Comprehensive relationships between kinetic energy and rainfall intensity based on precipitation measurements from an OTT Parsivel² optical disdrometer

Abstract

When raindrops collide with the topsoil surface, they cause soil detachment, which can be estimated by measuring the kinetic energy (KE) of the raindrops. Considering their direct measurements on terrestrial surfaces are challenging, empirical equations are commonly utilized for estimating the KE from rainfall intensity (I_r), which has a great influence on soil loss and can be easily obtained. However, establishing the optimal relationship between KE and I_r is difficult. In this study, we used a laser-based instrument (OTT Parsivel² Optical disdrometer) to collect datasets in Sangju City (South Korea) between June 2020 and December 2021 to examine the characteristics of KE–I_r relationships. We derived two different expressions for KE–I_r: KE expenditure (KE_exp; J m⁻²h⁻¹) and KE content (KE_con; J m⁻²mm⁻¹), using 37 rainfall events. Subsequently, the 37 rainfall events were categorized into three groups based on the magnitude of the mean rainfall intensity of each event. Overall, the KE values estimated through the equations derived based on 37 events were higher than those estimated by the equations derived based on the three rainfall event groups. Our findings should facilitate the development of more suitable physics-based soil erosion models at event scales.

1 Introduction

Soil is an essential element for sustaining life on Earth and largely determines the function of any ecological system. It influences the biogeochemical (Basu et al., 2021) and carbon dynamics (Majumder et al., 2018)–as well as climate change (Bonfante and Bouma, 2015)–by controlling the movement of minerals, energy, and water in the environment (Osman, 2014). The loss of millions of hectares of agriculture due to water-induced soil erosion is a global concern, with more than 36 billion tons of soil being eroded annually (Pimentel, 2006). Therefore, understanding the processes that contribute to soil loss is essential.

Water-induced soil erosion is a two-stage process that begins with rainsplash detaching soil particles from the topsoil surface and continues with surface runoff transporting the detached particles (Morgan, 2005). When raindrops impact the soil, they release their kinetic energy (KE), causing soil particles to be airborne and resulting in rainsplash erosion (Torres et al., 1992). The KE of raindrops is a function of their mass and velocity (Eq. 1) and the KE of a single raindrop, assuming it is spherical, is calculated as follows:where m denotes the raindrop mass (g), denotes the density of raindrop (i.e., 1 g ), v denotes raindrop velocity , and D denotes raindrop diameter (mm).

Researchers have differing views on the effectiveness of the KE and momentum for estimating soil detachment by raindrops. According to Rose (1960) and Paringit and Nadaoka (2003), momentum of rainfall substantially outperforms KE in calculating soil detachment; Lim et al. (2015) corroborated these findings. In contrast, Al-Durrah and Bradford (1982) employed KE to forecast the quantity of soil removed. van Dijk et al. (2002) assumed that the amount of energy available for separation and transmission by rain-splash is expressed by KE. Morgan (2005) concluded that rainfall erosivity is best expressed in terms of KE. In addition, KE is an essential parameter in numerous erosion models (e. g., the Universal Soil Loss Equation (Wischmeier and Smith, 1958), SLEMSA (Elwell, 1978), WaTEM/SEDEM (Verstraeten et al., 2002), and Surface Soil Erosion Model (Lee et al., 2013)) for characterizing the erosivity of raindrops.

In recent decades, numerous researchers have developed various approaches for determining the KE of rainfall. Attempts have been tried to directly measure raindrop impact (Madden et al., 1998); however, the instruments involved are both expensive and difficult to operate. Consequently, empirical equations are an alternative for estimating KE from rainfall intensity (I_r), which has a great influence on soil erosion and can be easily obtained. Raindrop size and velocity may be recorded using a number of ways, including the stain-paper methods (Wischmeier and Smith, 1958), video recorders with high frame rates (Kinnell, 1981), as well as more diverse approaches (Kathiravelu et al., 2016). With the advent of technological and electrical improvements, using laser-based instruments, we can more precisely measure the characteristics of raindrops than previous methods.

Several types of KE–I_r equations exist: Polynomial (Carter et al., 1974), power-law (Park et al., 1980), exponential (Brown and Foster, 1987), linear (Sempere-Torres et al., 1998), or logarithmic (Davison et al., 2005). KE of rainfall includes two distinct types, both of which are connected to I_r (Kinnell, 1981). Kinetic energy expenditure (KE_exp; J m⁻²h⁻¹) is the rate at which KE is spent per unit area over a certain period of time, and kinetic energy content (KE_con; J m⁻²mm⁻¹) is defined as the KE per unit area per unit depth. KE_con stands for the mean squared velocity of raindrops arriving to the ground. Linear (Torres et al., 1992) and power-law (Park et al., 1980) relationships (Eqs 2, 3, respectively) are used for connecting KE_exp to I_r; logarithmic (Wischmeier and Smith, 1958) and exponential (Kinnell, 1981) relationships (Eqs 4, 5, respectively) are used for connecting KE_con to I_r.where a, b, c, d, e, f, g, h, and k denote empirical values.

Many scholars have recommended different values in their empirical equations depending on the geographic areas, i.e., the empirical constants vary from one region to another. Differences in measuring methods, the range of I_r, or difficulties in interpretation may all contribute to this discrepancy (van Dijk et al., 2002). In South Korea, numerous studies have been conducted in different locations to obtain the KE–I_r equations, such as in Daejeon (Lim et al., 2015), Seoul (Lee, 2020), Ansung (Kim et al., 2010), and Daegwanryung (Lee and Won, 2013). However, each study produced different results. The precision of KE–I_r connection varies spatially (Fornis et al., 2005); thus, our primary aim is to establish the most appropriate KE–I_r connection in Sangju City (South Korea) and possibly in other parts of the Sangju region with similar climatic conditions. Specifically, we derived two different expressions of the KE–I_r, i.e., KE expenditure (KE_exp; J m⁻²h⁻¹) and KE content (KE_con; J m⁻²mm⁻¹), based on a dataset of raindrop sizes and terminal velocities complied using a laser optical disdrometer (constructed inside the Kyungpook National University) between June 2020 and December 2021. Our findings should facilitate the development of more suitable physics-based soil erosion models.

The remaining sections of the article are structured as follows. Section 2 presents the features of the study site and measuring equipment. Section 3 presents the methodology. Section 4 provides a comprehensive analysis and discussion of the findings. Finally, the results are summarized in Section 5.

2 Area description and measurement instruments

Rainfall intensity and raindrop size distribution observations were conducted at Kyungpook National University in Sangju City, South Korea. Sangju is located in the North Gyeongsang Province, central South Korea (Figure 1) and has an inland climate. In August, the average air temperature reaches 26°C (79°F), while in January, it drops to 3°C (27°F). There is a significant difference in air temperature between the north and south. This area receives an average of 1,050 mm of rainfall annually. Figure 2 depicts the 20-year average monthly rainfall depth (mm) throughout the 2002–2021 period at Sangju City.

FIGURE 1

FIGURE 2

The OTT Parsivel² disdrometer is an optical sensor that generates a 30-mm wide, 180-mm long, and 1-mm high laser beam, and the principle of the measurements is as follows. 1) The maximum voltage is produced at the receiver if no raindrop intersects the laser beam. 2) Raindrops block a section of the laser beam corresponding to their diameter as they pass across the beam; the corresponding lower output voltage dictates the particle size. 3) Particle speed is determined by the duration of the signal. When a raindrop enters the light strip, a signal is generated; it terminates when the raindrop fully exits the light strip.

The disdrometer classifies particles into relevant classes after obtaining the volume equivalent diameter (D) and the particle speed (V). It has measurement ranges of 0.2–8.0 mm for liquid precipitation particles and 0.2–25 mm for solid precipitation particles. Precipitation particles may travel at speeds ranging from 0.2 to 20.0 m/s and this categorization has a smaller scale for tiny, sluggish particles than it does for big, fast particles. Observed particles in a two-dimensional field are divided into 32 D-classes, with ten classes in the 0.00–1.25 mm range, five in the 1.25–2.25 mm range, five in the 2.5–5.0 mm range, five in the 10.0–20.0 mm range, and two in the 20.0–25.0 mm range. Similarly, particle velocity is classified into 32 V-classes: 0.0–1.0 m/s, 1.0–2.0 m/s, 2.0–4.0 m/s, 4.0–8.0 m/s, 8.0–16.0 m/s, and 16.0–20.0 m/s.

3 Methodology

The steps below outlines how we investigated the relationship between KE and I_r from a dataset measured by the OTT Parsivel² disdrometer. Figure 3 is a schematic that depicts our approach.

FIGURE 3

Firstly, the OTT Parsivel² (PARticle SIze and VELocity) disdrometer was erected on the roof of a building inside the Kyungpook National University an elevation of 80 m above sea level. Rainfall properties, such as rainfall depth and intensity, as well as raindrop size and velocity were measured using a disdrometer at a 10-s interval starting in June 2020. This devide was then connected to a laptop to automatically save the measured data.

Secondly, individual rainfall events were classified according to the criteria followed by (Fornis et al., 2005; Petan et al., 2010; Lim et al., 2015): 1) Two distinct events should be separated by at least 6 h without rainfall; 2) total rainfall accumulated should amount more than 3 mm, and 3) rainfall event duration and average intensity should be longer than 30 min and higher than 0.1 mm/h, respectively. Table 1 presents detailed information on the selected rainfall events. After processing, the data were thoroughly reviewed for outliers in the rainfall intensity distribution. According to Lim et al. (2015), raindrops interacting with the safety covers of the disdrometer and merging with other drops may generate interference in the lazer zone, resulting in thicker drops. It is possible for two raindrops to travel through the sensor simultaneously during storm events, resulting in an overestimated, abnormally high intensity (Petan et al., 2010). Thus, records with unusually high intensity values were considered outliners and were discarded.

Thirdly, we established a relationship between I_r and KE using 37 rainfall events, which consisted of 151,852 KE–I_r points. The required empirical parameters (Eqs 2–5) were calculated using the least square method in MATLAB to reduce the estimation standard error. To handle the challenges in nonlinear regression, we utilized the Levenberg-Marquardt technique, which is a form of iteration process that terminates iterative computations when the decreasing residual sum of squares is less than the stated convergence threshold (Lim et al., 2015). 

The most essential parameter for soil erosion, (mm/h), was determined using the following equation:where denotes the laser beam area (i.e., 0.0054 m²), denotes the time interval (i.e., 10 s), and denotes the number of raindrops corresponding to diameter .

Equations 7, 8 constitute two rainfall erosivity indices that were employed in this investigation, including time-related kinetic energy (KE_exp; J m⁻²h⁻¹) and volume-related kinetic energy (KE_con; J m⁻²mm⁻¹), respectively.where denotes the terminal velocity of the raindrops (m/s), corresponding to diameter .

Finally, several statistical metrics [namely, Root Mean Square Error (RMSE), Mean Absolute Error (MAE), and the coefficient of determination (R²)] were used for evaluating the observed and estimated KE values, which were predicted by KE–I_r relationships. The specifications of the assessment criteria are presented in Table 2. The validity of the empirical models was assessed visually using goodness-of-fit plots.

4 Results and discussion

4.1 KE– relationships: Formation

4.1.1 Kinetic energy expenditure (KE_exp; J m⁻²h⁻¹)

The power-law (Eq. 9), polynomial (Eq. 10), and linear (Eq. 11) equations provide the best fits for the KE_exp–I_r connection. The scatter plot in Figure 4 depicts the correlations between KE_exp and I_r, and Table 3 summarizes the statistical analysis results.

FIGURE 4

Among the three equations, the power-law equation yields the highest R² (i.e., 0.945) and lowest RMSE and MAE (i.e., 13.52 J m⁻²h⁻¹ and 3.90 J m⁻²h⁻¹, respectively). These three equations provide a statistically equivalent estimation of KE_exp with an I_r less than 30 mm/h; however, the disparities become more pronounced with increasing rainfall intensity. The linear equation tends to underestimate the KE_exp at I_r levels higher than 30 mm/h. The two remaining equations provide more accurate estimates at higher intensities, with the regression curve nearly crossing the points in the central. The polynomial equation better predicts KE_exp, whereas the power-law equation tends to underestimate KE_exp at I_r values higher than 90 mm/h. We note that when I_r is zero, the resulting KE_exp given by the polynomial and linear equations is negative, which is implausible. At this stage, the power-law equation provides a realistic output.

4.1.2 Kinetic energy content (KE_con, J m⁻²h⁻¹)

The fitted power-law (Eq. 12), logarithmic (Eq. 13), and exponential (Eq. 14) equations are as follows:

Figure 5 shows a scatter plot of the measured KE_con data with the fitted models. We note that KE_con exhibits an extreme variation, contrary to KE_exp. The equations initially exhibit firm slopes at low intensity (i.e., I_r < 5 mm/h). The exponential equation tends to remain steady at I_r values greater than 26.5 mm/h, whereas other equations follow upward trends as I_r increases. The observed KE_con values increase with increasing rainfall intensity in low rainfall intensity zones, but they start to stabilize at rainfall intensities exceeding 30 mm/h.

FIGURE 5

According to the statistical results in Table 4, the exponential equation outperforms others by yielding the highest R² (i.e., 0.62) and lowest RMSE and MAE (i.e., 3.08 and 2.38 J m⁻² mm⁻¹, respectively). At low rainfall intensities (i.e., I_r < 5 mm/h), the regression line for the logarithmic equation increases rapidly with increasing I_r. In comparison to the others, the power-law equation underestimates KE_con at high rainfall intensity. The KE_con estimates using the power-law and logarithmic equations suggest that KE_con has no upper limit, although various studies have suggested otherwise (Kinnell, 1981; Rosewell, 1986; Brown and Foster, 1987). The exponential equation achieves a maximum KE_con value and then remains stable regardless of the rainfall intensity; because KE_con has a maximum value, defined as the parameter g in Eq. 5, the exponential equation is more suitable for forecasting KE_con values than the others. This result is consistent with the findings of (Kinnell, 1981).

4.2 KE_con–I_r relationship: Revision

The KE_con–I_r data points are highly dispersed in the low range in the scatter plot, and this similar phenomena have also been documented by previous research (Fornis et al., 2005; Petan et al., 2010; Sanchez-Moreno et al., 2012; Lim et al., 2015). Consequently, a small number of points with high I_r has a minor impact on the fitted equations, resulting in significant uncertainty and incorrect forecasts (Salles et al., 2002). Thus, this section concentrates on the development of KE_con–I_r equations based on magnitudes of the mean rainfall intensity of each event. Figure 6 shows information on the three rainfall event groups.

FIGURE 6

Figure 7 shows the correlations between KE_con and I_r corresponding to the three rainfall event groups. Table 5 contains the fitted equations for each rainfall event group, and Table 6 summarizes the statistical findings compared to observational data. We note that the KE_con–I_r points in group 1 exhibit minor dispersion compared to the others, resulting in identical trends of the three equations; however, the R² values in group 1 (i.e., 0.59–0.64) are smaller than those in group 2 (i.e., 0.6–0.65) and group 3 (i.e., 0.63–0.72). The exponential equations outperformed the others in every rainfall event group, while yielding the highest R² and lowest MAE and RMSE.

FIGURE 7

Figure 8 shows the uncertainties of KE_con estimations from the equations derived from the total 37 rainfall events and grouped rainfall events. We note that the equations obtained from the total 37 rainfall events tended to overestimate KE_con, particularly in rainfall events with low intensity (i.e., group 1).

FIGURE 8

4.3 KE– relationships: A comment

Both KE_exp and KE_con are valid representations of the KE of rainfall and can be correlated to I_r. It is possible to discover an empirical connection using KE_con versus I_r, although this does not conform to stringent requirements. In terms of statistics, linking KE_con to I_r leads to erroneous findings.

According to Kenney (1982), this is a frequent problem in spurious self-correlation. The statistical relationship between KE_con and I_r is identical to the relationship between KE_exp/I_r and I_r, because KE_con is the quotient function of KE_exp divided by I_r over a given period, as demonstrated in Eq. 8. This procedure intentionally alters the correlation coefficient between the two variables. As a result, spurious self-correlation coefficients tend to be substantially higher than the actual correlation coefficients between variables. When ratios such as KE_exp/I_r were displayed against I_r, they exhibit spurious self-correlation. The correlation coefficient was lowered once the initial variables KE_exp and I_r were highly connected, and their coefficients of variation are identical (Lim et al., 2015). This explained why the KE_exp–I_r relationship provided a better representation of KE of rainfall than the KE_con–I_r relationship; this was also proven by the fact that the former had a higher R² than the latter.

Rainfall characteristics, such as raindrop size distribution, which may impact the raindrop terminal velocity, may influence the results (Fornis et al., 2005). Figure 5 illustrates that high KE_con values occurred at a low rainfall intensity period, whereas lower KE_con values occurred at a higher rainfall intensity zone. This implies that large, quick raindrops form at a low rainfall intensity period, whereas smaller, slower droplets arrive at a higher rainfall intensity zone. According to Blanchard (1953), the onset of a rainfall event signaled by the arrival of a few huge and fast raindrops and followed by a sequence of smaller ones is known as the “sorting phenomenon,” leading to significant KE even when the rainfall intensity is modest. Moreover, the effects of wind and turbulence were neglected in this research; wind currents may impact the velocity of raindrops due to contact of air masses with the mountainous geography of South Korea, leading to high KE_con values for low intensity.

In this study, the maximum of KE_con value given by Eq. 14 is equal to 26.5 J m⁻² mm⁻¹. This value is substantially lower than those reported in other countries, such as in Spain [i.e., 38.4 J m⁻² mm⁻¹, (Cerro et al., 1998)], Portugal [i.e., 35.9 J m⁻² mm⁻¹, (Coutinho and Tomás, 1995)], and Hong Kong [i.e., 36.8 J m⁻² mm⁻¹, (Jayawardena and Rezaur, 2000)] but is relatively close to that in Korean sites, such as in Daejeon [i.e., 25.75 J m⁻² mm⁻¹, (Lim et al., 2015)] and Daegwanryung [i.e., 30.03 J m⁻² mm⁻¹; (Lee and Won, 2013)]. As Rosewell (1986) noted, the maximum of KE_con may vary between sites because of alterations in storm energy. Additionally, Angulo-Martínez et al. (2016) investigated these KE_con-max variations and linked them to topographical, meteorological, and experimental characteristics. The evaporation of tiny droplets that fall at great distances from the cloud may modify the raindrop size distribution and liquid water content, according to Blanchard (1953); this modification may have influenced the amount of energy released during rainfall events. The stochasticity of rainfall controls the KE variability can be seen in Figures 4, 5, 7, since a single value of I_r can generate various KE outputs. This could also witness in recent studies (Fornis et al., 2005; Petan et al., 2010; Sanchez-Moreno et al., 2012; Lim et al., 2015).

5 Conclusion

Soil erosion models that employ the KE of rainfall as an erosivity parameter are the most common; however, direct KE measurements are uncommon. Empirical connections between I_r and KE are an alternative.

Further research should collect data at multiple geographical and temporal scales to develop more precise equations for calculating raindrop-induced soil erosion.

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