A.-C.: Everything you always wanted to know about copula modeling but Conclusions References 20 were afraid to ask, J. Hydrol. Eng., 12(4), 347–368, 2007. 3004, 3011, 3014 Genest, C. and Remillard, B.: Validity of the parametric bootstrap for goodness-of-fit testing in Tables Figures semiparametric models, Annales de l’Institut Henri Poincare: Probabilites et Statistiques, 44, 1096–1127, 2008. 3011, 3015 J I Genest, C., Remillard, B., and Beaudoin, D.: Goodness-of-fit tests for copulas: A review and a 25 power study, Insur. Math. Eco., 44, 199–214, 2009. 3011 J I Graham, L. P., Hagemann, S., Jaun, S., and Beniston, M.: On interpreting hydrological change from regional climate models, Climatic Change, 81, 97–122, 2007. 3002 Back Close ´ Gregoire, V., Genest, C., and Gendron, M.: Using copulas to model price dependence in energy Full Screen / Esc markets, Energy Risk, 5(5), 58–64, 2008. 3011 30 Joe, H.: Multivariate Models and Dependence Concepts, Chapman and Hall, New York, 1997. 3010 Printer-friendly Version Jung, G. and Kunstmann, H.: High-resolution Regional Climate Modelling for the Volta Basin of Interactive Discussion West Africa, J. Geophys. Res., 112, D23108, doi:10.1029/2006JD007951, 2007. 3003 3022 icsinPpr|Dsuso ae icsinPpr|Dsuso ae | Paper Discussion | Paper Discussion | Paper Discussion | Paper Discussion Kojadinovic, I. and Yan, J.: Fast large-sample goodness-of-fit tests for copulas, Stat. Sinica, 21(2), in press, 2011. 3011 HESSD Kojadinovic, I. and Yan, J.: A goodness-of-fit test for multivariate multiparameter copulas based on multiplier central limit theorems, Stat. Comput., 21(1), 17–30, 2011. 3011 8, 3001–3045, 2011 5 Kotlarski, S., Block, A., Bohm,¨ U., Jacob, D., Keuler, K., Knoche, R., Rechid, D., and Walter, A.: Regional climate model simulations as input for hydrological applications: evaluation of uncertainties, Adv. Geosci., 5, 119–125, 2005, Copula-based http://www.adv-geosci.net/5/119/2005/. 3006, 3029 stat. refinement of Kuhn, G., Khan, S., and Ganguly, A.: Geospatial-temporal dependence among weekly pre- precipitation in RCMs 10 cipitation extremes with applications to observations and climate model simulations in south simulations america, J. Adv. Water Resour., 30, 2401–2423, 2007. 3004 Kunstmann, H., Schneider, K., Forkel, R., and Knoche, R.: Impact analysis of climate change P. Laux et al. for an Alpine catchment using high resolution dynamic downscaling of ECHAM4 time slices, Hydrol. Earth Syst. Sci., 8, 1031–1045, doi:10.5194/hess-8-1031-2004, 2004. 3003 15 Laux, P.: Statistical modeling of precipitation for agricultural planning in West Africa, Disser- Title Page tation, Mitteilungen/Institut fur¨ Wasserbau, Heft 179, ISBN: 978-3-933761-83-5, Universitat¨ Stuttgart, 198 pp., 2009. 3020 Abstract Introduction Leung, L. R., Hamlet, A. F., Lettenmaier, D. P., and Kumar, A.: Simulations of the ENSO hy- droclimate signals in the Pacific Northwest Columbia River basin, B. Am. Meteorol. Soc., 80, Conclusions References 20 2313–2328, 1999. 3003 Nelsen, R. B.: An Introduction to Copulas, Springer-Verlag, New York, 1999. 3010 Tables Figures Renard, B. and Lang, M.: Use of a gaussian copula for multivariate extreme value analysis: Some case studies in hydrology, Adv. Water Resour., 30, 897–912, 2007. 3004 J I Romilly, P.: Time series modelling of global mean temperature for managerial decision-making, 25 J. Environ. Manage., 76, 61–70, 2005. 3005 J I Salvadori, G., De Michele, C., Kottegoda, N. T., and Rosso, R.: Extremes in Nature: an approach using copulas, Springer-Verlag, New York, 2007. 3010, 3012 Back Close Schmidli, J., Frei, C., and Luigi Vidale, P.: Downscaling from GCM precipitation: A benchmark Full Screen / Esc for dynamical and statistical downscaling methods, Int. J. Climatol., 26, 679–689, 2006. 3003 30 Schmidli, J., Goodess, C. M., Frei, C., Haylock, M. R., Hundecha, Y., Ribalaygua, J., and Smith, T.: Statistical and dynamical downscaling: An evaluation and comparison of scenarios for the Printer-friendly Version European Alps, J. Geophys. Res., 112, D04105, doi:10.1029/2005JD007026, 2007. 3002, 3003 Interactive Discussion 3023 icsinPpr|Dsuso ae icsinPpr|Dsuso ae | Paper Discussion | Paper Discussion | Paper Discussion | Paper Discussion Scholzel,¨ C. and Friederichs, P.: Multivariate non-normally distributed random variables in climate research – introduction to the copula approach, Nonlin. Processes Geophys., 15, 761– HESSD 772, doi:10.5194/npg-15-761-2008, 2008. 3004 Senatore, A., Mendicino, G., Smiatek, G., and Kunstmann, H.: Regional climate change projec- 8, 3001–3045, 2011 5 tions and hydrological impact analysis for a Mediterranean basin in Southern Italy, J. Hydrol., 399(1–2), 70–92, 2011. 3003 Sennikovs, J. and Bethers, U.: Statistical downscaling method of regional climate model results Copula-based for hydrological modelling, 18th World IMACS/MODSIM Congress, Cairns, Australia, 13– stat. refinement of 17 July, 2009. 3002 precipitation in RCMs 10 Serinaldi, F.: Analysis of inter-gauge dependence by kendall’s τ, upper tail dependence coeffi- simulations cient, and 2-copulas with application to rainfall fields, Stoch. Env. Res. Risk A., 22, 671–688, 2008. 3004 P. Laux et al. Serinaldi, F.: Copula-based miexd models for bivariate rainfall data: an empirical study in re- gression perspective, Stoch. Env. Res. Risk A., 23, 677–693, 2009. 3004 15 Sklar, K.: Fonctions de repartition a` n dimensions et leura marges. Publications de l’Institut de Title Page Statistique de L’Universite´ de Paris, 8, 229–231, 1959. Smiatek, G., Kunstmann, H., Knoche, H. R., and Marx, A.: Precipitation and temperature Abstract Introduction statistics in high-resolution regional climate models: Evaluation for the European Alps, J. Geophys. Res., 114, D19107, doi:10.1029/2008JD011353, 2009. 3002 Conclusions References 20 Themeßl, M. J., Gobiet, A., and Leuprecht, A.: Empirical-statistical downscaling and error correction of daily precipitation from regional climate models, Int. J. Climatol., Tables Figures doi:10.1002/joc.2168, in press, 2010. 3003 Villarini, G., Serinaldi, F., and Krajewski, W. F.: Modeling radar-rainfall estimation uncertainties J I using parametric and non-parametric approaches, Adv. Water Resour., 31(12), 1674–1686, 25 2008. 3004 J I von Storch, H. and Zwiers, F. W.: Statistical Analysis in Climate Research, Cambridge Univer- sity Press, Cambridge, 1999. 3003 Back Close Wang, W., Van Gelder, P. H. A. J. M., Vrijling, J. K., and Ma, J.: Testing and modelling autore- Full Screen / Esc gressive conditional heteroskedasticity of streamflow processes, Nonlin. Processes Geo- 30 phys., 12, 55–66, doi:10.5194/npg-12-55-2005, 2005. 3005 Wilby, R. L. and Wigley, T. M. L.: Downscaling General Circulation Model output: a review of Printer-friendly Version methods and limitations, Prog. Phys. Geogr., 21, 530–548, 1997. 3002 Interactive Discussion

3024 icsinPpr|Dsuso ae icsinPpr|Dsuso ae | Paper Discussion | Paper Discussion | Paper Discussion | Paper Discussion Wood, A. W., Maurer, E. P., Kumar, A., and Lettenmaier, D. P.: Long-range experimental hy- drologic forecasting for the eastern United States, J. Geophys. Res.-Atmos., 107, 4429, HESSD doi:10.1029/2001JD000659, 2002. 3003 Yang, W.: Discrete-Continous Downscaling Model for Geenerating Daily Precipitation Time Se- 8, 3001–3045, 2011 5 ries. Dissertation, Mitteilungen/Institut fur¨ Wasserbau, Heft 168, Universitat¨ Stuttgart, 2008. 3010 Yang, W., Andreasson,´ J., Graham, L. P., Olsson, J., Rosberg, J., and Wetterhall, F.: Improved Copula-based use of RCM simulations in hydrological climate change impact studies, Hydrol. Res., 41(3– stat. reﬁnement of 4), 211–229, 2010. 3003 precipitation in RCMs 10 Zhang, L. S. R. and Singh, V.: Bivariate rainfall frequency distributions using archimedean simulations copulas, J. Hydrol., 32(1–2), 93–109, 2008. 3004 P. Laux et al.

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Copula-based stat. reﬁnement of Table 1. Station informations corresponding to 3 chosen MM5 grid cells, shape (tail index), precipitation in RCMs and scale parameters of the ﬁtted Generalized Pareto Distribution (GPD) for positive pairs of modelled and observed precipitation respectively. simulations P. Laux et al. Station Altitude Location Precipitation (MM5 output) Precipitation (observed) [m a.s.l.] lat lon Shape Scale Shape Scale Garmisch-Partenkirchen 719 47.48 11.06 0.42 0.34 0.094 8.29 Grainau 760 47.47 11.02 0.29 0.40 0.13 6.757 Title Page Grainau (Eibsee) 1010 47.46 11.00 0.28 0.42 0.06 8.34 Bad Reichenhall 470 47.72 12.88 0.07 0.78 0.26 6.98 Abstract Introduction Schneizlreuth-Unterjettenberg 507 47.68 12.83 0.07 0.82 0.10 8.29 Schneizlreuth-Ristfeucht 523 47.67 12.77 0.08 0.80 0.17 8.06 Conclusions References Schneizlreuth-Weissbach 630 47.72 12.77 0.08 0.80 0.14 8.67 Anger-Oberhogl¨ 690 47.8 12.9 0.06 0.81 0.28 5.50 Tables Figures Bischofswiesen-Winkl 690 47.69 12.94 0.07 0.82 0.13 8.20 Inzell 690 47.76 12.76 0.05 0.84 0.06 10.22 Anger-Stoissberg 830 47.8 12.82 0.08 0.77 0.32 7.00 J I Rottach-Egern 747 47.68 11.77 0.05 0.76 0.15 7.93 Kreuth 895 47.61 11.65 0.04 0.77 0.16 9.65 J I Schwarzkopfhutte¨ 1336 47.66 11.91 0.06 0.74 0.09 10.09 Back Close

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Copula-based stat. reﬁnement of Table 2. Goodness-of-ﬁt (GOF) test using the Cramer-von-Mises´ test statistics and parametric precipitation in RCMs bootstrap procedure (see Sect. 3.2.2). p-values exceeding 0.01 are highlighted in bold. simulations

Station Normal Copula Gumbel-H. Copula Clayton Copula P. Laux et al. Sn p-value θN Sn p-value θGH Sn p-value θC Garmisch-Partenkirchen 0.09 0.00 0.08 0.12 0.00 1.09 0.17 0.00 0.01 Grainau 0.10 0.00 0.09 0.15 0.00 1.10 0.16 0.00 0.01 Grainau (Eibsee) 0.07 0.00 0.12 0.08 0.00 1.11 0.12 0.00 0.06 Title Page

Bad Reichenhall 0.10 0.00 0.19 0.04 0.02 1.15 0.37 0.00 0.13 Abstract Introduction Schneizlreuth-Unterjettenberg 0.14 0.00 0.15 0.10 0.00 1.13 0.29 0.00 0.08 Schneizlreuth-Ristfeucht 0.11 0.00 0.17 0.05 0.00 1.14 0.40 0.00 0.09 Conclusions References Schneizlreuth-Weissbach 0.08 0.00 0.13 0.06 0.00 1.11 0.21 0.00 0.07 Anger-Oberhogl¨ 0.10 0.00 0.16 0.05 0.01 1.14 0.34 0.00 0.11 Bischofswiesen-Winkl 0.07 0.00 0.16 0.04 0.07 1.14 0.23 0.00 0.11 Tables Figures Inzell 0.08 0.00 0.16 0.05 0.01 1.13 0.24 0.00 0.13 Anger-Stoissberg 0.12 0.00 0.14 0.06 0.00 1.12 0.40 0.00 0.07 J I Rottach-Egern 0.05 0.00 0.17 0.03 0.09 1.14 0.21 0.00 0.13 Kreuth 0.06 0.00 0.12 0.06 0.00 1.10 0.17 0.00 0.06 J I Schwarzkopfhutte¨ 0.05 0.01 0.15 0.03 0.07 1.12 0.22 0.00 0.12 Back Close

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Copula-based stat. reﬁnement of precipitation in RCMs simulations

P. Laux et al. Table 3. Pearson correlation coeﬃcients (all signiﬁcant at α = 0.01 level) between positive pairs of pseudo-observations (mean value) produced by Copula-based stochastic simulations without using large-scale information (uncond), including advection (advec), cyclonality (cyclo), and humidity (humi) of the troposphere, and the observed precipitation at station Garmisch- Title Page Partenkirchen and the corresponding grid cell precipitation of RCM. Abstract Introduction

uncond advec cyclo humi Conclusions References

observed 0.36 0.43 0.45 0.37 Tables Figures RCM 0.98 0.93 0.98 0.65

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3028 2 P. Laux et al.: Copula-based stat. refinement of precipitation in RCMs simulations rainfall within one corresponding modelled grid cell could be very high; rain gauges in the near surrounding can measure large differences in rainfall amounts; (ii) difficulties to capture the location of convective rainfall events by the climate models; this is mostly due to coarse resolution of the land surface model (LSM) and the partly chaotic nature of convection, and (iii) wrong or inadequate model parametrisations for convection. | Paper Discussion | Paper Discussion | Paper Discussion | Paper Discussion The dependence structures of multivariate distributions can be modelled using classical distributions such as a multi- HESSD variate normal. Alternatively, a Copula approach can be used to describe the dependence structure independently from the 8, 3001–3045, 2011 marginal distributions (e.g. Genest and Favre, 2007; Dupois, 2007), and thus, to use different marginal distributions at the same time without any transformations. Copula-based There is an increase in applications of Copulas in hydrometeorology over the past years. Copula-based models stat. refinement of have been introduced for multivariate frequency analysis, precipitation in RCMs risk assessment, geostatistical interpolation and multivari- simulations ate extreme value analyses (e.g. De Michele and Salvadori, 2003; Bárdossy, 2006; Genest and Favre, 2007; Renard and Lang, 2007; Schölzel and Friederichs, 2008; Bárdossy and P. Laux et al. Li, 2008; Zhang and Singh, 2008). For rainfall modelling, De Michele and Salvadori (2003) used Copulas to model intensity-duration of rainfall events. Favre et al. (2004) utilized Copulas for multivariate hydrological frequency anal- Title Page ysis. Zhang and Singh (2008) carried out a bivariate rainfall frequencyanalysis using Archimedean Copulas. Renard Abstract Introduction and Lang (2007) investigated the usefulness of the Gaussian Copula in extreme value analysis. Kuhn et al. (2007) em- Conclusions References ployed Copulas to describe spatial and temporal dependence of weekly precipitation extremes. Serinaldi (2008) studied Tables Figures the dependence of rain gauge data using the nonparametric Kendalls rank correlation and the upper tail dependence coefficient (TDC). Based on the properties of the Kendall cor- J I relation and TDC, a Copula-based mixed model for modelling the dependence structure and marginals is suggested. J I Recently, Copula-based models for estimating error fields of radar information are developed (Villarini et al., 2008; Back Close AghaKouchak et al., 2010a,b) The intermittend nature of daily and sub-daily rainfall time series (zero-inflated data) can be modelled by mixed distribu- Full Screen / Esc tions, which are distributions with a continuous part describing data larger than zero and a discrete part accounting for Printer-friendly Version probabilities to observe zero values (Serinaldi, 2009). While the univariate form of such distributionsFig. 1. wasBias studies of by mean many annualFig. 1: totalBias of precipitation mean annual total for precipitation the RCM for (MM5) the RCM with respect to the DWD (MM5) with respect to the DWD reference data set [%] (Kot- Interactive Discussion authors (e.g. ), the bi- or multivariatereference case received data less set atten- [%] (Kotlarski et al., 2005). tion (Serinaldi, 2009). larski et al., 2005). Conventional rainfall models operate under the assump- 3029 tion of either constant variance or season-dependent variances using an AutoRegressive Moving Average (ARMA) model. However,it could be shown that daily rainfall data are affected by non-linear characteristics of the variance often referred to as variance clustering or volatility, in which large changes tend to follow large changes, and small changes tend icsinPpr|Dsuso ae icsinPpr|Dsuso ae | Paper Discussion | Paper Discussion | Paper Discussion | Paper Discussion

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4 P. Laux et al.: Copula-based stat. reﬁnement of precipitation in RCMs simulations 8, 3001–3045, 2011

Copula-based stat. reﬁnement of precipitation in RCMs simulations

P. Laux et al.

Title Page

Abstract Introduction

Conclusions References

Tables Figures

J I Fig.Fig. 2: 2. AlpineAlpine region region showing showing the location the of location observation of stations observation used in this stations paper: 1used Garmisch-Partenkirchen, in this paper: 2 1Grainau, Garmisch-Partenkirchen, 3 Grainau (Eibsee), 4 Bad Reichenhall,2 Grainau, 5 3 Schneizlr Grainaueuth-Unterjettenberg,6 (Eibsee), 4 Bad Schneizlreuth-Ristfeucht, Reichenhall, 5 Schneizlreuth- 7 Schneizlreuth- J I Unterjettenberg,Weissbach, 8 Anger-Oberhögl, 6 Schneizlreuth-Ristfeucht, 9 Bischofswiesen-Winkl, 10 7Inzell, Schneizlreuth-Weissbach, 11 Anger-Stoissberg, 12 Rottach-Egern, 8 Anger-Oberh 13 Kreuth,ogl,änd 14 Schwarzkopfhütte. Back Close 9 Bischofswiesen-Winkl, 10 Inzell, 11 Anger-Stoissberg, 12 Rottach-Egern, 13 Kreuth, and 14 Schwarzkopfhutte.¨ Full Screen / Esc

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0.2 Fig. 4: Concept of the bias correction followed in this paper. Autocorrelation 0

−0.2 0 5 10 15 20 Lag iid residuals. An ARMA model is used to compensate for autocorrelation, and a GARCH model to compensate for the Fig. 3: Autocorrelation function for precipitation (1979- heteroskedasticity. 1993) of Garmisch-Partenkirchen, Germany (top), and its The term conditional in GARCH - Generalized Autore- corresponding squared time series (bottom). gressive Conditional Heteroskedasticity - implies explicitely the dependence on a past sequence of observations, and au- 4 P. Laux et al.: Copula-based stat. reﬁnement of precipitation in RCMs simulations

Fig. 2: Alpine region showing the location of observation stations used in this paper: 1 Garmisch-Partenkirchen, 2 Grainau, 3 Grainau (Eibsee), 4 Bad Reichenhall, 5 Schneizlreuth-Unterjettenberg,6 Schneizlreuth-Ristfeucht, 7 Schneizlreuth- Weissbach, 8 Anger-Oberh¨ogl, 9 Bischofswiesen-Winkl, 10 Inzell, 11 Anger-Stoissberg, 12 Rottach-Egern, 13 Kreuth, and 14 Schwarzkopfh¨utte. icsinPpr|Dsuso ae icsinPpr|Dsuso ae | Paper Discussion | Paper Discussion | Paper Discussion | Paper Discussion

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0.6 Copula-based 0.4 stat. reﬁnement of precipitation in RCMs 0.2 simulations Autocorrelation 0 P. Laux et al.

−0.2 0 5 10 15 20 Lag ACF of squared observed rainfall Title Page 1 Abstract Introduction

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0.4 J I 0.2 Fig. 4: Concept of the bias correction followed in this paper. Autocorrelation J I 0 Back Close −0.2 0 5 10 15 20 Lag iid residuals. AnFull ARMA Screen / Esc model is used to compensate for autocorrelation, and a GARCH model to compensate for the Fig. 3. Autocorrelation function for precipitation (1979–1993) of Garmisch-Partenkirchen, Ger-heteroskedasticity.Printer-friendly Version many (top), andFig. its 3: corresponding Autocorrelation squared time function series (bottom). for precipitation (1979- 1993) of Garmisch-Partenkirchen, Germany (top), and its The term conditionalInteractive Discussionin GARCH - Generalized Autore- corresponding squared time series (bottom). gressive Conditional Heteroskedasticity - implies explicitely 3031 the dependence on a past sequence of observations, and au- 4 P. Laux et al.: Copula-based stat. reﬁnement of precipitation in RCMs simulations

Fig. 2: Alpine region showing the location of observation stations used in this paper: 1 Garmisch-Partenkirchen, | 2 Paper Discussion | Paper Discussion | Paper Discussion | Paper Discussion Grainau, 3 Grainau (Eibsee), 4 Bad Reichenhall, 5 Schneizlreuth-Unterjettenberg,6 Schneizlreuth-Ristfeucht, 7 Schneizlreuth- Weissbach, 8 Anger-Oberh¨ogl, 9 Bischofswiesen-Winkl, 10 Inzell, 11 Anger-Stoissberg, 12 Rottach-Egern, 13 Kreuth, and 14 HESSD Schwarzkopfh¨utte. 8, 3001–3045, 2011

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ACF of observed rainfall stat. reﬁnement of 1 precipitation in RCMs simulations 0.8

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0.2 Title Page Autocorrelation 0 Abstract Introduction

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0.4 Back Close Fig. 4. ConceptFig. of the 4: biasConcept correction of the followed bias correction in this paper. followed in this paper. 0.2 Full Screen / Esc Autocorrelation 0 Printer-friendly Version −0.2 0 5 10 15 20 iid residuals. An ARMA model is used to compensate for Lag Interactive Discussion autocorrelation, and a GARCH model to compensate for the Fig. 3: Autocorrelation function for precipitation (1979- heteroskedasticity. 3032 1993) of Garmisch-Partenkirchen, Germany (top), and its The term conditional in GARCH - Generalized Autore- corresponding squared time series (bottom). gressive Conditional Heteroskedasticity - implies explicitely the dependence on a past sequence of observations, and au- P. Laux et al.: Copula-based stat. reﬁnement of precipitation in RCMs simulations 5 toregressive describes a feedback mechanism that incorpo- | Paper Discussion | Paper Discussion | Paper Discussion | Paper Discussion ACF of residuals rates past observations into the present. GARCH is a time 1 HESSD series modelling technique that includes past variances for 8, 3001–3045, 2011 predicting present or future variances. 0.8 A univariate model of an observed time series y can be 0.6 t Copula-based written as 0.4 stat. reﬁnement of precipitation in RCMs 0.2 yt = f(t−1,X)+εt (1) simulations Autocorrelation

0 In this equation, the term f(t − 1,X) represents the de- P. Laux et al. terministic component of the current value as a function of −0.2 0 5 10 15 20 the information known at time t-1, including past innovations Lag ACF of squared residuals Title Page {εt−1,εt−2,...}, past observations {yt−1,yt−2,...} and other relevant time series data X. Bollerlslev (1986) developed 1 Abstract Introduction 0.8 GARCH as a generalization of the ARCH volatility mod- Conclusions References elling technique (Engle, 1982). The distribution of the resid- 0.6 uals, conditional on the time t, is given by Tables Figures 0.4 2 2 J I Vart−1(yt)= Et−1(εt )= σt (2) 0.2 Autocorrelation J I where 0 Back Close −0.2 P Q 0 5 10 15 20 2 2 2 Lag Full Screen / Esc σt = κ+ Giσt−i + Aj εt−j (3) Xi=1 Xj=1 Fig. 5. AutocorrelationFig. 5: Autocorrelation function for ARMA-GARCH function for residuals ARMA-GARCH for precipitation residu- (1979–1993) of Printer-friendly Version One can see that σ2 is the prediction of the variance,Garmisch-Partenkirchen given (top), and its corresponding squared residuals (bottom). t als for precipitation (1979-1993) of Garmisch-Partenkirchen Interactive Discussion 2 the past sequence of variance predictions, σt−i, and past (top), and its corresponding squared residuals (bottom). 2 realizations of the variance itself, εt−j. When P =0, the 3033 GARCH(0,Q) model becomes the original ARCH(Q) model introduced by Engle (1982). This equation mimics the vari- 1.0 ance clustering of the variable (i.e. precipitation and temper- 0.8 ature). The lag lengths P and Q and the coefﬁcients Gi and Aj determine the degree of persistence. 0.6 L i H

A common assumption is that the innovations are serially H 0.4 independent, however, GARCH(P,Q) innovations, {εt}, are modelled as 0.2 εt = σtzt. (4) 0.0 0.0 0.2 0.4 0.6 0.8 1.0 σt is the conditional standard deviation given by the square Wi:n rootofeq. 3, and zt is the standardized iid random draw from 1.0 some speciﬁed probability distribution. Usually, a Gaussian 2 0.8 distribution is assumed such that ε ∼ N(0,σt ). Reﬂecting this, eq. 5 illustrates that a GARCH innovations process {εt} 0.6 L i H

simply rescales an iid process {zt} such that the conditional H standard deviation incorporates the serial dependence of eq. 0.4

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Wi:n This subsection describes the fitting of semi-parametric cumulative distribution functions (CDFs). First, the empirical Fig. 6: K-plot of the observed rainfall time series at station CDF of each parameter is estimated using a Gaussian kernel Garmisch-Partenkirchen (Germany) before ARMA-GARCH function (using a kernel width of 50 points) to eliminate the transformation (top), and after ARMA-GARCH transforma- staircase pattern. This provides a reasonably good fit to the tion (bottom). Superimposed on the graph are a straight line interior of the distribution of the residuals. This procedure, (blue) correspondingto the case of independenceand a curve however, tends to perform poorly when applied to upper and corresponding to perfect positive dependence (red). lower tails. P. Laux et al.: Copula-based stat. refinement of precipitation in RCMs simulations 5 toregressive describes a feedback mechanism that incorpo- ACF of residuals rates past observations into the present. GARCH is a time 1 series modelling technique that includes past variances for predicting present or future variances. 0.8 A univariate model of an observed time series yt can be 0.6 written as 0.4

0.2 yt = f(t−1,X)+εt (1) Autocorrelation In this equation, the term f(t − 1,X) represents the de- 0 terministic component of the current value as a function of −0.2 0 5 10 15 20 the information known at time t-1, including past innovations Lag ACF of squared residuals {εt−1,εt−2,...}, past observations {yt−1,yt−2,...} and other relevant time series data X. Bollerlslev (1986) developed 1 GARCH as a generalization of the ARCH volatility mod- 0.8 elling technique (Engle, 1982). The distribution of the resid- 0.6 uals, conditional on the time t, is given by 0.4 2 2 Vart−1(yt)= Et−1(εt )= σt (2) 0.2 Autocorrelation where 0

−0.2 P Q 0 5 10 15 20 2 2 2 Lag σt = κ+ Giσt−i + Aj εt−j (3) Xi=1 Xj=1 Fig. 5: Autocorrelation function for ARMA-GARCH residu- 2 One can see that σt is the prediction of the variance, given als for precipitation (1979-1993) of Garmisch-Partenkirchen 2 the past sequence of variance predictions, σt−i, and past (top), and its corresponding squared residuals (bottom). 2 realizations of the variance itself, εt−j. When P =0, the

GARCH(0,Q) model becomes the original ARCH(Q) model | Paper Discussion | Paper Discussion | Paper Discussion | Paper Discussion introduced by Engle (1982). This equation mimics the vari- 1.0 ance clustering of the variable (i.e. precipitation and temper- HESSD 0.8 ature). The lag lengths P and Q and the coefficients Gi and 8, 3001–3045, 2011 Aj determine the degree of persistence. 0.6 L i H Copula-based A common assumption is that the innovations are serially H 0.4 stat. refinement of independent, however, GARCH(P,Q) innovations, {εt}, are modelled as precipitation in RCMs 0.2 simulations εt = σtzt. (4) 0.0 P. Laux et al. 0.0 0.2 0.4 0.6 0.8 1.0 σt is the conditional standard deviation given by the square Wi:n rootofeq. 3, and zt is the standardized iid random draw from 1.0 some specified probability distribution. Usually, a Gaussian Title Page 2 0.8 distribution is assumed such that ε ∼ N(0,σt ). Reflecting Abstract Introduction this, eq. 5 illustrates that a GARCH innovations process {εt} 0.6

L Conclusions References i H

simply rescales an iid process {zt} such that the conditional H standard deviation incorporates the serial dependence of eq. 0.4 Tables Figures

3. 0.2 J I

3.1.2 Generalized Pareto Distribution 0.0 0.0 0.2 0.4 0.6 0.8 1.0 J I

Wi:n This subsection describes the fitting of semi-parametric cu- Back Close mulative distribution functions (CDFs). First, the empiriFig.cal 6. K-plotFig. of 6: the K-plot observed of rainfall the observed time series atrainfall station Garmisch-Partenkirchentime series at station (Germany) Full Screen / Esc CDF of each parameter is estimated using a Gaussian kernelbefore ARMA-GARCH transformation (top), and after ARMA-GARCH transformation (bottom). SuperimposedGarmisch-Partenkirchen on the graph are a straight (Germany) line (blue) before corresponding ARMA-GARCH to the case of indepen- Printer-friendly Version function (using a kernel width of 50 points) to eliminatedence the andtransformation a curve corresponding (top), to and perfect after positive ARMA-GARCH dependence (red). transforma- staircase pattern. This provides a reasonably good fit to the tion (bottom). Superimposed on the graph are a straight line Interactive Discussion interior of the distribution of the residuals. This procedure, (blue) correspondingto the case of independenceand a curve 3034 however, tends to perform poorly when applied to upper and corresponding to perfect positive dependence (red). lower tails. 6 P. Laux et al.: Copula-based stat. refinement of precipitation in RCMs simulations

The upper and lower tails therefore are ﬁtted separately Sklar (1959) proved that every multivariate distribution from the interior of the distribution. For this reason, the F (x1,...xn) can be expressed in terms of a Copula C and

peaks over threshold (POT) method is applied: A threshold its marginals FXi (xi): value of 0.1 is chosen, i.e. the upper and lower 10% of the

residuals are reserved for each tail. The extreme residuals F (x1,...xn)= C(FX1 (x1),...,FXn (xn)) (6) (beyond the threshold) are ﬁtted to a parametric GPD, which can be described as C : [0,1]n → [0,1]. (7)

−1− 1 Copulas allow to merge the dependence structure and the 1 (x−θ) k y = f(x|k,σ,θ)= 1+k (5) marginal distributions to form a joint multivariate distribu- σ σ tion. The Copula function is unique when the marginals are using a maximum likelihood approach. Given the ex- steady functions. As the Copula is a reflection of the depen- ceedances in each tail, the negative log-likelihood function dency structure itself, its construction is reduced to the study is optimized to estimate the tail index /shape parameter k of the relationship of the correlated iid variables, giving free- and the scale parameter σ of the GPD. The composite GPD dom for the choice of the univariate marginal distributions. function allows for interpolation in the interior of the CDF Further information about Copulas can be found e.g. in Joe but also for extrapolation in the lower and upper tails. (1997); Frees and Valdez (1998); Nelson (1999); Salvadori et al. (2007). icsinPpr|Dsuso ae icsinPpr|Dsuso ae | Paper Discussion | Paper Discussion | The Paper Discussion Copula | Paper Discussion approach allows to account for the fact that 1 the dependence structureHESSD between regional and local meteorological fields and between simulated and observed fields 8, 3001–3045, 2011 0.8 is more complex than it can be modelled by the multivariate normal distribution or ordinary dependency measures such 0.6 as e.g. the PearsonCopula-based correlation coefficient. The complex multivariate dependencestat. refinement structure is of analyzed between RCM

Probability 0.4 modelled precipitationprecipitation (MM5 in output) RCMs and station observed precipitation. simulations 0.2

Fitted Generalized Pareto CDF As there exist no uniqueP. Laux characterization et al. of the Copula for Empirical CDF 0 dry days, our work focusses on the positive pairs (RCM pre- 0 1 2 3 4 5 6 Exceedance cipitation >0, observed precipitation >0). For dry days, only the conditional marginalsTitle can Page be identiﬁed (Yang, 2008).

1 Abstract Introduction 3.2.1 Empirical Copula Conclusions References 0.8 The dependence structure of daily measured precipitation Tables Figures 0.6 and simulated precipitation is studied. Since the underlying (theoretical) Copula is not known in advance, it is necessary J I

Probability 0.4 to analyze the empirical Copula, which is purely based on the data (Deheuvels, 1979).J The ranksI of the residuals of mod- 0.2 Pareto Lower Tail elled and observed rainfall from day 1 to day n, obtained Kernel Smoothed Interior Back Close Pareto Upper Tail from the original data as well as the ARMA-GARCH time 0 0 2 4 6 8 series model, are {r1(1)Full,...,r Screen1( /n Esc)} and {r2(1),...,r2(n)}, re- Standardized Residuals spectively. The empirical Copula is defined as: Printer-friendly Version Fig. 7. EmpiricalFig. and 7: fitted Empirical exceedance and probability fitted exceedance of the upper probability tail of observed of the rainfall residuals (top), composite of the piecewise CDF of the modelled (solid lines) and observed (dashed lines) n upper tail of observed rainfall residuals (top), composite of Interactive Discussionr1(t) r2(t) rainfall residuals at Garmisch-Partenkirchen (bottom). Cn(u,v)=1/n 1 6 u, 6 v (8) n n the piecewise CDF of the modelled (solid lines) and observed Xt=1 (dashed lines) rainfall residuals3035 at Garmisch-Partenkirchen (bottom). where u indicates the percentile of the modelled rainfall residuals, v indicates the percentile of the measured rainfall residuals and 1(...) is denoting the indicator function. 3.2 Copula based joint distribution functions of modelled and observed rainfall 3.2.2 Estimation of the theoretical Copula

Copulas are functions that link multivariate distribu- A goodness-of-ﬁt test for Copulas is applied comparing the

tion F (x1,...xn) to their univariate marginals FXi (xi). empirical Copula Cn (eq.8) with the parametric estimate of a P. Laux et al.: Copula-based stat. reﬁnement of precipitation in RCMs simulations 7 icsinPpr|Dsuso ae icsinPpr|Dsuso ae | Paper Discussion | Paper Discussion | Paper Discussion | Paper Discussion

of Salvadori for the conditionalHESSD simulation using Copulas Salvadori et al. (2007). The simulation is based on conditional probabilities of the8, form: 3001–3045, 2011

Copula-based∂ P (V ≤ v|U = u)= C(u,v); (10) stat. reﬁnement∂u of precipitation∂ in RCMs P (U ≤ u|V = v)= C(u,v). (11) simulations∂v For the Gumbel-HougaardP. Copula Laux e.g.et al. it is:

∂ θ θ 1/θ C(u,v)= u−1e−((log(u) )+(log(v) ) ∂u Title Page (−log(u))−1+θ((−log(u)θ)+(−log(v))θ)−1+1/θ. (12) Abstract Introduction The concept pseudo-observation simulation from model data is as follows: a pair ofConclusions variates (u,v)Referenceswith Copula C(u,v) needs to be generated whichTables ﬁnally canFigures be transformed into (x,y), using the probability integral transformation

J −I1 U = FX (x) ⇔ X = FX (U) (13) −1 V = FY (y) ⇔J Y = FY I(V ). (14) The complete algorithm isBack divided intoClose three steps:

Full Screen / Esc 1. Computation u = FX (x), where x denotes one value of the modelled rainfall and FX (x) is the marginal distri- Fig. 8. EmpiricalFig. Copula 8: Empirical density for Copula modelled density (u)for and modelled observed (u) precipitation and ob- (v) forbution station of the variate X;Printer-friendly Version Garmisch-Partenkirchen,served precipitation using the positive (v) for pairs station of original Garmisch-Partenkirc data (top), andhen, the positive pairs of the ARMA-GARCHusing transformed the positiveiid residuals pairs of original (bottom). data (top), and the posi- 2. Generation of randomInteractive samples Discussion for the variate v∗ from tive pairs of the ARMA-GARCH transformed iid residuals the conditional CDF CV |U (v|u)= cu(v) and calcula- 3036 −1 ∗ −1 (bottom). tion of v = cu (v ), where cu denotes the generalized inverse of cu (Nelsen, 1999); 3. Calculation of the corresponding y-values using the theoretical Copula model Cθ derived under the null hypothe- −1 sis. The test is based on the Cram´er-von Mises statistic (Gen- probability integral transformation FY (v)= y est and Favre, 2007): The ﬁnal result for y is a sample of pseudo-observations n which lies in the original data space and can be compared 2 Sn = n {Cθ(ut,vt)−Cn(ut,vt)} . (9) with the observed data series. Xt=1

As the definition of Sn involves the theoretical Copula 3.3 Usability of weather patterns for conditional simu- function, the distribution of this statistic depends on the un- lations known value of θ under the null hypothesis that C is from the Especially for complex terrain, it is assumed that the direc- class Cθ (Grégoire et al., 2008). Therefore, the approximate p-values for the test statistic are obtained using a parametric tion of advection is of crucial importance for the observed bootstrap (Genest and Remillard, 2008; Genest et al., 2009) precipitation amounts. The combinations of terrain exposi- as well as a fast multiplier approach (Kojadinovic and Yan, tion and advection direction leads to luv and lee side effects, 2009a,b). i.e. the stations can lie in the rainshadoworcan be exposedto intense rainfall. As independent from the RCM simulations, 3.2.3 Copula-based rainfall simulations large-scale weather patterns are used to further improve the results of the bias correction. Besides the advection direc- After the estimation of the Copula-based joint distribution tion, large scale information about cyclonality and tropo- - that is FX (x), FY (y) and Cθ(u,v) are obtained - condi- spheric humidity is evaluated. The objective weather pattern tional random samples from this distribution are generated classification method of the German Weather Service is used through Monte Carlo simulations. We follow the procedure (Bissolli and Dittmann, 2001). The classification domain is 8 P. Laux et al.: Copula-based stat. refinement of precipitation in RCMs simulations icsinPpr|Dsuso ae icsinPpr|Dsuso ae | Paper Discussion | Paper Discussion | Paper Discussion | Paper Discussion

HESSD 8, 3001–3045, 2011

Copula-based stat. reﬁnement of precipitation in RCMs simulations

P. Laux et al.

Title Page

Abstract Introduction

Conclusions References

Tables Figures

J I Fig. 9. Copula-based stochastic simulations of 50 consecutive positive pairs (precipita- Fig. 9: Copula-basedtion > 0.01 stochastic mm) performing simulations 100 of realizations 50 consecut (illustratedive positive as pairs box-whiskers) (precipitation of V >(observed0.01 mm) rainfall performing 100 real-J I at gauge, illustrated as red line) asssuming that U (corresponding coarse scale MM5 precipita- izations (illustrated as box-whiskers) of V (observed rainfall at gauge, illustrated as red line) asssuming that U (corresponding tion, illustrated as black line) is known. The boxes have lines at the lower Q1 and upper quartile Back Close coarse scale MM5Q3 and precipitation, the median illustrated values Q2 as(middle black line) horizontalis known. lines). The The boxes whiskers have lines (vertical at the lines) lower are Q1 lines and upper quartile Full Screen / Esc Q3 and the medianextending values from Q2 (middle each end horizontal of the boxes lines). to show The wh theiskers extent (vertical of the rest lines) of are the lines data. extending The maximum from each end of the boxes to show thelength extent of the of the whiskers rest of is the determined data. The maximum by 1.5 (Q3 le–ngthQ1). of Outliers the whiskers (crosses) is determined are data with by 1.5 values (Q3–Q1). Outliers (crosses) are databeyond with valuesthe ends beyond of the the whiskers. ends of the whiskers. Printer-friendly Version

Interactive Discussion Germany, and the meteorological criteria for the classifica- 3. Grouping types due to the humidity of the tropo- tion are (i) the direction of advection of air masses, (ii) the3037 sphere: this leads to the discrimination of dry (D) and cyclonality, and (iii) the humidity of the troposphere. This wet (W). Therefore, a humidity index is calculated as leads to numerical indices from which the weather types are the weighted areal mean of the precipitable water inte- derived (Bissolli and Dittmann, 2001). There exist 40 prede- grated over the 950, 850, 700, 500, and 300 hPa levels. fined types, which can be used. Due to the limited occurrence For each group of weather types, a theoretical Copula frequencies of single weather types, their usability for con- model is estimated separately. For sake of simplicity, the ditional simulations is restricted. However, the usage of the Gumbel-Hougaard Copula model is used. numerical indices provides the possibility to group the types to different classes. For this study, the following grouping strategies are evalu- 4 Simulation Results ated: In this section simulation results of both, the obtained RCM 1. Grouping types due to the direction of the advection and corresponding observed precipitation time series are ex- of air masses at 700 hPa: the weather types (WTs) are emplarily presented. Based on iid residuals obtained by grouped into northeasterly, southeasterly, southwest- ARMA-GARCH models the empirical and theoretical Copu- erly, and northwesterly flow. las, and the marginal distributions are estimated and analyzed and locally refined and bias corrected pseudo-observations are generated. 2. Grouping types due to the cyclonality at 950 hPa and 500 hPa: this leads to four classes, namely anticy- 4.1 Analysis of ARMA-GARCH time series models clonal - anticyclonal(AA), anticyclonal- cyclonal (AC), cyclonal - anticyclonal (CA), and cyclonal - cyclonal The autocorrelationfunction and Ljung-Box Q-test is applied (CC); and to the original time series, the squared original time series as P. Laux et al.: Copula-based stat. refinement of precipitation in RCMs simulations 9 icsinPpr|Dsuso ae icsinPpr|Dsuso ae | Paper Discussion | Paper Discussion | Paper Discussion | Paper Discussion

HESSD 8, 3001–3045, 2011

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P. Laux et al.: Copula-based stat. reﬁnement of precipitation in RCMs simulations 9 stat. reﬁnement of precipitation in RCMs simulations

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Abstract Introduction

Conclusions References Fig. 10: Empirical Copula density for modelled (u) and observed precipitation (v) for Garmisch-Partenkirchen, using Tables Figures the weather type classiﬁcation following the advection of air masses. The advection types correpond to: (a) Northeast, (b) Southeast, (c) Southwest, (d) Northwest, and (e) no prevail- J I ing direction. The white areas originate from interpolation effects using an ordinary kriging algorithm. J I

Back Close order of ARMA-GARCH models (using orders for AR, MA, P, and Q between 1 and 3, respectively), the threshold value for a wet day (0.01mm, 0.1mm, and 1mm), and the peak- Full Screen / Esc over-threshold (POT) value for lower and upper tail (10% Fig. 10. Empirical CopulaFig. density 10: cont. for modelledFig. 10: (u Empirical) and observed Copulaand 20%). density It precipitation is foundfor modelled that the ( (u) largerv) and for theob- Garmisch- wet day threshold, served precipitationthe (v) higher for Garmisch-Partenkirchen, is the distortion of the upper using tail, which can be Printer-friendly Version Partenkirchen, using the weather type classificationthe weather type following classificationpartly explained the following advection by the the fat advection tail of behaviour.air masses. of air The POT The and the advectionwell types as the correpond resulting standardized to: (a) residualsNortheast, andmasses. standardi(b) TheSoutheast,zed advectionorder types of(c) the correpondSouthwest, ARMA-GARCH to: (a)(d) Northeast, areNorthwest, less sensitive. (b) and Interactive Discussion (e) no prevailingsquared residuals direction. of the The ARMA-GARCH white areas model.Southeast, originate Accord- (c) fromSouthwest, interpolationFigure (d) 7 Northwest, (top) shows eff andects the (e) empirical using no prevail anand- ordinary fitted exceedance kriging algorithm.ing to the autocorrelation function plots the originaling direction. time se- Theprobabilityfor white areas originate the upper from tail of interpolatio the observedn rainfall residuals ries show serial dependence and heteroskedasticity,effects3038 i.e. using non- an ordinaryat station kriging Garmisch-Partenkirchen. algorithm. Both, for observed and iid behaviour (figure 3). After application of the ARMA- modelled rainfall, the Generalized Pareto Distribution seems GARCH model, the time seriesof the residualscan be seen as to be a good choice to fit the upper tails of the data. Figure serially independent (figure 5). The Ljung-Boxorder Q-test of ARMA-GARCH con- 7 (bottom) models illustrates (using orders the composite for AR, ofMA, the three piecewise firms the results of the autocorrelation functionP, and (not Q shown between 1CDFs and 3, for respectively), modelled and the observed threshold rainfall value residuals. here). for a wet day (0.01mm, 0.1mm, and 1mm), and the peak- 4.2 Analysis of empirical and theoretical Copula Mod- over-threshold (POT) value for lower and upper tail (10% Fig.The 10: K-plots cont. indicate that there remains a positive depen- els dence in the upper tails of the residuals (figureand 6). 20%).Further It in- is found that the larger the wet day threshold, formation abouthow to calculate and to interpretethe higher the K-plo ists the distortionFigure 8 of (top) the upper shows tail, the which empirical can Copula be density be- can be obtained e.g. by Genest and Favre (2007).partly explained There- bytween the fat modelled tail behaviour. and measured The POT rainfall and for the station Garmisch- well as the resulting standardizedfore, a sensitivity residuals analysis and standardi is conductedzed accountingorder of for the th ARMA-GARCHe Partenkirchen. are less Only sensitive. the positive pairs of modelled and mea- squared residuals of the ARMA-GARCH model. Accord- Figure 7 (top) shows the empirical and fitted exceedance ing to the autocorrelation function plots the original time se- probabilityfor the upper tail of the observed rainfall residuals ries show serial dependence and heteroskedasticity, i.e. non- at station Garmisch-Partenkirchen. Both, for observed and iid behaviour (figure 3). After application of the ARMA- modelled rainfall, the Generalized Pareto Distribution seems GARCH model, the time seriesof the residualscan be seen as to be a good choice to fit the upper tails of the data. Figure serially independent (figure 5). The Ljung-Box Q-test con- 7 (bottom) illustrates the composite of the three piecewise firms the results of the autocorrelation function (not shown CDFs for modelled and observed rainfall residuals. here). 4.2 Analysis of empirical and theoretical Copula Mod- The K-plots indicate that there remains a positive depen- els dence in the upper tails of the residuals (figure 6). Further information abouthow to calculate and to interprete the K-plots Figure 8 (top) shows the empirical Copula density be- can be obtained e.g. by Genest and Favre (2007). There- tween modelled and measured rainfall for station Garmisch- fore, a sensitivity analysis is conducted accounting for the Partenkirchen. Only the positive pairs of modelled and mea- 12 P. Laux et al.: Copula-based stat. refinement of precipitation in RCMs simulations

HESSD 8,Fig. 3001–3045, 12: Empirical 2011 Copula density for modelled (u) and observed precipitation (v) for Garmisch-Partenkirchen, using the weather type classification following the humidity of the troposphere:Copula-based (a) D, and (b) W (D - dry, W - wet). stat. refinement of precipitation in RCMs simulations Table 3: Pearson correlation coefficients (all significant at α=0.01P. Laux level) et al. between positive pairs of pseudo-observations (mean value) produced by Copula-based stochastic simulations without using large-scale information (uncond), in- cludingTitle advectionPage (advec), cyclonality (cyclo), and humidity (humi) of the troposphere, and the observed precipitation at Abstract Introduction station Garmisch-Partenkirchen and the corresponding grid Conclusionscell precipitationReferences of RCM:

Tables Figures uncond advec cyclo humi observed 0.36 0.43 0.45 0.37 J RCMI 0.98 0.93 0.98 0.65 J I

Back Close based stochastic simulations without any large-scale information.Full Screen / Esc

Fig. 11. Empirical Copula density for modelledFig. 11: ( Empiricalu) and observed CopulaFig. precipitation 12:density Empirical for (v) modelled for Copula Garmisch- (u) density and ob- for modelledPrinter-friendly (u) and Version ob- Partenkirchen, using the weather type classification following the cyclonality in 950 hPa and served precipitation (v)served for Garmisch-Partenkirchen, precipitation (v) for Garmisch-Partenkirchen, using 5 Discussion using 500 hPa respectively: (a) AA, (b) AC, the(c) CA, weather and (d) typeCC classification (A –the anticyclonic, weather following typeC – cyclonic). classification the cyclonality following in theInteractive humidity Discussion of the 950 hPa and 500 hPa respectively:troposphere: (a) D,AA, and (b) (b) AC, W (c) (D CA,- dry, W - wet). 3039 It is shown that ARMA-GARCH models are able to model and (d) CC (A - anticyclonic, C - cyclonic). serial dependence and volatility in the precipitation time series. Consequently, they are generally useful to generate Table 3: Pearson correlation coefficientsiid (allrandom significant variables. at However, even high order models are α=0.01 level) between positive pairs of pseudo-observationnot able to fullys capture the fat tail behaviour. Filtering the (mean value) produced by Copula-based stochastic simulations without using large-scale information (uncond), including advection (advec), cyclonality (cyclo), and humidity (humi) of the troposphere, and the observed precipitation at station Garmisch-Partenkirchen and the corresponding grid cell precipitation of RCM:

uncond advec cyclo humi observed 0.36 0.43 0.45 0.37 RCM 0.98 0.93 0.98 0.65

based stochastic simulations without any large-scale information.

Fig. 11: Empirical Copula density for modelled (u) and observed precipitation (v) for Garmisch-Partenkirchen, using 5 Discussion the weather type classiﬁcation following the cyclonality in 950 hPa and 500 hPa respectively: (a) AA, (b) AC, (c) CA, It is shown that ARMA-GARCH models are able to model and (d) CC (A - anticyclonic, C - cyclonic). serial dependence and volatility in the precipitation time series. Consequently, they are generally useful to generate iid random variables. However, even high order models are not able to fully capture the fat tail behaviour. Filtering the 12 P. Laux et al.: Copula-based stat. reﬁnement of precipitation in RCMs simulations icsinPpr|Dsuso ae icsinPpr|Dsuso ae | Paper Discussion | Paper Discussion | Paper Discussion | Paper Discussion

HESSD 8, 3001–3045, 2011

Copula-based stat. reﬁnement of precipitation in RCMs simulations

P. Laux et al.

Title Page

Abstract Introduction

Conclusions References

Tables Figures

J I

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Full Screen / Esc

Printer-friendly Version Fig. 12. Empirical Copula density for modelled (u) and observed precipitation (v) for Garmisch- Fig. 12: Empirical Copula density for modelled (u) and ob- Interactive Discussion Partenkirchen, using the weather type classiﬁcation following the humidity of the troposphere: (a) D, and (b)servedW (D – precipitation dry, W –wet). (v) for Garmisch-Partenkirchen, using the weather type classiﬁcation3040 following the humidity of the troposphere: (a) D, and (b) W (D - dry, W - wet).

Table 3: Pearson correlation coefﬁcients (all signiﬁcant at α=0.01 level) between positive pairs of pseudo-observations (mean value) produced by Copula-based stochastic simulations without using large-scale information (uncond), including advection (advec), cyclonality (cyclo), and humidity (humi) of the troposphere, and the observed precipitation at station Garmisch-Partenkirchen and the corresponding grid cell precipitation of RCM:

uncond advec cyclo humi observed 0.36 0.43 0.45 0.37 RCM 0.98 0.93 0.98 0.65

based stochastic simulations without any large-scale information.

Fig. 11: Empirical Copula density for modelled (u) and observed precipitation (v) for Garmisch-Partenkirchen, using 5 Discussion the weather type classiﬁcation following the cyclonality in 950 hPa and 500 hPa respectively: (a) AA, (b) AC, (c) CA, It is shown that ARMA-GARCH models are able to model and (d) CC (A - anticyclonic, C - cyclonic). serial dependence and volatility in the precipitation time series. Consequently, they are generally useful to generate iid random variables. However, even high order models are not able to fully capture the fat tail behaviour. Filtering the P. Laux et al.: Copula-based stat. reﬁnement of precipitation in RCMs simulations 13 icsinPpr|Dsuso ae icsinPpr|Dsuso ae | Paper Discussion | Paper Discussion | Paper Discussion | Paper Discussion

1 0.3 Northeast 70 HESSD 0.8 Southeast 0.29 Southwest 0.28 0.6 Northwest 71 8, 3001–3045, 2011 no prevailing direction 0.27

0.4 72 0.26 Probability

Copula-based0.25 0.2 MM5 grid id (lat) 73 stat. reﬁnement0.24 of 0 0 20 40 60 80 100 0.23 74 precipitation in RCMs precipitation amount [mm]

0.22 44 45 46 47 48 1 simulations AA MM5 grid id (lon) AC 0.8 CA P. Laux et al. CC Fig. 14: Kendall’s τ between observed precipitation at sta- 0.6 tion Garmisch-Partenkirchen and its the surrounding RCM grid cells (∆ x=19.2 km). Garmisch-Partenkirchen is located 0.4 Probability in the middle grid cell (ID:46/72). Title Page 0.2

1.0 Abstract Introduction 0 0 20 40 60 80 100 precipitation amount [mm] 0.8 Conclusions References 1 0.6 D (dry) W (wet) 0.8 Tables Figures 0.4 obs uncond 0.6 observations advec 0.2 cyclo humJ i I 0.4

Probability 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.2 m odel J I

0 Back Close 0 20 40 60 80 100 Fig. 15: Probability plots of observed and modelled precip- precipitation amount [mm] itation time series at station Garmisch-Partenkirchen. If the Full Screen / Esc quantiles of the two distributions agree, the plotted points Fig. 13. Empirical CDFFig. of observed 13: Empirical precipitation CDF of observed in Garmisch-Partenkirchen, precipitation in conditioned on Garmisch-Partenkirchen, conditioned on the occurrence of fall on exactly on the thin dotted line. The red quadrates the occurrence of (i) fouri) di fourfferent different advection advection types types plus plus one one unspecified unspecified type, type,illustrates (ii) the the following agreement between observedPrinter-friendly and RCM rain- Version cyclonality types in 950 hPaii) the and following 500 hPa cyclonality respectively: types in 950 AA, hPa AC, and CA, 500 hPa and CCfall. (A The – anticyclonic, black quadrates correspond to the Copula-based stochastic simulations without additional large-scale infor- C – cyclonic), and (iii) therespectively: following AA, humidity AC, CA, types and CC of the (A - troposphere: anticyclonic, C - D, and W (D – dry, W – Interactive Discussion wet). Dry (wet) types arecyclonic), colored and red iii) (blue). the following humidity types of the tropo- mation. The Copula-based simulations including advection, sphere: D, and W (D - dry, W - wet). Dry (wet) types are cyclonality, and humidity of the troposphere are illustrated as colored red (blue). 3041 blue, green, and orange quadrates respectively.

tion stations can differ significantly. It is also found that the time series before fitting to a theoretical Copula model is re- correlation between station and grid cell is not necessarily ducing the correlation between RCM (here: MM5) and ob- highest for the corresponding grid cell. Assessing the de- served precipitation and thus the estimated Copula parameter pendence of the distance between observed rainfall and rain- θ, which is directly related to Kendall’s τ. fall of the surrounding RCM grid cells potentially allows for From the theoretical Copula models analyzed, the spatial interpolation in ungauged regions. However, further Gumbel-Hougaard Copula is found to be a suited choice to investigations will be necessary. model the joint distribution of modelled (gridded) and ob- The empirical Copula density plot is used to analyze the served precipitation. While the Copula parameter is rela- dependence structure between modelled and observed pre- tively stable for the joint distribution functions between dif- cipitation. As computational inexpensive they are suitable to ferent locations within the same grid cell, the shape and scale i) find a theoretical Copula model, and ii) screen variables parameters of the fitted marginal distributions of the observa- such as e.g. large-scale weather types which could addition- icsinPpr|Dsuso ae icsinPpr|Dsuso ae | Paper Discussion | Paper Discussion | Paper Discussion | Paper Discussion

HESSD P. Laux et al.: Copula-based stat. reﬁnement of precipitation in RCMs simulations 13 8, 3001–3045, 2011

Copula-based 1 0.3 stat. reﬁnement of Northeast 70 precipitation in RCMs 0.29 0.8 Southeast simulations Southwest 0.28 0.6 Northwest 71 P. Laux et al. no prevailing direction 0.27

0.4 72 0.26

Probability Title Page

0.25 0.2 MM5 grid id (lat) Abstract Introduction 73 0.24 Conclusions References 0 0 20 40 60 80 100 0.23 precipitation amount [mm] 74 Tables Figures

0.22 44 45 46 47 48 1 AA MM5 grid id (lon) J I AC Fig. 14. Kendall’s τ between observed precipitation at station Garmisch-Partenkirchen and J I 0.8 CA its the surrounding RCM grid cells (∆x = 19.2 km). Garmisch-Partenkirchen is located in the Fig. 14: Kendall’s τ between observed precipitation at sta- Back Close CC middle grid cell (ID:46/72). 0.6 tion Garmisch-Partenkirchen and its the surrounding RCM Full Screen / Esc grid cells (∆ x=19.2 km). Garmisch-Partenkirchen is located 0.4 Probability in the middle grid cell (ID:46/72). Printer-friendly Version 0.2 Interactive Discussion

1.0 3042 0 0 20 40 60 80 100 precipitation amount [mm] 0.8 1 0.6 D (dry) W (wet) 0.8 0.4 obs uncond 0.6 observations advec 0.2 cyclo hum i 0.4

Probability 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.2 m odel

0 0 20 40 60 80 100 Fig. 15: Probability plots of observed and modelled precip- precipitation amount [mm] itation time series at station Garmisch-Partenkirchen. If the Fig. 13: Empirical CDF of observed precipitation in quantiles of the two distributions agree, the plotted points Garmisch-Partenkirchen, conditioned on the occurrence of fall on exactly on the thin dotted line. The red quadrates i) four different advection types plus one unspeciﬁed type, illustrates the agreement between observed and RCM rain- ii) the following cyclonality types in 950 hPa and 500 hPa fall. The black quadrates correspond to the Copula-based respectively: AA, AC, CA, and CC (A - anticyclonic, C - stochastic simulations without additional large-scale infor- cyclonic), and iii) the following humidity types of the tropo- mation. The Copula-based simulations including advection, sphere: D, and W (D - dry, W - wet). Dry (wet) types are cyclonality, and humidity of the troposphere are illustrated as colored red (blue). blue, green, and orange quadrates respectively.

tion stations can differ significantly. It is also found that the time series before fitting to a theoretical Copula model is re- correlation between station and grid cell is not necessarily ducing the correlation between RCM (here: MM5) and ob- highest for the corresponding grid cell. Assessing the de- served precipitation and thus the estimated Copula parameter pendence of the distance between observed rainfall and rain- θ, which is directly related to Kendall’s τ. fall of the surrounding RCM grid cells potentially allows for From the theoretical Copula models analyzed, the spatial interpolation in ungauged regions. However, further Gumbel-Hougaard Copula is found to be a suited choice to investigations will be necessary. model the joint distribution of modelled (gridded) and ob- The empirical Copula density plot is used to analyze the served precipitation. While the Copula parameter is rela- dependence structure between modelled and observed pre- tively stable for the joint distribution functions between dif- cipitation. As computational inexpensive they are suitable to ferent locations within the same grid cell, the shape and scale i) find a theoretical Copula model, and ii) screen variables parameters of the fitted marginal distributions of the observa- such as e.g. large-scale weather types which could addition- P. Laux et al.: Copula-based stat. refinement of precipitation in RCMs simulations 13

1 0.3 Northeast 70 0.8 Southeast 0.29 Southwest 0.28 0.6 Northwest 71 no prevailing direction 0.27

0.4 72 0.26 Probability

0.25 0.2 MM5 grid id (lat) 73 0.24 0 0 20 40 60 80 100 0.23 precipitation amount [mm] 74

0.22 44 45 46 47 48 1 AA MM5 grid id (lon) AC 0.8 CA icsinPpr|Dsuso ae icsinPpr|Dsuso ae | Paper Discussion | Paper Discussion | Paper Discussion | Paper Discussion CC Fig. 14: Kendall’s τ between observed precipitation at sta- 0.6 tion Garmisch-Partenkirchen and its the surrounding RCM HESSD grid cells (∆ x=19.2 km). Garmisch-Partenkirchen is located 0.4 8, 3001–3045, 2011 Probability in the middle grid cell (ID:46/72). 0.2 Copula-based 1.0 0 stat. reﬁnement of 0 20 40 60 80 100 precipitation in RCMs precipitation amount [mm] 0.8 simulations 1 0.6 D (dry) W (wet) P. Laux et al. 0.8 0.4 obs uncond observations 0.6 advec Title Page 0.2 cyclo hum i 0.4 Abstract Introduction

Probability 0.0 0.0 0.2 0.4 0.6 0.8 1.0 Conclusions References 0.2 m odel Tables Figures 0 Fig. 15. Probability plots of observed and modelled precipitation time series at station 0 20 40 60 80 100 Garmisch-Partenkirchen.Fig. 15: Probability If the quantiles plots of theof observed two distributions and modelled agree, the plottedprecip- points fall on precipitation amount [mm] exactly on theitation thin dotted time line. series The at red station quadrates Garmisch-Partenkirchen. illustrates the agreement between If the observed J I and RCM rainfall. The black quadrates correspond to the Copula-based stochastic simulations quantiles of the two distributions agree, the plotted points J I Fig. 13: Empirical CDF of observed precipitationwithout in additional large-scale information. The Copula-based simulations including advection, fall on exactly on the thin dotted line. The red quadrates Garmisch-Partenkirchen, conditioned on the occurrencecyclonality, of and humidity of the troposphere are illustrated as blue, green, and orange quadrates Back Close i) four different advection types plus one unspeciﬁedrespectively. type, illustrates the agreement between observed and RCM rain- Full Screen / Esc ii) the following cyclonality types in 950 hPa and 500 hPa fall. The black quadrates correspond to the Copula-based stochastic simulations without additional large-scale infor- respectively: AA, AC, CA, and CC (A - anticyclonic, C - Printer-friendly Version cyclonic), and iii) the following humidity types of the tropo- mation. The Copula-based simulations including advection, sphere: D, and W (D - dry, W - wet). Dry (wet) types are cyclonality, and humidity of the troposphere are illustrated as Interactive Discussion colored red (blue). blue, green, and orange quadrates respectively. 3043

HESSD 8, 3001–3045, 2011 14 P. Laux et al.: Copula-based stat. refinement of precipitation in RCMs simulations Copula-based stat. refinement of an adequate fit for all groups of weather types. It is also 1 precipitation in RCMs CP2, CP39, well-known fromsimulations other studies that the domain size (here: SWCCW 0.8 NEAAD whole Central Europe domain) can strongly impact the clas- (dry) (wet) sification resultsP. (e.g. Laux Laux, et al. 2009), and thus the subsequent 0.6 conditional modelling.

Title Page 0.4 In this study a stationary approach for the Copula param- Probability eter θ is chosen.Abstract FurtherIntroduction improvements are expected by 0.2 accounting for the temporal variability of θ. Figure 17 il- Conclusions References lustrates the temporal variability of τ, which is empirically 0 0 10 20 30 40 50 60 linked with theTables Copula parameter.Figures As seen in section 4.2. and precipitation amount [mm] 4.4, all the empirical Copulas derived in this study show a strong asymmetryJ with respectI to the minor axis u =1−v of Fig. 16. EmpiricalFig. 16: CDF of Empirical observed precipitation CDF of in observed Garmisch-Partenkirchen, precipitation conditioned in on the occurrence of the Northeast advection type WT 2 (NEAAD), and the Southwest advection[0,1]2. This asymmetry can not be depicted by the common Garmisch-Partenkirchen, conditioned on the occurrence of J I type WT 39 (SWCCW). Copula families such as the Clayton, Normal or Gumbel- the Northeast advection type WT 2 (NEAAD), and the Back Close Hougaard Copulas, acting as basic set of possible candidates Southwest advection type WT 39 (SWCCW). for the performedFull GOF Screen tests. / Esc Nonmonotonic transformation to construct asymmetric multivariate Copulas from the Gaus- sian (B´ardossy,Printer-friendly 2006) could Version reﬂect the asymmetries in the 0.7 empirical CopulasInteractive and Discussion thus also improve the bias correction. 0.65 3044 0.6

0.55 τ 0.5 6 Conclusions 0.45

0.4 The presented Copula-based approach is potentially useful 0.35 for statistical downscaling, bias correction, and local reﬁne-

0 200 400 600 800 1000 ment of RCMs. The performance will be evaluated and com- time step (positive pairs) pared to established methods for bias correction. Asymmetries are found in the empirical Copula densities Fig. 17: Kendall’s τ of 101 consecutive positive pairs for which cannot be reproduced by the theoretical Copulas used modelled and observed precipitation at station Garmisch- in this study. Therefore, it is generally difficult to find a the- Partenkirchen. oretical Copula model which is not rejected by the applied GOF test. Fitting the marginal distributions is of crucial importance as ally improve the performance of the bias correction. it strongly impacts the simulation results (more than the Cop- The objective weather pattern classification method of ula parameter θ). the German Weather Service (Bissolli and Dittmann, 2001) Large-scale weather patterns could be used to further con- shows only moderate potential to further constraint the strain the model, and thus, increase the performance of the model. Including information about the humidity of the tro- simulation results. posphere can slightly increase the skill for bias correction compared to the Copula-based stochastic simulations without using large-scale information. This can be seen from the Acknowledgements. This research is funded by the Bavarian State conditional CDFs and the correspondingprobability plots for Ministry of the Environment and Public Health (reference number the different groups, which are not very discriminative (com- VH-ID:32722/TUF01UF-32722). Additional funding by the Fed- pare section 3.3 and 4.4). Using the single 40 weather types eral Ministry of Education and Research (research project: Land Use and Climate Change Interactions in Central Vietnam LUCCi, (without grouping) could potentially increase the discrimina- reference number 01LL0908C), by the HGF Impuls- und Vernet- tive power (see e.g. figure 16), but decreases the sample size zungsfond (research project: Regional Precipitation Observation by for certain WTs far too much to reliably estimate the Copula Cellular Network Microwave Attenuation and Application to Wa- parameter(s) and the marginal distributions. Another limita- ter Resources Management PROCEMA, Virtual Institute, reference tion of the approach shown in this paper is that the same the- number VH-VI-314), and by HGF (research project: Terrestrial oretical Copula model, i.e. the Gumbel-Hougaard Copula, Environmental Observatories TERENO (http://www.tereno.net)) is is fitted to each WT class. It is obvious from the empirical highly acknowledged. The help of Thomas Rummer, who created Copula density plots, that this does not necessarily provide Figure 2 is appreciated. 14 P. Laux et al.: Copula-based stat. refinement of precipitation in RCMs simulations

an adequate ﬁt for all groups of weather types. It is also 1 CP2, CP39, well-known from other studies that the domain size (here: SWCCW 0.8 NEAAD whole Central Europe domain) can strongly impact the clas- (dry) (wet) siﬁcation results (e.g. Laux, 2009), and thus the subsequent 0.6 conditional modelling.

0.4 In this study a stationary approach for the Copula param- Probability eter θ is chosen. Further improvements are expected by 0.2 accounting for the temporal variability of θ. Figure 17 illustrates the temporal variability of τ, which is empirically 0 0 10 20 30 40 50 60 linked with the Copula parameter. As seen in section 4.2. and precipitation amount [mm] 4.4, all the empirical Copulas derived in this study show a icsinPpr|Dsuso ae icsinPpr|Dsuso ae | Paper Discussion | Paper Discussion | Paper Discussion | Paper Discussion strong asymmetry with respect to the minor axis u =1−v of Fig. 16: Empirical CDF of observed precipitation in [0,1]2. This asymmetryHESSD can not be depicted by the common Garmisch-Partenkirchen, conditioned on the occurrence of Copula families such as the Clayton, Normal or Gumbel- the Northeast advection type WT 2 (NEAAD), and the 8, 3001–3045, 2011 Hougaard Copulas, acting as basic set of possible candidates Southwest advection type WT 39 (SWCCW). for the performed GOF tests. Nonmonotonic transformation Copula-based to construct asymmetric multivariate Copulas from the Gaus- stat. refinement of sian (Bárdossy,precipitation 2006) could in reflect RCMs the asymmetries in the 0.7 empirical Copulas andsimulations thus also improve the bias correction. 0.65 P. Laux et al. 0.6

0.55

τ Title Page 0.5 6 Conclusions 0.45 Abstract Introduction

0.4 The presentedConclusions Copula-basedReferences approach is potentially useful 0.35 for statistical downscaling, bias correction, and local refine- Tables Figures 0 200 400 600 800 1000 ment of RCMs. The performance will be evaluated and com- time step (positive pairs) pared to established methods for bias correction. J I Asymmetries are found in the empirical Copula densities Fig. 17. Kendall’s τ of 101 consecutive positive pairs for modelled and observed precipitation Fig. 17: Kendall’s τ of 101 consecutive positive pairs for J I at station Garmisch-Partenkirchen. which cannot be reproduced by the theoretical Copulas used modelled and observed precipitation at station Garmisch- in this study. Therefore,Back it is generallyClose difficult to find a the- Partenkirchen. oretical Copula model which is not rejected by the applied GOF test. Full Screen / Esc Fitting the marginal distributions is of crucial importance as Printer-friendly Version ally improve the performance of the bias correction. it strongly impacts the simulation results (more than the Cop- The objective weather pattern classification method of ula parameter θ). Interactive Discussion Large-scale weather patterns could be used to further con- the German Weather Service3045 (Bissolli and Dittmann, 2001) shows only moderate potential to further constraint the strain the model, and thus, increase the performance of the model. Including information about the humidity of the tro- simulation results. posphere can slightly increase the skill for bias correction compared to the Copula-based stochastic simulations without using large-scale information. This can be seen from the Acknowledgements. This research is funded by the Bavarian State conditional CDFs and the correspondingprobability plots for Ministry of the Environment and Public Health (reference number the different groups, which are not very discriminative (com- VH-ID:32722/TUF01UF-32722). Additional funding by the Fed- pare section 3.3 and 4.4). Using the single 40 weather types eral Ministry of Education and Research (research project: Land Use and Climate Change Interactions in Central Vietnam LUCCi, (without grouping) could potentially increase the discrimina- reference number 01LL0908C), by the HGF Impuls- und Vernet- tive power (see e.g. figure 16), but decreases the sample size zungsfond (research project: Regional Precipitation Observation by for certain WTs far too much to reliably estimate the Copula Cellular Network Microwave Attenuation and Application to Wa- parameter(s) and the marginal distributions. Another limita- ter Resources Management PROCEMA, Virtual Institute, reference tion of the approach shown in this paper is that the same the- number VH-VI-314), and by HGF (research project: Terrestrial oretical Copula model, i.e. the Gumbel-Hougaard Copula, Environmental Observatories TERENO (http://www.tereno.net)) is is fitted to each WT class. It is obvious from the empirical highly acknowledged. The help of Thomas Rummer, who created Copula density plots, that this does not necessarily provide Figure 2 is appreciated.