Outlier Detection and Smoothing Process for Water Level Data Measured by Ultrasonic Sensor in Stream Flows

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Article Outlier Detection and Smoothing Process for Water Level Data Measured by Ultrasonic Sensor in Stream Flows

Inhyeok Bae 1 and Un Ji 1,2,*

1 Smart City and Construction Engineering, Korea University of Science and Technology, Goyang-Si 10223, Korea; [email protected] 2 Department of Land, Water and Environment Research, Korea Institute of Civil Engineering and Building Technology, Goyang-Si 10223, Korea * Correspondence: [email protected]; Tel.: +82-31-9100-229

Received: 8 March 2019; Accepted: 3 May 2019; Published: 7 May 2019

Abstract: Water level data sets acquired by ultrasonic sensors in stream-scale channels exhibit relatively large numbers of outliers that are off the measurement range between the ultrasonic sensor and water surface, as well as data dispersion of approximately 2 cm due to random errors such as water waves. Therefore, this study develops a data processing algorithm for outlier removal and smoothing for water level data measured by ultrasonic sensors to consider these characteristics. The outlier removal process includes an initial cutoff process to remove outliers out of the measurement range and an outlier detection process using modified Z-scores based on the median absolute deviation (MAD) of a robust estimator. In addition, an exponentially weighted moving average (EWMA) method is applied to smooth the processed data. Sensitivity analyses are performed for factors that are subjectively set by the user, including the window size for the MAD outlier detection stage, the rejection criterion for the modified Z-score outlier removal stage, and the smoothing constant for the EWMA smoothing stage, based on four different water level data sets acquired by ultrasonic sensors in stream-scale experiments.

Keywords: data smoothing; exponentially weighted moving average; median absolute deviation; modiﬁed Z-scores; outlier detection; ultrasonic sensor; water level monitoring

1. Introduction Reliable streamflow measurement and data are essential for water management, flood control, river maintenance, river rehabilitation, stream restoration, etc. [1,2]. All design floods and dominant discharges for river management and restoration projects are determined by hydrological and hydraulic analyses based on observed discharge in the field [1]. Water level data observed at streams are used to estimate streamflow discharge through empirically determined functional relationships [2]. Therefore, measurement methods and devices must be selected to maximize water level measurement accuracy [3–7]. Most water level gauging stations in streams are equipped with gauges or sensors such as staff gauges, float-operated gauges, pressure transducers, bubble gauges, ultrasonic sensors, or flood crest gauges [3]. Various water level measurement methods and data recording systems have recently been proposed to leverage rapid advances in sensor and data processing technology [8]. Currently, stilling well and float systems that require specialized structural installations are the main water level observation methods. However, new electronic measurement methods such as ultrasonic sensors and recording devices do not require structural installations, and allow for more precise measurements through more frequent data sampling [4,5]. Staff gauges have accuracies of 1–3 cm depending on

Water 2019, 11, 951; doi:10.3390/w11050951 www.mdpi.com/journal/water Water 2019, 11, 951 2 of 16 installation location and wind conditions, and flood crest gauges have accuracies of 5–10 cm. However, ultrasonic sensors have accuracies of 1–2 mm [3]. The World Meteorological Organization [5] and Sauer and Turnipseed [4] have proposed using the larger of either 3 mm or 0.2% of the effective stage (the height of the water surface above the orifice, intake, or other point of exposure of the sensor to the water body) as the requisite water level measurement accuracy for stream gauging stations. However, higher accuracy is required to compute storage changes in reservoirs or discharge using slope ratings, and ultrasonic sensors can be used in water level measurements that require such high accuracy. Measurement methods that use ultrasonic sensors have reduced costs and have few limitations on installation locations compared to traditional water level measurement methods such as the stilling well and float system [9]. Recently, Mousa et al. [10], Rahmtalla et al. [11], Satria et al. [12], and Sunkpho and Ootamakorn [13] have used ultrasonic sensors as real-time water level monitoring methods for flood forecasting and warning systems. In addition, Bae et al. [14] applied ultrasonic sensors to monitor water levels during hydraulic experiments. Errors introduced by spurious water level data can invalidate discharge measurements, and such errors are generally produced by human mistakes, malfunctions of the automatic head recorder, or obstructions to normal flow originating from branches or other debris blocking the control section [15]. In particular, when the water level is measured by an ultrasonic sensor, errors can be due to various environmental factors associated with the sensor’s installation position, as well as irregular ultrasonic signals caused by irregular water surface flow. Data resulting from such errors are referred to as outliers, and unlike random errors, these outliers have no statistical significance. Therefore, suitable statistical methods must be employed to detect outliers and exclude them from analysis [1,15]. The characteristics of the outlier distribution can vary according to the measurement method, device, and other conditions, and therefore different processing techniques must be used according to the data characteristics [16]. When an ultrasonic sensor measurement method is used for water level observations, the outlier distribution characteristics of water level data must be understood so that a suitable statistical processing technique can be applied to remove the outliers from the data set. Water level data collected by ultrasonic sensors exhibit dispersions of approximately 20–50 mm [17], and the main source of error that causes this data dispersion is water waves [3]. The errors caused by water waves can be categorized as random errors [18]. The effects of random errors can be minimized by repeated measurements under the same conditions [15]. However, in river flows, water levels change over time and the repeated measurements are performed within relatively short time periods. Therefore, statistical methods must be used to estimate representative values of the observed data within a relatively short time range, in which the water level can be considered constant. No algorithm has yet been proposed that can automatically process a very large amount of water level data acquired by ultrasonic sensors and perform the outlier detection and data smoothing for different data application purposes. Therefore, the objective of this study is to develop the generalized algorithm of statistical processes for removing outliers and random errors from water level data acquired by ultrasonic sensors in actual stream conditions and smoothing water level data to reduce the dispersion caused by water waves categorized as random errors. Suitable statistical methods applied to the developed algorithm according to data characteristics of water levels measured by ultrasonic sensors in stream flows are an initial cutoff process to remove outliers out of the measurement range, an outlier detection process using modified Z-scores based on the median absolute deviation (MAD) of a robust estimator having a function of accurate estimation without being influenced by a large deviation outlier, and an exponentially weighted moving average (EWMA) method that can control the smoothing level by applying different smoothing constants to smooth the processed data. In particular, this study presents a specific process for determining the optimized window size within minimum and maximum ranges set by the user in the algorithm. In addition, the algorithm has a specialized treatment to solve a problem that MAD values become zero in situations where water level remains constant within data Water 2019, 11, 951 3 of 16

Water 2019, 11, x FOR PEER REVIEW 3 of 16 set assigned by the optimized window size. In this study, long-term water level data collected from stream-scalebe controlled experiment intentionally channel was used where to perform designed a sensitivity flow rate could analysis be controlled to determine intentionally the factors was relevant used to performoutlier removal a sensitivity and smoothing analysis to process determine in the the al factorsgorithm. relevant These tofactors outlier included removal the and window smoothing size processfor moving in the median algorithm. and median These factorsabsolute included deviation the (MAD) window of outlier size for detection, moving median a rejection and criterion median absolutefor modified deviation Z-scores (MAD) of outlier of outlier removal, detection, and aa rejectionsmoothing criterion constant for factor modified for Z-scoresthe exponentially of outlier removal,weighted andmoving a smoothing average constant(EWMA) factor of the for smoothing the exponentially process, weighted which should moving be averagedefined (EWMA)by users ofin the smoothingdata processing process, algorithm which shouldfor water be le definedvels monitored by users by in theultrasonic data processing sensors. algorithm for water levels monitored by ultrasonic sensors. 2. Materials and Methods 2. Materials and Methods 2.1. Data Collection 2.1. Data Collection The water level data were collected in two rounds using ultrasonic sensors at a large-scale The water level data were collected in two rounds using ultrasonic sensors at a large-scale outdoor outdoor flume at the Korea Institute of Civil Engineering and Building Technology—River flume at the Korea Institute of Civil Engineering and Building Technology—River Experiment Center Experiment Center (KICT-REC) (Figure 1). The stream-scale channel’s length was 594 m, and the (KICT-REC) (Figure1). The stream-scale channel’s length was 594 m, and the channel’s width was channel’s width was 11 m. The stream bed slope of the downstream section, which was the 11 m. The stream bed slope of the downstream section, which was the downstream water level downstream water level measurement point, was approximately 3/1000. The generated flow in the measurement point, was approximately 3/1000. The generated flow in the channel could be controlled channel could be controlled by adjusting the flow supply pump equipment. As shown in Figure 2, by adjusting the flow supply pump equipment. As shown in Figure2, unsteady flow experiments with unsteady flow experiments with two different forms of flow discharge hydrographs were designed. two different forms of flow discharge hydrographs were designed. While the maximum discharge of While the maximum discharge of the two hydrographs was designed with the same at 5 m3/s, the 3 thetotal two durations hydrographs of the was flow designed events with were the different same at. 5 Two m /s, ultrasonic the total durations sensors ofwere the flowinstalled events at were the diupstreamfferent. Twoweir ultrasonicof measurement sensors point were (A) installed and at at the the downstream upstream weir bridge of measurement of measurement point point (A) and (B) in at the downstreamchannel to monitor bridge ofwater measurement levels when point two (B) different in the channel hydrographs to monitor were water generated levels whenone after two dianotherfferent (Figure hydrographs 1). Therefore, were generated a total of one four after data another sets of (Figure ultrasonic1). Therefore, sensors awere total collected of four data for setsthis ofstudy: ultrasonic two data sensors sets were at measurement collected for thispoints study: (A) two and data (B) setsfor athydrograph measurement 1 and points two (A) data and sets (B) forat hydrographmeasurement 1 andpoints two (A) data and sets (B) at for measurement hydrograph points 2. By (A)converting and (B) forthe hydrographwater level 2.measured By converting at the theupstream water levelweir measuredof measurement at the upstreampoint (A) weirto flow of measurementdischarge using point the (A)previously to flow dischargedeveloped using relation the previouslybetween water developed level and relation flow between discharge, water changes level andin the flow flow discharge, discharge changes from inthe the upstream flow discharge to the fromchannel the can upstream be tracked to the over channel time. can The be trackedflow rate over over time. a weir The flowis a function rate over of a weirthe head is a function on the ofweir. the headTherefore, on the the weir. flow Therefore, rate measurement the flow rate in measurement a rectangular in aweir rectangular is based weir on isthe based Bernoulli on the Equation Bernoulli Equationprinciples. principles. The discharge The discharge coefficient coe ffiforcient a rectan for a rectangulargular wide wide weir weir used used in inthe the experiment experiment was determined and calibrated prior the experiments to present the rectangular wide weir equation for flow.flow. WaterWater level level measurements measurements at at the the downstream downstream bridge bridge of measurement of measurement point point (B) observe (B) observe the water the levelwater changes level changes in the downstreamin the downstream section section of the experimentalof the experimental channel. channel.

Figure 1. Stream-scaleStream-scale experiment experiment channel channel of of Korea Korea Institute of Civil Engineering and Building Technology—River ExperimentExperiment CenterCenter (KICT-REC).(KICT-REC).

Ultrasonic sensors generate successive signals from the transmitter, and when the signals reﬂect oﬀ the target object’s surface and return to the receiver, their round-trip times, in conjunction with the speed of sound, are converted to distance. The ultrasonic sensors used for the water level measurements in this experiment were HC-SR04 models with a measurable distance range from the sensor unit of 0.2–4.0 m, a measurement resolution of 3 mm, and a minimum sampling interval of 60 ms. In this

Water 2019, 11, x FOR PEER REVIEW 4 of 16

Water 2019, 11, 951 4 of 16 experiment, the raw data were sampled at intervals of 2.5 s. These sensors can be combined with data storage devices, communication modules, power supplies, level aligners, etc., according to the user’s needs [19], and in this study, the ultrasonic sensors were combined with an Arduino board and a WiFi module to collect data in real-time at the outdoor ﬂume. The raw data acquired by this system showed the distance values from the sensor to the water surface (Figure3). The number of data sets collected for each measurement location of the target hydrographs was between 77,040 and 96,954 (Table1). Figure3 shows all data, including the outliers, obtained from the ultrasonic sensor for each Figure 2. Target hydrographs designed for this study in the stream-scale experiment channel. Watermeasurement 2019, 11, x FOR point PEER of theREVIEW target hydrographs. 4 of 16 Ultrasonic sensors generate successive signals from the transmitter, and when the signals reflect off the target object’s surface and return to the receiver, their round-trip times, in conjunction with the speed of sound, are converted to distance. The ultrasonic sensors used for the water level measurements in this experiment were HC-SR04 models with a measurable distance range from the sensor unit of 0.2–4.0 m, a measurement resolution of 3 mm, and a minimum sampling interval of 60 ms. In this experiment, the raw data were sampled at intervals of 2.5 s. These sensors can be combined with data storage devices, communication modules, power supplies, level aligners, etc., according to the user’s needs [19], and in this study, the ultrasonic sensors were combined with an Arduino board and a WiFi module to collect data in real-time at the outdoor flume. The raw data acquired by this system showed the distance values from the sensor to the water surface (Figure 3). The number of data sets collected for each measurement location of the target hydrographs was between 77,040 and 96,954 (Table 1). Figure 3 shows all data, including the outliers, obtained from the ultrasonic sensor for each Figuremeasurement 2. Target point hydrographs of the target designed hydrographs. for this study in the stream-scale experiment channel. Figure 2. Target hydrographs designed for this study in the stream-scale experiment channel.

Ultrasonic sensors generate successive signals from the transmitter, and when the signals reflect off the target object’s surface and return to the receiver, their round-trip times, in conjunction with the speed of sound, are converted to distance. The ultrasonic sensors used for the water level measurements in this experiment were HC-SR04 models with a measurable distance range from the sensor unit of 0.2–4.0 m, a measurement resolution of 3 mm, and a minimum sampling interval of 60 ms. In this experiment, the raw data were sampled at intervals of 2.5 s. These sensors can be combined with data storage devices, communication modules, power supplies, level aligners, etc., according to the user’s needs [19], and in this study, the ultrasonic sensors were combined with an Arduino board and a WiFi module to collect data in real-time at the outdoor flume. The raw data acquired by this system showed the distance(a) values from the sensor to the water surface (Figure(b) 3). The number of data sets collected for each measurement location of the target hydrographs was between 77,040 and 96,954 (Table 1). Figure 3 shows all data, including the outliers, obtained from the ultrasonic sensor for each measurement point of the target hydrographs.

Figure 3. Measured raw data of the distance from the ultrasonic sensor to water surface for about 60 h.

(a) Case of hydrograph 1—downstream bridge; (b) case of hydrograph 1—upstream weir; (c) case of hydrograph 2—downstream(a) bridge; (d) case of hydrograph 2—upstream weir.(b)

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Table 1. Results of outlier detection for each data set and moving window size for median absolute deviation (MAD) process.

Optimal Number of Number of Number of Number Flow Sensor Acquisition Moving Outliers Outliers Total % of Data after of Raw Condition Location Time (h) Window Removed Removed Outlier Outlier Outlier Data Size (wo) with Cutoﬀ with MAD Removal Hydrograph Downstream 58 24 86,942 22,971 3653 26,624 30.6 60,318 1 bridge Hydrograph Upstream 52 13 77,040 10,674 3169 13,843 18.0 63,197 1 weir Hydrograph Downstream 61 11 87,916 9087 3601 12,688 14.4 75,228 2 bridge Hydrograph Upstream 67 19 96,954 12,650 4748 17,398 17.9 79,556 2 weir

2.2. Data Processing Methods

2.2.1. Robust Statistical Methods for Outlier Detection In water level data measured by ultrasonic sensors, outliers can occur when the ultrasonic wave is reflected before reaching the water surface, and such measurement values’ deviations can be quite large (Figure3). Therefore, in order to detect outliers in the water level data measured by ultrasonic sensors, a robust estimator must be used that is insensitive to outliers with large deviations. According to Huber [20], Leys et al. [21], and Rousseeuw and Croux [22], the sample median can be used as a robust estimator of location because the median is very insensitive to the effects of such large deviations and can thus minimize the effect of asymmetry due to outliers in the distribution of water level data measured by ultrasonic sensors. In addition, the MAD can be used as a robust scale estimator for estimating the dispersion of data centered on the sample median. Like the median, the MAD is very insensitive to outlier deviation [20–23]. When the collected data set was given as X = x , x , , xn , the MAD was defined as follows [24–26]: { 1 2 ··· } n o MAD(X) = Med X Med(X) (1) − where Med(X) is a function for finding the median of the data set X. Therefore, MAD(X) shows the median value of absolute deviations between the individual observations of data set X and Med(X). In order to compare the MAD value calculated in Equation (1) with the standard deviation, Equation (2) is used to convert MAD(X) to the normalized MAD (MADN) [24]:

MAD(X) MADN(X) = (2) 0.6745 where the constant value 0.6745 is the 75% percentile of a normal distribution, which corresponds to the MAD of the standard normal distribution with a standard deviation of 1 [24,26–28]. The MADN(X) corresponds to the standard deviation of the normal distribution for the center location of the data estimated by the sample median. To detect outliers, dimensionless variables must be used, as in Equation (3), to compare MADN(X) with the deviation of the individual observation xi centered on the location estimated by the sample median. Mi in Equation (3) is the modiﬁed Z-score. This is compared with the outlier rejection criterion β to determine if the data point is an outlier [26].

xi Med(X) M = − < β (3) i MADN(X)

Iglewicz and Hoaglin [26] suggested the rejection criterion, β = 3.5, based on the results of a simulation that used pseudo-normal observations for sample sizes of 10, 20, and 40. In addition, Leys et al. [21] reported that β can be determined diﬀerently according to the researcher’s perspective. Water 2019, 11, 951 6 of 16

2.2.2. Exponentially Weighted Moving Average for Data Smoothing Even if the outliers that occur in the measured water level data are entirely removed, there is approximately 2 cm of dispersion that occurs due to water waves in the flowing water. If a constant water level is repeatedly measured under the same conditions at short intervals using an ultrasonic sensor, a normal distribution is formed by the observed values due to random error, excluding outliers. Therefore, a moving average can be applied as a smoothing method. This is the simplest form of a low-pass filter that is commonly used in signal processing to smooth short-term variations in time series data. However, the actual water level can change over time during the monitoring period. Therefore the most recently observed data must be reflected more than older data in the smoothing process, and the EWMA can be used for such cases to apply lower weights to older data [29]. The EWMA exponentially reduces the weighting factors of older data as shown in Equation (4), when the data set is listed as a sequence for time t [29]:

St(x) = αxt + (1 α) St 1(x) (4) − · − where α shows the decreasing degree of weighting as a smoothing constant factor, with a range of 0 < α 1. As shown in Equation (4), a high level of smoothing can be obtained by increasing the ≤ weight of the past data with a smaller α, which makes the data dispersion smaller. However, if α becomes too small, the changes in data over a short time are almost ignored. Therefore, it is necessary that α vary according to the researcher’s subjective standards and the data usage purpose. Because the EWMA requires past observations to perform calculations, the initial observation value x1 can be problematic to establish. Therefore, some methods use the initial observation, as in S1(x) = x1, and other methods use the average of multiple initial values [29]. In this study, the initial observation has been applied for the EWMA.

2.2.3. Generalized Data Processing Algorithm for Water Levels Measured by Ultrasonic Sensors in Water Flows An integrated data processing algorithm for eliminating outliers and smoothing data variation in water level data collected using ultrasonic sensors was developed in this study. The proposed algorithm’s main processing steps were broadly divided into the outlier removal step and the data smoothing step (Figure4). The outlier processing step was further divided into the initial cuto ff of outliers and outlier removal using modified Z-scores. Input parameters and conditional factors such as the initial cutoff range (rmin and rmax), maximum and minimum values of window size (wmin and wmax), the smoothing constant factor (α), the resolution of the ultrasonic sensor (λ), and the rejection criterion (β) must be defined by users in the data processing algorithm for water levels monitored by ultrasonic sensors. The data set after completing outlier removal (Rwo ) and the data set after completing both outlier removal and smoothing (E) were provided as the final output. The developed algorithm was coded with Python 3 to perform data processing tests on the data sets collected in the experiments. The first step in the proposed outlier removal stage was the initial cutoff. Regarding the distance from the water surface to the ultrasonic sensor, the initial cutoff range can be determined by considering the distance from the sensor to the minimum water level, assuming that the sensor’s installation location is constant. When the minimum water level cannot be defined prior to the analysis, zero water levels can be assumed for the minimum water level. The zero water level means that there is no water flow and is the same as the bed level. In this study, data outside of this interval were determined to be clear outliers, and the proposed algorithm’s initial cutoff stage removed these outliers en masse. Therefore, when the user enters rmax and rmin of the initial cutoff range in this stage to define the post-processing input variables, multiple outliers outside of this interval defined as the initial cutoff range can be removed. Therefore, the number of outliers included in the raw data could be reduced, and the outlier detection performance could be improved through the sample median and MAD estimation in the next stage. If the users cannot define rmax and rmin of the initial cutoff range, Water 2019, 11, 951 7 of 16

the distance from the sensor to the bed for rmax and the zero value for rmin are used as a default in theWater algorithm. 2019, 11, x FOR PEER REVIEW 8 of 16

Figure 4. Flow chart of data processing for the outlier removal and the data smoothing for water level Figure 4. Flow chart of data processing for the outlier removal and the data smoothing for water level data measured by ultrasonic sensor. data measured by ultrasonic sensor.

Finally, the EMWA data smoothing stage minimized the dispersion of data due to random errors in the water level data from which outliers had been completely removed. This step used Equation

(4) on 𝑅, which is the output data after the outliers were removed. The initial stage of the algorithm

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After the initial cutoff stage, the outlier removal process was completed using modified Z-scores. This stage calculated the sample median and MAD of each data point in the dataset that passed through the initial cutoff, and then calculated the corresponding modified Z-scores, which were dimensionless values. When this value exceeded the user-entered β, the corresponding data point was determined to be an outlier and removed. In this step, when the median and MAD were estimated, a moving window was applied to consider that the water level data changed over time, and therefore, the optimal window size (wo) had to be determined. The algorithm proposed here calculated the relative difference in the moving median and the moving mean, as measures of the data set’s central tendency with outliers removed, for each window size w. The w that had the smallest maximum relative difference was determined to be wo. In order to determine w’s calculation range, input for wmin and wmax were needed. If the number of samples was two, the median and mean calculation results were the same. Therefore, the minimum window size wmin had to be more than three. When the algorithm is applied to real-time monitoring system, the optimal window size cannot be determined in real time. In this case, fixed window size should be applied or optimal window size could be updated based on the previous results of data processing with a certain period of time. If more than 50% of the values in the data set within the window size were the same values when calculating MAD during outlier removal, the calculated MAD value could become zero, and the MADN and Mi calculations in Equations (2) and (3) would then be impossible. Machado [30], Chu et al. [31], and Vivcharuk et al. [32] have presented methods that use the mean absolute deviation (MeanAD) as an alternative to a zero MAD. However, if all data within the window size have the same value, MeanAD would also be zero. Even in cases where MeanAD was not zero, if there were outliers with large deviations, the estimated scale would be large, and detecting outliers with relatively small deviations would be difficult. When measuring the water level of flowing water, cases often occur where the MAD and MeanAD values are zero in situations where the water level remains constant. Therefore, in this study, to resolve this problem, the minimum measurement unit of the ultrasonic sensor was entered by the user as input data. Therefore, a method was used to define the resolution λ as the MAD value when the calculated MAD was zero. Because λ was the same as the deviation of the minimum unit that can occur during measurement, it corresponded to the minimum size of MAD that could occur when MAD was not zero. This processing method minimized the effect of outliers with relatively large deviations when more than 50% of the values of the data set in the window size had the same value, and could thus detect outliers robustly. Finally, the EMWA data smoothing stage minimized the dispersion of data due to random errors in the water level data from which outliers had been completely removed. This step used Equation (4) on Rwo , which is the output data after the outliers were removed. The initial stage of the algorithm for this step required a smoothing constant α input, for which different values could be used according to the situation as explained in Section 2.2.2. In this study, the water level data measured by ultrasonic sensors was used to evaluate the data’s level of smoothing according to the smoothing constant (α), and the results are presented in Section3.

3. Results

3.1. Outlier Removal Processing The algorithm developed to remove outliers and smooth the water level data collected by ultrasonic sensors was applied to the water level data acquired by the stream-scale experiments described in Section2. The input variables rmax and rmin required for the initial cutoff stage were set at 2.6 m and 1.2 m, respectively, for the data set acquired at the downstream bridge, and 4.1 m and 1.3 m, respectively, for the data set acquired at the upstream weir. Table1 shows the number of outliers that were removed by the initial cutoff process. Because the outlier distribution characteristics in each data set were different, the ratios of outliers removed by the initial cutoff were different. Water 2019, 11, x FOR PEER REVIEW 9 of 16 for this step required a smoothing constant 𝛼 input, for which different values could be used according to the situation as explained in Section 2.2.2. In this study, the water level data measured by ultrasonic sensors was used to evaluate the data’s level of smoothing according to the smoothing constantWater 2019 (, 𝛼11),, 951and the results are presented in Section 3. 9 of 16

3. ResultsFurther outlier removal was performed by calculating modified Z-scores for data sets that passed through the initial cutoff stage. Here, the 0.003 m resolution of the experiment’s ultrasonic sensors 3.1. Outlier Removal Processing which was provided by the manufacture was used as λ to handle cases in which the MAD value was zero.The For algorithm the modified developed Z-scores’ to rejection remove criterion outliers βand, a value smooth of 3.5 the was water used, level following data collected the standard by ultrasonicproposed sensors by Iglewicz was andapplied Hoaglin to the (1993). water To level calculate data acquired the optimal by windowthe stream-scale size wo forexperiments removing describedoutliers, 3in and Section 50 were 2. The used input for wvariablesmin and w𝑟max ,and respectively. 𝑟 required The window for the initial size was cutoff increased stage were by 1, setand at the 2.6 maximum m and 1.2 value m, respectively, of the relative for di thefference data set between acquired the movingat the downstream median and bridge, moving and mean 4.1 was m andcalculated 1.3 m, respectively, for the data set for with the outliersdata set removedacquired forat the each upstream of a total weir. of 48 Table window 1 shows sizes. the Figure number5 shows of outliersthe maximum that were relative removed difference by betweenthe initial the movingcutoff process. median andBecause mean the and outlier the outlier distribution detection characteristicsratios for the 48 in windoweach data sizes. set were Because different, the optimal the ratios window of outliers size w removedo was set by as the the initial window cutoff size were with different.the minimum value that included the maximum relative differences between the moving median andFurther mean, the outlier optimal removal window was sizes performed for the fourby calcul data setsating ranged modified between Z-scores 11 and for 24.data The sets ratio that of passeddetected through outliers the according initial cutoff to the stage. increase Here, in the window 0.003 sizem resolution showed aof trend the experiment’s of exponential ultrasonic decrease. sensorsThe maximum which was relative provided difference by the decreased manufacture as the was window used sizeas 𝜆increased to handle and cases then in increasedwhich the slightly MAD valueafter thewas lowest zero. value.For the When modifiedw was Z-scores’ set too small, rejection cases criterion occurred 𝛽 where, a value there of was3.5 was a low used, ratio following of normal thevalues standard in the proposed data sample, by Iglewicz and the and estimation Hoaglin accuracy (1993). To of calculate the moving the medianoptimal andwindow MAD size decreased, 𝑤 for removingwhereasthe outliers, ratio of 3 inaccurateand 50 were outlier used detections for 𝑤 increased. and 𝑤 Finally,, respectively. Figure6 Theshows window the data size set wasRwo increasedthat passed by the 1, and outlier the removal maximum process. value The of the ratio relative of data difference detected as between outliers the in the moving raw data median set was and at movingleast 14.4% mean and was at mostcalculated 30.6% for (Table the 1data). Furthermore, set with outliers more removed than 70% for of each all outliers of a total were of removed48 window in sizes.the initial Figure cuto 5 ffshowsstage. the maximum relative difference between the moving median and mean and the outlierIn addition detection to the ratios window for the size 48 wind setting,ow thissizes. study Because also the analyzed optimal the window degree ofsize outlier 𝑤 was detection set as theaccording window to size changes within the the minimum rejection criterionvalue thatβ thatincluded the user the must maximum enter as relati an inputve differences value in the between outlier thestage moving using median modified and Z-scores. mean, Whilethe optimal the value window of 3.5 thatsizes was for proposedthe four data by Iglewicz sets ranged and Hoaglinbetween [2611] andis commonly 24. The ratio used, of a detected sensitivity outliers analysis according for the rejection to the increase criterion inβ is window required size to ensure showed that a thetrend value of exponentialof 3.5 is appropriate decrease. forThe outlier maximum detection relative of water difference level decreased data measured as the by window ultrasonic sizesensors. increased In and this thenstudy, increasedβ was increased slightly fromafter 1the to lowest 5 in units value. of 0.1, When and the𝑤 was corresponding set too small, ratios cases of removedoccurred outlierswhere there were wascalculated; a low ratio the results of normal are shown values in in Figure the 7data. For sample, the window and sizethe usedestimation to calculate accuracy the movingof the moving median medianand MAD, and theMAD optimal decreased, window whereas size found the ratio when of βinwasaccurate set at outlier 3.5 was detections used. As increased. shown in FigureFinally,7,

Figurewhen β6 increased,shows the thedata ratio set of𝑅 removed that passed data decreased the outlier exponentially, removal process. and whenThe ratioβ exceeded of data thedetected range asfrom outliers 3.0 to in 3.5, the the raw percentage data set was reduction at least in14.4% removed and at outliers most 30.6% was greatly(Table 1). reduced. Furthermore, Table2 moreshows than the 70%ratios of ofall removed outliers were outliers removed when diinﬀ theerent initial rejection cutoff criteria stage. β were applied to the four data sets.

Figure 5. Outlier detection rate and maximum relative diﬀerence between the moving mean and Figure 5. Outlier detection rate and maximum relative difference between the moving mean and median according to window size. median according to window size.

Water 2019, 11, x FOR PEER REVIEW 10 of 16

Water 2019, 11, 951 10 of 16 Water 2019, 11, x FOR PEER( aREVIEW) (b) 10 of 16

Figure 6. Final data(a) set after the outlier removal process. (a) Case of hydrograph(b) 1—downstream bridge; (b) case of hydrograph 1—upstream weir; (c) case of hydrograph 2—downstream bridge; (d) case of hydrograph 2—upstream weir.

In addition to the window size setting, this study also analyzed the degree of outlier detection according to changes in the rejection criterion 𝛽 that the user must enter as an input value in the outlier stage using modified Z-scores. While the value of 3.5 that was proposed by Iglewicz and Hoaglin [26] is commonly used, a sensitivity analysis for the rejection criterion 𝛽 is required to ensure that the value of 3.5 is appropriate for outlier detection of water level data measured by ultrasonic sensors. In this study, 𝛽 was increased from 1 to 5 in units of 0.1, and the corresponding ratios of removed outliers were calculated; the results are shown in Figure 7. For the window size

used to calculate the moving(c) median and MAD, the optimal window size(d) found when 𝛽 was set at 3.5 was used. As shown in Figure 7, when 𝛽 increased, the ratio of removed data decreased exponentially,FigureFigure 6. FinalFinal and datawhen data set set 𝛽 afterafter exceeded the outlier the rangeremoval from process. 3.0 to (( aa3.5,)) Case Case the of ofpercentage hydrograph reduction 1—downstream in removed bridge; (bb) case of hydrograph 1—upstream weir; (cc) case of hydrograph 2—downstream bridge; (dd) outliersbridge; was ( greatly) case of reduced. hydrograph Tabl 1—upstreame 2 shows weir;the ratios ( ) case of ofremoved hydrograph outliers 2—downstream when different bridge; rejection ( ) casecase of of hydrograph hydrograph 2—upstream 2—upstream weir. criteria 𝛽 were applied to the four data sets. In addition to the window size setting, this study also analyzed the degree of outlier detection according to changes in the rejection criterion 𝛽 that the user must enter as an input value in the outlier stage using modified Z-scores. While the value of 3.5 that was proposed by Iglewicz and Hoaglin [26] is commonly used, a sensitivity analysis for the rejection criterion 𝛽 is required to ensure that the value of 3.5 is appropriate for outlier detection of water level data measured by ultrasonic sensors. In this study, 𝛽 was increased from 1 to 5 in units of 0.1, and the corresponding ratios of removed outliers were calculated; the results are shown in Figure 7. For the window size used to calculate the moving median and MAD, the optimal window size found when 𝛽 was set at 3.5 was used. As shown in Figure 7, when 𝛽 increased, the ratio of removed data decreased exponentially, and when 𝛽 exceeded the range from 3.0 to 3.5, the percentage reduction in removed outliers was greatly reduced. Table 2 shows the ratios of removed outliers when different rejection criteria 𝛽 were applied to the four data sets. Figure 7. Relations between outlier percentage and rejection criterion. Figure 7. Relations between outlier percentage and rejection criterion. Table 2. Percentage of detected outliers according to rejection criterion. % of Outliers Rejection Case of Hydrograph 1 Case of Hydrograph 2 Criterion β Downstream Bridge Upstream Weir Downstream Bridge Upstream Weir 1.0 30.92 31.32 31.22 31.83 2.0 11.56 13.04 12.29 11.99 2.5 8.05 8.77 8.13 8.31 3.0 6.52 6.37 6.23 6.60 3.5 5.71 4.57 4.78 5.63 4.0 5.25 3.95 4.15 5.12 5.0 4.76 2.87 3.28 4.42

Figure 7. Relations between outlier percentage and rejection criterion.

Water 2019, 11, x FOR PEER REVIEW 11 of 16

Table 2. Percentage of detected outliers according to rejection criterion.

% of Outliers Rejection Case of Hydrograph 1 Case of Hydrograph 2 Criterion 𝜷 Downstream Downstream Upstream Weir Upstream Weir Bridge Bridge 1.0 30.92 31.32 31.22 31.83 2.0 11.56 13.04 12.29 11.99 2.5 8.05 8.77 8.13 8.31 3.0 6.52 6.37 6.23 6.60 3.5 5.71 4.57 4.78 5.63 4.0 5.25 3.95 4.15 5.12 Water 2019, 11, 951 11 of 16 5.0 4.76 2.87 3.28 4.42

3.2.3.2. Data Data Smoothing Smoothing

The ultrasonic sensor-acquired water level data 𝑅 with the outliers removed exhibited The ultrasonic sensor-acquired water level data Rwo with the outliers removed exhibited approximatelyapproximately 2 2cm cm of of data data dispersion dispersion due due to random to random error. error. To mitigate To mitigate this data this dispersion, data dispersion, data smoothingdata smoothing using usingEWMA EWMA was wasperformed performed in the in thealgo algorithm’srithm’s final ﬁnal stage. stage. Figure Figure 8 8shows shows the the data smoothingsmoothing results results when when the the smoothing smoothing constant constant factor factor 𝛼α was set at 0.1. The The dispersion dispersion in in the the water water levellevel data data due due to to random random error error was was reduced. reduced.

(a) (b)

FigureFigure 8. TheThe datadata set set after after the smoothingthe smoothing process process using EWMA.using EWMA. (a) Case ( ofa) hydrograph Case of hydrograph 1—downstream 1— downstreambridge; (b) case bridge; of hydrograph (b) case 1—upstreamof hydrograph weir; 1—upstream (c) case of hydrograph weir; (c) 2—downstreamcase of hydrograph bridge; 2— (d) downstreamcase of hydrograph bridge; 2—upstream(d) case of hydrograph weir. 2—upstream weir.

TheseThese smoothing smoothing process process results results can can vary a accordingccording to the smoothing constant factorfactor α𝛼.. Figure Figure9 9compares compares the the smoothing smoothing process process e ffeffectsects on on a a partial partial section section of of the the overall overall data data when whenα was𝛼 was 0.01, 0.01, 0.1, 0.1,0.5, 0.5, and and 0.9. 0.9. When Whenα was 𝛼 was0.01, 0.01, the weight the weight of the of previous the previous data increaseddata increased and the and water the water level data’slevel data’sdispersion dispersion was greatly was greatly reduced, reduced, although although short-term short-term changes changes in the datain the were data ignored. were ignored. When α Whenhad a 𝛼large had valuea large of value 0.5 or of more, 0.5 or the more, data’s the dispersion data’s disp wasersion reflected was veryreflected closely, very although closely, thealthough smoothing the smoothingeffect was greatlyeffect was reduced, greatly contrary reduce tod, thecontrary goal of to this the study’s goal of smoothing this study’s process. smoothing Thus, process. the smoothing Thus, effect based on EWMA was very sensitive to α. In order to quantitatively examine the smoothing effect according to the input variable α, the changes in the standard deviation according to α were calculated for the data set in Figure9 (Table3). When α was 1, the weight of the previous data in Equation (4) became zero, reflecting a case where there was no smoothing process effect. Therefore, the level of smoothing was calculated by a standard deviation reduction rate based on the standard deviation of 0.047 when α was 1 which represent. The smoothing effect on the water level data increased as α decreased. Water 2019, 11, x FOR PEER REVIEW 12 of 16 the smoothing effect based on EWMA was very sensitive to 𝛼. In order to quantitatively examine the smoothing effect according to the input variable 𝛼, the changes in the standard deviation according to 𝛼 were calculated for the data set in Figure 9 (Table 3). When 𝛼 was 1, the weight of the previous data in Equation (4) became zero, reflecting a case where there was no smoothing process effect. Therefore, the level of smoothing was calculated by a standard deviation reduction rate based on the standard deviation of 0.047 when 𝛼 was 1 which represent. The smoothing effect on the water level Water 2019, 11, 951 12 of 16 data increased as 𝛼 decreased.

Figure 9. Comparison of the smoothing levels with diﬀerent smoothing constant factors (case of Figure 9. Comparison of the smoothing levels with different smoothing constant factors (case of hydrograph 1—downstream bridge). hydrograph 1—downstream bridge). Table 3. Level of smoothing and standard deviation with diﬀerent smoothing constant factors (used Tabledata set 3. fromLevel 14 of/09 smoothing 11:16 to 14 and/09 standard 11:31 of case deviation of hydrograph with different 1—downstream smoothingbridge) constant. factors (used data set from 14/09 11:16 to 14/09 11:31 of case of hydrograph 1—downstream bridge). Smoothing Constant Factor α Standard Deviation (m) Level of Smoothing (%) Smoothing Constant Factor Standard Deviation Level of Smoothing 1.00 0.0047 0.0 𝜶 0.90 0.0043(m) 9.3(%) 1.000.80 0.0047 0.0039 17.6 0.0 0.900.70 0.0043 0.0035 25.3 9.3 0.800.60 0.0039 0.0032 32.517.6 0.50 0.0028 39.4 0.70 0.0035 25.3 0.40 0.0025 46.1 0.600.30 0.0032 0.0022 53.032.5 0.500.20 0.0028 0.0019 60.539.4 0.400.10 0.0025 0.0014 70.646.1 0.300.01 0.0022 0.0007 84.953.0 0.20 0.0019 60.5

Finally, Figure0.10 10 shows the results when the0.0014 data set Rwo with complete outlier70.6 removal and the EWMA-smoothed0.01 data set E were both converted0.0007 to flow discharge and water84.9 level. Figure 10a,b show the results when the distance from the ultrasonic sensor to the water surface in the upstream weirFinally, reservoir Figure was converted 10 shows to the a waterresults level when value the anddata then set converted𝑅 with intocomplete flow dischargeoutlier removal by the weirand theequation EWMA-smoothed describing the data relation set between𝐸 were waterboth converted stage and flowto flow discharge. discharge That and is, Figurewater 10level.a,b recreateFigure 10a,bFigure show2 0s target the results discharge when hydrographs the distance 1 from and 2.th Flowe ultrasonic discharge sensor in Figure to the 10 waterb for surface hydrograph in the 2 upstreamwas validated weir reservoir through was cross-verification converted to a from water the level discharge value and data then measured converted by into ADCP flow discharge (Acoustic byDoppler the weir Current equation Profiler) describing in the the experiment. relation between ADCP water is a hydroacousticstage and flowcurrent discharge. meter That to is, measure Figure 10a,bcurrent recreate velocities Figure over 2′ as depthtarget rangedischarge and hydrographs to produce flow 1 and discharge 2. Flow data. discharge During in theFigure experiment 10b for hydrographof hydrograph 2 was 2, ADCP validated was through applied cross-verificati to get the informationon from forthe flowdischarge rate in data the measured channel section by ADCP and (Acousticthe results Doppler were compared Current in Profiler) Figure 10inb the with experi the datament. of ADCP flow discharge is a hydroacoustic converted fromcurrent water meter levels to measuremeasured current by ultrasonic velocities sensors over in a the depth upstream range weirand reservoir.to produce ADCP flow measurement discharge data. performed During eachthe experimenteight times of for hydrograph five different 2, ADCP steps ofwas hydrograph applied to 2get and the the information results were for producedflow rate in by the averaging channel sectioneight-time and measurements. the results were For compared the comparison, in Figure flow 10b discharge with the datadata producedof flow discharge by ultrasonic converted sensor from was wateraveraged levels for measured the same periodby ultrasonic corresponding sensors toin eight-timethe upstream measurements weir reservoir. of ADCP ADCP for measurement each step of performedhydrograph each 2. The eight deviations times for from five flowdifferent discharges steps of measured hydrograph by ADCP 2 and were the results0.05 m were3/s, + produced0.14 m3/s, − by0.06 averaging m3/s, +0.07 eight-time m3/s, and measurements.0.23 m3/s followed For the by comparison, time sequence. flow Except discharge the lowest data flow produced condition, by − − the deviations of flow discharge converted by water levels measured by ultrasonic sensors range from 2.17% to 3.81% of flow discharges measured by ADCP. − Figure 10a,b show the very large difference in the degree of dispersion in the data that had undergone outlier removal and the data that had undergone the smoothing process. Furthermore, when the flow was large, this difference was even larger. In order to compare the data processing results by stage, Figure 11 shows the interval data from 14/09 11:16 to 14/09 11:31 in the hydrograph 1— Water 2019, 11, x FOR PEER REVIEW 13 of 16 ultrasonic sensor was averaged for the same period corresponding to eight-time measurements of ADCP for each step of hydrograph 2. The deviations from flow discharges measured by ADCP were −0.05 m3/s, +0.14 m3/s, −0.06 m3/s, +0.07 m3/s, and −0.23 m3/s followed by time sequence. Except the lowest flow condition, the deviations of flow discharge converted by water levels measured by ultrasonic sensors range from −2.17% to 3.81% of flow discharges measured by ADCP. Figure 10a,b show the very large difference in the degree of dispersion in the data that had undergone outlier removal and the data that had undergone the smoothing process. Furthermore, Water 2019, 11, 951 13 of 16 when the flow was large, this difference was even larger. In order to compare the data processing results by stage, Figure 11 shows the interval data from 14/09 11:16 to 14/09 11:31 in the hydrograph 1—downstreamdownstream bridge bridge data data set, set, which which included included the datathe data on the on distancethe distance from from the ultrasonic the ultrasonic sensor sensor to the towater the water surface surface that had that not had been not converted been converted to water to level water data. level The data. average The average values ofvalues the outlier of the removal outlier removaland smoothing and smoothing results for results this interval for this were interval almost were the almost same at the 1.46855 same mat and1.46855 1.46860 m and m, respectively.1.46860 m, respectively.However, the However, averagevalue the average for raw datavalue without for raw data data processing without wasdata 1.232processing m, demonstrating was 1.232 them, demonstratingdifference due the to unremoved difference due outliers. to unremoved outliers.

(a) (b)

FigureFigure 10. 10. FlowFlow discharge discharge in the upstream weirweir andand waterwater level level in in the the downstream downstream bridge bridge monitored monitored by byultrasonic ultrasonic sensors. sensors. (a) Flow(a) Flow discharge discharge converted converted by water by water level datalevel in data case in of hydrographcase of hydrograph 1—upstream 1— upstreamweir; (b) weir; ﬂow ( dischargeb) flow discharge converted converted by water by level water data level in data case in of case hydrograph of hydrograph 2—upstream 2—upstream weir; weir;(c) water (c) water level level converted converted by measured by measured data indata case in ofcase hydrograph of hydrograph 1—downstream 1—downstream bridge; bridge; (d) water (d) Water water2019level, 11 convertedlevel, x FOR converted PEER by measuredREVIEW by measured data in data case in of case hydrograph of hydrograph 2—downstream 2—downstream bridge. bridge. 14 of 16

(a)

Figure 11. Cont.

(b)

Figure 11. Comparison of raw data, outlier removal result, and smoothing result of water level data measured by ultrasonic sensor in downstream bridge for hydrograph 1. (a) Comparison section from 14/09 11:16 to 14/09 11:31; (b) comparison section from 14/09 11:28 to 14/09 11:31.

4. Discussion and Conclusion This study developed an outlier removal and smoothing process algorithm for water level data collected using ultrasonic sensors in a stream-scale experiment. The study included sensitivity analyses on factors that were subjectively determined by the user in each data processing step, including the window size for the MAD of outlier detection, the rejection criterion for modified Z- scores of outlier removal, and the smoothing constant for the EWMA. In the water level data set that was collected by the ultrasonic sensors in this study, there were many outliers with relatively large deviations distributed in the set and a relatively large random error due to the occurrence of water waves. Considering these data characteristics, the data processing algorithm developed in this study was broadly divided into an outlier removal process and a smoothing process. The algorithm incorporated an initial cutoff stage that set the initial cutoff range and determined that data outside this interval were outliers, removing them en masse. This process detected over 70% of all outliers in the water level data set captured by ultrasonic sensors. The outlier removal process after the initial cutoff used modified Z-scores based on the median and

MAD. The outlier removal stage included a process for determining the optimal window size 𝑤 for setting the moving median and MAD. To address cases when the MAD was zero, such as when the same value was distributed above a specific ratio in the data set for the given window size, the

Water 2019, 11, x FOR PEER REVIEW 14 of 16

Water 2019, 11, 951 14 of 16 (a)

(b)

FigureFigure 11. 11. ComparisonComparison of of raw raw data, data, outlier outlier removal removal result, result, and and smoothing smoothing result result of of water water level level data data measuredmeasured by by ultrasonic ultrasonic sensor sensor in in downstream downstream bridge bridge for for hydrograph hydrograph 1. 1. ( (aa)) Comparison Comparison section section from from 14/0914/09 11:16 11:16 to to 14/09 14/09 11:31; ( b) comparison section from 1414/09/09 11:28 to 1414/09/09 11:31.11:31.

4.4. Discussion Discussion and and Conclusion Conclusions ThisThis study study developed developed an an outlier outlier removal removal and and smoothing smoothing process process algorithm algorithm for for water water level level data data collectedcollected usingusing ultrasonicultrasonic sensors sensors in ain stream-scale a stream-scale experiment. experiment. The study The includedstudy included sensitivity sensitivity analyses analyseson factors on that factors were that subjectively were subjectively determined determ by theined user by in eachthe user data in processing each data step, processing including step, the includingwindow size the forwindow the MAD size offor outlier the MAD detection, of outlier the rejectiondetection, criterion the rejection for modified criterion Z-scores for modified of outlier Z- scoresremoval, of outlier and the removal, smoothing and constant the smoothing for the EWMA.constant for the EWMA. InIn the the water levellevel datadata set set that that was was collected collected by by the the ultrasonic ultrasonic sensors sensors in this in study,this study, there there were manywere manyoutliers outliers with relatively with relatively large deviations large deviations distributed distributed in the set in and the a relativelyset and a largerelatively random large error random due to errorthe occurrence due to the of wateroccurrence waves. of Considering water waves. these Consid data characteristics,ering these data the datacharacteristics, processing algorithmthe data processingdeveloped inalgorithm this study developed was broadly in this divided study into was an broadly outlier removaldivided processinto an andoutlier a smoothing removal process process. andThe a algorithm smoothing incorporated process. The an algorithm initial cuto incorporatedff stage that setan initial the initial cutoff cuto stageff range that andset the determined initial cutoff that rangedata outside and determined this interval that were data outliers, outside removing this interval them were en masse. outliers, This removing process detected them en over masse. 70% ofThis all processoutliers detected in the water over level 70% dataof all set outliers captured in the by ultrasonicwater level sensors. data set The captured outlier by removal ultrasonic process sensors. after Thethe initialoutlier cuto removalff used process modified after Z-scores the initial based cutoff on used the median modified and Z-scores MAD. Thebased outlier on the removal median stage and included a process for determining the optimal window size wo for setting the moving median and MAD. The outlier removal stage included a process for determining the optimal window size 𝑤 for settingMAD. Tothe address moving cases median when and the MAD. MAD To was address zero, such cases as when when the the MAD same was value zero, was such distributed as when above the samea specific value ratio was in distributed the data set forabove the givena specific window ratio size, in the the data proposed set for algorithm the given defined window the ultrasonicsize, the sensors’ resolution λ as the MAD value. EWMA was then used in the data smoothing process, which allowed different levels of smoothing results to be calculated according to the data processing goals by entering various values for the smoothing constant factor α. Regarding the sensitivity analysis of the algorithm, the optimal window size wo for the outlier removal process was determined based on the maximum relative difference between the moving median and moving mean after outlier removal, which was used as the evaluation standard for the measure of central tendency. When the window size was small, the number of detected outliers was higher, and the estimation accuracy of the sample median and MAD were reduced, resulting in a higher rate of inaccurate outlier detection in the modified Z-score stage. The ratio of removed outliers decreased exponentially as the window size w increased. The maximum relative difference between moving median and moving mean after outlier removal decreased as w until a fixed level, and then increased slightly after the lowest level. β was the rejection criterion for the modified Z-scores, which the algorithm used to remove outliers, and as its value increased, the ratio of detected outliers tended to decrease exponentially. If β exceeded the range from 3.0 to 3.5, the percentage decrease in removed Water 2019, 11, 951 15 of 16 outliers was greatly reduced. Finally, the level of smoothing by EWMA in the smoothing process was expressed as the rate of reduction in the standard deviation caused by a reduction in α, based on the standard deviation value when the smoothing constant factor α was 1.0. This study confirmed that the level of smoothing for water level data measured by ultrasonic sensors increased to a maximum 84.9% of standard deviation reduction as the smoothing constant factor α decreased.

Author Contributions: Conceptualization, I.B. and U.J.; Methodology, I.B. and U.J.; Software, I.B.; Formal Analysis, I.B. and U.J.; Investigation, I.B. and U.J.; Data Curation, I.B. and U.J.; Writing—original draft preparation, I.H.; Writing—review and editing, U.J.; Supervision, U.J.; Project administration, U.J.; Funding acquisition, U.J. Funding: This research was supported by Korea Institute of Civil Engineering and Building Technology (KICT) for the International Matching Joint Research Project, grant number 20180561, and KICT-REC (River Experiment Center) for REC International Experiment Days (RIED) 2017. Acknowledgments: The authors gratefully acknowledge the support of all participants of KICT, CER (Center for Ecohydraulics Research) at the University of Idaho, Nature and Technology, and TENELEVEN. Conﬂicts of Interest: The authors declare no conﬂict of interest.

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