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## Rainfall measurement uncertainties

At the present time, very long series (more than 30 years) are only available for daily rain gauge data. The first tipping-bucket rain gauge devices (Figure 1.1) were installed at the half of XX century.

Due to the poor knowledge of the rainfall variability at short time scales, the installed tipping-bucket

devices were unadapted to measure extreme rainfall intensities: a series of mechanical issues, that are negligible at the daily time step, become significant at the hourly time scales.

Each rain gauge is built for a specific time scale and a specific rainfall intensity range. The use of rain gauges to explore scales different to the design time scale is submitted to a series of uncertainties

that we will explore and quantify through numerical simulations in Section 2.

In this thesis a detailed analysis of the uncertainties related to the measurement network is necessary because, by means of scale-invariance, daily gauge series could be used to extrapolate the extreme behavior of infra-daily time series or of spatial time series. In these cases, the errors can easily propagate through scales seriously affecting the reliability of the used methods.

**Extreme rainfall modeling**

The second uncertainty source concerns the lenght of the rain gauge records. Except rare cases, very few hourly rain gauges feature more than 15 years of data. In these conditions, the estimation of extreme quantities (characterized by a return period higher than 30 years) must be done by extrapolation.

Even if the rain gauge correctly measures the rainfall intensity, the limited sample size affects the extreme estimation. For this reason, we will describe in Section 3 the theory of extremes, including the main methods to extract the maximum information from a sample, depending on the sample size.

**Scale-invariance of heavy time rainfall**

The estimation of rainfall extremes is possible at scales for which long series of direct measures are available. For infra-daily time scales, often, the sample size is not sufficient to make a robust estimation. This is the main reason leading us to study the extreme behavior at two or more scales, with the aim to determine whether scale-invariant relationships are present or not. If the scaleinvariant relationships are verified, one can, for instance, determine the extreme rainfall behavior at the hourly scales by extrapolation of the behavior of daily series.

To apply such scale-invariant relationships, we first verify that, for a given station, the law of extremes is the same whatever the time scale of analysis (Section 7). Secondly, we verify the range of validity of scale invariance and build a scale-invariant model useful to derive the magnitude of infra-daily rainfall extremes starting from daily observations (Section 8). The scale-invariant method can deal with the different behaviors of extremes found within the region of interest.

The scale-invariant model will be adopted to estimate the magnitude of rainfall observations at different accumulation times.

### Scale-invariance of heavy spatial rainfall

To evaluate the spatial rainfall scale-invariance one needs spatial rainfall records. The only direct

source of spatial rainfall data is radar scans. Spatial rainfall fields detected by radar show a regular

pattern that does not depend on the scale: in Section 9 we will show that radar fields are scaleinvariant.

When dealing with the study of extremes, nonetheless, two main inconvenients prevent the use of radar data: i) the problems in the quantitative evaluation of the rainfall depth by radar; 2) the limited sample size, since the data is restricted to few relevant events.

To overcome these limitations, in literature most of the approaches use interpolation of point data to derive a spatial rainfall database. However, we will demonstrate by numerical simulation in Section 2.4, the rainfall interpolation can’t be used for the study of extremes since the field maximum is sistematically underestimated.

**Scale-invariance of rainfall fields in space-time**

As said in the previous paragraph, no direct evaluation of the extreme behavior of spatial rainfall can be drawn based on the radar scan in the region. The problem of the storm magnitude estimation for each accumulation time and for each integration surface can’t be solved unless an approximate assumption is taken. In Section 6.2.4.1 we present an approach proposed by De Michele et al. (2001) to relate spatial and temporal scale of the interpolated extremes. Even if this model neglects the problems in the interpolation of point data highlighted in 1.1.4, it provides an effective method to represent, with a simple relation, the intuitive decrease of maxima with the surface and with the accumulation time.

With this solution, the storm magnitude at all spatial and temporal scales can be calculated, leading to the determination of the severity diagrams, useful for the event intercomparison or for the evaluation of the meteorological simulations as shown in Section 10.

**Geographical context**

The C´evennes-Vivarais region is located on the south-eastern side of Massif Central. It is easterly bounded by the Rhˆone river, westerly bounded by the Massif Central, and southerly by the Mediterranean Sea shore. The mountainous part of the region is included in the C´evennes National Park, created in 1970.

The size of the region is 160 km in the E-W direction and 200 km in the N-S direction, for a total surface of 32000 km2.

The elevation in the domain raises the 1699 m (Mt. Loz`ere). The main mountain ridge of the Massif Central is oriented SSW-NNE, and it can be approximately detected in Figure 1.2 by drawing a straight line joining Mont Aigoual (1565m height), Mont Loz`ere (1699 m), Mont Gerbier de Jonc (1551m) and Mont Mezenc (1753m). A number of fine-scale structures are present, including a series of “shoulders”, perpendicular to the main ridge and delineating the valleys, most of which are oriented NW-SE. These valleys present common features, such as their depth (about 500 meters) and length (in the order of 10 km). Their separation distance (20 km) is a significant feature for the triggering of rainfall events (Miniscloux et al., 2001).

The region presents several rivers, either tributary of the Rhˆone or directly flowing into theMediterranean Sea.

**Meteo-hydrological context**

In 1999, the the INSU-CNRS (Institut National des Sciences de l’Univers) and by the OSUG (Observatoire de Sciences de l’Univers de Grenoble) created the OHMCV (Observatoire Hydro-M´et´eorologique C´evennes-Vivarais), devoted to the study of extreme rainfall and flash-floods in this region prone to extremely intense events.

The technological development in the last decades is at the base of the recent improvement of the media diffusion of the OHMCV data: most of the measurements and data are available through a portal called SevNol (Syst`eme d’Extraction et de VisualisatioN des donn´ees de l’OHMCV en Ligne1).

OHMCV (Figure 1.4) is born to gather the measurement networks and to enforce the consultation among the socio-hydro-meteorological communities through a series of workshops and multidisciplinary projets.

In the recent years, the international HyMeX (HYdrological cycle in the Mediterranean EXperiment) programme, aiming at a better understanding of the hydrological cycle in Mediterranean Regions, has been defined. Similarly to the AMMA project, involving the African monsoon, it is a long-term project.

HyMeX share with OHMCV the integrated approach regrouping social sciences, economics, meteohydrology and hydraulics.

LTHE members are involved in the project HyMeX, with important tasks concerning the Working Group 3 on “Heavy rainfall, flash-floods and floods” and the Working Group 5, on “Societal and economic impacts“.

**Climatic features**

All along the thesis we adopt the thematic cartography to describe the features of the region. The map covers the C´evennes-Vivarais region, whose boundaries have conventionally been fixed in the extended Lambert II projection, as the rectangle of coordinates X=[650 km,810 km],Y=[1830 km, 2030 km].

The region is therefore 160×200 km2.

We can summarize the climatic features of the region by 2 main indicators. The average annual rainfall depth is the first indicator, it is independent of the scale of analysis and it gives a preliminary description of the main climatic patterns of the region. In Figure 1.5 the average annual rainfall, obtained by geostatistical interpolation (see Section 4), is reported. From the map it is evident that the drier sub-regions are located along the Mediterranean shore, while the highest rainfall depths are recorded along the Massif Central mountain ridge, oriented SW-NE, roughly obtainable by joining the Mt. Aigoual to the Serre de la Croix de Bauzon. These two locations feature the largest rainfall accumulation, with more than 2 m of equivalent rainfall depth per year.

Another relevant indicator of the climatic behavior of the region is the number of rainy days. From the map in Figure 1.6, we see that the Rhˆone Valley and the Mediterranean shore feature the driest climate, while over the mountainous region of the C´evennes plateau, in the NW corner, it rains almost 50% of the days.

The climatic pattern takes a different spatial organization depending on the analyzed season. In Appendix B we draw some maps describing the monthly rainfall regime in the region, with the aim to defined in the framework of the HyMeX project are reported. From the SevNol website show some peculiar characteristics of the rainfall regime. In sequence, we show the average monthly rainfall, its proportion with respect to the total annual depth in order to highlight the months in which a dry/wet regime is observed, followed by the montly intermittency (proportion of wet days to the total).

Further information on the climatological behavior in the C´evennes-Vivarais region, with particular interest on the orographic effect on the rainfall regime, can be found in Molini´e et al. (2010).

#### Flash-Floods

The C´evennes-Vivarais region is naturally prone to very intense storms, for two reasons:

• its position, located at few dozens of kilometers from the Mediterranean Sea, source of warm and humid air masses;

• its complex topography: i) the main mountain ridge oriented 90◦ to the SSE flux generating heavy precipitation events; ii) narrow parallel valleys evenly separated, ideal conditions for the band convection triggering.

The very fast storms that are produced in this zone are known as “C´evenols”. Three particular synoptic conditions have been identified, leading to a generalized southern flow that, depending on the flow incidence, may involve the Aude sub-region (as in the 1999 event) or the northern Gard subregion (as in the 2002 event). With this configuration, the low and warm layers coming from the Mediterranean are advected towards the C´evennes relief. These layers are extremely charged especially during the fall months, enforcing the atmosphere instability from the sea shore to the C´evennes foothills. The complex topography in the relief region is then a factor for the convection triggering and for the moist flux convergence (Ducrocq et al., 2008).

In some particular situations, the convection becomes stationary because of several concomitant ingredients: the relief forces the conditionally unstable and moist low-level jet to raise, generating stable systems that remain in the same location until changes in the synoptic conditions occur. A similar situation, not directly related to the orography, is called “cold-pool”, and is generated by the precipitation evaporation. This phenomenon is at the origin of the catastrophic flood occurred in 2002 over the Gard region (Delrieu et al., 2005).

**Table of contents :**

**1 General context **

1.1 Purpose of the thesis

1.1.1 Rainfall measurement uncertainties

1.1.2 Extreme rainfall modeling

1.1.3 Scale-invariance of heavy time rainfall

1.1.4 Scale-invariance of heavy spatial rainfall

1.1.5 Scale-invariance of rainfall fields in space-time

1.2 Geographical context

1.3 Meteo-hydrological context

1.3.1 Climatic features

1.4 Hydrological context

1.4.1 Ard`eche basin

1.4.2 C`eze basin

1.4.3 Gard basin

1.4.4 Vidourle basin

1.5 Measurement network

1.5.1 Point-rainfall data

1.5.2 Radar-estimated rainfall fields

1.5.3 Disdrometer data

**2 Uncertainties in the extreme rainfall measurement **

2.1 Introduction

2.2 Role of the sampling frequency on the statistics of extremes

2.2.1 Moving-window and fixed-window sampling

2.2.2 Uncertainties associated to the measurement resolution

2.3 Tipping-bucket rain gauge measurement

2.4 Ground measurement network for spatial estimations: limits

2.5 Conclusion

**Part II Theoretical background**

**3 Statistics of extreme point-rainfall intensities **

3.1 Introduction

3.2 Frequency and return period: definitions

3.3 Extreme Value Theory

3.4 The role of independence in Extreme value analysis

3.5 Block-maxima analysis

3.6 Analysis of exceedances: POT

3.6.1 GPD fitting on synthetic series

3.6.2 GPD fitting on real series

3.7 Comparison Block-Maxima – POT

3.8 Point process: an unified framework for extreme analysis

3.9 Stationarity of the rainfall series

3.10 Conclusion

**4 Geostatistics **

4.1 Introduction

4.2 Stationarity of random functions

4.3 Theoretical variogram

4.4 Sample variogram

4.4.1 Anisotropic variogram

4.4.2 Variogram of an intrinsic random function

4.4.3 Indicator variogram

4.4.4 Climatological variogram

4.4.5 Variogram models

4.5 Interpolation of point data

4.5.1 Kriging

4.5.2 Simple kriging

4.5.3 Ordinary kriging

4.5.4 Universal kriging

4.6 Conclusion

**5 Scale invariance and self-similarity **

5.1 Introduction

5.2 Scaling in geometry: fractals

5.3 Fractals in nature

5.4 The origin of scaling in nature: turbulence

5.5 Generalization of the fractal concept: Multifractals

5.5.1 Scaling of the Generalized Structure Function ζ

5.5.2 Moment scaling analysis

5.6 Towards an unified multifractal formalism

5.7 Multifractal spectrum estimation

5.7.1 Moment-based estimation

5.7.2 Wavelet Estimation

5.8 Scale-invariance of spatial fields

5.9 Scale-invariance of time series

5.9.1 Long-range correlation: the first evidence of long-memory systems

5.9.2 Scaling range of rainfall time series

5.9.3 Point-rainfall scale-invariant models

5.10 Scale invariance of space-time rainfall

5.10.1 The “Frozen Turbulence” hypothesis

5.10.2 The concept of dynamic scaling

5.11 Scale invariance of extreme point-rainfall: IDF curves

5.11.1 IDF scaling

5.12 Application of Multiplicative Cascades

5.12.1 Other downscaling techniques

5.12.2 Multiplicative Cascades

5.12.3 Bare and dressed quantities

5.12.4 Canonical and Micro-canonical cascades

5.12.5 Discrete and continuous cascades

5.12.6 Cascade Implementation

5.12.7 Implementation of the multiplicative cascades

5.13 Conclusion

**6 Spatial Rainfall Extremes **

6.1 Introduction

6.2 From point to spatial maxima: ARF

6.2.1 Background

6.2.2 Geostatistically-based ARF approaches

6.2.3 Stochastic approaches

6.2.4 Scale-invariant ARF

6.3 Max-stable spatial-maxima modeling

6.4 Conclusion

Part III Results

**7 Heavy tails of rainfall distributions **

7.1 Introduction

7.2 Article: Scaling properties of heavy rainfall at short durations: a regional analysis

8 A scale-invariant Intensity-Duration-Frequency model

8.1 Introduction

8.2 Article: Intensity-Duration-Frequency curves in a GEV scale-invariant framework

**9 Space-time scaling of rainfall events: the September 2002 storm **

9.1 Introduction

9.2 Outline of the study

9.3 Data

9.4 Scale Invariance: Generalized structure Function

9.5 Scale invariance in Space-Time

9.5.1 Connections with Geostatistics

9.5.2 Space-time scale-invariance

9.5.3 Evaluation of dynamic scaling

9.6 Final remarks

9.7 Conclusion and perspectives

**10 Qualification of Meso-scale meteorological simulations **

10.1 Introduction

10.2 Severity Diagrams

10.3 Rainfall intensity diagrams: an indicator of the true model resolution

10.4 Article. Severity diagrams: a new approach for the multi-scale evaluation of extreme rainfall events

10.5 Severity diagrams and ensemble simulations

10.5.1 Severity Diagram of observed fields

10.5.2 AROME ensemble 1: Variability of Boundary Conditions

10.5.3 AROME ensemble 2: Variability of Initial Conditions

10.6 Conclusion

**Part IV Conclusion and Perspectives**

**11 Conclusion **

**12 Perspectives**