# Spatio-Temporal Synthesis of Continuous Precipitation Series Using Vine Copulas

^{*}

## Abstract

**:**

## 1. Introduction

- Is an extension of the Alternating Renewal based process model to a multi-site application feasible?
- Is the proposed extension able to reproduce both average event properties and extreme event statistics of observed rainfall in a high temporal resolution?
- How is the proposed model performing compared to the current practice?
- Are copulas efficient tools for modeling continuous precipitation in a high temporal resolution?

## 2. Materials and Methods

#### 2.1. Methods

#### 2.1.1. Alternating Renewal Process

#### 2.1.2. Multivariate Synthesis of Rainfall Events

#### Occurrence Model

#### Multivariate Wet Spell Model

_{k}, represents the marginal densities of the n variables. If four variables are involved in the analysis, the density is presented as

_{14|23}, c

_{13|2}, and c

_{24|3}are conditional pairs, c

_{12}, c

_{23}, and c

_{34}are unconditional pairs, and f

_{1}, f

_{2}, f

_{3}, and f

_{4}are marginal densities of each of the variables. A graphical representation of this structure is shown in Figure 2, which includes a nested set of trees used to depict the decomposition. The Vine tree structure [53] indicates the order of dependency of the variables for a regular Vine structure.

_{11}and C

_{22}bivariate models in Figure 2. The dependence structure between WSDs from both stations is as well modeled in this first tree (C

_{12}). For the case presented in Figure 2, which involves two stations, the structure defines the model as a D-Vine as every node is connected to at most two edges. For a case involving three stations, the structure is specified in a different way. The structure again guarantees that the WSA and WSD are modeled unconditionally for each of the stations with bivariate models C

_{11}, C

_{22}and C

_{33}. The dependences between WSDs are as well unconditionally modeled in this first tree. However, as there are three stations involved, the WSD corresponding to one of the stations is connected to the WSD of the other two stations (e.g., C

_{12}and C

_{13}) in addition to the WSA (e.g., C

_{11}). One of the nodes is therefore connected to three edges.

#### Multivariate Dry Spell Model

#### Limitations of the Model

#### 2.1.3. Alternative Method: Simulated Annealing

#### 2.1.4. Assessment Criteria for Performance

_{i}and z

_{j}are the 5-min rainfall time series at stations i and j. The second criterion is the Pearson’s coefficient of correlation applied to time steps for which rainfall is registered in both stations.

_{i,j}, p00

_{i,j}, p10

_{i,j}and p01

_{i,j}are the joint probabilities of rain (1) or no-rain (0) at stations i and j, i.e., p11

_{i,j}indicates the probability of rain at both stations. These probabilities are calculated by the number of time steps for each of the cases over the total number of time steps. The Log-odds ratio takes into account the probabilities of either rain or no-rain in two stations simultaneously. High values of this ratio indicate high spatial interrelation between two stations, as for the correlation coefficient.

_{i}* and Z

_{i}are simulated and observed characteristics respectively (e.g., moments, extreme values with a duration associated to a return period). Positive errors resulting from Equation (6) indicate overestimation by synthetic series.

#### 2.2. Study Area and Data

^{2}, reduction factors are extrapolated for bigger areas, i.e., for three out of the eight cases, by simple linear regressions. The reduction factors used for the different case studies are presented in Figure 4. Note that for events with durations of 5 min the Reduction Factor values corresponding to 15 min are taken, since this is the minimum available duration.

## 3. Results and Discussion

#### 3.1. Model Set Up

#### 3.1.1. Identification of Events Occurring Simultaneously

#### 3.1.2. Modeling of Events Occurring Simultaneously

#### 3.2. Evaluation of Results

#### 3.2.1. Spatial Consistency Measures

#### 3.2.2. Event Based Measures

#### 3.2.3. Extreme Values Measures

## 4. Conclusions

- Synthetic time series are compared in a pair-wise way, i.e., every two stations, to estimate spatial consistency measures. Overall, these measures (as shown in Figure 8) are better reproduced by the Vine copula model compared to the alternative method. Furthermore, these measures are not involved within the Vine copula model, as is the case for the SA optimization criteria, so these outcomes underline the satisfactory performance of the proposed model.
- The capability of the two models to reproduce different event characteristics is additionally compared. The evaluation based on single sites depicts some deficiencies of the proposed model, however these results are improved when the evaluation is performed based on areal precipitation (see Figure 9). Overall, SA performs better for single stations but fails when the areal precipitation is considered. These results suggest that the Vine copula model is reliable in reproducing the simultaneous occurrence of rainfall in several stations, whereas the SA is unable to mimic this behavior. Unfortunately, the proposed method systematically underestimates total seasonal rainfall.
- Extreme events are evaluated by assessing Intensity Duration Frequency curves (IDFs). The proposed model shows acceptable results as the ranges of errors are smaller than the ones from the SA and KOSTRA (see Figure 10). Vine copulas outperform SA. However, these models would be preferable for short durations, and KOSTRA for long ones. The reduction factors applied to KOSTRA are provided for different durations and areas of application, though for large areas, extrapolation by simple linear regressions was necessary. These factors could be revised, as they result in the underestimation of extreme values for most of the analyzed cases. Myers and Zehr [74] present reduction factors developed for the city of Chicago that are as well provided for different return periods, with higher factors for lower return periods.
- Overall, the copula-based proposed extension of the ARP delivers satisfactory results both for the average event statistics and for the extremes. Furthermore, the model has the advantage that the number of parameters is not affected by the temporal resolution. However the model consists of several steps of simulation that follow each other and is hard to interpret, as mentioned in the literature review as one limitation of the existing models for multi-site synthesis. Moreover, the model was only applied to cases involving two and three stations. Despite the fact that it could be applied to data sets involving more stations, a limitation would be the length of data available in several stations simultaneously for setting up the different modules within the model, especially the Vine copulas. Another important aspect that is subject of further research is the regionalization of this multi-site model, i.e., setting up the model for locations without observations. Current work involves exploring the capability of the radar data, which is spatially and temporally available in very high resolutions, to extract some of the characteristic that are required for modeling rainfall in multiple sites simultaneously.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

ARP | Alternating renewal process |

BAWU | German state of Baden Württemberg |

C-Vine | Canonical Vine copula |

DEM | Digital Elevation Map |

DSD | Dry spell duration |

D-Vine | Drawable Vine copula |

DWD | Deutscher Wetterdienst (German National Weather Service) |

IDF | Intensity duration frequency curves |

IID | Independent identically distributed |

KOSTRA | Koordinierte Starkniederschlagsregionalisierung, official German statistics of extreme precipitation |

m.a.s.l. | Meters above sea level referred to tide gauge Amsterdam (DHHN92) |

mNN | Meters above German Standard Zero (Normalnull) |

NS | German state of Lower Saxony (Niedersachsen) |

p00 | Joint probability of no-rain (0) and no-rain (0) at two stations |

p01 | Joint probability of no-rain (0) and rain (1) at two stations |

p10 | Joint probability of rain (1) and no-rain (0) at two stations |

p11 | Joint probability of rain (1) and rain (1) at two stations |

RSE | Relative standard error |

SA | Simulated annealing |

Tr | Return period |

WSA | Wet spell amount |

WSD | Wet spell duration |

WSI | Wet spell intensity |

WSPeak | Wet spell peak amount |

WSTpeak | Wet spell time to peak |

## References

- Bárdossy, A. Generating Precipitation Time Series Using Simulated Annealing. Water Resour. Res.
**1998**, 34, 1737–1744. [Google Scholar] [CrossRef] - Beck, F. Generation of Spatially Correlated Synthetic Rainfall Time Series in High Temporal Resolution: A Data Driven Approach. Ph.D. Thesis, Universität Stuttgart, Stuttgart, Germany, 2013. [Google Scholar]
- Licznar, P.; Schmitt, T.G.; Rupp, D.E. Distributions of microcanonical cascade weights of rainfall at small timescales. Acta Geophys.
**2011**, 59, 1013–1043. [Google Scholar] [CrossRef] - Cowpertwait, P.S.P.; Lockie, T.; Davis, M.D. A stochastic spatial-temporal disaggregation model for rainfall. Res. Lett. Inf. Math. Sci.
**2004**, 6, 109–122. [Google Scholar] - Müller, H.; Haberlandt, U. Temporal rainfall disaggregation using a multiplicative cascade model for spatial application in urban hydrology. J. Hydrol.
**2018**, 556, 847–864. [Google Scholar] [CrossRef] - Katz, R.W.; Parlange, M.B. Generalizations of chain-dependent processes: Application to hourly precipitation. Water Resour. Res.
**1995**, 31, 1331–1341. [Google Scholar] [CrossRef] - Zhang, Z.; Switzer, P. Stochastic space-time regional rainfall modeling adapted to historical rain gauge data. Water Resour. Res.
**2007**, 43, W03441. [Google Scholar] [CrossRef] - Gyasi-Agyei, Y.; Charles, S. Modelling the dependence and internal structure of storm events for continuous rainfall simulation. J. Hydrol.
**2012**, 464, 249–261. [Google Scholar] [CrossRef] - Willems, P. A spatial rainfall generator for small spatial scales. J. Hydrol.
**2001**, 252, 126–144. [Google Scholar] [CrossRef] - Cowpertwait, P.S.P.; Kilsby, C.; O’Connell, P. A space-time Neyman-Scott model of rainfall: Empirical analysis of extremes. Water Resour. Res.
**2002**, 38. [Google Scholar] [CrossRef] - Burton, A.; Fowler, H.J.; Kilsby, C.G.; O’Connel, P.E. RainSim: A spatial–temporal stochastic rainfall modelling system. Environ. Model. Softw.
**2008**, 23, 1356–1369. [Google Scholar] [CrossRef] - Burton, A.; Fowler, H.J.; Kilsby, C.G.; O’Connell, P.E. A stochastic model for the spatial-temporal simulation of nonhomogeneous rainfall occurrence and amounts. Water Resour. Res.
**2010**, 46. [Google Scholar] [CrossRef] [Green Version] - Tarpanelli, A.; Franchini, M.; Brocca, L.; Camici, S.; Melone, F.; Moramarco, T. A simple approach for stochastic generation of spatial rainfall pattern. J. Hydrol.
**2012**, 472, 63–76. [Google Scholar] [CrossRef] - Rodriguez-Iturbe, I.; Cox, D.R.; Isham, V. A point process model for rainfall: Further developments. Proc. R. Soc. Lond. A
**1988**, 417, 283–298. [Google Scholar] [CrossRef] - Verhoest, N.; Troch, P.E.; Troch, F.P. On the applicability of Bartlett–Lewis rectangular pulses models in the modeling of design storms at a point. J. Hydrol.
**1997**, 202, 108–120. [Google Scholar] [CrossRef] - Cowpertwait, P.S.P.; Isham, V.; Onof, C. Point process models of rainfall: Developments for fine-scale structure. Proc. R. Soc. A
**2007**, 463, 2569–2587. [Google Scholar] [CrossRef] - Vandenberghe, S.; Verhoest, N.E.C.; Onof, C.; De Baets, B. A comparative copula-based bivariate frequency analysis of observed and simulated storm events: A case study on Bartlett-Lewis modeled rainfall. Water Resour. Res.
**2011**, 47. [Google Scholar] [CrossRef] [Green Version] - Pham, M.T.; Vanhaute, W.J.; Vandenberghe, S.; De Baets, B.; Verhoest, N.E.C. A copula-based assessment of Bartlett–Lewis type of rainfall models for preserving drought statistics. Hydrol. Earth Syst. Sci. Discuss.
**2013**, 10, 7469–7516. [Google Scholar] [CrossRef] - Kaczmarska, J.; Isham, V.; Onof, C. Point process models for fine-resolution rainfall. Hydrol. Sci. J.
**2014**, 59, 1972–1991. [Google Scholar] [CrossRef] [Green Version] - Vernieuwe, H.; Vandenberghe, S.; De Baets, B.; Verhoest, N.E.C. A continuous rainfall model based on vine copulas. Hydrol. Earth Syst. Sci.
**2015**, 19, 2685–2699. [Google Scholar] [CrossRef] [Green Version] - Haberlandt, U. Stochastic rainfall synthesis using regionalized model parameters. J. Hydrol. Eng.
**1998**, 3, 160–168. [Google Scholar] [CrossRef] - Bernardara, P.; De Michele, C.; Rosso, R. A simple model of rain in time: An alternating renewal process of wet and dry states with a fractional (non-Gaussian) rain intensity. Atmos. Res.
**2007**, 84, 291–301. [Google Scholar] [CrossRef] - Callau Poduje, A.C.; Haberlandt, U. Short time step continuous rainfall modeling and simulation of extreme events. J. Hydrol.
**2017**, 552, 182–197. [Google Scholar] [CrossRef] - Haberlandt, U.; Ebner von Eschenbach, A.-D.; Buchwald, I. A space-time hybrid hourly rainfall model for derived flood frequency analysis. Hydrol. Earth Syst. Sci.
**2008**, 12, 1353–1367. [Google Scholar] [CrossRef] [Green Version] - Sordo-Ward, A.; Bianucci, P.; Garrote, L.; Granados, A. The influence of the annual number of storms on the derivation of the flood frequency curve through event-based simulation. Water
**2016**, 8, 335. [Google Scholar] [CrossRef] - Kao, S.; Govindaraju, R. Trivariate statistical analysis of extreme rainfall events via the Plackett family of copulas. Water Resour. Res.
**2008**, 44, W02415. [Google Scholar] [CrossRef] - Zhang, L.; Singh, V. Gumbel-Hougaard copula for trivariate rainfall frequency analysis. J. Hydrol. Eng.
**2007**, 12, 409–419. [Google Scholar] [CrossRef] - Salvadori, G.; De Michele, C. Statistical characterization of temporal structure of storms. Adv. Water Resour.
**2006**, 29, 827–842. [Google Scholar] [CrossRef] - Grimaldi, S.; Serinaldi, F. Design hyetograph analysis with 3-copula function. Hydrol. Sci. J.
**2006**, 51, 223–238. [Google Scholar] [CrossRef] [Green Version] - Ganguli, P.; Reddy, M. Probabilistic assessment of flood risks using trivariate copulas. Theor. Appl. Climatol.
**2013**, 111, 341–360. [Google Scholar] [CrossRef] - Serinaldi, F.; Grimaldi, S. Fully nested 3-copula: Procedure and application on hydrological data. J. Hydrol. Eng.
**2007**, 12, 420–430. [Google Scholar] [CrossRef] - Genest, C.; Favre, A.; Beliveau, J.; Jacques, C. Metaelliptical copulas and their use in frequency analysis of multivariate hydrological data. Water Resour. Res.
**2007**, 43, W09401. [Google Scholar] [CrossRef] - Grimaldi, S.; Serinaldi, F. Asymmetric copula in multivariate flood frequency analysis. Adv. Water Resour.
**2006**, 29, 1155–1167. [Google Scholar] [CrossRef] - Chen, L.; Singh, V.; Shenglian, G.; Hao, Z.; Li, T. Flood Coincidence Risk Analysis Using Multivariate Copula Functions. J. Hydrol. Eng.
**2012**, 17, 742–755. [Google Scholar] [CrossRef] - Wang, C.; Chang, N.-B.; Yeh, G.-T. Copula-based flood frequency (COFF) analysis at the confluences of river systems. Hydrol. Process.
**2009**, 23, 1471–1486. [Google Scholar] [CrossRef] - Durante, F.; Salvadori, G. On the construction of multivariate extreme value models via copulas. Environmetrics
**2010**, 21, 143–161. [Google Scholar] [CrossRef] - Bezak, N.; Mikoš, M.; Šraj, M. Trivariate Frequency Analyses of Peak Discharge, Hydrograph Volume and Suspended Sediment Concentration Data Using Copulas. Water Resour. Manag.
**2014**, 28, 2195–2212. [Google Scholar] [CrossRef] - Ma, M.; Song, S.; Ren, L.; Jiand, S.; Song, J. Multivariate drought characteristics using trivariate Gaussian and Student t copulas. Hydrol. Process.
**2013**, 27, 1175–1190. [Google Scholar] [CrossRef] - Song, S.; Singh, V. Meta-elliptical copulas for drought frequency analysis of periodic hydrologic data. Stoch. Environ. Res. Risk A
**2010**, 24, 425–444. [Google Scholar] [CrossRef] - Wong, G.; Lambert, M.; Leonard, M.; Metcalfe, A. Drought Analysis using trivariate copulas conditional on climatic states. J. Hydrol. Eng.
**2010**, 15, 129–141. [Google Scholar] [CrossRef] - Pham, M.T.; Vernieuwe, H.; De Baets, B.; Verhoest, N.E.C. A coupled stochastic rainfall-evapotranspiration model for hydrological impact analysis. Hydrol. Earth Syst. Sci.
**2018**, 22, 1263–1283. [Google Scholar] [CrossRef] - De Michele, C.; Salvadori, G.; Passoni, G.; Vezzoli, R. A multivariate model of sea storms using copulas. Coast. Eng.
**2007**, 54, 734–751. [Google Scholar] [CrossRef] - Czado, C. Pair-copula constructions of multivariate copulas. In Workshop on Copula Theory and Its Applications; Jaworki, P., Durante, F., Härdle, W., Rychlik, T., Eds.; Lecture Notes in Statistics; Springer: Berlin/Heidelberg, Germany, 2010; p. 198. [Google Scholar] [CrossRef]
- Gräler, B. Modelling skewed spatial random fields through the spatial vine copula. Spat. Stat.
**2014**, 10, 87–102. [Google Scholar] [CrossRef] - Xiong, L.; Yu, K.; Gottschalk, L. Estimation of the distribution of annual runoff from climatic variables using copulas. Water Resour. Res.
**2014**, 50, 7134–7152. [Google Scholar] [CrossRef] [Green Version] - Gräler, B.; van den Berg, M.J.; Vandenberghe, S.; Petroselli, A.; Grimaldi, S.; De Baets, B.; Verhoest, N.E.C. Multivariate return periods in hydrology: A critical and practical review focusing on synthetic design hydrograph estimation. Hydrol. Earth Syst. Sci.
**2013**, 17, 1281–1296. [Google Scholar] [CrossRef] [Green Version] - Shafaei, M.; Fakheri-Fard, A.; Dinpashoh, Y.; Mirabbasi, R.; De Michele, C. Modeling flood event characteristics using D-vine structures. Theor. Appl. Climatol.
**2017**, 130, 713–724. [Google Scholar] [CrossRef] - Gaál, L.; Molnar, P.; Szolgay, J. Selection of intense rainfall events based on intensity thresholds and lightning data in Switzerland. Hydrol. Earth Syst. Sci.
**2014**, 18, 1561–1573. [Google Scholar] [CrossRef] [Green Version] - Callau Poduje, A.C. Spatio-temporal modeling of precipitation in a high temporal resolution for urban hydrological applications. In Mitteilungen des Instituts für Hydrologie und Wasserwirtschaft Heft 107; Institute of Hydrology and Water Resources Management: Leibniz Universität Hannover, Germany, 2018; ISSN 0343-8090. [Google Scholar]
- Bacchi, B.; Kottegoda, N.T. Identification and calibration of spatial correlation patterns of rainfall. J. Hydrol.
**1995**, 165, 311–348. [Google Scholar] [CrossRef] - Brechmann, E.C.; Schepsmeier, U. Modeling Dependence with C- and D-Vine Copulas: The R Package CDVine. J. Stat. Softw.
**2013**, 52, 1–27. [Google Scholar] [CrossRef] - Joe, H. Families of m-variate distributions with given margins and m(m − 1)/2 bivariate dependence parameters. In Distributions with Fixed Marginals and Related Topics; Rüschendorf, L., Schweizer, B., Taylor, M.D., Eds.; Institute of Mathematical Statistics: Bethesda, MD, USA, 1996. [Google Scholar]
- Bedford, T.; Cooke, R.M. Probability density decomposition for conditionally dependent random variables modeled by vines. Ann. Math. Artif. Intell.
**2001**, 32, 245–268. [Google Scholar] [CrossRef] - Aas, K.; Czado, C.; Frigessi, A.; Bakken, H. Pair-copula constructions of multiple dependence. Insur. Math. Econ.
**2009**, 44, 182–198. [Google Scholar] [CrossRef] [Green Version] - Brechmann, E.C.; Czado, C.; Aas, K. Truncated regular vines in high dimensions with applications to financial data. Can. J. Stat.
**2012**, 40, 68–85. [Google Scholar] [CrossRef] - Dißmann, J.F.; Brechmann, E.C.; Czado, C.; Kurowicka, D. Selecting and estimating regular vine copulae and application to financial returns. Comput. Stat. Data Anal.
**2013**, 59, 52–69. [Google Scholar] [CrossRef] [Green Version] - R Core Team. R: A Language and Environment for Statistical Computing; R Foundation for Statistical Computing: Vienna, Austria, 2016; Available online: https://www.R-project.org/ (accessed on 15 December 2016).
- Asquith, W. Lmomco—L-Moments, Censored L-Moments, Trimmed L-Moments, L-Comoments, and Many Distributions; R Package Version 2.2.3; Texas Tech University: Lubbock, TX, USA, 2016. [Google Scholar]
- Yan, J. Enjoy the Joy of Copulas: With a Package copula. J. Stat. Softw.
**2007**, 21, 1–21. [Google Scholar] [CrossRef] - Kojadinovic, I.; Yan, J. Modeling Multivariate Distributions with Continuous Margins Using the copula R Package. J. Stat. Softw.
**2010**, 34, 1–20. [Google Scholar] [CrossRef] - Hofert, M.; Maechler, M. Nested Archimedean Copulas Meet R: The nacopula Package. J. Stat. Softw.
**2011**, 39, 1–20. [Google Scholar] [CrossRef] - Hofert, M.; Kojadinovic, I.; Maechler, M.; Yan, J. Copula: Multivariate Dependence with Copulas. R Package Version 0.999-14. 2015. Available online: http://CRAN.R-project.org/package=copula (accessed on 15 December 2016).
- Schepsmeier, U.; Stoeber, J.; Brechmann, E.C.; Graeler, B.; Nagler, T.; Erhardt, T. VineCopula: Statistical Inference of Vine Copulas. R Package Version 2.0.1. 2016. Available online: https://CRAN.R-project.org/package=VineCopula (accessed on 15 December 2016).
- Wilks, D.S. Multisite generalization of a daily stochastic precipitation generation model. J. Hydrol.
**1998**, 210, 178–191. [Google Scholar] [CrossRef] - Mehrotra, R.; Srikanthan, R.; Sharma, A. A comparison of three stochastic multi-site precipitation occurrence generators. J. Hydrol.
**2006**, 331, 280–292. [Google Scholar] [CrossRef] - DWA. Hydraulische Bemessung und Nachweis von Entwässerungssystemen; Arbeitsblatt A 118; Deutsche Vereinigung für Wasserwirtschaft, Abwasser und Abfall e. V.: Hennef, Germany, 2006. [Google Scholar]
- Eggert, B.; Berg, P.; Haerter, J.O.; Jacob, D.; Moseley, C. Temporal and spatial scaling impacts on extreme precipitation. Atmos. Chem. Phys.
**2015**, 15, 5957–5971. [Google Scholar] [CrossRef] [Green Version] - Bartels, H.; Dietzer, B.; Malitz, G.; Albrecht, F.M.; Guttenberger, J. KOSTRA-DWD-2000, Starkniederschlagshöhen für Deutschland (1951–2000). Technical Report, DeutscherWetterdienst—Abteilung Hydrometeorologie, Offenbach am Main, 2005. Available online: https://www.dwd.de/DE/fachnutzer/wasserwirtschaft/kooperationen/kostra/fortschreibung_pdf.pdf%3F__blob%3DpublicationFile%26v%3D3 (accessed on 20 June 2018).
- Verworn, H.R. Flächenabhängige Abminderung statistischer Regenwerte. Korresp. Wasserwirtsch.
**2008**, 9, 493–498. [Google Scholar] - Tawn, J.A. Bivariate extreme value theory: Models and estimation. Biometrika
**1988**, 75, 397–415. [Google Scholar] [CrossRef] - Nelsen, R.B. An Introduction to Copulas, 2nd ed.; Springer: New York, NY, USA, 2006; ISBN 9780387286594. [Google Scholar]
- Rémillard, B.; Scaillet, O. Testing for equality between two copulas. J. Multivar. Anal.
**2009**, 100, 377–386. [Google Scholar] [CrossRef] - Carr, D.; Lewin-Koh, N.; Maechler, M.; Deepayan, S. Hexbin: Hexagonal Binning Routines. R Package Version 1.27.1. 2015. Available online: https://CRAN.R-project.org/package=hexbin (accessed on 15 December 2016).
- Myers, V.A.; Zehr, R.M. A Methodology for Point-to-Area Rainfall Frequency Ratio; NOAA Technical Report NWS 24; National Weather Service, NOAA, U. S. Department of Commerce: Washington, DC, USA, 1980.

**Figure 2.**Example of components of a Vine copula to model events occurring simultaneously in two stations. (

**a**) Marginal densities, unconditional and conditional pair copula densities (under N(0, 1) margins for visualization). (

**b**) Vine tree structure.

**Figure 3.**Location of rain gauge stations and groups used for developing the model. The red circles indicate the areas involved for each case.

**Figure 5.**Dependence between event characteristics according to their occurrence in one, two, or three stations simultaneously for all cases.

**Figure 6.**Pairs of pseudo-values of summer events occurring in three stations simultaneously for the BAWU_B case (Kendall’s Tau in red). Upper triangular matrix: Observed. Lower triangular matrix: Synthetic.

**Figure 7.**(

**a**) Cumulative distributions of WSD, (

**b**) bias correction of WSD, and (

**c**) cumulative distributions of DSD for summer events corresponding to BAWU_A case.

**Figure 8.**Bivariate spatial consistency measures of observed and simulated time series, small events are excluded: (

**a**) Continuity measure; (

**b**) Pearson’s coefficient of correlation; (

**c**) Joint probability of rain in both stations; (

**d**) Log-odds ratio.

**Figure 9.**Errors of mean values of event characteristics based on areal and single sites resulting from two models used for multisite synthesis for summer (left plots of violins) and winter (right plots of violins) events for all stations.

**Figure 10.**Comparison of models regarding their ability to reproduce extreme events for durations ranging from 5 min to 24 h for all stations.

**Figure 11.**Comparison of IDF curves obtained from observed, KOSTRA, KOSTRA Reduced, and simulated time series (all realizations are shown as symbols: circles for Vine and crosses for SA) for the cases: (

**a**) NS_B corresponding to the biggest area; (

**b**) BAWU_D involving the smallest area.

Number Of Stations in Which Event Occurs | 1 | 3 | 3 | 2 | 1 | 1 | 2 | 2 | 3 | 3 | 1 |
---|---|---|---|---|---|---|---|---|---|---|---|

Station A | 1 | 1 | 1 | 0 | 0 | 0 | 1 | 0 | 1 | 1 | 0 |

Station B | 0 | 1 | 1 | 1 | 1 | 0 | 0 | 1 | 1 | 1 | 0 |

Station C | 0 | 1 | 1 | 1 | 0 | 1 | 1 | 1 | 1 | 1 | 1 |

Consecutive | Consecutive |

Stat. | ID | Name | Latitude [°] | Longitude [°] | Elevation [mNN] | Data Availability [y] | Distance [km] | |
---|---|---|---|---|---|---|---|---|

A | B | |||||||

BAWU_A | ||||||||

A | 70354 | FREIBURG | 48.04 | 7.82 | 236 | 10 | - | - |

B | 70314 | EMMENDINGEN-MUNDING | 48.13 | 7.83 | 201 | 10 | 13 | - |

C | 70323 | BUCHENBACH | 47.97 | 8 | 445 | 10 | 13.6 | 22.3 |

BAWU_B | ||||||||

A | 90164 | HOHENSTEIN-BERNLOCH | 48.35 | 9.33 | 740 | 11 | - | - |

B | 90163 | MUENSINGEN-APFELSTET | 48.38 | 9.48 | 750 | 11 | 11.7 | - |

C | 90156 | LANGENENSLINGEN-ITTEN | 48.2 | 9.33 | 777 | 11 | 16.7 | 23.2 |

BAWU_C | ||||||||

A | 70145 | WEINGARTEN | 47.8 | 9.62 | 708 | 13 | - | - |

B | 70153 | DEGGENHAUSERTAL-AZEN | 47.8 | 9.42 | 440 | 13 | 15 | - |

BAWU_D | ||||||||

A | 73930 | MERGENTHEIM, BAD-NEUN | 49.48 | 9.77 | 250 | 14 | - | - |

B | 73942 | LAUDA-KOENIGSHOFEN-HE | 49.55 | 9.63 | 324 | 14 | 12.2 | - |

NS_A | ||||||||

A | 2925 | LEINEFELDE | 51.39 | 10.3 | 356 | 19 | - | - |

B | 1691 | GOETTINGEN | 51.5 | 9.95 | 167 | 19 | 27.2 | - |

NS_B | ||||||||

A | 4745 | SOLTAU | 52.96 | 9.79 | 75.6 | 8 | - | - |

B | 1336 | FALLINGBOSTEL, BAD | 52.85 | 9.68 | 70 | 8 | 15.5 | - |

C | 5146 | UELZEN | 52.95 | 10.53 | 50 | 8 | 49.1 | 58.1 |

NS_C | ||||||||

A | 3815 | OSNABRÜCK | 52.26 | 8.05 | 95.4 | 14 | - | - |

B | 4371 | SALZUFLEN, BAD | 52.11 | 8.75 | 134.6 | 14 | 50.7 | - |

C | 963 | DIEPHOLZ(WEWA) | 52.58 | 8.35 | 39 | 14 | 41 | 59 |

NS_D | ||||||||

A | 656 | BRAUNLAGE | 51.73 | 10.6 | 607 | 8 | - | - |

B | 3650 | NORTHEIM-IMBSHAUSEN | 51.77 | 10.05 | 212 | 8 | 38.2 | - |

C | 4651 | SEESEN | 51.9 | 10.18 | 186 | 8 | 34.4 | 17.5 |

**Table 3.**Percentages of observed events occurring in several stations simultaneously (1, 2, 3) and in each of the stations (A, B, C) for different seasons and case studies.

Number or ID Stations | BAWU_A | BAWU_B | BAWU_C | BAWU_D | NS_A | NS_B | NS_C | NS_D | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Summer | Winter | Summer | Winter | Summer | Winter | Summer | Winter | Summer | Winter | Summer | Winter | Summer | Winter | Summer | Winter | |

1 | 61 | 57 | 58 | 50 | 65 | 57 | 63 | 59 | 77 | 73 | 68 | 58 | 78 | 68 | 68 | 63 |

2 | 20 | 21 | 21 | 23 | 35 | 43 | 37 | 41 | 23 | 27 | 21 | 23 | 15 | 21 | 19 | 21 |

3 | 19 | 21 | 22 | 27 | - | - | - | - | - | - | 11 | 20 | 6.6 | 11 | 13 | 16 |

A | 56 | 59 | 64 | 76 | 71 | 77 | 74 | 74 | 73 | 83 | 56 | 70 | 47 | 61 | 65 | 85 |

B | 58 | 54 | 68 | 75 | 79 | 81 | 72 | 78 | 57 | 49 | 55 | 66 | 50 | 54 | 47 | 42 |

C | 70 | 81 | 63 | 58 | - | - | - | - | - | - | 44 | 54 | 42 | 47 | 52 | 48 |

Number Stations | BAWU_A | BAWU_B | BAWU_C | BAWU_D | NS_A | NS_B | NS_C | NS_D | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Summer | Winter | Summer | Winter | Summer | Winter | Summer | Winter | Summer | Winter | Summer | Winter | Summer | Winter | Summer | Winter | ||

p-values | 2 | 0.75 | 0.3 | 0.3 | 0.45 | 0.3 | 0.45 | 0.6 | 0.25 | 0.0 | 0.5 | 0.75 | 0.9 | 0.75 | 0.1 | 0.75 | 0.8 |

3 | 0.0 | 0.1 | 0.1 | 0.1 | - | - | - | - | - | - | 0.3 | 0.1 | 0.1 | 0.2 | 0.05 | 0.2 |

**Table 5.**Evaluation of statistics of different variables describing the external structure of rainfall events resulting from two models for multisite synthesis based on areal precipitation and presented for all cases as RSE.

RSE [-] | Mean | Standard Deviation | Skewness | Kurtosis | |||||
---|---|---|---|---|---|---|---|---|---|

Variable | Season | SA | Vine Copula | SA | Vine Copula | SA | Vine Copula | SA | Vine Copula |

DSD | Summer | 0.15 | 0.08 | 0.30 | 0.07 | 0.12 | 0.80 | 0.31 | 4.40 |

Winter | 0.20 | 0.07 | 0.36 | 0.05 | 0.14 | 0.45 | 0.18 | 1.71 | |

WSD | Summer | 0.14 | 0.06 | 0.05 | 0.15 | 0.18 | 0.27 | 0.28 | 0.54 |

Winter | 0.10 | 0.07 | 0.12 | 0.13 | 0.26 | 0.27 | 0.81 | 1.47 | |

WSA | Summer | 0.25 | 0.12 | 0.38 | 0.28 | 0.23 | 0.26 | 0.61 | 0.45 |

Winter | 0.26 | 0.13 | 0.34 | 0.25 | 0.34 | 0.46 | 1.11 | 2.81 | |

WSI | Summer | 0.24 | 0.05 | 0.17 | 0.27 | 1.21 | 0.64 | 7.29 | 1.95 |

Winter | 0.11 | 0.06 | 1.15 | 0.59 | 3.58 | 1.65 | 24.41 | 6.87 |

**Table 6.**Evaluation of performance based on wet time steps for different temporal resolutions resulting from two multisite models and presented for all stations as RSE for single stations and areal precipitation.

RSE [-] | Intermittence | Mean | Standard Deviation | Autocorrelation | |||||
---|---|---|---|---|---|---|---|---|---|

Temporal resolution | SA | Vine Copula | SA | Vine Copula | SA | Vine Copula | SA | Vine Copula | |

Single Stations | 5 min | 0.20 | 0.28 | 0.05 | 0.07 | 0.20 | 0.11 | 0.11 | 0.08 |

1 h | 0.20 | 0.24 | 0.05 | 0.06 | 0.12 | 0.11 | 0.23 | 0.22 | |

6 h | 0.10 | 0.17 | 0.05 | 0.08 | 0.06 | 0.14 | 0.59 | 0.67 | |

1 day | 0.06 | 0.14 | 0.09 | 0.12 | 0.07 | 0.22 | 0.82 | 0.65 | |

Areal | 5 min | 0.16 | 0.11 | 0.24 | 0.08 | 0.26 | 0.07 | 0.07 | 0.07 |

1 h | 0.15 | 0.10 | 0.24 | 0.09 | 0.28 | 0.12 | 0.27 | 0.28 | |

6 h | 0.21 | 0.05 | 0.28 | 0.14 | 0.34 | 0.20 | 0.44 | 0.71 | |

1 day | 0.26 | 0.06 | 0.31 | 0.18 | 0.37 | 0.28 | 0.65 | 0.64 |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Callau Poduje, A.C.; Haberlandt, U.
Spatio-Temporal Synthesis of Continuous Precipitation Series Using Vine Copulas. *Water* **2018**, *10*, 862.
https://doi.org/10.3390/w10070862

**AMA Style**

Callau Poduje AC, Haberlandt U.
Spatio-Temporal Synthesis of Continuous Precipitation Series Using Vine Copulas. *Water*. 2018; 10(7):862.
https://doi.org/10.3390/w10070862

**Chicago/Turabian Style**

Callau Poduje, Ana Claudia, and Uwe Haberlandt.
2018. "Spatio-Temporal Synthesis of Continuous Precipitation Series Using Vine Copulas" *Water* 10, no. 7: 862.
https://doi.org/10.3390/w10070862