Disdrometer Performance Optimization for Use in Urban Settings Based on the Parameters That Aﬀect the Measurements

S S symmetry

Article Disdrometer Performance Optimization for Use in Urban Settings Based on the Parameters that Aﬀect the Measurements

Ferran Mocholí Belenguer 1,* , Antonio Martínez-Millana 2 , Antonio Mocholí Salcedo 3 , Víctor Milián Sánchez 4 and María Josefa Palomo Anaya 4

1 Traﬃc Control Systems Group, ITACA Institute, Universitat Politècnica de València, 46022 Valencia, Spain 2 SABIEN Group, ITACA Institute, Universitat Politècnica de València, 46022 Valencia, Spain; [email protected] 3 Department of Electronic Engineering, ITACA Institute, Universitat Politècnica de València, 46022 Valencia, Spain; [email protected] 4 Chemical and Nuclear Engineering Department, Institute of Industrial, Radiological and Environmental Safety, Universitat Politècnica de València, 46022 Valencia, Spain; [email protected] (V.M.S.); [email protected] (M.J.P.A.) * Correspondence: [email protected]; Tel.: +34-610833056

Received: 15 November 2019; Accepted: 13 February 2020; Published: 20 February 2020

Abstract: There are currently different types of commercial optical disdrometers to measure the rainfall intensity, of which many are commonly used for monitoring road conditions. Having information about the amount of rain, the composition of the precipitation particles and visibility are essential to avoid accidents, which requires intelligent systems that warn drivers and redirect traffic. However, few studies related to Intelligent Transport Systems (ITS) have been performed regarding why these devices are not optimized for this type of applications. Therefore, this paper analyzes and evaluates the operating mode of these equipment through their theoretical model, which will allow the design of prototypes of disdrometers with different characteristics. In addition, this model will be implemented in a simulation program, through which an exhaustive study analyzing how the type of precipitation and its intensity affect the measures provided by the model will be conducted. In this way, the results will help optimize its operation to be thus used in urban settings, which will allow obtaining more accurate real-time information, better traffic management, and a reduction in the number of accidents.

Keywords: disdrometer; precipitation; rain; safety; traﬃc

1. Introduction For a long time, many series of rainfall records have been made throughout the world [1,2]. These records, which are expressed in mm or kg/m2 and collected during a certain time period, have been useful for meteorology and climatology and are of special interest to many sectors such as hydrology, agriculture, or vehicle traffic management systems [3]. However, although recording the amounts of precipitation of liquids and solids constitutes a knowledge base in several fields of study, the precipitation intensity (PI) has become a variable of the same importance at present. The precipitation intensity is defined by the World Meteorological Organization as the amount of precipitation collected per unit time, and it is widely known that the rainfall and snowfall intensity information is extremely relevant in cases of severe weather [4,5]. In fact, high precipitation intensities in the form of liquid or solid directly affect transport, since roads and other infrastructure can be blocked in cases of insufficient drainage and runoff [6]. Moreover, the probability of having an accident while it rains or snows is greater than when it does not [7], which is mainly due to the lack of visibility.

Symmetry 2020, 12, 303; doi:10.3390/sym12020303 www.mdpi.com/journal/symmetry Symmetry 2020, 12, 303 2 of 19

Hence, appropriate short-term forecasts could reduce these risks and allow the sending of warnings to drivers to take precautions or avoid certain routes. This is the reason why many types of instruments and measuring techniques have been developed and are in operation today. In this way, there are diﬀerent types of instrument that measure precipitation micro-physical parameters and divide into remote sensing and in-situ [8]. Depending on how they work, all types of rain gauges can be divided into two large groups:

1. Catching instruments. 2. Non-catching instruments.

The first type catches the precipitation through a well-defined hole size and measures the equivalent mass or volume of water that has accumulated in a given time interval. In this sense, the precipitation intensity is retrieved from the amount of precipitation. On the contrary, the non-catching instruments calculate the precipitation intensity by mathematically integrating all particles transiting through a cross section in a given time. Optical, acoustic, and electromagnetic rain sensors are typical examples of this group. Nevertheless, although many calibration practices have been developed, especially for the rain gauges of the first group, there is currently no standard or accepted calibration reference for any type of rain gauge. Therefore, CIMO-XIII (The Thirteenth Session of the Commission for Instruments and Methods of Observation) adopted the measurement range and uncertainties related to the precipitation intensity recommended by an expert group of researchers who published that information in the Guide to Instruments and Methods of Observation of the World Meteorological Organization [9]. These data are shown in Table1.

Table 1. Uncertainties adopted by CIMO-XIII.

Full range 0.02–2000 mm h 1 · − Rainfall threshold detected 0.02–0.2 mm h 1 · − Output averaging time One minute Uncertainty required in the measurement: 0.2–2 mm h 1 0.1 mm h 1 · − · − 2–2000 mm h 1 5% · −

Accordingly, as these are commonly used on roads to warn and therefore prevent accidents and there are no studies with the purpose of optimizing their operation for these applications, we think that there is clearly a need to have an adaptive and optimized system capable of measuring the intensity of precipitation meticulously depending on the type of rain, wind, and other parameters. In this way, these devices could be adapted according to the boundary conditions, always providing the most accurate measurements. In addition, minimum and speciﬁc requirements could be established so that they could be installed and used in ITS applications. These requirements would be as follows:

1. It should be a simple, low-cost system implemented with commercial materials. 2. It should use optoelectronic systems, since the analysis performed by many authors suggests that these systems provide better results [10,11]. 3. It should be capable of operating both at ﬁxed stations in the road network and onboard a vehicle to provide information to the driver and activate the fog lights or windscreen wipers when required.

There are some disdrometer models such as Parisivel that could meet these specifications, but Grossklaus et al. [10] presented a model of a disdrometer that satisfied these requirements in a better way. Indeed, a few years later, the company Eigenbrodt manufactured the optical disc ODM470 by following the specifications indicated by them. Thus, this manuscript will theoretically analyze the operation of this equipment and evaluate the effects of a series of parameters on it, Symmetry 2019, 11, x FOR PEER REVIEW 3 of 19 followingSymmetry 2020 the, 12 specifications, 303 indicated by them. Thus, this manuscript will theoretically analyze3 ofthe 19 operation of this equipment and evaluate the effects of a series of parameters on it, such as the precipitationsuch as the precipitation intensity, intensity, drop size drop distribution, size distribution, period period sampling sampling method, method, estimation estimation method, method, measurementmeasurement time, time, wind wind speed, speed, type type of of converter converter,, and and many many others. others. For For this this,, a a program program exclusively exclusively designeddesigned by by our our work work group group has has been been implemented, implemented, which which enables enables to to simulate simulate the the behavior behavior of of the the equipmentequipment by by varying varying its its pa parameters.rameters.

2.2. Materials Materials and and Methods Methods

2.1.2.1. Operating Operating P Principlerinciple TheThe principle principle of of operation operation of of the the aforementioned aforementioned disdrometer disdrometer is is the the slight slight decrease decrease in in light light intensityintensity when when the the raindrops raindrops pass pass through through a a cylindrical cylindrical sensitive sensitive volume volume of of length length 퐿L andand diameter diameter 퐷D [[1212].]. At oneone endend of of the the cylinder, cylinder, there there is ais light a light source, source, which which is usually is usually an IR an LED IRdiode LED diode with a with control a controlsystem system that enables that enables the diode the to diode maintain to maintain its level its of level illumination of illumination constant. constant. At the otherAt the end, other there end, is therea photodetector, is a photodetector, whichis which a photodiode is a photodiode with a signalwith a conditioning signal conditioning circuit incircuit most in cases. most To cases. achieve To achievea homogeneously a homogeneously illuminated illuminated sensitive sensitive volume, avolume, suitable a opticalsuitable lens optical system lens composed system composed of a collimator of a collimatorand Fresnell and lens Fresnell is also required. lens is This also scheme required. is shown This in scheme Figure 1 is, which shown can in simultaneously Figure 1, which measure can sithemultaneously size and ﬂight measure time of the the size droplets and flight through time the of the sensitive droplets volume. through the sensitive volume.

Figure 1. Implementation of an optical disdrometer. Figure 1. Implementation of an optical disdrometer. During a precipitation event, important fluctuations will be observed in the photodetector wheneverDuring a a particle precipitation interferes event with, important the light fluctuationsbeam emitted will by be the observed LED. The in amplitude the photodetector of these wheneverfluctuations a depends particle oninterferes the size with of the the particles, light beam while emitted the duration by the of the LED. fluctuations The amplitude is depending of these on fluctuationsraindrop fall depends speed. Moreover, on the size the of numberthe particles, of fluctuations while the duration is proportional of the fluctuations to the number is depending of particles onthat raindrop pass between fall speed the. LED Moreover, and the the photodetector. number of fluctuations The only constraints is proportional that may to the occur number are the of particlessystem dependence that pass between on the frequencythe LED and of samplingthe photodetector. of the signal, The and only the constraints resolution that in the may illumination occur are thelevel system measurement. dependence However, on the the frequency operating of ranges sampling of di offferent the signal instruments, and the and resolution our analysis in will the illuminationdemonstrate level that thesemeasurement. constraints However, will not be the an oper inconvenienceating ranges that of limitsdifferent the systeminstruments operation. and our analysis will demonstrate that these constraints will not be an inconvenience that limits the system operation.2.2. Drop Size Measurement If there is no drop inside the sensitive volume, the illumination that reaches the photodetector 2.2. Drop Size Measurement will induce an output signal, which will be taken as a reference. When raining, each drop through the sensitiveIf there volume is no producesdrop inside a reduction the sensitive of light volume, received the byillumination the photodetector, that reaches and the correspondingphotodetector willsignal induce is proportional an output to signal, the quotient which betweenwill be taken the area as ofa reference. the section-line When of raining, the drop each and drop the area through of the thesection-line sensitive of volume the sensitive produces volume. a reduction Hence, the of range light of amplitudes received by of the the signal photodetector, may vary and between the corresponding0 and the amplitude signal is of proportional the reference to signal. the quotient Depending between on the the area type of of the photodetector section-line of and thesignal drop andconditioning, the area of the the magnitude section-line to beof the measured sensitive can volume. be a voltage, Hence, a currentthe range intensity, of amplitudes or a frequency, of the signal which may vary between 0 and the amplitude of the reference signal. Depending on the type of

Symmetry 2019, 11, x FOR PEER REVIEW 4 of 19 Symmetry 2020, 12, 303 4 of 19 photodetector and signal conditioning, the magnitude to be measured can be a voltage, a current alsointensity conditions, or a frequency, the type of which microcontroller also conditions to manage the type the of operation microcontroller of the disdrometer. to manage the Nevertheless, operation theof the maximum disdrometer. sensor Nevertheless, response will the occur maximum for a particle sensor of response at least diameter will occurD asfor shown a particle in Figure of at 1least. diameterThe tests퐷 as performed shown in Fig by diurefferent 1. authors [10] show that the homogeneity and isotropy of light in the sensitiveThe tests volume performed are essential by different for the authors correct [1 interpretation0] show that of the the homogeneity data. Unfortunately, and isotropy most lightingof light sourcesin the sensitive have some volume anisotropy are essential and inhomogeneities. for the correct interpretation Thus, inhomogeneities of the data. along Unfortunate the lengthly,of most the opticallighting volume, sources such haveas some those anisotropy due to the and divergence inhomogeneities. of light, must Thus, be inhomogeneities minimized. In our along model, the welength consider of thethe optical ideal volume case of, asuch perfectly as those collimated due to the source, divergence so the errorsof light estimated, must be mustminimized. be improved In our tomodel, consider we consider this effect. the ideal case of a perfectly collimated source, so the errors estimated must be improvedAs indicated, to consider the this slight effect. decrease in light signal E is proportional to the quotient between the 퐸 area ofAs the indicated, section-line the slight of the decrease drop and in the light area signal of the section-line is proportional of the sensitiveto the quotient volume. between Moreover, the thearea optical of the volumesection-line sensitivity of the drop can beand accurately the area of calibrated the section using-line metal of the spheres sensitive of divolume.fferent Moreover, diameters. Figurethe optical2 shows volume the relative sensitivity theoretical can attenuationbe accurately as a calibrated function of using the quotient metal spheres between of the different droplet diameterdiameters. d Fig andure diameter 2 showsD theof therelative sensitive theoretical volume. attenuation as a function of the quotient between the droplet diameter d and diameter 퐷 of the sensitive volume.

Figure 2. TheoreticalTheoretical relationship between light attenuation and the droplet diameter.diameter. 2.3. Signal Duration Measurement 2.3. Signal Duration Measurement The drops that penetrate the sensitive volume are detected as soon as the signal variation of the The drops that penetrate the sensitive volume are detected as soon as the signal variation of the photodetector exceeds a deﬁned threshold. Thus, the microcontroller will continuously monitor the photodetector exceeds a defined threshold. Thus, the microcontroller will continuously monitor the value of this signal and indicate the instants of entry and exit of the drop in the measurement volume. value of this signal and indicate the instants of entry and exit of the drop in the measurement The precision in the measurement of these times only depends on the maximum measurement volume. speed of the system and the minimum variation of the signal that can be detected. Depending on the The precision in the measurement of these times only depends on the maximum measurement clock frequency, hardware and implemented programming, we will ﬁnally be able to obtain a value of speed of the system and the minimum variation of the signal that can be detected. Depending on the the time between measures (∆T ), which represents the maximum delay in the detection of a drop clock frequency, hardware and implementedsam programming, we will finally be able to obtain a value that is entering or leaving the sensitive volume and consequently the determination of the signal of the time between measures (Δ푇푠푎푚), which represents the maximum delay in the detection of a duration. In other words, the output time minus input time in the volume has an error of ∆Tsam. drop that is entering or leaving the sensitive volume and consequently the determination± of the However, this error can be considered unsystematic. signal duration. In other words, the output time minus input time in the volume has an error of 2.4.±Δ푇푠푎푚 Data. However, Processing this error can be considered unsystematic.

2.4. DataAs long Processing as the drops that impact tangentially on the sensitive volume are ignored, the errors in the diameter estimation will be conditioned by the characteristics of the measuring instrument. As long as the drops that impact tangentially on the sensitive volume are ignored, the errors in the diameter estimation will be conditioned by the characteristics of the measuring instrument. The

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The response of the photodetectors is established by the surface covered by each drop; the cause of its variation is linked to the square of the diameter of those drops as shown in Figure1. Meanwhile, conventional digital systems work with A/D converters, whose resolution is a function n of their number of bits (nbits), which results in 2 bits measurement levels. Then, the surface resolution (πD2) of the detector is given by . Most studies propose an 8-bit converter, but we will compare the 22+nbits results obtained when using 8-bit and 10-bit converters. Over time, some techniques have been proposed to estimate the precipitation intensity from the information provided by the sensors, but in all cases, the drops are classiﬁed according to their size. The two most commonly used techniques are presented below: time technique and counting technique.

2.5. Time Technique It is known that during sampling time T, a certain number of drops have fallen through the P sensitive volume with a cumulative transition time of t . Illingworth and Stevens [13] showed that P i ti for each droplet size, the fraction of time T indicated the number of drops directly within sensitive volume V. However, this relation is only valid if there are many raindrops evenly distributed in space and if sampling time T is large compared to the time of the droplet transition. In this case, the number density of raindrops is expressed as P ti(size) N = (1) (size) V T · This calculation does not depend on the wind speed because an increase in wind speed increases the number of drops that reach the disdrometer in the same proportion when the transit time decreases. Meanwhile, the number density of drops is deﬁned by considering only those droplets whose centers are within the volume. Thus, the time of transition of the raindrops that reach the sensitive volume is deﬁned as the time that passes from when the center of a drop enters the sensitive volume to when it leaves this volume. Then, the total duration ts of the signal must be converted to transition time t from the center of a drop as D t = t (2) s d + D where d is the diameter of the drop and D is the diameter of the sensitive volume.

2.6. Counting Technique The droplet size spectra can also be determined by using the number of drops that penetrate the sensitive volume. By knowing local wind speed U and assuming that the vertical velocity of the raindrops is V f all, we can calculate the effective velocities of the drops. Length L and diameter D of the sensitive volume define the area of the side view (L D) of this volume. Multiplying this area by · the effective rate of passage of the droplets of a certain size gives an imaginary amount of air VL that passes through the disdrometer during sampling time T. This volume contains all droplets of a given size that reach the sensitive volume of the disdrometer within a sampling period r h i2 V = L D T U2 + V (3) L(size) · · f all(size)

The number density of drops of identical size is obtained as the quotient of the total number of drops detected divided by volume VL Asize N(size) = (4) VL The counting technique requires more assumptions than the time technique. For example, in this technique, the local wind speed must be known. Nevertheless, the terminal fall rate of raindrops can be estimated by the formula [14] V (d) = 17.67 d0.67 (5) f all · Symmetry 20202019, 1211, 303x FOR PEER REVIEW 66 of of 19

푉 (푑) = 17.67 · 푑0.67 (5) where d is the diameter of the drop in centimeters,푓푎푙푙 and the resulting drop rate is obtained in m/s. Thiswhere parameterization 푑 is the diameter has of been the drop chosen in centimeters, against other and options the resulting because drop it is the rate most is obtained novel. However,in 푚 / 푠. Thisthe results parameterization do not significantly has been dichosenffer from against those other offered options by other because authors it is [the15], most so any novel. of them However, could thebeused. results do not significantly differ from those offered by other authors [15], so any of them could be used.Tostudy the advantages and disadvantages of the two techniques, numerical simulations of measurementsTo study the with advantages the same precipitation and disadvantages model wereof the performed, two techniques, and the numerical results are simulations shown below. of measurements with the same precipitation model were performed, and the results are shown below. 2.7. Precipitation Intensity Calculation 2.7. PrecipitationThe precipitation Intensity intensity Calculation (PI) can be obtained from the droplet size spectrum by applying The precipitation intensity (PI) can be obtained from the droplet size spectrum by applying NXsize PI =푁푠푖푧푒 N V M (6) size· f all(size)· drop(size) 푃퐼 = ∑size=1푁푠푖푧푒 · 푉푓푎푙푙(푠푖푧푒) · 푀푑푟표푝(푠푖푧푒) (6) 푠푖푧푒=1 where Mdrop(size) is the mean mass of the droplets that are classified within the same size. where 푀 is the mean mass of thekg droplets that are classified within the same size. The The precipitation푑푟표푝(푠푖푧푒) intensity has units of if N is given in m 3, V is measured in m , 푘푔 m2s size −3 − f all(size) 푚 s precipitation intensity has units of 2 if 푁푠푖푧푒 is given in 푚 , 푉푓푎푙푙(푠푖푧푒) is measured in , and and Mdrop(size) is measured in kg. The푚 value푠 of Mdrop(size) can be determined by assuming a spherical푠 푀shape푑푟표푝(푠푖푧푒 of the) is drops.measured Thus, in it푘𝑔 is. quiteThe value simple of to 푀 calculate푑푟표푝(푠푖푧푒) thecan mass be determined of a spherical by dropassuming from a its spherical straight shapesection. of However,the drops. large Thus, and it is real quite droplets simple have to calculate an oblong the shape mass dueof a tospherical their aerodynamic drop from its properties. straight Thissection. effect However, has been large analyzed and [real13], droplets which verifies have an that oblong the vertical shape cross-sectional due to their aerodynamic area of a drop properties. decreases whenThis effect it deforms has been (elongates), analyzed which [13] starts, which approximately verifies that for the a 1.5 vertical mm drop cross diameter.-sectional This area eff oectf a causes drop decreasesa lower estimate when it of deforms the drop (elongates) volume. Figure, which3 showsstarts approximately similar results tofor those a 1.5 obtainedmm drop by diameter Pruppacher. This effectand Pitter causes [16 a]. lower estimate of the drop volume. Figure 3 shows similar results to those obtained by Pruppacher and Pitter [16].

Figure 3. EffectEﬀect of the drop o oblateness.blateness.

Rain can wetwet thethe outerouter lenses lenses of of the the disdrometer disdrometer due due to splashesto splashes of largeof large drops. drops. The The droplets droplets that wetthat thewet lens the willlens cause will cause electronic electronic signals signals that will that be will characterized be characterized by their by small their amplitudes small amplitudes and long anddurations. long durations. This type This of disturbance type of disturbance can cause ancan overestimation cause an overestimation of the number of the density number of small density drops. of smallTo drops. prevent this effect, we propose to perform a special filtration. Using this filter, those signals whoseTo duration prevent exceedsthis effect, a defined we propose time to period perform are eliminated.a special filtration. This threshold Using this is set filter, as the those maximum signals possiblewhose duration duration exceeds of the a signal defined generated time period by a are drop eliminated. of the detected This thresho size withld is respect set as tothe its maximum terminal possible duration of the signal generated by a drop of the detected size with respect to its terminal

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velocity and applying a time factor of 1.5. The ﬁlter time is called Tmax and is adjustable in our model, although it will not act in any case, since the aforementioned splashes are not simulated.

2.8. Edge Effects In the work of Grossklaus et al. [17–19] a series of phenomena that affect rainfall estimation are presented, which are called the edge effects. One of these main effects is the equivocal detection of water drops, which results in the cases detailed below, but there are other edge effects such as those due to drops just grazing the sensitive volume which also systematically influence the disdrometer measurements. However, as all of them can be described statistically, a correction is possible. Hence, all of these effects and the techniques proposed for their correction are considered and implemented in our model. In normal precipitation, drops of a wide range of sizes may be mixed. Therefore, the sensitive volume of the disdrometer should be sufficiently large to provide a reasonable probability of correctly measuring large droplets, which significantly contribute to the rainfall intensity. However, only the amount of shadow produced is measured, regardless of whether it is caused by one or more drops on the sensitive volume at any given time. These coincidences have two effects:

1. Two small drops can be detected as a larger drop. In this case, a larger amount of water than the one that actually contains the two smaller drops is detected. 2. The time of the transition detected as a consequence of the coincidence of two drops is larger than that corresponding to a drop of larger diameter, which simulates a greater probability of large drops and consequently an exaggerated contribution to the determination of the precipitation intensity.

Nevertheless, as these effects have a systematic influence on the measurement performed by the disdrometer, a statistical analysis enables to correct these effects. This error variance of the disdrometer measurements is estimated, considering the following effects that contribute to this variance [10]:

1. “The iterative procedure to compensate for the fringe effects is based on the assumption that these effects occur according to their mean statistics. The Monte Carlo studies are used to quantify the portion of the total error variance produced by the random occurrence of the fringe effects. This will be referred to in the following as (fringe effects). 2. The Monte Carlo simulations also consider the inhomogeneity and anisotropy of the sensitive volume. The variance that is caused by this effect is therefore included in the variance obtained from the Monte Carlo model. 3. The total error variance also includes a sampling error due to the fact that even at a constant rainfall the number of drops penetrating the sensitive volume varies from one measurement to another. This sampling error has not been considered in the Monte Carlo simulations. This variance that is only caused by the sampling error will be called. The value of was determined analytically”.

2.9. Numerical Model To quantify the edge eﬀects, the measurements made by the disdrometer were numerically modeled. Hence, our research team (Group of Traﬃc Control System, ITACA Institute, Universitat Politècnica de València, Spain) designed a simulation program in VisualBasic. This program enables to optimize the starting geometric data, such as the diameter or length of the sensitive volume. The maximum and minimum values of the droplet size to be measured and the resolution in the measurement of the diameters (number of bits of the A/D converter) are also adjustable. Next, a series of data on the type of precipitation, such as the wind speed and precipitation intensity, must be provided. Since the droplet size distribution follows rules that depend on where the test is performed, it was decided to use a gamma function as law. This distribution was chosen because it was the same used by Grossklaus et al. [17–19] and Marshall and Palmer [20] in their studies, which has proven to describe satisfactorily the concentrations of the smaller droplets. From this gamma Symmetry 2020, 12, 303 8 of 19

function, the parameters N0 and µ must be entered, and the system calculates the value of γ for the indicated precipitation intensity. To facilitate its use, two distributions of standard precipitation droplets—namely, stratiform and convective—are placed. In this way, although many studies analyzed this gamma distribution [20–25], in our study, the parameters used to describe these spectra were taken from Ulbrich [21,22], which can be seen in Table2.

Table 2. Drop surface distribution parameters.

3 1 µ Type of Precipitation N0 (m− cm− − ) µ Convective 7.54 106 1.63 × Stratiﬁed 1.96 105 0.18 ×

The next information to be supplied is the sampling time, time resolution and number of tests to perform. Finally, to correct possible edge effects, the value of Tmax must also be provided and indicate whether the effects of an excess of time of presence of the drop and the effect of the size of the drop on the duration time of each signal should be compensated. With all of this information, the program begins by calculating the drop distribution that corresponds to the indicated intensity and type of precipitation. The distribution of droplet size in natural rainfall is irregular. However, simple mathematical formulas are often recommended to describe these distributions. According to Marshall and Palmer [20], the normally used parameterization only requires two parameters Λd N(d, ∆d) = N e− (7) 0· where N(d, ∆d) is the concentration of droplets with diameters between d and ∆d; N0 is the cut-off point for parameter d = 0 mm; and Λ is the slope parameter. Nevertheless, this formulation does not represent a maximum of medium-sized rain drops as occurs in actual measurements. Therefore, the distributions of three parameters have been introduced (gamma distributions use a third parameter µ that defines the shape of a drop size distribution). In this way, Figure4 represents the distribution of the size of drops for different types of precipitation, which are characterized by N0 and µ, as stated in different works such as that of Ulbrich or Martin [10]. Ulbrich [21,22] shows that there is a definite relationship between parameters N0 and µ. This relationship is given by 4 3.2 µ N = 6.0 10 e · (8) 0 × · Therefore, the gamma distribution can be described using only two parameters. With this information, a drop distribution is assigned according to the size to obtain the PI specified in the data. By applying an iterative procedure on the value of the gamma function, we determine the number of drops that correspond to the central value diameter of each existing band between the minimum and maximum drop values. At the beginning of each iteration, the analytical model is initialized with a drop size distribution that is assumed to be true. Then, the measurement of the simulated spectrum of the disdrometer is compared with its actual measurement. The convergence is achieved if these spectra differ by less than 2%. A new iteration starts again after the initial spectrum is modified according to the results of the previous iteration. In most cases, the convergence should be achieved in less than three iterations. The best results are achieved by using the same drop size distribution measured as the spectrum of the first iteration. Symmetry 2019, 11, x FOR PEER REVIEW 8 of 19

studies, which has proven to describe satisfactorily the concentrations of the smaller droplets. From

this gamma function, the parameters 푁0 and 휇 must be entered, and the system calculates the value of 훾 for the indicated precipitation intensity. To facilitate its use, two distributions of standard precipitation droplets—namely, stratiform and convective—are placed. In this way, although many studies analyzed this gamma distribution [20–25], in our study, the parameters used to describe these spectra were taken from Ulbrich [21,22], which can be seen in Table 2.

Table 2. Drop surface distribution parameters.

Type of Precipitation N0 (m−3cm−1−µ) µ Convective 7.54 × 106 1.63 Stratified 1.96 × 105 0.18

The next information to be supplied is the sampling time, time resolution and number of tests to

perform. Finally, to correct possible edge effects, the value of 푇푚푎푥 must also be provided and indicate whether the effects of an excess of time of presence of the drop and the effect of the size of the drop on the duration time of each signal should be compensated. With all of this information, the program begins by calculating the drop distribution that corresponds to the indicated intensity and type of precipitation. The distribution of droplet size in natural rainfall is irregular. However, simple mathematical formulas are often recommended to describe these distributions. According to Marshall and Palmer [20], the normally used parameterization only requires two parameters

−Λ푑 푁(푑, ∆푑) = 푁0 · 푒 (7)

where 푁(푑, ∆푑) is the concentration of droplets with diameters between 푑 and ∆푑; 푁0 is the cut-off point for parameter 푑 = 0 mm; and Λ is the slope parameter. Nevertheless, this formulation does not represent a maximum of medium-sized rain drops as occurs in actual measurements. Therefore, the distributions of three parameters have been introduced (gamma distributions use a third parameter 휇 that defines the shape of a drop size distribution). In this way, Figure 4 represents the

Symmetrydistribution2020, 12 of, 303the size of drops for different types of precipitation, which are characterized by9 of 19푁0 and 휇, as stated in different works such as that of Ulbrich or Martin [10].

Figure 4. Gamma function of the drop size distribution for diﬀerent µ values.

This number of drops is randomly assigned between the diameter values in the considered band. For each drop, the terminal and global velocities are determined (sum of the terminal and that due to the wind). This step also assigns the position of the space occupied by each of these drops at the initial time. With this information, it is possible to obtain the moments where the drop is in contact with the sensitive volume, which enables one to check whether at a given time, there are simultaneously several drops within the sensitive volume, and whether they can appear overlapping. From these data, the rain signal is generated and repeated as many times as the number of trials to simulate. In our case, we worked with 20 or 200 simulations depending on the case. At this point, the simulation of the behavior of the disdrometer is ready to begin. For this purpose, sampling period T is divided into subperiods of ∆Tsam; for each of these instants, the area covered by the drops in the sensitive volume is determined. Therefore, for each drop, we determine whether it is in contact, totally or partially, with the sensitive volume. If the drop is completely inside the sensitive volume, the surface covered by the drop corresponds to the circle that represents its straight section

!2 d S = π (9) 2

If the drop passes while rubbing the surface or only partially penetrates it, as shown in Figure5, the covered surface ST is  " #  r  q 2 2  2 π 1 xi xi xi 2 π 1 xi b xi b xi b  ST = 2R sin− + 1 + r  + sin− − − 1 −  (10)  2 − R R − R  2 r − r − R  where R2 + b2 r2 x = − (11) i 2b Symmetry 2019, 11, x FOR PEER REVIEW 10 of 19 Symmetry 2020, 12, 303 10 of 19 Symmetry 2019, 11, x FOR PEER REVIEW 10 of 19

Figure 5. Representation of the surface covered by the passage of a drop through the sensitive volume. FigureFigure 5. 5.Representation Representation of ofthe the surface surface covered covered by the by passage the passage of a drop of athrough drop through the sensitive the sensitivevolume. volume. From the measurements of the obstructed surface at each sampling instant and considering the From the measurements of the obstructed surface at each sampling instant and considering the resolution of the system, a series of discrete values is obtained for the drop size (function of the resolutionFrom ofthe the measurements system, a series of the of obstructed discrete values surface is at obtained each sampling for the instant drop size and (function considering of thethe maximum surface obstructed) and duration of the signal. We assign each drop to any of the discrete maximumresolution surface of the system,obstructed) a series and duration of discrete of thevalues signal. is obtained We assign for each the drop drop to size any (function of the discrete of the drop sizes. dropmaximum sizes. surface obstructed) and duration of the signal. We assign each drop to any of the discrete With this information, the filters to eliminate excessively large or small drops and temporary dropWith sizes. this information, the filters to eliminate excessively large or small drops and temporary filters that enable one to eliminate the signals corresponding to splashes on the lenses are applied. filtersWith that enablethis information, one to eliminate the filters the to signals elimi correspondingnate excessively to large splashes or small on the drops lenses and are temporary applied. Then, the corresponding corrections are applied to drops that pass while rubbing the sensitive Then,filters the that corresponding enable one to corrections eliminate arethe appliedsignals tocorresponding drops that pass to whilesplashes rubbing on the the lenses sensitive are volume.applied. volume. Finally, we can calculate the PI using the time and counting techniques. Finally,Then, the we cancorresponding calculate the corrections PI using the are time applied and counting to drops techniques. that pass while rubbing the sensitive volume. Finally, we can calculate the PI using the time and counting techniques. 3. Results Results 3. ResultsFigFigureure 66 showsshows an example of of the the signals signals generated generated by by the the system system for for 1 second 1 s as a as consequence a consequence of of a stratiform rain of a precipitation intensity of 10 mm/h. From this general signal, a series of time a stratiformFigure 6 rain shows of a an precipitation example of the intensity signals of generated 10 mm/h. by From the system this general for 1 second signal, as a a series consequence of time intervalsof a stratiform was was extracted, extracted,rain of a whereprecipitation where the signalsthe intensity signals that can of that 10 induce canmm / errors induceh. From in errorsthis the estimationgeneral in the signal, of estimation the a precipitationseries ofof time the precipitation are appreciated. Thus, Figure 7 shows the case of a signal corresponding to a droplet areintervals appreciated. was extracted,Thus, Figure where7 shows the the signals case of a that signal can corresponding induce errors to a in droplet the estimation that passes whileof the thatrubbingprecipitation passes the while sensitive are appreciated.rubbing volume the (A) Thus,sensitive and Fig the urevolume case 7 ofshows a(A) time andthe interval casethe caseof where a signalof a severaltime corresponding interval drops simultaneouslywhere to a severaldroplet drops simultaneously pass through the sensitive volume (B). Both types of signals will lead to errors passthat throughpasses while the sensitive rubbing volume the sensitive (B). Both volume types (A) of signalsand the willcase lead of a to time errors interval in the where precipitation several in the precipitation estimation, and their effects must be corrected to make the measurement as estimation,drops simultaneously and their eff passects through must be correctedthe sensitive to makevolume the (B). measurement Both types asof accuratesignals will as possible.lead to errors accuratein the precipitation as possible. estimation, and their effects must be corrected to make the measurement as accurate as possible.

Figure 6. Signals g generatedenerated by by the the passage passage of of the the droplets droplets through through the the sensitive sensitive volume volume for for 1 1second s with witha precipitation a precipitation intensity intensity of 10 of mm 10/h. mm/h. Figure 6. Signals generated by the passage of the droplets through the sensitive volume for 1 second with a precipitation intensity of 10 mm/h.

Symmetry 2020, 12, 303 11 of 19 Symmetry 2019, 11, x FOR PEER REVIEW 11 of 19

(A)

(B)

Figure 7.7. SuperpositionSuperposition ofof severalseveral dropletsdroplets simultaneously passing through the sensitive volume. volume. (A) Signals generated by the passage; ((B)) AA dropletdroplet thatthat rubsrubs thethe sensitivesensitive volume.volume.

To determinedetermine the values that optimize the operation of the system, the following analyses were performed forfor didifferentfferent precipitationprecipitation intensitiesintensities (PIs):(PIs): 1. TThehe drop drop distribution distribution was was obtained obtained according according to to their their size: size: maintaining maintaining a constant a constant number number of dropsof drops of each of each size, size, we randomlywe randomly generated generated different different spatial spatial and andtemporal temporal distributions distributions (200 analyses)(200 analyses) to estimate to estimate the variations the variations that thatcould could arise arise in t inhe thedetermination determination of ofthe the measured measured as asa functiona function of ofthe the position position of ofthe the drops drops at atthe the initial initial time. time. 2. TThehe effect effect of of considerin consideringg different different sampling sampling times times of of 1− 1120120 seconds s was analyzed.was analyzed. To minimizeTo minimize the − theeffect effect ofthe of the spatial spatial and and temporal temporal distribution distribution of theof the drops dro onps on the the measurement, measurement, the the average average of of200 200 measurements measurements with with di ffdifferenterent distributions distributions was was analyzed. analyzed. 3. The effect of the type of gamma distribution applied was analyzed by comparing the results of the stratiform and convective distribution.

Symmetry 2020, 12, 303 12 of 19

3. The effect of the type of gamma distribution applied was analyzed by comparing the results of the stratiform and convective distribution. 4.SymmetryThe 2019 eff, ect11, x of FOR the PEER wind REVIEW speed on the results of the counting and time estimation systems12 of was 19 analyzed. These measurements were performed for a PI of 5 mm/h with measurement periods of 4. T5he s, resolutioneffect of the of 0.1wind milliseconds speed on the and results an average of the of counting 20 measurements. and time estimation systems was analyzed. These measurements were performed for a PI of 5 푚푚/ℎ with measurement periods 5. The results were compared for 8- and 10-bit converters and an average of 20 measurements. of 5 seconds, resolution of 0.1 milliseconds and an average of 20 measurements. 5. InThe this results way, were the main compared objective for was8- and to analyze10-bit converters the different and aspects an average that canof 20 influence measurements. the quality of the readings offered by the optical disdrometers, such as the number of bits of the AD converters, In this way, the main objective was to analyze the different aspects that can influence the sampling time, type of measurement technique, intensity of wind and precipitation. The results quality of the readings offered by the optical disdrometers, such as the number of bits of the AD obtained from these analyses are shown in Figures8–14, which represent: converters, sampling time, type of measurement technique, intensity of wind and precipitation. The 1.resultsThe obtained values from of the these precipitation analyses are measured shown within Fig respectures 8−14 to, thewhich actual represent: precipitation for different types of precipitation and measurement techniques (Figures8 and9). 1. The values of the precipitation measured with respect to the actual precipitation for different 2. Estimation of precipitation measurement error for different sampling times, measurement types of precipitation and measurement techniques (Figures 8 and 9). technique, and type of precipitation (Figure 10). 2. Estimation of precipitation measurement error for different sampling times, measurement 3. techniqueEstimate of, and the type PI versus of precipitation the real PI when(Figure working 10). with 10-bit AD converters for different types 3. Eofstimate precipitation of the PI and versus measurement the real PI techniques when working (Figures with 11 10 and-bit 12AD). converters for different types 4. ofEstimate precipitation of the IPand versus measurement the real IP techniques when working (Figure withs 11 8-bit and AD 12) converters. for different types of 4. Eprecipitationstimate of the and IP measurementversus the real techniques IP when working (Figures with 13 and 8-bit 14 ).AD converters for different types of precipitation and measurement techniques (Figures 13 and 14).

Figure 8. 8.PI PI estimation estimation for for stratiform stratiform rainfall rainfall using using the (A ) the time (A technique) time technique and (B) counting and (B) technique. counting technique.

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B) Figure 9. PI estimation for convective rainfall using the (A) time technique and (B) counting technique. Figure 9. 9.PI PI estimation estimation for for convective convective rainfall rainfall using using the (A ) the time (A technique) time technique and (B) counting and (B) technique. counting technique.

(A)

Figure 10. Cont. (A)

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(B(B) )

FigureFigure 10.10. PIPI estimation estimation errorserrors ofof the the counting counting and and time time techniques techniques for for different different diﬀerent sampling sampling times times for for (A(A) )stratiform stratiform rain rain and ( (B)) convective convective rain. rain.

(A) (A)

(B)

FigureFigure 11.11.Estimated Estimated PI PI with with 10-bit 10- convertersbit converters for stratiform for stratiform rainfall, rainfall, diﬀerent different precipitation precipitation intensities (B) andintensities diﬀerent and wind different speeds: wind (A speeds:) time technique (A) time technique and (B) counting and (B) counting technique. technique. Figure 11. Estimated PI with 10-bit converters for stratiform rainfall, different precipitation intensities and different wind speeds: (A) time technique and (B) counting technique.

Symmetry 2020, 12, 303 15 of 19 SymmetrySymmetry 2019 2019, ,11 11, ,x x FOR FOR PEER PEER REVIEW REVIEW 1515 ofof 1919

(A(A) )

(B(B) )

FigureFigureFigure 12. 12. 12. Estimated EstimatedEstimated PI PI PI with with with 10-bit 10 10-bit- convertersbit converters converters for convective for for convective convective rainfall, rainfall,rainfall, diﬀerent different different precipitation precipitation precipitation intensities intensitiesandintensities diﬀerent and and wind different different speeds: wind wind (A speeds: speeds:) time technique( A(A) )time time technique technique and (B) countingand and ( B(B) )counting counting technique. technique technique. .

(A(A) ) Figure 13. Cont.

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(B) (B) Figure 13. Estimated PI with 8-bit converters for stratiform rainfall, different precipitation intensities Figure 13. Estimated PI with 8-bit converters for stratiform rainfall, diﬀerent precipitation intensities Figureand different 13. Estimated wind speeds: PI with ( A8-)bit time converters technique for and stratiform (B) counting rainfall, technique different. precipitation intensities andand different diﬀerent w windind speeds: speeds: (A ()A time) time technique technique and and (B) (countingB) counting technique technique..

(A) (A)

(B) (B) Figure 14. Estimated PI with 8-bit converters for convective rainfall, diﬀerent precipitation intensities

and diﬀerent wind speeds: (A) time technique and (B) counting technique.

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4. Discussion Traffic accidents have an adverse effect both on an individual and societal level resulting in costs due to person injuries and property damage, increased travel times, and emissions due to congestion. Indeed, the safety impact of weather conditions has been actively studied in different studies [26–28], which conclude that accident risks are significantly heightened during rainy, snowy and icy road conditions. According to Salli et al. [29] the accident risk is more than four times higher for these conditions compared to bare road surface. However, a study based on data from the National Highway Traffic Safety Administration showed that rain and wet roads cause more car accidents and injuries than snow, sleet, or fog. Therefore, checking road conditions, especially rain, and properly warning users of it could help reduce the risk of landslides, collisions, or other unwanted traffic accidents. Today, multiple options allow for quickly checking whether storms will slow down travel or keep the wheels running safely and smoothly. Sophisticated levels of information help time a drive and warn of pending weather-related dangers, learn whether an umbrella would be handy to bring along, or if a snowstorm is brewing. New technologies have simplified the search for this type of information and now this can be consulted by anyone. However, despite the fact that disdrometers have been used for many years on roads around the world, their performance has not been optimized for these scenarios. Hence the interest of this paper. Accordingly, this research has analyzed the different aspects that can influence the quality of the readings offered by the optical disdrometers (such as number of bits of the AD converters, sampling time, type of measurement technique, wind intensity, and precipitation), has conducted an in-depth analysis of their impact on results and has proposed what should be the optimal values to use in each case, which will help in the design and implementation of future optical dysdrometers. Thus, this study aims to be a guide for the new disdrometers to be installed, which will be able to obtain more precise and detailed information on the state of the roads. In this way, accident reduction, better traffic management, and better quality of information provided to users will be achieved. With the indicated computer model, a set of numerical simulations has been performed, where adequate measures were taken to limit the effect of multiple occupancies of the sensitive volume, tangential incidences, and presence of drops in the sensitive volume in the initial and final instants of the sampling period. In this way, under these conditions and considering the results, the following conclusions are extracted: 1. The use of the counting technique always results in an estimate of precipitation the time technique. 2. The time technique usually overestimates of the precipitation values, except for precipitation intensities above 50 mm/h. 3. The counting technique provides very close estimated PI values to the actual values, which are slightly below them. 4. The PI measurement with 8-bit A/D converters provides better results with the time technique than with the counting technique. The latter usually underestimates the PI. 5. The PI measurement with 10-bit A/D converters provides better results with the counting technique than with the time technique. In this case, the latter usually overestimates the PI values. 6. For measurements using the time technique, the results show that it is better to work with 8-bit A/D converters. 7. For measurements using the counting technique, the results show that it is better to work with 10-bit A/D converters. 8. The edge effects have a smaller influence on the convective precipitation. The reason is that for a given precipitation intensity, the numerical density in the convective rain is less than that in the stratiform rainfall, and multiple occupations are less frequent. In the counting technique, the effect of the wind on the PI estimation is practically negligible. In the time technique, there is a decrease in the PI estimation when the wind speed increases. Symmetry 2020, 12, 303 18 of 19

In conclusion, with the results of this study, the design and operation of the optical disdrometers could be optimized by knowing at any time, depending on the type of precipitation and the wind, which should be the type of estimation and the number of bits of the AD converter to be used. In addition, an estimate of the error made is also provided, which enables its correction. In this way, the results will allow to assess the convenience of using this type of system in ITS environments, which is our ﬁeld of work. However, knowing these data, its conﬁguration can also be optimized for other scenarios. Consequently, its application in urban and interurban environments would help to have more accurate information in real time, which would make it easier for management centers to apply the corresponding measures when there are adverse meteorological phenomena. These would result in an improvement of the information provided to users and therefore, in an increase in road safety and a decrease in the number of accidents due to these causes.

Author Contributions: Conceptualization, F.M.B., A.M.-M., and A.M.S.; Methodology, F.M.B., A.M.-M., and A.M.S.; Software, F.M.B., A.M.-M., and A.M.S.; Validation, F.M.B., A.M.-M., A.M.S., V.M.S., and M.J.P.A.; Formal analysis, F.M.B., A.M.-M., and A.M.S.; Investigation, F.M.B., A.M.-M., and A.M.S.; Resources, F.M.B., A.M.-M., and A.M.S.; Data curation, F.M.B., A.M.-M., and A.M.S.; Writing—original draft preparation, F.M.B.; Writing—review and editing, F.M.B.; Visualization, F.M.B. and M.J.P.A.; Supervision, F.M.B.; Project administration, F.M.B.; Funding acquisition, A.M.S. All authors have read and agreed to the published version of the manuscript. Funding: This research has been funded by the Universitat Politècnica de València through its internal project “Equipos de detección, regulación e información en el sector de los sistemas inteligentes de transporte (ITS). Nuevos modelos y ensayos de compatibilidad y verificación de funcionamiento”, which has been carried out at the ITACA Institute. Conflicts of Interest: The authors declare no conflict of interest.

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