Jerry Jose

Auguste Gires

Ioulia Tchiguirinskaia

Yelva Roustan

Daniel Schertzer

Scale invariant relationship between rainfall kinetic energy and intensity in Paris region: An evaluation using universal multifractal framework

Keywords: rainfall intensity, rainfall kinetic energy, disdrometer, multifractal, scale invariance, gamma DSD

To calculate the effect of rainfall in detaching particles and initiating soil erosion, or in eroding wind turbine leading edge, it is important to measure recorded drop size distributions (DSD) and fall velocity over long period. Commonly used relationships between kinetic energy (KE) and rainfall rate (R) exhibit strong dependence on the temporal resolution at which the analysis is carried out. Here we aim at developing a new scale invariant relationship relying on the framework of Universal Multifractals (UM), which is widely used to analyze and characterize geophysical fields that exhibit extreme variability across wide range of scales. Rainfall data is collected using three optical disdrometers working on different underlying technologies (one Campbell Scientific PWS100 and two OTT Parsivel 2 instruments) and operated by the Hydrology Meteorology and Complexity laboratory of École des Ponts ParisTech in Paris area (France). They provide access to the size and velocity of drops falling through sampling areas of few tens of cm 2 . Such data enables estimation of rainfall DSD, R and KE at various resolutions. The temporal variations of this geophysical data over wide range of scales are then characterized in the UM framework, which was never done for KE. A new power law relation is developed and tested against the theoretical

1. Introduction 1.1. On the importance of rainfall and kinetic energy Understanding the relation between rainfall rate (R) and kinetic energy (KE) is essential for accurate determination of various rainfall parameters and understanding their after effects on surrounding ecosystem [START_REF] Karlen | Soil quality: Humankind's foundation for survival[END_REF]. It has been well established that onsite erosion of soil -splash and runoff -depends on drop size distribution (DSD) and fall velocity of the spectrum [START_REF] Ellison | Ellison: Studies of raindrop erosion -Google Scholar[END_REF][START_REF] Fernández-Raga | Splash erosion: A review with unanswered questions[END_REF]. Hence, KE and R are primarily used to quantify rainfall erosivity and to estimate erosion rates in universal models towards sustainable land use planning [START_REF] Angulo-Martínez | Measurement uncertainty in rainfall kinetic energy and intensity relationships for soil erosion studies: An evaluation using PARSIVEL disdrometers in the Southern Appalachian Mountains[END_REF][START_REF] Shojaei | Prediction of factors affecting activation of soil erosion by mathematical modeling at pedon scale under laboratory conditions[END_REF][START_REF] Mohamadi | Effects of rainfall patterns on runoff and soil erosion in field plots[END_REF]. Erosion heavily affects agricultural sector: On on-site level, it impoverishes the top soil off nutrients and organic matter along with their water holding capacity; this in turn increases the use of fertilizers and hence causes pollution at the recipient end of off-site erosion. Further, on off-site level, soil transport by erosion can trigger flood events through silting up of basins and rivers [START_REF] Pimentel | Soil Erosion: A Food and Environmental Threat[END_REF][START_REF] Enne | Desertification in europe: Mitigation strategies, land-use planning[END_REF]. Rainfall erosivity is a key parameter in various erosion models such as US based Universal Soil Loss Equation (USLE, [START_REF] Smith | Rainfall Erosion[END_REF] and its revised version (RUSLE, [START_REF] Renard | Predicting soil erosion by water: a guide to conservation planning with the Revised Universal Soil Loss Equation[END_REF], South Korean based SEMMA models for calculating soil loss [START_REF] Park | Statistical Soil Erosion Model for Burnt Mountain Areas in Korea-RUSLE Approach[END_REF], and in European models such as EUROSEM, WaTEM etc. that account for sediment transportation along soil loss [START_REF] Morgan | The European Soil Erosion Model (EUROSEM): a dynamic approach for predicting sediment transport from fields and small catchments[END_REF][START_REF] Van Oost | Evaluating the effects of changes in landscape structure on soil erosion by water and tillage[END_REF][START_REF] Kirkby | The pesera coarse scale erosion model for europe. i. â model rationale and implementation[END_REF], an advance on USLE). Most of these frameworks use KE as the major quantifying factor for estimating erosivity at spatial scales as rainfall KE represents the total energy available for detachment and transport of soil on surface of impact.

As direct measurement of KE is limited to specific geographical locations possessing required instrumental capabilities, understanding the relationship between KE and the more commonly available rainfall parameter, R, is important in estimating the former in more places.

Accurate estimation of rainfall KE is also important in understanding and mitigating leading-edge erosion (LEE) on wind turbine blades. Erosion damage reduces aerodynamic performance of blades resulting in reduced annual energy production and increased downtime [START_REF] Keegan | On erosion issues associated with the leading edge of wind turbine blades[END_REF]. Though LEE involves a multitude of atmospheric factors, impact velocity and amount of precipitation have been established as some of the major external factors in erosion [START_REF] Herring | The increasing importance of leading edge erosion and a review of existing protection solutions[END_REF]. As in soil erosion, larger drops with greater mass and vertical terminal velocity causes a disproportionate amount of erosion in LEE also. Rainfall KE helps to quantify the impact of droplets hitting perpendicular to the surface and hence rainfall erosion on blades which is the accumulated aggregate of multiple impacts stochastically distributed over the surface of the coated laminate [START_REF] Bech | Extending the life of wind turbine blade leading edges by reducing the tip speed during extreme precipitation events[END_REF]. In Whirling arm rain erosion test (WA-RET) (ASTM-G73-10, 2017; [START_REF] Liersch | Investigation of the impact of rain and particle erosion on rotor blade aerodynamics with an erosion test facility to enhancing the rotor blade performance and durability[END_REF]DNVGL-RP-0171, 2014), the industrial standard for measuring durability of leading-edge structures, specimens are subjected in controlled velocity and rain conditions to assess the damage caused by droplet impacts [START_REF] Bech | Extending the life of wind turbine blade leading edges by reducing the tip speed during extreme precipitation events[END_REF]. Considering the rapid growth of offshore wind industry as sustainable clean energy solution in the Americas and Asia with monsoon seasons, proper representation of KE and R is important in quantifying LEE in wind turbines.

Review of existing relations and need for a scale invariant representation

Application of erosion frameworks such as USLE or RUSLE poses uncertainties because of their empirical basis which has single or limited measurement locations and specific methods of data collection. These frameworks take KE as the major quantifying factor for erosivity, and employ various corrections to mitigate overestimation at low inten-sity rainfall as smaller droplets are less effective in soil detachment [START_REF] Van Dijk | Rainfall intensity-kinetic energy relationships: a critical literature appraisal[END_REF].

Another commonly used erosivity index is rainfall momentum, but it has been shown that for natural rainfall they exhibit similar relationship with rainfall intensity [START_REF] Hudson | Soil conservation[END_REF].

These models are based on traditional two parameter exponential rain drop size distribution models developed from smaller sample collection methods such as flour pellets or stain paper [START_REF] Laws | The relation of raindrop-size to intensity[END_REF][START_REF] Marshall | The distribution of raindrops with size[END_REF]. Later studies using multi parameter radiometry and disdrometers have shown improvements in measurement if DSD is assumed to be a three parameter gamma distribution [START_REF] Ulbrich | Natural Variations in the Analytical Form of the Raindrop Size Distribution[END_REF]. Gamma distribution assumes fewer large drops in rain and represents a narrower DSD than the exponential one, and thereby reduces the overestimation of KE by the latter. For recording DSD and in turn obtaining empirical KE, disdrometers are commonly used in meteorological campaigns, where fall velocity is either directly measured or estimated from empirical relations [START_REF] Gunn | THE TERMINAL VELOCITY OF FALL FOR WATER DROPLETS IN STAGNANT AIR[END_REF][START_REF] Atlas | Doppler radar characteristics of precipitation at vertical incidence[END_REF].

There has been various studies towards accurate representation of KE -R relationship. [START_REF] Fox | TECHNICAL NOTE: The representation of rainfall drop-size distribution and kinetic energy[END_REF] demonstrates the impact of formulation of DSD in calculation of KE and hence on erosion. [START_REF] Smith | Rainfall Erosion[END_REF] proposed a logarithmic function based on DSD formulation by [START_REF] Laws | The relation of raindrop-size to intensity[END_REF] and terminal velocity by [START_REF] Gunn | THE TERMINAL VELOCITY OF FALL FOR WATER DROPLETS IN STAGNANT AIR[END_REF].

It was used in modelling erosion in USLE, which was later replaced by continuous exponential functions in revised USLE approach [START_REF] Renard | Predicting soil erosion by water: a guide to conservation planning with the Revised Universal Soil Loss Equation[END_REF][START_REF] Brown | storm Erosivity Using Idealized Intensity Distributions[END_REF].

Other proposed forms of KE -R equations were linear [START_REF] Kinnell | Rainfall Intensity-Kinetic Energy Relationships for Soil Loss Prediction1[END_REF][START_REF] Sempere-Torres | Experimental evidence of a general description for raindrop size distribution properties[END_REF], polynomial [START_REF] Carter | Activation of reovirion-associated poly(A) polymerase and oligomer methylase by cofactor-dependent cleavage of polypeptides[END_REF] and power-law [START_REF] Park | Splash erosion modelling: physical analyses [Impact of water drops on soil[END_REF]. Critical literature appraisal by [START_REF] Van Dijk | Rainfall intensity-kinetic energy relationships: a critical literature appraisal[END_REF] on various KE -R relations lists measurement techniques and procedures, sampling biases, interpretation methods and storm types as reasons for discrepancies and suggests another exponential based predictive equation. More recent reviews such as Angulo-Martínez and Barros (2015), [START_REF] Wilken | Uncertainties in rainfall kinetic energy-intensity relations for soil erosion modelling[END_REF], [START_REF] Mineo | Assessment of Rainfall Kinetic-Energy-Intensity Relationships[END_REF] shows lack of universality among various KE -R relationships (logarithmic, linear, power law and exponential) using disdrometer measurements and modelling at different geographic stations, meteorologic conditions and time aggregation used for calibration.

Most of the empirical formulations express KE as volume specific -kinetic energy per unit area and mm of rainfall or KE mm (J m -2 mm -1 ) due to prevalence of non automated measurements and lack of accuracy in determining exposure time. But expression of volume specific kinetic energy creates a statistical artefact (spurious ratio correlation) in KE -R relationship due to the inclusion of R (mm h -1 ) in the KE mm expression. [START_REF] Salles | Kinetic energy of rain and its functional relationship with intensity[END_REF] suggests usage of more consistent time specific KE or KE time (J m -2 h -1 ) that has been shown to produce less heteroscedasticity than corresponding KE mm -R scatter plots.

For representing erosion, KE time has been expressed as the rate of expenditure of rainfall kinetic energy [START_REF] Kinnell | Rainfall Intensity-Kinetic Energy Relationships for Soil Loss Prediction1[END_REF], rainfall or kinetic power [START_REF] Smith | The temporal and spatial variability of rainfall power[END_REF] and rainfall kinetic energy flux density [START_REF] Steiner | Reflectivity, Rain Rate, and Kinetic Energy Flux Relationships Based on Raindrop Spectra[END_REF]. The two expressions are related to each other through rain intensity and hydrological studies usually harmonize KE -R relationships with KE time .

KE time = R × KE mm (1)
In this paper the expressions for KE are derived and analyzed in the form of KE time unless otherwise specified.

Due to variations in methodologies used and DSD characteristics during measurement, different functional forms show different behaviour towards KE -R estimation. Though commonly preferred, exponential relations have been shown to underestimate KE time for lower intensities of rainfall [START_REF] Carollo | Modeling Rainfall Erosivity by Measured Drop-Size Distributions[END_REF]. Power-laws that predict kinetic energy well at lower intensities tend to overestimate the same for higher intensities. Logarithmic curves are limited in their usage though they fit low and high intensity KE rather decently [START_REF] Van Dijk | Rainfall intensity-kinetic energy relationships: a critical literature appraisal[END_REF]. Further due to the empirical formulation, when it comes to expressing KE time vs R, most of these mathematical equations fail to have much physical justification. Using generalized scaling formulation that expresses various existing DSD models as it's special cases (Torres et al., 1994;[START_REF] Sempere-Torres | Experimental evidence of a general description for raindrop size distribution properties[END_REF], [START_REF] Salles | Kinetic energy of rain and its functional relationship with intensity[END_REF] found that power law is the most suitable function to relate KE time and R from a microphysical point of view. The parameters (prefactor and exponent) of power law are related to rain type, geographical location and measuring technique. [START_REF] Shin | Universal Power Law for Relationship between Rainfall Kinetic Energy and Rainfall Intensity[END_REF] has proposed a representative power law based on the ideal assumption that the drop-size is uniformly distributed under the constant rainfall intensity.

Purpose of paper

Current literature on KE and R lacks a common consensus on the usage of expression that is valid across various scales of measurement, hydro-meteorological regimes or observation techniques. There is an increased focus on research characterizing rainfall microphysics at local and regional scales [START_REF] Petan | The rainfall kinetic energy-intensity relationship for rainfall erosivity estimation in the mediterranean part of Slovenia[END_REF]. Here we examine the variation of KE and R for over 7 years in Paris region using continuous data from three disdrometers from two different manufacturers. Using the framework of Universal Multifractal (UM) [START_REF] Schertzer | Physical modeling and analysis of rain and clouds by anisotropic scaling multiplicative processes[END_REF], efforts were made to characterize the variability of KE time and R on event based and year based analysis, and to formulate a scale invariant relation based on power law relationships. Multifractals allow characterization of complex geophysical fields with a limited number of scale invariant exponents (see Schertzer and Tchiguirinskaia, 2020, for a recent review). Multifractal behaviour of measured rainfall has been abundantly studied previously (see [START_REF] Gupta | Multiscaling properties of spatial rainfall and river flow distributions[END_REF][START_REF] Kumar | A multicomponent decomposition of spatial rainfall fields: 2. Self-similarity in fluctuations[END_REF][START_REF] Deidda | Multifractal modeling of anomalous scaling laws in rainfall[END_REF][START_REF] Olsson | Multifractal analysis of daily spatial rainfall distributions[END_REF][START_REF] García-Marín | Universal multifractal description of an hourly rainfall time series from a location in southern Spain[END_REF][START_REF] Langousis | Multifractal rainfall extremes: Theoretical analysis and practical estimation[END_REF][START_REF] Emmanouil | Quantitative assessment of annual maxima, peaks-over-threshold and multifractal parametric approaches in estimating intensity-duration-frequency curves from short rainfall records[END_REF] which was never done to the knowledge of the authors, is a first goal of the paper. The main goal of this paper is to explore the possibility of establishing a physically based scale invariant power law relationship between KE and R using the UM framework, without having to rely on strong assumptions on DSD shape; and compare its performance with more classical approach.

This article is structured as follows. In section 2 the methodology used is detailed.

First, the commonly used framework of gamma distributed DSD which yields a power-law relation is reminded along with the associated parameters' estimation techniques. Then, the process to identify power-law relation in the UM framework is explained after a required reminder on the theoretical underlying basis. Similarly, the associated parameters' estimation techniques, which will be implemented, are described. Section 3 includes details of data collection and quality control. Results are discussed in section 4 where validity of formulated relation is tested and contrasted over different types of rain events. The final section, section 5, concludes the study and summarizes the main observations. [START_REF] Uijlenhoet | A consistent rainfall parameterization based on the exponential raindrop size distribution[END_REF] :

N A (D) = v(D)N v (D) (2)
where v(D) represents the terminal fall velocity (m s -1 ) as a function of the equivalent spherical diameter of raindrop D (mm).

Traditional mathematical expressions describing N v (D) such as exponential [START_REF] Marshall | The distribution of raindrops with size[END_REF], weibull [START_REF] Best | The size distribution of raindrops[END_REF], gamma [START_REF] Ulbrich | Natural Variations in the Analytical Form of the Raindrop Size Distribution[END_REF] and lognormal [START_REF] Feingold | The Lognormal Fit to Raindrop Spectra from Frontal Convective Clouds in Israel[END_REF]) can be expressed as particular case of general formulation proposed by [START_REF] Sempere-Torres | Experimental evidence of a general description for raindrop size distribution properties[END_REF]. For the scope of this paper, gamma distribution of DSD, which has been recognized to better represent natural rain, will be considered [START_REF] Ulbrich | Natural Variations in the Analytical Form of the Raindrop Size Distribution[END_REF]. In this framework :

N v (D) = N 0 D µ e -(ΛD) (3)
where N v (D) is in m -3 mm -1 , D in mm, N 0 (in m -3 mm -1-µ ), µ and Λ (in mm -1 ) are distribution parameters measuring raindrop concentration, mean size and shape of spectrum respectively. Λ is usually expressed with the help of the median volume diameter D 0 as Λ = (3.67 + µ)/D 0 .

Theoretical power law relation between R and KE

Rainfall intensity or rain rate (R, in mm h -1 ) can be calculated from N A (D) using following expression [START_REF] Steiner | Reflectivity, Rain Rate, and Kinetic Energy Flux Relationships Based on Raindrop Spectra[END_REF] :

R = 3.6 × 10 -3 π 6 ∞ 0 D 3 N A (D)dD (4)
Kinetic energy per unit area per unit time (KE time in J m -2 h -1 ) of falling drops can also be expressed in terms of measured N A (D) as follows [START_REF] Steiner | Reflectivity, Rain Rate, and Kinetic Energy Flux Relationships Based on Raindrop Spectra[END_REF] :

KE = 3.6 × 10 -6 πρ 12 ∞ 0 D 3 v 2 (D)N A (D)dD (5)
where ρ is density of water in standard conditions in (kg m -3 ).

Though there are more sophisticated equations proposed in the literature for v(D), for simplicity in calculation, here we are following the widely used power law formulation by [START_REF] Atlas | Path-and Area-Integrated Rainfall Measurement by Microwave Attenuation in the 1-3 cm Band[END_REF]. [START_REF] Uijlenhoet | Raindrop size distributions and radar reflectivity-rain rate relationships for radar hydrology[END_REF] demonstrated it has the only functional form consistent with power law relationship between rainfall related parameters :

v(D) = c D g (6)
c = 3.78 m s -1 mm -g and g = 0.67 (with v in ms -1 and D in mm).

Most of the rainfall parameters can be approximated as moments of the DSD; and when DSD follows gamma model, the n th moment, M n can be computed as [START_REF] Atlas | Path-and Area-Integrated Rainfall Measurement by Microwave Attenuation in the 1-3 cm Band[END_REF] :

M n = ∞ 0 D n N(D)d(D) = N 0 Λ -(µ+n+1) Γ(µ + n + 1) (7)
where Γ(α) is the complete gamma function

Γ(α) = ∞ 0 x α-1 e -x dx, where α > 0. ( 8
)
Substituting equations 2, 3, 6 and 7, and solving the integral gives reduced expressions for KE and R in terms of gamma function;

R = 6 × 10 -4 πcN 0 Γ(4 + g + µ) Λ (4+g+µ) (9) KE = 3.6 × 10 -6 ρπc 3 12 N 0 Γ(4 + 3g + µ) Λ (4+3g+µ) (10)
This reduction using gamma function enables representation of KE and R in the form of a power law as follows :

KE = bR a (11)
where

a = 4 + 3g + µ 4 + g + µ b = 5 × 10 -4 ρc 2 6πcN 0 × 10 -4 1-a Γ(4 + 3g + µ) (Γ(4 + g + µ)) a (12)
The final expression follows the same pattern as that by [START_REF] Salles | Kinetic energy of rain and its functional relationship with intensity[END_REF] based on generalized DSD and that of [START_REF] Uijlenhoet | A consistent rainfall parameterization based on the exponential raindrop size distribution[END_REF] based on exponential DSD.

When µ = 0 gamma distribution for DSD becomes a simple Marshall and Palmer negative exponential parameterization representation of DSD [START_REF] Marshall | The distribution of raindrops with size[END_REF] with Eq.

11 becoming KE = 8.539R 1.287 . Same values were obtained for exponent 'a' and pre-factor 'b' using the general function proposed by [START_REF] Salles | Kinetic energy of rain and its functional relationship with intensity[END_REF] for Marshall and Palmer approximation. Closer value of exponential coefficient were also reported by [START_REF] Uijlenhoet | A consistent rainfall parameterization based on the exponential raindrop size distribution[END_REF] in a power law formulation based on [START_REF] Marshall | The distribution of raindrops with size[END_REF] approximation of DSD and power law dependence of rain drop terminal velocity, Eq. 6 [START_REF] Atlas | Path-and Area-Integrated Rainfall Measurement by Microwave Attenuation in the 1-3 cm Band[END_REF].

Many simplifications were followed in this formulation of KE-R relation as a power law (Eq. 11). Gamma DSD is notably assumed as well as a power law form for terminal fall velocity as function of diameter. This approximation shows limitations at higher values of diameter (D > 5 mm) and doesn't account for atmospheric turbulence or updrafts and downdrafts [START_REF] Adirosi | Raindrop size distribution: Fitting performance of common theoretical models[END_REF]. Effect of truncation errors in measurement is not considered. The formulation also ignores the effect of horizontal wind velocity and surface impact angle of rain drops. Some authors tend to consider normalized spectra of DSD [START_REF] Testud | The Concept of "Normalized" Distribution to Describe Raindrop Spectra: A Tool for Cloud Physics and Cloud Remote Sensing[END_REF] that requires only two parameters to describe the DSD; it was also not considered here for the sake of simplicity in deriving a theoretical relation between KE and R.

It should be noted that although widely accepted as the best representation of natural DSD, gamma function also has associated errors that exists outside experimental methodology and sampling [START_REF] Adirosi | Evaluation of Gamma Raindrop Size Distribution Assumption through Comparison of Rain Rates of Measured and Radar-Equivalent Gamma DSD[END_REF].

Estimation of power law coefficients from gamma DSD parameters

As mentioned before, most rainfall parameters can be expressed as moments of the DSD according to Eq. 7. The right hand side of the equation is specific to the non normalized three parameter gamma distribution considered here, as shown in Eq. 3. In practice DSD is not measured continuously but for discrete diameters D i . As a consequence, the estimated moments Mn of order n are computed thanks to the following discrete sum rather than previous integral :

Mn = N class ∑ i=1 D n i N i (D i )∆D i [mm n m -3 ] (13)
where D i is the diameter of droplet in class i, N i (D i ) is the drop size distribution (estimation discussed later in Eq. 26), ∆D i is the width of diameter class i and N class the total number of diameter classes.

Investigation of various DSD parameter estimating methods by [START_REF] Cao | Errors in Estimating Raindrop Size Distribution Parameters Employing Disdrometer and Simulated Raindrop Spectra[END_REF] highlighted the risk in usage of maximum likelihood and L-moment estimators for processing data with truncation in lower end of DSD spectra. As all disdrometers are limited by some minimum value of measurable diameter D min (> 0), it is advisable to use traditional method of moments [START_REF] Brawn | Estimation of an atmospheric gamma drop size distribution using disdrometer data[END_REF]. Although the choice of moments for proper parameters estimation with the method of moments is a relevant topic, it is outside the scope of this paper. Hence, authors relied on the existing literature. More precisely, although higher DSD moments are considered to be associated with higher errors, the tendency of lower errors with middle order moments was shown later in [START_REF] Smith | The Bias and Error in Moment Estimators for Parameters of Drop Size Distribution Functions: Sampling from Gamma Distributions[END_REF] (using radar measurements) and [START_REF] Cao | Analysis of video disdrometer and polarimetric radar data to characterize rain microphysics in oklahoma[END_REF] (using joint disdrometer-radar observations). [START_REF] Cao | Errors in Estimating Raindrop Size Distribution Parameters Employing Disdrometer and Simulated Raindrop Spectra[END_REF] evaluated the performance of various moment estimators using simulations of gamma DSD (with a more realistic estimation of errors) and found that second, third and fourth moments (M234) as the best overall performer for estimating R. [START_REF] Konwar | Microphysics of clouds and rain over the western ghat[END_REF] (M234) and Huang et al. ( 2021) (even higher order moments -2,3 and 6) are some recent examples of using middle order moments while analysing data from same disdrometer make as the one in current study (OTT Parsivel 2 ). Hence, it was chosen to employ these moments for computing gamma DSD parameters: µ, N 0 and Λ (refer Eq. 7). Specific moment equations are provided in the appendix (Appendix B). These parameters are then introduced in Eq. 12 to derive the expected power law parameters assuming a gamma DSD.

Variation of a and b with DSD parameters is examined with available data set in section 4 for different type of rainfall events.

Theoretical relation between multifractal fields in UM framework

Overview of the theoretical framework

The framework of Universal Multifractals (UM) enables to characterize the extreme variability of geophysical fields across scales, with the help of a limited number of parameters with physical meaning [START_REF] Schertzer | Physical modeling and analysis of rain and clouds by anisotropic scaling multiplicative processes[END_REF]. It relies on the assumption that these fields are generated through an underlying multiplicative cascade. Such behaviour is assumed to be inherited from the scale invariant features of the Navier-Stokes equations, but has not yet been formally showed. Consequently, statistical properties of fields are conserved in all scales. In this study, time specific kinetic energy and rainfall intensity from disdrometer measurements are analyzed using UM. Review by [START_REF] Schertzer | MULTIFRACTALS, GENERALIZED SCALE INVARI-ANCE AND COMPLEXITY IN GEOPHYSICS[END_REF] discusses the techniques and methodologies employed in more detail.

For analysis in UM framework, the field in consideration is normalized (divided by its mean value) and its total size in terms of numerical values is restricted to be a power of two. The field can be one or two dimensional; since current study involves only time series we will be limiting our discussion to single dimension. Let us consider a normalized conservative field (an additional parameter for non-conservative fields is introduced later) ε λ at resolution λ . Resolution λ is the ratio of L, the outer scale, to l, the observational scale. Let us first consider the occurrence pattern of the field, i.e. the same field with 1 for strictly positive values and 0 otherwise. If the field is fractal, then the probability p that a segment of length l intersects the field (p is the probability of occurrence of non zero elements in the total binary field) scales with resolution as follows:

p = λ D f λ D = λ -c f (14)
where c f = D -D f , the fractal co-dimension of the field. The fractal dimension D f indicates how the binary field (rain and no rain in this case) fills the available space in a scale invariant way. As D is constant, in order to fully characterize the field a value of D f for each threshold is required. This is the intuitive notion of multifractality. In order to be mathematically consistent, this characterization should actually be done with the help, not of a direct threshold at the maximum resolution, but with a scale-invariant threshold called singularity γ, and their corresponding codimension function c(γ) as :

p ε λ ≥ λ γ ≈ λ -c(γ) (15)
Above equation implies that statistical moments q of the field scale with resolution (Schertzer andLovejoy, 1987, 1988) are :

⟨ε λ q ⟩ ≈ λ K(q) (16)
where K(q) is the moment scaling function related to c(γ) by Legendre transform [START_REF] Parisi | On the singularity structure of fully developed turbulence in Turbulence and predictability in geophysical fluid dynamics and climate dynamics[END_REF]. For a conservative field in UM framework, K c (q) can be fully characterized with only two parameters, multi-fractality index α and mean intermittency codimension C 1 .

K c (q) = C 1 α -1 q α -q (17)
C 1 measures clustering of average intensity across scales; when the field is homogeneous For a non conservative field φ λ , i.e. a field whose average (⟨φ λ ⟩) changes with scales, a non-conservative parameter H is used in expression of scaling:

φ λ = ε λ λ -H (18)
where ε is a conservative field characterized with C 1 and α.

Positive H represents a smoother field which needs to be fractionally differentiated for obtaining a conservative field. Conversely negative H represents a field in need of fractional integration for retrieving conservative field. H is related to the spectral slope β , which gives a measurement of the correlation range. Larger β means higher contribution of large-scale phenomenon in variability of data.

β = 1 + 2H -K c (2) (19)

Estimation of scaling behaviour and UM parameters

The quality of scaling of multifractal fields can be examined using trace moment (TM)

where log-log plot of upscaled fields against resolution λ is taken for each moment q (Eq. 16). For multifractal fields, the plot is a straight line with moment scaling function K(q) as slope. Quality of the scaling is given by the estimate R 2 of the linear regression. The value for q = 1.5 is used as reference. Double trace moment (DTM) is a more robust version of TM tailored for UM fields where the moment scaling function K(q, η) of the field ε λ

(η)
(obtained in practice by raising ε to power η at the maximum resolution and then upscaling it) is expressed as a function of multifractality index α [START_REF] Lavallée | Nonlinear variability and landscape topography: analysis and simulation[END_REF] :

⟨ ε λ (η) q ⟩ ≈ λ K(q,η) = λ η α K(q) (20)
From the above equation, value of α can be obtained as the slope of the linear part when K(q, η) is represented for a given q as a function of η in log-log plot.

Power law relations in UM framework and practical implementation

If a field is UM, then a power law relation of it is also a UM field with coefficients depending upon initial UM parameters. The reasoning for power law comes from the DTM analysis in UM framework [START_REF] Tessier | Universal multifractals: Theory and observations for rain and clouds[END_REF][START_REF] Lovejoy | The remarkable wide range spatial scaling of TRMM precipitation[END_REF].

Consider two multifractal fields (ε 1 and ε 2 ) that are power law related by an exponent a and a prefactor b, as below :

ε 1 = bε a 2 (21)
From exponents in Eq. 20, K(q) of ε 1 can be expressed as follows

< (ε 1 ) q > ≈ < (ε 2 (a) ) q > ≈ λ K ε 2 (q,a) K ε 1 (q) = K ε 2 (q, a) = a α ε 2 K ε 2 (q) (22)
Expanding K(q) with UM parameters (as in Eq. 17)

C 1,ε 1 α ε 1 -1 (q α ε 1 -q) = a α ε 2 C 1,ε 2 α ε 2 -1 (q α ε 2 -q) (23)
From above equation, we can deduce that for if a power relation exists, the UM parameters are related as follow :

α ε 1 = α ε 2 = α C 1,ε 1 = a α C 1,ε 2 (24)
In this study, both KE and R time series were analysed using UM framework. As the UM model discussed here is solely non negative, it removes all possibility of having nonnegative values for both fields in analysis. Using their UM parameters in Eq. 24, it is possible to formulate power law relationship in the format of Eq. 21. It has the same shape as the theoretical power law obtained using moments of the DSD in section 2.1, in Eq. 11.

Details of disdrometer data collection and subsequent multifractal analysis are covered in upcoming sections.

Data collection and instrumentation

Measurement campaigns

For this study, continuously monitored data in natural conditions was obtained from three optical disdrometers operating with two different principles. Table 1: Short description of the precipitation measurement campaign selected

Overview of instrument functioning and outputs

The three optical disdrometers used here are two OTT Parsivel 2 (see [START_REF] Battaglia | PARSIVEL Snow Observations: A Critical Assessment[END_REF] or the device documentation OTT, 2014) and one PWS 100 (see [START_REF] Ellis | New laser technology to determine present weather parameters URL[END_REF] or the device documentation Campbell-Scientific-Ltd, 2012).

The OTT Parsivel 2 are occlusion based devices with a transmitter that creates laser sheet and an intercepting receiver directly aligned with it. The size (equivolumic diameter) and fall velocity are assessed from changes (decrease in amplitude of intensity and duration of the decrease) in received laser intensity due to the passing of rainfall drops through a sampling area of 54 cm 2 . An ellipsoidal shape model with a standard relation between the axis ratio and the equivolumic diameter are assumed for drops in the process.

The PWS100 consists of a transmitter that generates four horizontal parallel laser sheets, and two receptors which are not aligned with the transmitter. The signal received by each receptor corresponds to the light refracted by drops and contains four consecutive peaks associated with each laser sheet. From the delay between those peaks, fall velocity and diameter of the drops are estimated. PWS100 has a sampling area of 40 cm 2 . Computations assuming spherical shape of droplets are performed here with a later correction for oblateness before final result generation [START_REF] Gires | Method and device for measuring the equivalent diameter of a water drop[END_REF].

Both disdrometers have a collection time step of 30 s and provide main output as a matrix containing the number of drops (n i, j ) recorded during the time step ∆t according to classes of equivolumic diameter (index i defined by a centre D i and a width ∆D i both expressed in mm) and fall velocity (index j and defined by a centre v j and a width ∆v j , both expressed in ms -1 ). For Parsivel 2 there are 32 classes of ∆D i from 0.062 mm to 24.5 mm and ∆v j from 0.05 ms -1 to 20.8 ms -1 . And for PWS the same ranges from 0.05 to 27.2 (∆D i , mm) and 0.05 to 27.2 (∆v j , ms -1 ) in 34 classes. Width of diameter and velocity classes are not similar for all classes, as they are designed to be more and more refined towards smaller values.

From the raw matrix, the studied rainfall parameters -rain rate, drop size distribution and time specific kinetic energy were obtained using following expression for each time steps.

R = π 6∆t ∑ i, j n i, j D 3 i S e f f (D i ) (25) N(D i ) = 1 S e f f (D i )∆D i ∆t ∑ j n i, j v j (26) KE = ρ wat π 6∆t ∑ i, j n i, j D 3 i v 2 j S e f f (D i ) (27)
where S e f f (D i ) is the sampling area of disdrometer in mm 2 , ∆t is the time step duration in hr and ρ wat is the volumic mass of water (10 3 kg m -3 ). N(D i )∆D i gives the number of drops with a diameter in the class i per unit volume (in m -3 ). Details of the devices, their functioning and data collection can be found in [START_REF] Gires | Two months of disdrometer data in the paris area[END_REF].

Data quality and filtering

For the data presented in this paper, filters suggested by various authors [START_REF] Kruger | Two-Dimensional Video Disdrometer: A Description[END_REF][START_REF] Thurai | Drop Axis Ratios from a 2D Video Disdrometer[END_REF][START_REF] Jaffrain | Quantification of the Small-Scale Spatial Structure of the Raindrop Size Distribution from a Network of Disdrometers[END_REF][START_REF] Gires | Two months of disdrometer data in the paris area[END_REF] were used to remove possible non-meteorological measurements (from environmental factors such as splashing, horizontal wind etc.) on the basis of size and velocity of drops.

Using the disdrometer data set, two series of Multifractal analysis were performed -event based and year based. For event based analysis, individual rainfall events were identified with following criteria in rain intensity time series -rainfall events with a cumulative depth greater than 0.7 mm and separated by at least 15 minutes of dry weather before and after.

From all measured events with this condition, rain rate (R), drop size distribution (DSD)

and kinetic energy (KE) were calculated for the three disdrometers (denoted Pars 1, Pars 2 and PWS hereafter). From the results further filtering was done to remove events having a percentage of nan values (not a number -blank /missing data) > 1 % and R 2 value < 0.9, for both KE and R. Remaining nans were then replaced with 0. In year based methodology, continuous time series from Jan 2016 to Dec 2019 -including rain and no rain conditions -were used for multifractal analysis, and results were compiled according to the year of measurement.

There were minor lapses in continuity of data measurement from Jun 2013 to Dec 2015;

for that reason those time periods were not considered in continuous year based analysis to avoid possible measurement biases. However, since selection of individual events are not affected by that, event based analysis involves data from Jun 2013 to Dec 2019. Lesser number of events can be observed in the excluded years for year based analysis in Table 2. Between 2016 and 2019, there were also a few days of maintenance where data was not recorded -27 Sep 2017 to 26 Dec 2017, 01 to 07 Jan 2018 and 12 to 14 Apr 2019. After quality control for each disdrometer, a total of 214,665 time steps were analysed in 556 rain events (total 1610 events counting all three disdrometers; data was not always available for all the disdrometers, hence the lesser number) from 2013 to 2019 for event based analysis.

In year based analysis, a total of 3,919,680 time steps were considered from 2016 to 2019; percentages of rainy time steps were about 7%, 6.6% and 5.7% for Pars 1, Pars 2 and PWS respectively.

Results and discussions

Overview of analysis

For analysing the KE and R time series using UM framework, as explained in previous sections, two strategies were followed -event based analysis and year based analysis. For event based analysis, we identified a total of 556 rainfall events between 28 Sep 2013 and 31 Dec 2019 of which 493 events were common among all three disdrometers. For UM analysis, each time series was resized to the highest power of two in such a way that the trimmed series accommodated maximum rainfall cumulative depth and then the field was normalized. After resizing for UM analysis, length of individual events ranged from 64 to 2048 time steps, where each time step corresponds to 30 s, i.e. the recording time step of disdrometers used. KE and R from each events were analyzed as separate fields in UM framework. For year based methodology, similar procedure was followed for UM analysis on year long continuous time series, for the years from 2016 to 2019.

In coming subsections, estimation of UM parameters and power law relations are illustrated using one event data for event based analysis and one year data for year based analysis. Power law coefficients were also estimated using theoretical framework with DSD parameters as mentioned in section 2.1. Variation and correspondence among coefficients determined by UM and DSD parameters are discussed thereafter and validated with data.

Multifractal analysis of events

For illustration of the analysis carried out, one event from 2017 for Pars 1 disdrometer, that occurred on 16 September between 11:35:00 and 13:00:00 (local time) is presented here. Figure 2a displays the time series of R and KE for this event, as well as KE vs. R plots. The latter also shows power law fits with coefficients from UM analysis and DSD parameters (explained later in this section). For this event, 171 time steps were trimmed to 128 time steps along region of maximum rain occurrence of which 124 were rainy data points. Trimmed and normalized KE and R were then subjected to analysis using UM framework discussed in section 2.2. Initial analysis indicated values of non-conservative parameter H greater than 0.5 among many events. Hence to retrieve a conservative field on which the UM analysis can be implemented without bias, fluctuations of KE and R time series were used [START_REF] Lavallée | Nonlinear variability and landscape topography: analysis and simulation[END_REF]). Characterization of variability in KE field for the event considered here can be seen in Figure 2b with TM (Eq. 16 in log-log plot), DTM graphs (Eq. 20 in log-log plot) and UM parameter values. As shown in TM and DTM graphs, the field exhibits a very good multifractal behaviour with a single scaling regime from 30 s to 64 min. For example, TM coefficient of determination r 2 for q = 1.5 was greater than 0.99. Values of UM parameters α, C 1 , and H for KE of this particular event were 1.820, 0.311 and 0.547 respectively. It should be mentioned that H computed on the fluctuations of KE was found to be equal to 0.189, meaning the taking the fluctuations indeed enabled retrieval of a conservative field from the original smoother field. R also exhibited excellent scaling behavior and corresponding values of UM parameters α, C 1 , and H for this event were 1.655, 0.229, and 0.100 respectively. KE and R analyzed from every event exhibited similar multifractal characteristics with a unique scaling regime. The quality of scaling was examined using coefficient of determination, R 2 for q = 1.5 in TM analysis, and as previously mentioned events with values < 0.9 were discarded. 9.8 % percentage of total events were rejected on this basis. For the events with good scaling behaviour, robust retrieval of multifractal parameters α, C 1 ,

β and H was possible. For example, the assessed values of α and C 1 for both KE and R time series, exhibited a maximum standard deviation (using various values of q in DTM analysis) of 0.0584 and 0.0670 for measurement at EP-SIRTA and 0.0446 and 0.0443 for measurements at ENPC.

Figure 3 displays the values of multifractality index α and mean intermittency C 1 for all the studied events for both fields. It appears that the values of α for KE and R are rather well distributed along the bisector. This pattern suggests a power law relation between these two quantities as discussed in section 2.2 (Eq. 21) where ε 1 and ε 2 are KE and R respectively (i.e. KE = bR a as in Eq. 11). The exponent of the power law a was deduced from corresponding α and C 1 values of UM fields (KE and R) for every event subjected to UM analysis, using the expected relations for power law related UM fields, i.e. Eq. 24. The α used is the average of α KE and α R (which were anyway similar). Value of prefactor b was estimated by fitting Eq. 11 at maximum resolution with estimated values of a on event's KE -R graph. For the event used as illustration, we found a = 1.083 and b = 11.493. This power law fit from UM parameters is displayed in Figure . 2a as KE UM in the KE vs. R graph.

Variation of computed power law exponent a and prefactor b are also shown in Fig. 3 as time series of events for each year. Graphs of remaining years are provided in appendix A.

For 3.5 % of total events filtered, estimates of α was found to be greater than the theoretical maximum (> 2); however a values for those events were found to be consistent with the overall average. Despite appreciable variability in UM parameters across events, values of a and b showed overall stability in the short range of values specified in graphs, suggesting robustness of the estimates. Year wise average values over the events for a and b, and number of events for all three disdrometers are given in Table 2. Average values of power law parameters from the events that were common between the three disdrometers are also shown in the same table. Comparable values of a and b were observed in both cases. A clear range of variation can be observed between the two types of disdrometers. For both Parsivel 2 the average a and b were similar and around 1.22 and 8.17 respectively, while for PWS100 a values were consistently lower than that of Parsivel 1; preceding and succeeding years are adjusted accordingly, refer Table 1.

Multifractal analysis of continuous data (year based analysis)

505

To illustrate year based analysis, continuous time series and UM analysis for Pars 1 Values and trend of variation are similar to that observed for the event based analysis. This confirms the robustness of the discussed power law, which is valid and retrieved not only at the event scale, but also at the year scale. 1; preceding and succeeding years are adjusted accordingly

Power law coefficients from DSD parameters

To understand previous findings further, values of a and b were computed from theoretical framework discussed in section 2.1. Rainfall DSD was assumed to follow gamma distribution (Eq. 3) and theoretical values of a and b (denoted as a DSD and b DSD from here on for clarity) were computed as per Eq. 12 after estimating values of gamma DSD parameters (µ, N 0 and Λ) with the help of method of moments. The variation of rain rate R was accurately reproduced using M234 moment estimators (Fig. 8d for example). Theoretical power law relation (KE DSD = b DSD R a DSD ) was then compared with the one estimated using UM analysis (represented from here on as KE UM with coefficients a UM and b UM for clarity) for every events. Average values of DSD parameters, estimation error and corresponding a DSD and b DSD for Pars 1 events are given in Table 4 according to event's rain type (definition introduced later). RMSE DSD is the root mean square error (RMSE) between empirical DSD and theoretical gamma DSD with fitted parameters (using the value for the center of each diameter class D i ). It is used here as an indicator of the quality of fit of the assumed gamma DSD distribution and the empirical one. For RMSE estimation only the portion of DSD above 0.5 mm diameter class was considered as smaller drops have lesser contribution in overall KE and R, and are associated with greater uncertainties in measurement. Also, higher size drops (above 9 mm diameter class) were not observed during the events and are hence not shown in DSD figures discussed after (Fig. 6 and Fig. 8). Power law from UM analysis on the other hand still provides a close fit. To see if there is such a trend through all the events, coefficient of determination of both KE -R fits were plotted against each other (r 2 UM vs r 2 DSD ) and compared using corresponding values of RMSE DSD (Fig. 7a). Negative value of coefficient of determination for DSD (r 2 DSD) ) is due to the high difference between KE DSD and empirical KE in certain events.

From the plot it is evident that a generalized conclusion -theoretical values of a and b (a DSD & b DSD ) works well in cases where empirical DSD coincides with gamma DSDis not possible. However, there were many events with good DSD correspondence with gamma, where KE DSD fitted data better. To investigate further this issue and given that the DSD is available for all time steps, a DSD and b DSD were computed for each time step to study their variations within a rainfall event. Figure 8 displays their temporal evolution for the event in Fig. 2. Values of parameters, especially b DSD , shows considerable variation within an event. These variations basically come from variations in DSD parameters µ an N 0 as it can be seen on Fig. 8c and 8f (also evident from Eq. 12), which are reflecting physical variations in the rainfall process. It should be mentioned that during this event (and other events), gamma DSD parameters were able to properly reproduce observed rain rate (Fig. 8d), meaning that the assumption of gamma DSD distribution and the M234 moment estimator approach remains valid throughout the event(s). This suggests that the variability of a DSD and b DSD observed at event scale is also valid within events at much smaller scales. This could explain some of the bias previously observed with DSD approach developed in this paper.

As there were still considerable variation in r 2 values of r 2 KE DSD (Fig. 7a), especially if we compare with KE UM which shows better r 2 KE UM regardless the type of event, a question of possible bias arises due to difference in methods of estimation of power law coefficients.

Unlike KE DSD for which both coefficients are obtained from theoretical relation involving DSD parameters, for KE UM only coefficient a is fully estimated from UM analysis. Prefactor b UM is obtained by fitting the data at highest available resolution (30 s) using UM estimated a UM . Such discrepency may introduce a bias in the comparison between DSD and UM approach to retrieving a power law. Hence, to understand this further and to make a fair comparison, a new power law was considered where only a DSD is obtained using Eq. The constants of the power law relations (in literature) between KE and R are not universal in application and need tweaking as per the rainfall type, measurement location as well as techniques. Based on assumed dependence between DSD parameters and rain rate in formulation exponents; [START_REF] Salles | Kinetic energy of rain and its functional relationship with intensity[END_REF] suggest four range of values for exponent values of the universal power law, and [START_REF] Uijlenhoet | A consistent rainfall parameterization based on the exponential raindrop size distribution[END_REF] propose six differ-ent relationships in their research. Hence, in order to refine the analysis of this observed variability between events and to examine possible dependence of a and b on type of rain, events were sorted according to rain types. Table 4 shows, for Pars 1, averaged values of gamma DSD parameters, indicator of the quality of the fitting, and power law coefficients from both DSD and UM estimations across events sorted according to type of rainfall. Tables for Pars 2 and PWS are given in appendix -(Table B1). For defining types of rainfall from light to extreme, a classification based on intensity [START_REF] Tokay | Evidence from Tropical Raindrop Spectra of the Origin of Rain from Stratiform versus Convective Clouds[END_REF] was employed. µ, λ and N 0 tend to decrease with heavier events, with a stronger trend for N 0 .

It should be mentioned that an opposite trend is reported in reference used; this could be due to instrumental bias in DSD measurement as the impact disdrometer used in reference is known to under-represents smaller drops in intense rainfall. It should also be noticed that the capacity of the gamma distribution to model observed DSD diminishes with heavier rainfall (this is more visible on RMSE estimation over whole range of DSD and less prominent in displayed estimate here involving only higher drop sizes), suggesting a limit in validity for gamma distribution assumption. Also, an increase in mean diameter is noticed (not shown here). These findings are in agreement with previous studies reported in literature [START_REF] Francesco | Modeling Rainfall Erosivity by Measured Drop-Size Distributions[END_REF]. When it comes to power law coefficients, both a DSD and b DSD show increase with increasing rain rate, and this is more pronounced for b DSD . Given the observed decrease in validity of gamma distribution of DSD, these trends should be taken carefully because they are likely to be mere artifacts and not representative of the actual process at stake. In the case of UM estimated parameters, b UM follows similar trend with stronger magnitude (going from 8 to 33) while a UM shows slight reduction in value with increasing rain rate (from 1.25 to 1.05). Since the average rainfall criteria used for classification here is a somehow arbitrary and more biased towards lesser rainfalls, another classification relying on the maximum of 10 minute moving average was also employed. The results are tabulated in Table B2 (Appendix B). With this criteria which is more biased towards larger rainfall events, consistent and similar results are retrieved when it comes to values and variation of power law coefficients a and b, from DSD as well as UM, suggesting robustness of obtained trends. 11

from DSD moments from UM # events µ N 0 Λ RMSE DSD a DSD b DSD a UM b UM (m -3 mm -1-µ ) (mm -1 ) (m -3 mm -1 ) (Jm -2 mm -a h a-1 ) (Jm -2 mm -a h a-
. It appears that regardless of the duration as well as type of rainfall considered, tuned relations KE DSD f it and KE UM perform significantly better than the other fixed ones, yielding r 2 values greater than 0.9. In most cases, KE UM exhibits slightly better performances with a difference not significant. The analysis also confirmed the poor performance of KE DSD , i.e. the inability of the DSD approach to properly fit the prefactor 'b'. It should be stated that the power law obtained using UM analysis is not providing significant performance enhancement compared to that obtained from gamma DSD while the prefactor b is estimated from fitting of the data (KE DSD f it ). However, with UM analysis, it is possible to discard all assumptions of DSD following a gamma distribution thus eliminating the known inadequacy of gamma model and its sensitivity to sampling resolution [START_REF] Adirosi | Comparison of different fittings of experimental dsd[END_REF][START_REF] Adirosi | Evaluation of Gamma Raindrop Size Distribution Assumption through Comparison of Rain Rates of Measured and Radar-Equivalent Gamma DSD[END_REF][START_REF] Ignaccolo | Phase space parameterization of rain: The inadequacy of gamma distribution[END_REF][START_REF] Adirosi | Raindrop size distribution: Fitting performance of common theoretical models[END_REF][START_REF] Gatidis | A critical evaluation of the adequacy of the gamma model for representing raindrop size distributions[END_REF]. The inadequacy was observed in current analysis as well, as previously discussed and illustrated in Fig. 6 and Fig. 7.

Conclusion

We examined the relationship between rainfall intensity R and time specific kinetic energy KE using high resolution (30 s) optical disdrometer data from the past 7 years in Paris region. The variability across scales of both parameters was characterized using the framework of Universal Multifractals. Analyzed KE and R times series were found to convey excellent multifractal behaviour (which is novel for KE), with multifractality index α and mean intermittency C 1 suggesting power law relation between them; it can be written as KE = bR a . Such power law was found to be valid across analyzed data, i.e. independent of the event, on whether they are computed on event or yearly basis, and of the underlying corresponding drop size distribution. Some variability in the value of the exponent a and prefactor b is reported according the event and disdrometer type.

As shown by previous results, similar power-law can be theoretically obtained when relying on the common assumption of a gamma distribution for the DSD, and a powerlaw relation between fall velocity and equivolumic drop diameter. KE-R relation obtained through UM analysis was compared with results found using this common framework and biases were acknowledged. Despite some exceptions, in most cases when a gamma DSD approximation was relevant, estimations of power law parameters from the two approaches were found to be consistent. When not, UM approach provided slightly better fit in general but not in a significant manner, keeping in mind that the pre-factor needs to fitted to data in both cases to ensure a fair comparison. Thus, the newly discussed power law relationship between KE and R retrieved with the help of UM framework generalizes previous results and theoretical formulations without having to rely on the ad-hoc assumption of a gamma DSD. The main underlying assumption of UM framework, i.e. that there is an underlying multiplicative process, is actually physically based in the sense that comes from the scale invariance features of the Navier-Stockes equations. Here, a UM analysis confirmed the validity of this assumption. Deriving the power-law relation in a multifractal framework opens the path to new approaches for simulating KE from simple R measurements. Given that complete KE measurement is much less available than R one, this impact will be investigated further in future work.

For the future, it would hence be interesting to expand the data set across geographical and meteorological conditions to reduce the biases that might have accumulated from region of measurement. (m -3 mm -1-µ ) (mm -1 ) (m -3 mm -1 ) (Jm -2 mm -a h a-1 ) (Jm -2 mm -a h a- (m -3 mm -1-µ ) (mm -1 ) (m -3 mm -1 ) (Jm -2 mm -a h a-1 ) (Jm -2 mm -a h a-

  some examples among others). Wolfensberger et al. (2017) & Schertzer and Lovejoy (2011) used UM in climatological analysis of precipitation -modelled and actual -in relation to external geographical and meteorological descriptors. Checking the validity of UM framework on KE,

C 1

 1 is equal to zero. α measures how this clustering changes with respect to intensity levels;α ∈ [0, 2]. Larger values of both corresponds to stronger extremes. Simulations of such fields can be obtained by generating a levy noise with parameter α, 'coloring' it to introduce C 1 and then taking the exponential which yields solely non-negative values.

Figure 1 :

 1 Figure 1: Location of disdrometers in Paris area, at ENPC and EP SIRTA (basemap from openstreetmap.org)

Figure 2 :

 2 Figure 2: a) Time series of R, time series of KE, KE vs R graph b) and c) MF analysis graphs with KE and R (log-log plot of Eq. 16 and Eq. 20 for TM and DTM analysis respectively and log-log plot of exponents in Eq. 20 for UM parameters) for Pars 1 event 16 September 2017 11:35:00 to 13:00:00

506

  Figure 4: a) Time series of R, time series of KE, KE vs R graph b) and c) Multifractal analysis graphs with KE and R (log-log plot of Eq. 16 and Eq. 20 for TM and DTM analysis respectively and log-log plot of exponents in Eq. 20 for UM parameters) using the year based analysis, for Pars 1, for 2017

Figure 5 :

 5 Figure 5: a) Results from multifractal analysis on continuous year wise data set: a) α and C 1 (DTM); b) a and b values

  Figure 6: a) 2019 Pars 1 event where empirical DSD corresponds with gamma distribution and b) where it doesn't follow gamma distribution (DSD displayed only till diameter class around 9 mm as higher drops were not observed during the events)

Figure 7

 7 Figure 7: a) r 2 values between KE UM and KE DSD for Pars1, 2019; b) r 2 values between KE UM and KE DSD f it (b from fitting of data) for Pars1, 2019 (RMSE calculated by considering only parts of DSD where drop diameter > 0.5 mm)

  Fig. 6a and Fig. 6b.

Figure 8 :

 8 Figure 8: Variation of a DSD and b DSD for each time steps in an event (same event discussed in Fig. 2) a) Correspondence between empirical DSD (red) and calculated DSD (blue); d) Correspondence between empirical rainfall rate (R emp ) and that calculated from DSD moments (R DSD ) b) & e) Variation of a DSD and b DSD within the event c) & f) Variation of a DSD and b DSD with DSD parameters N 0 and µ

Figure 9 :

 9 Figure 9: Variation of power law coefficients from UM and DSD calculation, for events in year 2019 (similar variation for other years also)

Table 4 :

 4 Variation of DSD parameters and power law coefficients according to the type of rainfall for Pars 1 (R = average rain rate for rainy time steps) To evaluate the performance of established power law relationship across scales of measurement, KE was calculated from empirical R using KE = bR a with average values of a and b and compared with existing relations in literature. For this purpose, exponential equation used in RUSLE (KE BF = 29[1 -0.72exp(-0.05R)], Brown and Foster, 1987), exponential equation used in RUSLE2 (KE MG = 29[1 -0.72exp(-0.082R)], C. McGregor et al., 1995), universal exponential law proposed by van Dijk et al. (2002) (KE V D = 28.3[1 -0.52exp(-0.0421R)]) and ideal power law proposed by Shin et al.(2016) (KE Shin = 10.3R 11/9 ) were used alongside KE UM and KE DSD derived here (power law where fitted values of b were used -KE DSD f it -is also included). For UM and DSD power laws in this paper, average values among all calculated events were used for representation according to make of disdrometers. Average values of coefficient a and prefactor b were close for both Pars 1 and Pars 2 disdrometers and were taken common for the Parsivel 2 make (for both UM and DSD). It is also logical to keep values of power law coefficients separate between different disdrometers as varying KE and R estimation has been reported across types of disdrometers used[START_REF] Angulo-Martínez | Measurement uncertainty in rainfall kinetic energy and intensity relationships for soil erosion studies: An evaluation using PARSIVEL disdrometers in the Southern Appalachian Mountains[END_REF][START_REF] Angulo-Martínez | Comparison of precipitation measurements by ott parsivel 2 and thies lpm optical disdrometers[END_REF] Johannsen et al., 2020b). Figure10shows KE and R variation fitted with above mentioned equations for maximum resolution, 30 seconds. As expected KE UM provides better fit than KE DSD (and KE DSD f it shows closer fits). With respect to empirical data, UM power laws exhibit -for all three devices -slightly better or comparable coefficient of determination (r 2 ) with regards to commonly used relations. The exponential equations from literature appear very close to each other. The relatively lower values of coefficient of determination than that during event based fits are likely to be due to the effect of using average values instead of event specific values of a and b.

Figure 10 :

 10 Figure 10: Fitting of empirical KE -R using power laws from UM and DSD, and popular expressions from literature for a) and b) Parsivel 2 , and c) PWS

Figure 11 :

 11 Figure 11: r 2 values of various KE-R relations discussed for a) events analysed in 2017, 2018 and 2019 (for Pars 1), b) all events grouped according to type of rain -light, moderate and heavy Average value for each year is given in legends, with KE UM and KE DSD f it displayed in dotted lines

  The devices are part of the TARANIS observatory (exTreme and multi-scAle RAiNdrop parIS observatory, Gires et al., 2018) of the Fresnel Platform of École des Ponts ParisTech (https://hmco. enpc.fr/Page/Fresnel-Platform/en); and are operated by Hydrology Meteorology and Complexity laboratory of École des Ponts ParisTech (HM&Co-ENPC).Table 1 gives a brief outline of measurement campaigns. From November 2016 to September 2017

the instruments were moved to École Polytechnique (EP) on SIRTA (Site Instrumenté de Recherche par Télédétection Atmosphérique) for a joint intensive measurement campaign over the Ile-de-France region. The aforementioned location is about 38 km away from ENPC towards south west of Paris. To summarise, the measurements presented in the following sections involve two different types of optical disdrometers and data collected from two different locations of Paris region.

Table 2 :

 2 2 at around 1.17 and b values greater at 13.02. PWS generally registered slightly higher intensities than Pars 1 and Pars 2. It should be mentioned that a and b seem to show a very rough correlation where values of b decreases with increase in values of a. But between make of instruments, irrespective of the type and number of events, values of a remains rather constant. Differences in values obtained between Parsivel 2 and PWS is expected due to operational differences between disdrometers. Such effects are also reported in Johannsen et al. (2020a) who showed biases in measurement and subsequent R -KE relation due to difference in type of sensors used (three optical disdrometer were used). Angulo-Martínez and Barros (2015) also highlighted some differences among various Parsivel 2 .Figure 3: Year wise α and C 1 variation on event based analysis from 2016 to 2019 (remaining years are provided in appendix) a, b and no. of events analysed between 2013 and 2019 according to location of measurement and disdrometer used.

	29

* From Nov 2016 to Sep 2017 as shown in Table

Table 3 :

 3 a and b from year based analysis, from 2014 to 2019 according to disdrometer used for measurement.

				1.356	1.273	1.278
		b	10.232 6.102	8.033	8.247
	Pars 2	a b	1.303 7.605	1.290 8.338	1.286 7.463	1.273 7.695
	PWS	a b	1.171 12.616 13.614 12.792 14.981 1.164 1.190 1.190

* From Nov 2016 to Sep 2017 as shown in Table

Table B1 :

 B1 Variation of DSD parameters and power law coefficients according to the type of rainfall (R = average of rainy data points)

										1 )
		very light R < 1	190	2.53 1.36 ×10 7 6.59	19.78	1.196	10.180	1.246	8.097
		light	1 ≤ R < 2	160	1.61 1.20 ×10 8 4.70	24.81	1.229	12.630	1.202	9.262
	Pars 2	moderate 2 ≤ R < 5 heavy 5 ≤ R < 10	101 28	0.82 1.66 ×10 6 3.19 0.26 2.72 ×10 4 2.14	20.93 23.85	1.260 1.281	15.026 17.171	1.208 1.247	9.911 10.140
		very heavy 10 ≤ R < 20	8	-0.38 5.81 ×10 2 1.36	21.20	1.317	19.951	1.098	19.997
		extreme	R ≥ 20	6	-0.94 6.16 ×10 2 1.07	57.99	1.363	15.593	1.180	19.040
		very light R < 1	163	2.79 7.82 ×10 6 4.73	10.95	1.187	10.95	1.194	12.070
		light	1 ≤ R < 2	176	2.65 2.64 ×10 7 4.15	14.30	1.195	14.30	1.154	12.703
	PWS	moderate 2 ≤ R < 5 heavy 5 ≤ R < 10	152 46	1.87 4.73 ×10 3 2.85 1.16 2.82 ×10 3 1.96	14.98 18.90	1.216 1.253	20.024 24.439	1.169 1.193	13.136 13.933
		very heavy 10 ≤ R < 20	18	0.59 7.07 ×10 2 1.51	19.24	1.266	20.834	1.133	18.983
		extreme	R ≥ 20	6	0.53 4.09 ×10 2 1.28	20.63	1.26	21.031	1.165	18.016
						from DSD moments		from UM
				# events µ	N 0	Λ	RMSE DSD a DSD	b DSD	a UM	b UM

Table B2 :

 B2 1 ) Variation of DSD parameters and power law coefficients according to the type of rainfall (R = maximum value of 10 minute moving average)

		very light R < 1	13	3.82 1.30 ×10 7 8.19	8.19	1.161	9.471	1.245	8.396
		light	1 ≤ R < 2	82	2.63 7.76 ×10 6 6.39	15.96	1.191	10.962	1.271	8.747
	Pars 1	moderate 2 ≤ R < 5 heavy 5 ≤ R < 10	235 134	2.04 1.10 ×10 5 5.32 1.44 2.22 ×10 7 4.19	21.77 24.79	1.213 1.235	12.528 13.826	1.227 1.225	9.119 9.369
		very heavy 10 ≤ R < 20	54	0.51 4.55 ×10 3 2.67	20.75	1.269	16.214	1.197	11.697
		extreme	R ≥ 20	38	-0.22 1.77 ×10 3 1.67	26.84	1.314	26.84	1.147	16.642
		very light R < 1	13	4.27 2.02×10 7	8.94	6.75	1.154	9.097	1.217	7.833
		light	1 ≤ R < 2	82	2.97 2.96×10 6	7.04	15.74	1.183	10.165	1.253	8.245
	Pars 2	moderate 2 ≤ R < 5 heavy 5 ≤ R < 10	215 112	1.99 9.38×10 7 1.17 3.20×10 6	5.43 3.94	22.15 24.33	1.214 1.239	11.857 12.969	1.212 1.222	8.810 9.280
		very heavy 10 ≤ R < 20	42	0.39 5.92×10 3	2.74	25.49	1.276	15.369	1.196	10.696
		extreme	R ≥ 20	29	-0.63 1.78×10 3	1.55	35.88	1.343	18.133	1.192	13.445
		very light R < 1	6	3.84 3.39×10 4	6.15	6.78	1.16	13.92	1.223	13.322
		light	1 ≤ R < 2	61	3.26 5.29×10 4	5.27	10.13	1.175	14.687	1.187	11.997
	PWS	moderate 2 ≤ R < 5 heavy 5 ≤ R < 10	201 168	2.59 1.91×10 7 2.32 2.78×10 6	4.20 3.50	12.56 14.65	1.195 1.202	17.258 19.013	1.176 1.168	12.602 12.702
		very heavy 10 ≤ R < 20	75	1.72 3.31×10 3	2.65	16.90	1.223	20.846	1.155	14.401
		extreme	R ≥ 20	50	0.67 1.11×10 3	1.69	19.48	1.264	20.879	1.156	15.430

Acknowledgements

The authors greatly acknowledge partial financial support from the Chair of Hydrology for Resilient Cities (endowed by Veolia) of the École des Ponts ParisTech, EU NEW INTERREG IV RainGain Project, EU Climate KIC Blue Green Dream project, the Île-de-France region RadX@IdF Project, and the ANR JCJC RW-Turb project (ANR-19-CE05-0022-01)

Appendix A: Multifractal events year wise

DSD parameters and power-law coefficients