Optimization of Parameters Related to Grain Growth of Spring Wheat in Dryland Based on the Next-Generation APSIM

Abstract

To improve the applicability of crop models, this study compared two algorithms for optimizing the single objective parameters of the spring wheat in the dryland grain growth sub-model to identify the more efficient algorithm for application in future model parameter optimization. Based on field experiments from 2015 to 2021 in Gansu Province, this study combined weather data and yearbook yield data from 1984 to 2021 to optimize parameters related to grain growth of spring wheat in dryland based on the next-generation APSIM using two algorithms: the Nelder–Mead simplex algorithm and the DREAM-zs algorithm. The results were as follows: the optimization results of both algorithms were the same, but the DREAM-zs algorithm converged faster; the optimized parameters for the grain growth stage of Dingxi35 spring wheat were: a grain number per gram stem of 25 grains, an initial grain proportion of 0.05, and a maximum grain size of 0.049 g; after optimization, the root mean square error (RMSE) of observed and simulated yield values decreased from 186.84 kg/hm² to 115.71 kg/hm², and the normalized root mean square error (NRMSE) decreased from 10.33% to 6.40%. The optimized results were consistent with the growth and development process of wheat and had high applicability.

1. Introduction

Wheat is one of the three major crops and is a staple food in Europe, various Asian countries, and northern China. China is the world’s largest producer of wheat, with a long history of cultivation [1]. Due to the different temperature requirements of wheat cultivars, there are two types of wheat, winter wheat and spring wheat, and different types are grown in different regions. Spring wheat is grown in China’s Heilongjiang, Inner Mongolia, and Northwest China, where it has a shorter phenology; winter wheat is grown in North China, southern Xinjiang, Shaanxi, and the Yangtze River Basin, where it has a longer phenology [2,3]. Wheat growth is a complex process, and crop modeling guides researchers in making efficient decisions based on the growth theory of the crop.

Crop models are important tools for predicting crop growth and development and are an important part of intelligent agricultural production. The crop model is a more complex dynamic model, consisting mainly of equations and parameters [4]. There are two main ways to improve crop models: first, to improve the structure of the model by enhancing the understanding of the crop growth and development process; second, to optimize the parameters of the model [5]. For example, Liu et al. [6] added a framework for waterlogging to APSIM and calibrated it based on measured data from five countries. For wheat crops, which have been studied for many years and are more mature, the first path is more difficult to progress and tends to lead to less adaptability of models. Therefore, this study focuses on improving the crop model by optimizing the parameters of the model. At present, the parameters in the next-generation APSIM are not fully applicable to the loess hilly agricultural area in northwestern China. The parameters need to be optimized according to the local climate, soil and management measures so that the simulated yield is infinitely close to the actual local yield and the applicability of the model is improved.

In earlier studies, researchers usually used trial-and-error methods to adjust parameters based on the results of long-term field trials and extensive agricultural knowledge and experience at the test sites. For example, Kumar et al. [7] used a trial-and-error method to validate the genetic coefficients of the DSSAT model for cotton under different cropping environments based on field trial data. Patel et al. [8] used a trial-and-error method to calibrate and validate the DSSAT-CERES model for wheat based on field trial data. Although the trial-and-error method can minimize the error between the simulated and actual values, there are shortcomings in the process of manual parameter optimization, such as large data size, time-consuming data processing, and low optimization accuracy and efficiency. Currently, to address the drawbacks of manual parameter optimization, researchers use algorithms to optimize and calibrate the parameters of crop models. This study is a single-objective optimization problem. For the single-objective optimization problem, the most used algorithm is the swarm intelligence algorithm. For example, Wei et al. [9] used a particle swarm optimization algorithm (PSO) for the parameter estimation study of the tomato growth and development model. Trejo et al. [10] used various evolutionary and affine algorithms: differential evolution (DE), covariance matrix adaptation evolution strategy (CMA-ES), particle swarm optimization (PSO), and artificial bee colony (ABC) for crop growth SUCROS model for parameter estimation. Swarm intelligence algorithms use the principles of biological evolution to simulate an optimization search process of various animals or groups of things [11]. However, the individual update rules of the swarm intelligence algorithm are simple, and the individual learning ability is not strong enough. Moreover, the algorithm generates a large amount of intermediate data during the iteration process, and the useful information contained in this data is not fully utilized. These deficiencies seriously affect the performance of the swarm intelligence algorithm when faced with complex model parameter optimization problems. To compensate for the shortcomings of the swarm intelligence algorithm and improve the optimization performance to run crop models more conveniently, the use of Bayesian optimization methods is proposed for optimization. For example, Moon et al. [12] used Bayesian optimization to calibrate the WOFOST crop model for bell pepper, mainly using the hyperparameter optimization tool HyperOpt. Iizumi et al. [13] applied Bayesian methods to parameter estimation of a large-scale rice crop model, PRYSBI. The number of iterations for tuning parameters using Bayesian methods is low and convergence is fast. However, there are fewer studies applying Bayesian methods to APSIM models, and it is not possible to know whether the optimized data are reliable.

The effective application of the model relies on the fast and accurate estimation of parameters. Wheat yield is an important variable in the APSIM model, and its simulation accuracy directly affects the overall simulation effect of the crop growth model. In order to provide technical support for the accurate simulation and effective application of the APSIM model in the dryland spring wheat production process in the loess hilly region, the Nelder–Mead simplex and DREAM-zs algorithms were used in this study to optimize the parameters related to the grain growth of dryland spring wheat.

2. Materials and Methods

2.1. Next-Generation APSIM

To meet the simulation requirements of more complex agricultural systems and to increase the speed of running large simulations, the next-generation APSIM has been established and has a continuous iteration of updates [14]. The next-generation APSIM can run on Windows, Linux, and OSX operating systems. The next-generation APSIM retains the advantages of the APSIM 7.x framework, chooses C# as the programming language, and uses the MONO and GTK# frameworks to enable cross-platform development and running of models. The next-generation APSIM provides a library of common functions and algorithms for model simulation and supports users when designing new plant models using the Plant Modeling Framework (PMF) framework [15]. For example, Khaembah et al. [16] developed a potential yield model for forage beet using the PMF framework in the next-generation APSIM. In this study, APSIM version 2023.7.7270.0 was used. The parameters were optimized mainly for the sub-model of wheat grain organ function (Wheat.Grain) [17].

2.2. Study Field

The experimental area of this study was selected in Anding District, Dingxi City, Gansu Province, China, a typical agricultural area in the Loess Hills. The experimental area is located in the south-central part of Gansu Province, at 104°38′ E, 35°35′ N, with an average altitude of 2000 m. It belongs to the middle temperate inland area, and the geographical location of the experimental area is shown in Figure 1. The area has sufficient sunshine, high light intensity, and significant temperature difference between day and night. The average annual temperature is 7.72 °C, the average annual sunshine time is more than 2500 h, and the average annual rainfall is 494.86 mm. The seasonal climate is distinct, with little precipitation but unevenly distributed throughout the year, with little rainfall in winter and spring and an arid climate, and more and more concentrated rainfall in summer and autumn, which is a typical arid and semi-arid climate in temperate continental regions. The soil type of the field test site is typical loessial soil with a loose texture, a tilled layer capacity of 1.26 g/cm³, a pH value of 8.36, a soil organic matter content of 12.01 g/kg, a total nitrogen content of 0.61 g/kg, and a total phosphorus content of 1.77 g/kg. Soil bulk density, air-dried moisture, wilting coefficient, field capacity, saturated moisture, and soil water conductivity were determined before sowing, emergence, elongation, filling, and after harvesting of spring wheat. The sampling requirements were as follows: 0–50 mm, 50–100 mm, 100–300 mm, 300–500 mm, 500–800 mm, 800–1100 mm, 1100–1400 mm, 1400–1700 mm, and 1700–2000 mm soil samples were taken, air-dried, sieved through a 1 mm sieve, and bagged. During the phenology stage of spring wheat, soil moisture was measured every 15 days. Among them, 0–100 mm was measured by drying method, and below 100 mm was measured by neutron moisture analyzer, totaling 9 layers. The lower available moisture was also determined. The specific soil properties are shown in

Table 1. Soil properties in the experimental area.

Parameters	Soil Depth (mm)
	0–50	50–100	100–300	300–500	500–800	800–1100	1100–1400	1400–1700	1700–2000
Bulk density (g/cm3)	1.29	1.23	1.33	1.20	1.14	1.14	1.25	1.12	1.11
Air-dried moisture (mm/mm)	0.01	0.01	0.05	0.07	0.09	0.10	0.11	0.12	0.13
Wilting coefficient (mm/mm)	0.09	0.09	0.09	0.09	0.09	0.11	0.11	0.12	0.13
Field capacity (mm/mm)	0.27	0.27	0.27	0.27	0.26	0.27	0.26	0.26	0.26
Saturated moisture (mm/mm)	0.46	0.49	0.45	0.50	0.52	0.52	0.48	0.53	0.53
Lower available moisture (mm/mm)	0.09	0.09	0.09	0.09	0.10	0.12	0.13	0.18	0.22
Soil water conductivity (mm/h)	0.60	0.60	0.60	0.60	0.60	0.60	0.60	0.60	0.60

.

The day-by-day meteorological data for 1984–2021 were obtained from the NASA Weather Database. The day-by-day meteorological data include radiation (MJ/m²), daily maximum temperature (°C), daily minimum temperature (°C), and precipitation (mm). The specific annual average monthly average distribution of temperature and precipitation data are shown in Figure 2.

This study was conducted in 2015–2021 in the experimental field in Anjiapo village, Fengxiang Town, Anding District, Dingxi City, where the main crop grown in the area is spring wheat, which matures once a year. In the experimental field, the test area was 24 m² (6 m × 4 m), leaving 0.5 m at each edge, in a randomized position with three replications. The test crop was Dingxi35 spring wheat, which is suitable for cultivation in the dryland agricultural area of central Gansu Province, with an annual precipitation of about 450 mm and an elevation of 1700–2000 m. The traditional three-tillage and two-pursued tillage method was used to manipulate the planter for sowing, with a sowing depth of 0.07 m, a row spacing of 0.25 m, and a planting density of 400 plant/m². Overall, 105 kg/ha of pure N fertilizer and 105 kg/ha of P₂O₅ fertilizer were applied to the experimental area. The crop was sown according to the normal sowing schedule (NSW), with the normal sowing time around March 19. At harvest, 20 wheat plants were taken for data measurement and the yield per plot was converted to yield per unit area. Measured yield data for 1984–2014 are the yearbook yield data.

2.3. CroptimizR Package

The CroptimizR 0.5.1 package is a package for estimating crop model parameters in the R language [18]. The CroptimizR package provides a framework for linking crop models to optimization algorithms for parameter optimization and calibration of crop models. Specific features are reflected in the ability to fine-tune the selected optimization algorithm through the parameters defined in the package; the ability to set initial values and constraints on the parameters to be optimized, etc.

The CroptimizR 0.5.1 package currently provides two optimization algorithms: the Nelder–Mead simplex algorithm, which is a frequentist method, and the DREAM-zs algorithm, which is a Bayesian method. The frequentist methods and Bayesian methods are two different ways of solving statistical problems. The frequentist methods require a large number of independent experiments to interpret probabilities as statistical means, while Bayesian methods do not require a large number of experiments and are interpreting probabilities as belief degrees [19]. In the Bayesian method, the model parameters are considered random variables and both prior knowledge and data are used to calculate the posterior parameter distribution through Bayes’ theorem [20]. Bayesian methods are more useful when the number of trials considered in the study is very small. When the amount of data tends to infinity, the frequentist method and the Bayesian method obtain the same results, i.e., the frequentist method is the limit of the Bayesian method.

2.3.1. Frequentist Methods

The frequentist method does not assume any prior knowledge and does not refer to experience, but only makes probabilistic inferences according to the currently available data [21]. The frequentist method treats unknown parameters as deterministic and ordinary variables that take unknown values. When the amount of data tends to be infinite, the frequentist method can give precise estimates. However, it may produce biases when there is a lack of data.

2.3.2. Bayesian Methods

The main idea of the approach based on Bayesian theory is to treat the unknown parameters as if they were random variables, just like the variables used in the description of the experiment. Prior knowledge about the parameters is integrated into the model, and this knowledge is always updated as more and more data are observed [22]. The Bayesian estimation method is based on the Bayesian equation (Equation (1)).

$$P(\theta |X)=\frac{P(X|\theta )\times P(\theta )}{P(X)}$$

(1)

θ is a random variable that conforms to a certain probability distribution. In the Bayesian approach, there are two inputs and one output, the input is the prior and the likelihood, and the output is the posterior. The prior, P(θ), is a prediction of θ when no data are observed; the likelihood, P(X|θ), is what the data we observe should look like assuming θ is known; and the posterior, P(θ|X), is the final distribution of the parameters.

2.4. Nelder–Mead Simplex Algorithm

The Nelder–Mead simplex algorithm, first proposed in 1965, is a very popular direct search method for multidimensional unconstrained minimization [23]. The Nelder–Mead simplex algorithm is an iterative optimization strategy based on the concept of simplicity to construct solutions for unconstrained optimization problems [24]. For an n-dimensional optimization problem, an n + 1-dimensional simplex is initially constructed, the function values of the vertices of the simplex are calculated, and then new vertices and simplexes are constructed by analyzing and comparing the function values of the vertices until the convergence condition is reached. Through the study and analysis of the problem, the established functions (Equation (2)) are as follows:

$$f(x)={{\displaystyle \sum (Y_r-Y_s)}}^2$$

(2)

where Y_r is the actual yield of wheat and the data are obtained from the statistical yearbook and field trial data; Y_s is the simulated yield of wheat using APSIM 2023.7.7270.0 software, i.e., the simulated yield of wheat.

The flowchart for optimizing the model parameters with the Nelder–Mead simplex algorithm in the CroptimizR package is shown in Figure 3. The specific steps of the whole optimization process are as follows:

Step 1: Initialization. Initialize n + 1 points x₁, x₂, …, x_n+1.

Step 2: Sorting. Reorder the vertices according to the value of f(x), f(x₁) ≤ f(x₂) ≤ … ≤ f(x_n+1); determine whether the cutoff condition is satisfied, and the cutoff condition is that the number of iterations reaches a threshold.

Step 3: Calculate the center of gravity. The center of gravity of the first n points is calculated by discarding the lowest point x_n+1, and the formula for the center of gravity is given in Equation (3).

$$x_0=\frac{1}{n}{\displaystyle \sum _{i=1}^n{x}_i}$$

(3)

Step 4: Calculate the reflection point. The formula for calculating the reflection point x_r is shown in Equation (4). If the value of the reflected point f(x) is better than the second-best point f(x_n) but worse than the best point f(x₁), i.e., f(x₁) ≤ f(x_r) ≤ f(x_n), replace the best point x_n+1 with the reflected point x_r and return to perform the second step.

$$x_r=x_0+\rho (x_0-x_{n+1}) \rho =1$$

(4)

where x₀ is the center of gravity of the first n points, ρ is the reflection coefficient, and x_n+1 is the worst point.

Step 5: Calculate the extension point. If the reflection point is the optimal point, i.e., f(x_r) < f(x₁), the extension point x_e is calculated, and the formula is given in Equation (5); if the extension point is better than the reflection point, i.e., f(x_e) < f(x_r), the optimal point x_n+1 is replaced with the extension point x_e, and return to perform the second step; otherwise, the optimal point x_n+1 is replaced with the reflection point x_r, and return to perform the second step.

$$x_e=x_0+\gamma (x_r-x_0) \gamma =\frac{1}{2}$$

(5)

where x₀ is the center of gravity of the first n points, γ is the extension coefficient, and x_r is the reflection point.

Step 6: Calculate the contraction point. If the value of f(x) at the reflection point is worse than the second closest point f(x_n), i.e., f(x_n) < f(x_r) < f(x_n+1), calculate the contraction point x_c, and the formula is given in Equation (6). If the shrinkage point is better than the worst point, i.e., f(x_c) ≤ f(x_n+1), replace the worst point x_n+1 with the shrinkage point x_c and return to perform the second step; otherwise, perform the seventh step. If the shrinkage point is worse than the worst point, i.e., f(x_c) ≥ f(x_n+1), the inner shrinkage point x_cc is calculated, and the formula is given in Equation (7). If the inner shrinkage point is better than the lowest point, replace the lowest point x_n+1 with the inner shrinkage point x_cc; otherwise, perform the seventh step.

$$x_c=x_0+\alpha (x_r-x_0) \alpha =\frac{1}{2}$$

(6)

$$x_{cc}=x_0+\alpha (x_{n+1}-x_0) \alpha =\frac{1}{2}$$

(7)

where x₀ is the center of gravity of the first n points, α is the contraction coefficient, x_r is the reflection point, and x_n+1 is the worst point.

Step 7: Calculate the rollback point. The formula for calculating the rollback point x_i is shown in Equation (8). Replace all points other than the current optimal point with the rollback point x_i and return to perform the second step.

$$x_i=x_1+\sigma (x_i-x_1) \sigma =\frac{1}{2}$$

(8)

where x₁ is the best point, and σ is the rollback coefficient.

2.5. DREAM-zs Algorithm

The DREAM algorithm is a family of algorithms containing DREAM-zs, DREAM-D, DREAM-DZS, DREAM-ABC, DREAM-BMA, and MTDREAM-zs [25]. The DREAM-zs algorithm is based on the original DREAM algorithm, which is short for Differential Evolutionary Adaptive Metropolis. It is a multi-chain MCMC method and is considered to be an effective method for dealing with complex, high-dimensional and multi-modal target distributions. Different from the original DREAM algorithm is that the DREAM-zs algorithm considers the effect of existing samples on the Markov chain on newly generated candidates when sampling [26]. It was first proposed by Laloy et al. [27] in 2012 as an updated version of the DREAM algorithm and was developed to estimate the posterior probability function of the parameters [28].

The flowchart for optimizing the model parameters using the DREAM-zs algorithm in the CroptimizR package is shown in Figure 4. The objective function of this algorithm is given in Equation (2). The specific steps of the whole optimization process are as follows:

Step 1: Initialization. Let Z be an M₀ × d matrix, where M₀ is the sampling number of the prior distribution and d refers to the number of inverse parameters. X is an N × d matrix, where N is the number of Markov chains. The number of samples T for each Markov chain.

Step 2: Calculate the tth candidate sample zⁱ on the ith Markov chain, see Equation (9).

$$z^i={\theta }_{t-1}^i+{\zeta }_n+(1+{\lambda }_n)\gamma (\delta ,n){\displaystyle \sum _{j=1}^{\delta }(Z^{a_j}-Z^{b_j})}$$

(9)

where ${\theta }_{t-1}^i$ is the t − 1st sample point on the ith Markov chain; ζ_n is the random number of normal distribution N(0, β*); λ_n is the random number of uniform distribution u(−β, β); scaling factor γ(δ, n) = 2.38/(2δn)^0.5; and are the two samples in matrix Z; a_j, b_j∈{1, 2, …, N}, respectively; δ is the number of point pairs for calculating the scaling factor.

Step 3: Determine whether to replace zⁱ with ${\theta }_{t-1}^i$ in Equation (10). In the equation, the value of cross-probability P_CR is related to the candidate sample points $z_j^i$.

$$z_j^i=\left\{\begin{array}{l}{\theta }_{j,t-1}^i,u\le 1-P_{CR}\\ z_j^i,else\end{array}\right.j=1,2,\dots ,d$$

(10)

Step 4: Calculate the acceptance rate α based on the posterior probability density function, see Equation (11), and decide whether to accept zⁱ to be the sample at the next position in the Markov chain ${\theta }_t^i$.

$$\alpha =\left\{\begin{array}{l}\min (1.0,\frac{f(z^i|y)}{f({\theta }_{t-1}^i|y)}),f({\theta }_{t-1}^i|y)>0\\ 1,f({\theta }_{t-1}^i|y)=0\end{array}\right.$$

(11)

Step 5: Generate a random number v on the uniform distribution u(0, 1) and accept the candidate sample zⁱ if α ≥ v, i.e., ${\theta }_t^i=z^i$; otherwise reject zⁱ, ${\theta }_t^i={\theta }_{t-1}^i$.

Step 6: Repeat steps 2 to 5 until the convergence criterion proposed by Gelman et al. [29] is satisfied. When R_sat ≤ 1.2 is calculated based on the intra-chain variance W and inter-chain variance B, see Equation (12), the Markov chain is judged to be converged. Based on a random sample of smooth segments of the Markov chain, the posterior distribution and statistical indicators, such as the mean and standard deviation, are calculated for each parameter.

$$R_{sat}=\sqrt{\frac{T-1}{T}+\frac{N+1}{NT}\frac{B}{W}}$$

(12)

where T is the number of samples per Markov chain, and N is the number of Markov chains.

2.6. Parameters Optimization

The next-generation APSIM contains a large number of parameters, and although only the grain growth stage is involved in this study, there are still dozens of parameters. Before parameter optimization, a sensitivity analysis needs to be performed for all parameters involved. According to Zhao et al. [30], the varietal parameters that affect yield at the grain growth stage are the grain number per gram stem, maximum grain size, and potential filling rate, so the parameters to be optimized were selected, as shown in

Table 2. Basic parameters of crop properties in the study sites.

Name	Value	Unit	Definition in APSIM NG
Minimum leaf number	7	Leaves	[Phenology].MinimumLeafNumber.FixedValue
Sensitivity to vernalisation	5	—	[Phenology].VrnSensitivity
Sensitivity to photoperiod	3	—	[Phenology].PpSensitivity
Base phyllochron	35	oC.d	[Phenology].Phyllochron.BasePhyllochron.FixedValue
Water content	0.2	—	[Grain].WaterContent.FixedValue

. The filling rate was calculated from the number of grains, the potential size increment, and the duration of the filling process. The number of grains is calculated in Equation (13), the potential size increment is calculated in Equation (14), and the duration of the filling process is calculated in Equation (15). Equations (14) and (15) are divided into two phases: GrainDevelopment and GrainFilling. The GrainDevelopment phase is from Flowering stage to StartGrainFill stage. The GrainFilling phase is from StartGrainFill stage to EndGrainFill stage. Since this study is a single-objective optimization, the thermal time parameters of the phenology are not considered, and default values are used for these parameters. The parameters to be optimized were the grain number per gram stem, initial grain proportion and maximum grain size.

$$NumberFunction=GrainNumber=GrainsPerGramOfStem\times [StemPlusSpike].Wt$$

(13)

$$PotentialSizeIncrement=\left\{\begin{array}{l}[Grain].InitialGrain\mathrm{Pr}oportion\times [Grain].MaximumPotentialGrainSize,GrainDevelopment\\ (1-[Grain].InitialGrain\mathrm{Pr}oportion)\times [Grain].MaximumPotentialGrainSize,GrainFilling\end{array}\right.$$

(14)

$$FillingDuration=\left\{\begin{array}{l}[Pheno\log y].GrainDevelopment.T\arg et,GrainDevelopment\\ [Pheno\log y].GrainFilling.T\arg et,GrainFilling\end{array}\right.$$

(15)

In Equation (13), GrainsPerGramOfStem is the grain number per gram stem, with a default value of 26 grains; [StemPlusSpike].wt is the dry weight of stem at Flowering stage, measured in field trials using the drying method at 1.305 g; in Equation (14), [Grain].InitialGrainProportion is the initial grain proportion, with a default value of 0.05; [Grain].MaximumPotentialGrainSize is the maximum grain size, with a default value of 0.05 g; in Equation (15), [Phenology].GrainDevelopment.Target is the thermal time at the GrainDevelopment phase, with a default value of 120 oD, and [Phenology].GrainFilling.Target is the thermal time at the GrainFilling phase, with a default value of 545 oD.

Based on long-term trials at the field sites, the lower bound of the grain number per gram stem at the study sites was 24 grains and the upper bound was 29 grains; the lower bound of the initial grain proportion was 0.04, and the upper bound was 0.06; and the lower bound of the maximum grain size was 0.04 g, and the upper bound was 0.06 g.

The simulation was carried out in the next-generation APSIM model software with the software version APSIM 2023.7.7270.0. The start time of the simulation was set in the Clock module: 1 January 1984, and the end time: 31 December 2021. The Replacement module was set, and the module to be optimized was placed in it. Since this study optimizes the parameters of the wheat grain growth stage, the Wheat module is put in the Replacement module. In the Wheat module, some crop varieties of wheat have been encapsulated in it. However, the crop variety grown in the study area is not available in the model, so a new crop variety named “Dingxi35” needs to be created, and the specific crop attribute parameters are shown in Table 2, daily maximum temperature (°C), daily minimum temperature (°C), and precipitation (mm). The relevant parameters of the soil module need to be modified according to the soil conditions in the study area, see Table 1. Management measures are added: seeding measures, irrigation measures, fertilization measures and tillage measures, etc. After the above sections are added and modified, run the model and check the simulation results.

For the development of the parametric optimization program, the computer hardware is Windows 10, Intel(R) Core(TM) i7-6700HQ CPU, 16.0 GB RAM, RStudio 2022.07.0 Build 548 IDE, and the programming language version is R 4.2.3. The specific steps are as follows:

Step 1: Initialization. Install and load the required libraries. The libraries involved in the program are CroptimizR 0.5.1, CroPlotR 0.9.0, ApsimOnR 0.0.0.9000, dplyr 1.1.2, ggplot2 3.4.2, gridExtra 2.3, and tidyr 1.3.0. Then, define the path to the locally installed version of APSIM. The CroptimizR 0.5.1 package is used to estimate the crop model parameters; the CroPlotR 0.9.0 package enables the crop model output. The analysis process is standardized to facilitate the plotting of images and statistics; the ApsimOnR 0.0.0.9000 package is used to run the APSIM model, perform multiple simulations based on the parameters set, and return all outputs; the dplyr 1.1.2 package is a data analysis package that can perform very convenient data processing and analysis operations on DataFrame type data; the ggplot2 3.4.2 package is used to generate statistical or data graphs that provide the underlying syntax based on graph syntax, allowing the composition of graphs by combining independent components; the gridExtra 2.3 package is used to put together several graphs into a group of graphs, which can combine multiple graphs drawn by ggplot2 into one large graph; the tidyr 1.3.0 package is used for data processing, which can realize the mutual conversion between data long format and wide format.

Step 2: Set the variable list. Specify the name and observation parameters of the model simulator.

Step 3: Run the model for evaluation before optimization. Set the options related to the model, including the path and name of the model file; set the path of the meteorological file; set the names of the SQLite simulation value data table and the actual observation value data table; run the model.

Step 4: Based on the evaluation results, select the observation values. The observation value only keeps the name of the model simulator and the observation value of the observed parameters.

Step 5: Set the information of the parameters to be optimized. In the information of the list of parameters to be optimized, it is necessary to define the upper and lower limits of each parameter to be optimized.

Step 6: Set the options of the parameter optimization method. The options include: the number of iterations for minimization (each time starting from a different initial value of the parameter to be optimized), the maximum number of evaluations of the minimization criterion, the tolerance criterion between two iterations (a threshold for the relative difference of the parameter values between the first two iterations), the path for storing the optimization results and setting the random seed (so that each execution will give the same result, if you want randomization, do not set it).

Step 7: Optimize. The optimization process takes some time, and the optimization result will be output on the console when it is finished. And the graph and data will be stored in a folder. If the optimized parameter values are very close to the upper bound, this may indicate a large error in the observed or simulated output values.

Step 8: Run the model after optimization. Run the model using the parameter optimization results to replace the original default values in the model input file.

Step 9: Plot the results. Here we use the CroPlotR 0.9.0 package to compare the simulation results with the observed results. The CroPlotR 0.9.0 package, like the CroptimizR 0.5.1 package, can be used with any crop model. If the gap between the simulated and observed values of yield before and after optimization has been significantly reduced in the graph, the optimization is complete.

2.7. Model Testing

In this study, root mean squared error (RMSE) and normalized root mean squared error (NRMSE) were chosen as standard measures of the model’s error in predicting quantitative data [31]. The RMSE yields the degree of concentration of the data around the line of best fit by calculating the square root of the mean of the squared residuals. A lower RMSE indicates a better predictive performance of the model, see Equation (16). The NRMSE is obtained by normalizing the RMSE based on the RMSE, see Equation (17). Model performance evaluation criteria: very good when the NRMSE is less than or equal to 10%; good when the NRMSE is greater than or equal to 10% and less than or equal to 15%; acceptable when the NRMSE is greater than or equal to 15% and less than or equal to 20%; marginal when the NRMSE is greater than or equal to 20% and less than or equal to 25%; and poor when the NRMSE is greater than or equal to 25% [32].

$$RMSE=\sqrt{\frac{1}{n}{\displaystyle \sum _{i=1}^n{(Y_M-Y_s)}^2}}$$

(16)

$$NRMSE=\frac{\sqrt{\frac{1}{n}{\displaystyle \sum _{i=1}^n{(Y_M-Y_s)}^2}}}{Y_A}\times 100\%$$

(17)

where Y_M is the measured value, Y_S is the simulated value, and Y_A is the average of the measured values.

3. Results

3.1. Optimization Results

The next-generation APSIM wheat grain growth stage model parameter optimization program based on the CroptimizR 0.5.1 package was run. After the program was run, a review of the console output and result plots showed that: optimized using the Nelder–Mead simplex algorithm, the total time of model simulations was 2265 s, the total number of criterion evaluations was 255, the average time for the model to simulate all required situations was 8.9 s, and the total time of parameter estimation process was 2271 s. Optimized using the DREAM-zs algorithm, the total time of model simulations was 2144 s, the total number of criterion evaluations was 255, the average time for the model to simulate all required situations was 8.4 s, and the total time of the parameter estimation process was 2151 s. The optimization results of both algorithms are the same, but the DREAM-zs algorithm converges faster. A comparison of the specific optimization algorithm performance parameters is shown in

Table 3. Performance parameters of the two optimization algorithms.

Parameters	Nelder–Mead Simplex Algorithm	DREAM-zs Algorithm
Total number of criterion evaluation	255	255
Total time of model simulations (s)	2265	2144
Average time for the model to simulate all required situations (s)	8.9	8.4
Total time of parameter estimation process (s)	2271	2151

.

The results of the parameter optimization are shown in Figure 5. The numbers in the figure represent the number of repetitions minimized, and the size of the bubbles represents the final value of the minimization criterion. The white numbers are the minima that lead to the minimum criterion value among all repetitions. In this case, the minimum converges to a different parameter value, which indicates the existence of a local minimum.

3.2. Model Testing Results

The calibration significantly improved the model performance compared to when the default parameters of APSIM were used. The RMSE of the optimized model was reduced from 186.84 kg/hm² to 115.71 kg/hm², and the NRMSE was reduced from 10.33% to 6.40%. The model test results of the two algorithms before and after the final parameter optimization are shown in

Table 4. Model test results of the two algorithms before and after parameter optimization.

Model Parameter	Nelder–Mead Simplex Algorithm		DREAM-zs Algorithm
	RMSE (kg/hm2)	NRMSE (%)	RMSE (kg/hm2)	NRMSE (%)
Default value	186.84	10.33	186.84	10.33
Optimized value	115.71	6.40	115.71	6.40

. The fitting effect of simulated and measured values of wheat yield is shown in Figure 6, from which it can be seen that after parameter optimization, the simulated and measured values fit better, and the model performance is rated as very good.

4. Discussion

The growth of wheat is a complex process that is influenced by many factors and has a long cycle time. The crop model can solve the problems of long crop growth cycles in field trials, which cannot be repeated in a short period, according to the growth mechanism of the crop, and guides researchers to make efficient decisions. The next-generation APSIM-Wheat model is a branch of the next-generation APSIM model. The default data in the model is derived from long-term field trial data in Queensland and Western Australia. It is not fully applicable to the Loess Hills’ typical agricultural area in Northwest China due to regional differences in conditions such as climate, soil, crop properties, and management practices. The optimization program used in the previous study was an external program, which was not convenient for the model operation and had some errors. Therefore, in this study, based on field experiments, the next-generation APSIM model was automatically run using R language to evaluate and calibrate the parameter values of the grain growth stage in the next-generation APSIM-Wheat model more accurately. It not only improves the efficiency of the model, but also further promotes the applied extension research of the APSIM-Wheat model in loess hilly areas.

The goal of model parameter optimization is to find out if the result of the model simulation value fits well with the actual measurement value by calculating the minimum value of the objective function. Due to the complexity of the crop model and crop growth process, it is difficult for conventional optimization algorithms to make the final result reach the global optimum. In this study, the Nelder–Mead simplex algorithm and the DREAM-zs algorithm were used to optimize the three most sensitive parameters of the wheat grain growth stage, respectively. The Nelder–Mead simplex algorithm and the DREAM-zs algorithm represent two different statistical analysis methods, the Nelder–Mead simplex algorithm is representative of the frequentist method, and the DREAM-zs algorithm is representative of the Bayesian method. The swarm intelligence algorithm used in previous studies has limitations due to its time-consuming nature and its tendency to converge prematurely to a local optimum due to factors such as population size, iteration frequency, and iteration pattern [33]. The frequentist method uses the frequency of an event as the probability of that event occurring, while the Bayesian method has a prior assumption and continuously corrects the prior probability distribution as the sample increases. The Bayesian method uses more experience than the frequentist method, narrowing the scope and avoiding blindness. For crop model parameter optimization, the Bayesian method is theoretically superior when the number of samples is small. To verify the applicability of this theory in the parameter optimization study of the APSIM model, this study used two algorithms to optimize the model parameters by setting the same conditions of weather, soil, and management practices in the APSIM-Wheat model as in the field trial sites for simulation. The simulated results were validated with the field measurements by the equations, which showed that the RMSE and the NRMSE of the model were significantly reduced, and the simulated values were better fitted to the measured values. This study not only improved the model performance but also the optimization results, which were within the reasonable range of the dryland spring wheat growth and development study in Gansu Province [34], which was consistent with the growth state of dryland spring wheat in the study area. The method improved the efficiency of parameter optimization, which not only provided more data references for future studies but also laid the foundation for the future evaluation model of parameter optimization adaptation of the yield formation process in dryland spring wheat.

In addition to the selection of optimization algorithm, the error in optimization results may also come from meteorological data, soil data, and management measures; For example, whether the meteorological data are accurate for the test site, whether the soil data are suitable for the test site, and whether the management measures are correct. This study is a single-objective optimization, which considers the effect of the grain growth stage on crop yield. Since the grain growth stage of wheat is more closely related to the fertility stage, future research can conduct multi-objective collaborative optimization by adding the fertility stage and studying the collaborative optimization of fertility and grain growth stages to derive optimization results and minimize the error as much as possible. At present, this study is only limited to a single variety in a single region. Future studies can further improve the generalizability of this optimization method by selecting multiple representative varieties according to different agricultural regions and based on the data from multi-year field trials.

5. Conclusions

As an important tool for agricultural research, the accurate simulation of crop models is closely related to the value of each parameter. Optimizing model parameters based on actual data in field trials and improving the simulation accuracy of the model is important for the adaptation research of crop models and the intelligent production of crops. Using R language to run the next-generation APSIM model automatically to optimize the parameters not only shortens the optimization time and improves the optimization efficiency but also provides an efficient and feasible method for the optimization of other stages of the model in the future.

The final results of parameter optimization for the grain growth stage of Dingxi35 spring wheat in Loess Hills were: a grain number per gram stem of 25 grains; an initial grain proportion of 0.05; and a maximum grain size of 0.049 g. After parameter optimization, the simulated values fit better with the observed values, and the model performance was evaluated as very good.