**Estimation of Precipitation Evolution from Desert to Oasis Using Information Entropy Theory: A Case Study in Tarim Basin of Northwestern China**

### **Xiangyang Zhou 1,2, Zhipan Niu 2,3,\* and Wenjuan Lei <sup>4</sup>**


Received: 25 August 2018; Accepted: 10 September 2018; Published: 15 September 2018

**Abstract:** The cold-wet effect of oasis improves the extreme natural conditions of the desert areas significantly. However, the relationship between precipitation and the width of oasis is challenged by the shortage of observed data. In this study, the evolution of annual precipitation from desert to oasis was explored by the model establishment and simulation in Tarim Basin of northwestern China. The model was developed from the principle of maximum information entropy, and was calibrated by the China Meteorological Forcing Dataset with a high spatial resolution of 0.1◦ from 1990 to 2010. The model performs well in describing the evolution of annual precipitation from the desert to oasis when the oasis is wide enough, and the *R*<sup>2</sup> is generally more than 0.90 and can be up to 0.99. However, it fails to simulate the seasonal precipitation evolution because of the non-convergence solved by nonlinear fitting and the unfixed upper boundary condition solved by the least square method. Through the simulation with the parameters obtained from the nonlinear fitting, the basic patterns, four stages of precipitation evolution with the oasis width increasing, are revealed at annual scale, and the current stages of these oases are also uncovered. Therefore, the establishment of the model and the simulated results provide a deeper insight from the perspective of informatics to understand the regional precipitation evolution of the desert–oasis system. These results are not only helpful in desertification prevention, but also helpful in fusing multisource data, especially in extreme drought desert areas.

**Keywords:** evolution of precipitation; model simulation; information entropy theory; desert–oasis areas; Tarim Basin

### **1. Introduction**

Oasis serves to improve the extreme natural conditions of the arid regions by affecting the regional hydrometeorological factors [1–7], and the oasis–desert interactions are important for the stable co-existence of oasis and desert ecosystems [8] and water resources management [9]. Hence, quantifying the spatial evolution of precipitation with the oasis width increase is crucial to the regional eco-environmental security in the arid areas.

In fact, the spatial evolutions of precipitation are affected by the cold-wet effect of oasis. The cold effect is mainly resulted from the greater absorption of latent heat through evapotranspiration [10,11] and the higher surface albedo of the vegetation than the desert surface [3]. The wet effect is because of more water vapor source from the evapotranspiration of oasis [3,4]. As a result, the local water vapor can account for up to 20% of the precipitation in arid and semiarid regions [12,13] and 20–50% in humid regions [14–17]. In fact, the extreme arid environment leads to the much larger difficulties of data observation. Many studies are based on the field observation at a low spatial resolution [1–6,12–17], although some multiscale datasets are available in specific areas such as Heihe Basin located in the northwestern China [18]. Thus, generally, exploring the relationship between the precipitation and oasis width quantitatively is still challenged by the resolution of data.

Merging multisource data is a useful tool to solve the problem of data shortage and low resolution, especially fusing the calibrated remote sensing data with the land surface observed data. After the early attempts in the 1980s [19,20], the resolution of the assimilation data has improved substantially. For example, as for the ocean system, the current resolution is about 1/4–1/6 degrees [21,22]. In China, the current resolution is 0.1 degrees spatially [23,24], and its good spatial continuity has been demonstrated by several studies. These studies have uncovered the spatial distribution of precipitation and evaporation in the Nam Co basin in Tibetan Plateau [25,26], and the spatial patterns of permafrost based on the climatic factors of this dataset [27]. The higher resolution of fusion dataset provides an alternative source to analyze the spatial evolution of precipitation.

Entropy, combining the micro and macro status, has been widely used to simulate the evolution of a system. Information entropy is a better measure of variability than the variance and coefficient of variation when the probability distribution is not symmetric [28,29], because entropy may be related to higher order moments of a distribution [30]. Hence, it has been employed to evaluate the complexity of typical chaos [31] and uncertainty of precipitation and the potential water resources ability at different scales [32–35]. Spatially, entropy has been widely employed to evaluate the spatial distribution of rainfall gauge net [36,37], quantify the soil water dynamic processes [38–40], explore the multifractal generation [41,42] and simulate the shallow water and solution transportation based on the molecule collision [43,44]. These applications above indicate that entropy is not only a good index to describe the uncertainty of a time series, but also a good method to describe the spatial continuity.

Therefore, the objective of this study was to quantitatively explore the evolution of precipitation from desert to inner oasis based on the Tarim Basin, northwestern China. A model describing precipitation evolution was developed from the information entropy theories, and then the parameters of model were fitted by the China Meteorological Forcing Dataset with the spatial resolution of 0.1 degrees. Through the simulation, the typical precipitation evolution patterns weare revealed, and the current stage of these oases were also identified. The establishment of the model and the simulated results provide a deeper insight from the perspective of informatics to understand the regional precipitation evolution of the desert–oasis system. These results are useful for the desertification prevention and multisource data fusion, especially in extreme drought desert areas.

### **2. Study Area, Data and Methodology**

### *2.1. Study Area*

The Tarim Basin, covering an area of approximately 560,000 km2, is located in the south of Xinjiang Province in northwestern China [45,46], as shown in Figure 1. The Taklamakan Desert, which formed at least 5.3 Ma years ago, is in the center of the Tarim Basin [47]. The total area of the oasis is approximately 103,900 km<sup>2</sup> [11,48]. In the desert region, the annual precipitation is from less than 15 to 60 mm, increasing from the east to the west [49], and the increase rate is 10.15, 6.29, and 0.87 mm per decade in the mountain, oasis and desert areas, respectively, based on the observed data from 1960 to 2010 [50]. However, the annual potential evaporation is over 3200 mm based on the evaporation from water surface, and most of the basin is a generally unsuitable or extremely unsuitable area for human settlement [51]. Currently, the irrational reclamation of land and overuse of natural resources have led to the destruction of vegetation and shrinkage of the water area [52–54] and have threaten the security and sustainable development of oases [48].

**Figure 1.** Spatial distribution of oasis in Tarim Basin, obtained from Google Earth. The linkage for Google's permissions is: https://www.google.ca/permissions/geoguidelines.

### *2.2. Data*

In this study, the China Meteorological Forcing Dataset, developed by the Data Assimilation and Modeling Center for Tibetan Multispheres, Institute of Tibetan Plateau Research, Chinese Academy of Sciences [23,24], was used to test the performance of the model and to calibrate the parameters of the precipitation evolution form the desert toward oasis. The dataset was produced by merging a variety of data sources. More details on the dataset are given in the user's guide of the Dataset [55]. The spatial and temporal resolutions of the dataset are 0.1 degrees and 3 h, respectively, and the data length is 21 years (1990–2010). Based on the dataset, the spatial distributions of annual precipitation in Tarim basin are shown in Figure 2.

**Figure 2.** Spatial distribution of annual precipitation in Tarim Basin based on The China Meteorological Forcing Dataset.

### *2.3. Information Entropy and Principle of Maximum Entropy*

Assuming the probability density function of a continuous variable *x* is expressed by *f*(*x*), and the information entropy is calculated as [56,57]:

$$H(\mathbf{x}) = -\int\_{a}^{b} f(\mathbf{x}) \ln(\mathbf{x}) d\mathbf{x} \tag{1}$$

$$\int\_{a}^{b} f(\mathbf{x})d\mathbf{x} = 1\tag{2}$$

where *a* and *b* and the lower and upper boundaries of the variable *x*, respectively.

In addition, the known information of the variable represented by *gi*(*x*), such as mean and standard deviation, can be given by the following formula:

$$\int\_{a}^{b} g\_i(\mathbf{x}) f(\mathbf{x}) = N\_i \tag{3}$$

Hence, the general solution of *f(x)* can be obtained by solving the conditional extreme values using the method of Lagrange multipliers: the objective function is the maximum information entropy, and the constraint is the known information in Equation (3). More details about the mathematical derivation is presented in the literature [40,58]. The general form of *f(x)* is given as:

$$f(x) = \mathfrak{e}^{\frac{\mathfrak{n}}{\sum\_{i=1}^{n} g\_i(x)}} \tag{4}$$

### **3. Establishment of the Model**

Assuming the precipitation at the boundary area between oasis and desert is *P*0, the precipitation evolution from desert toward the oasis is represented by the integration of a function *p*(*z*), as given in Equation (5):

$$P(z) = P\_0 + \int\_0^Z p(z)dz\tag{5}$$

where *z* represents the distance from the desert–oasis boundary to a certain point of the oasis. *P(z)* represents the precipitation at the location with distance *z*. Equation (5) can also be transformed into the following form:

$$P(z) - P\_0 = \int\_0^Z p(z)dz\tag{6}$$

Usually, *P(z)* reaches a constant approximately when the oasis is wide enough. Here, Equation (6) can be written as:

$$\int\_0^{+\infty} p(z)dz = \mathbb{C} \tag{7}$$

where the constant *C* represents the maximum increment of precipitation with the increase of oasis width. Dividing both sides of the equation by the constant *C*, Equation (7) becomes:

$$\frac{1}{C} \int\_0^{+\infty} p(z) dz = 1\tag{8}$$

Hence, the item of *p*(z)/*C* can be considered as a probability density function, and its information entropy is calculated by Equations (9) and (10):

$$H = -\int\_0^Z \frac{1}{\mathbb{C}} p(z) \ln \frac{p(z)}{\mathbb{C}} dz \tag{9}$$

*Water* **2018**, *10*, 1258

$$H = -\frac{1}{\mathbb{C}} \int\_0^Z p(z) \ln p(z) dz + \frac{1}{\mathbb{C}} \ln \mathbb{C} \tag{10}$$

As for a specific oasis, the spatial distribution can be considered as stable approximately in a mid-long term. Correspondingly, the arithmetic mean and geometric mean of precipitation evolution can be considered as constant, as given by Equations (11) and (12), respectively.

$$\int\_0^\infty \frac{z}{\mathcal{C}} p(z) dz = \mu\_z \tag{11}$$

$$\int\_0^\infty \frac{\ln(z)}{C} \mathbf{p}(z) dz = \nu\_z \tag{12}$$

where *μ<sup>z</sup>* and *ν<sup>z</sup>* represent the arithmetic mean and geometric mean. Here, the Lagrange function is obtained based on the information entropy of precipitation evolution with the constraints of Equations (8), (11) and (12), as given by Equation (13).

$$\begin{split} L &= -\frac{1}{\mathsf{C}} \int\_{0}^{Z} p(z) \ln p(z) dz + \frac{1}{\mathsf{C}} \ln \mathsf{C} + \lambda\_{0} (\int\_{0}^{Z} \frac{1}{\mathsf{C}} p(z) dz - 1) \\ &+ \lambda\_{1} (\int\_{0}^{Z} \frac{z}{\mathsf{C}} p(z) dz - \mu\_{z}) + \lambda\_{2} (\int\_{0}^{Z} \frac{\ln(z)}{\mathsf{C}} p(z) dz - \nu\_{z}) \end{split} \tag{13}$$

Let *<sup>∂</sup><sup>L</sup> <sup>∂</sup>p*(*z*) = 0, then the general solution of *p*(*z*)*/C* is obtained by solving the conditional extreme values, as given in Equation (14):

$$p(z)/\mathbb{C} = e^{\lambda\_0 - 1 + \lambda\_1 z + \lambda\_2 z \ln z} \tag{14}$$

Equation (14) can also be written as:

$$p(z)/\mathbb{C} = \epsilon^{\lambda\_0 - 1} \epsilon^{\lambda\_1 z} z^{\lambda\_2} \tag{15}$$

Let *<sup>λ</sup>*<sup>2</sup> *<sup>=</sup> <sup>α</sup>* <sup>−</sup> 1, *<sup>λ</sup>*<sup>1</sup> *<sup>=</sup>* <sup>−</sup>1*/β*. It is easy to obtain that *<sup>e</sup><sup>λ</sup>* 0 <sup>−</sup><sup>1</sup> *<sup>=</sup>* <sup>−</sup>*α/*Γ*(α)*. Thus, the probability density function *p*(*z*)*/C* can also be expressed as:

$$p(z)/\mathbb{C} = \frac{1}{\Gamma(a)\beta^a} z^{a-1} e^{-z/\beta} \tag{16}$$

Substituting Equation (16) into Equation (5), the evolution of precipitation with the oasis width increase is:

$$P(z) = P\_0 + \mathcal{C} \int\_0^z \frac{1}{\Gamma(a)\beta^a} z^{a-1} e^{-\frac{z}{\beta}} dz \tag{17}$$

In fact, the right-hand term of Equation (16) is the probability density function of a gamma distribution. Therefore, Equation (17) shows that the precipitation evolution model is based on the scaled cumulative distribution function of a gamma distribution with an initial precipitation. In other words, the model is based on the linear transformation of the cumulative distribution function of a gamma distribution. The parameters of the model include the shape factor *α*, scale factor *β*, zooming constant *C* and initial precipitation *P*<sup>0</sup> at the boundary between desert and oasis.

Assuming the initial precipitation *P*<sup>0</sup> at the boundary between desert and oasis is 50 mm/year and the zooming constant *C* is 50, the illustration of the precipitation evolution are shown in Figure 3 with different shape and scale factors.

**Figure 3.** Illustration of precipitation evolution from desert to inner oasis. Assuming the initial precipitation *P*<sup>0</sup> at the boundary between desert and oasis is 50 mm/year and the zooming constant *C* is 50, the precipitation evolution is shown with different shape factor and different scale factor.

### **4. Calibration of the Model and the Simulated Results**

### *4.1. Calibration of the Model*

### 4.1.1. Performance of the Model

In this study, four main oases were used to test the performance of the model. These oases are Kashi Oasis located in the western basin, Akesu Oasis in the northern basin, Kuerle Oasis in the northeastern basin and Hetian Oasis in the southwestern basin. The evolution routes of precipitation from desert to oasis are shown in Figure 4. Generally, these routes are located in the middle of these oases.

**Figure 4.** Selected routes of precipitation transition from the desert to oasis in different location of the Tarim Basin, obtained from Google Earth. The permission of Google to use Google Earth Periodicals is given at: https://www.google.ca/permissions/geoguidelines.

Following the routes shown in Figure 4, the evolution of precipitation from desert to oasis is obtained, and the results are scattered in Figure 5. Here, the parameters of the model are obtained by nonlinear fitting. The correlation coefficient is used to evaluate the performance of the model given by Equation (17). The fitted results and the square of correlation coefficient, *R2*, of the four selected routes are also shown in Figure 5.

The model performs well in three oases: Kashi Oasis, Akesu Oasis and Kuerle Oasis, corresponding to *R*<sup>2</sup> of 0.99, 0.93 and 0.97, respectively. However, the model does not perform well in Hetian Oasis and the *R*<sup>2</sup> is 0.47. To improve the accuracy trend of precipitation evolution, the model should be calibrated further for the Hetian Oasis.

**Figure 5.** Performance of the model simulating precipitation from desert to oasis in different areas of Tarim Basin.

### 4.1.2. Calibration of Model for Hetian Oasis

Through combining the spatial distribution of oasis and the local hydrological cycle, two main reasons were found to lead to the poor performance of the model. The first is the much smaller width of Hetian Oasis. As shown in Figure 5, the Hetian Oasis is about 55 km wide, while Kashi Oasis is 200 km, Akesu Oasis is 120 km and Kuerle Oasis is 80 km. The second reason is that the prevailing wind direction of the Hetian Oasis is opposite to the cold effect of oasis [59–61]. Our previous study revealed that the opposite direction leads to water vapor from the local evapotranspiration accumulating over the buffer zone located in the down wind direction of the oasis [49], as shown in Figure 6a. Hence, the evolution of precipitation in the Hetian Oasis should be divided into two phases: the buffer zone

with abnormal high precipitation and the regular pattern of the precipitation evolution after the buffer zone, as shown in Figure 6b.

By overlapping the spatial distribution of water vapor content and oasis layout, it was found that the first four grids should be removed to calibrate the model. After their removal, the calibrated model performs much better, and the *R*<sup>2</sup> increased from 0.47 to 0.87, as shown in Figure 6b.

**Figure 6.** Spatial distributions of water vapor in Tarim Basin and the calibrated model in Hetian.

### 4.1.3. Parameter Inversion

Based on the results obtained by nonlinear fitting, the parameters of the model for all four oases are obtained, as shown in Table 1. According to Equation (17) and Figure 3, the meanings of these parameters are clear. *P*<sup>0</sup> is the initial value that represents the precipitation at the boundary area between oasis and desert. Here, AKesu Oasis exhibits largest value, followed by Kashi, Kuerle and Hetian respectively. The zooming constant, *C*, represents the maximum potential increment of precipitation with the oasis width increase. The results of zooming constant *C* suggest the maximum promotion on local precipitation in Kashi, middle level in AKesu and Kuerle, and the minimum promotion in Hetian Oasis. Shape and scale factors represent the increase rate of the precipitation. A larger value is usually accompanied with a smaller increase rate, and vice versa (Figure 3).

**Table 1.** Performance of the model and the parameters inversion for precipitation evolution from desert to oasis in different locations of the Tarim Basin.


### 4.1.4. Influence of the Local Terrain

Apart from the cold-wet effect of oasis, the significant uplift of the terrain would lead to a sharp decrease in temperature and result in a substantial increase in precipitation at the local scale. As given in our previous study [49], the increment of the elevation is about 80–100 m. Meanwhile, the widths of Kashi, Akesu, Kuerle and Hetian Oases are approximately 210, 120, 80 and 55 km, respectively, as shown in Figure 5. This indicates that the increase rate is ordered by: Hetian > Kuerle > Akesu > Kashi. However, this trend is opposite to the increment and increase rate of precipitation from desert to oasis. Hence, the increase of precipitation from desert to oasis is not dominated by the local terrain.

### 4.1.5. Performance of the Model at Seasonal Scale

The performance of the model at seasonal scale was also analyzed. The first problem confronted is that the result does not converge when solved by nonlinear fitting using MATLAB (MathWorks, Natick, MA, USA). Here, the results are based on least square method and the results are shown by Figure 7. Generally, the model seems to perform well with several exceptions, including the winter of Kuerle and Hetian Oases, and the summer of Hetian Oasis. The reason can be attributed to the transportation of water vapor. As discussed above, the wind blowing from the oasis to the desert would decrease the cold-wet effect of the oasis substantially [49].

However, the second problem is that the solutions of the model based on the least square method are substantially affected by the upper boundary conditions. Two examples based on different upper boundary conditions are given to illustrate this problem. As shown in Table 2, the initial precipitation *P*<sup>0</sup> is not affected by the upper boundary conditions. The other three parameters, however, are influenced significantly. For example, the difference of the parameter C estimated for the summer of the Kuerle Oasis can be up to 140 mm, although the *R*<sup>2</sup> is 0.99 for both solutions. Here, the failure of the model may be resulted from the resolution of the data or the more complex water vapor transportation. Hence, further studies should be carried out to explore the simulation of the season precipitation evolution from desert to the oasis.


**Table 2.** Comparisons on the sensitivity of parameter with different upper boundary conditions (UBC) at seasonal scale. *R2* are indicated in italics and Bold.

**Figure 7.** Performance of the information entropy based model at seasonal scale: (**a**) Kashi Oasis located in the western basin; (**b**) Akesu Oasis located in the northern basin; (**c**) Kuerle Oasis in the northeast basin; and (**d**) Hetian Oasis in the southwestern basin.

### *4.2. Simulated Results*

Based on the performance of the model at different scales, the parameters are reasonable at annual scale. The evolution of precipitation with the increase of oasis width was calculated by Equation (17) using the parameters given in Table 1. The results are shown in Figure 8, from which the basic patterns of precipitation evolution are revealed and the current stage of these oases are also obtained. More details on the two aspects are as follows.

### 4.2.1. Basic Patterns of Precipitation Evolution

According to the simulated results (Figure 8), it could be found that the basic patterns of precipitation evolution exhibit four stages: a small increase at first, then a much faster increase after a threshold of oasis width, followed by a small increase rate again when the oasis width reaches a relatively large width, and finally the increase rate decreases to close to zero when the oasis is wide enough. The threshold values of the four stages are also obtained (Table 3).

**Figure 8.** The simulated precipitation evolution from desert to oasis with the oasis width increase.


**Table 3.** Typical patterns of precipitation evolution with the increase of oasis width in different oases of the Tarim Basin based on the simulated results.


### 4.2.2. Current Stages of Different Oases

The widths of Kashi, Akesu, Kuerle and HetianOases are 200 km, 135 km, 80 km and 60 km, respectively, as shown in Figure 5. By contrast with Figure 8, it can be obtained that the four oases belong to different stages that are manifested by the following aspects.


### **5. Discussions**

### *5.1. The Simulated Results for Desertification Prevention*

In the Tarim Basin, affected by the substantial increase of local human activities, the destruction of oasis environments is increasing, resulting from the irrational reclamation of land and overuse of natural resources [62]. In fact, the irrigated oasis area has exceeded about 33.6% of the maximum carrying capacity of local precipitation [52]. The increase of the arable land is at the expense of the destruction of vegetation and shrinkage of the water area [53,54]. Even worse, the expansion of artificial oases has led to the degradation of natural oases and the oasis–desert ecotone, which may threaten the security and sustainable development of oases [48]. These problems require understanding the interaction between the oasis and local hydrological cycle, and to selecting the more appropriate areas for desertification prevention and layout of oasis expansion.

According to the simulated results shown in Figure 8 and the current stage of the oasis precipitation, it can be concluded that Kashi Oasis is at the beginning of the fourth stage, which implies that the increment of precipitation would be about 120 mm/year. The increment of precipitation in Akesu and Heian Oasis are the middle level because both oases are at the first half of the second stage. The least promotion would appear in the southwestern basin of Hetian Oasis because it is at the end of the first stage. Therefore, Kashi Oasis of the western basin is the most appropriate area for desertification prevention, while the southwestern basin, i.e., Hetian Oasis, presents a greater challenge.

### *5.2. The Model for Multisource Data Fusion*

The considerable challenge to the data assimilation is the resolution [22]. In this study, the spatial resolution of the assimilated data is 0.1 degrees [23,24], and many researchers have used this dataset to analyze the regional hydrological processes, such as the impact of lake effects on the temporal and spatial distribution of precipitation in the Nam Co basin of the Tibetan Plateau [25], evaporation from the lake [26] and the impact of climatic factors on permafrost in the Tibetan Plateau [27]. The good spatial continuity of the dataset is reflected by gradual changes of the local hydrological processes.

However, the performance of the dataset is still not clear in the desert areas. In this study, the dataset agrees well with simulated results of the model based on information entropy, which is not only demonstrated by the good performance of the model, but also reflected by the good spatial continuity of annual precipitation evolution in the desert–oasis system. Furthermore, the poor performance of the model without the calibration is also an indicator of the different interaction between the oasis and water vapor transportation in the Hetian Oasis. In fact, opposite direction between the water vapor transportation and cold effect of the oasis leads to water vapor from local evapotranspiration accumulating near the desert boundary, where the higher temperature reduces the promotion of oasis on local precipitation substantially [49]. Hence, the model is a useful tool to test the spatial continuity of the dataset obtained from merging multisource data, especially in those areas with extreme natural conditions, for example, desert areas.

### **6. Conclusions**

Based on the established model, parameter inversion, forward simulation, and the discussions on the potential applications of the method, the conclusions are as follows.


Therefore, the theory of the model and simulated results would provide a deeper insight from the perspective of informatics into understanding the precipitation evolution from desert to oasis, which is not only helpful in preventing desertification but also helpful in merging multisource data in the extreme drought desert areas.

**Author Contributions:** X.Z. as the first author was responsible for establishing the model, collecting the data to calibrate the model, plotting the figures and writing of the main body of the manuscript. Z.N., the second author and corresponding author, searched the background of the hydrological effect of oasis and proposed the idea of this study, and helped to revise this manuscript. W.L., the third author, contributed to the data analysis and manuscript approving. All authors have read and approved the final manuscript.

**Funding:** The research was funded by National Natural Science Foundation of China (grant number 41701558), Science and Technology Funding of Water Resources Department of Guizhou Province (grant number KT201707), Science and Technology Funding of Guizhou Province (grant number LH [2017]7290) and the open funding of Guizhou Provincial Key Laboratory of Public Big Data (grant number NO2017BDKFJJ021).

**Acknowledgments:** The forcing dataset used in this study was developed by Data Assimilation and Modeling Center for Tibetan Multi-spheres, Institute of Tibetan Plateau Research, Chinese Academy of Sciences.

**Conflicts of Interest:** The authors declare no conflict of interest.

### **References**


© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

### *Article* **Modeling the Application Depth and Water Distribution Uniformity of a Linearly Moved Irrigation System**

### **Junping Liu, Xingye Zhu \*, Shouqi Yuan and Alexander Fordjour**

Research Center of Fluid Machinery Engineering and Technology, Jiangsu University, Zhenjiang 212013, China; liujp@ujs.edu.cn (J.L.); shouqiy@ujs.edu.cn (S.Y.); fordjouralexander27@yahoo.com (A.F.)

**\*** Correspondence: zhuxy@ujs.edu.cn; Tel.: +86-511-887-80284

Received: 19 March 2019; Accepted: 17 April 2019; Published: 19 April 2019

**Abstract:** A model of a linearly moved irrigation system (LMIS) has been developed to calculate the water application depth and coefficient of uniformity (CU), and an experimental sample was used to verify the accuracy of the model. The performance testing of the LMIS equipped with 69-kPa and 138-kPa sprinkler heads was carried out in an indoor laboratory. The LMIS was towed by a winch with a 1.0 cycle/min pulsing frequency while operating at percent-timer settings of 30, 45, 60, 75, and 90%, corresponding to average moving speeds of 1.5, 2.3, 3.3, 4.0, and 4.7 m min−1, respectively. The application depth and CU obtained under various speed conditions were compared between the measured and model-simulated data. The model calculation accuracy was high for both operating pressures of 69 and 138 kPa. The measured application depths were much larger than the triangular-shaped predictions of the simulated application depth and were between the parabolic-shaped predictions and the elliptical-shaped predictions of the simulated application depth. The results also indicate that the operating pressure and moving speed were not significant factors that affected the resulting CU values. For the parabolic- and elliptical-shaped predictions, the deviations between the measured and model-simulated values were within 5%, except for several cases at moving speeds of 2.3 and 4.0 m min−1. The measured water distribution pattern of the individual sprinklers could be represented by both elliptical- and parabolic-shaped predictions, which are accurate and reliable for simulating the application performances of the LMIS. The most innovative aspect of the proposed model is that the water application depths and CU values of the irrigation system can be determined at any moving speed.

**Keywords:** linearly moved irrigation system; application depth; moving speed; uniformity coefficient

### **1. Introduction**

With the development of science and technology and the need for practical production, the rapid development of the linearly moved irrigation system (LMIS) has been popular in various countries. In China, the LMIS was introduced in the 1980s, with increasing use in recent years due to technological innovations, management convenience, economics and water savings [1–3]. Linear systems, also known as mobile side, have structures similar to center pivot; however, circular movement is replaced by linear movement so that the water application rate is constant along the entire length of the lateral move type. The basic elements to determine LMIS operation include the moving speed, V (m/min), and the maximum range of the sprinkler head, R max (m). The moving speed of LMIS directly affects how water is sprayed in the unit area; when the moving speed is slow, the unit area of spraying is greater, and when the moving speed is fast, the unit area of water spraying is lower. Effective irrigation is not the application of water without control or planning, but is the application of the correct amount of water at the right time, especially with uniformity, during the irrigation process. Thus, to achieve

water optimization in agriculture, it is important to frequently evaluate the performance of irrigation systems by using some parameters that express and quantify the operation quality. The application depth and water distribution uniformity coefficient are often used as indicators of problems concerning irrigation distribution.

Irrigation uniformity is defined as the variation in irrigation depths over an irrigated area and is an important performance characteristic of the sprinkler irrigation system [4–10]. The importance of sprinkler irrigation uniformity was recognized as early as 1942 [11]. Widespread research has been conducted on the factors affecting sprinkler irrigation uniformity. Such factors include nozzle size and pressure, the type of diffuser device, sprinkler spacing, riser height, field topography, discharge angle, number and configuration of the sprinklers and wind speed and direction, all of which can influence water application uniformity [12–23]. In recent years, several studies have been carried out on irrigation performance by center pivot to identify the main problems of irrigation efficiency [24–28]. However, there have been few studies concerning the application depth and uniformity of the LMIS at different operating speeds [29,30]. In the current studies, none of the existing methods can be conducted to calculate the application depth and uniformity of the LMIS. In some certain special cases, the surface runoff appears in the field as the application of the LMIS. It is difficult to determine a mismatch of the farmland soil infiltration capacity and the hydraulic performance of LMIS. Some targeted solutions for solving the problem are unavailable in the scientific research. Therefore, it is very important to study the calculation method of the application depth and the combination uniformity of the LMIS, which is of great theoretical value and practical significance. The objective of this study is to put forward a calculation method of application depth and uniformity of LMIS and to verify the accuracy of the calculation results through experimental tests.

### **2. Materials and Methods**

### *2.1. LMIS and Spray Sprinkler*

The sprinkler irrigation system used in this study was specifically manufactured as an experimental sample by the Research Center of Fluid Machinery Engineering and Technology (Jiangsu University, Zhenjiang City, Jiangsu Province, China). It was an intermittent linear moving system towed by a winch and installed with a fixed spraying plate package as shown in Figure 1. The fixed spray plate sprinkler studied in this research was the Nelson D 3000 sprayhead (Nelson Irrigation Co., Walla Walla WA, USA). A single-strut pressure regulator (Figure 2a) was part of the package for a low-pressure deflection sprinkler. The shapes of the nozzles were circular (Figure 2b). The nozzle diameter of the fixed spray plate sprinkler was 5.16 mm, corresponding to number #26 specified by the Nelson Company. The sprinkler deflects 36 water streams uniformly, centered from itself, to form a full-circle spray pattern. The pressure regulator and sprinkler were connected immediately with a dedicated screw-type connector (Figure 2c). The reason for using such sprinklers was that prior tests had shown that this package and pressure combination had a good coefficient of uniformity [31–33]. All sprinklers were on flexible drop hoses that set the D 3000 sprayhead approximately 1.0 m above the soil surface, and spaced 3.0 m apart. A pressure regulator of 10 or 20 psi (69 or 138 kPa) was installed just upstream of the spray sprinkler. The discharge of the LMIS was measured with an electrical flow gauge (Model E-mag/DN25, manufactured by Kaifeng Electronic Instrument Company, China) with an accuracy of 0.3%.

**Figure 1.** Description of the linearly moved irrigation system (LMIS).

**Figure 2.** Pressure regulator and sprinkler used in the LMIS (**a**), nozzles used in the sprinkler (**b**), and dedicated screw-type connector of pressure regulator (**c**).

### *2.2. Modeling for Calculation of Application Depth*

As the structure parameters of sprinkler type, installed height and spacing were determined, with the water distribution along the axis direction of the LMIS known. First, the LMIS was maintained at a constant working condition. Then, according to the maximum wetted radius, R max, and the water application rate of the sprinklers, the figure of water application depth could be obtained. The relationship between the water application and spraying distance may be described by a geometric pattern. An elliptical, parabolic or triangular pattern was chosen to represent it as shown in Figure 3, where the curve of abc represents the distribution line of sprinkler irrigation intensity, the vertical axis represents the water application rate p (mm/h), and the horizontal axis represents the spraying distance L (m). In theory, under the conditions of an indoor experiment without any wind effects, the value of L is two times the maximum wetted radius, R max.

**Figure 3.** Water application depth: (1) elliptical, (2) parabolic and (3) triangular.

The catch device was supposed to be set at point O, away from the LMIS, with a distance larger than R max, to study the average application depth of the system. The working phenomenon was as following: first, the LMIS approached point O, and point O started to receive the water sprayed out by the LMIS; then, the LMIS gradually receded from point O while moving until the system passed over the catch devices entirely and point O did not receive any water. The LMIS-fulfilled total water application depth was the sum of passing section area of point O. Figure 4 represents the depiction of water application distribution of point O. Combined with Figure 3, when the LMIS was passing through point O at a speed of v, it means that the curve of abc was passing through point O at a speed of v and the passing time could be calculated as t = Lac/v.

At this time, the curve of abc also represents the distribution line of sprinkler irrigation intensity, and the vertical axis also represents the water application rate p (mm/h); however, the horizontal axis represents the passing time t (s). When the LMIS was passing through any point of O, the receiving water of point O was the sum of the water application section area passing point O. Figure 4 represents the distribution section S, and S is the superimposed distribution map of all water sprayed to point O. When the travel speed is v, it can be seen that t = Lac/v. The relationship between the water application rate ρ and the passing through time T was as follows:

Elliptical shape:

$$\rho = \text{P}\_{\text{k}} \sqrt{1 - \frac{4\left(\text{T} - \text{t}/2\text{}\right)^{2}}{\text{t}^{2}}}, \text{ P}\_{\text{k}} = \frac{4\text{WDP}}{\pi \text{t}}\tag{1}$$

Parabolic shape:

$$\rho = -\frac{4\mathbf{P\_k}}{\mathbf{t^2}}(\mathbf{T} - \mathbf{t}/2)^2 + \mathbf{P\_{k\prime}}\ \mathbf{P\_k} = \frac{1.5WDP}{\mathbf{t}} \tag{2}$$

Triangular shape:

$$\begin{cases} \begin{array}{c} \rho = \frac{2\mathcal{P}\_{\text{k}}}{\text{t}} \text{T} \\ \rho = \frac{2\mathcal{P}\_{\text{k}}}{\text{t}} (\text{t} - \text{T}) & (\text{t}/2 \le \text{T} \le \text{t}) \end{array} , \begin{array}{c} \text{P}\_{\text{k}} = \frac{2\mathcal{W}\mathcal{D}P}{\text{t}} \end{array} \tag{3}$$

where

Pk = peak water application rate (mm h<sup>−</sup>1)

t = water application time (h).

**Figure 4.** Depiction of water application distribution of point O.

Then, the wetted area S was calculated as follows:

$$\mathbf{S} = \int\_0^t \boldsymbol{\varrho}(\mathbf{T}) \mathbf{d}\mathbf{t} \tag{4}$$

Supposing that the travel speed of LMIS was set at different values of v1, v2, and vn, respectively, t1 = Lac/v1, t2 = Lac/v2, and tn = Lac/vn. The parabolic pattern was selected and was assumed to be representative of LMIS irrigation water, whose pattern for the same water application depth and time has a peak rate between the elliptical and triangular patterns. Figure 5 shows an elliptical application shape. Notice the increase of water application depth (proportional to larger geometric areas) related to high speed, average speed, and low speed, as well as the constant peak water application rate.

**Figure 5.** Water application patterns with decreasing speed at the same irrigation point: (1) high speed, (2) average speed, and (3) low speed.

### *2.3. Modeling for Calculation of Uniformity*

The spraying irrigation state of LMIS can be regarded as a limit state of mobile spraying for fixed-type pipe with an infinite narrow width. Supposing that the LMIS traveled at a uniform speed, and the water distribution for every sprinkler was the same and does not change with time, it could be determined that the depth of irrigation water along the direction of the sprinkler is the same. Therefore, only the branch direction sprinkling uniformity degree needed to be considered, which can be representative of the entire area of the spraying uniformity. In this way, the concept of linear spraying uniformity is put forward. The uniformity of the LMIS is equal to the uniformity of the axial direction of the unit. The coefficient of uniformity CU (%), developed by Christiansen (1942), was calculated using the following equation:

$$\text{CU} = 1 - \frac{\Delta \overline{h}}{\overline{h}} \tag{5}$$

When the point was adopted by affirmative grid,

$$
\Delta \overline{h} = \frac{\sum\_{i=1}^{n} \left| h\_i - \overline{h} \right|}{n}, \overline{h} = \frac{\sum\_{i=1}^{n} h\_i}{n} \tag{6}
$$

where *hi* = water depth of calculated point *i*, mm/h; *h* = mean water depth of all calculated points, mm/h; and *n* = total number of calculated points used in the evaluation.

Figure 6 represents the calculated sketch of spraying uniformity for LMIS. The K direction is the running direction and the *j* direction is the axial of the unit. The water depth of h*jk* was as follows:

$$\begin{aligned} h\_{11} = h\_{12} = h\_{13} = \dots = h\_{1k} = h\_1 \\ h\_{21} = h\_{22} = h\_{23} = \dots = h\_{2k} = h\_2 \\ \dots \\ h\_{j1} = h\_{j2} = h\_{j3} = \dots = h\_{jk} = h\_j \\ \text{Then, } \overline{h} = \frac{h\_{11} + h\_{12} + h\_{13} + \dots b\_{1k} + b\_{21} + b\_{22} + \dots + b\_{2k} + h\_{j1} + h\_{j2} + \dots h\_{jk}}{n\_{i} n\_{j}} \\ \text{Then, } \overline{h} = \frac{h\_{1} + h\_{2} + h\_{3} + \dots + h\_{j}}{n\_{j}} = \sum\_{i=1}^{j} h\_{i} / n\_{j} \end{aligned} \tag{7}$$

Here, *h* has been simplified to the average value of the depth of the axial point irrigation water. In the same way:

$$\begin{split} \Delta \overline{h} &= \frac{|\overline{h} - h\_{11}| + |\overline{h} - h\_{12}| + \dots + |\overline{h} - h\_{1k}| + \dots + |\overline{h} - h\_{j1}| + \dots + |\overline{h} - h\_{jk}|}{n\_k \cdot n\_j} \\ &= \frac{n\_k |\overline{h} - h\_1| + n\_k |\overline{h} - h\_2| + \dots + n\_k |\overline{h} - h\_j|}{n\_k \cdot n\_j} = \sum\_{i=1}^j \underbrace{|\overline{h} - h\_i|}\_{n\_j} \end{split} \tag{8}$$

This type has also been turned into an axial Δ*h*. The spray uniformity coefficient can be obtained after substitution of the axial *h* and Δ*h* into Equation (5).

**Figure 6.** Calculated sketch of spraying uniformity for the LMIS.

### *2.4. Setup and Procedures of Indoor Experiment*

The study site was located at the indoor facilities of the Research Center of Fluid Machinery Engineering and Technology, Jiangsu University (Jiangsu Province, China). Figure 7 shows the test plot for the LMIS mobile water distribution. The structure, built with several metal 40-mm-long and 20-mm-wide rectangular bars, was 12.0 m wide and the spray sprinkler could be adjusted from 0 to 2.5 m above the surface. Because only a short width (12.0 m) was tested, pressure variation through the measured width of the system was assumed to be negligible. The frame corners were equipped with four wheels and a winch was used to tow the system. The water source was a reservoir with a capacity of 60 m3. A 3.0-kW electric centrifugal pump was connected to the water supplying pipe and a 36-mm external-diameter hose pipe was used to supply water to the spray sprinkler. Manometers and valves were installed as required to control water supply during the experiments (as shown in Figure 1). Catch cans were used to collect the applied water. They were constructed from transparent plastic, with an inverted conical shape. The catch can opening was 200 mm for the inside diameter and the catch can height was 250 mm. Nine rows of catch cans were distributed along the direction of the vertical moving unit, and the distance and spacing of catch cans were 1 m each. The LMIS and the catch cans were installed in a plot with cement flooring. After the spraying test, the weighing method was used to calculate the depth of irrigation water for every catch can. The measured irrigation depth of the point values was taken with an average irrigation depth of the 9 rows of catch cans.

**Figure 7.** Test plot for the LMIS mobile water distribution.

Experiments were carried out at a constant pressure of 69 and 138 kPa, respectively, maintained by a 10- and 20-psi pressure regulator. The pulsing frequency towed by the winch was 1.0 cycle/min while operating the winch at percent-timer settings of 30, 45, 60, 75%, and 90%, corresponding to the average moving speed of 1.5, 2.3, 3.3, 4.0, and 4.7 m min<sup>−</sup>1, respectively. These selected five values cover the range of duty cycles that are expected to be used in the field. A value of 100% is the same as normal operating conditions (constant sprinkler discharge) and zero percent is the same as operating the system

dry. All operating pressures and moving speeds were within the manufacturer's recommendations. The following standards [34–37] were adopted in the design of the experimental setup and in the experiment itself. The duration of each test was approximately 30 to 45 min. The applied water in the catch cans was read in a measuring cup with a volume of 500 mL and an accuracy of 5 mL. The read data were then converted to average irrigation depth by dividing the cross-sectional area of the catch can. A minimum of three replications were conducted for each pressure and moving speed combination, and data were averaged and used as the final experimental data. Flow rates from the two operating pressures used for the application depth tests were measured three times under the same operating conditions. A metal pipe was positioned over the sprinkler nozzles and the discharge water directed into a bucket for 2 min. Discharge volumes were then weighed with an electronic balance (Otimpa Corp, China) with 1 gram accuracy and converted into flow rate.

The average air temperature during testing was 20.8 ◦C and ranged from 17.8 ◦C to 22.4 ◦C. The average relative humidity was 41% and ranged from 36% to 47%. To verify the accuracy of the model, a moving superposed water quantity verification test was carried out. Taking into account the condition that the work at the beginning of the LMIS may not be stable, the moving procedure of the system was start-up and test data were collected after operating at the working pressure for more than 10 min.

### **3. Results and Discussion**

The stable working state of the LMIS means the spraying speed is stable, and the working pressure of the spray head on the system is stable. The wind speed during the entire tests ranged from 0.0 to 0.12 m s−<sup>1</sup> and was usually less than 0.1 m s−1. These data are not presented or discussed as the values were less than the lower threshold of the measuring equipment. The potential source of error in this study is any differences in evaporative conditions across the time period of the various tests. Although many studies have been conducted on evaporation during sprinkler irrigation, Schneider [38] noted that no more than 2% losses resulted from evaporation during use of sprinkler irrigation systems. For example, even under a condition of an average temperature of 26 ◦C, a relative humidity of 64%, and a wind speed of 6.4 m/s, the measured evaporation was only 0.8% of total sprinkler discharge [39]. Therefore, due to the indoor experimental conditions without any wind effects in the study, this particular potential source of error was minimized. An indoor experiment was conducted to verify the accuracy of the model using the Nelson D 3000 spray head with a nozzle diameter of 5.16 mm at working pressures of 69 kPa and 138 kPa and at an elevation of 1.0 m above the soil surface.

### *3.1. Coe*ffi*cient of Discharge*

The results of measured flow rates of sprinkler irrigation nozzles used in this study are shown in Table 1. Analysis of the measured data were performed to find the influence of the geometrical parameters as well as the operating pressure on discharge of the sprinkler head, which can be expressed in terms of the discharge coefficient, C. The coefficient of discharge of the sprinkler is generally expressed as follows:

$$C = \frac{Q}{d^2 \frac{\pi}{4} \sqrt{2gH}} \tag{9}$$

where *C* is the coefficient of discharge, *Q* is the flow rate of the sprinkler (m3 h<sup>−</sup>1), *d* is the diameter of the nozzle (m), *g* is the gravitational acceleration (m s<sup>−</sup>2), and *H* is the pressure head (m).

As shown in Table 1, when using the fixed spray plate sprinkler, the measured nozzle flow rates ranged from 0.77 to 0.84 m3 h−1, with a mean value of 0.808 m3 h−1, and 1.15 to 1.22 m3 h−<sup>1</sup> with a mean value of 1.183 m<sup>3</sup> h−<sup>1</sup> for the pressure regulators of specification 69 and 138 kPa, respectively. After calculating using Equation (8), the nozzle coefficients of discharge ranged from 0.880 to 0.960 with a mean value of 0.923, and 0.929 to 0.986 with a mean value of 0.956, for 69 and 138 kPa, respectively. From the aforementioned analysis, it was found that the coefficients of discharge fluctuated within a

small acceptable range under the same operating pressure, which could be attributed to the acceptable experimental error. Additionally, the coefficients of discharge obtained using the 138 kPa pressure regulator were higher than those obtained using the 69 kPa pressure regulator, which means that using a pressure regulator with a higher outlet pressure produced higher coefficients of discharge.


**Table 1.** Measured flow rates of sprinkler irrigation nozzles.

### *3.2. Measured Water Distributions*

Figure 8 represents the measured sprinkler intensity and fitting curve of radial points with a nozzle diameter of 5.16 mm and a height of 1.0 m at operating pressures of 69 kPa and 138 kPa, respectively.

**Figure 8.** Measured radial application rate and fitting curve for operating pressures of 69 kPa and 138 kPa.

As shown in Figure 8, the sprinkler nozzle wetting radius for 69 kPa and 138 kPa was 4.4 m and 5.7 m, respectively. The sprinkler head produced quite similar water application profiles under different operating pressures. The average values of the sprinkler head application rates varied from 3.2 mm h−<sup>1</sup> to 92.8 mm h−1. The water application rate increased to a maximum value first and then decreased approximately linearly as the distance from the sprinkler increased. The maximum application rate was determined for the two analyzed pressures: 63.2 mm h−<sup>1</sup> at distances of 1 m for 69 kPa, and 92.8 mm h−<sup>1</sup> at 2 m from the sprinkler for 138 kPa. Starting from this distance, the application rate decreased until it reached the minima. For 69 kPa at 4 m, and 138 kPa at 5 m, the minimum values were 5.3 and 4.3 mm h−1, respectively. The application rates had large variability for all measured points. The applied water depths were extremely similar to the radial water depths, indicating that the water distribution pattern of individual sprinklers had a primary influence on the overall water application depth and distribution uniformity. The radial water distribution curve appeared as a saddle type, and had two sprinkler intensity peaks close to the sprinkler. The least squares method was used for curve fitting to simulate the water distribution characteristics, and the fitting degree was as high as 0.89. Table 2 summarizes the measured application depths and corresponding CU values obtained with an LMIS average moving speed of 1.5, 2.3, 3.3, 4.0 and 4.7 m/min under different operating pressures of 69 kPa and 138 kPa. For the various LMIS speeds, the application depth of 69 kPa varied from 1.3 to 4.2 mm with an average of 2.38 mm, the CU varied from 84.5% to 88.9% with an average of 86.58%, and the standard deviation had an average of 0.16; the application depth of 138 kPa varied from 2.6 to 7.7 mm with an average of 4.08 mm, the CU varied from 85.9% to 89.8% with an average of 87.28%, and the standard deviation had an average of 0.68. It appeared that the application depth decreased as

the LMIS speed increased, which could be understood easily. The CU value did not linearly relate to the LMIS speed, which was interpreted as adjusting for unimportant variability on a small scale.

**Table 2.** Measured irrigation application depths and corresponding coefficient of uniformities under different operating pressures and LMIS speeds.


*3.3. Comparison of Application Depth and CU Values between Experimental Measured Data and Modeling Simulations*

As shown in Sections 2.2 and 2.3, in the modeling for calculation of application depth and CU values of the sprinkler irrigation, the data from Figure 8 were used as the basic foundation, and the fitting curves were supposed to be regulated as elliptical, parabolic, and triangular shapes, respectively.

### 3.3.1. Application Depth

The simulated application depths were obtained with a constant spacing of 3.0 m for comparison. The data from Table 2 were used as the experimental measured data. As a further illustration of model validation of the application depth calculation, Figure 9 presents a comparison of application depth between measured data and model-simulated results at an operating pressure of 69 kPa and 138 kPa, respectively.

**Figure 9.** Comparison of application depth between experimental measured data and model-simulated results: (**a**) 69 kPa, (**b**) 138 kPa.

As seen from Figure 9, there appears agreement between the measured and simulated mean application depths. The measured data and model-simulated application depths for all the elliptical-, parabolic-, and triangular-shaped predictions decreased with an increasing average moving speed, which was in accordance with the analysis that slower moving speeds will result in more spraying water. The measured application depths were much larger than the triangular-shaped prediction simulated application depths and were just between the parabolic-shaped prediction and the elliptical-shaped prediction simulated application depths.

Compared to the experimental measured data at an operating pressure of 69 kPa, the simulated result was from 28.0% to 33.3% lower with an average of 30.9% for the triangular-shaped prediction, from 4.0% to 11.1% lower with an average of 7.8% for the parabolic-shaped prediction, and from 4.7% to 13.1% higher with an average of 8.5% for the elliptical-shaped prediction, respectively. Compared to the experimental measured data at an operating pressure of 138 kPa, the simulated result was from 23.3% to 39.3% lower with an average of 32.0% for the triangular-shaped prediction, from 2.3% to 19.1% lower with an average of 10.2% for the parabolic-shaped prediction, and from 3.9% to 20.4% higher with an average of 8.7% for the elliptical-shaped prediction, respectively. Normally, the measured water depth is generally a slight deviation from the calculated value, caused by the fitting error, the testing error, evaporation and drift during the spraying process [40–42]. Therefore, it was determined that both the parabolic- and elliptical-shaped predictions were the acceptable shapes of the water distribution pattern, as they were closer to the measured application depth.

The measured application depth and model-simulated results based on overlapping the measured water distribution data of individual sprinklers were curved lines. Special attention was given to the development of empirical equations for water application depth with regard to moving speed. The curvilinear Equation (10) was regressed and Table 3 presents the equations of those profiles. The coefficient of determination (R2) for measured data ranged from 0.9698 to 0.9954 and for all the model simulations was 0.9946.

$$\mathbf{p} = \mathbf{A}\mathbf{v}^2 - B\mathbf{v} + \mathbf{C} \tag{10}$$

where p is the application depth (mm) and *v* is the LMIS speed (m min<sup>−</sup>1).


**Table 3.** Equations of water application depth with regard to moving speed.

### 3.3.2. Coefficient of Uniformity

The simulated CU values were obtained using MATLAB calculations based on the basic foundation of application depth values from Figure 9. The combined CU values were calculated using the aforementioned method. The data from Table 2 were used as the experimental measured data. As a further illustration of model validation of CU calculations, Figure 10 presents the comparison of CU between experimental measured data and model-simulated results at an operating pressure of 69 kPa and 138 kPa, respectively.

**Figure 10.** Comparison of CU between experimental measured data and model-simulated results. (**a**) 69 kPa, (**b**) 138 kPa.

As seen in Figure 10, there appears agreement between the measured and simulated CU values. The simulated CU for all the elliptical-, parabolic-, and triangular-shaped predictions maintains a constant value and does not vary with increasing speed. It was determined that a different moving speed has an influence on the depth of irrigation water along the direction of the sprinkler, which varies at a same percentage and does not affect CU values; only the branch direction data affect the sprinkling CU degrees, which can be representative of the entire area of the spraying uniformity. Therefore, when the branch water distribution shape was determined as an elliptical, parabolic, or triangular prediction, the simulated CU keeps a constant value at different moving speeds. However, the measured CUs were variable for all moving speeds. The range of combined CU values at different pressures were as follows: 84.5% at a moving speed of 2.3 m min–1 to 88.9% at a moving speed of 4.0 m min–1 (69 kPa), and 85.9% at a moving speed of 4.0 m min–1 to 89.8% at moving speed of 2.3 m min–1 (138 kPa). It was determined that the measured CU values with regard to moving speed fluctuated within a small range, and could be attributed to the acceptable fitting and experimental error.

Compared to the experimental measured data at an operating pressure of 69 kPa, the simulated CU value was 93.5% and of an absolute deviation rate from 5.2% to 10.7% with an average of 8.0% for the triangular-shaped prediction, 92.5% and of an absolute deviation rate from 4.0% to 9.5% with an average of 6.9% for the parabolic-shaped prediction, and 89.3% and of an absolute deviation rate from 0.4% to 5.7% with an average of 3.2% for the elliptical-shaped prediction, respectively. Compared to the experimental measured data at an operating pressure of 138 kPa, the simulated CU value was 92.0% and of an absolute deviation rate from 2.4% to 7.1% with an average of 5.4% for the triangular-shaped prediction, 90.4% and of an absolute deviation rate from 0.7% to 5.2% with an average of 3.6% for the parabolic-shaped prediction, and 87.9% and of an absolute deviation rate from 0.8% to 2.3% with an average of 1.6% for the elliptical-shaped prediction, respectively.

From the aforementioned analysis, it was determined that for the parabolic- and elliptical-shaped predictions, the deviations of the points are within 5% except for several cases at a moving speed of 2.3 and 4.0 m min–1. This deviation may result from inaccurate measurement of the effective wetted widths. Generally, the simplified model has a high computational accuracy, which could be brought into Equation (5) to further calculate the LMIS spraying uniformity. The comparison of measured and model-simulated data indicated that the measured water distribution pattern of individual sprinklers could be represented both as an elliptical- or parabolic-shaped prediction, which was accurate and reliable for simulating the application performance of the LMIS and verified that the calculation of application depth and CU values shown in this work is applicable in practice.

In short, a new model of the LMIS has been developed and a sprinkler irrigation system was specifically manufactured as an experimental sample to verify the accuracy of the model. The differences in model accuracy owing to different operating pressures were not significant, and the moving speed of the LMIS did not appear to influence model accuracy either. Comparisons between experimental data and model simulations revealed that the model can accurately predict water application depth and CU values along the LMIS. Although the model was developed and validated for linear moving systems, it could be readily used for center pivots, which constitute a simplification of the calculation approach adopted.

### **4. Conclusions**

This study presents a model of application depth and uniformity for the LMIS. Based on the obtained results and the conditions in which this trial was carried out, the following can be concluded:

At an operating pressure of 69 kPa and 138 kPa, the sprinkler nozzle wetting radius was 4.4 m and 5.7 m, respectively. The sprinkler head produced quite similar water application profiles, which appeared as a saddle type and had two sprinkler intensity peaks close to the sprinkler.

Compared to the experimental measured data at operating pressures of 69 kPa or 138 kPa, the simulated application depth was on average 30.9% or 32.0% lower for the triangular-shaped prediction, 7.8% or 10.2% lower for the parabolic-shaped prediction, and 8.5% or 8.7% for the

elliptical-shaped prediction, respectively. The curvilinear equations for the measured application depth and model-simulated results were regressed and the coefficient of determination (R2) was from 0.9698 to 0.9954. The simulated CU was an average deviation rate of 8.0% or 5.4% for the triangular-shaped prediction, 6.9% or 3.6% for the parabolic-shaped prediction, and 3.2% or 1.6% for the elliptical-shaped prediction, respectively.

This study indicates that a sprinkler irrigation system for the LMIS that is both elliptical and parabolic in shape could be considered as far as application depths and CU values are concerned. The model can therefore be further developed to provide a useful tool for LMIS design and management.

**Author Contributions:** J.L. was the supervision of this manuscript; X.Z. was writing the original draft and making all revisions; S.Y. was providing the funding acquisition; and A.F. was doing the literature resources.

**Funding:** This research was funded by the National Key Research and Development Program of China (No. 2016YFC0400202), the Key R&D Project of Jiangsu Province (Modern Agriculture) (No. BE2018313), and the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD).

**Acknowledgments:** The authors are greatly indebted to the supports from students of the Research Centre of Fluid Machinery and Engineering, Jiangsu University for their assistances in conducting the experiment.

**Conflicts of Interest:** The authors declare no conflict of interest.

### **References**


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