INTRODUCTION
Irrigated agriculture is essential in some regions of Brazil for maintaining high crop yields due to the irregular precipitation distribution. The conventional sprinkler irrigation system is widely used in Brazil (^{Alves et al., 2017}). However, this system does not operate at its maximum efficiency under adverse weather conditions, such as high temperature, low air humidity, and high wind speed, wasting water, bad fertilizers solubilization, and instability in electrical network (^{Molle et al., 2012}; ^{Sheikhesmaeili et al., 2016}).
For irrigation management, irrigation efficiency (Ei) is determined by application (E_{A}) and distribution (Ed) efficiencies and is calculated as the product of these two variables. Knowledge of Ei is of paramount importance for proper irrigation management because the gross water depth to be applied is adjusted from this variable (^{Kifle et al., 2017}).
In sprinkler irrigation systems, part of the water depth applied by emitters does not reach the soil surface and/or the shoot occupied by crops. This portion of water represents evaporation and wind drift (WDEL), which, in turn, are expressed as the percentage of the gross applied volume lost in a given irrigation event (^{Andrés & Cuchí, 2014}). The ratio between the water depth that reached the irrigation target (collected water depth) and applied water depth is defined as application efficiency (E_{A}). WDEL is obtained by subtracting E_{A} by the unit value.
Evaporation losses depend on the relative humidity, wind speed, ambient temperature, emitter working pressure, emitter installation height fromthe soil surface, and droplet diameter. Wind drift, on the other hand, depend mainly on wind speed, droplet diameter, and the distance traveled by droplets until they reach the shoot of crops or soil surface (^{Maroufpoor et al., 2017}).
The quantification of WDEL is considered of high relevancein both environmental and economic aspects. However, estimating these losses separately is a rather complicated task due to the difficulties of the methodologies required for their measurements (^{Beskow et al., 2011}). Several studies have been conducted in different regions of the world with the purpose of modeling or evaluating some existing models for WDEL estimation in sprinkler irrigation systems (^{Colombo et al., 2015}; ^{Faria et al., 2012}; ^{Stambouli et al., 2013}).
Another factor necessary to be considered in sprinkler irrigation systems assessments is the Ed. To calculate this, uniformity coefficients need to be used. Several methods can be found in the literature, but the Christiansen uniformity coefficient is the most widely used for conventional sprinkling (^{Cavero et al., 2016}). Ed is also influenced by weather conditions and characteristics of the irrigation equipment.
According to ^{Colombo et al. (2015)}, knowledge of the performance of an irrigation system, especially regarding the uniformity of distribution of the applied water depth, is essential for making decisions that allow the rational use of water, energy, and fertilizers. Thus, knowing the influence of weather variables is essential to predict or estimate irrigation uniformity, and even to identify the best time for the operation of a sprinkler irrigation system.
Therefore, this study aimed: i) to determine the distribution efficiency using the Christiansen uniformity coefficient (CUC) in a sprinkler irrigation system under different weather conditions and working pressures; ii) to compare the mean, median, and cumulative CUC values; and iii) to evaluate the predictive capacity of four application efficiencyestimationmodels.
MATERIAL AND METHODS
Location and characterization of the area
The study was conducted in an experimental irrigation area of the Department of Agricultural Engineering of the Federal University of Viçosa, Viçosa, MG, Brazil, with geographical coordinates of 20°45′ S and 42°51′ W, and an altitude of 651 m. The local climate is Cwa, according to the KöppenGeiger classification, i.e., a humid subtropical climate with dry winter and hot summer (^{Alvares et al., 2013}).
This study consisted of the evaluation of an irrigation system with six midisectorial or 360° sprinklers with a 4 mm nozzle. Sprinklers were spaced at 11 × 8 m to suit the experimental area configuration, supported by a 1.7 m rod above the soil surface and recommended working pressure for system operation of 196.13 kPa.
Distribution uniformity evaluation
Collectors were arranged between six sprinklers operating simultaneously. Collectors were of the Fabrimar^{®} brand and were installed equidistant at 2 m, resulting in 44 collectors (Figure 1). A rod was used to suspend the collector at the height of 70 cm from the soil surface, following the methodology proposed by ^{Merriam & Keller (1978)}.
The data on mean air temperature, relative humidity, solar radiation, and wind speed were collected throughout the testing period using an Irriplus E5000^{®} automatic weather station, located 30 m from the experimental area.
Eighty field tests with 60 minute durations were performed under different weather conditions (Figure 4) from April to July 2017. The reading of weather elements was performed every 20 minutes, and their mean was used to represent the weather conditions during irrigation.
Water depth was measured at the end of each test on each collector using a 15 mm Fabrimar^{®} graduated beaker. The Christiansen uniformity coefficient (CUC) (^{Christiansen, 1942}) was used to calculate the uniformity of distribution, according to [eq. (1)].
Where:
CUC is the Christiansen uniformity coefficient (%);
N is the number of observations;
D_{i} is the water depth applied at the ith point on the soil surface (mm), and
D_{m} is the mean applied water depth (mm).
The mean and median CUC values were calculated after CUC has been determined for each evaluation. Cumulative CUC was also determined by summing the 80 values of water depth measured in each collector. The calculation was performed with these cumulative values, according to Equation (1).
The mean, median, and cumulative CUC were determined for every ten successive evaluations to verify the sensitivity of differences in a few numbers of evaluations, resulting ineight replications in a randomized block design. These data were analyzed by the Tukey test at a 0.05 significance level.
Evaporation and wind driftevaluation
Before starting the evaluations, the flow versus working pressure curve was determined for the sprinkler used in this study (Figure 2A). For this, pressures ranging from 78 to 333 kPa were used. Thus, a linear regression model was adjusted to predict the flow as a function of working pressure.
Working pressure readings of the six sprinklers were taken in all evaluations at the beginning, middle, and end of the irrigations. Schrader valves (coarse nozzle) were installed in the riser pipe at 5 cm below the sprinkler to facilitate working pressure readings. The flow rate and, thus, the water depth applied in each evaluation were obtained with the mean pressure using the regression equation shown in Figure 2A.
The evaluation of evaporation and wind drift (WDEL) or application efficiency (E_{A}) was carried out usingtwenty evaluations with working pressures closer to 196.13 kPa (pressure recommended by the manufacturer). This procedure was used to standardize the droplet size, as operating pressures of the irrigation system showed high variations during the evaluation period (Figure 2B). Thus, it was also possible to correlate them better with input variables based on weather conditions of prediction models.
Pressure variation is related to the instability of the electrical network and water quality used in irrigation, as it had suspended solids. These impurities caused disc filter clogging and, consequently, reduced the working pressure during the test. Backwash was performed manually at the interval of each evaluation.
The arithmetic means water depth applied by sprinklers and the water depths measured in the collectors were used to calculate the evaporation andwind drift. Thus, WDEL was calculated using [eq.(2)].
Where:
WDELis the evaporation and wind drift (%),
D_{applied} is the applied water depth (mm), and
D_{collected} is the collected water depth (mm).
However, data processing was based on E_{A}. For this, WDEL values were subtracted from 100.
In addition to measurements, E_{A} was estimated by different models as a function of weather conditions in the tests. The mathematical models used in the present study were proposed by ^{Keller & Bliesner (1990)}, ^{Playán et al. (2005)}, ^{Tarjuelo et al. (2000)}, and ^{Beskow et al. (2008)}, represented, respectively, by equations shown in Table 1.
Author  Empirical equation 

Keller & Bliesner (1990) 

Tarjuelo et al. (2000) 

Playán et al. (2005) 

Beskow et al. (2008) 

E_{A} is the application efficiency (%); (e_{s}−e_{a}) is the vapor pressure deficit (kPa); WP is the sprinkler working pressure (kPa); U is the mean wind speed (m s^{−1}); T is the air temperature (°C); and RH is the relative air humidity (%).
The vapor pressure deficit (Δe) and vapor saturation pressure (e_{s}) were obtained by eqs (3) and (4):
Where:
Δe is the vapor pressure deficit (kPa);
e_{s} is the vapor saturation pressure (kPa);
T is the temperature (°C), and
RH is the relative humidity (%).
The following statistical indicators were used to verify the performance of the models usingthe observed values: root mean square error (RMSE), mean bias error (MBE), in addition to the parameters recommended by ^{Legates & McCabe (1999)}, Willmott agreement index (d) and mean absolute error (MAE). The magnitude of the coefficient of determination (R^{2}) and Pearson correlation coefficient (r) was used to correlate the measured with the estimated data.
RESULTS AND DISCUSSION
The Christiansen uniformity coefficient (CUC) values were increasingly ordered and associated with the sprinkler working pressure, ranging from 66.51 to 92.04% and 44.1 to 196 kPa, respectively (Figure 3).
The 72 evaluations of the irrigation system through CUC showed results higher than 80%, and one of the main factors influencing the distribution uniformity was sprinkler working pressure (Figure 3). In contrast, due to the characteristics of the study area and range of weather variation duringthe study, weather elements affected the distribution uniformity with less intensity (Figure 4).
The variable CUC had a low Pearson coefficient (r), i.e., a weak correlation with all the weather elements. The water distribution of an irrigation system mainly depends on the spacing, type, size, internal design, and working pressure of emitters (^{Zhu et al., 2015}).
The variation in the uniformity of water distribution evaluated at different times, working pressures, and weather conditions have been reported in other studies. ^{Li et al. (2015)} evaluated a sprinkler irrigation system in which the working pressure ranged from 200 to 350 kPa and found CUC values from 73.27 to 81.11%. ^{Robles et al. (2017)}, on the other hand, studied the effect of wind speed on distribution uniformity and found CUC values of 89 and 67% for low and high wind speed conditions, respectively. ^{Justi et al. (2010)} evaluated 25 irrigation events and found mean and maximum CUC values of 79.72 and 89.45%, respectively.
^{Sheikhesmaeili et al. (2016)} suggested being acceptable CUC values of at least 80% in sprinkler irrigation systems. Most evaluations presented CUC values above 80% (Figures 3 and 4), but values below 80% can be found due to working pressure and wind speed variations, as observed in this and other studies. This behavior discourages the use of only one evaluation to represent the uniformity of water distribution in conventional sprinkler irrigation systems.
A reference value needs to be adopted when an irrigation system has several CUC values. This value can be used to determine theirrigation efficiency and, therefore, provide adequate irrigation management, turning the net water depth into gross. The CUC_{mean} of 84.66% shown in Figure 5 represents the mean of all evaluations. CUC_{median} had a value of 86.04%, which is higher than the CUC_{mean}. In contrast, the CUC_{cumulative}, which represents CUC considering the sum of water depths distributed in the area over time, presented a value of 90.64%, which is higher than the others are.
Water distribution in the irrigation area varies over time, significantly changing its uniformity when considering multiple irrigations. Usually, a point of the area to be irrigated may receive water depths equal to, lower than or higher than that of the mean at different irrigation events because of the random pattern of the collected precipitation, which is influenced by some weather parameters and/or working pressure. Therefore, the same point that received a low irrigation depth and was in deficit compared to other areas can receive a water depth higher than the mean in the next irrigation event, partially or totally supplying the deficit that occurred in the first irrigation. Thus, when CUC_{mean} or CUC_{median} are adopted, it is disregarded that a region that receives different water depths may have in time a higher and more representative CUC of the area, especially in areas of the semiarid, which receive high and more frequent water depths.
Figure 6 shows three distinct distribution uniformities for an irrigation system. Evaluations 1 and 2 presented CUC values of 84.96 and 86.06%, respectively. The CUC_{cumulative} of all 80 evaluations was 90.64%.
Among the various causes that affect the distribution uniformity of a sprinkler irrigation system, special attention should be given to working pressure and wind speed and direction (^{Li et al., 2015}). Variation of these factors is undesirable, as they tend to reduce the distribution uniformity coefficient. Weather factors are dynamic, even choosing less windy times is not a guarantee of even distribution. Assuming that wind speed and direction vary, it is possible to understand that a conventional sprinkler irrigation system operating at different times and low irrigation frequency can redistribute water in the area to be irrigated, thus promotinga better irrigation uniformity over time, as shown by the CUC_{cumulative} (Figure 6).
Water redistribution by irrigation systems occurs in different applications or within the same application depending on the duration of the irrigation. It would be a new way of understanding the dynamics of irrigation and CUC use, often underestimated. In this sense, it has already been proposed to monitor the uniformity of water distribution in the soil profile, as it can redistribute water and, thus, present a CUC closer to reality (^{Rocha et al., 1999}).
The CUC_{cumulative} values for every ten evaluations ranged from 88.50 to 89.39%. On the other hand, the CUC_{mean} comprised values from 83.25 to 86.07%, and CUC_{median} values ranged from 83.91 to 87.51%. Based on the mean of the eight groups of evaluations (replications), the CUC_{cumulative} was higher than CUC_{median} and CUC_{mean} (Table 2). CUC_{mean} was the variable that most underestimated distribution uniformity.
Evaluation  CUC_{cumulative}  CUC_{mean}  CUC_{median} 

1–10  88.78  85.13  85.82 
11–20  88.94  84.09  87.51 
21–30  88.92  86.07  86.72 
31–40  89.39  83.53  83.91 
41–50  88.72  84.29  86.29 
51–60  88.82  85.42  87.20 
61–70  88.95  85.46  85.62 
71–80  88.50  83.25  84.21 
Mean  88.88a  84.66c  85.91b 
Means followed by the same lowercase letter for different CUC determinations do not differ from each other by the Tukey test at 0.05 significance level.
Under practical conditions, conducting eighty evaluations to propose a more representative CUC of the area becomes unfeasible. Therefore, it was suggested in Table 2 to perform at least ten evaluations using the CUC_{cumulative}, as CUC_{mean} and CUC_{median} underestimate distribution uniformity, besides presenting higher variations between evaluations.
Application efficiency (E_{A}) was determined, ordered in increasing order, and correlated by Pearson's coefficient (r) with weather elements (Figure 7).
The higher the wind speed, the lower the application efficiency, thus promoting higher wind drift (r = −0.70). The vapor pressure deficit, solar radiation, and relative humidity directly correlated with E_{A}, with values of −0.63, −0.85, and 0.62, respectively. Thus, these weather elements increased evaporation and wind drift (Figure 7). Evaporation and wind drift were also reported by ^{Cavero et al. (2016)} for the day and night periods. The positive correlation between weather data with E_{A} is paramount to search for models that best fit a particular location (^{Saraiva et al., 2013}).
Table 3 shows the statistical indices for application efficiency (E_{A}) estimations, as proposed by Keller & Bliesner, Playán, Tarjuelo, and Beskow in relation to the values measured in the field.
Model  RMSE  MAE  d  MBE  r 

Keller  15.98  14.85  0.35  14.64  0.72 
Playán  7.21  6.49  0.62  5.56  0.74 
Tarjuelo  8.93  8.00  0.52  7.28  0.83 
Beskow  2.98  2.54  0.94  0.77  0.90 
The results show that the Beskow method was the most efficient in the estimation of E_{A}, which is evidenced by lower values of RMSE, MAE, and MBE, besides higher d and r. On the other hand, the other evaluated methods presented low performance, with high RMSE and MAE values. ^{Beskow et al. (2008)}, analyzing the Playán and Tarjuelo models to estimate E_{A}, worked with a single sprinkler operating and several sprinklers operating simultaneously, and found unsatisfactory adjustments.
In addition to statistical indices, the analysis in Figure 8 shows the distribution of the measured and estimated E_{A} values by each method. Except for the Beskow method, all others had unsatisfactory performance in relation to the measured values, showing a tendency to overestimate them, which is also indicated by the high MBE values. The Beskow method presented good precision (R^{2} = 0.803) and accuracy (a = 18.939 and b = 0.781), but there was a small dispersion of data due to systematic and random errors.
The Beskow method showed better agreement, possibly due to the regional approximation and similarity of weather conditions of the present study. This method was developed under the conditions of Lavras, MG, Brazil (^{Beskow et al., 2008}), and the other models in the United States (^{Keller & Bliesner, 1990}) and Spain (^{Playán et al., 2005}; ^{Tarjuelo et al., 2000}). ^{Saraiva et al. (2013)} found similar results.
Figure 8 also shows that it is important to draw attention to the fact that the valuesestimated by the Keller & Bliesner method presented a very small variation. This method presented low sensitivity to weather variations, which possibly corroborated for this model to present the worst performance.
The ^{Keller & Bliesner (1990)} model underestimated evaporation and wind drift and/or overestimated application efficiency (Figures 8 and 9). ^{Faci et al. (2001)} observed similar resultsin working with spraytype sprinklers.
As shown in Table 3 and Figures 8 and 9, the used models tended to overestimate the observed field data, and one of the factors that may be related to this application efficiency overestimation is the nozzle size used for modelgeneration. The sprinkler nozzle used in the present study had 4 mm. According to ^{Beskow et al. (2008)}, smaller nozzles promote higher spraying of dropletsand, thus, have a larger area per unit of mass, which may result in higher evaporation loss and susceptibility to drift by the wind.
Even finding a model that fits well with field conditions, it requires caution when using a model to estimate E_{A} of a conventional sprinkler irrigation system. It is always important to analyze model limitations, such as nozzle diameter, working pressure, and jet slope used to generate the predictive model (^{Beskow et al., 2008}). Sprinkler height may also favor E_{A}, as the longer the riser pipe, the longer the droplet will be exposed to the environment (^{Faci et al., 2001}; ^{Playán et al., 2005}).
CONCLUSIONS
The distribution uniformity given by the Christiansen uniformity coefficient had high variation during the experimental period (66.51 to 92.04%). The highest and lowest CUC for the experimental area was the CUC_{cumulative} and CUC_{mean}, respectively. The Beskow model was the best alternative for predicting application efficiencies. The Keller & Bliesner model is not recommended for the study conditions to predict application efficiency, as it presented low sensitivity to input parameters.