# Multifractal comparison of rainfall measurement with the help of a disdrometer and a mini vertically pointing Doppler radar

Mateus Seppe Silva, Rodrigo Vieira Casanova Monteiro, Jerry Jose, Auguste Gires, Igor Paz, Ioulia Tchiguirinskaia, Daniel Schertzer

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## ► To cite this version:

Mateus Seppe Silva, Rodrigo Vieira Casanova Monteiro, Jerry Jose, Auguste Gires, Igor Paz, et al.. Multifractal comparison of rainfall measurement with the help of a disdrometer and a mini vertically pointing Doppler radar. Hydrological Sciences Journal, 2025, 70 (7), pp.1072-1090. 10.1080/02626667.2025.2464948. hal-05213530

HAL Id: hal-05213530

https://enpc.hal.science/hal-05213530v1

Submitted on 18 Aug 2025

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Distributed under a Creative Commons CC BY 4.0 - Attribution - International License
Multifractal comparison of rainfall measurement with
the help of a disdrometer and a mini vertically pointing
Doppler radar

Mateus Seppe Silva¹,², Rodrigo Vieira Casanova Monteiro¹,², Jerry Jose¹,
Auguste Gires¹*, Igor Paz², Ioulia Tchiguirinskaia¹, and Daniel Schertzer¹

¹École nationale des ponts et chaussées, Institut Polytechnique de Paris,
Champs-sur-Marne, France

² Military Institute of Engineering (IME), Rio de Janeiro, Brazil

Corresponding author: Auguste Gires, auguste.gires@enpc.fr

Abstract

Local rainfall measurements include a wide range of methods with
different concepts and assumptions. Understanding the differences
enables to determine the most reliable device for each raining
condition. Here, we compare two such devices: Parsivel□² (optical
disdrometer by OTT) and mini Doppler radar (part of mini
Meteorological Station by Thies). The performance assessment
followed two focuses: evaluating the scaling features using the
framework of Universal Multifractals (UM), and analyzing the
influence of physical parameters (drop size, wind velocity and rainfall
rate).

The devices were mounted on a meteorological mast, in Pays d'Othe
wind farm, 110 km southeast of Paris. The mini Doppler radar
returned very low rain measurements during conditions with bigger
drops, and showed non-detection in light rain involving heavy wind.
UM analysis revealed similar scaling behaviour for both devices,
which extended to finer scales of mini Doppler radar data till 4s
beyond which white noise was detected.
**keywords:** rainfall; wind; disdrometer; Doppler radar; multifractal; drop size

# 1 Introduction

Rainfall exhibits extreme variability over wide ranges of space-time scales. More precisely it is an intermittent process where the highest fluctuations are concentrated on small fractions of the embedding time-space. This makes rainfall difficult to characterize, model and even measure. Rainfall measurements are widely available, often as a single quantity of rain rate (mm h⁻¹) or cumulative rainfall depth (mm) over a given rainfall period. This quantity gives very useful information regarding the intensity and occurrence of rain in general, but, it does not capture the complexity of all the physical processes involved with rainfall. Size resolved measurement devices such as disdrometers hence augment our comprehension by providing information on the microphysical phenomena, by measuring raindrops of different sizes falling at different velocities. This is commonly done with the hypothesis of a drop size distribution (DSD). DSD is assumed to encompass smoothly the distribution of falling drops per unit volume of air and hence helps in having a better understanding of the various after-effects of rainfall such as erosion (see Serio et al., 2019, for a review), the effect of precipitation on vegetation (Zore et al., 2022), dispersal of pollutants and sediments (Blanco-Alegre et al., 2021), leading edge erosion on wind turbines (Herring et al., 2019) etc. DSD also enables retrieval of various rainfall related quantities related to DSD moments, such as rain rate, kinetic
energy etc. However, obtaining DSD estimates requires more
sophisticated and expensive instrumentation as opposed to the widely
available traditional bulk measurements. For an accurate estimation of
these bulk quantities, it is important to compare them with those involving
DSD characterization.

Commonly available rainfall measurement techniques which provide point
values of rain rates (or cumulative rainfall depths) are weighing-type rain
gauges involving floats or tipping buckets with manual or automated data
loggers. On measuring the microstructure of rain, i.e. the DSD,
disdrometers are the commonly used devices. They measure the size and
velocity of precipitation drops according to signal from raindrop impact
(impact disdrometers) or optical signal differences due to drops passing
through sampling area (optical disdrometers). Impact-based disdrometers
are known to have limitations in sampling, dead-time correction, noise
and quantification etc. (Hashiguchi et al., 2018; Joss and Waldvogel, 1969;
Sheppard and Joe, 1994), and rely on strong assumptions on the relation
between size and velocity of falling drops. Optical disdrometers possess a
comparatively wider range of spectrum and provide more accurate
measurements; they are also more expensive and more demanding in
terms of processing. Though considered durable, they also share some of
the limitations of impact disdrometers; for example, the widely used OTT
Parsivel®² is reported to have limitations owing to its design and limited
sampling area (Montero-Martínez et al., 2016; Tokay et al., 2014), and is
known to underestimate the number of small drops (i.e. typically smaller
than 0.3 mm). Another development in the disdrometer type is the 2-
dimensional video disdrometer (2DVD) which records individual
information of rain drops as opposed to the bin-based classification
commonly provided by optical disdrometers; 2DVD also uses a bigger
sampling area (Adirosi et al., 2020; Kruger and Krajewski, 2002). Several
studies have compared the performance between various disdrometers:
Johannsen et al. (2020) and Angulo-Martínez et al. (2018) compared
optical disdrometers from different manufacturers by focusing on
differences in DSD and fall velocity, Gires et al. (2016) compared various
rainfall parameters between two different optical disdrometers using
multifractal tools, Park et al. (2017), Thurai et al. (2011), and Liu et al.
(2019) provide comparisons between commonly used optical
disdrometers and 2DVD. Various weather stations include precipitation
measurements through a vertically pointing mini Doppler radar sensor
which provides local measurements by integrating the information over a
vertical column. It provides rainfall estimates by measuring the fall
velocities and associating them with equivolumic diameter with the help
of theoretical assumptions on the relation between DSD and average fall
velocity (Cluckie et al., 2000). Since the output is bulk information of
rainfall, characterization of DSD is not possible with the help of these
devices. Comparison of rainfall measurements between such an
instrument (meteorological station by ThiesCLIMA, 2013) and one of the
widely used disdrometer (OTT Parsivel<sup>2</sup>), while accounting for the
variability across scales and intermittency, is at the core of this study.

Rainfall varies across a wide range of spatio-temporal scales, as briefly
mentioned earlier, and many studies have tried to characterize the
process through scaling analysis over the years (Deidda et al., 1999; García-Marín et al., 2008; Gupta and Waymire, 1990; Kumar and Foufoula-Georgiou, 1993; Olsson and Niemczynowicz, 1996). A first step in scaling analysis is to study power-law spectra. However, this second-order statistics is insufficient for characterizing the varying intensities of the field, and their intermittency (and hence rare events). To overcome these limitations, the framework of Universal Multifractals (UM) considers an infinite number of fractal dimensions (hence "multifractal") to characterize all ranges of intensities across scales through scale invariant singularities. It relies on the idea that the underlying process is a stochastic multiplicative cascade (Schertzer and Lovejoy, 1987). The fields are assumed to inherit the scale invariant properties of governing equations of the atmosphere - Navier Stokes - and exhibit scale invariant properties. For a conservative multifractal field, variability and intermittency across scales can be characterized using UM with only two parameters having physical meaning (see Schertzer and Lovejoy, 2011) for a review). UM have been widely used as a powerful tool for analyzing and modelling various complex spatial and temporal phenomena, including rainfall (Gires et al., 2015; Gires et al., 2012; Hubert et al., 1993; Royer et al., 2008; Wolfensberger et al., 2017.), wind (Schmitt et al., 1994), temperature. In particular, Gires et al. (2016) and Jose et al. (2022) has previously used UM framework for rain analysis from one of the measurement devices in current study, Parsivel².

As previously stated, in this research, we have compared two different precipitation measuring devices - an optical disdrometer (OTT Parsivel²)
and a vertically pointing mini Doppler radar (meteorological station by
ThiesCLIMA). Parsivel□² uses occlusion based optical techniques and
provides a size resolved measurement while vertically pointing mini
Doppler radar accumulates vertical information through DSD assumptions.

Hence, we expect the former to be more direct and accurate (within
known limitations) in comparison. Quantifying their differences in bulk
rainfall measurement, across scales, forms the main objective of our work.

This was done in a two-folded approach: first by examining the scaling
properties of the rain field collected by the two devices, and then by
examining the influence of various meteorological parameters on rainfall
measurements (rain rate, mean diameter, wind). Towards this, rain events
were analysed for a time period of around 1.5 years (12/11/2020 to
05/31/2022) from two vertical heights at an operational wind farm, 110
km southeast of Paris, France. UM correlations between the atmospheric
fields are outside the scope of this paper and are studied elsewhere (Jose
et al., 2024). In the upcoming section, devices and data are presented
along with an overview of the measurement campaign in section 2. The
framework of UM is also introduced in this section, and the results of the
multifractal analysis are presented in section 4.1. This section is followed
by further statistical analysis based on meteorological parameters
subsection 5), and overall conclusions and discussions in section 6.
# 2 Data, Campaign and Methodology

## 2.1 Campaign and Measurement

The research was made by analyzing data from RW-Turb project (Rainfall, Wind and Turbulence, https://hmco.enpc.fr/portfolio-archive/rw-turb/), which aims at studying the effect of rainfall on wind power production. As part of this project, a joint measurement campaign between École nationale des ponts et chaussées and Boralex was set up at Pays d'Othe wind farm, 110 km Southeast of Paris. According to Meteo-France (https://meteofrance.com/comprendre-climat/france/le-climat-en-france-metropolitaine), the climate in the area is a degraded oceanic one which corresponds to a transition between oceanic climate and continental one. Average annual rainfall are reported in section 5.1. All details of the measurement campaign, i.e. site description, devices and data available can be found in (Gires et al., 2022) and only the main elements are reminded here. 3-month long data is also made available there for the public along with the scripts required for basic data processing. During this campaign, two sets of disdrometers (denominated in this work Parsivel 1 and Parsivel 2), meteorological stations (Station 1 and Station 2), and 3D sonic anemometers are attached on a meteorological mast (see Fig. 1, instruments at ~ 75 m and ~ 45 m height from the ground; denoted as location 1 and location 2 respectively) which is operated on the wind farm. Within a 1 km radius of the mast, there is a small slope in the North-South direction. All devices were calibrated in factory by the manufacturers, and were installed by experts. Only rainfall measurements from disdrometers and meteorological stations were analyzed for the
research presented in this paper. 2D wind measured by station was also
used for specific analysis. Figure 1 brings a quicklook of a day of the
measurement campaign, as an example of the information collected for
disdrometers. It displays the rain rate vs. time, the temporal evolution of
cumulative rainfall depth and the drop size distribution as well as wind
and temperature time series. Interested readers are directed to (Gires et
al., 2022) for detailed information on quicklook graphs; the data for all
campaign days are visible
at https://hmco.enpc.fr/calendars/Calendar_RW_Turb wind farm/.

2.1.1 *Disdrometer*

In this research, measurements from two OTT Parsivel☐² optical disdrometers were used (see references in introduction or the device documentation OTT (2014). Parsivel☐² works by shining a laser beam from a transmitter to a receiver through a sampling area of 54 cm☐². From the reduction in the intensity of the beam and its duration due to the passage of raindrops through this sampling area, Parsivel☐² estimates the size and fall velocity of hydrometeor. For this, it assumes an ellipsoidal shape model for drops using standard relation between the axis ratio and equivolumic diameter, i.e., the diameter of a spherical drop with the same volume. Then, the data is binned and the number of drops according to classes of size and velocity is reported for each 30 s time step. Parsivel☐² measures equivolumic diameter (Dᵢ: from 0.062 mm to 24.5 mm) and velocity (vᵢ: 0.05 m.s☐⁻¹ to 20.8 m.s☐⁻¹) in 32 classes (indexed as i and j
width of these classes are not uniform, as they are more and more refined towards smaller values where most drops concentrate.

With a sampling interval of 30 s (Δt), Parsivel□² provides a matrix containing the number of drops (nᵢⱼ) as per their equivolumic diameter (defined by a centre value of Dᵢ and a width ΔDᵢ) and fall velocity (defined by a centre value of vⱼ and a width Δvⱼ). From this raw matrix, the number of drops per unit volume in each diameter class or the drop size distribution (DSD, denoted as N(D)) can be obtained in a discrete way:

$$N(D_i) = \frac{1}{S_{eff}(D_i) \Delta D_i \Delta t} \sum_j \frac{n_{i,j}}{v_j} \quad (1)$$

where nᵢⱼ is the number of drops in the bin Dᵢⱼ□, Sₐₗₐₗ(Dᵢ) the sampling area of disdrometer in mm□² (which can be slightly adjusted according to drop size to account for side effects), Δt is the time step duration in seconds.

From this, N(Dᵢ)ΔDᵢ gives the number of drops with a diameter in the class i per unit volume (in m⁻³). The studied rainfall or rain rate (mm.h⁻¹) is obtained from raw matrix using the following expression for each time step:

$$R = \frac{\pi}{6 \Delta t} \sum_{i,j} \frac{n_{i,j} D_i^3}{S_{eff}(D_i)} \quad (2)$$

The mean equivolumic diameter (Dₘ) is obtained from the ratio of fourth and third moment of the DSD (Smith and Kliche, 2005). This is commonly used to characterize the shape of DSD spectrum. With the discrete DSD available from data, it is computed as:

$$D_m = \frac{\sum_{i=D_{min}}^{D_{max}} N(D_i) D_i^4 \Delta D_i}{\sum_{i=D_{min}}^{D_{max}} N(D_i) D_i^3 \Delta D_i} \quad (3)$$
Detailed information regarding functioning, data set, application of
Parsivel□² and comparison with other optical disdrometer can be found in
(Gires et al., 2018). Parsivel□² are widely used for rain measurements;
some examples of recent studies involving evaluations and comparison
are Tokay et al. (2014), Angulo-Martínez et al. (2018), Lin et al. (2021)
among others.

2.1.2 Meteorological Station (Vertically Pointing Doppler Radar)

Rain measurement from an optical disdrometer is compared with that of a
vertically pointing Doppler radar here, which is included in the
meteorological station manufactured by ThiesCLIMA (ThiesCLIMA, 2013).
The vertically pointing Doppler radar houses an array of sending aerials
that beams a very small electromagnetic signal (mW range) and
measures the transmission and reflection by rain droplets; from the
difference frequency of the successive pulses in transmitted and reflected
signals, the relative speed of raindrop movement (towards the module
since drops are falling) is obtained. From a power law relation between fall
velocity of drops and their diameter, individual volumes and rainfall
intensity are computed (Gunn and Kinzer, 1949). Unlike disdrometer
which provides the spectrum, the station only provides a single value
measurement of rain in mm.h⁻¹, which corresponds to an average
behaviour over a vertical column above the device during the time step.
The exact assumptions used in this estimation are not provided by the
manufacturer and we are limited to what is available in the manual.
For practical reasons, and since the rain measuring sensor and the wind
measuring sensor are located within the meteorological station, this
device is referenced throughout this work as "Station". Like the
disdrometer, the Station is also a punctual measurement device in the
sense that it measures rainfall only in a cone above the device with
limited spatial extension. It also measures wind speed and direction, air
temperature, relative humidity and brightness at the location. These
measurements are made available at a much finer observation scale than
the disdrometers, at 1 Hz as specified by the manufacturer. However,
later analysis suggested the presence of 'white noise' for time steps below
2 s limiting the resolution; this is discussed in the analysis section. From
the quicklook in Fig. 1a (first row), it can be seen that the rainfall intensity
and cumulative depth from Station follow the trends of disdrometers ones.

## 2.2 Data quality and selection

Fig. 2 gives an overview of the whole data used for analysis here. In
general, the Parsivels returned larger rain measurements when compared
to the stations. Investigating the possible reasons for these discrepancies
- which will be discussed later - was the main point of the analysis. It is
important to highlight that the difference between the upper set of
devices (Parsivel 1 and Station 1) was bigger than the difference of the
lower one (Parsivel 2 and Station 2), which is mostly due to the stations. In
other words, while the Parsivels reported similar rainfall amounts, the
stations returned further apart results. Fig. 2a and 2b show the rain rate
measured by each device for each time step (30 seconds), with Parsivels
in blue and stations in orange. It can be perceived that, as a general
observation, the Parsivels consistently recorded greater peak values than
the stations. Fig. 2c brings the total cumulative rain measured by the four
devices during the whole campaign, with the orange values representing
the amount measured by location 2 devices during the 3 months that
location 1 was out of power following a lightning impact on the mast.
When comparing devices in different sets and in the multifractal analysis,
those months were removed for data homogenization.

Also, it should be noted that the upper set (location 1) recorded stronger
winds, which is consistent with the physical expectations. In the upcoming
sections, we will discuss the influence of wind and the mean diameter of
the raindrops (D_m) with the help of some scatter plots, made at the time
steps of the Parsivel data. Additionally, we will study how the ratio
between the cumulative rain measurements for both devices behaved,
based on a division of the events (introduced below) by wind, D_m, and rain
classes.

For the analysis, an event was defined as a period of time where rain was
perceived, with preceding and succeeding dry periods of at least 15
minutes. If a dry period lasted less than that amount of time, the two
adjacent rain periods were considered as part of the same event. In total,
there were 184 events in the considered period, which started on
12/11/2020 and ended on 05/31/2022. The average duration of the events
was approximately of 3 hours and 23 minutes. Also, in order to avoid the
influence of snow in the analysis, we have established a minimum temperature filter for the events. Thus, events, for which temperatures were detected below 2°C, were not considered. In order to explore the data, the rain events of less than 1 hour of duration were identified by the Parsivel and then discarded for multifractal ensemble analysis, since the latter was performed on scales ranging from the measurement scale to one hour. This discarded data in UM analysis for technical reasons is included in the later statistical analysis involving whole data where the aim was to investigate the general differences in performance between both instruments. As different instruments are recording rain slightly differently, selecting rain events using only Parsivel 1 time series left some events with higher occurrences of zeroes in Station 1 and Station 2; the presence of zeroes distorts the UM analysis and is known to yield biased estimates (Gires et al., 2012). To account for this, another filter was applied to remove the events where rain occurrence in Station 1 was less than 50%. The total amount of events removed, adding the potential snow and the percentage filter, was 78. A table with some statistics of the events, including if they were removed or not, is can be found in supplementary material (Tab. S1).

In upcoming sections, the difference in device performance is analysed on the basis of events, an ensemble of events and time steps. For this, the events and time steps were classified into the following categories according to the parameters under consideration (Tab. 1). The criteria was selected on the basis of the spectrum of values observed during rain events in the measurement location.
Having this rather crude category division was helpful, despite its arbitrariness, to give a suggestive identification of patterns and relations between rain intensity, $D_m$ sizes, and wind velocity, pending a multi-physics analysis.

### 3 Methodology: UM Framework

For comparing the devices, the scaling properties of rain were analyzed using spectral analysis and the framework of Universal Multifractals (Schertzer and Lovejoy, 2011, 1987). Spectral analysis is widely used for characterizing scaling properties; here, the second-order statistics of rain in the frequency domain were examined for power-law scaling as follows:

$$E(k) \approx k^{-\beta} \qquad (4)$$

where k corresponds to the temporal frequency (or wave number, proportional to the invert of scale) and β is the spectral exponent.

However, to fully characterize the complexity of rain, across its intensities and spatiotemporal variation, information on higher and lower-order statistics is required. For this, we use UM which relies on the assumption that the studied field is being generated by an underlying multiplicative cascade process with conserved statistical properties across scales. The rainfall field is assumed to inherit the scale invariant properties of the governing Navier-Stokes equations.

Before describing UM framework, let us begin by considering the occurrence pattern of rain fields (1 for rain, 0 for no rain or dry). It is possible to characterize it using a unique fractal dimension $D_F$ valid across resolutions (λ, ratio of outer scale to observational scale). For a fractal
field, the probability p of having 1 (i.e. a rainy time step) in such a binary field scales according to the resolution:

$$p = \frac{\lambda^{D_F}}{\lambda^D} = \lambda^{-c_F} \qquad (5)$$

where $c_F=D-D_F$, is the fractal co-dimension of the field. $\lambda^{D_F}$ is the number of rainy pixels among $\lambda^D$ of the embedding of dimension $D$ (equal to 1 for time series as in this paper, or 2 for maps). To fully characterize the rain field one would need infinite number of $D_F$, corresponding to various intensities defining the occurrence of rain. In UM, this is done using the scale invariant notion of singularity $\gamma$ ($\gamma=\log_{\lambda}(\epsilon_{\lambda})$), and their corresponding codimension function $c(\gamma)$, which compiles information of corresponding intensities across all scales (Schertzer and Lovejoy, 1988; Schertzer and Lovejoy, 1987). Probability of exceeding an intensity ($\lambda^\gamma$) then becomes:

$$p(\epsilon_{\lambda} \ge \lambda^{\gamma}) \approx \lambda^{-c(\gamma)} \qquad (6)$$

where $\epsilon_{\lambda}$ is the studied field at any resolution $\lambda$, which has previously been normalized to one on average. In practice, only discrete resolution $\lambda$ are studied starting from the maximum one and iteratively dividing it by 2. For a multifractal field, the above equation implies that corresponding statistical moments $q$ scale with resolution (Schertzer and Lovejoy, 1987):

$$(\epsilon_{\lambda}^q)^{1/q} \approx \lambda^{K(q)} \qquad (7)$$

where $K(q)$ is the moment scaling function, characterizing moments across all the resolutions considered. $K(q)$ and $c(q)$ are equivalent functions (through Legendre transform relating moment orders and probability density functions) and they both characterize variability of the field across all scales considered. This also means that for every singularity $\gamma$, there is
a unique corresponding statistical order moment $q$ and vice versa (Parisi, 1985; Schertzer and Lovejoy, 1985).

In general, the underlying multiplicative cascade process converges to an universal behaviour through application of a generalized central limit theorem (Schertzer and Lovejoy, 1997). Thus, for a conservative field in UM, $K(q)$ (and $c(y)$), which in theory could require infinite parameters to be defined, can actually be fully characterized with the help of only two parameters, $\alpha$ and $C_1$ (Schertzer and Lovejoy, 2011, 1987).

$$ K(q)=\zeta \qquad (8) $$

$C_1$ measures the mean intermittency codimension and quantifies the clustering of the (average) field across scales. It is the fractal co-dimension of the singularity corresponding to $q=1$. For a one-dimensional field, like rainfall time series, the value of $C_1$ ranges from 0 to 1; $C_1 = 0$ corresponding to an homogeneous field with little variability. $\alpha$, the multifractality index, measures how fast the intermittency evolves when the intensity level is changed ($\alpha \in [0,2]$).

If a field is non-conservative (say $\phi_\lambda$), i.e. its average is not conserved across scales, then its scaling behaviour can be characterized as a combination of a conservative field, $\epsilon_\lambda$, and non-conservation parameter $H$ : $\phi_\lambda \approx \epsilon_\lambda \lambda^{-H}$. From this, the conservative field ($\epsilon_\lambda$) can be retrieved using fractional integration or differentiation depending on the value of $H$. $H$ is related to the spectral slope as follows:

$$ \beta=1+2H-K(2) \qquad (9) $$

where, $K(2)$ refers only to the conservative part of the field. In the current analysis of rain, fields were found to be non-conservative ($\beta$ typically
greater than 1) and the conservative part (i.e. with $\beta \sim 1$ for time series like here, and $H \sim 0$) was retrieved by taking the absolute value of the fluctuations, which are taken as the modulus of the local finite difference gradient operator. This enables to avoid biases in the UM analysis (Lavallée et al., 1993). UM parameters can be estimated using Trace Moment (TM) analysis (log-log plot of Eq. 7), and local derivatives of $K(q)$ at $q = 1$ ($K'(1)=C_1$; $K''(1)=C_1\alpha$). The quality of scaling can be determined by $r^2$ value of linear regression for TM curve at each moment $q$; as a reference, in this paper, the value at $q = 1.5$ is used. For our analysis, the double TM (DTM) technique was used for a more robust estimation of UM parameters, as it overcomes some limitations of simpler techniques being intrinsically designed for UM fields (Lavallée et al., 1993). Using DTM methodology, it is possible to directly determine UM parameters from the moment scaling function curve. For this, the field is re-normalized and raised to a power $\eta$ at the maximum resolution:

$$ \langle \varepsilon_{\lambda}^{\eta} \rangle = \frac{\langle \varepsilon^{\eta q} \rangle}{\langle \varepsilon_{\lambda}^{\eta} \rangle^{q}} \approx \lambda^{K(q, \eta)} \qquad (10) $$

Following Eq. 7, corresponding moment scaling function becomes:

$$ \lambda^{K(q, \eta)} \approx \frac{\lambda^{K|\eta q|}}{\lambda^{qK|\eta|}} = \lambda^{K|\eta q| - qK|\eta|} = \lambda^{\eta q K|q|} \qquad (11) $$

From log-log plot of Eq. 10, for given moments, $q$, corresponding $K(q, \eta)$ curves are obtained the same way as in TM. From log-log plot of $K(q, \eta)$ vs. $\eta$ (Eq. 11), the values of UM parameters can be obtained. This curve has a linear portion, and $\alpha$ is directly the slope and $C_1$ is derived from the intercept (Eq. 8). Here the linear portion of the curve is considered around $\eta=1$. The values at the moment $q=1.5$ are considered as reference here
and it should be mentioned that similar values can be obtained through
other moments as well (as implied from similar slopes of DTM curves).

# 4 Results and discussions

## 4.1 Multifractal comparison of device measurements

To compare the performance between disdrometers and stations, rain
events from both sets of devices on the meteorological mast were
subjected to UM analysis. Following the criteria and filters introduced in
section 2.2, a total of 106 (see below explanation for this number) rain
events were selected from 12/11/2020 to 05/31/2022 using Parsivel 1
time series.

### 4.1.1 UM analysis of rain events as an ensemble

An ensemble of all rain events was formed for each device with a sample
size of roughly 1 hr (64 minutes). This means that longer events were split
into various one hour samples. Both disdrometer and station data were
used with time steps of 30 s (the instrumental measurement time step of
disdrometers); for this, station data which is originally available at 1 s was
averaged to 30 s. The forming of ensembles here consists of splitting the
time series into arrays of length equal to sample size (roughly 1 hr, more
precisely of 128 = $2^7$ time steps) and combining them together as columns
of a single matrix. This ensemble of samples is then used in UM analysis.
When the size of an event was not a multiple of the sample size, only the
portion of the series till the largest multiple was selected, and the rest was
discarded. Similarly, events of length less than the sample size (1 hr) were
discarded as well.

The fractal dimension ($D_F$) of the support, corresponding to the
occurrence of rain, is retrieved from the slope of $N_\lambda=\lambda^{D_F}$ in log-log form
where $N_\lambda$ is the number of active structures (rain occurrence) at resolution
$\lambda$. Figure 3a shows the computation of $D_F$ for Station 1 ensemble, each
point of $\lambda$ corresponds to a power of 2, from 0 at the top right
(corresponding to 30 s) to the number of time steps of the ensemble (128,
corresponding to approximately 1 hr) at the bottom left. In a time series
with non-zero values in every scale ratio (i.e. always raining), the $D_F$
would be 1, and it is possible to say that the greater the number of zeros,
the smaller the $D_F$. Since only rain events are used in the analysis,
following the criteria in section 2.2 (based on Parsivel 1 data), $D_F$ close to
1 was obtained for all ensembles as expected (Fig. 3a, 3e, S1a, S1e). The
smallest value of $D_F$ among devices was 0.94 (which was 0.92 before the
removal of events using the zero-filter introduced before), for the Station
1 ensemble, as can be seen in Fig. 3a. Similarly, Station 2 showed lower
$D_F$ than Parsivel 2; this difference indicates that even after the removal of
zero events, Station still has lesser time steps of rain occurrence than
Parsivels. Following this preliminary analysis, this will be studied more
precisely in the following sections.

The rain ensembles from every device were then subjected to spectral
and UM analysis. Observed scaling behaviours are discussed below. Figure
3 displays the UM plots for Parsivel 1 and Station 1. From spectral analysis (log-log plot of Eq. 4) of the ensemble and Eq. 9, high values of $\beta$ (~ 1.2 and 1.75, both ≈ 1, Parsivel and Station) and $H$ (~ 0.2 and 0.45, Parsivel 1 and Station 1) were retrieved for both devices (Table 2), showing the fields are non-conservative, which is expected for rainfall. Hence, in order to retrieve the conservative part over which TM and DTM analysis should be implemented, absolute value of the fluctuations of the field were considered. This reduced the value of $\beta$ and $H$ to acceptable ones for UM analysis (0.89 and 0.08 for Parsivel and 0.81 and 0.03 for Station) and confirmed validity of analysis. Thus, absolute values of the fluctuations were used as proxy for the underlying conservative field for characterizing the rain fields through UM analysis. Values at both locations are reported in Table 2. From UM analysis of rain fluctuations, both devices showed excellent scaling, and no break was observed in the regime analysed (30 s to one hour). A break is a change in the scaling behaviour at some scale, which splits the behaviour of the field with different values of $\beta$ and UM parameters. Its existence is inferred from spectral plots (Fig. 3b, Fig. 3f) and TM analysis (Fig. 3c, Fig. 3g) which in our case are following a single slope, corresponding to a unique scaling regime over the whole range of studied scales. It appears that a slight break could be considered in the spectral curve for Parsivel 1 (Fig. 3b) for $\log(k)$ around 2, which corresponds to approximately 9 min in physical time scale. Given that such slight break was not observed on TM curves (Fig. 3g), it was not considered further. The quality of fit in both cases is evaluated using the goodness of fit $r^2$, value (at the moment $q=1.5$ for TM curve), and values
close to 1 ($r^2 \sim 0.99$) were obtained in both cases, for ensembles across all devices. Although scaling appeared similar for instruments of the same make (from values of $\beta$, *H* and $r^2$ of TM), and none of the devices had scaling breaks, it should be noted that it showed slightly different profiles between devices (Fig. 3b, 3c; Fig. 3f, 3g).

To characterize the variability in rain field for different instruments, UM parameters were estimated using DTM technique (see previous section), and compared. Figures 3d and 3h show the DTM curves for the instruments at location 1.

The UM estimates at both pairs of instruments are consistent with usual values for rain fields during rainfall event, which are $\alpha \sim 1.6-1.9$ and $C_1 \sim 0.05-0.2$ (Gires et al., 2012; Verrier et al., 2011). The values were moreover similar across all 4 devices, especially between similar make devices (Tab. 2). To discuss the differences, the Stations exhibited slightly smaller $C_1$ than Parsivels (0.13 and 0.12 for Station 1 and Station 2, respectively, compared to 0.15 for both Parsivels). Regarding $\alpha$, though there were slight differences in the value between devices, the difference in estimates remained similar between devices of the same make. For the upper set (location 1), the $\alpha$ values were very close, at 1.69 for Parsivel 1 and 1.65 for Station 1 (Tab. 2, ensemble analysis column). For Parsivel, this value decreased at location 2 while Station showed a slight increase (1.61 for Parsivel 2 and 1.74 for Station 2). Even so, the devices still showed really close results at the same location. Since $C_1$ remains moreover constant, it can be roughly inferred that Station exhibits slightly
more variability in rain measurement; however, current results are not
sufficient to make any conclusive general claims. Also, as seen before, the
rain measurements by Stations are more non-conservative than Parsivels
(higher β and H values of ~ 1.76 and 0.45) in the scaling regime
analyzed, and this remained consistent across locations.

4.1.2 UM analysis of individual rain events

For examining the inter-event performance between devices, the events
used in the ensemble were subjected individually to multifractal analysis.
As previously established on the ensemble analysis, UM analysis were
carried out on the absolute value of the fluctuations of the field, and
assessed UM parameters were then compared between the devices. As all
of them were under the same rain field, in an ideal situation, the UM
parameters should be equal. Knowing that it is not the case, scatter plots
of α and C₁ were created to compare the devices, between their make and
location (Fig. 4 and 5).

While comparing devices of the same make, the pattern for the α (Fig. 4a)
and C₁ (Fig. 4c) for Parsivels were found to be along the bisector; this
suggests similar variability across devices at both locations. Parsivel 1 had
51 events with bigger α (55 for Parsivel 2), and 60 with bigger C₁ (46 for
Parsivel 2). For the Stations, the α (Fig. 4b) plot has a higher concentration
of points in the region where Station 2 presents higher values (80 events
above the bisector and 26 below); C₁ plot (Fig. 4d) shows a reverse trend
with higher concentration in the Station 1 region (70 below the bisector
and 36 below). This is consistent with the higher value of α observed for
Station 2 ensemble in the previous section. It should be noted that unlike $\alpha$, which was spread out, $C_1$ values moreover followed the bisector for both devices; this means that the difference in $C_1$ between devices for events is not as large.

While comparing different devices at the same location (same vertical height), the scattering plot patterns for both sets were found to be similar, where the Parsivel side show a greater concentration of the points (Figure 5a). In general, $\alpha$ was spread out with higher concentration towards Parsivels (76 and 57 events respectively at locations 1 and 2, Figures 5a and 5b) with the devices in location 2 having a more evenly split distribution. $C_1$ showed the same trend (Figures 5c and 5d), however, they were more spread out than the scatter plot between devices of the same make (Figures 4c and 4d). Parsivel 1 had 70 events with larger $C_1$ (36 for Station 1), and Parsivel 2 had 84 events with larger $C_1$ (36 for Station 2). Figure 5 presents also patterns regarding the distribution of UM parameters associated with wind velocity, which prompted an analysis of those parameters according to the wind classes. This division is showed in Table 3.

To see the overall behaviour, average values of UM parameters were computed and compared with that of ensemble analysis (Tab. 2). The average values were, respectively for Parsivel and Station, at location 1 (location 2): for $\alpha$, 1.56 (1.55) and 1.41 (1.57); and for $C_1$, 0.16 (0.16) and 0.145 (0.135). The overall difference between average and ensemble
values is because of the slight difference in sizing of events; for ensemble
$N_{sam}$ of 1 hr was followed while individual analysis sized events to the
closest power of 2. The trend found in ensembles is retrieved here as well,
with Parsivel showing a decrease in $\alpha$ from location 1 to 2 while Station
shows the opposite trend. Moreover, it can be roughly said that the
difference in values are similar to what was observed in ensemble
analysis, for example, Station 2 and Parsivel 2 retain the position as the
devices with the highest and lowest values of $\alpha$.

While comparing the exact values against those from ensemble analysis
between devices of the same make, Station 1 presented a bigger
decrease in the $\alpha$ value (1.65 to 1.41) compared to other devices (the
Parsivel 1 for example, 1.69 to 1.56). This can be explained by the fact
that the ensemble tends to attenuate the bias of a high number of zeros,
while for individual events this does not happen. Hence, both Stations
presented a greater increase in $C_1$, and a greater decrease in $\alpha$, with
Station 1 having the lowest average, which is the consistent behaviour
expected with high zero occurrences. This higher drop for the Stations
triggered a deeper analysis of zero occurrences, which will be further
detailed in the upcoming section.

The UM values for events were also split by average wind values, using
the classes defined in Table 1. The results can be seen on Table 3. This
was done in order to evaluate the influence of wind on the multifractality
of the measured rain fields.
It is observable that Parsivels exhibit a similar behavior both regarding $\alpha$ and $C_1$ for all classes. The only major difference in average values is on $C_1$ values of Moderate Wind (Location 2 presents approximately 85% of the value of Location 1). It is notable that both devices presented the same high values of $\alpha$ for Very Heavy Wind and $C_1$ for Light Wind, which were far greater than for the other classes.

The Stations, on the contrary, exhibit different values in general. Station 2 showed systematically greater $\alpha$ values, while Station 1 showed greater $C_1$ in all classes. It is also noticeable that, in Very Heavy Wind, Station 1 presented its highest average for $C_1$ values, while Station 2 presented its lowest. This may be due to the low amount of events in this class for the Station 2, which were only 4, and the occurrence of an unusually small value of $C_1$ in one of those 4 events. The other single values of $C_1$ were compatible with some of the values perceived at Station 1, and presented an average without this outlier similar to the previous classes (0.13). This lack of standard for both Stations with similar wind classes suggests that the variability in the strength the wind may have a influence in the measured value of the rain field.

The $r^2$ values were showed so it can be seen that scalability was maintained through different wind classes, without a clear trend of values for any device. It is observable, however, that station values were smaller than parsivel values for both sets, which can be attributed to the fact that Station missed more time steps, even filtering the events with a great amount of zeros.
### 4.1.3 Small scale behaviour and actual sampling resolution of station

The Station's measurement frequency, according to the manufacturer, is 1 Hz; however, for the UM analysis presented so far, averaged data at 30 s was used to be comparable with Parsivel (measurement frequency of 30 s). There has been few studies exploring the finer scale behaviour of rainfall, such as (Gires et al., 2015). To understand the behavior at available finer scales, UM analysis was performed on Station data from 1 s to 1 hr. For temporal scales smaller than 30 s, a big spike was observed in the spectral analysis of 1 s data. Figure 6 shows the spectral plot for the original and fluctuations field. This kind of peak is usually a signal of repetition, and by looking into the data, it was confirmed that the device did not provide data at the claimed frequency of 1 Hz for the rainfall field, but probably at a maximum of 0.5 Hz. However, some repetitions were still observable at 2 s time steps. To precisely determine the finest frequency at which variations could be studied, an ensemble multifractal analysis was performed at different frequencies using the same events chosen in section 4.1.1, with the same parameters regarding minimum duration and sample size.

From progressive UM analysis starting at the lowest available frequency (1 Hz), it was found that the smallest temporal scale that exhibits good scaling was 4 seconds. Figure 7 displays the multifractal analysis (on rain fluctuations) for averaged data at this time step (scaling regime being 4 s to 1 hr). The spectral plot (on direct data, Fig. 7b) shows no visible breaks, with a good linear fit to the regression curve. The TM curve (Fig. 7c) also
exhibits good scaling, with a r² value of 0.97. The UM parameters α and C₁
(Fig. 7d) were 1.76 and 0.08 with α being closer to the value exhibited in
Fig. 3d for 30 s data (1.65), and C₁ smaller (0.13). After identifying the
relevant higher temporal scale for studying variability of Station data as 4
s, the scaling behaviour at scales finer than previously explored 30 s was
examined and showed that in general, the scaling behaviour previously
reported is simply extended at smaller scales. In order to refine this
general observation and explore potential small changes, an artificial
(since no change on scaling behaviour was seen in the spectral plot at Fig.
7b) scale break was introduced at 30 s (creating a finer second regime
from 4 s to 30 s) to understand the differences with previous analysed
coarser regimes (30 s to 1 hr). There was no improvement in the general
value of r², with both values (before and after the break, for both regimes)
having again the same value of 0.97. This is expected since the break was
artificially imposed. The values of α and C₁ at the finer regime (4 s to 30 s)
were found to be 1.85 and 0.07 (higher α and smaller C₁ than those
observed for 30 s to 1 hr regime), while the values at coarser regimes
remained the same. This suggests that the difference in the UM values
between Station data from 4 s (α, C₁: 1.76, 0.08, Fig. 7d) and 30s
(previously for ensembles and average of individual events, Tab. 2) comes
from scales finer than 30 s.
# 5 Influence of meteorological parameters on device measurements

Following the multifractal analysis of the field (see section 4.1), further analyses were done for understanding and quantifying the differences between the devices, by focusing on the following parameters: sensitivity of rain detection, $D_m$, wind speed, and rain rate. For this, an analysis was developed, mainly based on all the recorded time steps of each device, except when explicitly mentioned otherwise (as in rare situations when some considerations were made about event properties).

## 5.1 Sensitivity of rain detection

The closest station, available on Météo-France website, to the Pays d'Othe region is Melun (roughly 90 km away); it had an average of 657.9 mm annual rainfall depth from the years 1991 to 2020 (https://meteofrance.com/climat/normales/france; accessed: June 13, 2023). Among the devices on the mast, Parsivel 1 recorded a value close to 800 mm in 15 months, which would give 640 mm of yearly average (assuming rather homogeneous distribution over the year which is roughly the case), a reasonable value when compared to standard measurements. However, the other device at the same location, Station 1, recorded only 480 mm for 15 months, which gives an average of 384 mm yearly, a far underestimated value. This under-reporting by Station was also observed at the lower set of instruments (location 2).

While comparing the data collected by instruments, it was found that at some time steps, one device was not detecting any rain while the other
did, resulting in a difference in the volume of rain recorded. These time
steps of non-detection when one device recorded non-zero rain while the
other registered 0 mm are called "missed" points here for convenience.
During these 'missed' points, the devices were functioning and they are
not to be confused with 'the non-recording' of rain which is registered in
device data as 'NaN' (not a number). From 12/11/2020 to 05/31/2022, at
location 1, 322 hrs were 'missed' by Station (since Parsivel recorded rain);
similarly, Parsivel 'missed' 132 hrs (corresponding to time steps where
Station recorded rain). At location 2, Station 'missed' 240 hrs while
Parsivel 'missed' 318 hrs. It's valuable to point out that, to compare both
devices, Parsivel and Station rain array's were analysed to evaluate if
there was any temporal lag between them. After looking inside the arrays
at each 30 s time step, it was perceived that events on Station always
started two time steps (30 s) after Parsivel. This time shift, related to the
practical recording of the devices, was removed so that both devices
could be compared.

This comparison does not include the 3 months of missing data in location
1, due to a power shortage. This also does not take into account the
(varying) time delay between both locations - during which the rainfall
goes through both locations. Regardless, the sensitivity to rain detection
was captured in UM analysis. Figure 8 shows the scatter plots of D<sub>F</sub> values
of rain events between the Stations, and between Station 1 and Parsivel 1,
for the 184 events selected before applying the snow and zero occurrence
filter. Between the Stations, Station 2 presented a greater value in 142
events, while Station 1 in only 19 (both presented the same value on the
remaining 23). This is consistent with the earlier mentioned observation of Station 'missing' more hours at location 1 (322 hrs) than location 2 (240 hrs). Between the devices at location 1, Station had a bigger $D_F$ in 51 events, while the Parsivel in 126. This is consistent with the disdrometer 'missing' less than Station (132 hrs as opposed to 322 hrs).

**5.2 Influence of $D_m$ on measured rainfall rate**

In order to study the potential effect of underlying drop size distribution (DSD), the mean diameter of the rain drops ($D_m$) was used as criteria across devices for the same time steps. A scatter plot of rain rate at each time step was plotted between devices at the same location, coloured according to corresponding $D_m$ (Fig. 9). Even though $D_m$ is used for comparing both devices, note that the information is based solely on Parsivel (since the station provides only rain rate). The same procedure is done in wind analysis, since, in this case, only the Station measures wind. From Fig. 9, it is observable that the disdrometer measured greater values in the majority of time steps (at both locations), and that this difference increases with greater rain rates (for values > 20 mm.h$\square^{-1}$, there is a clear concentration below the bisector). It can also be seen that the majority of the points with high $D_m$ (> 4 mm) are in regions far from the bisector, and closer to the Parsivel axis implying that Station did not properly detect them.

To further understand this relation between $D_m$ and performance, a routine was developed. Firstly, the time steps were split according to
equal classes of drop size separated by width $\delta$. So, for a $D_m=d$, the time steps with $D_m$ values between $d$ and $d+\delta$ are grouped. Secondly, the cumulative rainfall depth (Depth) of the devices was computed for intervals of length $d+\omega$ ($\omega > \delta$), and the error or difference in Depth between the two devices was computed for the $D_m$ classes. So, for each $D_m=d$, the formula chosen to compute the relative error was

$$E=1-\frac{\text{Dept } h_{\text{Station}}}{\text{Dept } h_{\text{Pars}}}, \text{ for the interval } D_m=[d, d+\omega]. \text{ It is possible to obtain negative values for the curve, which would mean that the Station measured more rain in the interval, but that never occurred. An interval of } \omega \text{ was chosen here instead of individual points for two reasons. The first one was to find the optimal interval where both devices give comparable rainfall estimates since parameters like } D_m \text{ and wind speed are not constant during a rain event; the second is that it allows to preserve the smoothness of the curve, removing occasional spurious variations in time steps for a singular } D_m \text{ value. It generates a smoother behaviour that facilitates interpretation when comparing both devices. For selecting the value of } \delta, \text{ various values were systematically tested, and the one that yielded the best trade-off between the smoothness of the curve and preserving the distribution profile was selected.}$$

This same methodology was applied to other parameters as well, and we will discuss it with rain rate and wind velocity in upcoming sections. Fig. 10 brings this distribution for both locations with $\delta = 0.05$ mm (for
example, the value for $D_m = 1$ mm corresponded for all added values of rain with a $D_m$ between 1 and 1.05) and $\omega = 1$ mm.

Figure 11 shows the error curve for the $D_m$ using the routine ($\delta = 0.05$ mm and a $\omega = 1$ mm). A value in the X-axis means the start of a $\omega$ step, and the lowest point in the Y-axis is the optimal interval. Other values of $\omega$ were also tested, and the result for optimal intervals were the same. The best interval (smallest error) for location 1 was between 0.9 mm and 1.9 mm (which means that the optimal point at the curve was 0.9), with a ratio between Station and Parsivel of 83.14 % (error of 16.86 %). For location 2, the best was between 0.8 mm and 1.8 mm with a ratio of 83.75 % (error of 16.25 %). It can be observed that the performance of the Station in comparison to the Parsivel diminishes with the increase of the $D_m$ after the optimal region; the error values are increasing progressively (up to 80 %) with increase in $D_m$ class (referring to the classes defined in section 2.2 for ease of understanding). Also, for values under 0.8 mm (light diameter class), the error increases while the $D_m$ diminishes, although, when comparing Figures 11a and 11b, it is noticeable that this increase is smaller for location 2. This suggested the idea that the Station was missing some rain when under light rain (approximating smaller $D_m$ to light rain). Since the error in light rain is far smaller at location 2 than location 1 (Fig. 11 again), it was seen as an indication that this effect could be related to some boundary layer difference between both sets. Since the main difference between both locations is the wind, it was
suggested as hypothesis that stronger winds (which are smaller in smaller heights) combined with light rain was a cause of loss of information from the Station. This was further investigated on section 5.4.

This relation also suggests that the algorithm for conversion of the electromagnetic signal received by the mini Doppler to the value of rain rate works acceptably in standard DSD regions (between 0.8 and 2 mm), but not so much in less common DSD distributions (< 0.8 and > 2). In addition, small drops are more likely to have their fall velocity more affected by wind, meaning that the standard diameter - fall velocity relationship assumed in the device rationale is more likely not to be valid for all cases. It is also noticeable that the error presented in the standard DSD regions are still relatively high, at almost 20 %. It suggests that, in addition to missing out on higher $D_m$, the Station has an overall tendency, even in closer areas, to underestimate rain rate values compared to the Parsivel.

## 5.3 Sensitivity in the measurement of different rain rates

Another parameter used for comparing performance of devices was rain rate (RR, mm h⁻¹) itself. In the case of disdrometers, this is obtained by integrating the information across the drop size distribution and fall velocity into a single quantity; for stations, we are left with a single quantity for which any assumptions on DSD made by Doppler radar algorithms are not available. Based on the criteria presented in section 2.2, time steps of 30 s were categorized into light, average, heavy and
very heavy rain (Tab. 1). As expected from the discrepancies in detection and difference in $D_m$ before, both devices are not always in the same class of RR in every time step; this makes the filtering to rain classes more difficult. For example, if the Parsivel rain class for a time step is 'light', the corresponding rain class of station is not necessarily the same, and vice-versa. To be consistent, time steps were classified according to rain values of both devices s (Fig. 12a & Fig. 12b); and they were compared alongside for the same location (illustrated using location 1). The same figure displays the corresponding error ratio (following the routine in section 5.2), as function of the rain classes as well; it shows a growing pattern from 'light' to 'very heavy' rain (ranging from 0.21 to 0.73 at location 1, Fig. 12a). This may be due to the fact that larger rain rates can have a greater concentration of large drops, and as the Station under-measures larger diameters when compared to the Parsivel as observed in Fig. 10 and Fig. 11, it causes a larger difference between both devices.

In previously discussed Fig. 11, the percentage error between Station and Parsivel were higher in regions of larger $D_m$ than small and average values. To understand it further, the contribution of $D_m$ for cumulative depth was considered in two extremes of rain classes – 'light' and 'very heavy'. Figure 13 shows this information in a pie chart for rain classified according to Parsivel. The 'very heavy' rain class has a large contribution from 'large' and 'very large' $D_m$, while the 'light rain' mostly contains small and average $D_m$. This observation is in accordance with the under
detection of higher *D*<sub>m</sub> and consequent lower *RR* by Station, which
translates to reduced cumulative depth while compiling time steps of
higher rain classes (Fig. 12). It is also worth noting that the error between
devices (as seen in *D*<sub>m</sub> and confirmed through *RR*) is the smallest for *D*<sub>m</sub>
between 0.8 and 1.8, the size range that corresponds to the maximum
contribution to rainfall by volume (Fig. 10a) which is reassuring for usage
of Station for measuring rain.

The results found for the rain rate partially confirm what was found in the
last section for the *D*<sub>m</sub> influence, the under-measuring in the larger values
of *D*<sub>m</sub> compared to the Parsivel, and previously observed discrepancies
between devices in detection of rain. These observations are mirrored in
UM analysis as well, in which comparatively greater values of *H* and lower
values of *C*<sub>1</sub> for Station suggest a smoother field and lesser variability in
rain respectively (under-measuring of *D*<sub>m</sub> in Fig. 10, and consequently rain
in Fig. 12) recorded by them than the disdrometers. This means a
relatively more homogeneous field with smoother variation in which larger
peaks are attenuated (as observed in individual event time series)
dampening the overall variability.

**5.4 Influence of wind on rainfall measurement**

While discussing the sensitivity of rain detection between devices using
*D*<sub>F</sub>, a difference in performance was found between similar devices at
different locations (75 m and 45 m height). This suggested a possible
effect influence of wind in the occurrence of zeros, as it is known to
happen in rainfall measurements (Lanza et al., 2021). This is illustrated in
Fig. 14 with Stations where ~ 75 % of the time steps are with higher wind velocity at location 1 (at 80 m). For this comparison wind velocity was obtained from the Station sensors.

Figure 15 shows the application of the routine, previously used for $D_m$ and RR, for wind (time steps grouped according to wind class), for both devices. The chosen parameters were $\delta=0.25$ and a $\omega=1$. On cumulative rainfall depth, a large gap is observable in light ($< 5.4 \text{ m s}^{-1}$) and average ($5.4 \text{ m s}^{-1} < \text{wind} < 7.9 \text{ m s}^{-1}$) values of wind. In larger values ($> 10 \text{ m s}^{-1}$), devices show similar values. It should be noted that this value of wind is reported as a threshold for Parsivel to start presenting wind interference (Friedrich et al., 2013) and hence might influence the comparison in that region.

Figure 16 shows the curve error for both locations (following the routine in section 5.2). It is observable that the location 2 presents lower values of error (as expected from lesser difference in cumulative depth between devices in Fig. 15). Location 1 presents two peaks, one in light wind (around $3 \text{ m s}^{-1}$) and other in heavy wind (around $9 \text{ m s}^{-1}$), with all this region (before $10 \text{ m s}^{-1}$) being the one with highest values of errors. For winds greater than this threshold, the error value drops to the region of 0.2. For the location 2, the values were smaller, with the highest being around 0.2 (the lowest region for location 1). There are some sudden discontinuities for bigger values of wind speed in location 2, which are due to the fact that for those devices, there are far fewer time steps with such
strong wind, causing the values to be smaller and with greater
uncertainty. This lack of continuous trend among the values triggered an
analysis refined with the other parameters (rain rate and D_m), to search for
a clearer understanding of the wind effect.

Figure 17 brings the wind trends according to the rain classes. The curves show the ratio of cumulative depth per wind classes as a function of the rain classes, with values filtered by the Station rain rates. It was chosen to analyse for Station filtered classes since they have the largest contribution to the 'missing' points which would be avoided. For location 1, the curves for 'average', 'heavy' and 'very heavy' rain have a similar decreasing behaviour from 'light' to 'very heavy' wind (the ratio is closer to 1 while the wind speed is greater and the rain rate smaller). The values range from 2.23 to 1.54 for 'very heavy' rain, 1.98 to 1.35 for 'heavy' rain, and 1.56 to 0.96 for 'average' rain. However, for 'light' rain, the behaviour is the opposite, with an increasing trend and high value of ratios (4.01 and 3.78) for 'heavy' and 'very heavy' wind. These higher values show that the Station at location 1 is detecting a lesser amount of rain when the wind velocity gets heavier. This general decreasing trend in the behaviour for the wind classes (except 'light' wind) is consistent with what was presented in the Figures 15 and 16, which showed a decrease in the error between Parsivel and Station with an increase of wind speed. The behaviour of the 'light' rain can be understood better when considered alongside the D_m variation within the rain class discussed in Fig. 13. It showed that the 'light' rain mainly consists of small diameters, and it can
be inferred that in Station data, they are affected the most by wind. For
location 2, a similar behaviour was observed, where all rain classes
showed a decreasing trend except 'light'. However, the overall values of
the ratios are smaller, as expected from Figure 16). For 'light' rain class,
the ratio is significantly lesser here than in location 1 which can be
explained by the latter having heavier wind conditions (refer the scatter
plot in Fig. 14). The two greatest wind classes ('heavy' and 'very heavy')
account for 81.30 % of the 'missing' points for the Station (102.4 mm of
129.98 mm of the 'missed' cumulative depth). This concentration of
missed rain at stronger winds is consistent with larger ratios found for
'heavy' winds in 'light' and 'average' rain at Fig. 17 (zero occurrences as
mentioned before are associated with 'light' rain). The region of 'light'
wind corresponds to only 3.6 % of missed volume by the Station (4.68
mm). Those values suggest that Station struggles to register rain in
heavier wind conditions unless the values of precipitation is significant.

In Fig. 5, the 'average' wind velocity of the events are plotted in colour
plot for variations of UM parameters. It can be roughly said that higher
'average' wind are recorded on Location 1 than Location 2, but less than
what is mentioned above in terms of individual time steps (and also
shown in Fig. 14). Though no obvious trends can be seen in UM
parameters, based on statistical analysis above, meteorological station
struggles to register rain when wind conditions are stronger.
# 6 Conclusions

In this study, bulk rainfall output of two widely used precipitation measuring devices, working on different measuring principles, are compared - OTT Parsivel² optical disdrometer and ThiesCLIMA mini meteorological station (vertically pointing Doppler radar). Comparison has been done at two vertical locations using data spanning over one year and a half. Through multifractal analysis of rainfall events, individually and as an ensemble, variability and intermittency of measured rainfall was characterized across various temporal scales and not only at the recording time step.

In event based analysis, both devices exhibited good scaling and standard multifractal behavior for rain fields from 30 s to 1 hr, confirming earlier studies (Gires et al., 2016; Jose et al., 2022). Similar values were retrieved for UM parameters across devices of the same make at both vertical locations, for individual as well as ensemble analysis. In ensemble analysis, rain measured at the Station was found to be more non-conservative ($H \sim 0.4$) in comparison, and showed slightly more variability (higher $\alpha$ and similar $C_1$; Tab. 2). The difference in UM parameters between devices were more conspicuous while analysing rain events individually, with values showing significant scattering (Fig. 4 and Fig. 5). This also resulted in a difference in values of $\alpha$ and $C_1$ between the average from individual analysis and that from ensemble analysis (average of individual events < ensemble). Observed difference can be physically explained by the fact that individual events are more affected by the presence of zeroes than ensembles. Station was losing information
(Section 5.1) and recording zero rain in some time steps during while Parsivel recorded rain. This under-measuring resulted in a reduced estimate of $\alpha$ for the former for the same rain event. While analyzing the finer scales (< 30 s, up to 1 s) available at Station (which delivers data up until 1 Hz), repetition in data was detected through Spectral analysis (Fig. 6). From subsequent UM analysis, 4 s was found to be the finest resolution available at Station used in current study.

The overall measurement by devices were also examined time-step wise, and a routine was developed for examining following parameters - sensitivity of detection, influence of $D_m$ in this detection, the difference in measurement across rain types (based on rain rate), and possible influence of wind. While examining sensitivity of measurement, previously hinted under-reporting by Station was observable in cumulative rainfall depth. Using the routine proposed, the optimal range of $D_m$ where the Station provided comparable rainfall as Parsivel² was determined (between 0.8 and 2 mm, with ~ 20% of error; Fig. 11). Outside this range, error progressed to more than 70 %. Sensitivity in measuring various rain types was also explored by categorizing studied events into 4 classes of rain rate. Station was found to be under-reporting rain in higher rain rate events, which are outside of the optimal range of $D_m$ observed. Since wind velocity is different at two vertical locations (location 1 having ~ 75 % of time steps with higher velocity than location 2), the routine was applied to see its influence in rain detection. Location 2 showed lower (and more stable) values of error in rain measurement (between devices). At location 1, the moving mean error increased till 9 m s⁻¹, and reduced thereafter
to a stable value (Fig. 16). When performing the wind and rain classes
analysis, a big difference of cumulative rain between Parsivel and Station
(more than 4 times) was observed for for 'light' rain in 'very heavy' wind
at location 1 (Fig. 17). Many of the rain missing points of Station were on
those classes as well. It should be mentioned that these observations
further involve unexplored complication from airflow pattern around the
device geometry (Cauteruccio et al., 2020).

The correlation between physical parameters appear to be important to
describe the differences between a Parsivel Disdrometer and the Doppler
Station, but they may not be enough, since both devices did not show
similarity of behaviour in any range of value for wind, mean diameter or
rain rate. This work highlights the need for assessing various measuring
techniques of rainfall estimates through scale invariant tools. It also
illustrates the importance of measurement resolution and optimal
operating conditions for accurate estimates for bulk measurement
devices. The investigation should be expanded with more data, notably to
cover a wider range of rain types. It would also be interesting to collect
data at ground level with a similar configuration since rainfall
measurement devices are more commonly installed on the ground where
wind intensity is lower. We hope this will be achieved in the near future.
Another future investigation path could be to consider boundary layer
stability and its evolution along the daily cycle as another criteria. This
would be especially relevant for the upper location. Such analysis would
require additional data such as the ones provided by meteorological
soundings.
**Acknowledgements:**

The authors greatly acknowledge partial financial support from ANR JCJC RW-Turb project (ANR-19-CE05-0022-01)

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<table><caption>Table 1. Criteria used for categorizing rain events and time steps. For this criteria, the parameters were taken at the sampling interval of Parsivel (30 s).</caption><thead><tr><th>Category</th><th>Rain rate</th><th>Mean</th><th>Wind velocity</th></tr></thead></table>
<table><thead><tr><th rowspan="2"></th><th colspan="3">diameter</th></tr><tr><th>(RR, mm h□<sup>-1</sup>)</th><th>(D<sub>m</sub>, mm)</th><th>(w, m s□<sup>-1</sup>)</th></tr></thead><tbody><tr><td>Light</td><td>RR &lt; 0.8</td><td>D<sub>m</sub> &lt; 0.8</td><td>w &lt; 5.4</td></tr><tr><td>Moderate</td><td>0.8 ≤ RR &lt; 4</td><td>0.8 ≤ D<sub>m</sub> &lt; 2</td><td>5.4 ≤ w &lt; 7.9</td></tr><tr><td>Heavy</td><td>4 ≤ RR &lt; 8</td><td>2 ≤ D<sub>m</sub> &lt; 5</td><td>7.9 ≤ w &lt; 10.8</td></tr><tr><td>Very Heavy</td><td>RR ≥ 8</td><td>D<sub>m</sub> ≥ 5</td><td>w ≥ 10.8</td></tr></tbody></table>

Table 2. UM parameters of rain for Parsivel and Station at both locations. α, C₁ and r² are obtained on the underlying conservative field (i.e. the fluctuations), while β and H values shown are retrieved on direct data. As
expected, the latter two were reduced on fluctuations, which corresponds to a conservative field. Here, *H* was computed without *K*<sub>c</sub>(2) correction.

<table><thead><tr><th rowspan="2">Location</th><th rowspan="2">Device</th><th colspan="6">Ensemble analysis</th><th colspan="2">Average values</th></tr><tr><th>β</th><th>H</th><th>α</th><th>C<sub>1</sub></th><th>r<sup>2</sup></th><th>α</th><th>C<sub>1</sub></th></tr></thead><tbody><tr><td rowspan="2">1</td><td>Parsivel 1</td><td>1.20</td><td>0.2</td><td>1.69</td><td>0.1</td><td>0.99</td><td>1.56</td><td>0.16</td></tr><tr><td>Station 1</td><td>1.76</td><td>0.4</td><td>1.65</td><td>0.1</td><td>0.99</td><td>1.55</td><td>0.16</td></tr><tr><td rowspan="3">2</td><td>Parsivel 2</td><td>1.26</td><td>0.2</td><td>1.61</td><td>0.1</td><td>0.99</td><td>1.41</td><td>0.145</td></tr><tr><td rowspan="2">Station 2</td><td>1.74</td><td>0.4</td><td>1.74</td><td>0.1</td><td>0.99</td><td>1.57</td><td>0.13</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr></tbody></table>
<table><caption>Table 3. UM parameters of rain fluctuations for Parsivel and Station at both locations, split by wind classes. The number of events for wind class at location 1 (location 2) were: for Light Wind, 25 (42), for Moderate Wind, 39 (37), for Heavy Wind, 30 (23), for Very Heavy Wind, 12 (4).</caption><thead><tr><th rowspan="2">UM</th><th rowspan="2">Wind Velocity (m)</th><th colspan="2">Location 1</th><th colspan="2">Location 2</th></tr><tr><th>Parsivel</th><th>Station</th><th>Parsivel</th><th>Station</th></tr></thead><tbody><tr><td rowspan="4">α</td><td>s<sup>-1)</sup></td><td>1</td><td>1</td><td>2</td><td>2</td></tr><tr><td>Light Wind</td><td>1.48</td><td>1.49</td><td>1.50</td><td>1.63</td></tr><tr><td>Moderate Wind</td><td>1.55</td><td>1.39</td><td>1.58</td><td>1.57</td></tr><tr><td>Heavy Wind</td><td>1.58</td><td>1.39</td><td>1.54</td><td>1.48</td></tr><tr><td></td><td>Very Heavy Wind</td><td>1.71</td><td>1.37</td><td>1.76</td><td>1.51</td></tr><tr><td rowspan="4">C<sub>1</sub></td><td>Light Wind</td><td>0.203</td><td>0.15</td><td>0.202</td><td>0.15</td></tr><tr><td>Moderate Wind</td><td>0.165</td><td>0.144</td><td>0.139</td><td>0.124</td></tr><tr><td>Heavy Wind</td><td>0.134</td><td>0.136</td><td>0.138</td><td>0.132</td></tr><tr><td>Very Heavy Wind</td><td>0.152</td><td>0.158</td><td>0.144</td><td>0.115</td></tr><tr><td rowspan="4">r<sup>2</sup></td><td>Light Wind</td><td>0.96</td><td>0.94</td><td>0.96</td><td>0.93</td></tr><tr><td>Moderate Wind</td><td>0.96</td><td>0.92</td><td>0.96</td><td>0.94</td></tr><tr><td>Heavy Wind</td><td>0.96</td><td>0.95</td><td>0.96</td><td>0.95</td></tr><tr><td>Very Heavy Wind</td><td>0.97</td><td>0.96</td><td>0.95</td><td>0.94</td></tr></tbody></table>

## List of Figures

Figure 1: a) Quicklook for a sample day (12 Jan 2021) with relevant information used in the current analysis; from https://hmco.enpc.fr/calendars/Calendar_RW_Turb wind farm/: the temporal evolution of the rain rate (first row), drop size and horizontal
velocity (second row), velocity curve and DSD (third row). b) Meteorological mast at Pays d'Othe wind farm, instruments are shown in the inset; image from (Gires et al., 2022)

Figure 2: General overview of the data used in the current study, at locations 1 and 2, from 12/11/2020 to 05/31/2022.

Figure 3: UM analysis of rain at Station 1 (a-d) and Parsivel 1 (e-h), using ensembles of data at 30 s at a sample length of 1 hr (64 min). The spectral analysis and fractal dimension estimations are shown on direct data, the rest on fluctuations of the field. Plots of UM analysis at location 2 are given in supplementary material (Fig. S1).

Figure 4: Scatter plots of UM parameters for individual events analysis, comparing devices of the same make.

Figure 5: Scatter plots of UM parameters for individual events analysis, comparing devices at the same locations. The average wind velocity of the events are shown in colour gradient (common range for all graphs).

Figure 6: Spectral analysis of station 1 data at 1 Hz (1 s), original and fluctuations of the field; a similar result was found for Station 2 as well. The peak on the fluctuations for the greatest frequency can be seen, being a sign of repetition of data.

Figure 7: UM ensemble analysis of rain fluctuations at Station 1, for resolution ranging from 4 s to 4096 s (~1 hr). The spectral and fractal dimension plots are shown on direct data, the rest on fluctuations.

Figure 8: Scatter plot of $D_F$ for all rain events studied in section 4.1, between devices and locations.
Figure 9: Scatter Plots of rain rate (mmh⁻¹) with Dm variation on colour bar

Figure 10: Cumulative rainfall depth (mm) by Dm classes distribution

Figure 11: Moving mean error ratio for Dm, according to the routine

Figure 12: Cumulative depth and Error trend in timesteps compiled as per rain classes, at location 1

Figure 13: Percentage distribution for Dm classes for 'light' and 'very heavy' rain time steps, filtered by Parsivel values

Figure 14: Scatter plot of wind speed for 30 seconds time steps, between Station 1 and Station 2

Figure 15: Cumulative rainfall depth separated by wind values, with classes separated by a δ of 0.25 m s⁻¹

Figure 16: Moving mean error trend for wind, for both locations.

Figure 17: Ratio of cumulative rainfall depth (Parsivel/Station) organized according to wind and rain classes, for both locations. Classification of rain classes are on the basis of RR measured by Station.
<figure><img src="image_3.png" /></figure>

Figure 1: a) Quicklook for a sample day (12 Jan 2021) with relevant information used in the current analysis; from https://hmco.enpc.fr/calendars/Calendar_RW_Turb wind farm/: the temporal evolution of the rain rate (first row), drop size and horizontal velocity (second row), velocity curve and DSD (third row). b) Meteorological mast at Pays d'Othe wind farm, instruments are shown in the inset; image from (Gires et al., 2022)

<figure><img src="image_4.png" /><figcaption>(a) Rain measured at location 1 (b) Rain measured at location 2</figcaption></figure>
from 12/11/2020 to 05/31/2022; from 12/11/2020 to 05/31/2022;
blue colour corresponds to blue colour corresponds to
Parsivel and orange represents Parsivel and orange represents
Station.
Station.

TOTAL CUMULATIVE RAIN (mm)

<figure><img src="image_5.png" /></figure>

(c) Cumulative rainfall depth (mm) for Parsivels and Stations, at locations 1 and 2. The values in orange refer to a 3 month period in which location 1 was out of power.
Figure 2: General overview of the data used in the current study, at locations 1 and 2, from 12/11/2020 to 05/31/2022.

<figure><img src="image_6.png" /></figure>

(a) Fractal dimension estimation (b) Spectral analysis, β estimation
from Eq. 4 in log-log plot
$D_F$)

<figure><img src="image_7.png" /><figcaption>(c) TM analysis, Eq. 7 in log-log plot for various values of $q$</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>(e) $D_F$ estimation</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>(d) DTM analysis; log-log plot of exponents in Eq. 11</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>(f) Spectral analysis</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>(g) TM analysis</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>(h) DTM analysis</figcaption></figure>

Figure 3: UM analysis of rain at Station 1 (a-d) and Parsivel 1 (e-h), using ensembles of data at 30 s at a sample length of 1 hr (64 min). The spectral analysis and fractal dimension estimations are shown on direct
data, the rest on fluctuations of the field. Plots of UM analysis at location 2
can be found in the supplementary material.

<figure><img src="image_13.png" /><figcaption>(a) α: Parsivels</figcaption></figure>

<caption>(b) α: Stations</caption>

<caption>(c) $C_1$: Parsivels</caption>

<caption>(d) $C_1$: Stations</caption>

Figure 4: Scatter plots of UM parameters for individual events analysis,
comparing devices of the same make.

<figure><img src="image_14.png" /><figcaption>(a) α: Location</figcaption></figure>

<caption>(b) α: Location</caption>

<caption>(c) $C_1$: Location 1</caption>

<caption>(d) $C_1$: Location 2</caption>

1

2

Figure 5: Scatter plots of UM parameters for individual events analysis,
comparing devices at the same locations. The average wind velocity of
the events are shown in colour gradient (common range for all graphs).
<figure><img src="image_15.png" /><figcaption>(a) Original Field spectral plot
(b) Fluctuations Field spectral plot

Figure 6: Spectral analysis of station 1 data at 1 Hz (1 s), original and fluctuations of the field; a similar result was found for Station 2 as well. The peak on the fluctuations for the greatest frequency can be seen, being a sign of repetition of data.</figcaption></figure>
<figure><img src="image_16.png" /><figcaption>(a) Fractal Dimension</figcaption></figure>

TM Analysis

<figure><img src="image_17.png" /><figcaption>(c) TM Analysis</figcaption></figure>

<figure><img src="image_18.png" /><figcaption>(b) Spectral Analysis</figcaption></figure>

UM parameters for 1st scaling regime

<figure><img src="image_19.png" /><figcaption>(d) DTM analysis</figcaption></figure>

Figure 7: UM ensemble analysis of rain fluctuations at Station 1, for resolution ranging from 4 s to 4096 s (~1 hr). The spectral and fractal dimension plots are shown on direct data, the rest on fluctuations.

<figure><img src="image_20.png" /></figure>

<figure><img src="image_21.png" /></figure>
(a) Between devices at location 1

(b) Between Stations at location 1
and 2

Figure 8: Scatter plot of $D_F$ for all rain events studied in section 4.1,
between devices and locations.

<figure><img src="image_22.png" /><figcaption>(a) Scatter Plot for rain rate with $D_m$ (b) Scatter Plot for rain rate with $D_m$
colour bar, Location 1 colour bar, Location 2
Figure 9: Scatter Plots of rain rate (mmh⁻¹) with $D_m$ variation on colour bar</figcaption></figure>

<figure><img src="image_23.png" /><figcaption>(a) Cumulative rainfall depth (mm) (b) Cumulative rainfall depth (mm)
at location 1 at location 2
Figure 10: Cumulative rainfall depth (mm) by $D_m$ classes distribution</figcaption></figure>
<figure><img src="image_24.png" /><figcaption>(a) Moving mean error ratio trend (b) Moving mean error ratio trend for $D_m$, location 1 (c) Moving mean error ratio trend for $D_m$, location 2

Figure 11: Moving mean error ratio for $D_m$, according to the routine</figcaption></figure>

<figure><img src="image_25.png" /><figcaption>(a) Filtered by Parsivel values (b) Filtered by Station values

Figure 12: Cumulative depth and Error trend in timesteps compiled as per rain classes, at location 1</figcaption></figure>
<figure><img src="image_26.png" /><figcaption>Figure 13: Percentage distribution for $D_m$ classes for 'light' and 'very heavy' rain time steps, filtered by Parsivel values</figcaption></figure>

<figure><img src="image_27.png" /><figcaption>Figure 14: Scatter plot of wind speed for 30 seconds time steps, between Station 1 and Station 2</figcaption></figure>
<figure><img src="image_28.png" /><figcaption>(&) Cumulative rainfall depth for (b) Cumulative rainfall depth for location 1 and location 2

Figure 15: Cumulative rainfall depth separated by wind values, with classes separated by a $\delta$ of 0.25 m s$^{-1}$</figcaption></figure>

<figure><img src="image_29.png" /><figcaption>(a) Error curve for location 1
(b) Error curve for location 2

Figure 16: Moving mean error trend for wind, for both locations.</figcaption></figure>
CUMULATIVE RAIN RATIO

<figure><img src="image_30.png" /></figure>

CUMULATIVE RAIN RATIO

<figure><img src="image_31.png" /></figure>

(a) Ratio of cumulative depth (b) Ratio of cumulative depth
between the devices at location 1 between the devices at location 2
Figure 17: Ratio of cumulative rainfall depth (Parsivel/Station) organized
according to wind and rain classes, for both locations. Classification of rain
classes are on the basis of *RR* measured by Station.