A Comprehensive Model for Predicting Water Advance and Determining Infiltration Coefficients in Surface Irrigation Systems Using Beta Cumulative Distribution Function

Abstract

Surface irrigation systems are among the most common yet often inefficient methods due to poor design and management. A key factor in optimizing their design is the accurate prediction of the water advance and infiltration relationships’ coefficients. This study introduces a novel model based on the Beta cumulative distribution function for predicting water advance and estimating infiltration coefficients in surface irrigation systems. Traditional methods, such as the two-point approach, rely on limited data from only the midpoint and endpoint of the field, often resulting in insufficient accuracy and non-physical outcomes under heterogeneous soil conditions. The proposed model enhances predictive flexibility by incorporating the entire advance dataset and integrating the midpoint as a constraint during optimization, thereby improving the accuracy of advance curve estimation and subsequent infiltration coefficient determination. Evaluation using field data from three distinct sites (FS, HF, WP) across 10 irrigation events demonstrated the superiority of the proposed model over the conventional power advance method. The new model achieved average RMSE, MAPE, and R² values of 0.790, 0.109, and 0.997, respectively, for advance estimation. For infiltration prediction, it yielded an average error of 12.9% in total infiltrated volume—outperforming the two-point method—and also showed higher accuracy during the advance phase, with average RMSE, MAPE, and R² values of 0.427, 0.075, and 0.990, respectively. These results confirm that the Beta-based model offers a more robust, precise, and reliable tool for optimizing the design and management of surface irrigation systems.

1. Introduction

Water scarcity is one of the most critical global challenges of the 21st century. As the world’s population continues to grow, the demand for water for urban, industrial, and agricultural purposes is increasing significantly. Simultaneously, access to freshwater is declining due to factors such as climate change, pollution, and unsustainable water management practices. The agricultural sector, which is one of the largest consumers of water worldwide, heavily relies on irrigation. Water scarcity is particularly severe in agriculture, which accounts for approximately 70% of global freshwater withdrawals [1]. However, many irrigation systems are inefficient, suffering from significant water losses due to poor system design, outdated technology, and inadequate farm management. The average water use efficiency in irrigation in many regions reaches only 50%, meaning that just half of the water consumed is actually beneficial for crops. Therefore, improving irrigation efficiency is crucial for both water conservation and enhancing agricultural productivity [1,2].

Surface irrigation systems, including furrow, border, and basin irrigation, are among the most widely used irrigation methods worldwide. Despite their widespread use, surface irrigation systems are often criticized for their low efficiency, with losses attributed to runoff, deep percolation, and evaporation [3,4]. However, surface irrigation systems have significant potential for improvement, and when properly designed and managed, they can achieve high efficiency comparable to more modern irrigation methods such as drip or sprinkler irrigation [5,6]. Improving the efficiency of surface irrigation systems is crucial for reducing water waste, increasing crop yield, and ensuring the sustainability of agricultural practices. One of the primary challenges in achieving high efficiency in surface irrigation systems is the accurate estimation of infiltration [7]. The infiltration characteristics of a given soil play a critical role in determining the optimal design and performance of surface irrigation systems, as they directly influence water distribution and application uniformity [5]. By accurately determining infiltration parameters, irrigation managers can better control water application, minimize losses, and maximize efficiency.

The Kostiakov and Kostiakov–Lewis equations (Equations (1) and (2), respectively) are among the most widely used infiltration models. Due to their simplicity and ability to accurately estimate infiltration under various field conditions using a limited number of parameters, they are commonly employed in the design and evaluation of surface irrigation systems [8,9].

$$i=k{t}^a$$

(1)

$$i=k{t}^a+f_{o}t$$

(2)

where $i$ is the cumulative infiltration depth; $t$ is the infiltration opportunity time; $f_o$ is the final infiltration rate; $k$ and $a$ are the constant coefficients.

To apply these infiltration models in practice, it is necessary to determine the values of the parameters $k$ and $a$ for a specific field. Several methods have been proposed to estimate these coefficients, each with its own strengths and limitations. These methods include the volume balance approach combined with water advance data along the field [10], the dimensionless advance curve fitting method [11], the one-point method by Shepard, Wallender [12], the one-point method by Valiantzas, Aggelides [13], and the multi-step calibration method by Walker [14]. The foundation of most proposed methods is the volume balance equation, which is expressed as Equation (3):

$$V_x={\sigma }_z{{\mathrm{kt}}_x}^a={\displaystyle \frac{Q{t}_x}{x}}-{\sigma }_y{A}_o-{\displaystyle \frac{f_o{t}_x}{r+1}}$$

(3)

where $Q$ is the inflow discharge; $f_o$ is the water cross sectional at the upstream end of the field; ${\sigma }_y$ and ${\sigma }_z$ are the averaging coefficients; $t_x$ is the water advance time; $x$ is the water advance distance from the upstream end of the field and $V_x$ is the infiltrated water volume when water reaches $x$ distance.

In Equation (3), the volume of water entering the field is equal to the sum of the surface and subsurface water volumes. In the volume balance method, accurate estimation of surface flow volume is critically important, as it is directly influenced by variations in flow depth and the progression of water advance along the field. Consequently, significant efforts have been dedicated to developing methodologies for accurately estimating the curves representing flow depth variations and water advance dynamics [15,16].

One of the methods based on the volume balance equation is the two-point method proposed by Elliott and Walker [17]. This method has gained significant attention due to its relative simplicity and acceptable accuracy. In the two-point method, Elliott and Walker [17] use the water advance relationship along the field, expressed as a power equation (Equation (4)), to determine the coefficients of the Kostiakov or Kostiakov–Lewis infiltration equation.

$$x={p{t}_x}^r$$

(4)

where $p$ and $r$ are the constant empirical coefficients.

Elliott and Walker [17] utilized the water advance data from two points—the midpoint and the end of the field—to determine the coefficients of Equation (4), as follows:

$$\left\{\begin{array}{c}\left(L/2,t_{L/2}\right)\\ \left(L,t_L\right)\hfill \end{array}\right.\hspace{1em}\to \hspace{1em}r={\displaystyle \frac{\log \left[\left(L/2\right)/L\right]}{\log \left(t_{L/2}/t_L\right)}}\begin{array}{c}\hspace{1em}and\hspace{1em}\end{array}p={\displaystyle \frac{L}{{t_L}^r}}$$

(5)

To determine the coefficients $k$ and $a$ in the Kostiakov and Kostiakov–Lewis infiltration equations, they again utilized water advance data from two points—the midpoint and the end of the field—within the volume balance equation (Equation (3)), as follows:

$$\left\{\begin{array}{c}\left(L/2,t_{L/2}\right)\\ \left(L,t_L\right)\hfill \end{array}\right.\hspace{1em}\to \hspace{1em}\left\{\begin{array}{c}{V}_{L/2}={\displaystyle \frac{Q{t}_{L/2}}{L/2}}-{\sigma }_y{A}_o-{\displaystyle \frac{f_o{t}_{L/2}}{r+1}}\\ V_L={\displaystyle \frac{Q{t}_L}{L}}-{\sigma }_y{A}_o-{\displaystyle \frac{f_o{t}_L}{r+1}}\hfill \end{array}\right.$$

(6)

Finally, using Equation (6), the values of the coefficients $k$ and $a$ are determined as follows:

$$a={\displaystyle \frac{\log (V_L/V_{L/2})}{\log \left(t_L/t_{L/2}\right)}}\hspace{1em}and\hspace{1em}k={\displaystyle \frac{V_L}{{\sigma }_z{t_L}^a}}$$

(7)

where ${\sigma }_z$ is the shape factor and its value is equivalent to ${\sigma }_z=r\beta \left(r,a+1\right)$.

Due to the complexity of calculating the value of ${\sigma }_z$ using the Beta function, Kiefer [18] and Seyedzadeh, Panahi [19] proposed simplified relationships, expressed as Equations (8) and (9), respectively, to estimate this parameter.

$${\sigma }_z={\displaystyle \frac{a+r\left(1-a\right)+1}{\left(1+a\right)\left(1+r\right)}}$$

(8)

$${\sigma }_z\simeq {\displaystyle \frac{1}{1+ra}}$$

(9)

The provided relationships highlight the critical importance of the parameter $r$ in calculations for determining the coefficients of infiltration equations. However, this parameter is determined using data from only two points of water advance—the midpoint and the endpoint of the field. Studies by Vanani, Shayannejad [20], Elliott and Walker [17], Grassi [21], and Panahi, Seyedzadeh [22] have shown that the value of this parameter may exceed unity, leading to negative and unphysical values for $k$ and $a$. Using the endpoint of the field for determining the parameter $r$ is logical because, at this point, the error in measuring the advance time is minimal due to the longer duration of water advance time. Consequently, this measurement error is unlikely to cause significant inaccuracies in determining the value of $r$ or the coefficients of the infiltration equation. Based on this reasoning, for the second point in the two-point method, would it not be more appropriate to use endpoints of the field (e.g., $0.8L$ or $0.9L$)? What is the rationale behind selecting the midpoint of the field ($0.5L$) as the intermediate point in the two-point method?

In recent studies, Panahi, Seyedzadeh [23] and Seyedzadeh, Panahi [24] proposed a dynamic method based on the temporal characteristics of water advance along the field to determine the midpoint location in the two-point method. Their findings demonstrate that, in certain cases, employing alternative points rather than the field’s midpoint can enhance the accuracy of the two-point method in estimating the coefficients of infiltration equations.

Despite the widespread use and relative simplicity of the two-point method [17], its reliance on only two data points (typically the midpoint and endpoint of the field) to characterize the entire advance curve constitutes a significant limitation. This approach possesses a low degree of freedom (DoF = 1), rendering it highly sensitive to measurement inaccuracies at these specific locations and less adaptable to spatial variability in soil properties along the field. Consequently, under non-uniform field conditions, the method can yield non-physical results, such as negative infiltration coefficients [17,20,22]. The quest for more robust and flexible advance relationships is ongoing, as evidenced by comparative studies evaluating various predictive methods. For instance, a recent comparison of methods for predicting advance time highlighted the importance of accuracy and simplicity, with methods like Valiantzas [25]’s algebraic solution showing good performance but still being rooted in specific assumptions about the advance curve’s form [26]. These findings underscore the need for advance models that are not only accurate but also inherently more flexible to capture the complexities of real-world field conditions. To address these limitations, this study introduces a novel advance model based on the Beta cumulative distribution function. The proposed model is governed by two shape parameters (α and λ), providing a higher degree of freedom (DoF = 2). This enhanced flexibility allows the model to utilize the entire dataset of advance points, rather than just two, resulting in a more accurate representation of the advance curve, particularly in heterogeneous soils. Furthermore, the proposed Beta-based model offers a theoretical generalization; the power advance relationship utilized in the conventional two-point method is shown to be a specific case of the Beta model (when α = 1), establishing the proposed framework as more comprehensive. Therefore, while the two-point method may perform adequately under ideal, homogeneous conditions, even minor soil heterogeneity can compromise its accuracy, necessitating a more adaptable approach like the one presented herein.

Given the significance of the water advance relationship in the volume balance equation and its influence on determining the coefficients of the infiltration equation, the primary objective of this study is to develop a comprehensive model for the water advance curve along the field based on the cumulative beta function. This model will be evaluated in estimating the water advance curve and the infiltration coefficients relationship. Additionally, an analytical investigation will be conducted to identify the optimal midpoint location for the two-point method.

2. Materials and Methods

2.1. Theory Background

By dividing Equation (4) by the advance relationship obtained from substituting the data of the field’s endpoint, Equation (4) is normalized as follows:

$$\left\{\begin{array}{c}x=p{t_x}^r\\ L=p{t_L}^r\end{array}\hspace{1em}\to \hspace{1em}\right.{\displaystyle \frac{x}{L}}={\left({\displaystyle \frac{t_x}{t_L}}\right)}^r\hspace{1em}\to \hspace{1em}{x}_{Norm}={t_{Norm}}^r\hspace{1em}\to \hspace{1em}\left\{\begin{array}{c}0\le x_{Norm}\le 1\\ 0\le t_{Norm}\le 1\end{array}\right.$$

(10)

The value of $r$ is derived using logarithmic transformations. In Equation (10), substituting normalized advance data from various points along the field yields various values for $r$. To achieve a high-precision curve fit, the mean of all computed $r$ values can be selected as the final $r$. However, the resulting advance equation will intersect the endpoint of the field and an arbitrarily chosen midpoint, which may not provide sufficient accuracy for determining the coefficients of the infiltration equation. Consequently, the value of $r$ must be determined such that the advance equation passes through an optimally selected midpoint while simultaneously ensuring a robust fit for the complete dataset of water advance along the field. In other words, the value of $r$ is inherently dependent not only on the field’s endpoint but also on data from an additional point (located between the upstream and downstream boundaries of the field). To identify the optimal midpoint location, the relative sensitivity of $r$ with respect to the relative advance time is analyzed. Utilizing Equation (10) and its relative derivative, the relative sensitivity of the parameter $r$ is formulated as Equation (11).

$$r={\displaystyle \frac{Ln(x_{Norm})}{Ln(t_{Norm})}}\hspace{1em}\to \hspace{1em}{\displaystyle \frac{\frac{dr}{r}}{d{t}_{Norm}}}={\displaystyle \frac{Ln(t_{Norm})}{Ln(x_{Norm})}}\times {\displaystyle \frac{-Ln(x_{Norm})}{t_{Norm}{\left[Ln(t_{Norm})\right]}^2}}\hspace{1em}\to \hspace{1em}\left|{\displaystyle \frac{\frac{dr}{r}}{d{t}_{Norm}}}\right|={\displaystyle \frac{-1}{t_{Norm}Ln(t_{Norm})}}$$

(11)

Based on Equations (10) and (11), the variations in the values of $r$ for different values of $x_{Norm}$ and $t_{Norm}$ are illustrated in Figure 1.

According to Figure 1, if a constant value for $r$ (e.g., $r=0.6$) is assumed, the curve with the smallest slope exhibits the least sensitivity to $r$. Therefore, using this curve, the location of the midpoint can be determined in such a way that $r$ has minimal sensitivity. To identify the point corresponding to the lowest relative sensitivity of $r$, it is necessary to determine the extremum of Equation (11). Thus, the derivative of Equation (11) is set to zero as follows:

$${\displaystyle \frac{d(\frac{-1}{t_{Norm}Ln(t_{Norm})})}{d{t}_{Norm}}}=0\hspace{1em}\to \hspace{1em}{\displaystyle \frac{Ln(t_{Norm})+1}{{\left[t_{Norm}Ln(t_{Norm})\right]}^2}}=0\hspace{1em}\to \hspace{1em}{t}_{Norm}={\displaystyle \frac{1}{e}}=0.367879$$

(12)

Equation (12) indicates that if the midpoint in the two-point method is selected at a location where the water advance time to that point equals $0.368t_L$, the parameter $r$ will exhibit the lowest relative sensitive.

Based on Equations (10) and (12), the location of the point with the lowest relative sensitivity to $r$ is as follows:

$$x_{Norm}={t_{Norm}}^r\hspace{1em}\to \hspace{1em}{x}_{Norm}={\displaystyle \frac{1}{e^r}}$$

(13)

Based on studies conducted by various researchers and empirical evidence, it has been established that the value of the parameter $r$ lies between 0.5 and 1 $\left(0.5\le r\le 1\right)$. Using this finding and applying Equation (13), the average optimal location for the midpoint in the two-point method is calculated as expressed in Equation (14).

$$0.5\le r\le 1\hspace{1em}\to \hspace{1em}\overline{x_{Norm}}={\displaystyle \frac{{\displaystyle \underset{0.5}{\overset{1}{\int }}{e}^{-r}}dr}{\Delta r}}={\displaystyle \frac{e^{-0.5}-e^{-1}}{1-0.5}}\hspace{1em}\to \hspace{1em}\overline{x_{Norm}}=0.48\simeq 0.5$$

(14)

As can be seen in Equation (14), if the value of the parameter $r$ is empirically assumed to be between 0.5 and 1, the optimal location for the midpoint is approximately at the center of the field length $\left(x=0.5L\right)$. This result aligns with the midpoint location in the two-point method proposed by Elliott and Walker [17].

In the two-point method, the advance relationship introduced by Elliott and Walker [17] is predicated on the assumption of uniform soil texture along the field. However, in cases where soil heterogeneity is present, the accuracy of the advance relationship diminishes, thereby reducing the accuracy of the derived infiltration coefficients. The coefficients of the advance relationship, along with the infiltration coefficients, are determined based on water advance data collected at two specific points within the field. Consequently, the advance relationship is only strictly valid at these two points, and discrepancies in water advance predictions arise at other locations, leading to decreased accuracy in the estimation of infiltration coefficients. This limitation underscores the necessity for developing a more accurate and flexible advance relationship.

The beta distribution function is a continuous probability distribution extensively utilized in data analysis and statistical modeling, particularly for datasets confined to the interval [0, 1]. Governed by two parameters, the beta distribution exhibits significant flexibility in representing a wide range of distribution parameters within the [0, 1] interval. These properties render the beta distribution particularly suitable for modeling variables that are interpreted in relative or probabilistic terms. By leveraging Equation (10) and incorporating normalized parameters $\left(x_{Norm},\hspace{1em}{t}_{Norm}\right)$ within the [0, 1] interval, a new advance relationship can be formulated based on the cumulative beta distribution function, as expressed in Equation (15).

$$x_{Norm}=I_{t_{Norm}}\left(\alpha ,\lambda \right)$$

(15)

where $\alpha$ and $\lambda$ are the distribution parameters and $I_{t_{Norm}}$ is the beta distribution function.

In Equation (15), the value of $I_{t_{Norm}}$ can be calculated as follows:

$$I_{t_{Norm}}\left(\alpha ,\lambda \right)={\displaystyle \frac{{\displaystyle \underset{0}{\overset{t_{Norm}}{\int }}{u}^{\left(\alpha -1\right)}}{\left(1-u\right)}^{\left(\lambda -1\right)}du}{\beta \left(\alpha ,\lambda \right)}}\hspace{1em}and\hspace{1em}\beta \left(\alpha ,\lambda \right)={\displaystyle \frac{\mathsf{\Gamma }\left(\alpha \right)\mathsf{\Gamma }\left(\lambda \right)}{\mathsf{\Gamma }\left(\alpha +\lambda \right)}}$$

(16)

If in Equation (16) the value of $\lambda$ is taken as unity $\left(\lambda =1\right)$ and the value of $\alpha$ is taken as $r$ $\left(\alpha =r\right)$, Equation (16) is obtained as follows:

$$\left\{\begin{array}{c}\lambda =1\\ \alpha =r\end{array}\right.\hspace{1em}\to \hspace{1em}{x}_{Norm}=I_{t_{Norm}}\left(r,1\right)={\displaystyle \frac{{\displaystyle \underset{0}{\overset{t_{Norm}}{\int }}{u}^{\left(r-1\right)}}du}{\beta \left(r,1\right)}}={\displaystyle \frac{u^r}{\alpha \beta \left(r,1\right)}}\left|\begin{array}{c}{t}_{Norm}\\ 0\end{array}\right.\begin{array}{ccc}& \stackrel{\beta \left(r,1\right)={\displaystyle \frac{1}{r}}}{\to }& \end{array}{x}_{Norm}={t_{Norm}}^r$$

(17)

Equation (17) corresponds to the advance relationship proposed by Elliott and Walker [17]. Specifically, the advance relationship introduced by Elliott and Walker [17] represents a special case of the advance relationship developed in this study (Equation (15)). The advance relationship of Elliott and Walker [17] is characterized by two parameters $\left(p,r\right)$, the values of which are derived from data collected at two points: the end and midpoint of the field. However, by definition, any advance relationship must inherently satisfy the conditions at the upstream and downstream endpoints of the field. As a result, determining the parameters of the Elliott and Walker [17] advance relationship requires only one additional data point (midpoint). This implies that the degree of freedom associated with the Elliott and Walker [17] advance relationship is equal to one.

In contrast, the advance relationship proposed in this study (Equation (15)) also incorporates two parameters, $\alpha$ and $\lambda$. However, the values of these parameters cannot be uniquely determined using only the endpoint of the field and one additional point (e.g., the midpoint). Consequently, the degree of freedom for this relationship is equal to two. According to the higher degree of freedom of the proposed advance relationship compared to that of Elliott and Walker [17], it can be inferred that the proposed relationship exhibits greater flexibility and enhanced accuracy in predicting the water advance time. This improvement ultimately leads to a more precise determination of the infiltration coefficients.

In accordance with the properties of the beta function, the endpoint of the field is inherently satisfied by the proposed advance relationship, as is self-evident. Mathematically, this can be expressed as:

$$\left\{\begin{array}{c}x=L\\ t=t_L\end{array}\right.\hspace{1em}\to \hspace{1em}\left\{\begin{array}{c}{x}_{Norm}=1\\ t_{Norm}=1\end{array}\right.\hspace{1em}\to \hspace{1em}{I}_1\left(\alpha ,\lambda \right)=1$$

(18)

Therefore, as indicated by Equation (18), the values of $\alpha$ and $\lambda$ in Equation (15) are independent of the end point, and their determination necessitates an optimization process. The following methodology is employed to estimate the values of these parameters:

1.

The normalized advance time $\left(t_{Norm}=t_x/t_L\right)$ for each station is calculated using the water advance time at the endpoint of the field and the corresponding advance times at each station.

2.

The normalized advance distance $\left(x_{Norm}=x/L\right)$ for all stations is computed based on the distances of the individual stations and the total length of the field.

3.

Initial values, for example, 0.5, are assigned to the parameters $\alpha$ and $\lambda$ as starting points for the optimization process.

4.

The normalized advance time for each station is calculated using Equation (15) with the assumed coefficients $\alpha$ and $\lambda$.

The relationship for computing the normalized advance time based on Equation (15) is defined as follows:

$$t_{Norm}={I^{-1}}_{x_{Norm}}\left(\alpha ,\lambda \right)$$

(19)

where $I_{x_{Norm}}^{-1}$ represents the inverse of the beta distribution function. To calculate this parameter in Excel, the following command can be used:

$$t_{Norm}=\mathrm{BETA}.\mathrm{INV}\left(x_{Norm},\alpha ,\lambda \right)$$

(20)

5.

By multiplying the normalized advance time values for each station by the water advance time at the endpoint of the field, the advance time for each station $\left(t_x=t_{Norm}\times t_L\right)$ is derived.

6.

The advance times computed in step 5 must align with the field-observed advance times. Consequently, the sum of squared errors between the computed and observed advance times is minimized. For this purpose, an objective function is formulated as follows:

$$f\left(\alpha ,\lambda \right)=\min \left[{\displaystyle \sum _{i=1}^n{\left(t_{i-\exp }-t_{i-calc}\right)}^2}\right]$$

(21)

where $f\left(\alpha ,\lambda \right)$ represents the optimization objective function, $t_{i-\exp }$ denotes the field-measured advance time, and $t_{i-calc}$ corresponds to the computed advance time.

7.

Given the low sensitivity of the advance relationship at the field’s midpoint, it is critical to ensure that the advance relationship intersects the midpoint. As such, one of the optimization constraints necessitates that the field-observed advance time and the computed advance time at the midpoint of the field are equal.

8.

Ultimately, employing an optimization technique (such as the Solver tool in Excel), the objective function defined in step 6 is minimized, subject to the constraint established in step 7, to determine the optimal values of the parameters $\alpha$ and $\lambda$.

Following the determination of the advance relationship (Equation (15)), the infiltration coefficients are derived by integrating this relationship with the volume balance equation. To achieve this, the volume balance equation (Equation (3)) is reformulated as follows:

$$\left(k{\sigma }_z{t_x}^a+{\displaystyle \frac{f_o{t}_x}{r+1}}\right)x=Q{t}_x-{\sigma }_{y}x{A}_o\hspace{1em}\to \hspace{1em}\left\{\begin{array}{c}{V}_{1-x}=Q{t}_x-{\sigma }_{y}x{A}_o\hfill \\ V_{2-x}=\left(k{\sigma }_z{t_x}^a+{\displaystyle \frac{f_o{t}_x}{r+1}}\right)x\end{array}\right.$$

(22)

where the value of ${\sigma }_y$ is 0.77 and the value of $A_o$ can be calculated using Manning’s equation.

As indicated by Equation (22), the value of $V_{1-x}$ can be directly calculated using field data. But $V_{2-x}$ is a function of the coefficients of the infiltration relationship and, consequently, depends on the water advance relationship along the field. As a result, its value cannot be directly determined from field data. Since $V_{2-x}$ is derived using additional estimated parameters, its value will not match $V_{1-x}$. However, because the advance relationship passes through both the midpoint and endpoint of the field, the two sides of the equation are exactly equal at these specific points. Consequently, the coefficients $k$ and $a$ are determined by utilizing data from these two locations.

Method Application

To estimate the infiltration coefficients in this study, the following methodology is employed:

1.

Given $0\le t_{Norm}\le 1$, values of the parameter $t_{Norm}$ are selected within the range [0, 1]. Increasing the number of values considered enhances the precision of the final infiltration relationship.

2.

The computed advance times are derived by multiplying $t_{Norm}$ by the water advance time to the field’s endpoint $\left(t_x=t_{Norm}\times t_L\right)$.

3.

The normalized advance distance $\left(x_{Norm}\right)$ is calculated for the specified range of $t_{Norm}$ values by utilizing the assumed values of the parameter $t_{Norm}$, the derived values of the parameters $\alpha$ and $\lambda$, and Equation (15).

To calculate $x_{Norm}$ using Equation (15) in Excel, the following command can be applied:

$$x_{Norm}=\mathrm{BETA}.\mathrm{DIS}\left(t_{Norm},\alpha ,\lambda \right)$$

(23)

4.

The advance distances are calculated by multiplying the values of $x_{Norm}$ by the total length of the field $\left(x=x_{Norm}\times L\right)$.

5.

Using field data $\left(\alpha , \lambda , Q, A_o \mathrm{and} t_L\right)$ and with the following equation, the volume of infiltrated water is estimated for different advance distances:

$$V_{TS-1-i}=t_{L}Q\left[I_{x_{Norm-i}}^{-1}\left(\alpha ,\lambda \right)\right]-{\sigma }_y{x}_i{A}_o$$

(24)

where $V_{TS-1-i}$ is the experimental volume of water infiltrated from the upstream end of the field to the distance $x$.

6.

Initial estimates, such as 0.5, are assigned to the coefficients $k$ and $a$ as starting points for the optimization process.

7.

By applying the coefficients estimated in step 6 and the following equation, the parametric volume of infiltrated water $\left(V_{TS-2-i}\right)$ is computed for different advance distances:

$$V_{TS-2-i}={\displaystyle \sum _{k=1}^i(k(t_{x_i}}-t_{x_k}{)}^a+f_0(t_{x_i}-t_{x_k}))L\Delta I_{t_{Norm-k}}\hspace{1em},\hspace{1em}\Delta I_{t_{Norm-k}}\to 0$$

(25)

8.

The volume of infiltrated water up to the midpoint and endpoint of the field is determined using the values derived from Equations (24) and (25), as expressed in Equation (26):

$$\left\{\begin{array}{c}{V}_{TS-1-{\displaystyle \frac{L}{2}}}=t_{L}Q\left[I_{0.5}^{-1}\left(\alpha ,\lambda \right)\right]-{\displaystyle \frac{L}{2}}{\sigma }_y{A}_o\\ V_{TS-2-{\displaystyle \frac{L}{2}}}={\displaystyle \sum _{i=1}^{x\to L/2}{V}_{TS-2-i}}\end{array}\right.\hspace{1em}and\hspace{1em}\left\{\begin{array}{c}{V}_{TS-1-L}=t_{L}Q-L{\sigma }_y{A}_o\\ V_{TS-2-L}={\displaystyle \sum _{i=1}^{x\to L}{V}_{TS-2-i}}\end{array}\right.$$

(26)

9.

In Equation (26), the infiltrated water volumes up to the midpoint of the field must be equal, as must the infiltrated water volumes up to the endpoint of the field. Consequently, an objective function for estimating the coefficients $k$ and $a$ can be formulated by minimizing the sum of squared errors of the infiltrated water volumes across various advance distances, as follows:

$$f\left(a,k\right)=\min \left[{(V_{TS-1-{\displaystyle \frac{L}{2}}}-V_{TS-2-{\displaystyle \frac{L}{2}}})}^2+{(V_{TS-1-L}-V_{TS-2-L})}^2\right]$$

(27)

10.

The objective function in Equation (27) is optimized using numerical optimization techniques or the Solver tool in Excel to estimate the coefficients $k$ and $a$.

In the two-point method proposed by Elliott and Walker [17], once the coefficients of the advance relationship $\left(p,r\right)$ and the coefficients of the infiltration relationship $\left(a,k\right)$ are determined, the volume of infiltrated water up to different advance distances across the field can be computed using Equation (22) as follows:

$$V_{EW-1-i}=Q{t}_L{\left({\displaystyle \frac{x_i}{L}}\right)}^{{\displaystyle \frac{1}{r}}}-{\sigma }_y{A}_o{x}_i$$

(28)

$$V_{EW-2-i}=\left\{k{\sigma }_z{\left[t_L{\left({\displaystyle \frac{x_i}{L}}\right)}^{1/r}\right]}^a+{\displaystyle \frac{f_o}{1+r}}{t}_L{\left({\displaystyle \frac{x_i}{L}}\right)}^{1/r}\right\}{x}_i$$

(29)

where $V_{EW-1-i}$ and $V_{EW-2-i}$ are the experimental and parametric volume of the infiltrated water from upstream end of the field to the distance $x$.

2.2. Field Data

This study utilizes data from the doctoral thesis of Esfandiari [27], who conducted research across three field sites: the Field Services Unit Paddock (FS), the Horticulture Farm Paddock (HF), and the Walla Park Farm (WP). The FS and HF sites are located at the University of Sydney farm, while the WP site is situated in Quirindi, Sydney, Australia. The irrigation method employed in the studied fields was open-end furrow irrigation, with maize cultivated across all sites.

Esfandiari [27] utilized calibrated RBC flumes (Replogle, Bos, Clemmens) to measure the inflow and outflow rates of the furrows, enabling the determination of inflow and runoff volumes. To derive the advance and recession curves, monitoring stations were established at 7 m intervals in the FS field, 15 m intervals in the HF field, and 117 m intervals in the WP field. A total of 16 furrows were analyzed across 81 irrigation events. For this study, data from 6 furrows in 10 irrigation events were selected. The characteristics of the furrows used in this study are summarized in

Table 1. Furrow and flow parameters utilized in the present study [27].

Field Name	Furrow No.	Irrigation Event No.	Inflow Rate (L/s)	Cutoff Time (min)	Manning’s Coefficient	Field Slope (%)	Field Length (m)	Soil Texture
FS	1	2	0.74	57.7	0.030	0.170	59	Sand-Sandy Loam
		3	0.47	50.0	0.032
	2	2	0.63	49.2	0.020	0.200
		3	0.72	52.2	0.022
HF	1	2	0.70	46.0	0.016	0.066	125	Sandy Clay Loam-Clay
		3	2.02	28.0	0.016
	2	2	0.56	51.0	0.010	0.034
		3	2.17	30.0	0.014
WP	3	3	3.93	939.0	0.015	0.140	941	Clay
	6	4	4.69	1051.0	0.014

. For additional details, consult the original thesis by Esfandiari [27].

The furrows’ number and irrigation event identifiers presented in Table 1 adhere to the numbering system established by Esfandiari [27].

Following, the field sites, furrow numbers, and irrigation event numbers are encoded in a standardized format (e.g., FS23), where ‘FS’ represents the field site, ‘2’ corresponds to furrow number 2, and ‘3’ indicates irrigation event number 3.

2.3. Evaluation Indicators

To evaluate and compare the advance relationships derived from the proposed method in this study with those derived from the two-point method of Elliott and Walker [17] for estimating advance time, as well as to assess the infiltration relationships resulting from these methods, key statistical metrics—namely, the Root Mean Square Error (RMSE), Mean Absolute Percentage Error (MAPE), and the Coefficient of Determination (R²)—were utilized. The corresponding mathematical formulations for these metrics are presented in Equations (30), (31) and (32), respectively.

$$RMSE=\sqrt{{\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n{\left(P_i-M_i\right)}^2}}$$

(30)

$$MAPE={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n\left|\frac{M_i-P_i}{M_i}\right|}$$

(31)

$$R^2=1-{\displaystyle \frac{{\displaystyle \sum _{i=1}^n{\left(M_i-P_i\right)}^2}}{{\displaystyle \sum _{i=1}^n{\left(M_i-\overline{M}\right)}^2}}}$$

(32)

where $n$ is the number of measurement stations, $M$ is the observed or field-measured values, $P$ is the predicted or estimated values, $\overline{M}$ and $\overline{P}$ are the mean values of the observed and predicted values, respectively.

The RMSE index ranges within the interval [0, +∞). A minimum RMSE value of zero indicates an exact match between the measured and predicted values, signifying perfect model accuracy. Conversely, higher RMSE values reflect increasing deviations between the predicted and observed data, indicating reduced model performance [28].

The MAPE index represents the average percentage error of the measured data relative to the predicted data. This index is scale-independent, allowing for the comparison of results across different models. Similar to RMSE, the MAPE value ranges within the interval [0, +∞). A MAPE value of zero indicates perfect alignment between the predicted and measured values. As the MAPE value increases above zero, the predicted values deviate further from the measured data [28].

The R² index ranges within the interval [−∞, 1]. In cases such as out-of-sample validation, the value of this index may become negative. An R² value of zero indicates no correlation between the predicted data and the field-measured data. Conversely, an R² value of unity signifies a perfect match between the measured and predicted data. Thus, higher values of this index reflect the model’s greater capability to accurately estimate field-measured data [29].

Initially, the relative error index was considered for comparing the total infiltrated volume estimated by different methods. However, this index was found to be unsuitable for the comparative purpose of this study. Therefore, the statistical metrics RMSE, MAPE, and R² were ultimately selected for a more robust and standardized comparison of the infiltration relationships. To conduct this analysis, the actual volume of infiltrated water $\left(V_{1-x}\right)$ for each station during the advance phase was calculated using field data $\left(t_x, Q, x, A_o, \mathrm{and} {\sigma }_y\right)$ and Equation (22). Subsequently, the volumes of infiltrated water derived from the EW and TS methods $\left(V_{2-i}\right)$ were computed for each station during the advance phase using the infiltration coefficients from Table 3 and Equations (25) and (29). Finally, the accuracy of the derived infiltration relationships in estimating the volume of infiltrated water during the advance phase was evaluated using the RMSE, MAPE, and R² statistical metrics.

3. Results and Discussion

To evaluate and compare the advance relationship proposed by Elliott and Walker [17] (EW) with the advance relationship developed in this study (TS), the coefficients of the EW model were initially determined for various irrigation events using the two-point method. Following this, the coefficients of Equation (15) were computed for the same irrigation events using the methodology introduced in this study. Field-measured advance data, combined with statistical performance metrics (RMSE, MAPE, and R²) were employed to compare the performance of Equations (4) and (15). The derived coefficients for both advance relationships, along with the corresponding values of the evaluation metrics, are summarized in

Table 2. Coefficients of the advance relationships derived from the EW and TS models, accompanied by the corresponding values of the performance evaluation metrics.

Sites and Events	EW		TS		EW			TS
	p	r	α	λ	RMSE	MAPE	R2	RMSE	MAPE	R2
FS12	9.198	0.678	0.565	0.819	0.567	0.213	0.988	0.487	0.195	0.992
FS13	11.767	0.505	0.486	0.953	0.469	0.198	0.997	0.454	0.206	0.997
FS22	12.890	0.515	0.684	1.415	0.993	0.289	0.976	0.783	0.238	0.983
FS23	13.020	0.539	0.688	1.340	0.563	0.241	0.990	0.334	0.161	0.996
HF12	7.057	0.858	1.134	1.326	0.796	0.097	0.993	0.234	0.051	1.000
HF13	10.282	0.934	1.106	1.810	0.260	0.042	0.997	0.112	0.054	0.999
HF22	8.536	0.804	0.891	1.112	0.340	0.076	0.999	0.188	0.034	1.000
HF23	13.822	0.848	0.915	1.080	0.110	0.048	0.999	0.051	0.014	1.000
WP33	4.290	0.858	0.738	0.854	8.555	0.104	0.998	2.929	0.081	1.000
WP64	10.053	0.735	0.944	1.316	12.210	0.147	0.994	2.332	0.060	1.000

. Additionally, graphical comparisons of the field-measured advance data and the advance curves generated by Equations (4) and (15) are illustrated in Figure 2.

Figure 2 illustrates that the coefficients of EW advance relationship, determined using two points (midpoint and endpoint of the field), result in curves that pass through these specific points. In contrast, the TS advance relationship inherently ensures passage through the endpoint of the field. Additionally, the midpoint was incorporated as an optimization constraint in determining the coefficients of Equation (15), ensuring that the advance curve also passes through this point. A comparison of the advance relationships, as depicted in Figure 2, reveals that the curves generated by the TS relationship exhibit strong alignment with the field-measured advance data in the majority of irrigation events. Where discrepancies occur, the TS curves demonstrate significantly smaller deviations from the field data compared to the EW curves. For a comprehensive evaluation of the two methods, the results summarized in Table 2 can be referenced. The analysis indicates that, based on the RMSE and R² metrics, the TS advance relationships consistently provide more accurate predictions of field advance times across all irrigation events. Similarly, the MAPE metric confirms the superior accuracy of the TS relationships in all cases except for HF13. The average RMSE, MAPE, and R² values for the TS advance relationships are 0.790, 0.109, and 0.997, respectively, while the corresponding averages for the EW advance relationships are 2.486, 0.146, and 0.993, respectively.

Due to the critical role of the advance relationship in estimating the coefficients of the infiltration relationship, a comparative analysis was conducted to evaluate the influence of the EW and TS advance relationships on determining infiltration coefficients. Utilizing these advance relationships, the coefficients of the Kostiakov–Lewis infiltration equation were calculated. The derived coefficients for various irrigation events are summarized in

Table 3. Coefficients of the Kostiakov–Lewis infiltration equation derived from the EW and TS methods.

Sites and Events	EW		TS		${\mathit{f}}_{\mathit{o}}$ $\left(\mathbf{m}\mathbf{m}.{\mathbf{m}}^2/\mathbf{m}.\mathbf{h}\right)$
	$\mathit{k}\left(\mathbf{m}\mathbf{m}.{\mathbf{m}}^2/\mathbf{m}.{\mathbf{h}}^{\mathbf{a}}\right)$	a	$\mathit{k}\left(\mathbf{m}\mathbf{m}.{\mathbf{m}}^2/\mathbf{m}.{\mathbf{h}}^{\mathbf{a}}\right)$	a
FS12	4.127	0.0742	4.771	0.1367	30.0
FS13	4.806	0.3142	4.749	0.3213	22.8
FS22	9.616	0.4796	7.813	0.3750	24.0
FS23	7.516	0.4084	6.050	0.3077	30.0
HF12	4.265	0.2400	3.837	0.1759	3.6
HF13	2.847	0.0627	2.642	0.0374	8.4
HF22	3.927	0.1375	3.789	0.1165	5.4
HF23	5.586	0.1383	5.345	0.1203	17.4
WP33	85.535	0.0019	84.316	0.0149	5.4
WP64	73.065	0.2650	76.897	0.2197	3.0

.

To conduct a comparative analysis of the relationships obtained from the EW and TS methods, the average infiltration opportunity time was calculated based on the advance and recession times. Following this, the volume of infiltrated water was estimated by applying the derived infiltration relationships. The accuracy of the estimated infiltrated water volume was then assessed by comparing it to the field-measured infiltrated water volume using the relative error index. The outcomes of this assessment are summarized in

Table 4. Field-measured and model-predicted volumes of infiltrated water derived from the infiltration relationships, along with a comparative analysis of prediction errors across different methods for estimating infiltrated water volume.

		Sites and Events
		FS12	FS13	FS22	FS23	HF12	HF13	HF22	HF23	WP33	WP64
Infiltrated Volume (m3)	Exp	2.182	1.290	1.690	1.865	1.312	1.314	1.224	1.686	150.776	152.321
	EW	1.957	1.339	1.676	1.980	1.048	1.330	1.362	2.695	140.397	185.628
	TS	1.994	1.335	1.588	1.903	0.990	1.305	1.341	2.666	141.812	175.644
Error (%)	EW	10.3	3.8	0.8	6.2	20.1	1.3	11.3	59.9	6.9	16.5
	TS	8.6	3.5	6.0	2.0	24.5	0.6	9.6	58.2	5.9	10.2

.

Table 4 indicates that the infiltration relationships derived from the EW method yield lower errors in only three irrigation events, whereas the TS method demonstrates superior accuracy in the remaining events. Additionally, the table highlights that, with the exception of WP33 and WP64, both the field-measured and estimated volumes of infiltrated water are relatively small across all irrigation events. As a result, the relative error index amplifies minor discrepancies between the field-measured and estimated values, presenting them as disproportionately large. Moreover, the relative error index does not provide insight into whether the error distribution is standardized or non-standardized. Consequently, the relative error index is not an appropriate index for comparing infiltration relationships derived from different methodologies. For this reason, the RMSE, MAPE, and R² metrics were utilized to assess the performance of the infiltration relationships. The results of this evaluation are summarized in

Table 5. Evaluation of infiltration relationships derived from the EW and TS methods for estimating the volume of infiltrated water during the advance phase.

Sites and Events	EW			TS
	RMSE	MAPE	R2	RMSE	MAPE	R2
FS12	0.033	0.115	0.975	0.028	0.090	0.977
FS13	0.018	0.064	0.993	0.017	0.073	0.993
FS22	0.027	0.158	0.982	0.025	0.151	0.987
FS23	0.015	0.114	0.994	0.011	0.075	0.997
HF12	0.035	0.138	0.960	0.027	0.112	0.980
HF13	0.033	0.128	0.955	0.029	0.116	0.972
HF22	0.008	0.028	0.999	0.007	0.021	0.999
HF23	0.009	0.014	0.999	0.008	0.016	0.999
WP33	2.675	0.064	0.996	2.176	0.055	0.997
WP64	3.519	0.069	0.993	1.939	0.043	0.998

.

Table 5 demonstrates that, based on the RMSE and R² metrics, the infiltration relationships derived from the TS method achieve superior accuracy in estimating the depth of infiltrated water during the advance phase across all irrigation events. Additionally, when evaluated using the MAPE metric, the TS method outperforms the EW method in all irrigation events except for HF23. The average RMSE, MAPE, and R² values for the TS method are 0.427, 0.075, and 0.990, respectively, whereas the corresponding averages for the EW method are 0.637, 0.089, and 0.985, respectively. The results in Table 5 further indicate that both methods yield infiltration relationships with acceptable accuracy, showing no statistically significant differences between the two approaches. To further assess the performance of the EW and TS methods, the field-measured and computed volumes of infiltrated water during the advance phase were analyzed using boxplot diagrams. The findings of this analysis are presented in Figure 3.

According to Figure 3 and the discrepancies between field-measured and calculated values, it is observed that for site FS, the boxes are predominantly located to the right of the axis in most irrigation events, indicating an underestimation of infiltration depth by the infiltration relationships. In contrast, for site HF, most boxes are located to the right of the axis, suggesting an overestimation of infiltration depth by the infiltration relationships. However, for site WP, the boxes are distributed almost symmetrically on both sides. When comparing the infiltration relationships derived from the EW and TS methods, it is evident that the boxes corresponding to the TS method are shorter in all irrigation events, and the results from this method exhibit a more standardized distribution in most irrigation events.

Therefore, it can be concluded that the advance relationship proposed in this study demonstrates superior accuracy in estimating advance time and determining infiltration coefficients. It is recommended that this method be adopted for practical applications.

Limitations of the Study and Future Research Paths

While the proposed model based on the Beta cumulative distribution function demonstrates significant improvements over the traditional two-point method by utilizing the entire advance dataset and providing greater flexibility, its performance is inherently tied to the quality and resolution of the input data. The accuracy of the optimized parameters and, consequently, the derived infiltration coefficients, depends on the number and precision of the measured advance points.

Future research should focus on the following directions to build upon the findings of this study:

• Applying and validating the proposed model on a wider range of surface irrigation methods (e.g., border irrigation, basin irrigation) and field conditions (e.g., different slopes, soil types, and management practices).
• Developing user-friendly computational tools (open-source software) or simplified procedural guidelines to make the advanced optimization process presented in this study more accessible for irrigation practitioners and engineers for routine design and evaluation.
• Exploring the integration of this accurate advance prediction model into real-time irrigation management systems for automated inflow cutoff decisions, aiming to maximize application efficiency during actual irrigation events.
• Conducting a thorough sensitivity analysis to determine the minimum number and optimal placement of advance measurement points required to achieve a desired level of accuracy, thus optimizing the field data collection effort.

4. Conclusions

This study introduced a novel advance model based on the Beta cumulative distribution function for estimating water advance and infiltration coefficients in surface irrigation. The proposed model offers greater flexibility and accuracy than the traditional two-point method by utilizing the entire advance dataset and incorporating the midpoint as a constraint during optimization.

Evaluation based on 10 field irrigation events showed that the Beta-based model outperformed the power advance relationship, with average RMSE, MAPE, and R² values of 0.790, 0.109, and 0.997, respectively. Subsequent infiltration estimation using the Kostiakov–Lewis equation further confirmed the superiority of the proposed method, yielding more accurate results both for total infiltrated volume and station-wise infiltration during the advance phase.

The Beta advance model provides a robust, physically consistent alternative to conventional methods and is particularly advantageous under heterogeneous soil conditions. It is recommended for use in modern surface irrigation design and management to improve water efficiency and application uniformity.