Predicting sugar beet leaf area index: evaluating performance of double sigmoid functions under different irrigation and plant density scenarios

Plant growth is very important in order to evaluate and manage irrigation and agricultural operations, with the aim of increasing production. Hence, plant indices, such as the leaf area index (LAI), are crucial to determining the photosynthetic surface area of a plant, its water consumption, and its yield¹. LAI measures the total one-sided leaf area per unit ground area, reflecting canopy density². Due to the difficulty of measuring LAI directly on a large scale³, different sigmoid functions can be used to estimate LAI.

A double sigmoid function generally has two sigmoidal-shaped curves, providing a better fit for modeling certain biological and ecological processes. Nonlinear regression models best describe growth curves, with parameters estimated by minimizing the residual sum of squares⁴. Therefore, LAI can be modeled as a double sigmoid curve, formed by the subtraction (or interaction) of two underlying sigmoid functions. Several singular sigmoid functions have one inflection point and have been used to model growth dynamics, including the Logistic⁵, Gompertz⁶, Beta⁷, Richards⁸, Von Bertalanffy⁹, Weibull¹⁰, and Hill¹¹. Double sigmoid functions exhibit two inflection points where the growth rate transitions from an increasing to a decreasing trend¹². By fitting growth data to sigmoid functions, researchers can better understand and predict plant performance under varying environmental and management conditions.

Despite their widespread application in modeling crop growth, sigmoid functions have several limitations that need to be acknowledged. Firstly, these models assume smooth and predictable growth patterns with clear inflection points. However, actual crop development often shows irregularities due to environmental stress, nutrient variability, or pest incidence, leading to inaccurate predictions. Secondly, parameter estimation in nonlinear regression is highly sensitive to initial values, making the calibration process computationally challenging and prone to convergence issues. Thirdly, double sigmoid models, while more flexible, involve additional parameters that increase the risk of overfitting to calibration data, reducing model robustness when applied to independent datasets or under stress conditions. Finally, these models primarily represent time-dependent growth without directly accounting for dynamic environmental drivers such as humidity, wind speed, and soil water status^13,14,15,16. Addressing these limitations may require hybrid modeling strategies or integrating additional meteorological and soil parameters into future predictive.

A study was conducted on five maize cultivars to evaluate the performance of different nonlinear functions (Richards, Logistic, Weibull, and Gompertz) in fitting leaf growth data¹⁶. The results indicate that the Richards, Logistic, and Gompertz functions demonstrate superior performance compared to Weibull in predicting leaf growth in maize. In another study¹⁷, reported that the growth behavior of a lettuce canopy was described using three nonlinear functions: Gompertz, Logistic, and grey Verhulst¹⁸. The functions were applied to top projected canopy area (TPCA), top projected canopy perimeter (TPCP), and plant height (PH). The grey Verhulst model showed better fitting for TPCA and TPCP growth, while the Logistic model fit well the PH changes. Also¹⁹, demonstrated that the Logistic model was more suitable for describing height growth of maize than the Gompertz function, as it achieved a coefficient of determination exceeding 99%. In a recent study by²⁰, it was demonstrated that the XGBoost function and Light GBM function outperformed the Gompertz function and Logistic function in predicting maize plant growth. The accuracy of 49 function compositions of the double sigmoid functions in describing rapeseed dry matter based on days after planting (DAP) and growing degree days (GDD) was examined¹². The results indicated that the Beta-Richards and Richards-Gompertz functions accurately represented the measured dry matter during the growing season based on DAP and GDD, respectively.

Sugar beet is primarily grown in arid and semi-arid regions as an irrigated crop. Adequate water supply and management are crucial for the successful production of sugar beet and white sugar due to the crop’s long growing season²¹. The water requirement for sugar beet cultivation is influenced by various factors such as climatic conditions, irrigation practices, growing season, crop density, genotype, and the use of nitrate fertilizer²². A water-saving technique commonly used in arid regions is deficit irrigation²³, which involves applying less water than the full crop requirement for evapotranspiration²⁴. Optimizing crop density is a key factor in enhancing both the quantity and quality of sugar beets. Crop density directly impacts root size, sugar content, yield, and mineral composition in the roots²⁵.

Based on our knowledge, few studies were conducted on determining the LAI of sugar beet. Given the significance of sugar beet in sugar production, this study was aimed to (1) estimate the LAI of sugar beet using different double sigmoid functions (Logistic-Logistic, Gompertz-Gompertz, Logistic-Hill, Weibull-Logistic, and …) under varying irrigation treatments and crop densities under direct sowing and transplant cultivations, (2) assess the LAI of sugar beet based on days after planting (DAP) and growing degree days (GDD), and (3) select the best double sigmoid functions. The findings of this study can help in estimating the LAI of sugar beet throughout its growth cycle.

Field experiment

The impact of plant density, planting method and irrigation regime on sugar beet yield over two years (2017 and 2018) at the Research Station, School of Agricultural, Shiraz University, Iran, situated 16 km north of Shiraz was studied by^26,27. In this study LAI data, depth of irrigation water, and rainfall amount were obtained from^26,27. The total amount of rainfall during growing period was 22.5 mm in 2017 occurred in the first week and 55 mm in 2018 occurred in first and second week of planting period²⁷.

The experiment used a split-split plot experimental design arranged in a complete randomized block framework with three replications. The main plots were assigned to three irrigation levels: full irrigation (100%, I100), 75% of full irrigation (I75), and 50% of full irrigation (I50). Two planting methods — direct sowing (D) and transplant cultivation (T)— were used in the subplots. The sub-subplots consisted of four plant density treatments of 180,000 plants ha^− 1 (P180), 135,000 plants ha^− 1 (P135), 90,000 plants ha^− 1 (P90), and 45,000 plants ha^− 1 (P45). Data from 2017 were employed to calibrate the double sigmoid function, and its predictive performance was validated using data from 2018. For more detailed information, please refer to^26,27.

Mathematical functions

Among the various mathematical models used to describe plant growth curves, sigmoid functions (such as the Logistic, Gompertz and Richards functions and …) are particularly effective. They accurately capture the three distinct phases of biological growth: the lag phase (characterized by slow initial development), the log phase (marked by rapid growth), and the stationary phase (where growth is slow and reaches plateau). The various sigmoid equations used in this research are as follows:

where y represents the leaf area index (dependent growth parameter), bounded between a minimum value y_min and maximum value y_max. The independent variable t corresponds to the growth period, measured either in days after planting (DAP) or growing degree days (GDD). The curve’s shape and transition rates are governed by three constant coefficients: “a” controls the inflection point position, “b” determines the growth rate, and “v” adjusts the curve’s asymmetry. By combining singular sigmoid functions, various double sigmoid functions can be created to predict LAI as follows:

where ʘ is the operator × or / and (:{text{f}}_{1}left(text{x}right)) and (:{text{f}}_{2})(x) are one of singular sigmoid functions (Eqs. 1–7), respectively. The double sigmoid functions for predicting LAI do not exhibit a purely increasing trend; the LAI rises up to a certain point (LAI_max), then stabilizes, and after a while begins to decrease, LAI_min is the LAI at initial growth stage, and LAI_end is the LAI at end of growing season. The double sigmoid functions are presented in Table 1. The reason for 15 selected widely used double sigmoid functions [similar to 12] was to avoid unrealistic shapes or excessive parameterization. The order of combination was also crucial as the first function primarily influences the early growth trajectory, while the second governs the senescence phase.

Determining coefficients of equations

The coefficients (a, b, c, d and v) of the functions in Table 1 were calculated using Excel software, based on the 2017 dataset (calibration year). Solver, a powerful tool in Excel, was utilized to solve the nonlinear equations. By minimizing the error value and employing the GRG (Generalized Reduced Gradient) method as the problem-solving technique, Solver determined the optimal coefficients for the functions. The objective function was minimized by sum of square error (SSE) as follows:

where (:{text{P}}_{text{i}}) and (:{text{M}}_{text{i}}) are the predicted and measured values by the double sigmoid function, respectively and n is the number of observations.

All the calculations were conducted based on days after planting (DAP), however, cumulative heat units was used in the sigmoid growth function to make the results more applicable for various environmental conditions. The cumulative heat units, determined as growing degree days (GDD), are calculated as follows:

where (:{text{T}}_{text{a}text{v}text{e}}) is the mean daily air temperature (°C) and (:{text{T}}_{text{b}text{a}text{s}text{e}}) is the base temperature (°C) or the air temperature threshold below which plant growth does not occur. In this study, base temperature was 2.6 °C²⁸. As temperature is used to calculated GDD, the results will be more applicable for different regions with different climates.

Linking double sigmoid function coefficients to agronomic variables

According to²⁹, the values of the coefficients of double sigmoid functions (a, b, c, d and v) are influenced by factors related to agronomic management practices. The values of constants in Eqs. (1-7) were fitted to quadratic functions of plant density and the total amount of seasonal applied water and rainfall, using SPSS software.

where CON is the coefficients of a, b, c, d, and v in Eqs. (1–7), I and R are the seasonal applied water and rainfall (m), respectively, P is the plant density (plant (:{text{m}}^{-2})), and A₁, B₁, C₁, D₁, E₁, F₁, G₁, H₁, and I₁ are the coefficients of multiple regression.

Statistical parameters

To evaluate the model, normalized root mean square error (NRMSE), the index of agreement (d), and mean residual error (MRE) were used. These parameters are calculated as follows³⁰:

where (:{text{L}text{A}text{I}}_{{text{P}}_{text{i}}}), (:{text{L}text{A}text{I}}_{{text{m}}_{text{i}}}) and (:{text{L}text{A}text{I}}_{{text{m}}_{text{a}text{v}text{g}}}) are the predicted, measured and mean of measured leaf area index, respectively, and n is the number of observations. MRE and NRMSE with smaller values and d with higher value indicates higher precision and accuracy of the model. Values of NRMSE less than 0.1 indicate excellent estimation accuracy, values between 0.1 and 0.2 show good estimation accuracy, values between 0.2 and 0.3 indicate fair estimation accuracy, and values of NRMSE greater than 0.3 show poor estimation accuracy³¹.

Fitting functions to leaf area index based on GDD

The coefficients of the sigmoid functions (using solver tool) were determined for all 15 functions under various irrigation levels and planting densities in both direct sowing and transplant cultivation. Among 15 functions, the Logistic-Logistic, Gompertz-Gomperz, Hill-Hill, Logistic-Richards, and Logistic-Hill functions performed more accurately considering the statistical indicators (Table S1). Based on the statistical parameters in Tables S1, the Logistic-Logistic, Gompertz-Gompertz, Hill-Hill, Logistic-Richards and Logistic-Hill double sigmoid functions yielded the most favorable outcomes for both direct sowing and transplant cultivation. The estimated coefficient values and statistical results of the top five performing double sigmoid functions are provided in Tables 2 and 3 for I100, four plant densities, and two cultivation methods. Also, the estimated coefficient values of the top five performing double sigmoid functions for all irrigation treatments, plant densities and cultivation methods are provided in Tables S2-S6.

In direct cultivation (Table 2), the Logistic-Richards function for direct sowing showed the best overall performance, with the lowest NRMSE (0.04) and a high index of agreement (0.99), indicating minimal deviation between the predicted and measured values. Similarly, the Logistic-Hill function also performed strongly under direct sowing conditions (NRMSE = 0.04, d = 0.99, and MRE= −0.004) (Table 2).

Under optimal irrigation water conditions (Table 2), all models showed strong fits with NRMSE values mostly below 0.05. The Logistic-Richards, Logistic-Logistic, and Logistic-Hill functions performed slightly better under high planting densities, with minimal errors (NRMSE = 0.008 at 180,000 plants ha^− 1). The Gompertz-Gompertz and Hill-Hill functions had higher NRMSE values in comparison with latter models, especially at 135,000 and 45,000 plants ha^− 1 (0.04 and 0.04, respectively). Overall, all functions provided adequate fits under well-watered conditions, with Logistic-Richards and Logistic-Hill showing more consistent accuracy across planting densities.

At 75% irrigation level (data not shown), model accuracy decreased slightly, but NRMSE values remained below 0.1 for all functions. Logistic-Richards, Logistic-Logistic, and Logistic-Hill maintained high accuracy (NRMSE range: 0.03–0.07). Hill-Hill had minor increases in error, while Gompertz-Gompertz showed higher sensitivity to planting density under water deficit condition, with NRMSE values reaching 0.09 at 45,000 plants ha^− 1.

Under severe irrigation stress, 50% level (data not shown), model performance varied significantly. Logistic-Richards consistently outperformed all others with low error values across all densities (NRMSE range: 0.04–0.09). Logistic-Hill showed comparable performance, especially at intermediate densities. Logistic-Logistic and Gompertz-Gompertz had substantial performance degradation, with Logistic-Logistic reaching a NRMSE of 0.150 at 90,000 plants.

ha^− 1 and Gompertz-Gompertz peaking at 0.12 at 45,000 plants ha^− 1. Hill-Hill showed moderate robustness, but Logistic-based models outperformed it under stress. The findings highlight the Logistic-Richards model’s adaptability in modeling asymmetric growth patterns typical under drought stress conditions. To maintain clarity and conciseness, the model comparison was conducted specifically under the direct sowing condition.

In transplant cultivation (Table 3), the Logistic-Hill function again performed competitively, achieving the lowest NRMSE (0.06) and a high index of agreement (0.99) among the functions considered, with a near-zero MRE (−0.004). Although the Logistic-Logistic function achieved the same index of agreement (0.99) under transplant conditions, it presented a higher NRMSE (0.06) and a positive MRE (0.04), indicating a slight overestimation trend. The performance of models under different irrigation water levels, planting densities, and transplant cultivation were similar to that in direct cultivation (Table 3).

The goodness-of-fit parameters presented in Table 2 strongly support this observation. The Logistic–Richards function demonstrated the best performance with the lowest NRMSE (0.04) and the highest index of agreement (d = 0.99), indicating its superior accuracy and reliability under direct cultivation, while the Logistic-Hill performed the best (NRMSE = 0.064, d = 0.99, MRE=−0.004) under transplant condition (supplementary figure, Fig. S1). The Hill–Hill function also performed well, with NRMSE = 0.05 and d = 0.99, showing excellent predictive agreement across various planting densities under direct cultivation. The Logistic–Hill and Logistic–Logistic functions followed closely, with NRMSE values of 0.04 and 0.06 under direct cultivation and 0.064 and 0.063 under transplant condition, respectively (Fig. S1). All functions exhibited minimal mean relative error (MRE), typically ranging from − 0.01 to − 0.004, indicating no systematic bias in their predictions. The consistency of performance across planting densities (180,000 to 45,000 plants ha^− 1) under full irrigation (I100) further underscores the robustness of these models.

The Beta–Beta, Logistic–Gompertz, and Beta–Logistic functions showed moderate fitting performance for both cultivation methods (Fig. 1). The Beta–Beta function had NRMSE values ranging from 0.09 to 0.11, with index of agreement (d) values of 0.99 to 0.99 and close-to-zero MRE values (−0.004 to 0.0006). The Logistic–Gompertz function had NRMSE values of 0.08 and 0.09, with d values between 0.99 and 0.99, but positive MRE values (0.02 and 0.01) indicating a tendency to overestimate LAI. The Beta–Logistic function performed the weakest, with NRMSE values of 0.15 and 0.12, d values of 0.98 and 0.99, and the largest MRE of 0.04, showing overestimation of LAI. The Beta–Beta, Logistic–Gompertz, and Beta–Logistic functions exhibited greater deviation and less reliability in capturing LAI dynamics, compared to top-performing functions like Logistic–Richards and Hill–Hill.

Moreover, variation of measured and predicted leaf area index by double sigmoid functions corresponding to 100% irrigation (I100) and a planting density of 180,000 plants ha^− 1 (P180) for both cultivation methods during 2017 growing season (calibration year) are presented in Fig. 1. According to Fig. 1, the Logistic–Gompertz function modeled LAI with high accuracy under transplant conditions, especially when GDD was used as the time scale. Under direct sowing, the function also performed well, though it slightly overestimated early vegetative growth. The Beta–Logistic function slightly underfitted the initial LAI rise in direct sowing, while overestimating peak values in transplant cultivation. These observations suggest that such combinations are sensitive to cultivation-specific growth dynamics and may not generalize well without calibration. Among 15 selected double sigmoid functions, certain functions consistently showed poor performance, especially when specific functions were used as the first stage in the double sigmoid sequence. These included Weibull–Weibull, Von Bertalanffy–Von Bertalanffy, Logistic–Beta, Logistic–Weibull, Weibull–Logistic, and Von Bertalanffy–Logistic. The Weibull–Weibull function had high errors (NRMSE = 0.25 for direct sowing, 0.36 for transplanting) and slight under and overestimation (MRE= −0.005 for direct sowing, 0.11 for transplanting). The Von Bertalanffy–Von Bertalanffy function showed fair and poor results (NRMSE = 0.28 for direct sowing, 0.38 for transplanting) and slight overestimation (MRE = 0.04 for direct sowing, 0.005 for transplanting). The Logistic–Beta function performed fair and poor (NRMSE = 0.26 for direct sowing, 0.36 for transplanting) with slight overestimation (MRE = 0.01 for direct sowing, and 0.01 for transplanting). The Logistic–Weibull function was more accurate for direct sowing (NRMSE = 0.21, MRE= −0.01) but less accurate for transplanting (NRMSE = 0.38, MRE = 0.13). The Weibull–Logistic function was more accurate for direct sowing (NRMSE = 0.12, MRE = 0.005) in comparison with that for transplanting (NRMSE = 0.45, MRE = 0.11). The Von Bertalanffy–Logistic function provided good fits for both planting methods (NRMSE = 0.17 for direct sowing, and 0.25 for transplanting). The sensitivity of model performance to sigmoid component order and form was evident, with some combinations producing biologically implausible growth curves. The most significant issues arose when Weibull, Von Bertalanffy, or Beta functions were used as the initial component. These combinations often led to underestimation of early vegetative growth or produced unrealistic shapes with sudden transitions or overly flattened peaks. For instance, Weibull–Logistic and Von Bertalanffy–Logistic underestimated LAI during early GDD accumulation in direct sowing, while Logistic–Beta did not align well with the decline phase in transplant and also for direct sowing. The over- or underestimation of dual functions emphasizes the importance of function order, as improper placement can distort growth phase representation, especially when the mathematical structure lacks flexibility or biological relevance.

Figure 2 (a-j) present the relationship between the measured and predicted LAI of the five best-performing function pairs previously described in detail during the calibration phase using independent data (2018 growing season as validation). Under direct sowing conditions, the Logistic-Richards function (Fig. 2d) outperformed the other models, achieving the highest coefficient of determination ((:{text{R}}^{2})= 0.98), lowest normalized root mean square error (NRMSE = 0.1474), and a high index of agreement (d = 0.98). Under transplant cultivation, the Hill–Hill model exhibited the best performance among all evaluated functions, achieving high (:{text{R}}^{2}) = 0.99, d = 0.99, NRMSE = 0.127, and MRE = − 0.05, indicating accurate and consistent prediction of LAI. According to Fig. 2a-j, the slope of linear regression (0.95 − 0.94 for direct and 0.94 − 0.92 for transplant cultivations) indicate that the all functions accurately predicted LAI values (close to 1.0) and small values of MRE showed a slight underprediction by the functions.

Similar to Fig. 2, the statistical parameters between the measured and predicted LAI based on days after planting (DAP) was calculated for all functions (Data are not shown) for calibration and validation data. During the calibration phase, the Logistic–Logistic, Gompertz–Gompertz, Hill–Hill, Logistic–Richards, and Logistic–Hill functions demonstrated strong predictive accuracy. For direct sowing, NRMSE values ranged from 0.04 to 0.07, d values from 0.995 to 0.998, and MRE values from − 0.002 to − 0.08. For transplant cultivation, the corresponding values ranged from 0.06 to 0.09 for NRMSE, 0.996 to 0.998 for d values, and − 0.004 to − 0.01 for MRE. In the validation phase, performance remained strong though with slightly increased error margins. For direct sowing, NRMSE and d, ranged from 0.12 to 0.14, from 0.983 to 0.988, respectively and MRE from − 0.008 to − 0.04, respectively. For transplant cultivation, NRMSE, d, and MRE values were varied between 0.21 and 0.22, 0.973 to 0.977, and 0.01 and 0.02, respectively.

Adjusted coefficients of sigmoid double functions

The coefficients of the selected functions were determined based on key influencing factors such as seasonal applied irrigation water, rainfall, and plant density. Using the 2017 dataset from the calibration phase, the coefficients of the functions were adjusted for the five best-performing models (Logistic–Logistic, Gompertz–Gompertz, Hill–Hill, Logistic–Richards, and Logistic–Hill) to reflect these environmental and management variables (Fig. 3a-h). These updated coefficients were then used to predict LAI values for the 2018 dataset in the validation phase. However, after incorporating the adjusted coefficients into the Logistic–Hill model, it was found that this function failed to accurately predict the LAI under transplant cultivation. According to Fig. 3d, the Logistic-Richards function exhibited the best performance under direct sowing conditions, achieving the highest coefficient of determination ((:{text{R}}^{2}) = 0.99) and index of agreement (d = 0.99), along with the lowest normalized root mean square error (NRMSE = 0.08) among all evaluated functions. Furthermore, the slope of the linear regression line was 0.99 (close to 1.0) and MRE was − 0.003, indicating excellent agreement between the predicted and measured LAI values under direct sowing. Under transplant cultivation, the Hill–Hill function exhibited the best performance compared to the other functions, with (:{text{R}}^{2}) = 0.98, d = 0.98, NRMSE = 0.16, and MRE = − 0.03. Additionally, as shown in Fig. 3g, the slope of the regression line for this function was 0.9683, which is close to the 1:1 line, indicating a more accurate prediction of LAI relative to the other evaluated functions.

Validation of functions

Figure 4 (a-h) shows the relationship between the measured and predicted leaf area index based on growing degree days in validation year (2018 growing season), after adjusting the constant coefficients of double sigmoid functions. Among the four evaluated functions under direct sowing conditions (Fig. 4d), the Logistic–Richards function demonstrated superior performance, with (:{text{R}}^{2}) = 0.98, NRMSE = 0.15, MRE = − 0.05, and d = 0.98. Additionally, as shown in Fig. 4d, the slope of the regression line for this function was 0.94, indicating close agreement with the 1:1 line. Under transplant cultivation, the Hill–Hill model yielded the best performance among the four functions, with statistical parameters of (:{text{R}}^{2}) = 0.98, NRMSE = 0.17, MRE = − 0.05, and d = 0.98. However, as illustrated in Fig. 4f, the slope of the Hill–Hill function was 0.9272, while the slope of the Gompertz–Gompertz function was 0.9852, suggesting that the Gompertz–Gompertz function provided a closer fit to the 1:1 line, therefore, it is more accurate in terms of slope alignment.

To evaluate the impact of coefficient correction on function performance, statistical indicators were compared between the calibration and validation phases for both direct sowing and transplant cultivation. In the calibration phase for direct sowing, the Logistic–Richards function without coefficient correction showed a very low NRMSE (0.04). However, after applying coefficient correction, the NRMSE increased to 0.08, indicating a slight reduction in calibration accuracy. Similarly, under transplant cultivation, the Logistic–Logistic function (without correction) performed better during calibration (NRMSE = 0.06, d = 0.99, MRE = 0.045) than the Logistic–Logistic function (with correction) with higher error (NRMSE = 0.19, d = 0.97, MRE = 0.11). In the validation phase, a similar trend was also observed. Under direct sowing, the Logistic–Richards function showed nearly identical performance with and without correction: NRMSE = 0.15, d = 0.98 (without correction) vs. NRMSE = 0.17, d = 0.98 (with correction), indicating that coefficient correction had a negligible effect on function accuracy. Under transplant cultivation, the Hill–Hill function had slightly lower error without correction (NRMSE = 0.13, d = 0.99, and MRE= −0.05) compared to the corrected version (NRMSE = 0.17, d = 0.98, and MRE=−0.05). Overall, the results suggest that correcting the coefficients did not significantly improve—and in some cases slightly reduced—the predictive accuracy of the functions, especially during calibration. The functions calibrated without coefficient adjustment generally performed as well as or better than those with correction across both cultivation methods and evaluation phases.

Modeling the leaf area index (LAI) of sugar beet is essential for understanding crop growth dynamics, optimizing management practices, and improving yield predictions. Sigmoid functions, particularly single and double sigmoid models, are widely used to capture the characteristic S-shaped growth patterns of LAI over the growing season³². Single sigmoid functions, such as the Logistic or Gompertz models, are commonly employed to describe the cumulative LAI development during the growing season¹¹. These models are characterized by an initial slow growth phase, a rapid exponential increase, and a plateau as the crop canopy closes and reaches its maximum LAI⁷. However, single sigmoid models may have limitations in accurately capturing the asymmetric nature of LAI dynamics³³. Double sigmoid functions address these limitations by allowing for two distinct inflection points: one for the rising phase and another for the senescence phase^5,12 This approach provides greater flexibility in fitting the observed LAI pattern, especially in crops like sugar beet with pronounced differences between the rates of leaf development and decline stags. Studies have shown that double sigmoid functions improved the fit to observed LAI data and enhanced the accuracy of phenological phase detection^34,35. Phenological phase detection for sugar beet involves identifying stages such as emergence, leaf development, canopy closure, root bulking, and sugar accumulation^36,37. Since LAI reflects canopy expansion and closure, modeling its growth using double sigmoid functions allows precise identification of transitions between these phases, which is essential for optimizing irrigation, nutrient management, and yield forecasting. This study assessed the effectiveness of double sigmoid functions in modeling LAI based DAP and GDD and for direct sowing and transplant cultivation with varying irrigation levels and planting densities to make the result more applicable for different regions. The top five best functions were Logistic-Logistic, Gompertz-Gompertz, Hill-Hill, Logistic-Richards, and Logistic-Hill functions, which performed the best in predicting sugar beet LAI in comparison with other functions using both DAP and GDD. However, caution is advised when using Weibull, Von Bertalanffy, and Beta functions in sigmoid double modeling, particularly in the initial stage, as they may compromise model performance and accuracy in capturing early and late LAI dynamics. Logistic-Richards achieved the best fit for direct sowing calibration data (NRMSE = 0.04 and d = 0.99), while Hill-Hill performed best under transplant cultivation using independent 2018 validation data (R² = 0.99, NRMSE = 0.13, d = 0.99). The superior performance of LR and HH models reflects the strength of double sigmoid functions in modeling nonlinear and asymmetric growth patterns. The Logistic-Richards function effectively handled variations in early growth rate and plateau behavior in direct sowing, while the Hill-Hill function was more adaptable under transplanting, where early establishment delays slightly modified the LAI curve.

The Logistic–Richards function, combining Richards and Logistic functions, is effective for direct sowing due to its adaptability to varying growth rates. Although, lack of improvement from coefficient correction suggested that the original models were well-optimized for the datasets, it also highlights their limited generalizability without recalibration. The correction method may not have captured complex interactions between environmental factors and LAI, as seen in the Logistic–Hill function’s failure in transplant cultivation after correction. This sensitivity underscores the need for tailored adjustment strategies for different models in response to environmental recalibration.

Growing Degree Days (GDD) is widely used because thermal time strongly correlates with crop development; however, it remains a temperature-centric metric and does not explicitly capture other environmental drivers that influence LAI dynamics. While GDD may indirectly reflect seasonal differences in rainfall or irrigation (as soil moisture affects canopy temperature and energy balance), critical meteorological variables such as relative humidity, vapor pressure deficit (VPD), wind speed, solar radiation, and leaf temperature directly affect stomatal conductance, transpiration, and leaf expansion. Future work should consider integrating additional meteorological and soil parameters into the model structure or applying advanced approaches such as nonlinear corrections, machine learning-based parameter estimation, or hybrid modeling frameworks that link empirical growth curves with mechanistic crop processes to improve adaptability across diverse environments.

The results of predicting the trend of changes in leaf area index under different irrigation treatments and crop densities using various double sigmoid functions are consistent. For instance, the Logistic–Richards model exhibited superior accuracy in predicting LAI under direct sowing across irrigation and planting density treatments when calibrated with GDD. The model’s normalized root mean square error (NRMSE) ranged from 0.008 to 0.096, while the index of agreement (d) consistently exceeded 0.991, indicating a strong agreement between observed and predicted values. Mean relative error (MRE) remained low (generally below 0.09) across treatments, further confirming the robustness of the model. The highest accuracy was observed under full irrigation at the highest planting density (I1D1D), where NRMSE was 0.008 and d reached 0.99, reflecting the model’s ability to capture the smooth and symmetrical LAI trajectory under optimal growing conditions. Agronomically, this performance is attributed to more complete physiological development under full irrigation and dense planting, which supports rapid canopy closure, minimal soil evaporation, and extended photosynthetic activity. These conditions align closely with the assumptions of sigmoid models, where leaf expansion follows a biphasic pattern—rapid early growth followed by gradual senescence. Conversely, performance slightly declined under severe water deficit (I3) and low-density treatments (D3 and D4), with NRMSE values approaching 0.09. These conditions introduce irregularities such as delayed early growth, premature senescence, and reduced LAI peaks due to limited water availability and increased bare soil exposure. Lower planting densities further amplified microclimatic variability by increasing soil heat flux and canopy temperature, accelerating leaf loss under stress. Despite these challenges, the model maintained strong predictive ability (d > 0.982) even in stressed scenarios, underscoring its robustness. The results emphasize that the Logistic–Richards model, calibrated with GDD, effectively represents LAI dynamics for sugar beet under direct sowing across diverse irrigation and density treatments. However, its underperformance in extreme deficit conditions highlights a structural limitation as fixed-parameter sigmoid models cannot fully capture abrupt physiological changes caused by severe stress. Future refinements could involve integrating water stress indices, dynamic senescence parameters, or additional meteorological variables such as vapor pressure deficit, wind speed, and leaf temperature to improve adaptability under climate variability. Such improvements would enhance the model’s applicability for precision irrigation scheduling and crop management in arid and semi-arid regions.

The similar performance of double sigmoid functions under direct sowing and transplant cultivation can be attributed to several factors. First, both planting methods were subjected to comparable environmental conditions during the main growth stages and received the same irrigation schedules, which are primary determinants of leaf expansion and canopy development. Adequate water availability during early vegetative growth likely minimized physiological differences between the two systems. Second, maximum LAI were achieved within a similar timeframe in both cultivation methods, reducing variation in LAI dynamics. Sugar beet’s high morphological plasticity enables transplanted plants to compensate for early growth delays by adjusting leaf size and number, ultimately matching the photosynthetic capacity of direct-sown plants. Additionally, high planting densities (e.g., 180,000 plants ha⁻¹) accelerated canopy closure for both methods, resulting in similar light interception and LAI patterns across treatments. These factors collectively explain the comparable statistical performance of double sigmoid models for direct and transplanted crops despite differences in initial establishment.

It was reported by³⁸ process-based models such as AquaCrop, DSSAT, APSIM, and SSM simulate crop development and yield by incorporating detailed representations of soil, crop, and atmospheric processes. While these models are valuable for scenario analysis, they require extensive data inputs and calibration, reducing transparency and practicality in data-scarce environments. For instance, DSSAT and APSIM require over 200 input parameters, whereas simpler models like SSM use about 55. Despite their complexity, increased parameterization does not always ensure superior robustness; simpler models such as SSM and CropSyst showed lower variability in yield prediction (CV: 8.2% and 14.3%) compared to APSIM (15.0%) and DSSAT (18.5%). Sugar beet LAI was predicted by sugar beet simulation model (SSM) developed by³⁹ model and results showed that the SSM model was slightly underestimated the higher values of LAI resulting in decreasing model accuracy (NRMSE = 23.2% and d = 0.0.95). In contrast to latter models, the double sigmoid approach offers a transparent and computationally efficient alternative for LAI estimation. Our best models achieved high accuracy (R² > 0.98; NRMSE as low as 0.04) using only DAP or GDD, without requiring detailed soil or weather datasets. This simplicity makes double sigmoid models practical for operational use and precision agriculture in data-limited regions. However, their empirical nature limits the ability to capture dynamic soil-water-plant interactions.

It was reported by¹⁶ that the Richards ((:{text{R}}^{2}) = 0.95–0.99), Logistic ((:{text{R}}^{2}) = 0.95–0.98), and Gompertz ((:{text{R}}^{2}) = 0.95–0.98) functions were found to be the most suitable functions to fit maize leaf growth across various cultivars. In contrast, the Weibull (R² = 0.88–0.93) and Morgan-Mercer-Flodin (MMF), (R² = 0.95–0.98) functions exhibited acceptable fits but were limited by highly correlated parameter estimates, reducing their suitability for precise maize leaf growth modeling. Also¹², reported that the Beta-Richards and Richards-Gompertz functions provided the best fit for the measured dry matter during the growing season based on DAP and GDD, respectively. The combined use of two Von Bertalanffy or Weibull functions did not effectively capture the growth dynamics of rapeseed plants, exhibiting limited accuracy in modeling their development. Scientists such as^14,29,40 found that the Logistic function was effective in predicting the height of lettuce plants, the dry matter yield of maize, and yield and dry matter of rapeseed, respectively. A study on growth modeling of beef tomatoes (‘9930) in Taiwan was conducted by⁴¹ over two seasons using Gompertz and Logistic functions based on growing degree days (GDD) and days after transplanting (DAT). The GDD-based models showed slightly better performance in estimating plant height, Leaf Area Index (LAI), and dry matter components compared to DAT-based models. The Logistic function better matched the expected biological stages than the Gompertz function. In Taiwan, tomato scions grafted onto eggplant rootstocks were studied over six seasons to enhance summer cultivation, with seedling growth modeled using Gompertz, Richards, and Logistic curves. The results showed that, the performance of the three nonlinear models did not vary greatly in the same growing season. However, a significant difference across growing seasons was observed (R² = 0.74–0.85 for calibration, 0.72–0.80 for validation) suggesting to apply models for each season, separately³². The obtained results in the current study did not show any priority by application of DAP and GDD, while using GDD makes the results more applicable for different regions.

This study analyzed 15 different double sigmoid functions to model sugar beet leaf area index (LAI) based on growing degree days (GDD) and days after planting (DAP) under varying irrigation treatments (I100, I75, I50) and plant densities (P180, P135, P90, P45) across direct sowing and transplant cultivation. The Logistic-Richards and Hill-Hill functions showed the best fit during the growing season, based on both GDD and DAP, as indicated by superior statistical indicators in both calibration and validation phases. The Von Bertalanffy, Weibull and Beta functions were not effective in describing the leaf area index of sugar beet variation. In terms of irrigation water levels in calibration stage, functions under full irrigation (I100) under direct sowing demonstrated optimal performance, with the Logistic-Richards function achieving a low NRMSE of 0.008 at P180. Under water stress conditions (I75, I50), the Logistic-Richards function maintained robust fits (NRMSE = 0.03–0.09). Other functions, such as Gompertz-Gompertz in calibration phase under direct sowing, exhibited higher errors under water stress (NRMSE up to 0.12), highlighting the resilience of the Logistic-Richards function. Plant density had a significant impact on the function accuracy, with higher density (P180) resulting in the best fits (e.g., NRMSE = 0.008 for Logistic-Richards in calibration phase and under direct sowing), while lower density (P90, P45) led to increased errors (NRMSE = 0.04–0.05) due to sparser canopies. Adjusting function coefficients to account for environmental factors such as seasonal applied water, rainfall, and planting density generally led to decreased predictive accuracy under direct and transplant cultivation. The findings of this study provide a reliable tool for estimating sugar beet LAI throughout the growing season, supporting precision agriculture applications such as irrigation optimization and planting schedule adjustments. However, caution is advised when applying these functions under severe water stress or low plant density, where deviations may occur. Finally, the Logistic–Richards function, with its superior accuracy and robustness, is recommended for practical use in sugar beet management based on GDD and DAP. Future research should validate these functions under diverse field conditions and across multiple growing seasons to confirm their robustness. Incorporating additional environmental factors, such as light intensity, soil water content, or nutrient availability, may improve model performance, particularly for coefficient-corrected versions. Advanced modeling technique, such as machine learning, could offer greater flexibility in capturing complex LAI dynamics.