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Introduction
Characterization of forage biomass in pasture land is complicated by the temporal and spatial variability that result from variation in vegetation patch sizes. These factors, along with topography, water location and distribution of fertilizer, result in non-uniform grass for biomass production. Although a normal distribution is assumed when estimating forage biomass, there are several problems that result from this assumption. For example, cattle do not graze pastures uniformly, but graze selectively in a manner that creates patches, some of which are more heavily grazed than others (Senft et al., 1987). The heterogeneity of plant biomass is related to the level of grazing intensity as this affects the biomass frequency distribution, which is often skewed to the right when plants are grazed (Shiyomi et al., 1984; Shiyomi et al., 1998).
Skewed distributions can be modelled by other distributions, such as the beta, log-normal, Johnson’s SB or Gamma distribution. The Gamma distribution was used by Shiyomi et al. (1998) and Tsutsumi et al. (2002) to model biomass production in pasture, but most of these distributions have inherent limitations when used to model forage biomass. For example, most of these distributions do not exist in a closed form, so that they are difficult to integrate, and some require advance knowledge of the minimum and maximum values for parameter estimation and rescaling estimated values (Remington et al., 1994).
The Weibull distribution can be used to overcome these difficulties because of its flexibility and simplicity. It is used to describe a wide variety of biomass distributions in grazed pasture (Remington et al., 1994) and in early first harvest of short-term forage crops in Asturias (N Spain) fertilized with three N treatments (Gorgoso et al., 2012). In other agronomic studies about rot incidence of strawberry also was applied succesfully (East, 2011). Remington et al. (1992) used the Weibull distribution to describe the amounts of aerial dry matter and maximum plant heights of blue grama (Bouteloua gracilis (H.B.K.) Lagg. ex Steud) and buffalograss (Buchloe dactyloides (Nutt.) Engelm).
The objective of this study was to evaluate the use of the three-parameter Weibull distribution to describe the frequency distribution of aerial dry matter production in the first harvest of Lolium multiflorum Lam., fertilized with different N doses. The null hypothesis was that the distributions studied follow a Weibull distribution was tested.
Matherials and Methods
Trial plots and biomass analysis
Forty kg ha-1 of a mixture of three Italian ryegrass cultivars, Lolium multiflorum westerwoldicum ‘Jivet’ and ‘Barspirit’ (tetraploids) and Lolium multiflorum ‘Barprisma’, (diploid) were sown manually in three 100 m2 (20 m x 5 m) plots at the end of October of 2009, 2010 and 2011. The mixture comprised 50, 20 and 30 % (by weight) of the three cultivars, respectively. The plots were in Carreño, Asturias, Spain. The treatment was the application of three doses of N fertilizer, and the amounts of N supplied with the doses were 0, 40 and 80 kg ha-1. The fertilizer was applied manually in March 2010, 2011 and 2012, as calcium ammonium nitrate (27 % N, 3 % Mg O and 7,5 % Ca O).
In each N treatment, between 28 and 30 forage samples were collected at random, within a square metallic frame of surface area 0.25 m2. The different plants within the frame were cut to soil level and the samples were weighed in the field. Aliquots (100 g) of each sample were oven-dried at 70 °C for 48 h to determine aerial dry matter production.
Modelling the aerial dry matter
The Weibull distribution
The three-parameter Weibull distribution were obtained by integrating the Weibull density function; both have the following expression for a random variable x:
where F(x) is the cumulative relative frequency of biomass production equal to or smaller than random variable x, a is the situation parameter, b is the scale parameter and c is the shape parameter.
In this study, biomass production intervals of 30 g m-2 were established for the fits of the Weibull function for all treatments. The values of the three parameters are equal to or higher than zero and the values allow the function to have different shapes.
Weibull distribution fits
The Weibull distribution parameters were estimated by the method of moments because is more suitable than the maximum likelihood method for a number of samples less or equal to 30 (Grender et al., 1990).
The method of moments is based on the relationship between the Weibull distribution parameters (a, b and c) and the first and second moments of the aerial dry matter (DM) distribution (mean value of aerial dry matter and variance, respectively):
where
Location parameter a of the Weibull distribution was considered in all cases as the aerial DM in the sample with the minimum value in each treatment.
Goodness of fits
The Shapiro-Wilk test was used to test the normality of data in each treatment, at an error probability level of 5 %.
The Kolmogorov-Smirnov (K-S) test was used to determine the goodness of fit of the Weibull distribution in each treatment. This test compares the cumulative estimated frequency F0(xj) with the cumulative observed frequency Fn(xi). The statistic Dn of the KS test for a given cumulative distribution was used to evaluate and compare the results:
where supx is the supremum of the set of distances. Thus, this value was calculated as follows (Cao, 2004):
Biomass production intervals of 30 g m-2 were chosen in all fits. The following expression was used to compare the value obtained with the critical value proposed by Miller (1956), and error probability was established at 5 %:
where Ln is the natural logarithm, a is the significance level and n is the number of samples considered in each treatment. When Dn > Dn,a , the test rejects the null hypothesis that the distribution follows a Weibull distribution.
Results and Discussion
The null hypothesis of a normal biomass distribution was not rejected in seven out of nine cases (only the samples of the distribution Lm_2012_0 and Lm_2012_40 cannot be considered normal). (Table 1).
Table 1 Shapiro-Wilk test statistic and skewness and kurtosis coefficients for the observed distributions.
| Distribution | Shapiro-Wilk test statistic | Skewness | Kurtosis |
| Lm_2010_0 | 0.953 | 0.404 | -0.683 |
| Lm_2010_40 | 0.980 | -0.295 | -0.437 |
| Lm_2010_80 | 0.972 | 0.534 | 0.363 |
| Lm_2011_0 | 0.943 | -0.511 | -0.819 |
| Lm_2011_40 | 0.947 | -0.147 | -1.272 |
| Lm_2011_80 | 0.960 | 0.044 | -1.175 |
| Lm_2012_0 | 0.905* | -0.718 | -0.691 |
| Lm_2012_40 | 0.921* | -0.115 | -1.485 |
| Lm_2012_80 | 0.938 | -0.031 | -1.378 |
*Significant at a=0.05
Absolute skewness ranged from a low of -0.718 in the Lm_2012_0 distribution and a maximum of 0.534 in Lm_2010_80 (Table 1). The Weibull function also enables distributions to be represented with different values of the kurtosis coefficient, which is indicative of the shape of the function. Estimates of the Weibull parameters for the levels of fertilizer are indicated (Table 2). Location parameter a and scale parameter b units correspond to the aerial DM units (g m-2); the sum of both parameters (a+b) corresponds to the 63rd percentile of the Weibull distribution (Johnson and Kotz, 1971), and the shape parameter c is unity.
Table 2 Weibull parameters (a, b and c) for the estimated biomass (g m-2) and different levels of N fertilizer.
| Year | N applied (kg ha-1) | Code | a | b | a + b | c |
| 2010 | 0 | Lm_2010_0 | 173.2 | 202.6 | 375.8 | 1.63 |
| 40 | Lm_2010_40 | 257.5 | 312.6 | 570.1 | 2.47 | |
| 80 | Lm_2010_80 | 347.5 | 213.4 | 560.9 | 1.65 | |
| 2011 | 0 | Lm_2011_0 | 200.0 | 265.5 | 465.5 | 2.17 |
| 40 | Lm_2011_40 | 550.0 | 151.2 | 701.2 | 1.73 | |
| 80 | Lm_2011_80 | 830.0 | 149.3 | 979.3 | 1.75 | |
| 2012 | 0 | Lm_2012_0 | 550.0 | 250.8 | 800.8 | 2.23 |
| 40 | Lm_2012_40 | 950.0 | 151.2 | 1101.2 | 1.73 | |
| 80 | Lm_2012_80 | 1230.0 | 143.3 | 1373.3 | 1.67 |
The results of the Kolmogorov-Smirnov test for goodness of fit (α=0.05) (Table 3) showed that there was not enough evidence to reject the null hypothesis that the 9 distributions of biomass follow a three parameter Weibull distribution.
Table 3 Results of the Kolmogorov-Smirnov test (KS) for the nine distributions.
| Distribution | Data | Dn | Dn,α | Distribution | Data | Dn | Dn, α |
| Lm_2010_0 | 28 | 0.140 | 0.257 | Lm_2011_80 | 30 | 0.150 | 0.248 |
| Lm_2010_40 | 28 | 0.130 | 0.257 | Lm_2012_0 | 30 | 0.179 | 0.248 |
| Lm_2010_80 | 29 | 0.139 | 0.252 | Lm_2012_40 | 30 | 0.188 | 0.248 |
| Lm_2011_0 | 30 | 0.179 | 0.248 | Lm_2012_80 | 30 | 0.151 | 0.248 |
| Lm_2011_40 | 30 | 0.181 | 0.248 |
D n : KS statistic; D n , a : critical value of Miller (1956) at a=0.05.
The Weibull distributions indicated an increase in the 63rd percentile of aerial dry matter (a + b), with a higher level of fertilizer in all cases in concordance with the mean values obtained in each treatment for aerial dry matter, except in Lm_2010_80, where the 63rd percentile is lower than Lm_2010_40, for which the value of parameter c was maximal (2.47). Situation parameter a also was related to the dose of fertilizer. Shape parameter c was always lower than 3.6 and all the Weibull distributions were skewed to the right. The nine observed distributions of biomass production (g m-2) in relative frequencies in each interval considered (30 g m-2) and the distributions described by the three parameter Weibull function are shown (Figure 1).
In relation to the modeling the aerial dry matter (DM), it is possible that the observed aerial DM production follows other more flexible distributions than the normal due to the asymmetry of the distributions. Mielke (1986) stated that if the absolute value of the skewness exceeds 0.01, normality cannot be reliably assumed in constructing confidence intervals or in hypothesis testing. Using this as a guideline, it can be seen that all of the frequency distributions were skewed. The results of the Kolmogorov-Smirnov test for goodness of fit (α=0.05) are consistent with those obtained by Remington et al. (1992) in a comparison of the results of the Weibull distribution with the normal distribution for describing amounts of aerial DM and maximum plant heights of blue grama (Bouteloua gracilis) and buffalo grass (Bouteloua dactyloides).
In our study, the normal distribution described only 50 % of the distributions of biomass and 0 % of the maximum height data using the Anderson-Darling test statistic. However, the Weibull distribution was able to describe all the observed biomass and maximum height data. Remington et al. (1994) also used the Weibull distribution to model the distribution of Agropyron cristatum biomass as 12 different distributions depending on the intensity of grazed pasture (no grazing, lightly grazed, moderately grazed and heavily grazed) in each of three months (June, August and October). In this case, the Weibull distribution was also capable of describing 100% of the observed biomass distributions, and the Kolmogorov-Smirnov test (α=5 %) was used to test the goodness of fit for 25 % (three distributions) that did not pass the Anderson-Darling test for normality (α=5 %).
In future studies including more treatments, it would be possible to relate the three parameters of the Weibull function to the main variables that affect aerial DM (fertilizer dose, soil depth, slope, climate variables) and to use regression models to predict the Weibull parameters and thus the biomass distributions. When the function is fitted by the method of moments, these variables can be related to the first and the second moments of the distributions (mean and variance, respectively).
Conclusions
The Weibull distribution is highly flexible for describ the aerial dry matter distributions of early first harvest of Lolium multiflorum in Asturias. The skewness and kurtosis of the distributions differed as a result of the different amounts of N supplied, site and meteorological conditions and sampling locations. The parameters of the Weibull distribution can be easily related to the values of the described variable (aerial dry matter) and are simple to interpret.
