Variability, Prediction, and Simulation of Rainfall Erosivity Risk in the State of Sinaloa, Northwest Mexico

Abstract

Observed rainfall erosivity risk (ORE) index is defined as the erosivity risk in the event of extreme rainfall events. ORE measures the kinetic energy of raindrops generated during a period of maximum precipitation intensity with the formula $\mathrm{O}\mathrm{R}\mathrm{E}=\left(\mathrm{E}\mathrm{D}·\mathrm{T}\mathrm{E}\mathrm{I}\right)/10$, where ED = erosivity density, TEI = total erosivity index, and ORE is measured in MJ mm ha⁻¹ h⁻¹ yr⁻¹. The goal of this study is to model ORE, estimate its spatiotemporal variability, and predict (PRE) and simulate ORE for the state of Sinaloa (1969–2018). Five indices of rainfall erosivity were calculated: the modified Fournier index, precipitation concentration index, ED, TEI, and rainfall erosivity factor. The nonparametric trend in ORE was calculated. Using multiple nonlinear regressions (MNR), PRE (dependent variable) was calculated as a function of cumulative annual, annual average, seasonal average, and seasonal cumulative rainfall (independent variables). To simulate PRE, cumulative distribution functions, adjusted return periods (ARPs), and the 99th percentile were used. ORE ranged from 51.39 MJ mm ha⁻¹ h⁻¹ yr⁻¹ in 1970 (Culiacán) to 92679.40 MJ mm ha⁻¹ h⁻¹ yr⁻¹ in 1998 (Sta. C. de Alaya). The only year that had very high ORE at all nine stations was 1998. The only significant trend was ORE = 34.64 MJ mm ha⁻¹ h⁻¹ yr⁻¹ (Culiacán). The nine PRE models were significantly predictive (Spearman correlation > 0.280). Guatenipa, Rosario, and Siqueros registered very high PRE, since one to eight extreme erosivity events per century are predicted on average. A new methodology is proposed for calculating ORE and PRE, which can be used to develop alternatives for managing and protecting agricultural land in the state considered “the breadbasket of Mexico”.

1. Introduction

Hydric erosion is an anthropogenic and natural problem that can cause significant environmental, agricultural, and economic damage in any region of the world [1,2,3]. According to [4], damage of this nature affects around 28% of the world’s degraded lands, a percentage that could increase due to intense rainfall erosivity (IRE) [5], which is one of the six factors contributing to the universal soil loss equation (USLE) [6]. IRE is a natural erosive process that depends largely on the intensity and magnitude of rainfall [7] and can affect the displacement and fertility of soil aggregates. This has a harmful impact on approximately 85% of soil cover worldwide [1], undermining the yield of agricultural crops and the food sovereignty of the population [1,8,9]. Some of the most widely used IRE indices globally are the modified Fournier index (MFI) [7,10,11], the precipitation concentration index (PCI) [12,13], the total erosivity index (TEI) [14,15] (expressed as mm), and the rainfall erosivity factor (R-factor) [16,17], which is a multi-annual average index that measures the kinetic energy and intensity of rainfall to describe the effect of rainfall on sheet and rill erosion, expressed as MJ mm ha⁻¹ h⁻¹ yr⁻¹ [18]. Another example is erosivity density (ED) [18,19], which is defined as the erosivity per rainfall unit (mm), and is expressed as MJ ha⁻¹ h⁻¹ [18]. According to [16], choosing the best IRE index for a particular region is a complex issue [20], since the spatiotemporal variability of cumulative annual rainfall (CAR) exhibits different regional behaviors in different parts of the world [21]. Furthermore, refs. [16,22] argue that IRE depends not only on CAR variability but also on rainfall: annual average rainfall (AAR), seasonal average rainfall (SAR), and seasonal cumulative rainfall (SCR). Ref. [23] adds that the intrinsic randomness of extreme rainfall in semi-arid regions makes it essential to study IRE using cumulative distribution functions (CDFs); that is, by integrating the probability density function (PDF) [24]. Comparing the twentieth and twenty-first centuries, it has been observed that extreme rainfall events have recently shown an increase in adjusted return periods (ARPs) [25], providing evidence of a higher observed risk of IRE (ORE) [26,27]. In Mexico, although an index for estimating ORE has not been developed, progress has been made in creating regionalized IRE indices. For example, refs. [28,29] used 14 R-factor (dependent variable) predictive models for all of Mexico, using CAR as the only independent variable. Of the 14 models, 2 apply to the state of Sinaloa.

In this study, daily rainfall (DaR) was measured at nine weather stations in Sinaloa (1969–2018). DaR was obtained from the National Meteorological Service and National Water Commission database [30]. Five IRE indices were calculated: MFI, PCI, TEI, R-factor, and ED. In a novel proposal, the ORE index, defined as $\mathrm{O}\mathrm{R}\mathrm{E}=\left(\mathrm{E}\mathrm{D}·\mathrm{T}\mathrm{E}\mathrm{I}\right)/10$, was calculated. The nonparametric trend of ORE was also calculated. Because multiple linear regression (MLR) often presents various limitations [31], ORE (PRE, predicted rainfall erosivity risk; dependent variable) was predicted using multiple nonlinear regression (MNR) with CAR, AAR, SAR, and SCR as the independent variables. PRE was simulated using ARPs for six CDFs [log-normal, normal, generalized extreme values (GEV), gamma (2), beta4, and logistic] and the 99th percentile (P99).

The goal is to create a novel methodology for calculating the ORE index, estimating the spatiotemporal variability of ORE, and predicting and simulating PRE for the state of Sinaloa in northwestern Mexico.

This innovative methodology can be applied to any region of the world, especially for developing agricultural soil management and protection plans based on the variability of ORE, and to predict and simulate PRE. Due to its greater sensitivity to IRE, the use of the ORE index should be prioritized in cyclonic basins, where the highest percentage of CAR is usually associated with tropical cyclones.

2. Materials and Methods

2.1. Study Area

Sinaloa is in the northwest of Mexico (Figure 1), and although it is mostly semi-arid, it is considered one of the most important agricultural states in the country [32], being called “the breadbasket of Mexico” [33]. Historically, Sinaloa’s high agricultural production is fundamentally due to its fertile soils (gleysol, fluvisol, vertisol, and phaeozem), the latter two types being the most predominant [34].

2.2. Obtaining and Running Quality Control of Daily Rainfall (DaR) Data

Initially, DaR was obtained from 70 weather stations for Sinaloa for the period 1951–2018. The data were obtained from the SMN–CONAGUA climate database [30] at https://smn.conagua.gob.mx/es/climatologia/informacion-climatologica/informacion-estadistica-climatologica (accessed on 12 February 2025).

It was decided that the data should meet three basic conditions: (a) ≥50 years of available data, (b) ≤10% missing data, and (c) stations ~50 km apart. Using these criteria, nine stations with data covering the period 1969–2018 were chosen: Culiacán (0.36% missing data), El Playón (2.87%), Guatenipa (1.29%), Ixpalino (3.52%), Mocorito (9.36%), Sanalona II (1.36%), Rosario (2.94%), Sta. C. de Alaya (4.67%), and Siqueros (5.39%). Using the nearest neighbor method, missing data were estimated and the series were homogenized using the standard normal homogeneity test. These two tests were carried out using the climatol package [35,36].

2.3. Cumulative Annual Rainfall (CAR), Annual Average Rainfall (AAR), Seasonal Average Rainfall (SAR), Seasonal Cumulative Rainfall (SCR), and 99th Percentile (P99)

To include average and seasonal variation in IRE, in this study the following scores were calculated: CAR, AAR, SAR, and SCR. The P99s were also calculated for CAR, AAR, SAR and SCR.

2.4. Temporal Variability of Intense Rainfall Erosivity Indices (IRE)

For the calculation of IRE, when sub-hourly precipitation data are not available [37], DaR must be calculated as a starting point. In this study, to obtain monthly cumulative rainfall (mR), Equation (1) was applied (

Table 1. Equations for calculating intense rainfall erosivity (IRE) indices. Source: authors.

Rainfall Erosivity Index
$\mathrm{m}\mathrm{R}={\displaystyle \sum _{\mathrm{i}=1}^{30}}\mathrm{D}\mathrm{a}\mathrm{R}$	(1)
$\mathrm{M}\mathrm{F}\mathrm{I}={\displaystyle \sum _1^{12}}\frac{{\mathrm{m}\mathrm{R}}^2}{\mathrm{C}\mathrm{A}\mathrm{R}}$	(2)
$\mathrm{PCI}=100·{\displaystyle \sum _1^{12}}\frac{{\mathrm{mR}}^2}{{\mathrm{CAR}}^2}$	(3)
$\mathrm{R}\text{-}\mathrm{factor}=0.001680·{\mathrm{CAR}}^2+6.6847·\mathrm{CAR}$	(4)
$\mathrm{R}\text{-}\mathrm{factor}=0.000442·{\mathrm{CAR}}^2+6.8938·\mathrm{CAR}$	(5)
$\mathrm{T}\mathrm{E}\mathrm{I}=\mathrm{M}\mathrm{F}\mathrm{I}·\mathrm{P}\mathrm{C}\mathrm{I}$	(6)
$\mathrm{E}\mathrm{D}=\frac{\mathrm{R}\text{-}\mathrm{factor}}{\mathrm{CAR}}$	(7)
$\mathrm{O}\mathrm{R}\mathrm{E}=\frac{\mathrm{E}\mathrm{D}·\mathrm{T}\mathrm{E}\mathrm{I}}{10}$	(8)

DaR = daily rain (mm day⁻¹), mR = monthly cumulative rainfall (mm month⁻¹), MFI = modified Fournier index (mm), CAR = cumulative annual rainfall (mm yr⁻¹), PCI = precipitation concentration index (%), R-factor = rainfall erosivity factor (MJ mm ha⁻¹ h⁻¹ yr⁻¹), TEI = total erosivity index (mm), ED = erosivity density (MJ ha⁻¹ h⁻¹), ORE = observed rainfall erosivity risk (MJ mm ha⁻¹ h⁻¹ yr⁻¹).

). It was decided to use the MFI and PCI indices [38,39] (Equations (2) and (3)), due to their extensive use worldwide [3,38]. These indices are employed to evaluate the erosivity of monthly and annual rainfall. The R-factor, defined as the capacity of rainfall to cause soil loss [40], was calculated using Equations (4) and (5) created by [28,41], and which work specifically for the historical variation in CAR of Sinaloa [42]. In particular, Equation (4) was applied to the Culiacán, El Playón, Guatenipa, Mocorito, Sanalona II, and Sta. C. de Alaya stations, and Equation (5) to the Ixpalino, Rosario, and Siqueros stations. Using Equation (6), TEI was calculated [43], cited by [15,44], who evaluated total erosivity through the product of MFI and PCI (Equation (6)). In addition, using Equation (7), ED [45] was estimated, which is defined as the erosivity per unit of precipitation [18,45]. Equation (8) is proposed for the first time, defining ORE, which includes the intrinsic variation in the five previous IRE indices (Equations (2)–(7)). ORE is defined as the erosivity risk when extreme rainfall events occur.

2.5. Spatial Variability—Trend of Observed Rainfall Erosivity Risk (ORE)

Average ORE was obtained by interpolation using the weighted inverse distance method (power = 2), which is one of the most accurate interpolation methods [46,47]. The nonparametric trend in ORE was calculated using the Mann–Kendall test [48] and Sen’s slope test [49]. The trend was calculated using the trend package, version 1.1.6. The trend was considered significant when the value was ≥1.96.

2.6. Predicted Rainfall Erosivity Risk (PRE)

To obtain PRE, ORE was calculated as $\mathrm{O}\mathrm{R}\mathrm{E}=\mathrm{E}\mathrm{D}·\mathrm{T}\mathrm{E}\mathrm{I}$. At each weather station, PRE (dependent variable) was obtained using MNR (second-degree polynomials), with CAR, AAR, SAR, and SCR as the independent variables. The result was then divided by 10.

2.7. Validation of Predicted Rainfall Erosivity Risk (PRE) Models

Normality Analysis of Residuals from Multiple Linear Regression (MLR)

To obtain PRE, MLR was initially applied. Because no weather station recorded normality in the residuals for all four tests (Shapiro–Wilk, Anderson–Darling, Lilliefors, and Jarque–Bera) (p-value < 0.05), it was decided to apply MNR. This methodology was also applied by [50].

2.8. Fitting Multiple Nonlinear Regressions (MNRs) Between Observed and Predicted Rainfall Erosivity Risk (ORE vs. PRE)

The square root was calculated for each coefficient of determination (R²) of each MNR of each dispersion diagram (ORE vs. PRE); that is, the Spearman correlation coefficient (Sr) was calculated. Each Sr was compared with a Spearman critical correlation coefficient (Scr = 0.280; n = 50, period 1969–2018). Sr was considered statistically significant when Sr > Scr. The mean relative error (MRE) and the root mean square log error (RMSLE) were also calculated.

2.9. Simulation of Predicted Rainfall Erosivity Risk (PRE)

2.9.1. Cumulative Distribution Functions (CDFs)

Equations (9)–(14) in

Table 2. Equations used to calculate the cumulative distribution functions (CDFs) and adjusted return periods (ARPs) of cumulative annual rainfall (CAR), annual average rainfall (AAR), seasonal average rainfall (SAR), and seasonal cumulative rainfall (SCR). Source: authors.

Variable Name	Mathematical Equations Used to Calculate Probability Density Functions (PDFs), Cumulative Distribution Functions (CDFs), and Adjusted Return Periods (ARPs)
Log-normal (x; μ, σ)	$\mathrm{f}\left(\mathrm{x}\right)=\frac{1}{\mathsf{\sigma }\mathrm{x}\sqrt{2\mathsf{\pi }}}{\mathrm{e}}^{{-\frac{1}{2}\left(\frac{\mathrm{l}\mathrm{n}\left(\mathrm{x}\right)-\mathsf{\mu }}{\mathsf{\sigma }}\right)}^2}$	(9)
$\mathrm{Normal}\hspace{0.17em}(\mathrm{z};\mathsf{\alpha },\mathsf{\beta })$	$\mathrm{f}\left(\mathrm{z}\right)=\frac{1}{\mathsf{\beta }\sqrt{2\mathsf{\pi }}}{\mathrm{e}}^{{-\frac{1}{2}\left(\frac{\mathrm{z}-\mathsf{\alpha }}{\mathsf{\beta }}\right)}^2}$	(10)
$\mathrm{G}\mathrm{E}\mathrm{V}\hspace{0.17em}(\mathrm{x};\mathsf{\beta },\mathrm{k},\mathsf{\mu })$	$\mathrm{f}\left(\mathrm{x}\right)=\frac{1}{\mathsf{\beta }}{\left(1+\mathrm{k}\frac{\mathrm{x}-\mathsf{\mu }}{\mathsf{\beta }}\right)}^{-\frac{1}{\mathrm{k}}-1}\exp \left\{-{\left[1+\mathrm{k}\left(\frac{\mathrm{x}-\mathsf{\mu }}{\mathsf{\beta }}\right)\right]}^{-\frac{1}{\mathrm{k}}}\right\}$	(11)
$\mathrm{Gamma}\hspace{0.17em}\left(2\right)(\mathrm{x};\mathsf{\alpha },\mathsf{\lambda })$	$\mathrm{f}\left(\mathrm{x}\right)=\frac{{\lambda }^{\alpha }}{\mathsf{\Gamma }\left(\mathsf{\alpha }\right)}{\left(\mathrm{x}\right)}^{\mathsf{\alpha }-1}{\mathrm{e}}^{-\frac{x}{\lambda }};$ $\mathrm{where}\hspace{0.17em}\mathsf{\Gamma }\left(\mathsf{\alpha }\right)\mathrm{is}\mathrm{the}\mathrm{gamma}\mathrm{function}$	(12)
$\mathrm{Beta}4\hspace{0.17em}(\mathrm{x};\mathsf{\alpha },\mathsf{\beta },\mathrm{c},\mathrm{d})$	$\mathrm{f}\left(\mathrm{x}\right)=\frac{1}{\mathrm{B}\left(\mathsf{\alpha },\mathsf{\beta }\right)}\frac{{\left(\mathrm{x}-\mathrm{c}\right)}^{\mathsf{\alpha }-1}{\left(\mathrm{d}-\mathrm{x}\right)}^{\mathsf{\beta }-1}}{{\left(\mathrm{d}-\mathrm{c}\right)}^{\mathsf{\alpha }+\mathsf{\beta }-1}};$ $\mathrm{where}\hspace{0.17em}\mathrm{B}\left(\mathsf{\alpha },\mathsf{\beta }\right)\mathrm{is}\mathrm{the}\mathrm{beta}\mathrm{function}$	(13)
$\mathrm{Logistic}\hspace{0.17em}(\mathrm{x};\mathsf{\mu },\mathsf{\sigma })$	$\mathrm{f}\left(\mathrm{x}\right)=\frac{1}{\mathsf{\sigma }}\frac{{\mathrm{e}}^{-\frac{(\mathrm{x}-\mathsf{\mu })}{\mathsf{\sigma }}}}{{\left[1+{\mathrm{e}}^{-\frac{(\mathrm{x}-\mathsf{\mu })}{\mathsf{\sigma }}}\right]}^2}$	(14)
Cumulative distribution function (CDF)	$\mathrm{F}\left(\mathrm{x}\right)={\int }_{-\mathrm{\infty }}^{\mathrm{x}}\mathrm{f}\left(\mathrm{y}\right)\mathrm{d}\mathrm{y}$; where f(x) and f(z) are the probability density functions (PDFs)	(15)
Adjusted return period (ARP)	$\mathrm{A}\mathrm{R}\mathrm{P}=\frac{1}{1-\mathrm{F}\left(\mathrm{x}\right)}$	(16)

$\mathrm{x}$ and $\mathrm{z}$ represent precipitation (CAR, AAR, SAR, and SCR).

show the six PDFs used in this study: log-normal, normal, GEV, gamma (2), beta4, and logistic. Six CDFs of CAR, AAR, SAR, and SCR were calculated using Equation (15). Using Equation (16), ARPs were calculated for each CDF. In all CDFs, except in the normal CDF (Equation (10)), $\mathsf{\mu }$ and $\mathsf{\sigma }$ represent the location and scale parameters. For the normal CDF, the location and scale parameters are denoted by the symbols $\mathsf{\alpha }$ and $\mathsf{\beta }$, respectively. For the GEV function (Equation (11)) in particular, the shape, scale, and location parameters are indicated by the symbols $\mathsf{\beta }$, k, and $\mathsf{\mu }$, respectively. For the gamma (2) function (Equation (12)), the shape and scale parameters are indicated by the symbols $\mathsf{\alpha }$ and $\mathsf{\lambda }$, respectively. For the beta4 function (Equation (13)), the symbols $\mathsf{\alpha }$, $\mathsf{\beta }$, c, and d represent the parameters of shape 1, shape 2, lower limit, and upper limit, respectively. All parameters in Equations (9)–(14) were automatically calculated using Xlstat software and the maximum likelihood estimation method.

2.9.2. Adjusted Return Periods (ARP)

First, the P99 values of CAR, AAR, SAR, and SCR were calculated. Then, ARP was calculated with P99. Finally, each P99 was substituted into PRE.

2.10. Software Used and Significance of Statistical Analyses

For this study, the following programs were used: Office 365, Xlstat version 2024, Surfer 16.3.408, CorelDRAW version 2019, Past version 4.02, and RStudio 2024.12.1 build 563.

The significance level for all statistical analyses was α = 0.05.

3. Results

3.1. Maximum, 99th Percentile (P99), and Average Values of Cumulative Annual Rainfall (CAR), Annual Average Rainfall (AAR), Seasonal Average Rainfall (SAR), and Seasonal Cumulative Rainfall (SCR)

As shown in

Table 3. Maximum, 99th percentile (P99), and average values of cumulative annual rainfall (CAR), annual average rainfall (AAR), seasonal average rainfall (SAR) and seasonal cumulative rainfall (SCR) (mm yr⁻¹). Source: authors.

Variable	Inferential Statistic	Rainfall Amount (mm yr−1) Weather Station
		Culiacán	El Playón	Guatenipa	Ixpalino	Mocorito	Rosario	Sanalona II	Sta. C. de Alaya	Siqueros
CAR	Maximum	2088.20	881.30	1766.80	1617.60	1494.30	1975.10	1471.20	1707.10	1613.60
	P99	1588.94	850.92	1720.50	1381.52	1329.32	1423.97	1703.79	1529.97	1376.98
	Average	699.36	463.03	1040.62	790.94	705.86	930.35	872.08	775.63	743.38
AAR	Maximum	89.11	73.44	139.36	94.65	96.47	118.45	114.57	112.13	94.23
	P99	84.32	70.91	137.42	94.09	94.16	107.43	117.92	102.69	92.56
	Average	55.74	37.95	85.00	64.52	57.57	76.11	71.31	62.91	60.78
SAR	Maximum	277.13	231.03	332.73	305.13	296.10	403.83	328.47	334.60	293.03
	P99	267.40	220.37	331.77	287.31	283.31	321.31	385.08	331.19	288.67
	Average	174.02	109.13	238.77	189.03	175.69	231.94	220.69	189.48	186.12
SCR	Maximum	831.40	693.10	998.20	915.40	888.30	1211.50	985.40	1003.80	879.10
	P99	802.20	661.10	995.31	861.94	849.93	963.45	1155.25	993.56	866.01
	Average	522.05	327.38	716.30	567.08	527.07	695.82	662.08	568.43	558.35

, the maximum precipitation values were recorded in the central part of the state (Culiacán with CAR = 2088.20 mm yr⁻¹), the mountainous area (Guatenipa with AAR = 139.36 mm yr⁻¹), and southern Sinaloa (Rosario with SAR = 403.83 mm yr⁻¹ and SCR = 1211.50 mm yr⁻¹). The largest P99s were associated with Guatenipa (CAR = 1720.50 mm yr⁻¹; AAR = 137.42 mm yr⁻¹) and Sanalona II (SAR = 385.08 mm yr⁻¹; SCR = 1155.25 mm yr⁻¹). The highest average rainfall was recorded in the mountainous part of Guatenipa: CAR = 1040.62 mm yr⁻¹, AAR = 85.00 mm yr⁻¹, SAR = 238.77 mm yr⁻¹, and SCR = 716.30 mm yr⁻¹.

3.2. Intense Rainfall Erosivity (IRE) Indices

MFI (Figure 2a) ranged from 50.77 mm in 1988 (El Playón) to 1640.29 mm in 1998 (Sta. C. de Alaya), averaging MFI = 212.11 mm. PCI (Figure 2b) ranged from 14.15% in 1997 (Mocorito) to 59.15% in 1998 (Sta. C. de Alaya), with an overall average PCI of 26.39%. The R-factor (Figure 2c) ranged from 966.64 MJ mm ha⁻¹ h⁻¹ yr⁻¹ in 1999 (El Playón) to 21284.76 MJ mm ha⁻¹ h⁻¹ yr⁻¹ in 1998 (Culiacán), with an average total R-factor of 6115.82 MJ mm ha⁻¹ h⁻¹ yr⁻¹. TEI (Figure 2d) ranged from 65.36 mm in 1970 (Culiacán) to 97019.79 mm in 1998 (Sta. C. de Alaya), averaging TEI = 6223.01 mm. ED (Figure 2e) ranged from 6.92 MJ ha⁻¹ h⁻¹ yr⁻¹ in 1999 (El Playón) to 10.19 MJ ha⁻¹ h⁻¹ yr⁻¹ in 1998 (Culiacán), with an average ED of 7.73 MJ ha⁻¹ h⁻¹ yr⁻¹. ORE (Figure 2f) ranged from 51.39 MJ mm ha⁻¹ h⁻¹ yr⁻¹ in 1970 (Culiacán) to 92679.40 MJ mm ha⁻¹ h⁻¹ yr⁻¹ in 1998 (Sta. C. de Alaya), with an average ORE of 5021.05 MJ mm ha⁻¹ h⁻¹ yr⁻¹.

In

Table 4. Classification of observed rainfall erosivity risk (ORE) in the state of Sinaloa (MJ mm ha⁻¹ h⁻¹ yr⁻¹). Source: authors.

Observed Rainfall Erosivity Risk Index (ORE, MJ mm ha−1 h−1 yr−1)	Category	Index Value
0–5000	Very low	1
5001–10,000	Low	2
10,001–15,000	Medium	3
15,001–20,000	High	4
>20,000	Very high	5

, and for the first time for Sinaloa, a classification for ORE is proposed, ranging from ‘very low’ to ‘very high.’ In this study, the total percentages for each ORE classification were 76.55% (very low), 20.22% (low), 1.22% (medium), 0.01% (high), and 2.00% (very high). Medium ORE was recorded in 1971 (Mocorito; ORE = 10,513.84 MJ mm ha⁻¹ h⁻¹ yr⁻¹), 1995 (Ixpalino; ORE = 11,959.73 MJ mm ha⁻¹ h⁻¹ yr⁻¹), 2011 (Rosario; ORE = 10,587.97 MJ mm ha⁻¹ h⁻¹ yr⁻¹ and Siqueros; ORE = 11,467.94 MJ mm ha⁻¹ h⁻¹ yr⁻¹), and 2013 (Culiacán; ORE = 12,231.84 MJ mm ha⁻¹ h⁻¹ yr⁻¹). Rosario in 2006 was the only station and year that recorded a high ORE (ORE = 15,252.82 MJ mm ha⁻¹ h⁻¹ yr⁻¹). The only year that recorded very high ORE for all nine stations was 1998: Culiacán (ORE = 66,218.78 MJ mm ha⁻¹ h⁻¹ yr⁻¹), El Playón (ORE = 24,691.69 MJ mm ha⁻¹ h⁻¹ yr⁻¹), Guatenipa (ORE = 88,484.40 MJ mm ha⁻¹ h⁻¹ yr⁻¹), Ixpalino (ORE = 47,862.81 MJ mm ha⁻¹ h⁻¹ yr⁻¹), Mocorito (ORE = 50,230.01 MJ mm ha⁻¹ h⁻¹ yr⁻¹), Sanalona II (ORE = 62,767.80 MJ mm ha⁻¹ h⁻¹ yr⁻¹), Rosario (ORE = 38,880.83 MJ mm ha⁻¹ h⁻¹ yr⁻¹), Sta. C. de Alaya (ORE = 92,679.40 MJ mm ha⁻¹ h⁻¹ yr⁻¹), and Siqueros (ORE = 32,561.62 MJ mm ha⁻¹ h⁻¹ yr⁻¹).

3.3. Spatial Variability—Trend of Observed Rainfall Erosivity Risk (ORE)

The average annual value of ORE (Figure 3a) ranged from ORE = 3520.63 MJ mm ha⁻¹ h⁻¹ yr⁻¹ (very low) to ORE = 6038.81 MJ mm ha⁻¹ h⁻¹ yr⁻¹ (low), with an average of ORE = 5021.05 MJ mm ha⁻¹ h⁻¹ yr⁻¹ (low). The highest magnitudes of ORE were recorded in central Sinaloa (Culiacán: ORE = 5093.94 MJ mm ha⁻¹ h⁻¹ yr⁻¹; Guatenipa: ORE = 6038.81 MJ mm ha⁻¹ h⁻¹ yr⁻¹; Sanalona II: ORE = 5822.32 MJ mm ha⁻¹ h⁻¹ yr⁻¹; and Sta. C. de Alaya: ORE = 5656.34 MJ mm ha⁻¹ h⁻¹ yr⁻¹). The Mann–Kendall trend results (Figure 3b) ranged from ORE = −0.27 (Siqueros, trend not significant) to ORE = 2.59 (Culiacán, trend significant), with an average of ORE = 0.65 (trend not significant). The magnitude of the trend (Sen’s slope, Figure 3c) ranged from ORE = −5.26 MJ mm ha⁻¹ h⁻¹ yr⁻¹ (Rosario, trend not significant) to ORE = 34.64 MJ mm ha⁻¹ h⁻¹ yr⁻¹ (Culiacán, trend significant), averaging ORE = 9.03 MJ mm ha⁻¹ h⁻¹ yr⁻¹ (trend not significant).

3.4. Validation of the Models of Predictive Rainfall Erosivity Risk (PRE)

Normality Analysis of the Residuals of Multiple Linear Regression (MLR)

No MLR showed normality in the residuals (

Table 5. p (normal) values of normality analysis of residuals of multiple linear regression (MLR) (dimensionless). Source: authors.

N/Weather Station	p-Value (Dimensionless)
	Culiacán	El Playón	Guatenipa	Ixpalino	Mocorito	Sanalona II	Rosario	Sta. C. de Alaya	Siqueros
Shapiro–Wilk W	0.944	0.932	0.932	0.459	0.902	0.872	0.841	0.873	0.704
p (normal)	0.019	0.007	0.007	0.000	0.001	0.000	0.000	0.000	0.000
Anderson–Darling A	0.796	1.241	1.007	7.556	1.264	1.492	2.368	2.122	3.125
p (normal)	0.036	0.003	0.011	0.000	0.002	0.001	0.000	0.000	0.000
p (Monte Carlo)	0.036	0.002	0.009	0.000	0.003	0.001	0.000	0.000	0.000
Jarque–Bera JB	17.700	8.313	10.900	1925.000	20.450	63.790	52.500	26.190	471.900
p (normal)	0.000	0.016	0.004	0.000	0.000	0.000	0.000	0.000	0.000
p (Monte Carlo)	0.004	0.023	0.013	0.000	0.004	0.001	0.001	0.002	0.000

Bold: values that did not register normality (p-value < 0.05).

), since no station recorded a p-value ≥ 0.05 simultaneously in the four normality tests. Specifically, the p-value ranged from 0.000 (Culiacán, Ixpalino, Mocorito, Sanalona II, Rosario, Sta. C. de Alaya, and Siqueros) to 1925.000 (Ixpalino).

3.5. Models of Predicted Rainfall Erosivity Risk (PRE)

Nine MNRs (quadratic polynomials) were estimated to model PRE (Equations (17)–(25), respectively) for the nine weather stations. In the nine models, CAR, AAR, SAR, and SCR significantly explained the variability of PRE.

$$\mathrm{P}\mathrm{R}\mathrm{E}\left(\mathrm{C}\mathrm{u}\mathrm{l}\mathrm{i}\mathrm{a}\mathrm{c}\mathrm{á}\mathrm{n}\right)=110311-\left(211517·\mathrm{C}\mathrm{A}\mathrm{R}\right)+\left(2.54\times {10}^6·\mathrm{A}\mathrm{A}\mathrm{R}\right)-\left(1.01\times {10}^6·\mathrm{S}\mathrm{A}\mathrm{R}\right)+\left(335007·\mathrm{S}\mathrm{C}\mathrm{R}\right)+\left(79.86·{\mathrm{C}\mathrm{A}\mathrm{R}}^2\right)-\left(11501.60·{\mathrm{A}\mathrm{A}\mathrm{R}}^2\right)+\left(4244.68·{\mathrm{S}\mathrm{A}\mathrm{R}}^2\right)-(471.18·{\mathrm{S}\mathrm{C}\mathrm{R}}^2)$$

(17)

$$\mathrm{P}\mathrm{R}\mathrm{E}\left(\mathrm{E}\mathrm{l}\mathrm{P}\mathrm{l}\mathrm{a}\mathrm{y}\mathrm{ó}\mathrm{n}\right)=-1623.23-\left(248284·\mathrm{C}\mathrm{A}\mathrm{R}\right)+\left(2.98\times {10}^6·\mathrm{A}\mathrm{A}\mathrm{R}\right)+\left(902966·\mathrm{S}\mathrm{A}\mathrm{R}\right)-\left(300889·\mathrm{S}\mathrm{C}\mathrm{R})+(217.45·{\mathrm{C}\mathrm{A}\mathrm{R}}^2\right)-\left(31321.70·{\mathrm{A}\mathrm{A}\mathrm{R}}^2\right)-\left(3519.95·{\mathrm{S}\mathrm{A}\mathrm{R}}^2\right)+(391.21·{\mathrm{S}\mathrm{C}\mathrm{R}}^2)$$

(18)

$$\mathrm{P}\mathrm{R}\mathrm{E}\left(\mathrm{G}\mathrm{u}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{n}\mathrm{i}\mathrm{p}\mathrm{a}\right)=86127.20-\left(614643·\mathrm{C}\mathrm{A}\mathrm{R}\right)+\left(7.37\times {10}^6·\mathrm{A}\mathrm{A}\mathrm{R}\right)-\left(572892·\mathrm{S}\mathrm{A}\mathrm{R}\right)+\left(190966·\mathrm{S}\mathrm{C}\mathrm{R}\right)+\left(246.13·{\mathrm{C}\mathrm{A}\mathrm{R}}^2\right)-\left(35437.90·{\mathrm{A}\mathrm{A}\mathrm{R}}^2\right)-\left(2003.48·{\mathrm{S}\mathrm{A}\mathrm{R}}^2\right)+(222.67·{\mathrm{S}\mathrm{C}\mathrm{R}}^2)$$

(19)

$$\mathrm{P}\mathrm{R}\mathrm{E}\left(\mathrm{I}\mathrm{x}\mathrm{p}\mathrm{a}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{o}\right)=70342.20+\left(147091·\mathrm{C}\mathrm{A}\mathrm{R}\right)-\left(1.77\times {10}^6·\mathrm{A}\mathrm{A}\mathrm{R}\right)+\left(528185·\mathrm{S}\mathrm{A}\mathrm{R}\right)-\left(175503·\mathrm{S}\mathrm{C}\mathrm{R}\right)-\left(61.12·{\mathrm{C}\mathrm{A}\mathrm{R}}^2\right)+\left(8855.73·{\mathrm{A}\mathrm{A}\mathrm{R}}^2\right)-\left(579.85·{\mathrm{S}\mathrm{A}\mathrm{R}}^2\right)+(64.10·{\mathrm{S}\mathrm{C}\mathrm{R}}^2)$$

(20)

$$\mathrm{P}\mathrm{R}\mathrm{E}\left(\mathrm{M}\mathrm{o}\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{i}\mathrm{t}\mathrm{o}\right)=69447.40+\left(370022·\mathrm{C}\mathrm{A}\mathrm{R}\right)-\left(4.44\times {10}^6·\mathrm{A}\mathrm{A}\mathrm{R}\right)+\left(42711.40·\mathrm{S}\mathrm{A}\mathrm{R}\right)-\left(14236.10·\mathrm{S}\mathrm{C}\mathrm{R}\right)-\left(165.26·{\mathrm{C}\mathrm{A}\mathrm{R}}^2\right)+\left(23803.80·{\mathrm{A}\mathrm{A}\mathrm{R}}^2\right)+\left(3644.06·{\mathrm{S}\mathrm{A}\mathrm{R}}^2\right)-(404.73·{\mathrm{S}\mathrm{C}\mathrm{R}}^2)$$

(21)

$$\mathrm{P}\mathrm{R}\mathrm{E}\left(\mathrm{S}\mathrm{a}\mathrm{n}\mathrm{a}\mathrm{l}\mathrm{o}\mathrm{n}\mathrm{a}\hspace{0.17em}\mathrm{I}\mathrm{I}\right)=10582.20+\left(164803·\mathrm{C}\mathrm{A}\mathrm{R}\right)-\left(1.98\times {10}^6·\mathrm{A}\mathrm{A}\mathrm{R}\right)+\left(269350·\mathrm{S}\mathrm{A}\mathrm{R}\right)-\left(89818.90·\mathrm{S}\mathrm{C}\mathrm{R}\right)-\left(77.31·{\mathrm{C}\mathrm{A}\mathrm{R}}^2\right)+\left(11127.90·{\mathrm{A}\mathrm{A}\mathrm{R}}^2\right)+\left(687.21·{\mathrm{S}\mathrm{A}\mathrm{R}}^2\right)-(76.25·{\mathrm{S}\mathrm{C}\mathrm{R}}^2)$$

(22)

$$\mathrm{P}\mathrm{R}\mathrm{E}\left(\mathrm{R}\mathrm{o}\mathrm{s}\mathrm{a}\mathrm{r}\mathrm{i}\mathrm{o}\right)=14890.30+\left(27281.70·\mathrm{C}\mathrm{A}\mathrm{R}\right)-\left(327103·\mathrm{A}\mathrm{A}\mathrm{R}\right)+\left(835391·\mathrm{S}\mathrm{A}\mathrm{R}\right)-\left(278482·\mathrm{S}\mathrm{C}\mathrm{R}\right)-\left(8.68·{\mathrm{C}\mathrm{A}\mathrm{R}}^2\right)+\left(1242.67·{\mathrm{A}\mathrm{A}\mathrm{R}}^2\right)-\left(3454.40·{\mathrm{S}\mathrm{A}\mathrm{R}}^2\right)+(383.96·{\mathrm{S}\mathrm{C}\mathrm{R}}^2)$$

(23)

$$\mathrm{P}\mathrm{R}\mathrm{E}\left(\mathrm{S}\mathrm{t}\mathrm{a}.\mathrm{C}.\mathrm{d}\mathrm{e}\hspace{0.17em}\mathrm{A}\mathrm{l}\mathrm{a}\mathrm{y}\mathrm{a}\right)=-25668.70+(6216.09·\mathrm{C}\mathrm{A}\mathrm{R})-(74919.80·\mathrm{A}\mathrm{A}\mathrm{R})+(13436.60·\mathrm{S}\mathrm{A}\mathrm{R})-(4294.68·\mathrm{S}\mathrm{C}\mathrm{R})-(2.26·{\mathrm{C}\mathrm{A}\mathrm{R}}^2)+(324.61·{\mathrm{A}\mathrm{A}\mathrm{R}}^2)-(696.26·{\mathrm{S}\mathrm{A}\mathrm{R}}^2)+(77.31·{\mathrm{S}\mathrm{C}\mathrm{R}}^2)$$

(24)

$$\mathrm{P}\mathrm{R}\mathrm{E}\left(\mathrm{S}\mathrm{i}\mathrm{q}\mathrm{u}\mathrm{e}\mathrm{r}\mathrm{o}\mathrm{s}\right)=-13636.90+(114790·\mathrm{C}\mathrm{A}\mathrm{R})-(1.38\times {10}^6·\mathrm{A}\mathrm{A}\mathrm{R})-(32899.70·\mathrm{S}\mathrm{A}\mathrm{R})+(11024.20·\mathrm{S}\mathrm{C}\mathrm{R})-(45.34·{\mathrm{C}\mathrm{A}\mathrm{R}}^2)+(6517.30·{\mathrm{A}\mathrm{A}\mathrm{R}}^2)-(158.55·{\mathrm{S}\mathrm{A}\mathrm{R}}^2)+(17.70·{\mathrm{S}\mathrm{C}\mathrm{R}}^2)$$

(25)

3.6. Fitting Multiple Nonlinear Regressions (MNRs) Between Observed and Predicted Rainfall Erosivity Risk (ORE vs. PRE)

The nine models recorded significant fit, as shown by the fact that in the nine scatter diagrams for ORE vs. PRE (Figure 4a–i) R² ≥ 0.078; that is, Sr ≥ 0.280 (

Table 6. Measures of fit between observed and predicted rainfall erosivity risk (ORE vs. PRE) in the state of Sinaloa. Mean relative error (MRE, %), root mean square log error (RMSLE, dimensionless), coefficient of determination (R², dimensionless), and Spearman’s correlation coefficient (Sr, dimensionless). Source: authors.

Weather Station	MRE (%)	RMSLE (Dimensionless)	R2 (Dimensionless)	Sr (Dimensionless)
Culiacán	20.457	0.100	0.992	0.996
El Playón	40.640	0.205	0.898	0.948
Guatenipa	16.412	0.087	0.997	0.998
Ixpalino	36.048	0.198	0.894	0.946
Mocorito	24.500	0.151	0.978	0.989
Sanalona II	22.265	0.126	0.988	0.994
Rosario	23.211	0.209	0.948	0.974
Sta. C. de Alaya	22.898	0.123	0.997	0.998
Siqueros	21.991	0.127	0.955	0.977

Bold: significant Spearman correlation coefficient (Sr).

). Specifically, the fit ranged from R² = 0.894 (Sr = 0.946; Ixpalino) to R² = 0.997 (Sr = 0.998; Guatenipa and Sta. C. de Alaya).

As shown in Table 6, MRE ranged from MRE = 16.412% (Guatenipa) to MRE = 40.640% (El Playón). RMSLE ranged from RMSLE = 0.087 (Guatenipa) to RMSLE = 0.209 (Rosario).

3.7. Simulation of Predicted Rainfall Erosivity Risk (PRE)

p-Values of the Cumulative Distribution Functions (CDFs) That Recorded the Best Fit

In

Table 7. Cumulative distribution functions (CDFs) with the highest p-values for cumulative annual rainfall (CAR), annual average rainfall (AAR), seasonal average rainfall (SAR), and seasonal accumulated rainfall (SCR) (dimensionless). Source: authors.

Weather Station	Log-Normal	Normal	GEV	Gamma (2)	Beta4	Logistic
	Maximum p-Value (Dimensionless)
Culiacán	SCR = 1.00 SAR = 1.00 CAR = 0.96				AAR = 1.00
El Playón	AAR = 0.97			SCR = 0.99 SAR = 0.99 CAR = 0.99
Guatenipa	AAR = 0.98 SAR = 0.90 SCR = 0.90		CAR = 0.96
Ixpalino	AAR = 0.98				CAR = 1.00 SAR = 0.99 SCR = 0.99
Mocorito	CAR = 0.96	SAR = 0.96		AAR = 0.98	SCR = 0.96
Sanalona II	AAR = 0.89 SAR = 0.84 SCR = 0.84		CAR = 0.94
Rosario			CAR = 0.62 AAR = 0.84		SAR = 0.97 SCR = 0.98
Sta. C. de Alaya			AAR = 0.87	CAR = 0.85		SAR = 0.77 SCR = 0.77
Siqueros	CAR = 0.97 AAR = 0.95 SAR = 1.00 SCR = 1.00

, the p-value ranged from 0.62 (GEV; CAR; Rosario) to 1.00 (log-normal and beta4; SCR–SAR–CAR–AAR; Culiacán–Ixpalino–Siqueros), respectively. The overall mean was 0.93. The CDF that fit the best was log-normal, predicting 16 occurrences, and the CDF that fit the poorest was the normal distribution, predicting 1 occurrence.

3.8. Parameters of the Cumulative Distribution Functions (CDFs) That Recorded the Best Fit

In

Table 8. Magnitudes of the parameters of six cumulative distribution functions (CDFs): location ($\mathsf{\mu }$, $\alpha$), scale ($\mathsf{\sigma }$, $\beta$, $\mathrm{k},\mathrm{a}\mathrm{n}\mathrm{d}$ $\mathsf{\lambda }$), shape ($\mathsf{\beta }$ and $\mathsf{\alpha }$), shape 1 ($\mathsf{\alpha }$), shape 2 ($\mathsf{\beta }$), lower limit (c), and upper limit (d). Source: authors.

Weather Station	Parameters of Cumulative Distribution Functions (CDFs)
	Log-Normal $\mathsf{\mu }\mathbf{=}\mathbf{location}$ $\mathsf{\sigma }\mathbf{=}\mathbf{scale}$	Normal $\mathsf{\alpha }\mathbf{=}\mathbf{location}$ $\mathsf{\beta }\mathbf{=}\mathbf{scale}$	GEV $\mathsf{\beta }\mathbf{=}\mathbf{shape}$ $\mathbf{k}\mathbf{=}\mathbf{scale}$ $\mathsf{\mu }\mathbf{=}\mathbf{location}$	Gamma (2) $\mathsf{\alpha }\mathbf{=}\mathbf{shape}$ $\mathsf{\lambda }\mathbf{=}\mathbf{scale}$	Beta4 $\mathsf{\alpha }\mathbf{=}\mathbf{shape}\mathbf{1}$ $\mathsf{\beta }\mathbf{=}\mathbf{shape}\mathbf{2}$ $\mathbf{c}\mathbf{=}\mathbf{upper}\mathbf{limit}$ $\mathbf{d}\mathbf{=}\mathbf{lower}\mathbf{limit}$	Logistic $\mathsf{\mu }\mathbf{=}\mathbf{location}$ $\mathsf{\sigma }\mathbf{=}\mathbf{scale}$
Culiacán	SCR ($\mathsf{\mu }$ = 6.24; $\mathsf{\sigma }$ = 0.21) SAR ($\mathsf{\mu }$ = 5.14; $\mathsf{\sigma }$ = 0.21) CAR ($\mathsf{\mu }$ = 6.51; $\mathsf{\sigma }$ = 0.28)				AAR ($\mathsf{\alpha }$ = 1.86, $\mathsf{\beta }$ = 3.44, c = 32.08, and d = 99.44)
El Playón	AAR ($\mathsf{\mu }$ = 3.57; $\mathsf{\sigma }$ = 0.39)			SCR ($\mathsf{\alpha }$ = 6.36; $\mathsf{\lambda }$ = 51.46) SAR ($\mathsf{\alpha }$ = 6.36$;\mathsf{\lambda }$ = 17.15) CAR ($\mathsf{\alpha }$ = 6.96; $\mathsf{\lambda }$ = 66.53)
Guatenipa	SCR ($\mathsf{\mu }$ = 6.56; $\mathsf{\sigma }$ = 0.19) SAR ($\mathsf{\mu }$ = 5.46; $\mathsf{\sigma }$ = 0.19) AAR ($\mathsf{\mu }$ = 4.42; $\mathsf{\sigma }$ = 0.21)		CAR ($\mathsf{\beta }$ = −0.05, $\mathrm{k}$ = 198.89, and $\mathsf{\mu }$ = 930.73)
Ixpalino	AAR ($\mathsf{\mu }$ = 4.14; $\mathsf{\sigma }$ = 0.23)				SCR ($\mathsf{\alpha }$ = 3.39, $\mathsf{\beta }$ = 10.01, c = 322.66, and d = 1289.52) SAR ($\mathsf{\alpha }$ = 3.99, $\mathsf{\beta }$ = 14.96, c = 103.83, and d = 508.10) CAR ($\mathsf{\alpha }$ = 1.98, $\mathsf{\beta }$ = 10.63, c = 456.90, and d = 2592.85)
Mocorito	CAR ($\mathsf{\mu }$ = 6.52; $\mathsf{\sigma }$ = 0.28)	SAR ($\mathsf{\mu }$ = 175.69; $\mathsf{\sigma }$ = 44.66)		AAR ($\mathsf{\alpha }$ = 15.74; $\mathsf{\lambda }$ = 3.66)	SCR ($\mathsf{\alpha }$ = 25.72, $\mathsf{\beta }$ = 28.63, c = −417.42, and d = 1579.81)
Sanalona II	SCR ($\mathsf{\mu }$ = 6.47; $\mathsf{\sigma }$ = 0.21) SAR ($\mathsf{\mu }$ = 5.38; $\mathsf{\sigma }$ = 0.21) AAR ($\mathsf{\mu }$ = 4.24; $\mathsf{\sigma }$ = 0.23)		CAR ($\mathsf{\beta }$ = −0.09, $\mathrm{k}$ = 180.90, and $\mathsf{\mu }$ = 775.51)
Rosario			AAR ($\mathsf{\beta }$ = −0.01, $\mathrm{k}$ = 12.95, and $\mathsf{\mu }$ = 68.67) CAR ($\mathsf{\beta }$ = 0.10, $\mathrm{k}$ = 191.77, and $\mathsf{\mu }$ = 819.65)		SCR ($\mathsf{\alpha }$ = 3.11, $\mathsf{\beta }$ = 12.46, c = 349.52, and d = 2086.65) SAR ($\mathsf{\alpha }$ = 3.51, $\mathsf{\beta }$ = 17.75, c = 113.05, and d = 834.01)
Sta. C. de Alaya			AAR ($\mathsf{\beta }$ = −0.05, $\mathrm{k}$ = −0.10, and $\mathsf{\mu }$ = 14.30)	CAR ($\mathsf{\alpha }$ = 12.04; $\mathsf{\lambda }$ = 64.42)		SCR ($\mathsf{\mu }$ = 562.12; $\mathsf{\sigma }$ = 68.04) SAR ($\mathsf{\mu }$ = 187.38; $\mathsf{\sigma }$ = 22.68)
Siqueros	SCR ($\mathsf{\mu }$ = 6.29; $\mathsf{\sigma }$ = 0.27) SAR ($\mathsf{\mu }$ = 5.19; $\mathsf{\sigma }$ = 0.27) AAR ($\mathsf{\mu }$ = 4.08; $\mathsf{\sigma }$ = 0.24) CAR ($\mathsf{\mu }$ = 6.57; $\mathsf{\sigma }$ = 0.27)

, the parameters of location ($\mathsf{\mu }$, $\alpha$), scale ($\mathsf{\sigma }$, $\beta$, $\mathrm{k}$, $\mathsf{\lambda }$), shape ($\mathsf{\beta }$, $\mathsf{\alpha }$), shape 1 ($\mathsf{\alpha }$), shape 2 ($\mathsf{\beta }$), lower bound (c) and upper bound (d) are presented, respectively, for the six CDFs (log-normal, normal, GEV, gamma (2), beta4, and logistic). These parameters ranged from $\mathsf{\mu }$ = 3.57 (log-normal; AAR; Culiacán) to $\mathsf{\mu }$ = 562.12 (logistic; SCR; Sta. C. de Alaya); from $\mathsf{\sigma }$ = 0.19 (log-normal; SCR–SAR; Guatenipa) to $\mathsf{\sigma }$ = 68.04 (logistic; SCR; Sta. C. de Alaya); from k = −0.10 (GEV; AAR; Sta. C. de Alaya) to k = 198.89 (GEV; CAR; Guatenipa); from $\mathsf{\lambda }$ = 3.66 (gamma (2); AAR; Mocorito) to $\mathsf{\lambda }$ = 3.66 (gamma (2); AAR; Mocorito); from $\mathsf{\beta }$ = −0.09 (GEV; CAR; Sanalona II) to $\mathsf{\beta }$ = 17.75 (beta4; SAR; Rosario); from $\mathsf{\alpha }$ = 6.36 (gamma (2); SCR–SAR; El Playón) to $\mathsf{\alpha }$ = 17.74 (gamma (2); AAR; Mocorito); from c = −417.42 (beta4; SCR; Mocorito) to c = 456.90 (beta4; CAR; Ixpalino); and from d = 99.44 (beta4; AAR; Culiacán) to d = 2592.85 (beta4; CAR; Ixpalino).

3.9. Adjusted Return Periods (ARPs) of Cumulative Annual Rainfall (CAR), Annual Average Rainfall (AAR), Seasonal Average Rainfall (SAR) and Seasonal Cumulative Rainfall (SCR)

As shown in

Table 9. Adjusted return periods (ARPs, years) for CAR, AAR, SAR, and SCR (mm yr⁻¹). Source: authors.

Weather Station	CAR (mm yr−1)	ARP (Years)	AAR (mm yr−1)	ARP (Years)	SAR (mm yr−1)	ARP (Years)	SCR (mm yr−1)	ARP (Years)	Weather Station	CAR (mm yr−1)	ARP (Years)	AAR (mm yr−1)	ARP (Years)	SAR (mm yr−1)	ARP (Years)	SCR (mm yr−1)	ARP (Years)
Culiacán	300	1.00	40	1.12	150	1.37	450	1.37	Sanalona II	650	1.16	40	1.01	150	1.04	600	1.56
	600	1.53	50	1.60	180	2.54	550	2.78		850	2.08	60	1.35	200	1.56	700	2.81
	900	6.94	60	2.82	210	6.46	650	8.23		1050	5.62	80	3.73	250	4.13	800	6.37
	1200	55.05	70	6.53	240	20.76	750	31.86		1250	20.34	100	18.11	300	17.13	900	17.13
	1500	523.55	80	23.67	270	78.43	850	146.92		1450	93.46	120	120.62	350	94.08	1000	52.12
El Playón	200	1.04	20	1.08	50	1.06	100	1.01	Rosario	800	1.49	60	1.17	200	1.48	500	1.12
	400	1.67	40	2.67	100	1.88	300	1.88		1100	4.45	80	2.94	250	2.99	700	2.27
	600	4.98	60	11.74	150	6.09	500	9.94		1400	14.72	100	12.11	300	8.08	900	7.93
	800	22.80	80	58.34	200	29.47	700	97.86		1700	45.13	120	57.89	350	28.52	1100	46.73
	1000	138.73	100	284.38	250	184.58	900	1380.83		2000	125.49	140	292.47	400	131.41	1300	514.87
Guatenipa	600	1.01	30	1.00	150	1.01	300	1.00	Sta. C. de Alaya	500	1.10	40	1.06	100	1.02	200	1.00
	900	1.45	60	1.07	200	1.25	500	1.04		700	1.67	60	1.90	150	1.19	400	1.09
	1200	4.57	90	2.83	250	2.71	700	1.96		900	3.76	80	6.87	200	2.74	600	2.74
	1500	22.09	120	23.48	300	10.11	900	10.11		1100	11.95	100	41.01	250	16.82	800	33.99
	1800	133.12	150	351.43	350	55.04	1000	30.29		1300	49.82	120	397.73	300	144.41	1000	624.62
Ixpalino	550	1.10	55	1.38	120	1.01	450	1.17	Siqueros	400	1.02	55	1.63	60	1.00	250	1.00
	750	1.98	65	2.26	170	1.51	550	1.93		700	1.87	65	2.89	120	1.07	450	1.33
	950	4.81	75	4.61	220	5.11	650	4.53		1000	9.40	75	6.15	180	2.01	650	4.13
	1150	14.94	85	11.09	270	37.24	750	15.23		1300	76.87	85	14.87	240	7.16	850	22.59
	1350	59.88	95	30.03	320	642.58	850	78.71		1600	744.02	95	39.34	300	36.23	1050	158.42
Mocorito	400	1.03	50	1.48	100	1.05	200	1.01
	600	1.49	60	2.49	150	1.39	400	1.21
	800	3.62	70	5.32	200	3.41	600	3.38
	1000	12.36	80	13.94	250	20.81	800	46.78
	1200	50.33	90	43.56	300	372.22	1000	7070.50

, the minimum value of ARP = 1.00 years was recorded in Culiacán (CAR = 300 mm yr⁻¹), Guatenipa (AAR = 30 mm yr⁻¹; SCR = 300 mm yr⁻¹), Siqueros (SAR = 60 mm yr⁻¹; SCR = 250 mm yr⁻¹), and Sta. C. de Alaya (SCR = 200 mm yr⁻¹). The maximum values of ARP were recorded in Siqueros (744.02 years for CAR = 1600 mm yr⁻¹), Sta. C. de Alaya (397.73 years for AAR = 120 mm yr⁻¹), Ixpalino (642.58 years for SAR = 320 mm yr⁻¹), and Mocorito (7070.50 years for SCR = 1000 mm yr⁻¹).

3.10. Application of Simulation and Adjusted Return Periods (ARPs) of Predicted Rainfall Erosivity Risk (PRE)

The simulated PRE magnitudes were Culiacán (PRE = −19,469,339.90; very low), El Playón (PRE = 6699.09; low), Guatenipa (PRE = 1,539,852.24; very high), Ixpalino (PRE = −150,626.44; very low), Mocorito (PRE = −717,272.64; very low), Rosario (PRE = 34,381.22; very high), Sanalona II (PRE = −2,194,060.00; very low), Sta. C. de Alaya (PRE = 6219.15; low), and Siqueros (PRE = 57,879.95; very high). For Rosario, ARP = 16.15 years; prob. = 0.06, ARP = 21.44 years; prob. = 0.05, ARP = 13.40 years; prob. = 0.07, and ARP = 13.12 years; prob. = 0.08. For Siqueros, ARP = 136.46 years; prob. = 0.00, ARP = 30.83 years; prob. = 0.03, ARP = 26.24 years; prob. = 0.04, and ARP = 26.24 years; prob. = 0.04.

4. Discussion

For CAR (Table 3), the minimum magnitudes recorded in El Playón and the maximum magnitudes recorded in Rosario, Guatenipa, and Sta. C. de Alaya are similar to those reported by [15]; these authors point out that of 21 weather stations analyzed for the period 1982–2014, El Playón was the station that recorded the lowest magnitude (CAR = 424.64 mm year⁻¹). They also note that from these 21 stations, Guatenipa (CAR = 1014.11 mm year⁻¹), Rosario (CAR = 929.06 mm year⁻¹), and Sta. C. de Alaya (CAR = 822.63 mm year⁻¹) are ranked 6th, 10th, and 15th, respectively.

The lower magnitudes of the erosivity indices (Figure 2a–f) are consistent with what was pointed out by [51], who note that the most intense meteorological droughts since the 1930s in Sinaloa were recorded for the summers of the period 1996–2002. Furthermore, the results in Figure 2a–f are in agreement with [52,53,54], who point out that in Sinaloa, some of the most extreme tropical cyclones in terms of daily rainfall were Henriette and Ismael (1995), Isis and Javier (1998), John, Lane, and Paul (2006), and Juliette, Lorena, and Manuel (2013). This parameter produces high erosivity risks [55].

Figure 2c,e and Figure 3a are consistent with [18,45,56], who state that ORE and spatial erosivity patterns depend mainly on the variation in R-factor and ED, because these two indices can be considered to indicate the risk of soil erosion. The ORE trend (Figure 3b,c) is similar to that reported by [15], who state that out of 21 weather stations, Culiacán (DGE) presented the highest Mann–Kendall trend for TEI, with a magnitude of 2.84.

The results for the R-factor (Figure 2c) and ORE (Figure 2f) show similar magnitudes when precipitation events are not extreme; however, ORE exhibits greater sensitivity to the occurrence of highly erosive events such as tropical cyclones [56,57,58]. This is due to the fact that the ORE formulation intrinsically includes the sensitivity of five IRE indices which are widely used worldwide: (1) PCI [15,58], (2) MFI [15,58], (3) R-factor [59], (4) TEI [15], and (5) ED [59].

PRE (Equations (17)–(25)) was validated since, when working with MLR, the first validation required is the analysis of normality of the residuals [50]. Because the MLR residuals did not present normality, the recommendation of [60,61,62] was used; that is, the validations for MNR were MRE, RMSLE, R², and the Sr hypothesis test (Figure 4 and Table 6).

Of 36 p-values (Table 7), only SAR for Mocorito was normally distributed, which is in agreement with [63], who argue that hydroclimatological variables only occasionally fit a normal CDF, with extreme values of CDF being the most common [64,65,66,67,68].

The results in Table 9 are similar to those reported by [69,70,71], as these authors also found that most rainfall ARPs are in the range of 1 year to 100 years with a return level ranging from 0 to 500 mm [72]. Knowledge of the very high PRE simulations of this study can help prevent damage to agricultural soils in Guatenipa, Rosario, and Siqueros, specifically through the implementation of comprehensive soil management plans [73].

Equations (17)–(25) may not be highly efficient, due to multiple uncertainties inherent to MNR, for example: (a) the very complexity of the MNR itself [74], (b) the possibility that the most appropriate function was not chosen for the MNR [75], and (c) the variability of CAR, AAR, SAR, and SCR, associated with the effects of global, regional, and local climate change [76].

The ARP results may not be highly accurate because the R-factor, which was obtained indirectly using CAR (Equations (4) and (5)), does not include the recent variation (1982–2014) in significant trends of CAR in northern [Guasave station (DGE)] and southern Sinaloa (Potrerillos station) [15]. Furthermore, the R-factor was not calculated using sub-hourly precipitation data, as recommended by [59].

5. Conclusions

Using five IRE indices (MFI, PCI, TEI, R-factor, and ED), the calculation and classification of ORE for nine weather stations in the state of Sinaloa is proposed in a novel manner. The use of the ORE index can present a valuable opportunity to increase the accuracy of the erosivity risk, which is associated with highly erosive weather events, which are not covered by other conventional IRE indices such as MFI, PCI, TEI, R-factor, or ED. The central part of Sinaloa (Culiacán, Sanalona II, Guatenipa, and Sta. C. de Alaya) recorded the highest magnitude and trend of ORE. PRE (dependent variable) was obtained using MNR, where CAR, AAR, SAR, and SCR were the independent variables. Using six CDFs, six ARPs were also obtained for CAR, AAR, SAR, and SCR. PRE was simulated using P99. Three stations (Guatenipa, Rosario, and Siqueros) recorded very high PRE values because extreme erosivity events occur here with a frequency of one to eight times every 100 years. This methodology can be applied to any region of the world, particularly where it is desired to calculate the variability of ORE and to predict and simulate PRE, by integrating the most widely used IRE indices in the world into a summary statistic. In future research, it is recommended to calculate the R-factor using sub-hourly rainfall data wherever possible. It is also recommended to evaluate the performance of ORE under various rainfall, topographic, and soil conditions.