Integrating image processing and machine learning for phase specific estimation of Manning roughness coefficient in furrow irrigation

Abstract

Accurate estimation of Manning’s roughness coefficient (n) is vital for hydrological modeling and optimizing furrow irrigation systems, yet traditional methods remain limited due to spatial-temporal variability, the need for expert users, the inherent limitations of hydraulic equations, and labor-intensive measurements. This study introduces a novel algorithm that integrates high-resolution image processing with machine learning techniques to dynamically predict n during advance and storage phases in bare furrows. Three scenarios were evaluated: (i) full-field data (including inflow/outflow rates, advance/recession times, infiltration, slope, soil moisture, furrow images and etc.), (ii) images only, and (iii) images plus selected field data (inflow rate, slope, cross-sectional area, clod size and irrigation events). Manning’s n was computed using the SIPAR_ID model and Manning’s equation in advance and storage phases, yielding ranges of 0.017–0.636 (advance phase) and 0.015–0.317 (storage phase), with respective means of 0.083 and 0.054. The Random Forest algorithm in Scenario (i) achieved near-perfect performance (99% precision, recall, and F1-score), while Scenario (iii) preserved 95–96% accuracy with significantly reduced data inputs. Notably, excluding hydraulic variables (Scenario ii) led to a ~ 50% performance drop, highlighting their importance. This approach offers a robust, cost-effective solution for n estimation, bridging the gap between precision, practicality, and real-time application in sustainable water management and precision agriculture.

Introduction

Effective water management in furrow irrigation systems depends on various factors, including the inflow rate, soil type, furrow slope, infiltration rate, crop type, surface roughness of the furrows, and irrigation method used¹. Among these factors, roughness is of particular interest because of its spatial and temporal heterogeneity and its critical role in surface irrigation simulation models^2–4. Various coefficients, such as Chezy, Manning, and Darcy, are commonly used to interpret roughness, each represented as a constant in their respective equations. Among these equations, the Manning equation (Eq. (1)) has been widely accepted by hydraulic and irrigation engineers for modeling flow in open channels and surface irrigation systems, for the most part, because it is simple and quite accurate in predicting roughness⁵.

Nonetheless, the use of this equation in surface irrigation, especially in furrow irrigation, has various limitations because of its foundational assumption. First, the Manning equation assumes steady and uniform flow^6,7, meaning that the flow remains constant over time and that the cross-sectional area of the flow (A_flow) does not vary along the flow path. In contrast, in irrigation furrows, the flow is often unsteady and nonuniform due to water infiltration and changes in furrow shape⁸. Second, the Manning equation assumes a prismatic flow cross-Sects^9–11. , whereas furrow irrigation typically involves nonprismatic cross-sections caused by plowing and the presence of clods. Additionally, the Manning equation is designed primarily for deep flows, making its use questionable in shallow surface irrigation, where flow characteristics vary significantly due to bed roughness^8,12.

Despite these limitations, the Manning equation is still widely used because of its simplicity and the absence of better alternatives. It is generally applied to describe flow conditions in surface irrigation once infiltration stabilizes (i.e., reaches the basic infiltration rate), which typically occurs during the storage phase^5,13. However, in other irrigation phases, such as the advance phase, where infiltration into the soil remains unstable and determining the flow depth, A_flow, and wetted perimeter (P_w) becomes nearly impossible, the accuracy of the Manning equation becomes even more uncertain. Therefore, applying the Manning equation to estimate the roughness coefficient during these phases may lead to unreliable results. To overcome these limitations, alternative methods and assumptions that allow for a more accurate estimation of the roughness coefficient are needed.

Numerous methodologies have been proposed for estimating Manning’s n, which are classified into five main categories: image-based methods, empirical tables, empirical relations, direct methods, and analytical or inverse solutions. Some of these methods have facilitated the creation of computer models such as SIPAR_ID ¹⁴, EVALUE¹⁵, and SIRMOD¹⁶, which simulate surface irrigation flow and estimate infiltration parameters and roughness coefficients. Other methods rely on analytical approaches or direct application of the Manning Eqs. 1^7–19. A review of several studies^{1,2,12,19–23} revealed that estimating Manning’s n requires extensive field data collection, making the process time-consuming and costly. This often leads to treating the roughness coefficient as a constant value via empirical tables and relations, despite the potential for significant estimation errors²³. However, over- or underestimating the Manning roughness coefficient can cause substantial errors in surface irrigation design and simulation²⁴. Focusing on changes in the roughness coefficient throughout irrigation is highly important for better understanding irrigation dynamics. Employing simplified approaches and assumptions is recommended to reduce the complexity of roughness estimation.

Machine vision systems, which utilize image processing and machine learning techniques, offer promising alternatives. These systems are widely used across industries and are increasingly applied in the agricultural sciences²⁵. In these systems, various features are first extracted from images via image processing techniques. In the next step, a relationship is established between these features and the values of the investigated parameter by applying machine learning methods (such as logistic regression (LR)²⁶, linear discriminant analysis (LDA)²⁷, k-nearest neighbor (KNN)²⁸, decision trees (CART)²⁹, random forest (RF)³⁰, support vector machine (SVM)³¹, multilayer perceptron (MLP) neural networks³², and simple Bayesian (NBGaussian)³³, allowing for the estimation of the desired parameter on the basis of the extracted relationship.

These techniques enable the extraction of useful information, such as plant size and number³⁴, soil moisture and infiltration rates^35–37, surface cracks^38,39, and soil characteristics^40–42. Given that the integration of machine learning and image processing techniques has proven successful in detecting clod size and number, soil type, surface roughness, and other related features, these techniques can likely also be effectively applied to estimate the Manning roughness coefficient. This is particularly relevant since the Manning roughness coefficient is affected by many of these parameters, either directly or indirectly.

In light of the challenges associated with the accurate estimation of Manning’s roughness coefficient (n) in furrow irrigation- particularly its spatial and temporal variability- this study addresses a significant knowledge gap in current methodologies. Traditional approaches often rely on expert judgment, empirical observations, and time-consuming field measurements, which limit their repeatability and practicality under varying field conditions. To bridge this gap, we introduce a novel framework that integrates high-resolution soil surface images and field-measured hydraulic parameters via image processing and machine learning techniques. The scientific contribution of this work lies in presenting a dynamic, data-driven, and practical method for estimating Manning’s n that enables faster, more accessible, and repeatable assessments without requiring advanced technical expertise or specialized instruments. This research hypothesizes that the simultaneous use of image processing and machine learning techniques can accurately estimate Manning’s n in furrow irrigation during both the advance and storage phases. Specifically, it is hypothesized that the Manning equation can be applied during the storage phase, whereas the SIPAR_ID model, which uses input velocity and flow depth data at specific points in time during the advance phase¹⁴, can be used for estimation in that phase. The main goal of this study was to assess the performance of the proposed method in Manning’s n estimation in furrow irrigation, during both the advance and storage phases. Certain variables (such as vegetation cover) that could complicate data collection and factors varying light conditions during imaging were eliminated to simplify the data collection and algorithm training process. Thus, bare furrows under consistent lighting conditions were used for the analysis. We assessed the performance of image processing and machine learning methods by averaging image data, field data, and hydraulic data both separately and in combination. In the next step, after the effectiveness of the developed algorithm was validated, various machine learning techniques (such as LR, LDA, KNN, CART, RF, SVM, MLP, and NBGaussian) were used to evaluate the accuracy of roughness coefficient estimation, across different scenarios, each considering a varying number of input data. The most significant scientific achievement of this study is the introduction of image processing and machine learning as a dynamic approach for estimating Manning’s n in furrow irrigation, capturing both temporal and spatial variations for the first time under various field conditions.

Materials and methods

Field experiments

The pilot fields were established at the University of Tehran’s experimental farm in Karaj, with geographic coordinates of 50°57’30.2” E and 35°48’18.5” N, where the annual average rainfall is 251 mm. The study was conducted on two experimental fields with distinct soil textures: silty clay loam and clay loam. The first field, designated Field F, has been under continuous cultivation for many years. In contrast, the second field, designated Field E, had remained fallow for several years. The physical and chemical properties of the soils from both fields are presented in Table 1.

Table 1.

Soil properties of the pilot fields.

Field	Depth (m)	Soil texture	ECe (dS/m)	pH	FC (g/g)	PWP (g/g)	Bulk density (g/cm3)
F	0–0.2	Clay loam	1.58	7.23	0.182	0.087	1.50
	0.2–0.4	Clay loam	1.03	7.37	0.175	0.081	1.45
	0.4–0.6	Sandy loam	0.93	7.11	0.142	0.600	1.47
E	0–0.2	Silty clay loam	1.63	7.20	0.172	0.080	1.51
	0.2–0.4	Silty clay loam	1.23	7.25	0.155	0.069	1.48
	0.4–0.6	Sandy clay loam	0.90	7.23	0.150	0.066	1.49

ECe: electrical conductivity of soil saturated extract, FC: field capacity, PWP: permanent wilting point.

This study aimed to explore the influence of key factors on Manning’s n, an essential parameter in surface irrigation. The research analyzed six different inflow rates, representing various furrow conditions and the maximum nonerosive rate. Furthermore, the study considered the effects of three irrigation events and their distinct phases: the advance phase, which covers the time water takes to travel from the field’s entrance to its end, and the storage phase, which spans from the completion of the advance phase until the cessation of water flow. In addition, this research examined how irrigation intervals (5-day vs. 10-day) and soil textures (Fields E and F) affect Manning’s roughness coefficient. These parameters were quantified through image-derived texture analysis and integrated into machine learning models as categorical variables, improving roughness prediction accuracy. The Field E experiments involved three low inflow rates (0.27 L/s) and three high inflow rates (0.57 L/s) at both intervals. Similar tests were conducted in field F, but the 10-day interval included six low and six high inflow rates to enhance the algorithm’s learning process. Data collection limitations prevented further expansion in Field E (due to the labor-intensive and costly nature of data collection). Overall, during the three events, 90 furrows were analyzed, 18 of which were in Field F and 12 of which were in Field E. The advance and storage phases were analyzed separately because of their critical influence on Manning’s n.

Upstream of the field, a tank with a volume of 2000 L was placed to mitigate fluctuations in water pressure and to allow water to flow by gravity, and the water was transferred to the experimental furrows by polyethylene pipes (two pipes with a diameter and a length of 40 mm and 35 m, respectively). In both fields, furrows with widths of 75 cm and lengths of 20 m were created (the length was considered as short as possible so that the images taken from the surface of the furrow would be a good representation of the overall condition of the furrow). Among these furrows, 10 stations in the furrow at a distance of 2 m from each other were considered to measure the advance and recession times. Type 2 Washington State College (WSC) flumes⁴³ were installed at both the inlet and outlet of each furrow to measure the inflow and outflow rates (Fig. 1). The water depth in the flumes was recorded every five minutes during the first hour and then at ten-minute intervals until the irrigation process was completed. Four rulers were placed along the furrows at 4-meter intervals to measure the water depth during irrigation, providing the data necessary to calculate the energy gradient, cross-sectional area, and wetted perimeter of the flow. All the furrows were irrigated for approximately 180 min.

Fig. 1.

Experimental farm and set up.

Given that, for furrows longer than 40 ft (12.2 m), the bed slope can be used in place of the energy gradient⁴⁴ - a substitution justified by hydraulic equivalence under uniform flow conditions where gravitational forces balance boundary friction- the bed slope of the furrows was determined and applied (Tables 2 and 3). This substitution maintains engineering accuracy when flow remains subcritical and channel geometry is consistent, conditions satisfied in the experimental setup.

Table 2.

Summary of data for pilot furrows in field E.

Furrow	IE	Qin (L/s)	S0 (m/m)	Fflow (cm2)	f0 (m3/m/min)	Tadv (min)	Trec (min)	Vinf (l)	Sd	Sw	M (%)
E10-1-1	1	0.34	0.0076	0.020	0.00013	27.5	194.6	860.2	3.0	2.5	9.32
E10-2-1	2	0.21	0.0076	0.019	0.00010	17.2	189.2	456.9	2.0	1.5	8.44
E10-3-1	3	0.16	0.0076	0.019	0.00015	18.4	191.0	513.4	2.0	1.5	12.56
E10-1-2	1	0.58	0.0072	0.023	0.00089	9.5	194.1	3919.5	2.0	1.5	9.96
E10-2-2	2	0.34	0.0072	0.022	0.00028	11.7	187.9	1197.6	1.0	0.5	11.78
E10-3-2	3	0.34	0.0072	0.023	0.00039	11.1	190.3	1337.0	1.0	0.5	10.55
E10-1-3	1	0.24	0.0076	0.026	0.00028	87.4	188.7	1660.8	2.0	1.5	4.45
E10-2-3	2	0.20	0.0076	0.027	0.00011	17.5	189.8	982.5	2.0	1.5	9.65
E10-3-3	3	0.17	0.0076	0.024	0.00022	14.6	189.2	1054.6	1.0	0.5	10.00
E10-1-4	1	0.09	0.0064	0.025	0.00005	36.3	188.6	332.5	3.0	2.5	7.90
E10-2-4	2	0.33	0.0064	0.023	0.00015	11.1	188.7	1214.3	2.0	1.5	9.45
E10-3-4	3	0.31	0.0064	0.022	0.00019	6.9	193.1	966.9	1.0	0.5	9.39
E10-1-5*	–
E10-2-5	2	0.19	0.0057	0.026	0.00032	26.3	187.0	1337.3	2.0	0.5	7.60
E10-3-5	3	0.19	0.0057	0.023	0.00018	15.6	189.0	931.1	1.0	0.5	13.96
E10-1-6	1	0.22	0.0050	0.025	0.00022	33.2	191.6	1038.3	3.0	1.5	2.06
E10-2-6	2	0.33	0.0050	0.025	0.00014	11.5	187.7	1052.2	2.0	1.5	9.90
E10-3-6	3	0.31	0.0050	0.022	0.00024	9.3	189.6	1300.9	2.0	1.5	16.24
E5-1-1	1	0.11	0.0068	0.022	0.00009	39.8	188.2	535.6	3.0	1.5	2.03
E5-2-1	2	0.20	0.0068	0.023	0.00016	11.0	187.2	665.2	2.0	1.5	15.33
E5-3-1	3	0.23	0.0068	0.023	0.00015	9.8	189.9	569.6	1.0	0.5	16.41
E5-1-2	1	0.31	0.0072	0.031	0.00027	20.0	188.6	1293.7	2.0	1.5	1.51
E5-2-2	2	0.35	0.0072	0.029	0.00022	7.8	186.6	880.2	2.0	1.5	15.84
E5-3-2	3	0.35	0.0072	0.028	0.00020	6.5	189.2	844.5	1.0	0.5	10.17
E5-1-3	1	0.14	0.0066	0.026	0.00032	79.7	186.4	1285.4	2.0	1.5	2.09
E5-2-3	2	0.20	0.0066	0.027	0.00024	14.5	187.3	931.2	2.0	1.5	28.68
E5-3-3	3	0.21	0.0066	0.024	0.00018	8.4	187.4	627.4	1.0	0.5	15.11
E5-1-4	1	0.30	0.0070	0.029	0.00028	15.1	186.7	1125.1	3.0	2.5	1.56
E5-2-4	2	0.35	0.0070	0.030	0.00024	8.2	187.3	899.7	1.0	0.5	31.69
E5-3-4	3	0.34	0.0070	0.026	0.00020	6.5	188.2	753.3	1.0	0.5	14.23
E5-1-5	1	0.20	0.0062	0.030	0.00043	37.8	191.3	1729.7	2.0	1.5	2.35
E5-2-5	2	0.21	0.0062	0.027	0.00013	13.1	190.9	515.4	1.0	1.5	14.38
E5-3-5	3	0.20	0.0062	0.029	0.00014	7.9	187.8	528.7	1.0	0.5	13.51
E5-1-6	1	0.33	0.0064	0.028	0.00020	10.8	192.4	877.5	2.0	1.5	2.95
E5-2-6	2	0.34	0.0064	0.026	0.00013	7.6	191.2	649.5	1.0	0.5	13.65
E5-3-6	3	0.34	0.0064	0.026	0.00019	5.3	188.3	740.3	1.0	0.5	11.99

IE: irrigation event, Q_in: inflow rate, S₀: slope of furrow bottom, F_flow: flow cross sectional area, f₀: basic infiltration rate, T_adv: advance time, T_rec: recession time, V_inf: infiltration volume, S_d: size of clods before irrigation, S_w: size of clods after irrigation, M: initial soil moisture (gravimetric), * No data.

E10-1-2: E is the field name, 10 is the ten-day irrigation interval, 1 is the first irrigation event, 2 is the furrow number.

Table 3.

Summary of data for pilot furrows in field F.

Furrow label	IE	Qin (L/s)	S0 (m/m)	Fflow (cm2)	f0 (m3/m/min)	Tadv (min)	Trec (min)	Vinf (l)	Sd	Sw	M (%)
F10-1-1*	–
F10-2-1	2	0.79	0.0075	0.021	0.000137	4.3	188.5	779.6	2.0	0.5
F10-3-1*	–
F10-1-2*	–
F10-2-2	2	0.42	0.0077	0.026	0.000116	5.9	188.5	436.8	1.0	0.5
F10-3-2*	–
F10-1-3	1	0.66	0.0072	0.025	0.00015	5.2	190.1	581.2	2.0	0.5	8.3
F10-2-3	2	0.82	0.0072	0.019	0.00013	3.6	190.6	480.0	1.0	0.5	10.6
F10-3-3	3	0.71	0.0072	0.019	0.0002	4.9	186.7	1235.2	1.0	0.5	20.7
F10-1-4	1	0.34	0.0065	0.02	0.00005	7.0	187.6	437.7	2.0	0.5	7.5
F10-2-4	2	0.43	0.0065	0.019	0.00031	6.7	188.7	1157.4	2.0	0.5	10.9
F10-3-4	3	0.33	0.0065	0.016	0.00018	8.8	188.4	949.7	1.0	0.5	19.8
F10-1-5	1	0.760	0.0075	0.024	0.00013	4.38	199.4	732.1		0.5	7.7
F10-2-5	2	0.37	0.0075	0.026	0.00047	7.1	195.0	1540.4	1.0	0.5	6.5
F10-3-5	3	0.75	0.0075	0.022	0.000327	5.2	96.5	843.6	1.0	0.5	5.7
F10-1-6	1	0.5	0.0073	0.026	0.00066	13.5	199.8	2719.8		0.5	6.6
F10-2-6	2	0.9	0.0073	0.023	0.00042	3.9	193.8	1785.0	1.0	0.5	6.6
F10-3-6	3	0.31	0.0073	0.022	0.00012	8.3	99.5	264.3	1.0	0.5	8.2
F10-1-7	1	1.28	0.0078	0.024	0.00068	5.1	186.0	5062.2	2.0	0.5	8.6
F10-2-7	2	0.78	0.0078	0.023	0.00008	3.9	190.1	562.2	1.0	0.5	9.2
F10-3-7	3	0.66	0.0078	0.022	0.000078	4.7	140.3	574.8	1.0	0.5	25.7
F10-1-8	1	0.41	0.0100	0.035	0.000557	10.7	185.3	2134.6	2.0	0.5	9.2
F10-2-8	2	0.42	0.0100	0.031	0.000057	6.7	188.6	392.1	1.0	0.5	9.3
F10-3-8	3	0.33	0.0100	0.03	0.000106	7.3	142.2	314.4	1.0	0.5	21.8
F10-1-9	1	0.76	0.0090	0.027	0.000158	4.0	190.5	980.1	2.0	0.5	6.8
F10-2-9	2	0.87	0.0090	0.026	0.00045	3.3	220.6	1955.9	1.0	0.5	18.8
F10-3-9	3	0.68	0.0090	0.02	0.0003	5.0	202.4	1137.8	1.0	0.5	5.6
F10-1-10	1	0.26	0.0076	0.022	0.000084	13.0	189.8	641.5	2.0	0.5	6.9
F10-2-10	2	0.44	0.0076	0.022	0.000171	6.5	219.4	701.6	1.0	0.5	30.2
F10-3-10	3	0.33	0.0076	0.016	0.000162	7.8	201.5	717.6	1.0	0.5	7.1
F10-1-11	1	0.6	0.0090	0.033	0.000192	3.8	192.5	888.7	2.0	0.5	7.4
F10-2-11	2	0.91	0.0090	0.031	0.000536	3.9	190.7	1887.8	1.0	0.5	9.1
F10-3-11	3	0.64	0.0090	0.023	0.00017	4.4	188.1	850.8	1.0	0.5	8.1
F10-1-12	1	0.43	0.0092	0.031	0.000338	11.2	192.8	1559.6	2.0	0.5	8.1
F10-2-12	2	0.54	0.0092	0.026	0.000512	8.0	189.2	2051.8	1.0	0.5	8.4
F10-3-12	3	0.4	0.0092	0.021	0.000228	7.3	188.1	979.3	1.0	0.5	7.2
F5-1-1	1	0.37	0.0110	0.023	0.000182	11.3	122.0	692.3	2.0	0.5	4.8
F5-2-1	2	0.43	0.0110	0.022	0.000167	6.6	187.9	628.4	1.0	0.5	9.5
F5-3-1	3	0.43	0.0110	0.022	0.000235	3.9	189.1	972.8	1.0	0.5	8.0
F5-1-2	1	0.27	0.0125	0.028	0.000268	14.9	122.8	842.1	2.0	0.5	6.9
F5-2-2	2	0.11	0.0125	0.027	0.000144	8.2	187.9	584.3	1.0	0.5	9.0
F5-3-2	3	0.31	0.0125	0.03	0.000131	4.4	188.6	465.9	1.0	0.5	8.1
F5-1-3	1	0.48	0.0110	0.038	0.000337	10.8	186.8	1621.6	2.0	0.5	6.7
F5-2-3	2	0.33	0.0110	0.037	0.000118	7.6	192.7	718.9	1.0	0.5	8.8
F5-3-3	3	0.33	0.0110	0.038	0.000101	7.7	189.2	464.4	1.0	0.5	7.4
F5-1-4	1	0.245	0.0138	0.024	0.000282	23.3	190.8	1294.2	2.0	0.5	6.4
F5-2-4	2	0.16	0.0138	0.021	0.000201	18.9	191.8	821.5	1.0	0.5	10.3
F5-3-4	3	0.19	0.0138	0.021	0.00011	11.6	187.8	535.8	1.0	0.5	10.3
F5-1-5	1	0.51	0.0133	0.03	0.000516	13.4	189.5	1923.2	2.0	0.5	5.4
F5-2-5	2	0.28	0.0133	0.036	0.000086	9.3	191.2	354.2	1.0	0.5	9.5
F5-3-5	3	0.37	0.0133	0.033	0.000294	7.8	194.5	1352.6	1.0	0.5	9.0
F5-1-6	1	0.16	0.0140	0.037	0.000289	25.4	189.1	1133.4	2.0	0.5	5.3
F5-2-6	2	0.18	0.0140	0.043	0.000068	17.0	190.6	348.8	1.0	0.5	9.3
F5-3-6	3	0.1	0.0140	0.038	0.000102	24.4	192.3	459.6	1.0	0.5	7.2

IE: irrigation event, Q_in: inflow rate, S₀: slop of furrow bottom, F_flow: flow cross sectional area, f₀: basic infiltration rate, T_adv: advance time, T_rec: recession time, V_inf: infiltration volume, S_d: size of clods before irrigation, S_w: size of clods after irrigation, M: initial soil moisture (gravimetric), * No data.

F10-1-3: F is the field name, 10 is the ten-day irrigation interval, 1 is the first irrigation event and 3 is the furrow number.

Before irrigation, the soil moisture content was assessed via the gravimetric method. From each furrow, three samples were collected from the topsoil layer (0–10 cm), and the average moisture content was then calculated for further analysis.

The cross-sectional area (A_f) of each furrow was assessed with a profile meter⁴⁵ at three predetermined points: the upstream Sect. (2 m from the starting point), the middle Sect. (10 m from the starting point), and the downstream Sect. (18 m from the starting point). These measurements were taken before and after each irrigation event via a profile meter. Furthermore, the flow cross-sectional area (A_flow) and wetted perimeter (P_w) were calculated by determining the flow depth at different intervals with rulers and a profile meter. The average values for both the cross-sectional area and wetted perimeter were documented before and after irrigation, and these values were utilized in later analyses⁴⁶.

A summary of the data collected from the experimental furrows in both fields is provided in Tables 2 and 3.

Determination of manning roughness coefficient

Manning’s n was independently calculated for each examined phase, and the method used to determine the roughness coefficient in these phases is detailed below:

Advance phase

Given the assumption of the Manning equation, its application in irrigation is not appropriate. This is due to infiltration during irrigation, which leads to a reduction in the flow cross-sectional area and, consequently, causes flow nonuniformity, making the equation unsuitable for such scenarios. However, owing to the lack of a suitable and practical equation and the simplicity of calculations, researchers use this relationship to explain the hydraulic conditions of surface irrigation flow after water infiltration stabilizes in the soil (reaching the basic infiltration rate).

Rather than directly applying Manning’s equation, this study employs the SIPAR_ID model to estimate Manning’s n as an effective hydraulic resistance coefficient during the advance phase. This coefficient encapsulates the combined influence of bed roughness and transient flow conditions, enabling practical simulation of advance-phase hydraulics even under deviations from the equation’s theoretical assumptions. Despite the absence of a theoretically superior alternative, Manning’s n remains a pragmatic and widely communicable metric for characterizing hydraulic resistance in transient scenarios. The SIPAR_ID model was selected for its proven capacity to simultaneously calibrate infiltration and resistance parameters using field-measured advance-phase data, as demonstrated in prior studies^46–48. In this model, the input consists of advance phase data, together with the inflow rate and flow depth at a given location and at multiple time points, to evaluate the infiltration parameters and Manning roughness coefficient¹⁴. Thus, it is likely useful for the estimation of Manning’s n in the advance phase. In this study, inflow rates and flow depth data were input into the model from the beginning of irrigation to the conclusion of the advance phase, allowing for the estimation of infiltration parameters and the Manning roughness coefficient throughout the phase.

Storage phase

When water is delivered to the furrow’s end, the storage phase begins and continues until the inflow stops. Almost one hour after irrigation began, the water infiltration rate into the soil leveled off, and the outflow from the furrows, which had a steady inflow, also became uniform. This finding indicates that sufficient time was available for infiltration in all furrows after the steady inflow reached the base infiltration rate (Tables 2 and 3). By the end of this period, uniform flow conditions were achieved along the furrow. Using Manning’s equation (Eq. (1)), the storage period was determined as the duration from when the base infiltration rate was established until the irrigation cutoff was reached.

1

where Q is the flow rate (m³/s), n is the Manning roughness coefficient (s/m^1/3), A is the flow cross-sectional area (m²), R is the hydraulic radius (m), and S₀ is the bottom slope (or hydraulic gradient) of the furrow (m/m).

Determining the accuracy of manning roughness coefficient Estimation

To evaluate the accuracy of Manning’s n estimation with the SIPAR_ID software during the advance phase, three statistical indicators were compared between the simulated and experimental advance times: the coefficient of determination (R²) (Eq. 2), the root mean square error (RMSE) (Eq. 3), and the relative error (RE) (Eq. 4). In contrast, during the storage phase, it was assumed that measurement errors in the field data were minimal. Therefore, the Manning equation was directly applied, and the calculated Manning roughness coefficients were taken as the effective values for this phase.

2

3

4

where Xi represents the measured data, Yi denotes the estimated data, N indicates the total number of experimental data points, and and represent the average measured and estimated data, respectively.

Image preparation

To estimate the Manning roughness coefficient, images were taken from the surface of irrigation furrows via a Canon PowerShot SX540 HS digital camera with a 20-megapixel sensor and a maximum image size (resolution) of 3888 × 5184. To cover the entire considered surface properly and stabilize the photography and lighting conditions in all the furrows, a cart with a height of one meter was made. The standardized imaging protocol in this study was intentionally adopted to isolate the algorithm’s core performance from confounding variables during development. In this study, 90 furrows were examined, and images of 9 specific points (at distances of 2, 4, 6, 8, 10, 12, 14 and 16 m from the beginning of the furrow) in each furrow were taken during two phases: advance and recession of irrigation. Therefore, a total of 1620 images were collected. A total of 1486 images were considered for the training and testing process after the noisy images and the images related to the furrows with missing data were removed (Tables 2 and 3). After that, 80% of the images were used for training (1190 images), and the remaining 20% of the images were used for testing (296 images).

Determining the index of size and number of clods

In addition to flow rate, initial soil moisture, and soil texture, other parameters, such as clod size and number, also impact the flow velocity in furrows, which in turn affects the Manning roughness coefficient. Although it is challenging to quantify the size and number of clods, an effort was made to consider the influence of this parameter as much as possible in the developed algorithm by defining indicators.

In addition, the flow rate, initial soil moisture, soil texture, and other parameters, such as the size and number of clods, influence the flow velocity in the furrow and consequently the Manning roughness coefficient. This is because it is difficult to determine the size and number of clods quantitatively; therefore, an effort was made to consider the influence of this parameter as much as possible in the developed algorithm by defining indicators. As a result, indices, namely, Sd and Sw, which represent the size and number of clods before and after irrigation in each furrow, respectively, were defined. The images taken from the surface of the furrows were classified into three groups, namely, high, medium and low clods, before and after irrigation according to the comments and opinions of three experts (Fig. 2). To quantify this index, the indices of the size and number of clods before irrigation (S_d) for large, medium and small clods equal 3, 2 and 1, respectively, and the values of the S_w index for large, medium and small clods equal 2.5, 1.5 and 0.5, respectively.

Fig. 2.

Index of size and number of clods before irrigation (a. high, b. medium, c. low).

Development of an algorithm for roughness coefficient determination

After collecting field data, preparing images, and determining Manning’s n, the images were analyzed via the Python programming language. A set of features was extracted from the images, and the effectiveness of different machine learning methods in classifying roughness was evaluated. Therefore, 80% of the images were used for training (1190 images), and 20% of the images were used for testing (296 images).

Feature extraction

Feature extraction is crucial in simplifying the extensive data contained within images by identifying essential patterns or attributes. This significantly aids in processes such as classification, segmentation, and modeling⁴⁹. Effective feature extraction not only reduces the dimensionality of the data but also enhances the performance of predictive models and analytical algorithms⁵⁰.

Different features in three ranges related to shape and size, texture and color were extracted from the images taken from the surface of the furrows (Fig. 3). For this purpose, seven features related to shape and size were extracted by using a statistical torque called Hu’s momentum⁵¹. Moreover, using the co-occurrence matrix method⁵² and the local binary pattern (LBP)⁵³, which have been widely used by other scholars^54–57, 16 and 156 textural features were extracted from the images, respectively. Finally, 512 color features were prepared via the HSV color space and its normalized histogram.

Fig. 3.

Examples of images taken from the soil surface in different furrows.

Examined scenarios

The Manning roughness coefficient depends on various factors, including soil characteristics, hydraulic flow conditions, and other related variables. Failing to consider any of these factors can introduce errors in estimating Manning’s n. In this study, the goal was to evaluate the efficiency of using image processing techniques combined with machine learning methods to estimate Manning’s n in furrow irrigation. For this purpose, it is crucial to provide an algorithm with all the parameters that influence the roughness coefficient as inputs to allow accurate estimation. Some of these features, such as soil bed roughness, the presence or absence of clods, and soil moisture, are visible in surface soil images. However, certain hydraulic features, such as flow velocity and cross-sectional area, are not detectable in the images. Therefore, in addition to images, it is likely that both hydraulic and nonhydraulic parameters must be provided to the algorithm for accurate estimation. To achieve this goal, various scenarios were defined to assess the algorithm using different sets of data.

The first scenario was designed to incorporate all hydraulic and nonhydraulic data collected in the field, along with furrow surface images. This approach allowed for training the algorithm with the largest available dataset and determining the highest possible accuracy.

Since determining some parameters in the first scenario was time-consuming or expensive (such as the flow cross-section, advance time, recession time, etc.), these parameters were excluded from the learning process for other scenarios. In the next step, another scenario was defined to evaluate the algorithm’s performance solely on the basis of images. Other scenarios, such as using images and hydraulic parameters, using images and nonhydraulic parameters, and using hydraulic and nonhydraulic parameters without images, were also examined. However, owing to the algorithm’s low efficiency in certain cases, the results of these investigations were not presented.

Finally, the accuracy of the developed algorithm was checked in each of the three selected scenarios to determine the best approach. A scenario in which the algorithm with the least amount of input data achieves the highest accuracy in estimating the roughness coefficient is selected. The scenarios are as follows:

The first scenario uses images and all measured field data, which include images plus the inflow rate, outflow rate, advance time, recession time, infiltrated water volume, irrigation event number, initial soil moisture, slope, final (basic) infiltration rate, index of size and number of clods before and after irrigation, electrical conductivity (EC) and pH of the soil, cross-sectional area of the furrow, cross-sectional area of the flow and wetted perimeter.

The second scenario uses only images taken from the surface of the furrows.

The third scenario uses images and field data, which include the inflow rate, slope, cross-sectional area of the furrow, number of irrigation events, and index of size and number of clods before irrigation.

Categorization

The Manning roughness coefficient is a continuous parameter and depends on various factors. The change in each of these factors can lead to a change in its value, which makes it difficult to estimate the roughness coefficient via machine vision. Therefore, in this study, for convenience, the roughness coefficient was categorized into different classes. For this purpose, the roughness coefficients before irrigation (roughness coefficient in the advance phase) were sorted from low to high, and the roughness values were placed in the positive and negative ranges of 10% in one group (class). Similarly, the roughness coefficients after irrigation (roughness coefficient in the storage phase) were also divided into different groups. Accordingly, the roughness coefficients after irrigation were classified into nine classes (0–8), and those before irrigation were classified into ten classes (9–18) (Table 4).

Table 4.

The value of manning roughness coefficient in the considered classes.

	After irrigation
Manning roughness coefficient	0.0175	0.0242	0.0327	0.04	0.05	0.064	0.0834	0.124	0.259
Class	0	1	2	3	4	5	6	7	8

	Before irrigation
Manning roughness coefficient	0.0212	0.028	0.0378	0.048	0.057	0.071	0.0921	0.1176	0.165	0.443
Class	9	10	11	12	13	14	15	16	17	18

According to the classification, if the algorithm correctly recognizes the class of each furrow, it is able to estimate the Manning roughness coefficient with a maximum error margin of 10%.

Classification methods

A variety of classification algorithms have been introduced within machine learning disciplines and information science, each tailored to solve specific problems on the basis of different statistical and mathematical principles^58,59. Below is a summary of several commonly utilized methods across various domains, emphasizing their features, advantages, and drawbacks.

Naive Bayes (NB)

The naive Bayes algorithm is grounded in Bayes’ theorem and operates under the presumption that all the features are independent of one another⁶⁰. Although this assumption may not always be accurate, naive Bayes is still effective, especially in applications such as text classification and spam detection⁶⁰. It performs well in situations where datasets are noisy and only limited training data are available, as it can generate reliable predictions even with minimal data⁶¹. However, the independence assumption can result in less-than-optimal performance when significant correlations exist among features. Various adaptations, including Gaussian, multinomial, complement, Bernoulli, and categorical models, have been developed for different data types⁵⁹.

Linear discriminant analysis (LDA)

LDA is a simple classifier that reduces data dimensions, increasing computational efficiency (particularly for high-dimensional datasets)⁶². It assumes that the data follow a normal distribution with equal covariance matrices across classes. While this makes LDA appropriate for linearly separable datasets, its effectiveness diminishes when the homoscedasticity assumption is not met. LDA is often compared with ANOVA and regression techniques because of their common statistical foundations.

Logistic regression (LR)

Logistic regression is a widely utilized probabilistic model for binary classification that estimates outcome probabilities by fitting data to a logistic or sigmoid function⁶³. One drawback of logistic regression (LR) is its reliance on the assumption of a straightforward linear relationship between independent variables and the log-odds of the dependent variable. This assumption may not be valid for more complex datasets. Regularization methods such as L1 and L2 are frequently used to prevent overfitting in high-dimensional datasets⁶².

K-nearest neighbors (KNN)

The K-nearest neighbor (KNN) algorithm is a nonparametric method that assigns data points to a class by considering the majority class among their closest neighbors⁶⁴. Its implementation is straightforward, and it works well for nonlinear decision boundaries. However, its performance heavily relies on the selection of ‘k’ (the number of neighbors) and the distance metric used. Additionally, KNN can become computationally intensive for large datasets since it requires storing all instances during prediction⁶².

Support vector machine (SVM)

SVM creates a hyperplane to differentiate classes in high-dimensional space, which is functional for both linear and nonlinear classification tasks, particularly when utilizing kernel functions such as polynomials, radial basis functions (RBFs), and sigmoid kernels. While SVMs are generally robust, they may struggle with noisy datasets where class boundaries overlap⁶⁵. Careful selection of kernels and hyperparameters is essential for achieving optimal results.

Decision tree (DT)

Decision trees represent intuitive and easily interpretable methodologies for classification tasks and regression tasks. Data are divided through a recursive process of partitioning feature values, resulting in a tree-like structure of decisions²⁹. However, decision trees are prone to overfitting, especially in noisy datasets. Techniques such as pruning have been developed to streamline tree structures and enhance generalization⁶⁶. Recent developments, particularly in analysis, have broadened the scope of decision trees^67,68.

Random forest (RF)

Random Forest is a machine learning technique that builds multiple decision trees during training and combines their outputs to improve accuracy while minimizing overfitting. ³⁰. By utilizing bagging (bootstrap aggregation) and random feature selection, the RF promotes diversity among its decision trees, resulting in more robust predictions^69,70. However, as the number of trees increases, RFs can become computationally demanding⁶².

Multilayer perceptron (MLP)

An MLP is a deep learning architecture composed of several layers of neurons. It employs backpropagation to modify the weights of connections between neurons, allowing it to learn intricate patterns within data⁵⁸. The MLP is capable of modeling nonlinear relationships but necessitates careful tuning of hyperparameters, including the number of layers, neurons, and iterations. Moreover, it is sensitive to feature scaling and may require significant computational resources when processing large datasets⁵⁹.

In this study, the effectiveness of these classification techniques for predicting Manning’s roughness coefficient was assessed. Given the time-consuming and expensive nature of field data collection, compiling a large dataset was impractical. To evaluate the accuracy of these algorithms, a cross-validation approach was implemented to ensure thorough training and testing.

Cross-validation is a statistical technique employed to assess and compare learning algorithms by partitioning the data: the first subset is utilized for training or developing the model, whereas the other subset serves to validate the model’s performance. In k-fold cross-validation, the dataset is initially divided into k equally sized subsets or folds. Following this division, the process undergoes k iterations of training and validation. In each iteration, a unique subset is reserved for validation, while the remaining k-1 subsets are utilized for training the model. ^71,72.

Finally, the average of these k validation times is chosen as the final estimate. In this research, the value of K was set to 10. Given that the training and testing process is reiterated K times and the average results are presented, training and testing have the necessary validity, and there is no need to perform algorithm evaluation operations using validation data.

After training and testing all the classification methods via the mentioned approach, the best classification for each scenario was determined.

After choosing the best method, the data of each scenario is divided into two sets, training (80% of the data) and testing (20% of the data), and the algorithm is trained again on these datasets. Finally, by examining different indicators, the classification accuracy of the algorithm (using the selected classification method) is determined, and the best scenario is selected.

Evaluating the accuracy of different methods via statistical indices

To check the status of the predictions, a confusion matrix was used. A confusion matrix represents the summary of the prediction in the form of a matrix. It shows the number of correct and incorrect predictions per class. In addition, it aids in identifying the classes that the model mistakenly categorizes as other classes⁷³. To evaluate the performance of the proposed algorithm, four statistical indicators, namely, recall (Eq. 5), precision (Eq. 6), accuracy (Eq. 7), and the harmonic mean of recall and precision (Eq. 8), which are widely used in classification problems⁴⁹, were used:

5

6

7

8

where TP denotes the number of samples classified as belonging to class A while these samples are correctly considered as part of this class, TN is the number of samples that are classified as non-A class, while these samples are not correctly considered as class A, and FP is the number of samples that are classified as part of class A, while these samples are not part of class A. FN is the number of samples that are classified as non-A class, while these samples are class A.

Results and discussion

Manning roughness coefficient values

The results obtained from Tables 5 and 6 reveal valuable insights into the hydraulic behavior of furrow irrigation in Fields E and F. Specifically, Manning’s roughness coefficient (n) was estimated for both the advance and storage phases, whereas the accuracy of the SIPAR_ID model in simulating the advance phase was carefully evaluated.

Table 5.

The evaluation indices for estimating advance times during the advance phase (using the SIPAR_ID model) for fields E and F are presented.

Furrow	Evaluation Index			Furrow	Evaluation Index
	R 2	RMSE (min)	RE (%)		R 2	RMSE (min)	RE (%)
E10-1-1	0.996	0.29	2.94	F10-1-1	–	–	–
E10-2-1	0.997	0.36	6.47	F10-2-1	0.998	0.26	1.74
E10-3-1	0.98	0.79	10.4	F10-3-1	–	–	–
E10-1-2	0.997	0.574	7.19	F10-1-2	–	–	–
E10-2-2	0.998	0.28	3.34	F10-2-2	0.999	0.21	3.38
E10-3-2	0.994	0.46	8.6	F10-3-2	–	–	–
E10-1-3	0.996	0.37	3.2	F10-1-3	0.999	0.16	1.6
E10-2-3	0.9975	0.2875	2.58	F10-2-3	0.995	0.46	8.3
E10-3-3	0.996	0.33	3.7	F10-3-3	1	0.08	0.9
E10-1-4	0.997	0.32	3.76	F10-1-4	0.99	0.74	12.02
E10-2-4	0.995	0.41	5.1	F10-2-4	0.998	0.25	3.25
E10-3-4	0.992	0.54	9.2	F10-3-4	0.999	0.18	2.16
E10-1-5	–	–	–	F10-1-5	0.999	0.21	2.1
E10-2-5	0.998	0.29	3.6	F10-2-5	0.99	0.57	7.55
E10-3-5	0.999	0.212	1.8	F10-3-5	0.998	0.28	2.5
E10-1-6	0.986	0.69	8.4	F10-1-6	0.98	0.82	12.4
E10-2-6	0.998	0.22	2.1	F10-2-6	0.999	0.15	1.14
E10-3-6	0.997	0.31	2.6	F10-3-6	0.999	0.19	1.6
E5-1-1	0.986	0.66	10.2	F10-1-7	0.997	0.52	3.65
E5-2-1	0.997	0.35	3.8	F10-2-7	0.999	0.17	1.3
E5-3-1	0.996	0.38	4.7	F10-3-7	0.999	0.28	2.1
E5-1-2	0.995	0.39	5.45	F10-1-8	0.994	0.43	6.38
E5-2-2	0.994	0.45	5.15	F10-2-8	0.999	0.17	1.45
E5-3-2	0.997	0.29	2.78	F10-3-8	0.999	0.13	1.36
E5-1-3	0.98	0.83	10.8	F10-1-9	0.999	0.22	3.9
E5-2-3	0.998	0.22	0.29	F10-2-9	1	0.1	1.3
E5-3-3	0.998	0.25	2.8	F10-3-9	0.999	0.15	1.9
E5-1-4	0.993	0.51	6.9	F10-1-10	0.991	0.56	9.1
E5-2-4	0.997	0.29	5.33	F10-2-10	0.999	0.16	1.8
E5-3-4	0.997	0.23	3.56	F10-3-10	0.998	0.22	2.47
E5-1-5	0.91	2.6	6	F10-1-11	0.998	0.29	2.7
E5-2-5	0.998	0.25	4.27	F10-2-11	0.999	0.26	3.24
E5-3-5	0.998	0.2	2.8	F10-3-11	0.998	0.51	9.47
E5-1-6	0.998	0.25	3	F10-1-12	0.997	0.29	2.75
E5-2-6	0.996	0.37	4.06	F10-2-12	0.997	0.31	4.38
E5-3-6	0.997	0.31	2.77	F10-3-12	0.992	0.68	12.1
				F5-1-1	0.994	0.42	3.8
				F5-2-1	0.999	0.14	1.4
				F5-3-1	0.999	0.19	1.9
				F5-1-2	0.993	0.47	4.23
				F5-2-2	0.999	0.15	1.8
				F5-3-2	0.996	0.32	4.97
				F5-1-3	0.994	0.43	1.2
				F5-2-3	0.997	0.26	3.3
				F5-3-3	0.993	0.46	6.5
				F5-1-4	0.998	0.269	1.85
				F5-2-4	0.998	0.26	3.23
				F5-3-4	0.997	0.31	4.32
				F5-1-5	0.998	0.22	2.35
				F5-2-5	0.999	0.17	1.1
				F5-3-5	1	0.13	1.7
				F5-1-6	0.999	0.19	3.13
				F5-2-6	0.99	0.56	6.97
				F5-3-6	0.996	0.36	5

Table 6.

Manning roughness coefficient during both the advance and storage phases for fields E and F.

Furrow	Advance phase	Storage phase	Furrow	Advance phase	Storage phase
E10-1-1	0.074	0.059	F10-1-1	–	–
E10-2-1	0.0432	0.0384	F10-2-1	0.031	0.023
E10-3-1	0.0237	0.02241	F10-3-1	–	–
E10-1-2	0.101	0.0853	F10-1-2	–	–
E10-2-2	0.0378	0.036	F10-2-2	0.04	0.034
E10-3-2	0.0172	0.0157	F10-3-2	–	–
E10-1-3	0.133	0.124	F10-1-3	0.0608	0.0345
E10-2-3	0.0898	0.066	F10-2-3	0.0392	0.027
E10-3-3	0.0439	0.0251	F10-3-3	0.0558	0.0334
E10-1-4	0.457	0.281	F10-1-4	0.18	0.081
E10-2-4	0.059	0.045	F10-2-4	0.026	0.015
E10-3-4	0.0538	0.0373	F10-3-4	0.0488	0.0432
E10-1-5	–	–	F10-1-5	0.09	0.026
E10-2-5	0.236	0.1411	F10-2-5	0.041	0.024
E10-3-5	0.1871	0.1123	F10-3-5	0.0456	0.0356
E10-1-6	0.178	0.083	F10-1-6	0.1	0.0667
E10-2-6	0.0882	0.064	F10-2-6	0.031	0.037
E10-3-6	0.0798	0.0635	F10-3-6	0.099	0.039
E5-1-1	0.636	0.317	F10-1-7	0.074	0.05
E5-2-1	0.0821	0.0786	F10-2-7	0.028	0.0267
E5-3-1	0.0228	0.0198	F10-3-7	0.051	0.026
E5-1-2	0.148	0.08	F10-1-8	0.15	0.075
E5-2-2	0.0386	0.0193	F10-2-8	0.055	0.037
E5-3-2	0.04	0.022	F10-3-8	0.065	0.052
E5-1-3	0.1799	0.179	F10-1-9	0.0416	0.036
E5-2-3	0.0539	0.0443	F10-2-9	0.038	0.0427
E5-3-3	0.0526	0.0448	F10-3-9	0.0487	0.0338
E5-1-4	0.159	0.0896	F10-1-10	0.1132	0.055
E5-2-4	0.05	0.0434	F10-2-10	0.052	0.031
E5-3-4	0.0471	0.0293	F10-3-10	0.054	0.047
E5-1-5	0.158	0.1179	F10-1-11	0.088	0.038
E5-2-5	0.0515	0.0212	F10-2-11	0.053	0.041
E5-3-5	0.0275	0.0166	F10-3-11	0.0425	0.024
E5-1-6	0.112	0.0722	F10-1-12	0.1163	0.087
E5-2-6	0.056	0.0387	F10-2-12	0.0539	0.0371
E5-3-6	0.0336	0.0227	F10-3-12	0.0476	0.039
			F5-1-1	0.075	0.03386
			F5-2-1	0.06	0.0255
			F5-3-1	0.056	0.0318
			F5-1-2	0.11	0.09
			F5-2-2	0.0629	0.0376
			F5-3-2	0.0473	0.029
			F5-1-3	0.07	0.0516
			F5-2-3	0.031	0.0168
			F5-3-3	0.027	0.0178
			F5-1-4	0.1	0.086
			F5-2-4	0.0471	0.023
			F5-3-4	0.047	0.022
			F5-1-5	0.148	0.074
			F5-2-5	0.0362	0.024
			F5-3-5	0.0268	0.015
			F5-1-6	0.101	0.0546
			F5-2-6	0.054	0.047
			F5-3-6	0.0627	0.061

In terms of model performance, SIPAR_ID demonstrated remarkable accuracy. The coefficient of determination (R²) ranged from 0.91 to 0.999 in Field E and from 0.98 to 1.00 in Field F, indicating a near-perfect match between the observed and simulated advance times. The Root Mean Square Error (RMSE) varied between 0.2 and 2.6 min in Field E and between 0.08 and 0.8 min in Field F. The relative error (RE) also remained within acceptable bounds, ranging from 0.29% to 10.8% in Field E and from 0.9% to 12.4% in Field F. Averaging across both fields, the R² reached 0.995, with an RMSE of just 0.363 min and an RE of 4.2%, clearly highlighting the robustness and precision of the SIPAR_ID model in predicting Manning roughness under varying field conditions.

These findings align closely with the results reported by Nie et al. (2014, 2018) and Ramezani Etedali et al. (2011), confirming that the SIPAR_ID model is a reliable tool for simulating surface irrigation hydraulics^47,48,75. Looking deeper into the hydraulic resistance behavior, the average Manning’s n values during the advance and storage phases in Field E were found to be 0.110 and 0.073, respectively. In Field F, the corresponding values were 0.064 and 0.041. These results clearly show that as the number of irrigation events increased, the roughness coefficient consistently decreased in both phases. This behavior is consistent with previous studies by Bahmani et al. (2020) and Zarakani et al. (2020), and it emphasizes the dynamic nature of hydraulic roughness in response to repeated irrigation^76,77. The underlying cause of this reduction in roughness is closely tied to soil surface transformations triggered by wetting and drying cycles between irrigation events. As shown in Fig. 4, these cycles contribute to the gradual disintegration of clods and smoothing of the soil surface. With time, the soil becomes more compacted and less resistant to water flow, thus decreasing Manning’s n value. This observation is not only supported by field measurements but also theoretically grounded in earlier works by Amiri et al. (2016), Harun-ur-Rashid (1990), Kassem & Ghonimy (2011), who emphasized the sensitivity of hydraulic roughness to soil moisture regimes and structural changes^1,17,79.

Fig. 4.

An image of one of the experimental furrows before (a) and after irrigation (b) (F10-1-8, 2018/07/09).

Evaluation of the efficiency of the developed algorithm

First scenario

The values of the accuracy, recall, and precision indices in evaluating the performance of different classifiers in the classification of roughness samples (algorithm training and testing were performed via the K-fold method, and K was set to 10) are shown in Table 7. The results indicated that in the first scenario, the random forest (RF) method performed the best in data classification. It achieved an accuracy, recall, and precision of 99%. In comparison, the decision tree (CART) and K-nearest neighbor (KNN) methods had the highest accuracies in estimating the Manning roughness coefficient class after the RF method.

Table 7.

Mean values of precision, accuracy and recall of the K-fold method in assessing the performance of various machine learning methods across different scenarios.

Scenario	LR	LDA	KNN	CART	RF	NB	SVM	MLP
First
Accuracy	0.61	0.71	0.91	0.95	0.99	0.44	0.69	0.84
Recall	0.66	0.73	0.93	0.95	0.99	0.51	0.73	0.87
Precision	0.61	0.73	0.92	0.95	0.99	0.53	0.68	0.86
Second
Accuracy	0.32	0.39	0.44	0.42	0.60	0.23	0.30	0.37
Recall	0.23	0.39	0.41	0.39	0.60	0.24	0.17	0.33
Precision	0.25	0.39	0.40	0.37	0.55	0.29	0.19	0.33
Third
Accuracy	0.44	0.50	0.71	0.85	0.85	0.30	0.45	0.55
Recall	0.46	0.51	0.73	0.86	0.88	0.36	0.44	0.56
Precision	0.43	0.53	0.72	0.85	0.85	0.37	0.41	0.56

Adab et al. (2020) investigated the efficiency of the random forest method in estimating soil moisture and reported that providing auxiliary data to the algorithm, in addition to images, can increase the accuracy and efficiency of this method³⁵. This finding aligns with the results of the present study, which also utilized quantitative data in conjunction with images.

First, the best method for estimating the roughness classifications was determined; then, by using 80% of the data for training, the random forest method was employed to develop the algorithm, with the remaining 20% set aside for testing purposes. The results are presented in Table 8. The average values of the precision, recall and F1-score indices for this classification method were the same and equal to 99%, which indicates the high accuracy of the random forest classifier in detecting different classes of the Manning roughness coefficient. The superior performance of Random Forest aligns with its well-documented capacity to process complex datasets while minimizing overfitting through ensemble-based learning⁸⁰. Unlike conventional linear approaches, this method excels at modeling nonlinear feature dependencies (a critical advantage for hydraulic roughness estimation). Field-collected data, which inherently contain measurement noise and variability, were handled more reliably by Random Forest compared to both k-nearest neighbors and single decision trees³⁰. While support vector machines demonstrated moderate accuracy in preliminary tests, their computational demands and sensitivity to parameter tuning reduced their practical utility for this application⁸¹. An important characteristic of RF is that it is relatively resistant to overfitting problems^35,82, which makes the algorithm resistant to errors caused by noisy data. According to the results, except for classes 4, 7, 12 and 13, the other classes were classified correctly with 100% accuracy.

Table 8.

Mean values of precision, recall and F1-score indicators in different classes of the developed algorithm using random forest category in the first scenario (Test data).

Class	Precision	Recall	F1-score	Data number for algorithm evaluation
0	1	1	1	16
1	1	1	1	5
2	1	1	1	11
3	1	1	1	15
4	0.91	1	0.95	31
5	1	1	1	28
6	1	1	1	8
7	1	0.86	0.92	21
8	1	1	1	9
9	1	1	1	13
10	1	1	1	3
11	1	1	1	29
12	1	0.88	0.93	16
13	0.94	1	0.97	31
14	1	1	1	13
15	1	1	1	13
16	1	1	1	21
17	1	1	1	6
18	1	1	1	5
Average	0.99	0.99	0.99	Total	296

The confusion matrix for the random forest method in the first scenario is displayed in Table 9. The results revealed that, except for class 7, in which 3 of the 21 data points were classified in the incorrect class (class 4), no incorrect classification was observed in the other classes. Table 10 shows an example of the images of classes 4 and 7. Classes 4 and 7 are both related to the storage phase (i.e., the furrows are wet); hence, the algorithm correctly separates the storage and advance phases. As shown in Table 10 (i.e., the images), the furrow bed in class 4 (e.g., F10-1-7) appears smoother than that in class 7 (e.g., E10-1-3). It was expected that the algorithm would correctly recognize this difference and provide an accurate classification. However, the inclusion of field data in network training means that the difference in substrates is not the sole factor influencing the selection of the roughness class. In fact, the complexity of analyzing the field images and data caused the algorithm to encounter difficulties in classifying some cases. It is difficult to find the cause of this incorrect classification simply by examining the images of the classes. It is also difficult to find the exact cause due to many complexities and, of course, the nature of machine learning methods, which act as black boxes⁸³. Moreover, inaccurate estimation of the Manning roughness coefficient may also cause inaccurate training of the algorithm and, consequently, incorrect classification. The inaccurate estimation of the Manning roughness coefficient may stem from uncertainties associated with the inverse solution of the equations employed in this study for roughness coefficient estimation. Consequently, multiple estimations with varying parameter values may yield identical results in the inverse solution process.

Table 9.

Confusion matrix of the developed algorithm in the first scenario (Test data).

	Class (real)
	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18
Class (estimated)
0	16	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
1	0	5	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
2	0	0	11	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
3	0	0	0	15	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
4	0	0	0	0	31	0	0	0	0	0	0	0	0	0	0	0	0	0	0
5	0	0	0	0	0	28	0	0	0	0	0	0	0	0	0	0	0	0	0
6	0	0	0	0	0	0	8	0	0	0	0	0	0	0	0	0	0	0	0
7	0	0	0	0	3	0	0	18	0	0	0	0	0	0	0	0	0	0	0
8	0	0	0	0	0	0	0	0	9	0	0	0	0	0	0	0	0	0	0
9	0	0	0	0	0	0	0	0	0	13	0	0	0	0	0	0	0	0	0
10	0	0	0	0	0	0	0	0	0	0	3	0	0	0	0	0	0	0	0
11	0	0	0	0	0	0	0	0	0	0	0	29	0	0	0	0	0	0	0
12	0	0	0	0	0	0	0	0	0	0	0	0	14	2	0	0	0	0	0
13	0	0	0	0	0	0	0	0	0	0	0	0	0	31	0	0	0	0	0
14	0	0	0	0	0	0	0	0	0	0	0	0	0	0	13	0	0	0	0
15	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	13	0	0	0
16	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	21	0	0
17	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	6	0
18	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	5

Table 10.

A sample of the images taken from the furrow surface in the first scenario classified in the classes 4 and 7.

In addition to class 7, in class 12, out of the total of 16 test data, the algorithm classified 2 data points in the wrong class and placed them in class 13 (Table 9). In this class, the algorithm correctly recognized the irrigation phase and was able to classify the roughness coefficient in the desired class with good accuracy (Table 10). In this class, according to Table 11, the visual difference that can be analyzed by the human eye is not visible, and no specific reason can be mentioned for this incorrect classification (e.g., F10-1-3 and F10-3-4 have different roughness values, but their images are very similar). Notably, owing to the great similarity of the images taken from the furrow surface, this amount of error might be sensible, although the field data collection greatly helped to reduce this error.

Table 11.

A sample of images taken from the furrow surface in the first scenario classified in the classes 12 and 13.

Finally, if all the data collected from the field and the images were used simultaneously, the model would classify the roughness coefficient values with an appropriate accuracy. However, since the determination of some input values to the model, such as the flow cross-sectional area and wetted perimeter, is time-consuming and difficult, using this scenario for roughness estimation is not desirable.

Second scenario

Figure 5 presents the results of the values of the precision, accuracy and recall indices in evaluating the accuracy of different machine learning methods via k-fold cross-validation in the second scenario. The results showed that in this scenario, as in the first scenario, the random forest (RF) method was the most accurate, and after that, the k-nearest neighbor (KNN) method and the decision tree (CART) were ranked next (Table 7).

Fig. 5.

Comparing the values of the Precision, Accuracy and Recall indices in evaluating the performance of different machine learning methods in the second scenario, calculated by the k-fold validation method.

The average values of the precision, accuracy and recall indices every 10-fold for the random forest method were measured as 60, 60 and 55%, respectively, which reflected the low accuracy of the different machine learning methods in this scenario. Accordingly, the images taken from the furrow, despite the abundance of information such as the irrigation phase, number of irrigation events, size and number of clods, cross-sectional area of the furrow, structure of the soil, and size of the soil cracks, do not provide any information related to the flow rate, slope, flow cross-sectional area, wetted perimeter, etc., because these parameters greatly influence the Manning roughness coefficient^13,19,21.

Table 12 shows the confusion matrix of the random forest method in classifying roughness data in the second scenario. The results showed that the algorithm classified the data into different classes, but in most classes, the data were not successful in correct classification. It is difficult to investigate the exact cause of misclassification, but this low accuracy is probably due to the influence of the lack of required data. In fact, two samples may be recognized as similar in terms of the shape and bed of the furrow in the images, whereas according to the inflow rate, slope, etc., they are in different classes in terms of the value of the roughness coefficient. Therefore, not providing enough field data can lead to a decrease in classification accuracy.

Table 12.

Confusion matrix of the developed algorithm in the second scenario (Test data).

	Class (real)
	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18
Class (estimated)
0	11	0	0	0	0	0	0	0	0	0	0	1	1	1	0	1	1	0	0
1	0	3	0	0	0	2	0	0	0	0	0	0	0	0	0	0	0	0	0
2	0	0	5	0	2	4	0	0	0	0	0	0	0	0	0	0	0	0	0
3	0	0	1	3	7	4	0	0	0	0	0	0	0	0	0	0	0	0	0
4	0	0	1	0	24	5	0	0	0	1	0	0	0	0	0	0	0	0	0
5	0	0	3	0	3	21	0	0	0	0	1	0	0	0	0	0	0	0	0
6	0	0	0	0	0	0	8	0	0	0	0	0	0	0	0	0	0	0	0
7	0	0	0	0	3	1	0	12	4	1	0	0	0	0	0	0	0	0	0
8	0	0	0	0	0	0	0	0	6	3	0	0	0	0	0	0	0	0	0
9	0	0	0	0	0	0	0	0	3	10	0	0	0	0	0	0	0	0	0
10	0	0	0	0	0	0	0	0	0	0	2	0	1	0	0	0	0	0	0
11	2	0	0	0	0	0	0	0	0	0	0	20	2	5	0	0	0	0	0
12	1	0	0	0	0	0	0	0	0	0	0	1	7	5	2	0	0	0	0
13	1	0	0	0	0	0	0	0	0	0	0	1	0	29	0	0	0	0	0
14	0	0	0	0	0	0	0	0	0	0	0	1	0	3	9	0	0	0	0
15	2	0	0	0	0	0	0	0	0	0	0	0	1	0	1	8	1	0	0
16	1	0	0	0	0	0	0	0	0	0	0	5	0	2	0	1	11	1	0
17	1	0	0	0	0	0	0	0	0	0	0	0	0	1	0	0	0	4	0
18	0	0	0	0	0	0	0	0	0	0	0	0	1	0	0	0	1	0	3

Table 13 shows an example of the images taken from the furrows in class 16 and classes 0, 11, 13 and 15. As stated, the developed algorithm classified samples into classes 16, 15, 13, 11 and 0. Moreover, all the samples should be placed in class 16. An examination of the images of the furrow surface in the wrong classes revealed that these classes do not differ much from each other in terms of the shape of the furrow bed. As an example, the image presented in Table 13, which is related to class 15, such as its corresponding image in class 16, is related to the advance and first irrigation phases. The second image in class 16, which corresponds to the image of the furrow in class 13, also has relatively similar bed conditions, and the phase and event of irrigation were selected correctly. The fourth image in class 16 is related to the third irrigation and before irrigation, but since the furrow was not completely dry and the color of the soil was darker, the algorithm recognized the image corresponding to it in class 0 and the storage phase. In other words, the training of the algorithm was relatively suitable, but since important and influential information on the roughness coefficient, such as the inflow rate and slope, was not available, the classification faced problems and was not able to achieve high accuracy in estimating the Manning roughness coefficient, despite extracting suitable features from the images.

Table 13.

An example of the images taken from the furrow surface in the second scenario classified in the classes 16, 15, 13, 11 and 0.

Third scenario

The validation results for all ten experimental folds revealed that the RF method, with values of the precision, accuracy and recall indices equal to 85, 88 and 85%, and the CART method, with values of these indices equal to 85, 86 and 85%, had the most suitable performance (Table 7).

By training and testing the RF and CART methods, which yielded the best results, the CART method provided the most accurate data classification, with average precision, recall and F1-score indices of 95%, 96% and 95%, respectively, in all 19 classes (Table 14). These results revealed that other field parameters, such as the flow cross-sectional area and wetted perimeter, salinity, acidity, initial soil moisture, index of size and number of clods after irrigation (and other parameters not considered compared with those in the first scenario), are of relatively low importance and can be neglected to speed up the estimation and classification of roughness samples. Moreover, measuring these parameters in the field is difficult and time-consuming, and ignoring them leads to a reduction in the cost and time required for data collection. The indicators showed that the efficiency of the algorithm in classifying samples decreased slightly (approximately 3–4%) compared with that of the first scenario, which can be ignored owing to the increase in the speed of data collection and the implementation of the algorithm and the ability to obtain the results.

Table 14.

Mean values of the precision, recall and F1-Score indicators in different classes of the developed algorithm using random forest category in the third scenario (Test data).

Class	precision	recall	F1-score	Data number for algorithm evaluation
0	0.89	1	0.94	17
1	1	0.8	0.89	5
2	0.77	1	0.87	10
3	0.93	0.87	0.9	15
4	0.94	0.97	0.96	34
5	1	0.96	0.98	28
6	1	1	1	8
7	1	0.85	0.92	20
8	0.86	1	0.92	6
9	1	1	1	14
10	1	1	1	3
11	1	1	1	23
12	1	1	1	15
13	0.94	0.86	0.9	35
14	0.93	1	0.97	14
15	0.94	1	0.97	16
16	0.89	0.89	0.89	18
17	1	1	1	8
18	1	1	1	7
Average	0.95	0.96	0.95	Total	296

In addition to the high performance under limited data conditions (Scenario III), the algorithm’s ability to estimate Manning’s n dynamically (considering phase-specific variations) marks a significant improvement over static or constant approaches traditionally used in furrow irrigation models.

Table 15 shows the confusion matrix of the CART method in the classification of roughness samples. The results revealed that the classification of classes 1, 3, 4, 5, 7, 13 and 16 was associated with errors in some samples. The algorithm classified 30 samples in class 13 out of a total of 35 samples, one sample in class 14, one sample in class 4, one sample in class 3 and two samples in class zero, which indicates that the distribution of incorrect classifications is greater than that of other classes.

Table 15.

Confusion matrix of the developed algorithm in the fourth scenario (Test data).

	Class (real)
	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18
Class (estimated)
0	17	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
1	0	4	0	0	1	0	0	0	0	0	0	0	0	0	0	0	0	0	0
2	0	0	10	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
3	0	0	2	13	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
4	0	0	1	0	33	0	0	0	0	0	0	0	0	0	0	0	0	0	0
5	0	0	0	0	0	27	0	0	1	0	0	0	0	0	0	0	0	0	0
6	0	0	0	0	0	0	8	0	0	0	0	0	0	0	0	0	0	0	0
7	0	0	0	0	0	0	0	17	0	0	0	0	0	0	0	1	2	0	0
8	0	0	0	0	0	0	0	0	6	0	0	0	0	0	0	0	0	0	0
9	0	0	0	0	0	0	0	0	0	14	0	0	0	0	0	0	0	0	0
10	0	0	0	0	0	0	0	0	0	0	3	0	0	0	0	0	0	0	0
11	0	0	0	0	0	0	0	0	0	0	0	23	0	0	0	0	0	0	0
12	0	0	0	0	0	0	0	0	0	0	0	0	15	0	0	0	0	0	0
13	2	0	0	1	1	0	0	0	0	0	0	0	0	30	1	0	0	0	0
14	0	0	0	0	0	0	0	0	0	0	0	0	0	0	14	0	0	0	0
15	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	16	0	0	0
16	0	0	0	0	0	0	0	0	0	0	0	0	0	2	0	0	16	0	0
17	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	8	0
18	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	7

Bolded values in the table indicate the number of correctly classified samples for each respective class.

Table 16 shows a sample of the images taken from the furrows in class 13 and other classes in which the sample of class 13 is incorrectly classified. In class 13, the algorithm failed to recognize the correct irrigation phase, and except for the samples classified in class 14 (samples F5-2-6 and F10-3-8), the other samples in the incorrect class were placed in the class with a storage phase; that is, class 13 was related to the advance phase and dry furrows. The algorithm classified 4 out of 35 tested samples in classes with a storage phase (e.g., E5-2-3 (class 13) and E5-2-2 (class 0)). In contrast, this type of classification was also observed in class 7. In other misclassified classes, the irrigation phase has been correctly detected. The cause of this type of misclassification is not clear owing to the complexity of the algorithm, but as mentioned earlier, the influence of uncertainty in estimating the Manning roughness coefficient and the occurrence of such errors cannot be ignored.

Table 16.

A sample of images taken from the furrow surface in the fourth scenario classified in the classes 13, 14, 4, 3 and 0.

Although the proposed algorithm was calibrated under specific imaging conditions—namely uniform lighting and fixed camera height—the core concept combining image processing with machine learning remains inherently flexible. Controlled setups in this study helped establish baseline performance metrics and validate the approach. For real-world deployment, robustness under varied field conditions must be prioritized. To address this, future implementations will follow two key strategies: (1) integrating deep learning architectures capable of extracting invariant features under diverse lighting and image quality, and (2) expanding training datasets using farm-acquired images that include variation in illumination, camera angle, resolution (smartphone to DSLR), and vegetative cover. Moreover, ongoing development must address limitations introduced by discretizing continuous Manning’s n values, and consider additional hydraulic parameters to improve predictive accuracy. Enhancing algorithm responsiveness—especially estimation speed—will be critical for integration into real-time platforms. Ultimately, this phased transition not only preserves scientific integrity, but also ensures adaptability to dynamic farm contexts, positioning the framework for global relevance in precision agriculture and compatibility with established hydraulic modeling systems (such as SIRMOD and WinSRFR).

Conclusion

This study introduces a novel algorithm that couples image processing with machine learning techniques to dynamically and efficiently estimate Manning’s n during both the advance and storage phases.

Among the three input scenarios evaluated, the first—combining comprehensive field measurements with furrow imagery—achieved the highest prediction accuracy (95–96%). Excluding hydraulic and soil features in the second scenario led to a significant drop in classification precision, highlighting the importance of these parameters in model accuracy. Scenario III—utilizing images alongside selected field parameters such as inflow rate, slope, cross-sectional area, clod index, and irrigation event number—maintained acceptable accuracy with only a 3–4% reduction, while reducing field data collection requirements by approximately 80%. This approach offers a practical balance between precision and feasibility for real-world applications.

Although the algorithm performed well under controlled imaging conditions (furrows without vegetation, fixed camera height and lighting), its generalizability to variable farm environments is constrained. Limitations related to vegetative cover, natural lighting variability, and discretized roughness classification may compromise model robustness. To address these limitations, future efforts will focus on integrating deep learning architectures for illumination-invariant feature extraction and expanding training datasets using diverse imagery captured under natural field conditions—ranging from smartphones to DSLR cameras. Moreover, integration with real-time soil moisture sensors, crop growth models, weather data, and other relevant inputs will enable closed-loop, adaptive surface irrigation control. While the precise and dynamic estimation of Manning’s n is a critical component, its value is magnified when combined with other parameters to support fully adaptive surface irrigation platforms. These systems would enable automatic, data-driven water delivery aligned with real-time field conditions, maximizing resource efficiency and minimizing waste. This multi-layered integration has the potential to enhance scenario-driven irrigation strategies and refine scheduling decisions with field-scale precision granularity.

From both scientific and practical standpoints, the proposed framework serves as a foundation for next-generation irrigation tools. It not only advances research in dynamic hydraulic modeling, but also bridges the gap between academic development and on-farm applicability—empowering farmers with accessible, intelligent solutions. These advancements position the methodology to support global precision agriculture and integration with established hydraulic modeling platforms.

Author contributions

HRR, HE and AL conceptualized the research topic and formulated the objectives and methodology of the research. HRR, FK and MSA carried out the field experiments as well as the data curation of the research work. H.R.R. and M.O. developed the machine learning algorithm, including image processing pipelines, feature extraction methodologies, and model optimization. HRR, HE and AL participated in the writing of the manuscript. All the authors read and approved the final manuscript for publication.

Funding

This study was funded by the University of Tehran Science and Technology Park under Grant no. 5888656.

Data availability

All data generated and analyzed during this study are included in this article.

Declarations

Competing interests

The authors declare no competing interests.

Disclosure of the use of AI in scientific writing

During the preparation of this work, the author(s) used ChatGPT to assist in grammar correction and editing of manuscript content. After using this tool, the author(s) reviewed and edited the content as needed and take(s) full responsibility for the content of the publication.