Evaluation of Annual Rainfall Erosivity Index Based on Daily,Monthly,and Annual Precipitation Data of Rainfall Station Network in Southern Taiwan

Abstract

The erosivity factor in the universal soil loss equation (USLE) provides an effective means of evaluating the erosivity power of rainfall. The present study proposes three regression models for estimating the erosivity factor based on daily, monthly, and annual precipitation data of rainfall station network, respectively. The validity of the proposed models is investigated using a dataset consisting of 16,560 storm events monitored by 55 rainfall stations in southern Taiwan. The results show that, for 49 of the 55 stations, a strong positive correlation $(r^{2} > 0.5)$ exists between the annual rainfall amount and the annual rainfall erosivity factor. In other words, the estimation model based on the annual precipitation data provides a reliable means of predicting the long-term annual rainfall erosivity in southern Taiwan. Furthermore, the root mean square error (RMSE) and mean absolute percentage error (MAPE) analysis results show that the estimation models based on annual and monthly precipitation data have a more accurate prediction performance than that based on daily precipitation data.

1. Introduction

Water erosion is one of the most important worldwide environmental concerns, particularly in tropical and subtropical regions of the world such as Taiwan. One of the most important active agents of soil erosion is rain due to its potential for producing soil disaggregation and subsequent removal. The effects of raindrop impact and surface runoff on soil erosion are generally estimated using the universal soil loss equation (USLE) [1]; namely,

\begin{matrix} A = R K L S C P, \end{matrix}

(1)

where A is the rate of soil loss (ton ha⁻¹ yr⁻¹), R is the annual rainfall erosivity factor (MJ mm ha⁻¹ h⁻¹ yr⁻¹), K is the soil erodibility factor (t ha yr ha⁻¹ MJ⁻¹ mm⁻¹), L is the slope length factor, S is the slope steepness factor, C is the cover and management factor, and P is the supporting practices factor [2, 3]. Amongst these factors, the erosivity factor (R) is recognized as one of the most effective measures for describing the rainfall erosivity power on a regional scale [4, 5]. In both the original USLE model [1] and the revised-USLE (RUSLE) model [6], R is calculated as the product of the storm rainfall energy (E) and the maximum 30-min rainfall intensity (

I_{30}

). However, Wischmeier and Smith [7] also defined R as the average of the annual summations of the

E I_{30}

values for all storm events yielding more than 12.7 mm of rainfall.

The rainfall erosivity index, R, describes the erosive impact of rainfall and runoff on both the detachment and the entrainment of soil and is given as [1]

\begin{matrix} R_{j} = E_{j} \times I_{j 30} = \sum_{i = 1}^{T_{i}} (e_{i} P_{j i}) \times I_{j 30}, \end{matrix}

(2)

where

E_{j}

is the kinetic energy (MJ/mm),

I_{j 30}

is the maximum 30-min rainfall intensity (mm/h),

e_{i}

is the unitary kinetic energy (MJ/mm·ha),

P_{j i}

is the rainfall amount (mm), and

T_{i}

is the total rainfall duration. Note that the subscripts i and j denote the number of rainfall data instances and the number of rainfall events, respectively. Summing the rainfall erosivity index of all the rainfall events over one year, the annual rainfall erosivity index can be obtained as

\begin{matrix} R_{y} = \sum_{j = 1}^{Y} R_{j}, \end{matrix}

(3)

where Y is the number of rainfall events in the year. In addition, the unitary kinetic energy

e_{i}

is deduced from the relationship between the raindrop diameter and the rainfall intensity as follows [8]:

\begin{matrix} E_{i} = \{\begin{cases} 0.119 + 0.0873 l o g I_{i} & f o r I_{i} < 76 m m / h r, \\ 0.283 & f o r I_{i} \geq 76 m m / h r . \end{cases} \end{matrix}

(4)

The rainfall erosivity index, R, has been widely tested and applied in many countries and regions around the world whose rainfall intensity is characterized mainly as moderate to high [9–15]. In computing the rainfall erosivity factor, the maximum 30-min rainfall intensities for the storm and heavy storm events are generally computed on the basis of hyetograph data or high-resolution rainfall data (pluviograph data). Generally speaking, pluviograph data for at least 20 yrs are required to compute the rainfall erosivity for a given study area using the (R) USLE formulation [2]. However, such large volumes of data are not available for all regions of the world. Furthermore, even if sufficient pluviograph data are available, computing the rainfall erosivity is a complicated and tedious task. To overcome this problem, various simplified models have been proposed for estimating the rainfall erosivity factor using more readily available precipitation data.

Among such models, those based on annual precipitation data are particularly common since annual rainfall data are available in most regions of the world and tend to be fairly reliable. Furthermore, various studies have shown that a good correlation exists between the annual rainfall erosivity and the annual precipitation amount at many locations around the world [16, 17]. Accordingly, annual precipitation data have been used to obtain simple estimates of the rainfall erosivity in many countries [2, 11, 18–34].

Several researchers have used both annual precipitation data and maximum daily and hourly precipitation data to estimate the rainfall erosivity factor in the Mediterranean region [35, 36]. However, the models used in these studies estimate the mean annual rainfall erosivity over several yrs rather than the rainfall erosivity in a particular yr. Many regression models based on variations in the observed rainfall erosivity or seasonal erosivity have been proposed for predicting the daily rainfall erosivity [15, 33, 37–44] or monthly rainfall erosivity [45, 46]. It has been shown that the use of daily or monthly rainfall records provides a better understanding of the rainfall erosivity of individual storms than annual precipitation data [34]. In constructing daily or monthly prediction models, it is necessary to compute the rainfall erosivity on a daily or monthly basis, respectively. However, calculating the daily and monthly rainfall erosivity is more challenging than computing that for a particular storm. For example, if it rains from May 31 to June 1, the observed rainfall erosivity for this storm has just one value. However, the corresponding data should be divided into two different values (i.e., daily or monthly segments) when constructing daily or monthly models. Thus, the annual sum of the reclassified rainfall erosivity is different from the observed value due to the use of different boundary conditions. Moreover, the daily or monthly rainfall parameters used in daily and monthly models, respectively, provide an inadequate description of the kinetic energy and rainfall intensity terms in the rainfall erosivity index [33, 47].

Although annual regression models are a gross oversimplification of the observed variation in the rainfall erosivity and their estimated values are rough [33, 48], they nevertheless represent a viable alternative to detailed quantitative assessments in providing a long-term assessment of the annual mean rainfall erosivity using the USLE formulation [49]. Thus, as discussed above, numerous researchers have proposed methods for estimating the rainfall erosivity based on annual precipitation data and/or other rainfall parameters. However, such models require careful optimization and calibration for each specific location and include site-specific coefficients. The proposed study is to find out the suitable models among daily, monthly, and annual precipitation data.

The present study proposes three regression models for estimating the rainfall erosivity and finding out the suitable models in southern Taiwan based on daily, monthly, and annual precipitation data of rainfall station network, respectively, even without 30-min rainfall data. The detailed goals of this study can be summarized as follows: (a) to construct new models for the large-scale estimation of the erosivity factor in southern Taiwan and (b) to analyze the spatial distribution of the daily, monthly, and annual rainfall erosivity in southern Taiwan.

2. Materials and Methods

2.1. Study Area

This study considered the regions of Kaohsiung City and Pingtung County in southern Taiwan. The two regions cover areas of 2961 km² and 2784 km², respectively, and contain a total of 55 rainfall stations (see Figure 1). Both regions commonly experience extreme rainfall events during the summer months. For example, in August 2009, Typhoon Morakot resulted in catastrophic damage that left 665 people dead, 34 others missing, and roughly US$ 4.4 billion in damages.

Figure 1

Geographic locations of 55 rainfall stations in southern Taiwan.

2.2. Rainfall Data

Table 1 summarizes the basic geographic and rainfall data of the 28 rainfall stations in Kaohsiung City and 27 rainfall stations in Pingtung County over the 10 yr period extending from 2002 to 2011. Traditionally, the high-resolution rainfall data recorded by each station in Table 1 are used to calculate the rainfall erosivity factor in accordance with (2)–(4) [7]. In the present study, the reliability of these data was evaluated using the 10-min rainfall data obtained for the corresponding period from the Central Weather Bureau (CWB) of Taiwan. In the present study, 16,560 storm events were selected from the 550 observed annual rainfall datasets presented in Table 1 (i.e., 55 stations × 10 yrs). The corresponding daily, monthly, and annual rainfall data were then used for analysis purposes and for constructing the estimation models.

Table 1

Geographic and rainfall data (2002~2011) for 55 rainfall stations in southern Taiwan.

Number	Rainfall station	Latitude	Longitude	Reference period (yr)	Elevation (m)	Storm events	Annual rainfall (mm)
1	ZuoYing	120°17′N	22°40′E	10	13	230	1602
2	FongSen	120°23′N	22°32′E	10	61	241	1616
3	SaYe	120°16′N	22°50′E	10	35	366	1736
4	GangShan	120°17′N	22°45′E	10	31	259	1617
5	GuTingKeng	120°24′N	22°53′E	10	87	259	1421
6	MuJha	120°27′N	22°58′E	10	94	224	2154
7	CiShan	120°29′N	22°52′E	10	63	421	1996
8	FongSyong	120°21′N	22°45′E	10	55	290	1772
9	Jiashian	120°35′N	23°04′E	10	60	232	2650
10	SiBu	120°26′N	22°43′E	10	30	255	1903
11	FongShan	120°21′N	22°38′E	10	27	377	1787
12	DaLiao	120°25′N	22°36′E	10	24	302	1723
13	YueMei	120°32′N	22°58′E	10	112	212	2271
14	MeiNong	120°31′N	22°53′E	10	46	307	2227
15	JiDong	120°33′N	22°50′E	10	95	314	2257
16	JhuZihJiao	120°20′N	22°48′E	10	51	310	1799
17	JianShan	120°22′N	22°48′E	10	270	313	1823
18	SinFa	120°39′N	23°03′E	10	470	256	3032
19	DaJin	120°38′N	22°53′E	10	190	427	2710
20	YuYouShan	120°42′N	23°00′E	10	1637	381	4070
21	GaoJhong	120°43′N	23°08′E	10	760	241	2785
22	FuSing	120°48′N	23°13′E	10	700	380	2377
23	SiaoGuanShan	120°48′N	23°09′E	10	1781	355	2995
24	SiNan	120°48′N	23°04′E	10	1792	274	3750
25	MeiShan	120°49′N	23°16′E	10	860	319	2589
26	NanTienChih	120°54′N	23°16′E	10	2700	291	3661
27	PaiYun	120°57′N	23°27′E	10	3340	238	2642
28	NanSi	120°53′N	23°26′E	10	1949	438	2672
29	ALi	120°44′N	22°44′E	10	1040	263	2733
30	MaJia	120°41′N	22°40′E	10	740	248	3491
31	LiGang	120°29′N	22°47′E	10	42	310	2016
32	PingTung	120°30′N	22°39′E	10	25	292	2124
33	SinWei	120°32′N	22°45′E	10	56	352	2222
34	LinLuo	120°33′N	22°39′E	10	54	234	2230
35	NaJhou	120°30′N	22°29′E	10	20	306	1597
36	ChaoJhou	120°32′N	22°32′E	10	12	354	1848
37	FangLiao	120°35′N	22°21′E	10	69	305	1376
38	MaoBiTou	120°44′N	21°55′E	10	49	342	1419
39	JyuCheng	120°44′N	22°04′E	10	54	320	1610
40	LaiYi	120°37′N	22°31′E	10	74	363	2448
41	ChiShan	120°36′N	22°35′E	10	48	284	2630
42	SanDiMan	120°38′N	22°42′E	10	59	245	2575
43	LongCyuan	120°36′N	22°40′E	10	61	333	2438
44	LiLi	120°37′N	22°25′E	10	91	228	1944
45	ChunMi	120°37′N	22°22′E	10	86	387	1677
46	FangShan	120°39′N	22°14′E	10	36	197	1627
47	FongGang	120°41′N	22°11′E	10	63	307	1534
48	ShangDeWun	120°42′N	22°45′E	10	820	395	1700
49	GuSia	120°38′N	22°46′E	10	140	395	2582
50	WeiLiaoShan	120°41′N	22°49′E	10	1018	229	3548
51	SyuHai	120°53′N	22°11′E	10	20	227	2022
52	MouDan	120°50′N	22°11′E	10	285	273	2118
53	MouDanChihShan	120°50′N	22°09′E	10	504	251	2268
54	DanMenShan	120°44′N	22°06′E	10	260	364	1457
55	ShouKa	120°51′N	22°14′E	10	489	244	2184
	Total			550		16560

2.3. Validation of Models

The present study developed three regression models based on the daily, monthly, and annual rainfall data, respectively, for estimating the annual rainfall erosivity factor (R) in southern Taiwan. The estimated values of R were then compared with the observed erosivity factors calculated using (2)–(4) [7]. For each model, the differences between the estimated and observed values at each rainfall station were evaluated in terms of the root mean square error (RMSE) and mean absolute percentage error (MAPE) computed as mentioned by Lee and Heo [17] as follows:

\begin{matrix} R M S E = \sqrt{{(R_{o b e} - R_{e s t})}^{2},} \\ M A P E = |\frac{(R_{o b e} - R_{e s t})}{R_{o b e}}| \times 100 (%), \end{matrix}

(5)

where

R_{o b e}

denotes the observed rainfall erosivity factor and

R_{e s t}

is the estimated rainfall erosivity factor.

In order to develop an accurate model for estimating the rainfall erosivity, it must first be determined whether or not a significant relationship exists between the rainfall parameters and the rainfall erosivity. In identifying appropriate parameters for predicting the annual rainfall erosivity, the present study considered four different rainfall parameters, namely, the event rainfall amount ( $P_{j}$ ), the daily rainfall amount ( $P_{d}$ ), the monthly rainfall amount ( $P_{m}$ ), and the annual rainfall amount ( $P_{y}$ ). The correlation coefficients between these parameters and the rainfall erosivity were calculated for each of the 55 rainfall stations. In addition, the coefficient of variation ( $C V)$ of the observed annual rainfall erosivity and annual rainfall was also computed for each station in accordance with

\begin{matrix} C V = \frac{σ}{u}, \end{matrix}

(6)

where σ is the standard deviation and u is the mean value.

Figure 2 summarizes the methods to develop the regional erosivity models from daily, monthly, and annual precipitation data.

Figure 2

Flowchart of calculation.

3. Results and Discussion

3.1. Relationship between Rainfall Parameters and Rainfall Erosivity

Figures 3(a)~3(d) show the relationships between the event rainfall amount and the event rainfall erosivity, the daily rainfall amount and the daily rainfall erosivity, the monthly rainfall amount and the monthly rainfall erosivity, and the annual rainfall amount and the annual rainfall erosivity, respectively. In general, the results show that the rainfall erosivity varies from one geographic location to another, even under the same annual rainfall conditions. Figure 3(a) is one scatter plot of rainfall ( $P_{j}$ ) and rainfall erosivity ( $R_{j}$ ) that shows a significant nonlinear relationship ( $R_{j} = 0.73 P_{j}^{1.54}, r^{2} = 0.80$ ) between the event rainfall amount ( $P_{j}$ ) and the event rainfall erosivity ( $R_{j}$ ). Similarly, Figure 3(b) shows a significant relationship ( $R_{d} = 0.50 P_{d}^{1.66}, r^{2} = 0.82$ ) between the daily rainfall amount ( $P_{d}$ ) and the daily rainfall erosivity ( $R_{d}$ ). Figures 3(c) and 3(d) show that the monthly rainfall amount ( $P_{m}$ ) and monthly rainfall erosivity ( $R_{m}$ ) and the annual rainfall amount ( $P_{y}$ ) and the annual rainfall erosivity ( $R_{y}$ ) are also related; that is, $R_{m} = 0.60 P_{m}^{1.49}$ , $r^{2} = 0.91$ , and $R_{y} = 2.74 P_{y}^{1.20}$ , $r^{2} = 0.73$ , respectively. In other words, irrespective of the time interval considered, a relationship exists between the rainfall amount and the rainfall erosivity. Comparing the four intervals, it is seen that the strongest correlation exists between the monthly rainfall amount and the monthly rainfall erosivity.

Figure 3

Scatter plots of (a) rainfall event amount ( $P_{j}$ ) and rainfall event erosivity ( $R_{j}$ ), (b) daily rainfall amount ( $P_{d}$ ) and daily rainfall erosivity ( $R_{d}$ ), (c) monthly rainfall amount ( $P_{m}$ ) and monthly rainfall erosivity ( $R_{m}$ ), and (d) annual rainfall amount ( $P_{y}$ ) and annual rainfall erosivity ( $R_{y}$ ).

Table 2 shows the mean, minimum, maximum, and $C V$ values of the annual rainfall amount and annual rainfall erosivity at each of the 55 rainfall stations over the considered time period (2002~2011). From inspection, the average annual mean rainfall over the 55 stations is equal to 2237 mm. Moreover, the minimum annual rainfall of 491 mm was recorded at the SyuHai station in 2002, while the maximum annual rainfall of 6224 mm was recorded at the YuYouShan station in 2005. The annual mean rainfall erosivity over all 55 rainfall stations is equal to 31118 MJ mm ha⁻¹ h⁻¹ yr⁻¹. In addition, the minimum rainfall erosivity of 2271 MJ mm ha⁻¹ h⁻¹ yr⁻¹ was measured at the DanMenShan station in 2002, while the maximum rainfall erosivity of 142370 MJ mm ha⁻¹ h⁻¹ yr⁻¹ was measured at the YuYouShan station in 2005.

Table 2

Annual rainfall and annual rainfall erosivity data (2002~2011) for 55 rainfall stations in southern Taiwan.

Number	Rainfall station	Annual rainfall (mm)				Annual rainfall erosivity (MJ mm ha⁻¹ h⁻¹ yr⁻¹)				Regression models	$r^{2}$
Number	Rainfall station	Max	Min	Mean	CV	Max	Min	Mean	CV	Regression models	$r^{2}$
1	ZuoYing	2373	700	1602	0.35	33723	7937	22454	0.35	$R_{y}$ = $33.47 P^{0.88}$	0.62
2	FongSen	2253	889	1616	0.31	39689	7512	22072	0.43	$R_{y}$ = $66.47 P^{0.78}$	0.29
3	SaYe	2373	1306	1736	0.23	33723	16132	23135	0.26	$R_{y}$ = $0.90 P^{1.36}$	0.71
4	GangShan	2671	909	1617	0.38	46687	8273	20975	0.61	$R_{y}$ = $2.23 P^{1.25}$	0.74
5	GuTingKeng	2656	537	1421	0.49	38869	5033	17198	0.63	$R_{y}$ = $1.65 P^{1.26}$	0.95
6	MuJha	4461	1412	2154	0.43	59574	16691	23355	0.52	$R_{y}$ = $1.17 P^{1.31}$	0.93
7	CiShan	3057	1091	1996	0.35	40927	13247	24545	0.40	$R_{y}$ = $15.17 P^{0.97}$	0.73
8	FongSyong	2814	881	1772	0.34	41035	10802	25122	0.42	$R_{y}$ = $9.42 P^{1.05}$	0.64
9	Jiashian	4461	1506	2650	0.39	76637	15779	41555	0.53	$R_{y}$ = $3.10 P^{1.20}$	0.75
10	SiBu	2981	982	1903	0.34	49680	10644	27468	0.50	$R_{y}$ = $0.59 P^{1.42}$	0.86
11	FongShan	2672	867	1787	0.33	37336	11659	24753	0.38	$R_{y}$ = $6.92 P^{1.09}$	0.77
12	DaLiao	2454	821	1723	0.31	32686	7103	22962	0.38	$R_{y}$ = $3.68 P^{1.17}$	0.66
13	YueMei	3598	1410	2271	0.28	41677	17889	25374	0.33	$R_{y}$ = $17.90 P^{0.95}$	0.74
14	MeiNong	3399	1083	2227	0.35	59758	16033	31859	0.49	$R_{y}$ = $5.62 P^{1.12}$	0.81
15	JiDong	3221	1171	2257	0.30	47197	18988	32550	0.29	$R_{y}$ = $684.83 P^{0.50}$	0.30
16	JhuZihJiao	2848	876	1799	0.36	49183	7989	24302	0.55	$R_{y}$ = $9.36 P^{1.05}$	0.53
17	JianShan	3063	632	1823	0.39	60047	6044	26262	0.58	$R_{y}$ = $1.27 P^{1.32}$	0.86
18	SinFa	4589	1630	3032	0.34	75976	21836	47935	0.44	$R_{y}$ = $4.47 P^{1.15}$	0.76
19	DaJin	4105	1322	2710	0.32	66543	13987	40181	0.42	$R_{y}$ = $0.95 P^{1.34}$	0.83
20	YuYouShan	6224	1767	4070	0.34	142370	19412	72039	0.50	$R_{y}$ = $0.43 P^{1.44}$	0.88
21	GaoJhong	4565	1188	2785	0.40	69908	12387	40858	0.54	$R_{y}$ = $1.28 P^{1.30}$	0.72
22	FuSing	4231	1077	2377	0.46	81114	7355	33084	0.73	$R_{y}$ = $0.81 P^{1.35}$	0.65
23	SiaoGuanShan	4164	1697	2995	0.26	93739	14726	41586	0.54	$R_{y}$ = $0.16 P^{1.55}$	0.66
24	SiNan	5557	1800	3750	0.32	98956	13550	46899	0.59	$R_{y}$ = $1.50 P^{1.25}$	0.54
25	MeiShan	4270	1122	2589	0.39	69331	10799	33285	0.65	$R_{y}$ = $0.25 P^{1.49}$	0.77
26	NanTienChih	4986	1615	3661	0.33	101544	9538	41011	0.73	$R_{y}$ = $0.02 P^{1.74}$	0.73
27	PaiYun	3844	1294	2642	0.32	33848	6862	18175	0.48	$R_{y}$ = $0.45 P^{1.33}$	0.79
28	NanSi	3572	1230	2672	0.33	48871	6949	24998	0.56	$R_{y}$ = $0.033 P^{1.72}$	0.86
29	ALi	4049	1661	2733	0.34	59940	15606	35257	0.51	$R_{y}$ = $0.35 P^{1.45}$	0.86
30	MaJia	5530	1820	3491	0.35	122022	25807	64926	0.50	$R_{y}$ = $6.25 P^{1.12}$	0.64
31	LiGang	3142	1201	2016	0.32	39945	12491	27605	0.33	$R_{y}$ = $23.31 P^{0.93}$	0.62
32	PingTung	3351	990	2124	0.34	65988	10229	32927	0.50	$R_{y}$ = $1.84 P^{1.27}$	0.76
33	SinWei	3999	1312	2222	0.37	87155	13318	36733	0.60	$R_{y}$ = $2.38 P^{1.24}$	0.64
34	LinLuo	3267	1139	2230	0.30	53448	17799	32012	0.34	$R_{y}$ = $51.92 P^{0.83}$	0.61
35	NaJhou	2340	773	1597	0.33	33690	9998	20852	0.38	$R_{y}$ = $43.40 P^{0.83}$	0.54
36	ChaoJhou	2581	811	1848	0.32	52474	6426	27321	0.49	$R_{y}$ = $0.97 P^{1.35}$	0.66
37	FangLiao	2311	534	1376	0.47	28044	4763	16425	0.50	$R_{y}$ = $8.67 P^{1.03}$	0.81
38	MaoBiTou	2050	998	1419	0.21	99535	10631	22740	1.19	$R_{y}$ = $0.002 P^{2.19}$	0.49
39	JyuCheng	2218	986	1610	0.26	62730	7558	23929	0.70	$R_{y}$ = $0.07 P^{1.71}$	0.64
40	LaiYi	3439	1556	2448	0.24	82126	21753	43166	0.47	$R_{y}$ = $10.17 P^{1.06}$	0.36
41	ChiShan	3714	1347	2630	0.28	67586	15140	41204	0.38	$R_{y}$ = $4.87 P^{1.14}$	0.61
42	SanDiMan	3773	1655	2575	0.26	75709	20727	42442	0.44	$R_{y}$ = $1.27 P^{1.32}$	0.64
43	LongCyuan	3653	1249	2438	0.31	65942	12170	34742	0.44	$R_{y}$ = $2.27 P^{1.23}$	0.67
44	LiLi	2864	1137	1944	0.31	39750	12995	26451	0.38	$R_{y}$ = $17.46 P^{0.96}$	0.56
45	ChunMi	2615	847	1677	0.33	32659	7211	19299	0.45	$R_{y}$ = $1.82 P^{1.24}$	0.75
46	FangShan	2281	1074	1627	0.27	66431	8886	20485	0.81	$R_{y}$ = $0.17 P^{1.56}$	0.58
47	FongGang	1972	849	1534	0.27	37847	5489	17706	0.49	$R_{y}$ = $0.77 P^{1.36}$	0.68
48	ShangDeWun	2671	909	1700	0.33	42345	11248	54945	0.16	$R_{y}$ = $17.34 P^{0.99}$	0.59
49	GuSia	3495	1598	2582	0.27	48113	16361	35280	0.32	$R_{y}$ = $2.78 P^{1.20}$	0.73
50	WeiLiaoShan	5666	1377	3548	0.40	121787	14042	61679	0.52	$R_{y}$ = $2.66 P^{1.23}$	0.87
51	SyuHai	2939	491	2022	0.37	53235	4231	28585	0.49	$R_{y}$ = $2.14 P^{1.21}$	0.88
52	MouDan	2661	1396	2118	0.20	50943	13706	22589	0.50	$R_{y}$ = $0.50 P^{1.41}$	0.56
53	MouDanChihShan	3377	1577	2268	0.26	53046	13279	24154	0.52	$R_{y}$ = $0.99 P^{1.30}$	0.50
54	DanMenShan	2129	558	1457	0.41	30074	2271	16433	0.49	$R_{y}$ = $10.52 P$	0.48
55	ShouKa	2606	1735	2184	0.16	28325	10158	24854	0.20	$R_{y}$ = $1.24 P^{1.25}$	0.52

An inspection of Table 2 shows that the correlation coefficients ( $r^{2}$ ) between the mean annual rainfall and the rainfall erosivity range from 0.29 to 0.95. Moreover, 49 of the 55 stations have a correlation coefficient ( $r^{2}$ ) greater than 0.5, which are satisfied by a significance test (two-tailed test) with a 99% confidence level (P value < 0.01). The $C V$ values of the annual rainfall range from 0.16 to 0.49, while those of the annual rainfall erosivity range from 0.16 to 1.19. Of all the stations, the GuTingKeng station has the highest $C V$ (0.49) for the annual rainfall, while the MaoBiTou station has the highest $C V$ (1.19) for the annual rainfall erosivity.

GIS (Geographic Information System) was used to interpolate and plot the spatial variability of the annual rainfall erosivity factor ( $R_{y}$ ) over the study area using the Kriging interpolation method [16]. Figures 4(a) and 4(b) show the results obtained for the annual rainfall and annual erosivity, respectively. An inspection of Figure 4(a) shows that the mean annual total rainfall ranges from 1376 to 4070 mm yr⁻¹. Based on the regression relationship for the annual rainfall ( $R_{y} = 2.74 P_{y}^{1.20}$ ), the rainfall gradient values range from 14785 to 72039 MJ mm ha⁻¹h⁻¹yr⁻¹. Moreover, the Kriging interpolation results show that the annual rainfall erosivity has a west-east gradient with values ranging from 15000 to 70000 MJ mm ha⁻¹ h⁻¹ yr⁻¹. It is seen that the spatial distributions of the annual rainfall and annual rainfall erosivity, respectively, are similar. Different interpolation methods may result in different spatial distributions of the rainfall erosivity. However, Angulo-Martínez and Beguería [33] found that all common interpolation methods are capable of capturing the regional distribution of the R factor given the use of a spatially dense rainfall database with a high temporal resolution.

Figure 4

(a) Annual precipitation and (b) annual rainfall erosivity maps in southern Taiwan. Note that isohyet and isoerodent intervals are 500 mm and 10000 MJ mm ha⁻¹ h⁻¹ yr⁻¹, respectively.

The rainfall erosivity map presented in Figure 4(b) is of great relevance for soil erosion evaluation and control. It has implications not only for agriculture but also for many activities related to land use planning. Furthermore, it can be used as a guide for soil conservation practices and landscape modeling since the R factor is usually an important part of erosion models such as the USLE [16].

The higher erosivity observed in the tropic region is caused by the high amount of precipitation, intensity, and kinetic energy of rain. The main generating mechanism of rainfall is convection effect in most tropical regions. As a result, the regions receive more rain with higher intensities than the temperate regions, dominated by midlatitude cyclones [41].

The regression models to estimate rainfall erosivity for specific locations are unable to accurately predict actual rainfall erosivity for other locations due to site-specific conditions. Therefore, simplified methods based on annual precipitation for estimating rainfall erosivity should be used with caution according to location or time period. Their results deserve careful attention as applying simplified methods to estimating annual rainfall erosivity.

3.2. Applicability of Three Regression Models

The applicability of the daily, monthly, and annual regression models developed in the previous subsection (i.e., $R_{d} = 0.50 P_{d}^{1.66}$ , $R_{m} = 0.60 P_{m}^{1.49}$ , and $R_{y} = 2.74 P_{y}^{1.20}$ , resp.) was evaluated by comparing the results obtained from each model for the rainfall erosivity factor with the observed rainfall erosivity factor computed using the method presented by Wischmeier and Smith [7]. Table 3 presents the RMSE and MAPE analysis results for the estimated and observed rainfall erosivity values. It is seen that when estimating the $R_{y}$ factor using (3) based on the daily rainfall data, the MAPE ranges from 2% to 49% ( $C V = 0.50$ ) and the RMSE varies from 1031 to 16350 MJ mm ha⁻¹ h⁻¹ yr⁻¹ ( $C V = 0.55$ ). Similarly, when estimating the $R_{y}$ factor using the monthly rainfall data, the MAPE ranges from 1% to 37% ( $C V = 0.87$ ), while the RMSE varies from 190 to 17345 MJ mm ha⁻¹ h⁻¹ yr⁻¹ ( $C V = 0.96$ ). Finally, when estimating the $R_{y}$ factor based on the annual rainfall data, the MAPE ranges from 2% to 30% ( $C V = 0.59$ ) and the RMSE varies from 454 to 16030 MJ mm ha⁻¹h⁻¹yr⁻¹ ( $C V = 0.85$ ). Overall, it can be seen that the estimation errors of the three models range from 11% to 49%. For the daily model, the error rate exceeds 10% at 11 of the 55 stations. However, for the monthly and annual regression models, the error rate is less than 10% for 34 and 24 of the 55 stations, respectively.

Table 3

Comparison of estimated rainfall erosivity factor $R_{y}$ and observed rainfall erosivity factor $R_{y}$ (unit: MJ mm ha⁻¹ h⁻¹ yr⁻¹).

Rainfall station	$R_{y}$	${R_{y}}^{*}$	${R_{y}}^{* *}$	${R_{y}}^{* * *}$	RMSE^*	RMSE^**	RMSE^***	MAPE^*	MAPE^**	MAPE^***
ZuoYing	22454	26889	18927	19208	4435	3527	3246	20	16	14
FongSen	22072	25814	16576	19409	3742	5496	2663	17	25	12
SaYe	23135	27360	20807	19680	4225	2328	3455	18	10	15
GangShan	20975	19239	18767	19412	1736	2208	1563	8	11	7
GuTingKeng	17198	25627	20154	16625	8429	2956	573	49	17	3
MuJha	23355	32082	22104	25164	8727	1251	1809	37	5	8
CiShan	24545	27016	26060	24999	2471	1515	454	10	6	2
FongSyong	25122	33138	22358	21677	8016	2764	3445	32	11	14
Jiashian	41555	50785	40394	35132	9230	1161	6423	22	3	15
SiBu	27468	37029	25515	23613	9561	1953	3855	35	7	14
FongShan	24753	30880	22707	21887	6127	2046	2866	25	8	12
DaLiao	22962	29446	21045	20955	6484	1917	2007	28	8	9
YueMei	25374	35168	31863	29181	9794	6489	3807	39	26	15
MeiNong	31859	43738	30787	28519	11879	1072	3340	37	3	10
JiDong	32550	38068	29408	28979	5518	3142	3571	17	10	11
JhuZihJiao	24302	21529	23043	22071	2773	1259	2231	11	5	9
JianShan	26262	32897	27935	22429	6635	1673	3833	25	6	15
SinFa	47935	42159	48644	41295	5776	709	6640	12	1	14
DaJin	40181	47608	39683	36092	7427	498	4089	18	1	10
YuYouShan	72039	63584	74076	58778	8455	2037	13261	12	3	18
GaoJhong	40858	42227	43422	37294	1369	2564	3564	3	6	9
FuSing	33084	32053	34319	30836	1031	1235	2248	3	4	7
SiaoGuanShan	41586	37535	47868	40691	4051	6282	895	10	15	2
SiNan	46899	48164	64244	53285	1265	17345	6386	3	37	14
MeiShan	33285	31932	38104	34153	1353	4819	868	4	14	3
NanTienChih	44225	45067	43594	41766	842	631	2459	2	1	6
PaiYun	50065	33522	34348	35002	16543	15717	15063	33	31	30
NanSi	24998	29183	28933	32542	4185	3935	7544	17	16	30
Ali	35257	30350	39240	36458	4907	3983	1201	14	11	3
MaJia	64926	48576	60070	48896	16350	4856	16030	25	7	25
LiGang	27605	23403	27095	25301	4202	510	2304	15	2	8
PingTung	32927	40740	28894	26936	7813	4033	5991	24	12	18
SinWei	36733	41425	31463	28437	4692	5270	8296	13	14	23
LinLuo	32012	37356	30299	28563	5344	1713	3449	17	5	11
NaJhou	20852	25828	19294	19137	4976	1558	1715	24	7	8
ChaoJhou	27321	17221	24589	22796	10100	2732	4525	37	10	17
FangLiao	16425	13834	16895	15745	2591	470	680	16	3	4
MaoBiTou	22740	19711	23591	21800	3029	851	940.1	13	4	4
JyuCheng	23929	16625	16332	19319	7304	7597	4610	31	32	19
LaiYi	43166	33328	35225	31933	9838	7941	11233	23	18	26
ChiShan	41204	50232	42742	34806	9028	1538	6398	22	4	16
SanDiMan	42442	52468	37553	33943	10026	4889	8499	24	12	20
LongCyuan	34742	28349	33969	31908	6393	773	2834	18	2	8
LiLi	26451	20600	25869	24221	5851	582	2230	22	2	8
ChunMi	19299	25072	20443	20280	5773	1144	981	30	6	5
FangShan	20485	22368	18648	19564	1883	1837	921	9	9	4
FongGang	17706	21686	17516	18225	3980	190	519	22	1	3
ShangDeWun	54945	53340	53694	46476	1605	1251	8469	3	2	15
GuSia	35280	26490	35055	34041	8790	225	1239	25	1	4
WeiLiaoShan	61679	64075	61211	49864	2396	468	11815	4	1	19
SyuHai	28585	23858	24509	27186	4727	4076	1399	17	14	5
MouDan	22589	21230	27870	26842	1359	5281	4253	6	23	19
MouDanChihShan	24154	21203	26077	29144	2951	1923	4990	12	8	21
DanMenShan	16433	12021	16171	17130	4412	262	697	27	2	4
ShouKa	24854	31692	28602	28691	6838	3748	3837	28	15	15
Max	72039	64075	74076	58778	16543	17345	16030	49	37	30
Min	16425	12021	16171	15745	842	190	454	2	1	2
Mean	32106	32960	31611	29242	5804	3059	4222	19	10	12
CV	0.39	0.36	0.41	0.34	0.61	1.07	0.87	0.56	0.87	0.59

$^{*}$ Estimated annual mean R factor using daily rainfall model. ( $R_{d} = 0.5 P_{d}^{1.66}$ ).

^**Estimated annual mean R factor using monthly rainfall model. ( $R_{m} = 0.6 P_{m}^{1.49}$ ).

^***Estimated annual mean R factor using annual rainfall model. ( $R_{y} = 2.74 P_{y}^{1.20}$ ).

Figure 5 presents scatter plots showing the relationship between the $R_{y}$ factors estimated at the 55 stations using the three regression models and the observed $R_{y}$ factors computed using the method proposed by Wischmeier and Smith (1978). Note that, in the figures presented on the left, the x-axis represents the observed annual mean rainfall erosivity factor while the y-axis represents the estimated annual mean $R_{y}$ factor. Furthermore, the individual data points indicate the annual average rainfall erosivity at the different rainfall stations. The three figures presented on the right of Figure 5 show the residual distribution as a function of the annual rainfall erosivity for each of the three models. However, that of the daily regression model deviates farther from the normal line in Figure 5(a). Figures 5(b) and 5(c) show that a better agreement exists between the estimated and observed values of the rainfall erosivity when using the monthly and annual regression models, respectively. Overall, the results presented in Figure 5 show that, in terms of the error rate, the three regression models can be ranked as follows: daily > annual > monthly. In other words, the regression models based on annual and monthly rainfall data are more accurate than that based on daily rainfall data.

Figure 5

Validation results for (a) daily, (b) monthly, and (c) annual regression models based on relationship between estimated $R_{y}$ and observed $R_{y}$ .

According to Table 3 and Figure 5, the data of estimated annual mean rainfall erosivity was underestimated by daily rainfall models, respectively. Nevertheless, Liu et al. [50] indicated that much precipitation information could be provided by daily rainfall data rather than monthly and annual ones. Different from the results of [50] in China, the rainfall event in southern Taiwan might be consistent for several days, and an underestimate could be therefore produced as daily rainfall data was used to estimate erosivity.

3.3. Spatial Distribution Comparison of Three Regression Models

Figure 6 presents the annual rainfall erosivity ( $R_{y}$ ) and MAPE maps for each of the three regression models. Note that the observed annual rainfall erosivity map is also presented for comparison purposes. The annual rainfall erosivity ( $R_{y}$ ) map was based on annual average rainfall erosivity recorded at the 55 rainfall stations, and it leads to the comparison between spatial distribution of the annual rainfall amount and the geographic distribution of annual rainfall erosivity. It is seen that the spatial distributions of the estimated and observed values of $R_{y}$ are similar. However, for each regression model, the estimated value of $R_{y}$ is slightly lower than the observed value due to statistical errors. In addition, a comparison of Figures 6(b)~6(d) confirms that the monthly rainfall model yields a better estimation performance than the daily or annual model.

Figure 6

Rainfall erosivity maps and mean absolute percentage error (MAPE) maps for three estimation models: (a) observed, (b) daily, (c) monthly, and (d) annual models.

4. Conclusions

The rainfall erosivity factor (R) is one of the key factors in the USLE model and has gained increasing importance as the environmental effects of climate change have become more severe. This study has proposed three models for estimating the value of R based on daily, monthly, and annual precipitation data of rainfall station network, respectively. The validity of the three models has been evaluated using the rainfall data collected over a period of ten yrs (2002~2011) at 55 rainfall stations in southern Taiwan. The results have shown that, of the three models, the annual and monthly models yield a better agreement with the observed rainfall erosivity factor than the daily model.

Footnotes

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The authors would like to thank the financial support provided by the National Science Council in Taiwan (NSC 102-2625-M-020-002).

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