# Multifractal Comparison of Reflectivity and Polarimetric Rainfall Data from C- and X-Band Radars and Respective Hydrological Responses of a Complex Catchment Model

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## Abstract

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## 1. Introduction

^{2}). The study was carried out in the framework of the RainGain project [37].

^{2}upstream catchment of the Bièvre River in the Ile-de-France region. This catchment is an example of a peri-urban area, some parts of which are highly urbanised. Veolia is the company chosen since 1991 by the local authority SIAVB (“Syndicat Intercommunal pour l’Assainissement de la Vallée de la Bièvre”—Bièvre Valley Intermunicipal Sanitation Office) to design, install and operate a real-time control system of the river over the area. Hence, the study is performed with Veolia’s operational model, Optim Sim. Instead of suffering limitations, the aforementioned methodological goal and the present study benefit from cross-fertilisation of research and operational hydrology, whereas they have both suffered from a long-lasting divorce [38]. Furthermore, the potential decentralisation of meteorological and hydrological data collection, processing and distribution should enable most of businesses to optimise their operational management, because often more than 60% of their activities are weather-sensitive.

## 2. Materials and Methods

#### 2.1. Case Study Site

#### 2.2. Selected Rainfall Events and Three Data Types

- The SIAVB network of six tipping bucket rain gauges, being distributed over the catchment;
- The Météo-France polarimetric C-band radar of Trappes, being located in a direct proximity (~0–20 km) of the catchment;
- The ENPC polarimetric X-band radar of Champs-sur-Marne, with distances ranging between 25 to 45 km.

#### 2.3. Radar Data Processing

#### 2.3.1. From Météo-France

#### 2.3.2. From ENPC

^{−1}), this change increases the rainfall totals by about 40%. This simple analysis highlights how critical the choice of a and b parameters is, furthermore considering the wide range of possible values discussed in the literature (examples: [41,45,46]). By comparing the middle and right graphs, the second observation is that simply filtered KDP results in a slight increase in the rainfall estimates. Similar behaviour was observed for two other events. Selecting the appropriate radar algorithms remains a challenging, methodological, observational question that has far-reaching implications on water management. In Section 3.1.1, we innovatively use multifractal analysis to help to choose which X-band radar product will be used for the hydrological modelling. To the knowledge of the authors, this is the first time that such methodology has been developed.

#### 2.4. Hydrological Model

^{2}. This model is integrated in the Optim Sim platform, an offline tool developed by Veolia that imitates the actual regulation of the storage basins either at the local or catchment scale. It should be mentioned that the settings of neither InfoWorks CS model nor Optim Sim are fully up to date. For instance, a recent modification of the networks that consisted in the removal of the Vilgénis basin to restore the natural flow of the river (near “Vilgénis” rain gauge P6 on Figure 1) is not yet taken into account in the models. This difficulty pushes us in this study to focus on events without active regulation, i.e., when gates positions remain unchanged and therefore to focus on the differences associated with rainfall. There are two simulation modes in Optim Sim: the “replay mode” and the “forecasting mode”. The first one enables to replay past events by extracting data from the linked database of the six rain gauges of the SIAVB. The rain rates for each sub-catchment are obtained using the Thiessen polygons technique. Then the simulations and observations (flood depth or flow at the measuring points) can be compared and analysed. It should be noted that, during Optim Sim simulations in “replay mode”, the positions of gates are not necessarily the same as the ones resulting from the regulation during the real events. Since no other source of rainfall data can be used with this mode, we used the “forecasting mode”. This second mode simulates the behaviour of the river by introducing rainfall data from different possible sources, notably from (fully distributed) C-band radar measurements, although being converted in semi-distributed ones (i.e., a unique rainfall time series per catchment). It is also possible to use the forecasting mode for replays by merely setting past dates, thereby making it possible to compare the simulations with measured flow observations. Rainfall data can be input in three ways: an average intensity (mm/h) over all sub-catchments; a single average intensity time series over all sub-catchments; and an average intensity time series over each sub-catchment. In the last case, which we chose, a text file with a column containing rainfall intensity in mm/h with 5-min time steps has to be generated for each sub-catchment.

#### 2.5. Short Recap of Multifractals and Classical Error Metrics

- The correlation coefficient ($Corr\in \left[-1,\text{}1\right]$) measures the strength and the direction of the linear relationship between the both time series:$$Corr\left(A,B\right)=\frac{{{\displaystyle \sum}}_{i}\left[\left({A}_{i}-{\overline{A}}_{i}\right)\xb7\left({B}_{i}-{\overline{B}}_{i}\right)\right]}{\sqrt{{{\displaystyle \sum}}_{i}{\left({A}_{i}-{\overline{A}}_{i}\right)}^{2}}\xb7\sqrt{{{\displaystyle \sum}}_{i}{\left({B}_{i}-{\overline{B}}_{i}\right)}^{2}}},$$
- The Nash–Sutcliffe Efficiency ($Nash\in \text{}\left(-\infty ,1\right]$) is the most commonly used indicator to quantify performance of models in urban hydrology. It measures how well the model outputs ($A$) reproduce the observations ($B)$ in comparison to a model that only uses the mean of the observed data. It is calculated as:$$Nash\left(A,B\right)=1-\text{}\frac{{{\displaystyle \sum}}_{i}{({B}_{i}-{A}_{i})}^{2}}{{{\displaystyle \sum}}_{i}{({B}_{i}-{\overline{B}}_{i})}^{2}},$$
- The Root-Mean-Square Error (RMSE) is used in urban hydrology to quantify errors between two time series. It measures the deviation of predictions from observed value (calculated as the square root of average square deviation):$$RMSE\left(A,B\right)=\sqrt{\frac{1}{N}{{\displaystyle \sum}}_{i}{\left({A}_{i}-{B}_{i}\right)}^{2}}.$$

## 3. Results

#### 3.1. Direct Comparison of Rainfall

#### 3.1.1. Multifractal Analysis

^{2}, being used as a metrics of scaling behaviour, are lower for X-band radar data. Somewhat similar scaling behaviour is found for the other two rainfall events.

^{2}(see Figure 5b,d, for the event of 12–13 September 2015), being used as a metrics of the scaling behaviour, are lower for the X-band radar data, mainly because changes in scaling may occur over smaller scales that are not available with the C-band measurements. We tested that two scaling regimes with distinct linear fits in a log–log plot yield two distinct pairs of UM parameters for the X-band spatial rainfall. While the multifractality parameter estimated over large scales (1–64 km) becomes closer to that obtained for the C-band radar data (on the same range), the DTM estimator gives spurious multifractality parameter (>2) over the small scales (0.25–1 km, unknown for the C-band radar). This could be due to the emergent properties of rainfall extremes at small scales. Indeed, a well-known basic property of precipitation is that small-scale extremes (short duration or/and size, e.g., heavy rainfall episodes) can drastically influence much larger scales (e.g., yearly statistics and even climate) to the point of creating heavy tails for the probability of larger scale extremes. This basic feature remains out of reach from (quasi-) linear models (e.g., the still used Scott–Newman model and variants), whereas it is generic in multifractal cascade models.

#### 3.1.2. Rainfall Estimates over the Catchment

#### 3.2. Hydrological Comparison

^{3}/s, being well reproduced with the three rainfall products, which is confirmed by the error metrics (Figure 12): $Nash\approx 1$, $Corr\approx 1$ and $RMSE<0.12$. For the other observation points, the three rainfall products reproduce the observed flow dynamic (correlation values are approximately 0.8), with important differences in the obtained volumes (RMSE errors are between 0.35 and 0.6 for “Arcades de Buc” and “Moulin Vauboyen” and around 1.4 for “Pont Cambaceres”; Nash values are between −1.5 and 0.2). The fact that regulation at the catchment scale is not mimicked makes comparison with observations not relevant, as discussed above. For this event, there is a tendency of the X-band data to generate slightly smaller flows than the other two products, well visible on all curves of the figure. This tendency is more pronounced for the first half of the event (until 13 September 2015 at 07:00) than for the second one.

^{3}/s regulated target value because additional water coming from an overflow at Val d’Or is input in the model for this event. This should be investigated more precisely in future work. The differences between the three rainfall products on the simulated flows are more pronounced for this event than for the other two, as it can be noticed in Figure 12 (with bigger gaps between the three points for this event). It should also be noted that there are some changes according to the observation point and time. For instance, at the beginning of the event (before 16 September 2015 at ~ 05:00), rain gauges yield a greater flow at “Arcades de Buc” and “Moulin Vauboyen” whereas it is at “Pont Cambaceres” for the X-band radar. At this location, the C-band rainfall data generate a peak that is found neither with other rainfall data nor on observations. In the period between 08:00 and 16:50 on 16 September 2015, C-band data yield greater flows at “Arcades de Buc” and “Moulin Vauboyen” (more pronounced for the beginning of this period) whereas it is the rain gauges at “Pont Cambaceres”. Here the simulated flows are much smaller with X-band data. As for the 12–13 September event, it is not relevant to compare with actual measurements due to the problematic simulation of regulation at the catchment scale that was actually deployed.

## 4. Discussion

^{2}spatial resolution, having the highest quality indicator over the pixel’s vertical and are adjusted to rain gauges. In addition, the SIAVB rain gauge network provided the third set of rainfall data studied. Furthermore, the semi-distributed model InfoWorks CS was used with sub-catchments of average size of 2 km, which highlights its limitations to take into account the high spatial variability of the rainfall.

## Supplementary Materials

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Loukas, A.; Llasat, M.-C.; Ulbrich, U. Preface: “Extreme events induced by weather and climate change: Evaluation, forecasting and proactive planning”. Nat. Hazards Earth Syst. Sci.
**2010**, 10, 1895–1897. [Google Scholar] [CrossRef] [Green Version] - World Meteorological Organization (WMO). Guide to Meteorological Instruments and Methods of Observation WMO-No. 8. 2014. Available online: http://www.wmo.int/pages/prog/www/IMOP/CIMO-Guide.html (accessed on 14 February 2017).
- National Research Council of the National Academies. Urban Meteorology: Forecasting, Monitoring, and Meeting Users’ Need; National Academy Press: Washington, WA, USA, 2012. [Google Scholar]
- SIAVB. Available online: http://www.siavb.fr/riviere.aspx (accessed on 4 November 2016).
- Diss, S.; Testud, J.; Lavabre, J.; Ribstein, P.; Moreau, E.; Parent du Chatelet, J. Ability of a dual polarized X-band radar to estimate rainfall. Adv. Water Resour.
**2009**, 32, 975–985. [Google Scholar] [CrossRef] - Tabary, P.; Boumahmoud, A.-A.; Andrieu, H.; Thompson, R.J.; Illingworth, A.J.; Le Bouar, E.; Testud, J. Evaluation of two “integrated” polarimetric Quantitative Precipitation Estimation (QPE) algorithms at C-band. J. Hydrol.
**2011**, 405, 248–260. [Google Scholar] [CrossRef] - Emmanuel, I.; Andrieu, H.; Tabary, P. Evaluation of the new French operational weather radar product for the field of urban hydrology. Atmos. Res.
**2012**, 103, 20–32. [Google Scholar] [CrossRef] - Figueras i Ventura, J.; Boumahmoud, A.-A.; Fradon, B.; Dupuy, P.; Tabary, P. Long-term monitoring of French polarimetric radar data quality and evaluation of several polarimetric quantitative precipitation estimators in ideal conditions for operational implementation at C-band. Q. J. R. Meteorol. Soc.
**2012**, 138, 2212–2228. [Google Scholar] [CrossRef] - Figueras i Ventura, J.; Tabary, P. The New French Operational Polarimetric Radar Rainfall Rate Product. J. Appl. Meteorol. Climatol.
**2013**, 52, 1817–1835. [Google Scholar] [CrossRef] - Tabary, P. The new French operational radar rainfall product, Part I: Methodology. Weather Forecast.
**2007**, 22, 393–408. [Google Scholar] [CrossRef] - Ciach, G.J.; Habib, E.; Krajewski, W.F. Zero-covariance hypothesis in the error variance separation method of radar rainfall verification. Adv. Water Resour.
**2003**, 26, 573–580. [Google Scholar] [CrossRef] - Gires, A.; Tchiguirinskaia, I.; Schertzer, D.; Schellart, A.; Berne, A.; Lovejoy, S. Influence of small scale rainfall variability on standard comparison tools between radar and rain gauge data. Atmos. Res.
**2014**, 138, 125–138. [Google Scholar] [CrossRef] - Wyss, J.; Williams, E.R.; Bras, R.L. Hydrologic modeling of New England river basins using radar rainfall data. J. Geophys. Res.
**1990**, 95, 2143–2152. [Google Scholar] [CrossRef] - Sun, X.; Mein, R.G.; Keenan, T.D.; Elliot, J.F. Flood estimation using radar and raingauge data. J. Hydrol.
**2000**, 239, 4–18. [Google Scholar] [CrossRef] - Germann, U.; Berenguer, M.; Sempere-Torres, D.; Zappa, M. REAL—Ensemble radar precipitation estimation for hydrology in a mountainous region. Q. J. R. Meteorol. Soc.
**2009**, 135, 445–456. [Google Scholar] [CrossRef] [Green Version] - Lobligeois, F. Mieux Connaître la Distribution Spatiale des Pluies Améliore-t-il la Modélisation des Crues? Diagnostic sur 181 Bassins Versants Français. Ph.D. Thesis, Agro ParisTech, Paris, France, 2014. [Google Scholar]
- Einfalt, T.; Denoeux, T.; Jacquet, G. A radar rainfall forecasting method designed for hydrological purposes. J. Hydrol.
**1990**, 114, 229–244. [Google Scholar] [CrossRef] - Vieux, B.E.; Bedient, P.B. Assessing urban hydrologic prediction accuracy through event reconstruction. J. Hydrol.
**2004**, 299, 217–236. [Google Scholar] [CrossRef] - Einfalt, T.; Arnbjerg-Nielsen, K.; Golz, C.; Jensen, N.-E.; Quirmbach, M.; Vaes, G.; Vieux, B. Towards a roadmap for use of radar rainfall data in urban drainage. J. Hydrol.
**2004**, 299, 186–202. [Google Scholar] [CrossRef] - Berenguer, M.; Corral, C.; Sánchez-Diezma, R.; Sempere-Torres, D. Hydrological Validation of a Radar-Based Nowcasting Technique. J. Hydrometeorol.
**2005**, 6, 532–549. [Google Scholar] [CrossRef] [Green Version] - Liguori, S.; Rico-Ramirez, M.A.; Schellart, A.N.A.; Saul, A.J. Using probabilistic radar rainfall nowcasts and NWP forecasts for flow prediction in urban catchments. Atmos. Res.
**2012**, 103, 80–95. [Google Scholar] [CrossRef] - Ichiba, A. X-band Radar Data and Predictive Management in Urban Hydrology. Ph.D. Thesis, Université Paris-Est, Paris, France, 2016. [Google Scholar]
- Peleg, N.; Blumensaat, F.; Molnar, P.; Fatichi, S.; Burlando, P. Partitioning the impacts of spatial and climatological rainfall variability in urban drainage modeling. Hydrol. Earth Syst. Sci.
**2017**, 21, 1559–1572. [Google Scholar] [CrossRef] - Schilling, W. Rainfall data for urban hydrology: What do we need? Atmos. Res.
**1991**, 27, 5–21. [Google Scholar] [CrossRef] - Aronica, G.; Cannarozzo, M. Studying the hydrological response of urban catchments using a semi-distributed linear non-linear model. J. Hydrol.
**2000**, 238, 35–43. [Google Scholar] [CrossRef] - Berne, A.; Delrieu, G.; Creutin, J.-D.; Obled, C. Temporal and spatial resolution of rainfall measurements required for urban hydrology. J. Hydrol.
**2004**, 299, 166–179. [Google Scholar] [CrossRef] - Segond, M.L.; Neokleous, N.; Makropoulos, C.; Onof, C.; Maksimović, Č. Simulation and spatio-temporal disaggregation of multi-site rainfall data for urban drainage applications. Hydrol. Sci. J.
**2007**, 52, 917–935. [Google Scholar] [CrossRef] - Schellart, A.N.A.; Shepherd, W.J.; Saul, A.J. Influence of rainfall estimation error and spatial variability on sewer flow prediction at a small urban scale. Adv. Water Resour.
**2012**, 45, 65–75. [Google Scholar] [CrossRef] - Gires, A.; Onof, C.; Maksimović, Č.; Schertzer, D.; Tchiguirinskaia, I.; Simoes, N. Quantifying the impact of small scale unmeasured rainfall variability on urban runoff through multifractal downscaling: A case study. J. Hydrol.
**2012**, 442, 117–128. [Google Scholar] [CrossRef] - Ochoa-Rodriguez, S.; Wang, L.-P.; Gires, A.; Pina, R.D.; Reinoso-Rondinel, R.; Bruni, G.; Ichiba, A.; Gaitan, S.; Cristiano, E.; van Assel, J.; et al. Impact of spatial and temporal resolution of rainfall inputs on urban hydrodynamic modeling outputs: A multi-catchment investigation. J. Hydrol.
**2015**, 531, 389–407. [Google Scholar] [CrossRef] - Simões, N.E.; Ochoa-Rodriguez, S.; Wang, L.-P.; Pina, R.D.; Marques, A.S.; Onof, C.; Leitão, J.P. Stochastic Urban Pluvial Flood Hazard Maps Based upon a Spatial-Temporal Rainfall Generator. Water
**2015**, 7, 3396–3406. [Google Scholar] [CrossRef] [Green Version] - Gires, A.; Giangola-Murzyn, A.; Abbes, J.-B.; Tchiguirinskaia, I.; Schertzer, D.; Lovejoy, S. Impacts of small scale rainfall variability in urban areas: A case study with 1D and 1D/2D hydrological models in a multifractal framework. Urban Water J.
**2014**, 12, 607–617. [Google Scholar] [CrossRef] - El-Tabach, E.; Tchiguirinskaia, I.; Mahmood, O.; Schertzer, D. Multi-Hydro: A spatially distributed numerical model to assess and manage runoff processes in peri-urban watersheds. In Proceedings of the Final Conference of the COST Action C22, Road map towards a flood resilient urban environment, Paris, France, 26–27 November 2009; Pascheet, E., Evelpidou, N., Zevenbergen, C., Ashley, R., Garvin, S., Eds.; Hamburger Wasserbau-Schriftien: Hamburg, Germany, 2009. [Google Scholar]
- Fewtrell, T.J.; Duncan, A.; Sampson, C.C.; Neal, J.C.; Bates, P.D. Benchmarking urban flood models of varying complexity and scale using high resolution terrestrial LiDAR data. Phys. Chem. Earth
**2011**, 36 Pt A/B/C, 281–291. [Google Scholar] [CrossRef] - Giangola-Murzyn, A.; Gires, A.; Hoang, C.T.; Tchiguirinskaia, I.; Schertzer, D. Multi-Hydro modelling to assess flood resilience across scales, case study in the Paris region. In Proceedings of the 9th International Conference on Urban Drainage Modelling, Belgrade, Serbia, 4–7 September 2012. [Google Scholar]
- Tramblay, Y.; Bouvier, C.; Crespy, A.; Marchandise, A. Improvement of flash flood modelling using spatial patterns of rainfall: A case study in southern France. In Proceedings of the Sixth World FRIEND Conference, Montpellier, France, 7–10 October 2014; IAHS Publ.: Fez, Morocco, 2010; p. 340. [Google Scholar]
- RainGain Project. Available online: www.raingain.eu (accessed on 21 July 2016).
- Schertzer, D.; Tchiguirinskaia, I.; Lovejoy, S.; Hubert, P. No monsters, no miracles: In nonlinear sciences hydrology is not an outlier! Hydrol. Sci. J.
**2010**, 55, 965–979. [Google Scholar] [CrossRef] - Réseau Hydrographique. Available online: Https://www.data.gouv.fr/fr/datasets/reseau-hydrographique-idf/ (accessed on 14 December 2015).
- Gires, A.; Tchiguirinskaia, I.; Schertzer, D. Multifractal comparison of the outputs of two optical disdrometers. Hydrol. Sci. J.
**2016**, 61, 1641–1651. [Google Scholar] [CrossRef] - Marshall, J.S.; Palmer, W.M. The distribution of raindrop with size. J. Meteorol.
**1948**, 5, 165–166. [Google Scholar] [CrossRef] - Parent du Châtelet, J. ARAMIS, le réseau Français de radars pour la surveillance des précipitations. La Météorologie
**2003**, 40, 44–52. [Google Scholar] [CrossRef] - Gourley, J.J.; Tabary, P.; Parent du Chatelet, J. A fuzzy logic algorithm for the separation of precipitating from nonprecipitating echoes using polarimetric radar observations. J. Atmos. Ocean. Technol.
**2007**, 24, 1439–1451. [Google Scholar] [CrossRef] - Selex. Selex METEOR manual; Selex ES GmbH: Neuss, Germany, 2015. [Google Scholar]
- Fulton, R.A.; Bredienbach, J.P.; Seo, D.-J.; Miller, D.A.; O’Bannon, T. The WSR-88 rainfall algorithm. Weather Forecast.
**1998**, 13, 377–395. [Google Scholar] [CrossRef] - Matrosov, S.Y.; Clark, K.A.; Martner, B.E.; Tokay, A. X-Band Polarimetric Radar Measurements of Rainfall. J. Appl. Meteorol.
**2002**, 41, 941–952. [Google Scholar] [CrossRef] - Wallingford Software. InfoWorks CS Help Documentation; HR Wallingford Group: Wallingford, UK, 2009. [Google Scholar]
- Clarke, D.L. Analytical Archaeology; Methuen: London, UK, 1968. [Google Scholar]
- Schertzer, D.; Lovejoy, S. Physical modeling and Analysis of Rain and Clouds by Anisotropic Scaling Multiplicative Processes. J. Geophys. Res.
**1987**, 92, 9693–9714. [Google Scholar] [CrossRef] - Gupta, V.K.; Waymire, E. A Statistical Analysis of Mesoscale Rainfall as a Random Cascade. J. Appl. Meteorol.
**1993**, 32, 251–267. [Google Scholar] [CrossRef] - Harris, D.; Menabde, M.; Seed, A.; Austin, G. Multifractal characterization of rain fields with a strong orographic influence. J. Geophys. Res.
**1996**, 101, 26405–26414. [Google Scholar] [CrossRef] - Marsan, D.; Schertzer, D.; Lovejoy, S. Causal space-time multifractal processes: Predictability and forecasting of rain fields. J. Geophys. Res.
**1996**, 101, 26333–26346. [Google Scholar] [CrossRef] - Olsson, J.; Niemczynowicz, J. Multifractal analysis of daily spatial rainfall distributions. J. Hydrol.
**1996**, 187, 29–43. [Google Scholar] [CrossRef] - De Lima, M.I.P.; Grasman, J. Multifractal analysis of 15-min and daily rainfall from a semi-arid region in Portugal. J. Hydrol.
**1999**, 220, 1–11. [Google Scholar] [CrossRef] - Deidda, R. Rainfall downscaling in a space-time multifractal framework. Water Resour. Res.
**2000**, 36, 1779–1794. [Google Scholar] [CrossRef] [Green Version] - Pathirana, A.; Herath, S. Multifractal modeling and simulation of rain fields exhibiting spatial heterogeneity. Hydrol. Earth Syst. Sci.
**2002**, 6, 695–708. [Google Scholar] [CrossRef] - Biaou, A.; Hubert, P.; Schertzer, D.; Tchiguirinskaia, I.; Bendjoudi, H. Fractals, multifractals et prévision des précipitations. Sud Sci. Technol.
**2003**, 10, 10–15. [Google Scholar] - Pathirana, A.; Herath, S.; Yamada, T. On the modeling of temporal correlations in spatial-cascade rainfall downscaling. In Weather Radar Information and Distributed Hydrological Modeling; Tachikawa, Y., Vieux, B.E., Georgakakos, K.P., Nakakita, E., Eds.; IAHS Publication: Sapporo, Japan, 2003; Volume 282, pp. 74–82. [Google Scholar]
- Ferraris, L.; Gabellani, S.; Parodi, U.; Rebora, N.; von Hardenberg, J.; Provenzale, A. Revisiting multifractality in rainfall fields. J. Hydrometeorol.
**2003**, 4, 544–551. [Google Scholar] [CrossRef] - Ferraris, L.; Gabellani, S.; Rebora, N.; Provenzale, A. A comparison of stochastic models for spatial rainfall downscaling. Water Resour. Res.
**2003**, 39, 1368–1384. [Google Scholar] [CrossRef] - Macor, J.; Schertzer, D.; Lovejoy, S. Multifractal Methods Applied to Rain Forecast Using Radar Data. La Houille Blanche
**2007**, 4, 92–98. [Google Scholar] [CrossRef] - Royer, J.F.; Biaou, A.; Chauvin, F.; Schertzer, D.; Lovejoy, S. Multifractal analysis of the evolution of simulated precipitation over France in a climate scenario. C. R. Geosci.
**2008**, 340, 431–440. [Google Scholar] [CrossRef] - Nykanen, D.K. Linkages between Orographic Forcing and the Scaling Properties of Convective Rainfall in Mountainous Regions. J. Hydrometeorol.
**2008**, 9, 327–347. [Google Scholar] [CrossRef] - De Montera, L.; Barthes, L.; Mallet, C.; Gole, P. The effect of rain-no rain intermittency on the estimation of the universal multifractals model parameters. J. Hydrometeorol.
**2009**, 10, 493–506. [Google Scholar] [CrossRef] - Langousis, A.; Veneziano, D.; Furcolo, P.; Lepore, C. Multifractal rainfall extremes: Theoretical analysis and practical estimation. Chaos Solitons Fractals
**2009**, 39, 1182–1194. [Google Scholar] [CrossRef] - Tchiguirinskaia, I.; Schertzer, D.; Hoang, C.-T.; Lovejoy, S. Multifractal study of three storms with different dynamics over the Paris region. In Proceedings of the Weather Radar and Hydrology, Exeter, UK, 18–21 April 2011; Moore, J., Cole, S., Illingworth, A., Eds.; IAHS Publ.: Exeter, UK, 2011; 351, pp. 421–426. [Google Scholar]
- Hoang, C.-T.; Tchiguirinskaia, I.; Schertzer, D.; Lovejoy, S. Caractéristiques multifractales et extrêmes de la précipitation à haute resolution, application à la détection du changement climatique. J. Water Sci.
**2014**, 27, 205–216. [Google Scholar] [CrossRef] - Schertzer, D.; Lovejoy, S. Universal Multifractals do Exist! J. Appl. Meteorol.
**1997**, 36, 1296–1303. [Google Scholar] [CrossRef] - Schertzer, D.; Lovejoy, S. Multifractals, generalized scale invariance and complexity in geophysics. Int. J. Bifurcat. Chaos
**2011**, 21, 3417–3456. [Google Scholar] [CrossRef] - Lavallée, D.; Lovejoy, S.; Ladoy, P. Nonlinear variability and landscape topography: Analysis and simulation. In Fractals in Geography; De Cola, L., Lam, N., Eds.; Prentice-Hall: Englewood Cliffs, NJ, USA, 1993; pp. 171–205. [Google Scholar]
- Hoang, C.T. Prise en Compte des Fluctuations Spatio-Temporelles Pluies-Débits Pour une Meilleure Gestion de la Ressource en eau et une Meilleure Évaluation des Risques. Ph.D. Thesis, Université Paris-Est, France, 2011. [Google Scholar]
- Hittinger, F. Intercomparaison des incertitudes dans l’Analyse de Fréquence de Crues classique et l’Analyse Multifractale de Fréquence de Crues. Master’s Thesis, Ecole Nationale Supérieure d’Hydraulique et de Mécanique de Grenoble, Grenoble, France, 2008. [Google Scholar]
- Hoang, C.T. Analyse fréquentielle classique et multifractale des 10 séries pluviométriques à haute résolution. Master’s Thesis, Université P. & M. Curie, Paris, France, 2008. [Google Scholar]
- Gires, A.; Tchiguirinskaia, I.; Schertzer, D.; Lovejoy, S. Influence of the zero-rainfall on the assessment of the multifractal parameters. Adv. Water Resour.
**2012**, 45, 13–25. [Google Scholar] [CrossRef] - Sevruk, B.; Hamon, W.R. International Comparison of National Precipitation Gauges with a Reference Pit Gauge; Secretariat of the World Meteorological Organization: Geneva, Switzerland, 1984. [Google Scholar]
- Fankhauser, R. Influence of systematic errors from tipping bucket rain gauges on recorded rainfall data. Water Sci. Technol.
**1998**, 37, 121–129. [Google Scholar] [CrossRef] - Habib, E.; Krajewski, W.; Kruger, A. Sampling errors of Tipping-Bucket rain gauge measurements. J. Hydrol. Eng.
**2001**, 6, 159–166. [Google Scholar] [CrossRef] - Ciach, G.J. Local random errors in Tipping-Bucket rain gauge measurements. J. Atmos. Ocean. Technol.
**2003**, 20, 752–759. [Google Scholar] [CrossRef] - Einfalt, T.; Jessen, M.; Mehlig, B. Comparison of radar and raingauge measurements during heavy rainfall. Water Sci. Technol.
**2005**, 51, 195–201. [Google Scholar] [PubMed] - Gabella, M.; Orione, F.; Zambotto, M.; Turso, S.; Fabbo, R.; Pillon, A. A Portable Low Cost X-band RADAR for Rainfall Estimation in Alpine Valleys; FORALPS Technical Report; Universita degli Studi di Trento: Trento, Italy, 2008; pp. 1–52. [Google Scholar]
- Allegretti, M.; Bertoldo, S.; Prato, A.; Lucianaz, C.; Rorato, O.; Notarpietro, R.; Gabella, M. X-Band Mini Radar for Observing and Monitoring Rainfall Events. Atmos. Clim. Sci.
**2012**, 2, 290–297. [Google Scholar] [CrossRef] - Borup, M.; Grum, M.; Linde, J.J.; Mikkelsen, P.S. Dynamic gauge adjustment of high-resolution X-band radar data for convective rain storms: Model-based evaluation against measured combined sewer overflow. J. Hydrol.
**2016**, 539, 687–699. [Google Scholar] [CrossRef] - Bringi, V.N.; Chandrasekar, V. Polarimetric Doppler Weather Radar: Principles and Applications; Cambridge University Press: Cambridge, UK, 2001. [Google Scholar]
- Anagnostou, M.N.; Kalogiros, J.; Anagnostou, E.N.; Tarolli, M.; Papadopoulos, A.; Borga, M. Performance evaluation of high resolution rainfall estimation by X-band dual-polarization radar for flash flood applications in mountainous basins. J. Hydrol.
**2010**, 394, 4–16. [Google Scholar] [CrossRef] - Otto, T.; Russchenberg, H.W.J.; Leijnse, H. Advances in polarimetric X-band weather radar. In Proceedings of the 9th European Radar Conference, Amsterdam, The Netherlands, 31 October–2 November 2012; pp. 174–177, ISBN 978-2-87487-029-3. [Google Scholar]
- Otto, T.; Russchenberg, H.W.J. High-resolution polarimetric X-band weather radar observations at the Cabauw Experimental Site for Atmospheric Research. Geosci. Data J.
**2014**, 1, 7–12. [Google Scholar] [CrossRef] - Chandrasekar, V.; Baldini, L.; Bharadwaj, N.; Smith, P.L. Calibration procedures for global precipitation-measurement ground-validation radars. URSI Radio Sci. Bull.
**2015**, 88, 45–73. [Google Scholar] [CrossRef] - Da Silva Rocha Paz, I.; Ichiba, A.; Skouri-Plakali, I.; Lee, J.; Gires, A.; Tchiguirinskaia, I.; Schertzer, D. Challenges with space-time rainfall in urban hydrology highlighted with a semi-distributed model using C-band and X-band radar data. In EGU General Assembly 2017, Vienna, Austria, 23–28 April 2017; Geophysical Research Abstracts: Vienna, Austria, 2017; p. 19, EGU2017-327. [Google Scholar]

**Figure 1.**Illustration of the Bièvre catchment area with its representation in 27 sub-catchments used in InfoWorks CS. Five of them—BISAN 1, BISAN 2, BISAN 3, BINSAN 2 and VAL D’OR—belong to the CASQY-Bièvre catchment while the others belong to the SIAVB-Bièvre catchment. Locations of six rain gauges over the SIAVB-Bièvre catchment and four measurement points are shown. The altitude scale is also represented, in meters.

**Figure 2.**Illustration of rainfall measurement devices available over the Bièvre catchment. The square area (red dashed line) is the 128 × 128 km

^{2}area, covered by the two radars.

**Figure 3.**Disdrometer data output for the 16 September event. (

**a**) Temporal evolution of the rain rate with 30 s time steps; (

**b**) temporal evolution of the drop size distribution with 30 s time steps; (

**c**) scatter plot of the Marshall–Palmer relation (Equation (1)) and corresponding orthogonal regression for the four selected periods.

**Figure 4.**Time evolution of accumulated X-band rainfall during the event of 12–13 September 2015 over six catchments containing the rain gauge: GEN (P1 Geneste/GEN), MARAM (P2 Trou Salé/TROU), SYGAM (P3 Loup Pendu/LOUP), VAUHAM (P4 Sablons/SABLO), JOUY3 (P5 Vauboyen/VAUB) and VERR2 (P6 Vilgénis/VILG). The three DPSRI rainfall products at 1.5 km were obtained with: FIR filter and Z–R parameters a = 200 and b = 1.6 (

**left**); FIR filter and Z–R parameters a = 150 and b = 1.3 (

**centre**); and simple filter and Z–R parameters a = 150 and b = 1.3 (

**right**).

**Figure 5.**TM (Equation (5) in log–log plot) analysis for the 12–13 September 2015 event: X-band DPSRI rainfall products at 1.5 km obtained with (

**a**) FIR filter and Z–R parameters a = 200 and b = 1.6 for low intensities; (

**b**) FIR filter and Z–R parameters a = 150 and b = 1.3 for low intensities; (

**c**) simple filter and Z–R parameters a = 150 and b = 1.3 for low intensities; and (

**d**) C-band rainfall fields.

**Figure 6.**UM parameter estimates for three rainfall events (12–13 September 2015, 16 September 2015 and 5–6 October 2015 from top to bottom) using X-band and C-band radar rainfall (from left to right). Distributions (%) of parameter estimates (0.2-length bins) over individual time steps using the fixed interval of $\eta \in \text{}\left[0.6579,\text{}1.5199\right]$ method (red dashed lines) and the inflection point method (continuous green lines) are displayed. The vertical black lines indicate the “best fit” estimates obtained on the ensemble averaged DTM curves.

**Figure 7.**C-band (

**left**) and X-band (

**right**) pixel maps of the rainfall totals for the three events studied: 12–13 September 2015 (

**top**); 16 September 2015 (

**centre**) and 5–6 October 2015 (

**bottom**). Six circles indicate the rain-gauged values.

**Figure 8.**Statistical comparison of the temporal rainfall series over the six sub-catchments for which rain gauge measurements were available. For all quantitative parameters, the rain gauge information was used as the reference data.

**Figure 9.**Flow simulated at the four studied locations with X-band, C-band and rain gauged data for the 12–13 September event, along with observations. Simulations are carried out without the implementation of the tool mimicking regulation at the basin scale.

**Figure 10.**Flow simulated at the four studied locations with X-band, C-band and rain gauged data for the 16 September event, along with observations. Simulations are carried out without the implementation of the tool mimicking regulation at the basin scale.

**Figure 11.**Flow simulated at the four studied locations with X-band, C-band and rain gauged data for the 5–6 October event, along with observations. Simulations are carried out without the implementation of the tool mimicking regulation at the basin scale.

**Figure 12.**Statistical comparison of all hydrological modelling results, with respect to the local flow measurements.

**Figure 13.**Flow simulated at the four locations studied for the 16 September event: without the implementation of the tool optimising regulation at the catchment scale using X-band, C-band and rain gauge data; and with the optimising tool using X-band data.

Event | Radar | Start Time | Duration (hours) | Time Steps | $\mathit{\alpha}$ | ${\mathit{C}}_{1}$ |
---|---|---|---|---|---|---|

12–13 September 2015 | C-band | 04:05 | 44 | 528 (5 min) | 1.25 | 0.22 |

12–13 September 2015 | X-band | 04:05 | 44 | 773 (3.4 min) | 1.54 | 0.18 |

16 September 2015 | C-band | 00:05 | 16.8 | 202 (5 min) | 1.02 | 0.12 |

16 September 2015 | X-band | 00:05 | 16.8 | 296 (3.4 min) | 1.51 | 0.11 |

5–6 October 2015 | C-band | 09:10 | 31 | 372 (5 min) | 1.58 | 0.15 |

5–6 October 2015 | X-band | 09:10 | 31 | 545 (3.4 min) | 1.79 | 0.15 |

© 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**

Paz, I.; Willinger, B.; Gires, A.; Ichiba, A.; Monier, L.; Zobrist, C.; Tisserand, B.; Tchiguirinskaia, I.; Schertzer, D.
Multifractal Comparison of Reflectivity and Polarimetric Rainfall Data from C- and X-Band Radars and Respective Hydrological Responses of a Complex Catchment Model. *Water* **2018**, *10*, 269.
https://doi.org/10.3390/w10030269

**AMA Style**

Paz I, Willinger B, Gires A, Ichiba A, Monier L, Zobrist C, Tisserand B, Tchiguirinskaia I, Schertzer D.
Multifractal Comparison of Reflectivity and Polarimetric Rainfall Data from C- and X-Band Radars and Respective Hydrological Responses of a Complex Catchment Model. *Water*. 2018; 10(3):269.
https://doi.org/10.3390/w10030269

**Chicago/Turabian Style**

Paz, Igor, Bernard Willinger, Auguste Gires, Abdellah Ichiba, Laurent Monier, Christophe Zobrist, Bruno Tisserand, Ioulia Tchiguirinskaia, and Daniel Schertzer.
2018. "Multifractal Comparison of Reflectivity and Polarimetric Rainfall Data from C- and X-Band Radars and Respective Hydrological Responses of a Complex Catchment Model" *Water* 10, no. 3: 269.
https://doi.org/10.3390/w10030269