Comparison of the Two-Point Method and All-Time-Step Advance Approach to Estimate Infiltration Parameters for Surface Irrigation

Background and objectives: Surface irrigation remains the most prevalent irrigation method globally, particularly in regions with limited access to advanced technology, due to its operational simplicity, low infrastructure costs, and minimal energy demands. Despite its widespread use, the water application efficiency of surface irrigation systems is frequently undermined by inaccuracies in estimating hydraulic parameters, most notably infiltration coefficients. These coefficients, which quantify the rate at which water penetrates soil over time, are foundational to designing irrigation systems that balance water application with soil absorption. Accurate infiltration modeling is critical to minimizing environmental and agronomic challenges such as surface runoff (which wastes water and transports nutrients) and deep percolation (which depletes groundwater and leaches fertilizers).
Traditional estimation techniques, such as the two-point method proposed by Elliott and Walker (1982), derive infiltration parameters using data from two discrete phases: when water reaches the midpoint and the endpoint of the field. While this method is computationally efficient and field-friendly, it oversimplifies the infiltration process by ignoring temporal variability—such as fluctuations in soil moisture, micro-topography, or hydraulic resistance—that occur during irrigation. To address this limitation, this study introduces an all-time-step approach (Method 1), which integrates flow depth measurements from every advance phase along the field. By leveraging continuous data from all stations, Method 1 captures dynamic infiltration patterns, enabling more robust calibration of the Kostiakov and Kostiakov-Lewis equations. The primary objective of this study is to rigorously compare the accuracy of the traditional two-point method with a novel all-time-step approach for determining infiltration coefficients in surface irrigation. The innovation lies in the development of the all-time-step method, which leverages continuous flow depth measurements from every advance phase along the field, capturing dynamic infiltration patterns that are often overlooked by traditional methods. This approach significantly improves the calibration of the Kostiakov and Kostiakov-Lewis equations, leading to improved irrigation scheduling, optimized water-use efficiency, and enhanced support for sustainable agricultural practices.
Materials and methods: The study utilized experimental data from five open-ended furrows evaluated by Ramsey (1976) at the University of Arizona’s research farm, a semi-arid site characterized by sandy loam soils. These furrows, designed to replicate real-world agricultural conditions, spanned up to 330 ft (100.6 m) and were monitored under variable hydraulic parameters. Key measured variables included furrow geometry (cross-sectional profiles and longitudinal slopes of 0.1–0.1068%), inflow rates (0.023–0.065 m³/s adjusted to simulate diverse irrigation scenarios), and flow depths recorded at 30-foot (9.1 m) intervals during both the advance phase (when water advance reaches the downstream end of the field) and storage phase.
Surface storage volumes were calculated using the trapezoidal method, a numerical integration technique that divides the furrow into discrete segments and computes cumulative cross-sectional areas across all stations. This approach accounts for irregular furrow shapes and spatial variations in flow depth, ensuring precise estimation of stored water. Infiltrated volumes were derived by subtracting surface storage from the total inflow volume, while average infiltration depth and opportunity time (the duration water remains in contact with the soil at each station) were computed to calibrate the Kostiakov and Kostiakov-Lewis models.
Two calibration methodologies were rigorously evaluated:
Method 1 (All Advance Time Steps): Leveraged nonlinear regression in Microsoft Excel to optimize model coefficients using data from all advance phases (11 time steps per furrow), ensuring high temporal resolution.
Method 2 (Two-Point or Two advance Time Steps): Implemented via WinSRFR software (version 5.1.1), this approach relied on midpoint and endpoint advance data with volume balance equations, adhering to Elliott and Walker’s (1982) methodology.
Model performance was assessed using Nash-Sutcliffe Efficiency (NSE) (predictive accuracy), coefficient of determination (R²), Mean Absolute Percentage Error (MAPE), and boxplot analysis. These metrics evaluated deviations between observed and predicted infiltration depths across advance and storage phases, emphasizing practical applicability in irrigation planning. For instance, boxplots highlighted systematic overestimation trends, guiding adjustments for field-specific conditions like soil heterogeneity or slope variability.
Results: The Kostiakov-Lewis equation derived via Method 1 demonstrated superior accuracy in estimating average infiltration depth, achieving a MAPE of 12.6%, significantly lower than the TP method’s 26.1%. For instance, in Furrow 5, Method 1 reduced MAPE to 7.3%, highlighting its precision. In contrast, the Kostiakov model under Method 1 exhibited higher errors (MAPE = 44.3%), underscoring the importance of the steady-term parameter in the Kostiakov-Lewis equation for short-term advance-phase dynamics. During the storage phase, however, the Kostiakov model (Method 1) outperformed others, with a median absolute error of 10.8 mm.m²/m compared to 14.7 mm.m²/m for the TP method, suggesting its adaptability to prolonged infiltration.
Statistical evaluation revealed that the Kostiakov-Lewis model (Method 1) consistently achieved NSE values exceeding 0.7 across all furrows, indicating “good to excellent” alignment with field data. The Kostiakov model (Method 1) showed moderate performance, with NSE ranging from 0.66 to 0.96. Box plot analysis further exposed systematic overestimation of infiltration depths by all models, though Method 1’s Kostiakov equation yielded the lowest median error (10.9 mm.m²/m), making it reliable for operational irrigation planning. Contradictions emerged in model performance: while the Kostiakov-Lewis equation excelled in advance-phase accuracy, it lagged in storage-phase simulations, whereas the simpler Kostiakov model demonstrated versatility across phases, emphasizing the need for context-specific model selection.
Conclusion: Based on the hypothetical point estimation, the Kostiakov-Lewis infiltration equation derived from the all-time-steps method had the highest accuracy in estimating the average depth of infiltrated water during the average opportunity time. However, based on the estimation of infiltrated water depths during the storage phase, the Kostiakov equation derived from the same method showed the highest accuracy. Therefore, the equation with the least error in estimating the average depth of farm infiltrated water does not necessarily have the highest accuracy in farm irrigation planning. Using all time steps of the advance phase to determine the infiltration relationship can lead to increased accuracy of the infiltration equation. In the present study, the Kostiakov infiltration equation derived from the all-time-steps method demonstrated the highest accuracy.