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Estimating the rainfall erosivity factor from monthly precipitation data in the Madrid Region (Spain)

Estimating the rainfall erosivity factor from monthly precipitation data in the Madrid Region... The need for continuous recording rain gauges makes it difficult to determine the rainfall erosivity factor (Rfactor) of the Universal Soil Loss Equation in regions without good spatial and temporal data coverage. In particular, the R-factor is only known at 16 rain gauge stations in the Madrid Region (Spain). The objectives of this study were to identify a readily available estimate of the R-factor for the Madrid Region and to evaluate the effect of rainfall record length on estimate precision and accuracy. Five estimators based on monthly precipitation were considered: total annual rainfall (P), Fournier index (F), modified Fournier index (MFI), precipitation concentration index (PCI) and a regression equation provided by the Spanish Nature Conservation Institute (RICONA). Regression results from 8 calibration stations showed that MFI was the best estimator in terms of coefficient of determination and root mean squared error, closely followed by P. Analysis of the effect of record length indicated that little improvement was obtained for MFI and P over 5year intervals. Finally, validation in 8 additional stations supported that the equation R = 1.05·MFI computed for a record length of 5 years provided a simple, precise and accurate estimate of the R-factor in the Madrid Region. Keywords: Rainfall erosivity; R-factor; Universal Soil Loss Equation; Modified Fournier index; Soil erosion. INTRODUCTION Erosion models are a powerful tool for soil loss evaluation and land management. One of the most widely-used models is the Universal Soil Loss Equation (USLE) developed by Wischmeier and Smith (1961, 1965, 1978). Compared to more complex physically-based erosion models, USLE is a simplistic empirical model. Despite the simplicity of USLE, determination of the rainfall erosivity factor (R-factor) is no easy task. The Rfactor for a single storm was defined by Wischmeier (1959) as the product of the total kinetic energy (E) multiplied by the maximum 30-minute intensity (I30). The R-factor at a particular location is then obtained as the average of annual E·I30 values over long time intervals (over 20 years) to include apparent cyclical rainfall patterns (Wischmeier and Smith, 1978). Although there are several equations for computing the kinetic energy of a storm (Brown and Foster, 1987; Wischmeier and Smith, 1978), all of them require continuous recording rain gauges with time resolution of at least 15 minutes. This need for continuous recording makes it difficult to determine the Rfactor in many regions where good spatial and temporal data coverage is scarce. This is the case for the Madrid Region (Spain), where the R-factor has been only computed by the Spanish Nature Conservation Institute (ICONA) at 16 rain gauge stations based on rainfall data recorded from 1950 to 1985 (ICONA, 1988). There have been many attempts worldwide to establish correlations between the R-factor calculated by the prescribed method and more readily available rainfall data, such as daily and monthly precipitation (Angulo-Martínez and Beguería, 2009; Bonilla and Vidal, 2011; Colotti, 2004; Diodato, 2004; Diodato and Bellochi, 2007; Lee and Heo, 2011; Loureiro and Coutinho, 2001; Petkovsek and Mikos, 2004; Renard and Freimund, 1994; Salako, 2008; Yu and Rosewell, 1996; Yu et al., 2001). Nevertheless, most of the obtained equations have limited application out of the areas in which they were developed without a thorough validation analysis. Therefore, the objectives of this study were: (1) to identify a readily available estimate of the rainfall erosivity factor for the Madrid Region, and (2) to evaluate the effect of rainfall record length on the precision and accuracy of the estimates. MATERIAL AND METHODS Rainfall erosivity estimators Based on the literature review conducted, five estimators of rainfall erosivity were selected for this study: total annual rainfall (P), Fournier index (F), modified Fournier index (MFI), Oliver's precipitation concentration index (PCI) and a regression model proposed by the Spanish Nature Conservation Institute (RICONA). Other factors such as Hudson's KE > 25 index (Hudson, 1971), Lal's AIm index (Lal, 1976), Onchev's P/St universal index (Onchev, 1985), and Burst factor (Smithen and Schulze, 1982) were not considered since they still require continuous recording. A further description of selected estimators is provided below. Fournier (1960) conducted a regression analysis between sedimentation in rivers and several rainfall variables, finding a high correlation between the total annual erosion and the distribution coefficient of rainfall or, most commonly named, Fournier index: F= p2 P (1) where F is the Fournier index, p is the precipitation in the wettest month and P is the total annual rainfall. Due to the simplicity in calculating the Fournier index, Arnoldus (1980) attempted to correlate this index with the known values of the rainfall erosivity factor for 164 rainfall stations in the United States and 14 stations in West Africa. The results were not satisfactory because of the different behavior of each index: while the R-factor adds all erosive storms, the Fournier index only captures those storms out of the month with the highest precipitation within the denominator (total annual rainfall). Thus, Arnoldus (1980) proposed a modification of the Fournier index in which the storms that occur outside the month of maximum rainfall increase the overall value of the index, obtaining significantly higher coefficients of determination (r2 > 0.80): each estimator were then obtained by averaging annual values over time intervals. These series were used to evaluate the effect of record length through regression analyses. The effect of record length on each estimator was studied using the following time intervals: 1, 2, 5, 10, 15 and 20 years, described by Eq. 8: XN = MFI = i =1 pi2 P (2) Xi i =1 (8) in which MFI is the modified Fournier index, pi is the monthly rainfall and P is the total annual rainfall. Colotti (2004) reported that the Food and Agriculture Organization (FAO) of the United Nations used the modified Fournier index as an erosion estimate according to the following general equation: where XN represents the values of the estimator (RICONA, P, F, MFI and PCI) for a record length of N consecutive years, and Xi is the annual value of the estimator in year i. Eq. 8 was applied to all consecutive 1, 2, 5, 10, 15 and 20-year intervals within the period covered by each rainfall station. Study area R = a MFI + b (3) where R is the rainfall erosivity factor, MFI is the modified Fournier index, and a and b are two regional fitting parameters. Ferro et al. (1991) stated that the FF index, which represents the average value of the modified Fournier index for an interval of N years, is better correlated with the R-factor than both the mean annual precipitation and the modified Fournier index: FF = 12 pi2 j 1 , = Pj N j =1 i =1 MFI j j =1 (4) where pi,j is the rainfall (mm) in month i of year j, Pj is the annual precipitation (mm) of year j, and N is the number of years considered. The FF index was also successfully used by Ferro et al. (1999) to estimate the R-factor in southern Italy and southeastern Australia. Oliver (1980) proposed an index of rainfall concentration in order to estimate the aggressiveness of storms from the temporal variability of monthly precipitation: The Madrid Region is located in central Spain (Figure 1), between the Atlantic Ocean and the Mediterranean Sea, in a high plateau around 600 m above sea level (a.s.l.). The study area lies between 39º53'N­41º10'N and 3º03'W­4º34'W, covering a total extension of 8022 km2. The region is in the shape of a triangle (Figure 1), with the Central System mountain range on the NW side (elevations above 2400 m a.s.l.). From the base of the Central System, it begins a grade that ends in the Tagus valley (SE corner), with elevations below 500 m a.s.l. The area is dominated by a Mediterranean climate, characterized by seasonal temperatures, summer drought and erratic rainfall. The average annual rainfall of the area ranges from 1500 mm in the NW to 400 mm in the SE. Database PCI = 100 i =1 2 P pi2 (5) where PCI is the precipitation concentration index, pi is the monthly rainfall and P is the total annual rainfall. In Spain, ICONA (1988) proposed the following regression equation to estimate the rainfall erosivity factor in the Madrid Region: 0,563 RICONA = e-0,834 PMEX 1,314 MR-0,388 F24 (6) The rain gauge network provided by the Spanish Meteorological Agency (AEMET) in the Madrid Region consists of more than 170 rainfall stations. However, ICONA (1988) only determined the R-factor according to the prescribed method at 16 stations. To date, no more attempts to determine the R-factor in additional stations have been published. In this study, 8 stations were used for regression analysis (model calibration) between the single computed R-factor reported by ICONA (1988) and the five estimators presented previously (RICONA, P, F, MFI and PCI). The equations obtained from these 8 calibration stations were then validated in 8 additional stations. Locations of the 16 stations are shown in Figure 1. Table 1 presents the station name, elevation, available R-factor, analysis period covered and number of complete years. For each station, monthly rainfall and maximum monthly rainfall in 24 hours were provided by AEMET. Statistical models where RICONA is the rainfall erosivity factor (MJ·cm·ha­1·h­1·year­1) as estimated by ICONA, PMEX is the maximum monthly precipitation (mm), MR is the total rainfall from October to May (mm), and F24 is the ratio of the square of the maximum annual rainfall in 24 hours (mm) to the sum of the maximum monthly rainfall in 24 hours (mm): Two statistical models were selected for this study. First, a simple linear regression with intercept term, as defined by equation 9: R = 0 + 1 X + F24 ( P24h, annual ) = P24h,i i =1 12 (9) (7) Annual values for the period covered by each rainfall station were calculated for the five estimators. A series of values for where R is the rainfall erosivity factor, 0 is the intercept term, 1 is the slope, X is the estimator (RICONA, P, F, MFI and PCI) and represents the error. It should be noted that the intercept term 0 is just a fitting parameter which has no physical meaning since no erosion should occur for zero rainfall. Therefore, a simple linear regression through the origin (no intercept term) was defined as second statistical model: R = 2 X + (10) in which R is the rainfall erosivity factor, 2 is the slope, X is the estimator and represents the error. The assumptions for both models were that errors are independent of each other and normally distributed with a mean of zero and constant variance. Based on available rainfall data, a series of values for each estimator (RICONA, P, F, MFI and PCI) and record length (1, 2, 5, 10, 15 and 20 years) were determined. These values were then correlated with the single computed R-factor reported by ICONA (1988) for each station. Calibration Validation Fig. 1. Location of the study area and selected rainfall stations. Table 1. Characteristics of rainfall stations selected for calibration and validation. Code Station name Elevation (m.a.s.l.) 1890 960 645 1100 667 687 617 540 960 850 582 600 530 725 852 1028 R-factor (MJ·cm·ha­1·h­1·year­1) (ICONA, 1988) 194 89 63 124 65 74 53 105 85 87 65 62 66 99 69 130 Period covered 1947­2007 1947­2004 1947­2007 1956­2007 1947­2007 1947­2007 1951­2007 1952­1999 1970­2005 1970­2007 1951­2007 1955­2004 1972­2007 1976­2005 1968­2007 1976­2005 Complete years 61 58 61 52 61 61 57 47 35 38 57 50 35 27 36 27 50 km Calibration 2462 Navacerrada 3112 Puentes Viejas Dam 3119 Fuente El Saz 3190 Hoyo de Manzanares 3195 Madrid Retiro 3196 Madrid (Cuatro Vientos Airport) 3200 Getafe (Air Base) 3341 San Juan Dam Validation 3116 El Atazar Dam 3121E El Vellón Dam 3129 Madrid (Barajas Airport) 3169 Alcalá de Henares (Canaleja) 3182E Arganda 3193O Majadahonda (MAFRE) 3223 Pezuela de las Torres 3274 San Lorenzo de El Escorial (Royal Seat) Table 2. Regression results for R-factor (MJ·cm·ha­1·h­1·year­1) in the 8 calibration stations. Estimator / Record length RICONA (MJ·cm·ha­1·h­1·year­1) 1 year 2 years 5 years 10 years 15 years 20 years Annual rainfall, P (mm) 1 year 2 years 5 years 10 years 15 years 20 years Fournier index, F (mm) 1 year 2 years 5 years 10 years 15 years 20 years Modified Fournier index, MFI (mm) 1 year 2 years 5 years 10 years 15 years 20 years R = 0.82·F + 63.83 R = 1.45·F + 39.14 R = 2.42·F R = 2.51·F R = 2.51·F R = 2.51·F R = 0.73·MFI + 29.51 R = 0.90·MFI + 13.34 R = 1.05·MFI R = 1.05·MFI R = 1.05·MFI R = 1.05·MFI 0.26 0.46 0.71 0.87 0.90 0.92 0.62 0.78 0.91 0.94 0.96 0.96 38 33 24 16 15 13 27 21 13 11 10 9 458 226 89 43 28 20 458 226 89 43 28 20 R = 0.11·P + 28.67 R = 0.12·P + 20.08 R = 0.15·P R = 0.15·P R = 0.15·P R = 0.15·P 0.70 0.79 0.87 0.90 0.91 0.93 24 20 16 14 13 12 458 226 89 43 28 20 R = 0.30·RICONA + 64.13 R = 0.45·RICONA + 47.59 R = 0.81·RICONA R = 0.83·RICONA R = 0.84·RICONA R = 0.84·RICONA 0.41 0.63 0.66 0.80 0.89 0.91 34 27 26 20 15 14 458 226 89 43 28 20 Regression equation r2 RMSE (MJ·cm·ha­1·h­1·year­1) No. data points RESULTS AND DISCUSSION Regression analysis (model calibration) A preliminary analysis showed that regression results barely changed whether the period covered was that employed by ICONA (1950­1985) or a broader period. Thus, the decision was to consider the complete rainfall data set that was available for each rainfall station. Regression results obtained from the 8 calibration stations are presented in Table 2. These results include regression equation, coefficient of determination (r2) and root mean squared error (RMSE) for each estimator and record length. PCI was found to be poorly correlated with Rfactor so no results are presented hereafter. Assumptions of the statistical models were successfully validated. It was observed that the regression model through the origin (0 = 0) provided r2 and RMSE values extremely close to those provided by the regression model with intercept term (0 0), especially for record lengths over 5 years. In fact, the intercept term was found not to be statistically significant in the regression analysis, so the simpler regression model without intercept term was proposed in Table 2 for record lengths of 5 years or more. Overall, rather good results were provided by RICONA, P, F and MFI, especially for record lengths over 5 years. MFI was the estimator with the best results in terms of r2 and RMSE, closely followed by P. It should be pointed out that the equation obtained for the total annual rainfall (R = 0.15·P) closely agreed with previous results proposed by Van der Knijff et al. (2000) for Southern Europe (R = 0.13·P). Lastly, the slope of 0.84 obtained for RICONA when long record lengths were considered suggested that the available regression equation proposed by ICONA (Eq. 6) may overpredict the R-factor by approximately 16%. A possible explanation is that Eq. 6 was developed for a broad area of Spain (not only the Madrid Region) and it may not perfectly fit the climatic pattern of the Madrid Region. Effect of record length on estimate precision and accuracy As can be inferred from Table 2, record length had a direct effect on regression models: as record length increased, r2 increased and RMSE decreased. The reason is that as record length increased, the annual values of the estimators were averaged over a longer time interval as defined by Eq. 8. Thus, this `smoothing' effect translated into less dispersion and, consequently, a better fitting. In order to further analyze the effect of record length on the precision and accuracy of the estimates, two additional statistics were evaluated: the coefficient of variation (CV) and the mean absolute percentage error (MAPE). CV is defined as the ratio of the standard deviation to the mean of a sample, expressed as a percentage. This statistic represents the variability of an index about its mean value and represents the precision of the estimator. MAPE is a measure of the error estimating the R-factor (accuracy) and is determined as follows: MAPE (%) = ^ R - Rj R (11) j =1 where R is the known value of the rainfall erosivity factor ^ (Table 1), R is the estimated value from the regression model, and N is the number of data points for a given record length. CV results are presented in Figure 2. As can be seen in the figure, P and MFI were the estimators with the lowest CV for any record length. CV for RICONA and F was almost twice that for P and MFI. It can also be observed that CV values barely changed for P and MFI over 5 years of record length whereas RICONA and F required at least 10 years to become relatively constant. Fig. 3. Mean absolute percentage error for each estimator and record length (model calibration). time intervals up to 10 years were considered, but there was almost no improvement beyond 10 years. In fact, both P and MFI provided quite good results (in terms of CV and MAPE) for a record length of only 5 years. According to these results, a record length of 5 years could be considered adequate for P and MFI, while a record length of 10 years could be proposed for RICONA and F. Therefore, regression results in Table 2 were reduced to one single equation for each estimator, as shown in Table 3. Validation The simplified regression models obtained from the 8 calibration stations (Table 3) were used to estimate the R-factor in 8 additional stations. Results are shown in Figure 4, in which the y-axis represents measured R-factor reported by ICONA (Table 1) and the x-axis represents predicted R-factor. As can be seen in the figure, the four estimators provided fairly good results for the proposed record lengths, observing only some dispersion for the station with the highest R-factor (San Lorenzo de El Escorial, R = 130 MJ·cm·ha­1·h­1·year­1). RMSE and MAPE were also evaluated for the validation stations. Figure 5 shows that RMSE values obtained from validation were rather close to those previously obtained from calibration, P and MFI being the estimators with the lowest RMSE. With respect to MAPE, Figure 6 indicates that validation results were similar to those observed from calibration, except for RICONA, which experienced an important increase in estimate error due to the differences observed in one of the stations (San Lorenzo de El Escorial). Fig. 2. Coefficient of variation for each estimator and record length (model calibration). Figure 3 shows that MFI was overall the estimator with the lowest MAPE. Again, small reductions in MAPE were obtained for record lengths over 5 to 10 years. For P and MFI (the estimators with the lowest MAPE values), MAPE was below 13% for a record length as short as 5 years. In addition, a TukeyKramer honest significance difference (HSD) test revealed that P and MFI were statistically different from RICONA and F for a 5% significance level when a 5-year record length was considered. The above results showed that record length definitely increased both the precision and accuracy of the estimates when Table 3. Simplified regression models obtained from the 8 calibration stations. Estimator RICONA P F MFI Proposed record length 10 years 5 years 10 years 5 years Regression equation R = 0.83·RICONA,10 R = 0.15·P5 R = 2.51·F10 R = 1.05·MFI5 r2 0.80 0.87 0.87 0.91 RMSE (MJ·cm·ha­1·h­1·year­1) 20 16 16 13 CV (%) 14 12 13 12 MAPE (%) 12 13 14 11 R-ICONA (10 years) Annual rainfall, P (5 years) 175 Measured R-factor 125 100 75 50 25 0 0 25 50 75 100 125 150 175 Predicted R-factor Measured R-factor Predicted R-factor Fournier index, F (10 years) Modified Fournier index, MFI (5 years) Measured R-factor Measured R-factor Predicted R-factor Predicted R-factor Fig. 4. Scatter plot of R-factor measured by ICONA vs. R-factor predicted by estimates for proposed record lengths in MJ·cm·ha­1·h­1·year­1 (model validation). 20 18 16 14 12 10 8 6 4 2 0 R-ICONA (10 years) Annual rainfall P (5 years) Calibration Fournier index F (10 years) Validation Mod. Fournier index MFI (5 years) Fig. 5. Comparison of root mean squared error (RMSE) results for calibration and validation. RMSE (MJ·cm·ha-1·h-1·year-1) 18% 16% 14% 12% 10% 8% 6% 4% 2% 0% R-ICONA (10 years) Annual rainfall P (5 years) Calibration Fournier index F (10 years) Validation Mod. Fournier index MFI (5 years) Fig. 6. Comparison of mean absolute percentage error (MAPE) results for calibration and validation. MAPE (%) CONCLUSIONS coefficient of variation and mean absolute percentage error) for different time intervals. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Hydrology and Hydromechanics de Gruyter

Estimating the rainfall erosivity factor from monthly precipitation data in the Madrid Region (Spain)

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de Gruyter
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0042-790X
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Abstract

The need for continuous recording rain gauges makes it difficult to determine the rainfall erosivity factor (Rfactor) of the Universal Soil Loss Equation in regions without good spatial and temporal data coverage. In particular, the R-factor is only known at 16 rain gauge stations in the Madrid Region (Spain). The objectives of this study were to identify a readily available estimate of the R-factor for the Madrid Region and to evaluate the effect of rainfall record length on estimate precision and accuracy. Five estimators based on monthly precipitation were considered: total annual rainfall (P), Fournier index (F), modified Fournier index (MFI), precipitation concentration index (PCI) and a regression equation provided by the Spanish Nature Conservation Institute (RICONA). Regression results from 8 calibration stations showed that MFI was the best estimator in terms of coefficient of determination and root mean squared error, closely followed by P. Analysis of the effect of record length indicated that little improvement was obtained for MFI and P over 5year intervals. Finally, validation in 8 additional stations supported that the equation R = 1.05·MFI computed for a record length of 5 years provided a simple, precise and accurate estimate of the R-factor in the Madrid Region. Keywords: Rainfall erosivity; R-factor; Universal Soil Loss Equation; Modified Fournier index; Soil erosion. INTRODUCTION Erosion models are a powerful tool for soil loss evaluation and land management. One of the most widely-used models is the Universal Soil Loss Equation (USLE) developed by Wischmeier and Smith (1961, 1965, 1978). Compared to more complex physically-based erosion models, USLE is a simplistic empirical model. Despite the simplicity of USLE, determination of the rainfall erosivity factor (R-factor) is no easy task. The Rfactor for a single storm was defined by Wischmeier (1959) as the product of the total kinetic energy (E) multiplied by the maximum 30-minute intensity (I30). The R-factor at a particular location is then obtained as the average of annual E·I30 values over long time intervals (over 20 years) to include apparent cyclical rainfall patterns (Wischmeier and Smith, 1978). Although there are several equations for computing the kinetic energy of a storm (Brown and Foster, 1987; Wischmeier and Smith, 1978), all of them require continuous recording rain gauges with time resolution of at least 15 minutes. This need for continuous recording makes it difficult to determine the Rfactor in many regions where good spatial and temporal data coverage is scarce. This is the case for the Madrid Region (Spain), where the R-factor has been only computed by the Spanish Nature Conservation Institute (ICONA) at 16 rain gauge stations based on rainfall data recorded from 1950 to 1985 (ICONA, 1988). There have been many attempts worldwide to establish correlations between the R-factor calculated by the prescribed method and more readily available rainfall data, such as daily and monthly precipitation (Angulo-Martínez and Beguería, 2009; Bonilla and Vidal, 2011; Colotti, 2004; Diodato, 2004; Diodato and Bellochi, 2007; Lee and Heo, 2011; Loureiro and Coutinho, 2001; Petkovsek and Mikos, 2004; Renard and Freimund, 1994; Salako, 2008; Yu and Rosewell, 1996; Yu et al., 2001). Nevertheless, most of the obtained equations have limited application out of the areas in which they were developed without a thorough validation analysis. Therefore, the objectives of this study were: (1) to identify a readily available estimate of the rainfall erosivity factor for the Madrid Region, and (2) to evaluate the effect of rainfall record length on the precision and accuracy of the estimates. MATERIAL AND METHODS Rainfall erosivity estimators Based on the literature review conducted, five estimators of rainfall erosivity were selected for this study: total annual rainfall (P), Fournier index (F), modified Fournier index (MFI), Oliver's precipitation concentration index (PCI) and a regression model proposed by the Spanish Nature Conservation Institute (RICONA). Other factors such as Hudson's KE > 25 index (Hudson, 1971), Lal's AIm index (Lal, 1976), Onchev's P/St universal index (Onchev, 1985), and Burst factor (Smithen and Schulze, 1982) were not considered since they still require continuous recording. A further description of selected estimators is provided below. Fournier (1960) conducted a regression analysis between sedimentation in rivers and several rainfall variables, finding a high correlation between the total annual erosion and the distribution coefficient of rainfall or, most commonly named, Fournier index: F= p2 P (1) where F is the Fournier index, p is the precipitation in the wettest month and P is the total annual rainfall. Due to the simplicity in calculating the Fournier index, Arnoldus (1980) attempted to correlate this index with the known values of the rainfall erosivity factor for 164 rainfall stations in the United States and 14 stations in West Africa. The results were not satisfactory because of the different behavior of each index: while the R-factor adds all erosive storms, the Fournier index only captures those storms out of the month with the highest precipitation within the denominator (total annual rainfall). Thus, Arnoldus (1980) proposed a modification of the Fournier index in which the storms that occur outside the month of maximum rainfall increase the overall value of the index, obtaining significantly higher coefficients of determination (r2 > 0.80): each estimator were then obtained by averaging annual values over time intervals. These series were used to evaluate the effect of record length through regression analyses. The effect of record length on each estimator was studied using the following time intervals: 1, 2, 5, 10, 15 and 20 years, described by Eq. 8: XN = MFI = i =1 pi2 P (2) Xi i =1 (8) in which MFI is the modified Fournier index, pi is the monthly rainfall and P is the total annual rainfall. Colotti (2004) reported that the Food and Agriculture Organization (FAO) of the United Nations used the modified Fournier index as an erosion estimate according to the following general equation: where XN represents the values of the estimator (RICONA, P, F, MFI and PCI) for a record length of N consecutive years, and Xi is the annual value of the estimator in year i. Eq. 8 was applied to all consecutive 1, 2, 5, 10, 15 and 20-year intervals within the period covered by each rainfall station. Study area R = a MFI + b (3) where R is the rainfall erosivity factor, MFI is the modified Fournier index, and a and b are two regional fitting parameters. Ferro et al. (1991) stated that the FF index, which represents the average value of the modified Fournier index for an interval of N years, is better correlated with the R-factor than both the mean annual precipitation and the modified Fournier index: FF = 12 pi2 j 1 , = Pj N j =1 i =1 MFI j j =1 (4) where pi,j is the rainfall (mm) in month i of year j, Pj is the annual precipitation (mm) of year j, and N is the number of years considered. The FF index was also successfully used by Ferro et al. (1999) to estimate the R-factor in southern Italy and southeastern Australia. Oliver (1980) proposed an index of rainfall concentration in order to estimate the aggressiveness of storms from the temporal variability of monthly precipitation: The Madrid Region is located in central Spain (Figure 1), between the Atlantic Ocean and the Mediterranean Sea, in a high plateau around 600 m above sea level (a.s.l.). The study area lies between 39º53'N­41º10'N and 3º03'W­4º34'W, covering a total extension of 8022 km2. The region is in the shape of a triangle (Figure 1), with the Central System mountain range on the NW side (elevations above 2400 m a.s.l.). From the base of the Central System, it begins a grade that ends in the Tagus valley (SE corner), with elevations below 500 m a.s.l. The area is dominated by a Mediterranean climate, characterized by seasonal temperatures, summer drought and erratic rainfall. The average annual rainfall of the area ranges from 1500 mm in the NW to 400 mm in the SE. Database PCI = 100 i =1 2 P pi2 (5) where PCI is the precipitation concentration index, pi is the monthly rainfall and P is the total annual rainfall. In Spain, ICONA (1988) proposed the following regression equation to estimate the rainfall erosivity factor in the Madrid Region: 0,563 RICONA = e-0,834 PMEX 1,314 MR-0,388 F24 (6) The rain gauge network provided by the Spanish Meteorological Agency (AEMET) in the Madrid Region consists of more than 170 rainfall stations. However, ICONA (1988) only determined the R-factor according to the prescribed method at 16 stations. To date, no more attempts to determine the R-factor in additional stations have been published. In this study, 8 stations were used for regression analysis (model calibration) between the single computed R-factor reported by ICONA (1988) and the five estimators presented previously (RICONA, P, F, MFI and PCI). The equations obtained from these 8 calibration stations were then validated in 8 additional stations. Locations of the 16 stations are shown in Figure 1. Table 1 presents the station name, elevation, available R-factor, analysis period covered and number of complete years. For each station, monthly rainfall and maximum monthly rainfall in 24 hours were provided by AEMET. Statistical models where RICONA is the rainfall erosivity factor (MJ·cm·ha­1·h­1·year­1) as estimated by ICONA, PMEX is the maximum monthly precipitation (mm), MR is the total rainfall from October to May (mm), and F24 is the ratio of the square of the maximum annual rainfall in 24 hours (mm) to the sum of the maximum monthly rainfall in 24 hours (mm): Two statistical models were selected for this study. First, a simple linear regression with intercept term, as defined by equation 9: R = 0 + 1 X + F24 ( P24h, annual ) = P24h,i i =1 12 (9) (7) Annual values for the period covered by each rainfall station were calculated for the five estimators. A series of values for where R is the rainfall erosivity factor, 0 is the intercept term, 1 is the slope, X is the estimator (RICONA, P, F, MFI and PCI) and represents the error. It should be noted that the intercept term 0 is just a fitting parameter which has no physical meaning since no erosion should occur for zero rainfall. Therefore, a simple linear regression through the origin (no intercept term) was defined as second statistical model: R = 2 X + (10) in which R is the rainfall erosivity factor, 2 is the slope, X is the estimator and represents the error. The assumptions for both models were that errors are independent of each other and normally distributed with a mean of zero and constant variance. Based on available rainfall data, a series of values for each estimator (RICONA, P, F, MFI and PCI) and record length (1, 2, 5, 10, 15 and 20 years) were determined. These values were then correlated with the single computed R-factor reported by ICONA (1988) for each station. Calibration Validation Fig. 1. Location of the study area and selected rainfall stations. Table 1. Characteristics of rainfall stations selected for calibration and validation. Code Station name Elevation (m.a.s.l.) 1890 960 645 1100 667 687 617 540 960 850 582 600 530 725 852 1028 R-factor (MJ·cm·ha­1·h­1·year­1) (ICONA, 1988) 194 89 63 124 65 74 53 105 85 87 65 62 66 99 69 130 Period covered 1947­2007 1947­2004 1947­2007 1956­2007 1947­2007 1947­2007 1951­2007 1952­1999 1970­2005 1970­2007 1951­2007 1955­2004 1972­2007 1976­2005 1968­2007 1976­2005 Complete years 61 58 61 52 61 61 57 47 35 38 57 50 35 27 36 27 50 km Calibration 2462 Navacerrada 3112 Puentes Viejas Dam 3119 Fuente El Saz 3190 Hoyo de Manzanares 3195 Madrid Retiro 3196 Madrid (Cuatro Vientos Airport) 3200 Getafe (Air Base) 3341 San Juan Dam Validation 3116 El Atazar Dam 3121E El Vellón Dam 3129 Madrid (Barajas Airport) 3169 Alcalá de Henares (Canaleja) 3182E Arganda 3193O Majadahonda (MAFRE) 3223 Pezuela de las Torres 3274 San Lorenzo de El Escorial (Royal Seat) Table 2. Regression results for R-factor (MJ·cm·ha­1·h­1·year­1) in the 8 calibration stations. Estimator / Record length RICONA (MJ·cm·ha­1·h­1·year­1) 1 year 2 years 5 years 10 years 15 years 20 years Annual rainfall, P (mm) 1 year 2 years 5 years 10 years 15 years 20 years Fournier index, F (mm) 1 year 2 years 5 years 10 years 15 years 20 years Modified Fournier index, MFI (mm) 1 year 2 years 5 years 10 years 15 years 20 years R = 0.82·F + 63.83 R = 1.45·F + 39.14 R = 2.42·F R = 2.51·F R = 2.51·F R = 2.51·F R = 0.73·MFI + 29.51 R = 0.90·MFI + 13.34 R = 1.05·MFI R = 1.05·MFI R = 1.05·MFI R = 1.05·MFI 0.26 0.46 0.71 0.87 0.90 0.92 0.62 0.78 0.91 0.94 0.96 0.96 38 33 24 16 15 13 27 21 13 11 10 9 458 226 89 43 28 20 458 226 89 43 28 20 R = 0.11·P + 28.67 R = 0.12·P + 20.08 R = 0.15·P R = 0.15·P R = 0.15·P R = 0.15·P 0.70 0.79 0.87 0.90 0.91 0.93 24 20 16 14 13 12 458 226 89 43 28 20 R = 0.30·RICONA + 64.13 R = 0.45·RICONA + 47.59 R = 0.81·RICONA R = 0.83·RICONA R = 0.84·RICONA R = 0.84·RICONA 0.41 0.63 0.66 0.80 0.89 0.91 34 27 26 20 15 14 458 226 89 43 28 20 Regression equation r2 RMSE (MJ·cm·ha­1·h­1·year­1) No. data points RESULTS AND DISCUSSION Regression analysis (model calibration) A preliminary analysis showed that regression results barely changed whether the period covered was that employed by ICONA (1950­1985) or a broader period. Thus, the decision was to consider the complete rainfall data set that was available for each rainfall station. Regression results obtained from the 8 calibration stations are presented in Table 2. These results include regression equation, coefficient of determination (r2) and root mean squared error (RMSE) for each estimator and record length. PCI was found to be poorly correlated with Rfactor so no results are presented hereafter. Assumptions of the statistical models were successfully validated. It was observed that the regression model through the origin (0 = 0) provided r2 and RMSE values extremely close to those provided by the regression model with intercept term (0 0), especially for record lengths over 5 years. In fact, the intercept term was found not to be statistically significant in the regression analysis, so the simpler regression model without intercept term was proposed in Table 2 for record lengths of 5 years or more. Overall, rather good results were provided by RICONA, P, F and MFI, especially for record lengths over 5 years. MFI was the estimator with the best results in terms of r2 and RMSE, closely followed by P. It should be pointed out that the equation obtained for the total annual rainfall (R = 0.15·P) closely agreed with previous results proposed by Van der Knijff et al. (2000) for Southern Europe (R = 0.13·P). Lastly, the slope of 0.84 obtained for RICONA when long record lengths were considered suggested that the available regression equation proposed by ICONA (Eq. 6) may overpredict the R-factor by approximately 16%. A possible explanation is that Eq. 6 was developed for a broad area of Spain (not only the Madrid Region) and it may not perfectly fit the climatic pattern of the Madrid Region. Effect of record length on estimate precision and accuracy As can be inferred from Table 2, record length had a direct effect on regression models: as record length increased, r2 increased and RMSE decreased. The reason is that as record length increased, the annual values of the estimators were averaged over a longer time interval as defined by Eq. 8. Thus, this `smoothing' effect translated into less dispersion and, consequently, a better fitting. In order to further analyze the effect of record length on the precision and accuracy of the estimates, two additional statistics were evaluated: the coefficient of variation (CV) and the mean absolute percentage error (MAPE). CV is defined as the ratio of the standard deviation to the mean of a sample, expressed as a percentage. This statistic represents the variability of an index about its mean value and represents the precision of the estimator. MAPE is a measure of the error estimating the R-factor (accuracy) and is determined as follows: MAPE (%) = ^ R - Rj R (11) j =1 where R is the known value of the rainfall erosivity factor ^ (Table 1), R is the estimated value from the regression model, and N is the number of data points for a given record length. CV results are presented in Figure 2. As can be seen in the figure, P and MFI were the estimators with the lowest CV for any record length. CV for RICONA and F was almost twice that for P and MFI. It can also be observed that CV values barely changed for P and MFI over 5 years of record length whereas RICONA and F required at least 10 years to become relatively constant. Fig. 3. Mean absolute percentage error for each estimator and record length (model calibration). time intervals up to 10 years were considered, but there was almost no improvement beyond 10 years. In fact, both P and MFI provided quite good results (in terms of CV and MAPE) for a record length of only 5 years. According to these results, a record length of 5 years could be considered adequate for P and MFI, while a record length of 10 years could be proposed for RICONA and F. Therefore, regression results in Table 2 were reduced to one single equation for each estimator, as shown in Table 3. Validation The simplified regression models obtained from the 8 calibration stations (Table 3) were used to estimate the R-factor in 8 additional stations. Results are shown in Figure 4, in which the y-axis represents measured R-factor reported by ICONA (Table 1) and the x-axis represents predicted R-factor. As can be seen in the figure, the four estimators provided fairly good results for the proposed record lengths, observing only some dispersion for the station with the highest R-factor (San Lorenzo de El Escorial, R = 130 MJ·cm·ha­1·h­1·year­1). RMSE and MAPE were also evaluated for the validation stations. Figure 5 shows that RMSE values obtained from validation were rather close to those previously obtained from calibration, P and MFI being the estimators with the lowest RMSE. With respect to MAPE, Figure 6 indicates that validation results were similar to those observed from calibration, except for RICONA, which experienced an important increase in estimate error due to the differences observed in one of the stations (San Lorenzo de El Escorial). Fig. 2. Coefficient of variation for each estimator and record length (model calibration). Figure 3 shows that MFI was overall the estimator with the lowest MAPE. Again, small reductions in MAPE were obtained for record lengths over 5 to 10 years. For P and MFI (the estimators with the lowest MAPE values), MAPE was below 13% for a record length as short as 5 years. In addition, a TukeyKramer honest significance difference (HSD) test revealed that P and MFI were statistically different from RICONA and F for a 5% significance level when a 5-year record length was considered. The above results showed that record length definitely increased both the precision and accuracy of the estimates when Table 3. Simplified regression models obtained from the 8 calibration stations. Estimator RICONA P F MFI Proposed record length 10 years 5 years 10 years 5 years Regression equation R = 0.83·RICONA,10 R = 0.15·P5 R = 2.51·F10 R = 1.05·MFI5 r2 0.80 0.87 0.87 0.91 RMSE (MJ·cm·ha­1·h­1·year­1) 20 16 16 13 CV (%) 14 12 13 12 MAPE (%) 12 13 14 11 R-ICONA (10 years) Annual rainfall, P (5 years) 175 Measured R-factor 125 100 75 50 25 0 0 25 50 75 100 125 150 175 Predicted R-factor Measured R-factor Predicted R-factor Fournier index, F (10 years) Modified Fournier index, MFI (5 years) Measured R-factor Measured R-factor Predicted R-factor Predicted R-factor Fig. 4. Scatter plot of R-factor measured by ICONA vs. R-factor predicted by estimates for proposed record lengths in MJ·cm·ha­1·h­1·year­1 (model validation). 20 18 16 14 12 10 8 6 4 2 0 R-ICONA (10 years) Annual rainfall P (5 years) Calibration Fournier index F (10 years) Validation Mod. Fournier index MFI (5 years) Fig. 5. Comparison of root mean squared error (RMSE) results for calibration and validation. RMSE (MJ·cm·ha-1·h-1·year-1) 18% 16% 14% 12% 10% 8% 6% 4% 2% 0% R-ICONA (10 years) Annual rainfall P (5 years) Calibration Fournier index F (10 years) Validation Mod. Fournier index MFI (5 years) Fig. 6. Comparison of mean absolute percentage error (MAPE) results for calibration and validation. MAPE (%) CONCLUSIONS coefficient of variation and mean absolute percentage error) for different time intervals.

Journal

Journal of Hydrology and Hydromechanicsde Gruyter

Published: Mar 1, 2015

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