Hyperspectral Estimation of Nitrogen Content in Wheat Based on Fractional Difference and Continuous Wavelet Transform

Abstract

Nitrogen content is a crucial index for crop growth diagnosis and the exact estimation of nitrogen content is of great significance for grasping crop growth status in real-time. This paper takes winter wheat as the study object and the precision agriculture demonstration area of the Jiaozuo Academy of Agricultural and Forestry Sciences in Henan Province as the research area. The hyperspectral reflectance data of the wheat canopy in different growth periods are obtained with the ASD ground object hyperspectral instrument, and the original canopy spectral data are preprocessed by fractional differential and continuous wavelet transform; then, the vegetation indices are established, correlation analysis with nitrogen content is conducted, and the fractional differential spectra are selected; finally, based on the wavelet energy coefficient and the vegetation indices with strong correlations, the methods of support vector machine (SVM), ridge regression, stepwise regression, Gaussian process regression (GPR), and the BP neural network are used to construct the estimation model for nitrogen content in wheat at different growth stages. By adopting the ${\mathrm{R}}^2$ and root mean square error (RMSE) indices, the best nitrogen content estimation model at every growth stage is selected. The overall analysis of the nitrogen content estimation effect indicated that for the four growth periods, the maximum modeling and validation ${\mathrm{R}}^2$ of the nitrogen content estimation models of the SVM, ridge regression, stepwise regression, GPR, and BP neural network models reached 0.95 and 0.93, the average reached 0.76 and 0.71, and the overall estimation effect was good. The average values of the modeling and validation ${\mathrm{R}}^2$ of the nitrogen content estimation model at the flag picking stage were 0.85 and 0.81, respectively, which were 37.10% and 44.64%, 1.19% and 3.85%, and 14.86% and 17.39% higher than those at the jointing stage, flowering stage, and filling stage, respectively. Therefore, the model of the flag picking stage has higher estimation accuracy and a better estimation effect on the nitrogen content. For the different growth stages, the optimal estimation models of nitrogen content were different. Among them, continuous wavelet transform combined with the BP neural network model can be the most effective method for estimating the N content in wheat at the flagging stage. The paper provides an effective method for estimating the nitrogen content in wheat and a new idea for crop growth monitoring.

1. Introduction

The accurate and rapid acquisition of crop biochemical information is the basis for agricultural scientific production. As an important nutrient element in crops, nitrogen is involved in almost all physiological processes in crops, and it plays a major role in crop growth [1]. Nutritional diagnosis of nitrogen content is significant for achieving high crop yield and realizing the rational use of nitrogen fertilizer [2]. The traditional chemical analysis method [3] is time-consuming, laborious, and has high costs when obtaining nitrogen content, and it is destructive and irrecoverable. It can only obtain point source information and is difficult to expand on a macro scale. For the past few years, hyperspectral remote sensing techniques [4] have been developing continuously. Because of its advantages for high and continuous spectral resolution [5], it has been widely used in agriculture and has been extensively studied.

Compared with traditional diagnostic methods, spectral diagnosis has the advantages of high efficiency and non-destructiveness, and it has been widely used by scholars [6]. For example, by taking Ginkgo biloba leaves with different degrees of yellowing as the research object, Hu Bin [7] used the traditional image processing method and hyperspectral technology, respectively, to establish a regression model for chlorophyll content in Ginkgo biloba leaves. The traditional image method is easy to operate, but the accuracy is relatively low, while spectral analysis achieves better results than the image method, making it suitable for assessing the health problems of individual trees. Xiao Qinlin et al. [8] used seven different spectral pretreatment methods combined with a self-designed convolution neural network (CNN) to evaluate the chlorophyll content in different cotton varieties and, finally, the method combining the first derivative and standard normal transformation (SNV) was chosen as the optimal pre-processing method. Wang et al. [9] analyzed the relevance between four vegetation indices and leaf nitrogen content, leaf nitrogen accumulation, and aboveground nitrogen accumulation, respectively. By taking hybrid rice as the study subject, a nitrogen nutrition diagnostic model was built with vegetation indices as independent variables, and the optimal band distribution for screening was within the range of red and near-infrared bands. Spectral images enable the acquisition of the fine spectral features of vegetation, and substances that cannot be distinguished visually can be detected, thus providing an approach for the analysis of crop nutrients.

Due to the influence of light, shading, climatic conditions, the filming environment, and other factors, the spectra of the features will change, and accurately obtaining the “pure” spectral characteristics has been the focus of research. Ma Chunyan et al. [10] constructed models for estimating the nitrogen content in the flag, upper 1, upper 2, upper 3, and upper 4 leaves after the first derivative, second-order differential, and continuous removal processing of the raw spectra. L. F. Wang [11] constructed partial least squares models to estimate maize nitrogen content based on full spectrum, principal component analysis, and continuous projection algorithms, respectively. Zhang Feng [12] used six spectral preprocessing methods of multivariate scattering correction, first-order derivative, second-order derivative, the SG smoothing algorithm, normalization, and trend correction for the spectral preprocessing of the acquired spectral data to optimize the data. It was found that multivariate scattering correction optimized the experimental results in different machine learning models, and multivariate scattering correction was the best preprocessing method for detecting different disease periods of classified early potato blight. Qingliang Jiao et al. [13] proposed a CNN that can simultaneously accomplish spectral denoising, baseline correction, and spectral peak localization. The method can obtain high-quality spectra and accurate results from quantitative analysis of spectra, which is superior to the traditional preprocessing algorithms. Sun Jingjing et al. [14] compared the characteristics of different preprocessing methods such as SG smoothing, SG first and SG second derivatives, multivariate scattering correction, standard normal transformation, and the de-trending method based on the spectral data of corn grain, and discussed the impact of each preprocessing method.

In previous studies, hyperspectral vegetation indices have shown great advantages in reflecting the nutrient content and growth of crops, and the large-scale simulation and estimation of the nutrient content in crops can be performed by establishing numerical models between the content of the nutrients in crops and hyperspectral vegetation indices. Xu Jatong et al. [15] investigated the connection between the total phosphorus content in leaves and stems of different crops and six hyperspectral vegetation indices, and they established a simulation model using multivariate nonlinear regression. The study results showed that there was a relevance between the total phosphorus content in leaves and stems of crops and hyperspectral vegetation indices, and the simulation effect of these vegetation indices on the total phosphorus content in leaves was better than that of stems. Feng et al. [16] studied the capability of the vegetation index and red side parameters to estimate the chlorophyll content in winter wheat, and it was found that the vegetation index combined with the red edge parameters achieved a better estimation result.

Compared to the original spectrum, the preprocessed spectral curves for the first-order derivative, second-order derivative, multivariate scattering correction, and standard normal transformation are smoother and retain the useful information from the spectrum; meanwhile, they do not significantly change the spectral absorption characteristics and are smoother and more concentrated compared with the original spectrum, with small spectral variations.

Fractional order differentiation can dig deeper into the potential information on the spectrum and has been studied and used by many scholars. For example, Huang Hua et al. [17] combined the near-infrared transmission spectrum based on fractional differential technology and PCA-SRDA to conduct multi-model fusion, and the integrated learning model of apple recognition can effectively realize the traceability of the apple origin. Xiaoyan Lu et al. [18] estimated the SPAD values of tobacco based on hyperspectral fractional order differentiation, and the results showed that the RF-SPAD model constructed with the characteristic wavelength of order 1.9 obtained the highest accuracy, which was significantly better than that of the integer order model.

With the advantages of multiple wavelet basis functions, multi-resolution, and time–frequency locality, the continuous wavelet transform has received more and more attention for its excellent performance in image processing, analysis, decomposition, compression, and denoising. Chen Qian et al. [19] used continuous wavelet transform to decompose the raw spectrum and the first-order guide spectrum of the red edge band, respectively, carried out a correlation analysis between the obtained wavelet coefficients and the measured chlorophyll content data, and used the selected wavelet coefficients with good correlation to construct the winter wheat leaf chlorophyll content estimation model based on PLS, RF, BPNN, and XGBoost methods. Tan Xianming et al. [20] adopted common vegetation indices, a free combination of wavelengths, and continuous wavelet transform to construct a maize canopy chlorophyll intensity estimation model, and the results indicated that the constructed estimation model had higher stability and accuracy than the model built using vegetation indices.

In this study, the canopy spectral data and nitrogen content in winter wheat were obtained simultaneously at the jointing stage, flag picking stage, flowering stage, and filling stage. The fractional-order differential and continuous wavelet decomposition were performed on the wheat canopy spectral data, and the vegetation index was constructed. Then, the correlation between the fractional order differential spectra, wavelet energy coefficients, vegetation index, and wheat nitrogen content was analyzed, and the support vector machine (SVM), ridge regression, stepwise regression, Gaussian process regression (GPR), and BP neural network were used to construct nitrogen content estimation models. Finally, model accuracy verification was carried out to obtain the best nitrogen content estimation model.

2. Materials and Methods

2.1. Overview of the Study Area

As shown in Figure 1, the research area is located in Xuliang town, Boai county, Jiaozuo city, Henan Province (35.18° N, 113.03° E). The area is on the piedmont plain, with a high terrain in the west and south and a low terrain in the east and north. It is 80~480 m above sea level, and the highest peak is at the village of Guodinghou Liang, 486 m above sea level. There were 48 plots in the experimental area, with a single plot area of 6 × 8 m, 16 treatments, and 3 replicates. The wheat variety was Zhengmai 369 and was randomly arranged by block. The planting time is in early October and the sampling and collection dates for the spectral data were 13 April (jointing stage), 26 April (flag picking stage), 14 May (flowering stage), and 25 May (filling stage).

2.2. Data Acquisition

In 2019 and 2020, the crop canopy ASD hyperspectral reflectance data and the nitrogen content measured data in winter wheat were obtained simultaneously at the jointing stage, flag picking stage, flowering stage, and filling stage, and the data for 48 plots in each growth stage were collected as sample data.

2.2.1. Acquisition of Spectral Data

The wheat canopy spectrum acquisition instrument was the FieldSpec^®4 Hi-Res portable ground feature spectrometer Produced by ASD (Analytical Spectral Devices, Inc., Longmont, CO, USA) in the United States [21], whose spectral band ranges from 350 nm to 2500 nm. The sampling interval between 350 and 1000 nm was 1.4 nm, the sampling interval between 1000 and 2500 nm was 2 nm, and the resampling interval was 1 nm. The field angle was 25° [22]. The spectral data were collected between 10:00 and 14:00 on a cloudless day. During the measurement process, the probe was placed vertically downward and 0.3–0.6 m above the crop canopy, while spectral calibration was performed every 15 min using a whiteboard. Three points were chosen at random from each cell, and 10 spectral curves were automatically collected from each sample point. The average reflectance of 3 sample points in each cell was taken as the spectral reflectance value of the cell. The noise interference in the spectral band of 1800~2500 nm was relatively large, with a low signal-to-noise ratio. Therefore, the hyperspectral reflectance data in the range of 350~1800 nm was selected in this study.

2.2.2. Leaf Nitrogen Content Acquisition

Corresponding to the canopy spectrum measurement, a small number of wheat leaves were chopped from each plot, put into aluminum boxes and placed in the oven for 1 hour to dry, and then they were placed into an air blast drying oven for drying until the quality remained unchanged. After drying, the samples were ground and screened. The Kjeldahl nitrogen determination method [23] was used in this study to measure the wheat leaf nitrogen content, and the calculation formula is shown below:

$$\mathrm{X}=\frac{{\mathrm{V}}_1-{\mathrm{V}}_2\times \mathrm{c}\times 0.0140}{\mathrm{m}\times {\mathrm{V}}_3/100}\times \mathrm{F}\times 100\%$$

(1)

where X is the nitrogen content, c is the solution concentration, ${\mathrm{V}}_1$ is the amount of sulfuric acid consumed during the titration of the sample, ${\mathrm{V}}_2$ is the amount of sulfuric acid consumed by the titration of the blank sample, ${\mathrm{V}}_3$ is the volume of the digestive liquid, m is the sample mass, and F is the coefficient for converting nitrogen into protein.

2.3. Data Preprocessing

2.3.1. Construct Vegetation Indices

The vegetation index refers to various values obtained by a linear or nonlinear combination of spectral reflectance of different bands, which can reflect crop growth and nutrition status and can reduce the influence of errors caused by external environmental factors. It is often used to estimate the physiological and biochemical parameters of crops [24]. This study selected 35 vegetation indices [25] based on previous studies, and the details of these indices are listed in

Table 1. The vegetation indices and their calculation formulas.

Vegetation Index	Calculation Formula
Anthocyanin Reflectance Index (ARI)	$\left(1/{\mathrm{R}}_{550}\right)-\left(1/{\mathrm{R}}_{700}\right)$
Anthocyanin Reflectance Index (ARI2)	${\mathrm{R}}_{803}\left[\left(1/{\mathrm{R}}_{549}\right)-\left(1/{\mathrm{R}}_{702}\right)\right]$
Atmospherically Resistant Vegetation Index (ARVI)	$\frac{{\mathrm{R}}_{872}-\left[{\mathrm{R}}_{661}-\left({\mathrm{R}}_{488}-{\mathrm{R}}_{661}\right)\right]}{{\mathrm{R}}_{872}+\left[{\mathrm{R}}_{661}-\left({\mathrm{R}}_{488}-{\mathrm{R}}_{661}\right)\right]}$
Chlorophyll Absorption Ratio Index (CARI)	$\left({\mathrm{R}}_{700}-{\mathrm{R}}_{670}\right)-0.2\times \left({\mathrm{R}}_{700}-{\mathrm{R}}_{550}\right)$
Carotenoid Reflectance Index 1 (CRI1)	$\left(1/{\mathrm{R}}_{508}\right)-\left(1/{\mathrm{R}}_{549}\right)$
Carotenoid Reflectance Index 2 (CRI2)	$\left(1/{\mathrm{R}}_{508}\right)-\left(1/{\mathrm{R}}_{702}\right)$
Enhanced Vegetation Index (EVI)	$2.5\times \left[\left({\mathrm{R}}_{872}-{\mathrm{R}}_{661}\right)/\left({\mathrm{R}}_{872}+6{\times \mathrm{R}}_{661}-7.5{\times \mathrm{R}}_{488}+1\right)\right]$
Green Atmospherically Resistant Index (GARI)	$\left[{\mathrm{R}}_{872}-\right[{\mathrm{R}}_{559}-\left({\mathrm{R}}_{488}-{\mathrm{R}}_{661}\right)\left]]/[{\mathrm{R}}_{872}+\right[{\mathrm{R}}_{559}-\left({\mathrm{R}}_{488}-{\mathrm{R}}_{661}\right)]]$
Difference Vegetation Index (GNDVI)	$\left({\mathrm{R}}_{872}-{\mathrm{R}}_{559}\right)/\left({\mathrm{R}}_{872}+{\mathrm{R}}_{559}\right)$
Green Ratio Vegetation Index (GRVI)	$\left({\mathrm{R}}_{872}\right)/\left({\mathrm{R}}_{559}\right)$
Hyperspectral Normalized Difference Vegetation Index (HNDVI)	$\left({\mathrm{R}}_{827}-{\mathrm{R}}_{668}\right)/\left({\mathrm{R}}_{827}+{\mathrm{R}}_{668}\right)$
Modified Chlorophyll Absorption Ratio Index (MCARI)	$\left[\left({\mathrm{R}}_{702}-{\mathrm{R}}_{671}\right)-0.2\times \left({\mathrm{R}}_{702}-{\mathrm{R}}_{549}\right)]\times [\left({\mathrm{R}}_{702}\right)/{\mathrm{R}}_{671}\right]$
Modified Chlorophyll Absorption Ratio Index Improved (MCARI2)	$\frac{1.5\times \left[2.5\times \left({\mathrm{R}}_{803}-{\mathrm{R}}_{671}\right)-1.3\times \left({\mathrm{R}}_{803}-{\mathrm{R}}_{549}\right)\right]}{\sqrt{{\left(2\times {\mathrm{R}}_{827}+1\right)}^2-\left(6\times {\mathrm{R}}_{803}-5\times \sqrt{{\mathrm{R}}_{671}}\right)-0.5}}$
Red Edge Normalized Vegetation Index (MRENDVI)	$\left({\mathrm{R}}_{752}-{\mathrm{R}}_{702}\right)/\left({\mathrm{R}}_{752}+{\mathrm{R}}_{702}\right)$
MERIS Terrestrial Chlorophyll Index (MTCI)	$\left({\mathrm{R}}_{742}-{\mathrm{R}}_{702}\right)/\left({\mathrm{R}}_{702}+{\mathrm{R}}_{661}\right)$
Modified Triangular Vegetation Index (MTVI1)	$1.2\times \left[1.2\times \left({\mathrm{R}}_{800}-{\mathrm{R}}_{550}\right)-2.5\times \left({\mathrm{R}}_{670}-{\mathrm{R}}_{550}\right)\right]$
Normalized Difference Vegetation Index (NDVI)	$\left({\mathrm{R}}_{800}-{\mathrm{R}}_{670}\right)/\left({\mathrm{R}}_{800}+{\mathrm{R}}_{670}\right)$
Normalized Difference Water Index (NDWI)	$\left({\mathrm{R}}_{872}-{\mathrm{R}}_{1245}\right)/\left({\mathrm{R}}_{872}+{\mathrm{R}}_{1245}\right)$
Non-Linear Index (NLI)	$\left[{\left({\mathrm{R}}_{872}\right)}^2-{\mathrm{R}}_{661}]/[{\left({\mathrm{R}}_{872}\right)}^2+{\mathrm{R}}_{661}\right]$
Normalized Pigment Chlorophyll Index (NPCI)	$\left({\mathrm{R}}_{680}-{\mathrm{R}}_{430}\right)/\left({\mathrm{R}}_{680}+{\mathrm{R}}_{430}\right)$
Normalized Phaeophytinization Index (NPQI)	$\left({\mathrm{R}}_{415}-{\mathrm{R}}_{435}\right)/\left({\mathrm{R}}_{415}+{\mathrm{R}}_{435}\right)$
Optimized Soil-Adjusted Vegetation Index (OSAVI)	$\left({\mathrm{R}}_{800}-{\mathrm{R}}_{670}\right)/\left({\mathrm{R}}_{800}-{\mathrm{R}}_{670}+0.16\right)$
Photochemical Reflectance Index (PRI)	$\left({\mathrm{R}}_{531}-{\mathrm{R}}_{570}\right)/\left({\mathrm{R}}_{531}+{\mathrm{R}}_{570}\right)$
Photochemical Reflectance Index Improved (PRI4)	$\left({\mathrm{R}}_{529}-{\mathrm{R}}_{671}\right)/\left({\mathrm{R}}_{529}+{\mathrm{R}}_{671}\right)$
Plant Senescence Reflectance Index (PSRI)	$\left({\mathrm{R}}_{680}-{\mathrm{R}}_{500}\right)/\left({\mathrm{R}}_{750}\right)$
Red Edge Position Index (REP)	$700+\frac{40\times \left[\frac{{\mathrm{R}}_{670}+{\mathrm{R}}_{780}}{2}-{\mathrm{R}}_{700}\right]}{\left({\mathrm{R}}_{740}-{\mathrm{R}}_{700}\right)}$
Renormalized Difference Vegetation Index (RDVI)	$\left({\mathrm{R}}_{872}-{\mathrm{R}}_{661}\right)/\sqrt{{\mathrm{R}}_{872}+{\mathrm{R}}_{661}}$
Ratio Vegetation Index (RVI)	${\mathrm{R}}_{765}/{\mathrm{R}}_{720}$
Soil Adjusted-Vegetation Index (SAVI)	$\frac{1.5\times \left({\mathrm{R}}_{872}-{\mathrm{R}}_{661}\right)}{\left({\mathrm{R}}_{872}+{\mathrm{R}}_{661}\right)+0.5}$
Structure Insensitive Pigment Index (SIPI)	$\left({\mathrm{R}}_{803}-{\mathrm{R}}_{447}\right)/\left({\mathrm{R}}_{803}-{\mathrm{R}}_{681}\right)$
Transformed Chlorophyll Absorption Reflectance Index (TCARI)	$3\times \left[\left({\mathrm{R}}_{700}-{\mathrm{R}}_{670}\right)-0.2\times \left({\mathrm{R}}_{700}-{\mathrm{R}}_{550}\right)\times \left(\frac{{\mathrm{R}}_{700}}{{\mathrm{R}}_{670}}\right)\right]$
Triangular Greenness Index (TGI)	$-0.5\times \left[\left(670-480\right)\left({\mathrm{R}}_{670}-{\mathrm{R}}_{550}\right)-\left(670-550\right)\left({\mathrm{R}}_{670}-{\mathrm{R}}_{480}\right)\right]$
Triangular Vegetation Index (TVI)	$60\times \left({\mathrm{R}}_{800}-{\mathrm{R}}_{550}\right)-100\times \left({\mathrm{R}}_{670}-{\mathrm{R}}_{550}\right)$
Visible Atmospherically Resistant Index (VARI)	$\left({\mathrm{R}}_{559}-{\mathrm{R}}_{661}\right)/\left({\mathrm{R}}_{559}+{\mathrm{R}}_{661}-{\mathrm{R}}_{488}\right)$
Water Index (WI)	${\mathrm{R}}_{900}/{\mathrm{R}}_{970}$

.

2.3.2. Continuous Wavelet Transform

Wavelet transform is a linear transform method, which uses rich wavelet basis functions to decompose complex signals into wavelets with different frequencies. With this method, the weak information portion of the signal can be extracted effectively, fully highlighting the local features, and supporting multi-resolution analysis [26]. In this study, continuous wavelet transform was used to breakdown the hyperspectral data into a series of wavelet coefficients at different scales and wavelengths. Ten decomposition scales of wavelet coefficients were selected, and the conversion equation is as follows:

$$\mathrm{f}\left(\mathrm{a},\mathrm{b}\right)={{\displaystyle \int }}_{-\infty }^{+\infty }\mathrm{f}\left(\mathsf{\lambda }\right){\mathsf{\phi }}_{\mathrm{a},\mathrm{b}}\left(\mathsf{\lambda }\right)\mathrm{d}\mathsf{\lambda }$$

(2)

$${\phi }_{a,b}\left(\lambda \right)=\frac{1}{\sqrt{a}}\phi \left(\frac{\lambda -b}{a}\right)$$

(3)

where f(λ) is the spectral reflectance, a is the scale factor, b is the panning factor, λ is the corresponding band, and ${\phi }_{a,b}$ is the wavelet basis function.

2.3.3. Fractional Order Differential

Differential transformation is a spectral transformation method for the inversion of wheat biochemical parameters from hyperspectral information. Its advantages are that it can enhance the signal-to-noise ratio of spectral data and weaken the environmental noise. However, the commonly used differential transformation mostly differentiates the spectral reflectance using the integer order differential. The fractional order differential [27] is a mathematical expansion of the integer order differential, which has the advantages of being “memory” based and “global”. Compared with the integer order differential, it can explain more subtle changes and the overall information on the data. The fractional order differential mainly includes three definition forms, namely, Caputo, Riemann–Liouville, and Grünwald–Letnikov. The specific calculation formula is shown below:

$$\frac{d^{\alpha }f\left(\lambda \right)}{d{\lambda }^{\alpha }}\approx f\left(\lambda \right)+\left(-\alpha \right)f\left(\lambda -1\right)+\frac{\left(-\alpha \right)\left(-\alpha +1\right)}{2}f\left(\lambda -2\right)+\dots +\frac{\Gamma \left(-\alpha +1\right)}{n!\Gamma \left(-\alpha +1\right)}f\left(\lambda -n\right)$$

(4)

where f(λ) is the spectral reflectivity, λ is the corresponding waveform, Γ is the gamma function, α is the order, and n is the difference between the upper and lower limits. When α = 0, 1, and 2, it corresponds to the raw spectra, first-order derivative, and second-order derivatives, respectively. When α is a decimal, it corresponds to the fractional differential spectrum.

2.3.4. Correlation Analysis

Correlation analysis is the analysis of the consistency of trends between two or more datasets, mostly using the magnitude of the correlation coefficient to evaluate the closeness and extent of the relationship between the datasets [28]. The Pearson correlation coefficient [29] is one of the simplest and most widely used correlation coefficients at present. Its value range is −1~1. The closer the absolute value of the correlation coefficient ($\left|r\right|$) is to 1, the higher the correlation between the variables. The calculation formula for the Pearson correlation coefficient is shown below:

$${\rho }_{X,Y}=\frac{cov\left(X,Y\right)}{{\sigma }_X{\sigma }_Y}=\frac{E\left(XY\right)-E\left(X\right)E\left(Y\right)}{\sqrt{E\left(X^2\right)-E^2\left(X\right)}\sqrt{E\left(Y^2\right)-E^2\left(Y\right)}}$$

(5)

where ${\rho }_{X,Y}$ is the correlation coefficient, and $cov$ and $\sigma$ denotes the covariance and standard deviation, respectively.

2.4. Model Construction

2.4.1. SVM

SVM is a machine learning method based on the theory of statistical learning that seeks to minimize the structured risk to minimize the empirical risk and confidence range, thereby improving the generalization ability of the learning machine [30]. Its basic idea for regression is as follows: given training samples (x, y), it establishes a regression model $\mathrm{f}\left(\mathrm{x}\right)={\mathrm{w}}^{\mathrm{T}}\mathrm{x}+\mathrm{b}$ to make f(x) and y infinitely close, where w and b are the model parameters to be determined. Let ε denote the maximum error between the predicted value f(x) and y of the model, only the absolute value of the error between f(x) and y greater than ε is calculated as a deviation. Then, f(x) is taken as the center to build a 2ε-wide separation band. If the training sample falls within the range of the separation band, the calculation result is considered to be correct.

2.4.2. Ridge Regression

Ridge regression is essentially a modified least squares estimation method that obtains more realistic and reliable regression coefficients by giving up the unbiased nature of least squares, but at the cost of information loss and reduced accuracy. It makes the regression coefficients estimable by introducing an array of k-units. Though the introduction of the unit matrix leads to information loss, it allows a reasonable estimation of the regression model [31]. The process of ridge regression involves two steps: (1) find the best k-value in combination with the ridge tracking plot and choose the smallest k-value when the standardized regression coefficients of the respective variables tend to stabilize; (2) input the k-values for regression modeling. The calculation formula for ridge regression is shown below:

$$\mathrm{w}={\left({\mathrm{X}}^{\mathrm{T}}\mathrm{X}+\mathrm{kI}\right)}^{-1}{\mathrm{X}}^{\mathrm{T}}\mathrm{y}$$

(6)

where X is the eigenvalue matrix, y is the target value matrix, and K is the ridge parameter.

2.4.3. Stepwise Regression

The basic idea of stepwise regression [32] is to reduce the degree of multicollinearity by removing variables that are less significant and strongly correlated with other variables. Variables are introduced into the model sequentially, and after each explanatory variable is introduced, an F-test is performed, and a t-test is conducted for each of the selected explanatory variables. If the initially introduced explanatory variables become non-significant due to the introduction of later explanatory variables, they are removed to ensure that only significant variables are included in the regression equation before each new variable is introduced. This iterative process continues until no significant explanatory variables are included in the regression equation or no insignificant explanatory variables are removed from the regression equation, to ensure that the final set of explanatory variables is optimal.

2.4.4. GPR

GPR is a nonparametric model for regression analysis of data using a Gaussian process prior [33]. In regression prediction, only a single point value is usually predicted. Gaussian process regression can be regarded as a probability prediction, and in predicting the accurate point, the upper and lower limits are predicted, thus adding more reference value to the prediction. The calculation formulas for GPR are represented below:

$$y^{\left(i\right)}=f\left(x^{\left(i\right)}\right)+{\epsilon }^{\left(i\right)},i=1,\cdots ,n$$

(7)

$$f\left(x\right)~GP\left(m\left(x\right),K\left(x,x^{\prime }\right)\right)$$

(8)

where ${\epsilon }^{\left(i\right)}$ represents the noise variable that obeys $\mathrm{N}\left(0,{\mathsf{\sigma }}^2\right)$, m(x) is the mean function, and K(x, x^′) is the covariance function, also known as the kernel function. It can be seen that a Gaussian regression process is determined by a mean function and a covariance function.

2.4.5. BP Neural Network

The feedback neural network [34] is a neural network system that connects the output to the input layer after a time shift. Its basic idea is to make the actual output value of the system as close as possible to the expected output value by using gradient search. The input layer of the BP neural network usually uses two activation functions to send the input information into the hidden layer. In this study, the activation function is a sigmoid function [35], and its calculation formula is shown below:

$$\mathrm{f}\left(\mathrm{u}\right)={\left(1+{\mathrm{e}}^{-\mathrm{u}}\right)}^{-1}$$

(9)

$$\mathrm{O}=\mathrm{simoid}\left( \overrightarrow{\mathrm{W}}· \overrightarrow{\mathrm{X}}\right)$$

(10)

$$\mathrm{y}=\mathrm{simoid}\left( \overrightarrow{\mathrm{W}}·\overrightarrow{\mathrm{O}}\right)$$

(11)

$$\mathsf{\epsilon }=\left(1-\mathrm{y}\right)\left(\mathrm{T}-\mathrm{y}\right)$$

(12)

$${\epsilon }_i=a_i\left(1-a_i\right){\displaystyle \sum }_m{k}_{ki}{\epsilon }_k$$

(13)

where O is the output of the hidden layer, y is the output of the output layer, i is the hidden layer node, and ε is the computational error.

2.5. Accuracy Evaluation Index

• In this paper, 48 cell data were randomly assigned, among which 36 cell data were used as training sets and 12 cell data were used as test sets. The coefficient of determination and root mean square error of the training set and test set were used as the evaluation indexes for the model accuracy.
• The coefficient of determination

Indicates the closeness between the estimated and measured values of the model, and its value range is 0~1. The larger the value of ${\mathrm{R}}^2$, the higher the estimation accuracy of the mode. The specific calculation formula is shown in (14):

$${\mathrm{R}}^2=\frac{{\sum }_{i=1}^n{(y_i-\overline{y})}^2}{{\sum }_{i=1}^n{(x_i-\overline{y})}^2}$$

(14)

3.

The root mean square error

(RMSE) reflects the error between the estimated and measured values of the model. The smaller the RMSE, the higher the estimation accuracy of the model. The specific calculation formula is shown below:

$$\mathrm{RMSE}=\sqrt{\frac{{\sum }_{\mathrm{i}=1,\mathrm{j}=1}^{\mathrm{n}}{\left({\mathrm{x}}_{\mathrm{i}}-{\mathrm{y}}_{\mathrm{i}}\right)}^2}{\mathrm{n}}}$$

(15)

where ${\mathrm{x}}_{\mathrm{i}}$ is the measured value of wheat nitrogen content, ${\mathrm{y}}_{\mathrm{i}}$ is the estimated value of wheat nitrogen content, $\overline{\mathrm{y}}$ is the average value of wheat nitrogen content measured and estimated, and n is the total number of samples.

3. Results

3.1. Estimation of Nitrogen Content Based on Vegetation Index

3.1.1. Correlation Analysis of Vegetation Index and Nitrogen Content

Based on the canopy spectra collected during the four growth periods of winter wheat, 35 vegetation indices, as shown in Table 1, were constructed, respectively, and the correlation analysis was carried out with the nitrogen content in the corresponding period. The correlation coefficient is shown in Figure 2. Five vegetation indices with good correlation were selected in the different growth periods, and their correlation matrix with nitrogen content was drawn. The results are illustrated in Figure 3.

The analysis of Figure 2 and Figure 3 shows that 16 vegetation indices passed the 0.001 extremely significant level test in the four growth stages of wheat, including REP, RVI, MTCI, GRVI, PRI, MCARI2, EVI, SAVI, TVI, RDVI, MTVI1, GNDVI, NLI, MRENDVI, OSAVI, and GARI, and the vegetation index having the strongest associations with nitrogen content was REP. During the jointing period, the correlation matrices of the selected vegetation indices and nitrogen content are shown in Figure 3a, and five vegetation indices, including MTCI, REP, RVI, GRVI, and PRI, were selected as independent variables to construct the nitrogen content estimation model, with the absolute values of the correlation coefficients ranging from 0.52 to 0.55. During the flag picking period, the correlation matrices of the selected vegetation indices and nitrogen content are shown in Figure 3b, and five vegetation indices, including GARI, MRENDVI, NPQI, REP, and RVI, were selected as independent variables to construct the nitrogen content estimation model, with the absolute values of the correlation coefficients ranging from 0.72 to 0.8. During the flowering period, the correlation matrices of the selected vegetation indices and nitrogen content are shown in Figure 3c, and five vegetation indices, including GNDVI, GRVI, MRENDVI, REP, and RVI, were selected as independent variables to construct the nitrogen content estimation model, with the absolute values of the correlation coefficients ranging from 0.7 to 0.83. During the filling period, the correlation matrices of the selected vegetation indices and nitrogen content are shown in Figure 3d, and five vegetation indices, including GNDVI, GRVI, MTCI, REP, and RVI, were selected as independent variables to construct the nitrogen content estimation model, with the absolute values of the correlation coefficients ranging from 0.75 to 0.77.

3.1.2. Construction and Analysis of Nitrogen Content Estimation Model

The first five vegetation indexes with a high correlation coefficient were taken as the independent variables in the model to build the estimation model; 75% of the sample data were randomly selected for training the model, and 25% of the sample data were used for model accuracy verification. The training set and test set consisted of 36 and 12 cells, respectively. The modeling accuracy and validation accuracy of the different model construction methods used in different periods are listed in

Table 2. The modeling accuracy of the different modeling methods in different periods.

		SVM		RE		SR		GPR		BPNN
		${\mathbf{R}}^2$	$\mathbf{R}\mathbf{M}\mathbf{S}\mathbf{E}$	${\mathbf{R}}^2$	$\mathbf{R}\mathbf{M}\mathbf{S}\mathbf{E}$	${\mathbf{R}}^2$	$\mathbf{R}\mathbf{M}\mathbf{S}\mathbf{E}$	${\mathbf{R}}^2$	$\mathbf{R}\mathbf{M}\mathbf{S}\mathbf{E}$	${\mathbf{R}}^2$	$\mathbf{R}\mathbf{M}\mathbf{S}\mathbf{E}$
Jointing	Train	0.52	0.37	0.41	0.43	0.40	0.39	0.71	0.34	0.67	0.32
	Test	0.42	0.43	0.41	0.37	0.35	0.49	0.53	0.35	0.55	0.40
Flag picking	Train	0.75	0.23	0.83	0.20	0.81	0.21	0.91	0.21	0.81	0.22
	Test	0.72	0.35	0.78	0.21	0.80	0.23	0.89	0.41	0.80	0.27
Flowering	Train	0.76	0.22	0.82	0.17	0.84	0.16	0.89	0.23	0.90	0.14
	Test	0.71	0.30	0.72	0.26	0.78	0.27	0.87	0.14	0.85	0.25
Filling	Train	0.62	0.42	0.66	0.38	0.65	0.39	0.77	0.39	0.67	0.37
	Test	0.60	0.44	0.64	0.45	0.64	0.44	0.64	0.41	0.67	0.46

. Then, the training and test sets were fitted as scatter plots to visually judge the accuracy of several modeling methods, as shown in Figure 4.

The analysis of Table 2 and Figure 4 reveals that when vegetation indices were combined with SVM to estimate nitrogen content, the highest accuracy was obtained at the flowering stage with an overall sample ${\mathrm{R}}^2$ value of 0.72, which was 0.23, 0.01, and 0.11 higher than that at the jointing, flag picking, and filling stages, respectively. When vegetation indices were combined with ridge regression to estimate nitrogen content, the highest accuracy was obtained at the flag picking stage, with an overall sample ${\mathrm{R}}^2$ value of 0.82, which was 0.43, 0.03, and 0.17 higher than that at the jointing, flowering, and filling stages, respectively. When vegetation indices were combined with stepwise regression to estimate nitrogen content, the highest accuracy was obtained at the flag picking stage, with an overall sample ${\mathrm{R}}^2$ value of 0.81, which was 0.42, 0.01, and 0.17 higher than that at the jointing, flowering, and filling stages, respectively. When the vegetation indices were combined with GPR to estimate nitrogen content, the highest accuracy was obtained at the flowering stage with an overall sample ${\mathrm{R}}^2$ value of 0.88, which was 0.21, 0.05, and 0.14 higher than that at the jointing, flag picking, and filling stages, respectively. When the vegetation indices were combined with the BP neural network to estimate nitrogen content, the highest accuracy was obtained at the flowering stage, with an overall sample ${\mathrm{R}}^2$ value of 0.86, which was 0.23, 0.06, and 0.20 higher than that at the jointing, flag picking, and filling stages, respectively. These results indicated that when vegetation indices were combined with the five modeling approaches to estimate nitrogen content at different wheat growth stages, the highest accuracy was achieved at the flowering stage, with an overall sample ${\mathrm{R}}^2$ mean of 0.81, which was 58.82%, 2.53%, and 22.73% higher than that at the jointing stage, flag picking stage, and filling stage, respectively.

A comprehensive analysis of the modeling results for the different fertility periods using vegetation indices based on the five methods of SVM, ridge regression, stepwise regression, GPR, and BP neural network shows that the overall estimation effect of the jointing stage was not good, and only GPR and BP neural network models performed well. The overall sample ${\mathrm{R}}^2$ value of the GPR model was 0.67, which was 0.18, 0.28, 0.28, and 0.04 higher than that of SVM, ridge regression, stepwise regression, and the BP neural network models, respectively. The estimation effect of nitrogen content in the flag picking stage was better than that in the jointing stage, and the overall sample ${\mathrm{R}}^2$ value was above 0.8 except for the SVM model. Among these models, the GPR model performed the best, with an overall sample value of 0.83, which was 0.12, 0.01, 0.02, and 0.03 higher than that of SVM, ridge regression, stepwise regression, and BP neural network models, respectively. The estimation effect of nitrogen content at the flowering stage was similar to that at the flag picking stage. The overall sample value of the GPR model and the BP neural network model can reach 0.85. Among them, the overall sample ${\mathrm{R}}^2$ value of the GPR model was 0.88, which was 0.16, 0.09, 0.08, and 0.02 higher than that of SVM, ridge regression, stepwise regression, and the BP neural network model, respectively. The effect of the nitrogen content estimation at the filling stage was superior to that at the jointing stage, and the effect was lower than that at the flag picking stage and the flowering stage. Among them, the GPR model achieved the best effect, with an overall sample ${\mathrm{R}}^2$ value of 0.74, which was 0.13, 0.09, 0.10, and 0.08 higher than SVM, ridge regression, stepwise regression, and the BP neural network model, respectively. The overall research results indicated that the GPR model was more accurate in estimating nitrogen content after building vegetation indices for wheat canopy spectra. The average value of the overall sample A in the four growth periods was 0.78, which was 23.81%, 18.18%, 18.18%, and 5.41% higher than that of SVM, ridge regression, stepwise regression, and the BP neural network models, respectively.

3.2. Estimation of Nitrogen Content Based on the Wavelet Energy Coefficient

3.2.1. Correlation Analysis between the Wavelet Energy Coefficient and Nitrogen Content

The spectral data of the wheat canopy at different growth stages were processed by the continuous wavelet, and the decomposition scale was set from 1 to 10. The wavelet energy coefficients under 10 decomposition scales can be obtained after decomposition at each growth stage and, then, the correlation analysis was carried out with the nitrogen content at the corresponding growth stage. The correlation diagram is presented in Figure 5. Meanwhile, 10 bands with a strong correlation were selected to draw the correlation matrix with nitrogen content, as shown in Figure 6.

The correlation analysis between the wavelet energy coefficient and nitrogen content was carried out at the jointing stage. The analysis of Figure 5a shows that the absolute value of the correlation coefficient $\left|r\right|$ between the wavelet energy coefficient and nitrogen content first increased and then decreased with the increase in the decomposition scale at the jointing stage. When the decomposition scale was 2, the correlation peak appeared near the band of 410 nm, and the maximum $\left|r\right|$ value reached 0.67. Ten wavelet energy coefficients with high correlation coefficients were selected, and their correlation matrix with nitrogen content is shown in Figure 6a at the following orders and bands: 2-416 nm, 2-415 nm, 6-445 nm, 6-447 nm, 6-441 nm, 6-450 nm, 6-438 nm, 6-452 nm, 2-787 nm, and 2-418 nm.

Then, the correlation analysis between the wavelet energy coefficient and nitrogen content was carried out during the flag picking period. The analysis of Figure 5b shows that the absolute value of the correlation coefficient $\left|r\right|$ between the wavelet energy coefficient and nitrogen content first increased and then decreased with the increase in the decomposition scale in the flag picking period. When the decomposition scale was 5, the correlation peak appeared near the band of 450 nm, and the maximum $\left|r\right|$ value reached 0.94. Ten wavelet energy coefficients with high correlation coefficients were selected, and their correlation matrix with nitrogen content is shown in Figure 6b at the following scales and bands: 5455 nm, 6448 nm, 5446 nm, 6383 nm, 5438 nm, 1686 nm, 1687 nm, 4421 nm, 3417 nm, and 1685 nm.

Next, the correlation analysis between the wavelet energy coefficient and nitrogen content was conducted at the flowering stage. The analysis of Figure 5c shows that the absolute value of the correlation coefficient $\left|r\right|$ between the wavelet energy coefficient and nitrogen content first increased and then decreased with the increase in the decomposition scale at the flowering stage. When the decomposition scale was 6, the correlation peak appeared near the band of 440 nm, and the maximum $\left|r\right|$ value reached 0.87. Ten wavelet energy coefficients with high correlation coefficients were selected, and their correlation matrix with nitrogen content is shown in Figure 6c at the following scales and bands: 6446 nm, 6451 nm, 6457 nm, 6403 nm, 1682 nm, 2678 nm, 2677 nm, 2679 nm, 1681 nm, and 3487 nm.

Finally, the correlation analysis between the wavelet energy coefficient and nitrogen content was carried out during the filling period. The analysis of Figure 5d shows that the absolute value $\left|r\right|$ of the correlation coefficient between the wavelet energy coefficient and nitrogen content gradually decreased with the increase in the decomposition scale during the filling period. When the decomposition scale was 1, the correlation peak appeared near 670 nm, and the maximum $\left|r\right|$ value reached 0.85. Ten wavelet energy coefficients with high correlation coefficients were selected, and their correlation matrix with nitrogen content is shown in Figure 6d in the following orders and bands: 1-670 nm, 2-826 nm, 2-669 nm, 4-811 nm, 6-437 nm, 1-753 nm, 5-818 nm, 3-773 nm, 3-775 nm, and 2-754 nm.

3.2.2. Construction and Analysis of Nitrogen Content Estimation Model

Based on the above correlation analysis results, the estimation model was constructed with the selected ten wavelet energy coefficients with a strong correlation as the independent variables in the model. Then, the model was trained with 36 cell sample data, and the model accuracy was verified with 12 cell sample data. The modeling and verification accuracy of the different model construction methods in the different periods is presented in

Table 3. The modeling accuracy of the different modeling methods in the different periods.

		SVM		RE		SR		GPR		BPNN
		${\mathbf{R}}^2$	$\mathbf{R}\mathbf{M}\mathbf{S}\mathbf{E}$	${\mathbf{R}}^2$	$\mathbf{R}\mathbf{M}\mathbf{S}\mathbf{E}$	${\mathbf{R}}^2$	$\mathbf{R}\mathbf{M}\mathbf{S}\mathbf{E}$	${\mathbf{R}}^2$	$\mathbf{R}\mathbf{M}\mathbf{S}\mathbf{E}$	${\mathbf{R}}^2$	$\mathbf{R}\mathbf{M}\mathbf{S}\mathbf{E}$
Jointing	Train	0.60	0.33	0.74	0.29	0.67	0.30	0.68	0.32	0.70	0.30
	Test	0.61	0.40	0.70	0.38	0.65	0.32	0.64	0.35	0.65	0.32
Flag picking	Train	0.94	0.12	0.93	0.13	0.92	0.14	0.94	0.16	0.95	0.12
	Test	0.82	0.21	0.87	0.20	0.88	0.17	0.83	0.24	0.93	0.15
Flowering	Train	0.86	0.16	0.88	0.15	0.87	0.15	0.90	0.19	0.89	0.14
	Test	0.86	0.18	0.83	0.30	0.79	0.26	0.78	0.25	0.82	0.29
Filling	Train	0.82	0.29	0.80	0.30	0.85	0.27	0.80	0.34	0.86	0.25
	Test	0.81	0.35	0.75	0.38	0.72	0.46	0.78	0.29	0.82	0.60

. Meanwhile, the training set and test set were fitted into scatter plots to intuitively judge the accuracy of several modeling methods, and the results are shown in Figure 7.

The analysis of Table 3 and Figure 7 reveal that when continuous wavelet transform was combined with SVM, ridge regression, stepwise regression, GPR, and the BP neural network to estimate the nitrogen content in leaves, the highest accuracy was obtained in the flag picking period. The overall sample ${\mathrm{R}}^2$ value was 0.90, which was 0.31, 0.04, and 0.10 higher than that in the jointing, flowering, and filling stages, respectively. When continuous wavelet transform was combined with ridge regression, the overall sample ${\mathrm{R}}^2$ value at the flag picking stage was 0.91, which was 0.22, 0.08, and 0.13 higher than that at the plucking, flowering, and filling stages, respectively. When continuous wavelet transform was combined with stepwise regression, the overall sample ${\mathrm{R}}^2$ value at the flag picking stage was 0.91, which was 0.25, 0.08, and 0.13 higher than that at the jointing, flowering, and filling stages, respectively. When continuous wavelet transform was combined with GPR, the overall sample ${\mathrm{R}}^2$ value at the flag picking stage was 0.91, which was 0.24, 0.05, and 0.11 higher than that at the jointing, flowering, and filling stages, respectively. When continuous wavelet transform was combined with the BP neural network, the overall sample ${\mathrm{R}}^2$ value at the flag picking stage was 0.94, which was 0.27, 0.10, and 0.17 higher than that at the jointing, flowering, and filling stages, respectively. These results indicated that when continuous wavelet transform was combined with the five modeling methods to estimate nitrogen content at different wheat growth stages, the highest accuracy was achieved at the flag picking stage with a sample ${\mathrm{R}}^2$ mean of 0.91, which was 40.00%, 8.33%, and 15.19% higher than that at the jointing, flowering, and filling stages, respectively.

Meanwhile, a comprehensive analysis of the modeling results of combining continuous wavelet transform with SVM, ridge regression, stepwise regression, GPR, and the BP neural network for different fertility periods revealed that the overall estimation effect of the jointing stage was not good. The overall sample ${\mathrm{R}}^2$ value of all models except the SVM model was above 0.66, and the effect of ridge regression was relatively good, with an overall sample ${\mathrm{R}}^2$ value of 0.68, which was 0.09, 0.02, 0.01 and 0.01 higher than that of SVM, stepwise regression, GPR, and the BP neural network model, respectively. The estimation effect of nitrogen content in the flag picking period was significantly better than that in the other three periods. The overall ${\mathrm{R}}^2$ value of the five models was above 0.90, and the BP neural network model obtained the best effect. The overall sample ${\mathrm{R}}^2$ value was 0.94, which was 0.04, 0.03, 0.03, and 0.03 higher than that of SVM, ridge regression, stepwise regression, and the GPR model, respectively. The effect of nitrogen content estimation at the flowering stage was superior to that at the jointing stage, and the effect was worse than that at the flag picking stage. The effect of the five models was equivalent, and the overall sample ${\mathrm{R}}^2$ value was above 0.83. The overall sample ${\mathrm{R}}^2$ value of the SVM and GPR models was 0.86, which was 0.03, 0.03, and 0.02 higher than that of the ridge regression, stepwise regression, and BP neural network models, respectively. The estimation of nitrogen content at the filling stage was superior to that at the jointing stage and worse when compared to that at the flag picking and flowering stages. The overall sample ${\mathrm{R}}^2$ values of all five models were above 0.77. Among these models, the SVM and GPR models were more accurate with an overall sample ${\mathrm{R}}^2$ value of 0.80, which was 0.02, 0.02, and 0.03 higher than that of the ridge regression, stepwise regression, and BP neural network models, respectively. These results indicated that the GPR model was more accurate in estimating the nitrogen content after the wavelet transform of the wheat canopy spectrum. The average value of the overall sample ${\mathrm{R}}^2$ in the four growth periods was 0.81, which was 2.53%, 1.25%, 1.25%, and 1.25% higher than that of the SVM, ridge regression, stepwise regression, and BP neural network models, respectively.

3.3. Estimation of Nitrogen Content Based on Fractional Order Differentiation

3.3.1. Correlation Analysis of Fractional Order Differential Spectra and Nitrogen Content

The original canopy hyperspectral reflectance data for the different fertility periods were subjected to fractional order differential transformations. The order range was set from 0 to 2, the step size was set at 0.1, and 20 pieces of differential spectral reflectance for the different orders were obtained for each fertility period. Then, a correlation analysis was carried out between the nitrogen content and the fractional order differential spectra for the different fertility periods, and the plotted correlation relationship was obtained, as shown in Figure 8. Ten differential spectra with strong correlations were selected, and their correlation matrix with nitrogen content is plotted in Figure 9.

A correlation analysis was conducted between the fractional differential spectrum and nitrogen content in the jointing stage, as shown in Figure 8a. The maximum absolute value of the correlation coefficient $\left|r\right|$ between the differential spectra and nitrogen content was greater than 0.53 for all orders except orders 1 and 2. When the order was 1 at the jointing stage, the correlation peak appeared near the band of 770 nm, and the maximum value of $\left|r\right|$ reached 0.65. In addition to the integer order, i.e., order 1 and order 2, the number of spectral bands tested at the 0.01 highly significant level exceeded 722. When the order was 0.4, up to 735 bands can be reached. The orders and bands for the ten pieces of differential spectra with high correlation coefficients include: 1-772 nm, 1-782 nm, 1-775 nm, 1-1590 nm, 2-704 nm, 2-705 nm, 0.9-907 nm, 0.9-915 nm, 1-1627 nm, and 2-719 nm. The correlation matrix between the fractional differential spectrum with nitrogen content is shown in Figure 9a.

In the flag picking stage, the fractional differential spectrum was exploited to conduct correlation analysis with nitrogen content. As shown in Figure 8b, the maximum absolute value of the correlation coefficient $\left|r\right|$ between the differential spectra and nitrogen content was greater than 0.68 for all orders. When order 2 was used in the flag picking stage, the correlation peak appeared near the band of 730 nm, and the maximum value of $\left|r\right|$ reached 0.77. In addition to the integer order, i.e., order 1 and order 2, the number of spectral bands tested at the 0.01 highly significant level exceeded 880. When the order was 1.2, 1.4, and 1.5, up to 892 bands can be reached. The orders and bands of the ten pieces of differential spectra with high correlation coefficients include: 2-733 nm, 1-753 nm, 1-756 nm, 1-749 nm, 1.1-724 nm, 1.1-725 nm, 1.1-723 nm, 1.1-729 nm, 2-728 nm, and 1-747 nm. The correlation matrix of the differential spectra with nitrogen content is shown in Figure 9b.

At the flowering stage, a correlation analysis was conducted between the fractional differential spectra and nitrogen content. As shown in Figure 8c, the maximum absolute value of the correlation coefficient $\left|r\right|$ between differential spectra and nitrogen content was greater than 0.67 for all orders. When the order was 1, the correlation peak appeared near the band of 770 nm, and the maximum value of $\left|r\right|$ reached 0.76. In addition to the integer order, i.e., order 1 and order 2, the number of spectral bands tested at the 0.01 highly significant level exceeded 975, and when the order was 0.7, 1013 bands could be reached at most. The order and bands of the ten differential spectra with high correlation coefficients include: 1-774 nm, 1-769 nm, 1.1-729 nm, 1.1-731 nm, 1.1-732 nm, 1-754 nm, 1.1-728 nm, 1-748 nm, 1.1-733 nm, and 2-731 nm. The correlation matrix of the differential spectra with nitrogen content is shown in Figure 9c.

At the filling stage, the fractional differential spectra were exploited to carry out a correlation analysis with the nitrogen content. As shown in Figure 8d, the maximum absolute value of the correlation coefficient $\left|r\right|$ between the differential spectra and nitrogen content was greater than 0.72 for all orders. When the order was 1, the correlation peaks appeared near the band of 750 nm, and the maximum value of $\left|r\right|$ reached 0.77. In addition to the integer order, i.e., order 1 and order 2, the number of spectral bands tested at the 0.01 highly significant level exceeded 1150. When the order was 0.6, up to 1202 bands can be reached. The order and bands of the ten differential spectra with high correlation coefficients include: 1-752 nm, 1-741 nm, 1-754 nm, 1-745 nm, 1-734 nm, 1.1-730 nm, 1.1-723 nm, 1.2-367 nm, 1.1-726 nm, and 1.9-558 nm. The correlation matrix between the differential spectra and nitrogen content is shown in Figure 9d.

3.3.2. Model Construction and Analysis of Nitrogen Content Estimation

For winter wheat at different growth periods, 36 cell data were taken as the training dataset and 12 cell data as the test set. The modeling results for the fractional differential spectra combined with SVM, ridge regression, stepwise regression, GPR, and the BP neural network were analyzed, respectively, to obtain the best estimation model, and the result is shown in

Table 4. The accuracy of the different modeling methods in the different periods.

		SVM		RE		SR		GPR		BPNN
		${\mathbf{R}}^2$	$\mathbf{R}\mathbf{M}\mathbf{S}\mathbf{E}$	${\mathbf{R}}^2$	$\mathbf{R}\mathbf{M}\mathbf{S}\mathbf{E}$	${\mathbf{R}}^2$	$\mathbf{R}\mathbf{M}\mathbf{S}\mathbf{E}$	${\mathbf{R}}^2$	$\mathbf{R}\mathbf{M}\mathbf{S}\mathbf{E}$	${\mathbf{R}}^2$	$\mathbf{R}\mathbf{M}\mathbf{S}\mathbf{E}$
Jointing	Train	0.59	0.34	0.65	0.32	0.60	0.35	0.68	0.32	0.73	0.35
	Test	0.60	0.34	0.58	0.48	0.46	0.47	0.59	0.46	0.66	0.57
Flag picking	Train	0.79	0.23	0.75	0.24	0.76	0.23	0.82	0.24	0.91	0.16
	Test	0.76	0.32	0.73	0.27	0.71	0.28	0.76	0.22	0.90	0.21
Flowering	Train	0.72	0.25	0.74	0.23	0.74	0.23	0.84	0.20	0.88	0.15
	Test	0.70	0.32	0.70	0.25	0.65	0.25	0.79	0.35	0.81	0.26
Filling	Train	0.68	0.40	0.73	0.34	0.69	0.35	0.78	0.33	0.76	0.31
	Test	0.67	0.37	0.71	0.42	0.64	0.48	0.66	0.51	0.63	0.66

. After fitting the training dataset with the test set, the scatter diagram is presented in Figure 10.

The analysis of Table 4 and Figure 10 revealed that when the fractional order differential was combined with SVM, ridge regression, stepwise regression, GPR, and the BP neural network to estimate leaf nitrogen content, all had the highest accuracy at the flag picking stage. When fractional order differentiation was combined with the SVM model, the overall sample ${\mathrm{R}}^2$ value at the flag picking stage was 0.78, which was 0.20, 0.07, and 0.10 higher than that at the jointing, flowering, and filling stages, respectively. When the fractional order differential was combined with the ridge regression model, the overall sample ${\mathrm{R}}^2$ value at the flag picking stage was 0.74, which was 0.17, 0.02, and 0.02 higher than at the jointing, flowering, and filling stages, respectively. When the fractional order differential was combined with stepwise regression, the overall sample ${\mathrm{R}}^2$ value at the flag picking stage was 0.74, which was 0.23, 0.03, and 0.07 higher than that at the jointing, flowering, and filling stages, respectively. When the fractional order differential was combined with GPR, the overall sample ${\mathrm{R}}^2$ value at the flag picking stage was 0.80, which was 0.17, 0.06, and 0.07 higher than that at the jointing, flowering, and filling stages, respectively. When the fractional order differential was combined with the BP neural network, the overall sample ${\mathrm{R}}^2$ value at the flag picking stage was 0.90, which was 0.22, 0.06, and 0.25 higher than that at the jointing, flowering, and filling stages, respectively. These results indicated that when using fractional order differential and the five modeling methods to estimate nitrogen content at different stages of wheat growth, the accuracy at the flag picking stage was the highest, and the average value of the overall sample ${\mathrm{R}}^2$ was 0.79, which was 31.67%, 6.76%, and 14.49% higher than that at the jointing stage, flowering stage, and filling stage, respectively.

Based on the comprehensive analysis of the modeling results from combining the fractional order differential with SVM, ridge regression, stepwise regression, GPR, and the BP neural network at different growth stages, the overall estimation effect of the jointing stage was not good. Only GPR and the BP neural network models achieved a good effect, and the overall sample ${\mathrm{R}}^2$ value was above 0.6. The overall sample ${\mathrm{R}}^2$ value for the BP neural network model was 0.68, which was 0.10, 0.11, 0.17, and 0.05 higher than that of the SVM, ridge regression, stepwise regression, and GPR models, respectively. The estimation effect of nitrogen content in the flag picking stage was significantly better than that at the jointing stage. The overall sample ${\mathrm{R}}^2$ value of the five models was more than 0.74, and that of the GPR and BP neural network models was higher than 0.80. The BP neural network model achieved the highest estimation accuracy. The overall sample ${\mathrm{R}}^2$ value was 0.90, which was 0.12, 0.16, 0.16, and 0.10 higher than that of the SVM, ridge regression, stepwise regression, and GPR models, respectively. The estimation effect of nitrogen content in the flowering stage was better than that in the jointing stage and was inferior to that in the flag picking stage. The overall sample ${\mathrm{R}}^2$ value of the five models was above 0.71, and the accuracy of the BP neural network model was the highest. The overall sample ${\mathrm{R}}^2$ value was 0.84, which was 0.13, 0.12, 0.13, and 0.10 higher than that of the SVM, ridge regression, stepwise regression, and GPR models, respectively. The estimation effect of nitrogen content in the filling stage was better than that in the jointing stage, which was second to that in the flag picking stage and the flowering stage, and only the overall ${\mathrm{R}}^2$ value of the ridge regression and GPR models was above 0.72. The overall sample ${\mathrm{R}}^2$ value of the GPR model was 0.73, which was 0.05, 0.01, 0.06, and 0.08 higher than that of the SVM, ridge regression, stepwise regression, and BP neural network models, respectively. The overall results indicated that the model accuracy of nitrogen content estimation using the BP neural network model after fractional order differentiation of wheat canopy spectra was higher, and its mean sample ${\mathrm{R}}^2$ value was 0.77 at four growth stages, which was 11.59%, 11.59%, 16.67%, and 6.94% higher than that of the SVM, ridge regression, stepwise regression, and BP neural network models, respectively.

4. Discussion

A comprehensive analysis of the nitrogen content in winter wheat based on the SVM, ridge regression, stepwise regression, GPR, and BP neural network methods for four growth stages showed that the maximum modeling and validation ${\mathrm{R}}^2$ of the estimated models reached 0.95 and 0.93, respectively, and the average modeling and validation ${\mathrm{R}}^2$ reached 0.76 and 0.71, respectively, showing a good overall estimation effect of nitrogen content. The average value of the overall sample ${\mathrm{R}}^2$ of the nitrogen content estimation model based on the wavelet energy coefficient in the four growth stages of wheat was 27.24%, 15.11%, 4.20%, and 19.09% higher than the nitrogen content estimation model based on vegetation indices, and 10.10%, 15.40%, 13.44%, and 13.91% higher than the nitrogen content estimation model based on the fractional differential, indicating that the nitrogen content estimation model based on the wavelet energy coefficient achieved higher accuracy. This is because hyperspectral data has high resolution and rich information, which can express the detailed information on crops in more detail and can be applied to the estimation of crop phenotypic parameters. Assimilating the effective information in the spectral data into the wheat growth process is beneficial for determining the nitrogen content change in the wheat growth process. However, the canopy structure of wheat, the soil background, and atmospheric influence factors affect the performance of wheat canopy spectral reflectance, making it difficult to use the canopy spectral reflectance to effectively obtain wheat nitrogen content. After fractional differential, continuous wavelet transform and vegetation index processing can effectively eliminate noise and refine the spectral information, and the estimation results on wheat nitrogen content can be improved.

Comparing the effect of nitrogen content estimation in wheat at different growth stages, it can be seen that the accuracy of nitrogen content estimation is the higher in the flag picking stage, the average value of the modeling and validation ${\mathrm{R}}^2$ is 0.85 and 0.81, respectively, which is 1.19% and 3.85% higher than that in the flowering stage, 14.86% and 17.39% higher than that in the filling stage, and 37.10% and 44.64% higher than that in the jointing stage. The absolute value of the overall correlation coefficient showed an increasing trend followed by a decreasing trend. This is due to the rapid growth of wheat from the jointing stage to the flag picking stage, the simultaneous advancement of the organs such as ear, leaf, and stem, the doubling of leaf area and the length and volume of the stem and ear, and the accumulation of a large amount of dry matter, resulting in a large increase in nitrogen content, and the correlation also increases. From the flag picking stage to the filling stage, the leaves and stems of wheat gradually turn yellow and aged, the vegetation coverage decreases, the photosynthetic capacity decreases, the nitrogen content decreases, and the correlation also decreases. This is consistent with the change in biochemical parameters of wheat at different growth stages pointed out by Yu Jiaoyang [36].

Comparing the three preprocessing methods of fractional order differentiation, continuous wavelet transform, and vegetation index used in this paper, it can be seen that the continuous wavelet transform is significantly better than the other two methods in spectrum preprocessing. The correlation between different vegetation indexes and the nitrogen content of wheat at different growth stages was analyzed. It was found that the correlation between REP and nitrogen content was the strongest, indicating that REP was of great significance for estimating the nitrogen content of vegetation indexes. The correlation between the wavelet energy coefficient and nitrogen content in different growth periods was also analyzed. The analysis results indicated that the correlation between the wavelet energy coefficient and nitrogen content at a low decomposition order is higher than that at a high decomposition order. The corresponding layers of 10 wavelet energy coefficients selected in each growth period are all in 1–6 layers, and the sensitivity of the wavelet energy coefficient in 7–10 layers to nitrogen content is not strong. Besides, the correlation between fractional differential spectra and nitrogen content in leaves at different growth stages was analyzed. It can be seen that the fractional differential spectra with a strong correlation with nitrogen content were in the integer order. The maximum correlation coefficient between the wavelet energy coefficient and wheat nitrogen content obtained by continuous wavelet transform pretreatment increased by 21.82%, 17.50%, 4.82%, and 10.39%, respectively, compared with the maximum correlation coefficient between the vegetation index and wheat nitrogen content in four growth periods, and it increased by 3.08%, 22.08%, 14.47%, and 10.39%, respectively, compared with the maximum correlation coefficient between the spectral band and wheat nitrogen content obtained by fractional differential pretreatment. The research by Miao Mengke [37] also shows that the wavelet characteristic spectrum has a stronger physiological response to wheat and has good potential in identifying physiological abnormalities. In the process of hyperspectral data collection, due to the impact of the soil background and environmental factors, the original spectrum has noise, which affects sensitive spectral information extraction. Wavelet transform can be exploited to refine the spectral information, deeply mine potentially sensitive spectral information, highlight effective information related to biomass [38], and make the screened wavelet energy coefficient more relevant to nitrogen content.

Through comprehensive analysis of the nitrogen content estimation effects of different methods at different growth stages, the prediction accuracy of the BP neural network model was the highest after the spectral data were pretreated with fractional differential at the wheat jointing stage, and the ${\mathrm{R}}^2$-values for the training and test sets were 0.73 and 0.66, respectively. In the flag picking period, continuous wavelet transform was used to preprocess the spectral data, and the BP neural network was adopted for modeling and prediction, which achieved the highest accuracy. The ${\mathrm{R}}^2$ values for the training and test sets are 0.95 and 0.93, respectively. The GPR algorithm is the best method for modeling and prediction in the flowering period after building the vegetation index preprocessing spectral data, and the ${\mathrm{R}}^2$ values for the training and test sets are 0.89 and 0.87, respectively. During the filling period, continuous wavelet transform was used to preprocess the spectral data, and the SVM method was used for modeling and prediction, which achieved the highest accuracy. The ${\mathrm{R}}^2$ values for the training and test sets were 0.82 and 0.81, respectively. Finally, the best scheme was selected to estimate the nitrogen content in wheat leaves at different growth stages.

A comprehensive comparison of the results from using different methods to estimate the nitrogen content in wheat at different growth stages revealed that during the jointing period the highest accuracy of 0.73 and 0.66 was obtained by the BP neural network obtained for the training and test sets, respectively, after preprocessing the spectra with fractional order differential. During the flag picking period, the highest model accuracy of 0.95 and 0.93 was obtained by modeling with the BP neural network for the training and test sets, respectively, after pre-processing the spectra with continuous wavelet transform. During the flowering period, the highest accuracy of 0.89 and 0.87 was obtained by the GPR model for the training and test sets, respectively, after constructing the vegetation index. During the filling period, the highest model accuracy of 0.82 and 0.81 was obtained by the SVM model for the training and test sets, respectively, after continuous wavelet transform preprocessing of the spectra. The selection of appropriate spectral preprocessing and modeling methods at different growth stages of wheat can lead to more desirable results for nitrogen content estimation.

5. Conclusions

Taking the precision agriculture demonstration area of the Jiaozuo Academy of Agricultural and Forestry Sciences in Henan Province as the research area, this paper exploits hyperspectral technology to build the nitrogen content estimation model for wheat. Based on the three spectral preprocessing methods of fractional differential, continuous wavelet transform, and vegetation index, this paper constructs five nitrogen content estimation models of SVM, ridge regression, stepwise regression, GPR, and BP neural network at different growth stages of wheat. Through the comparison of the above methods, it was found that different pretreatment and modeling approaches have a big impact on the accuracy of wheat nitrogen content estimation. The research results indicate that continuous wavelet transform can be used as a reliable spectral preprocessing method. In the flag picking stage of wheat, continuous wavelet transform combined with the BP neural network can be used to construct a nitrogen content estimation model with better accuracy, which provides a new idea for nitrogen content estimation for small sample crops.

Future research work will be conducted in the following aspects: the sample set used in this paper is small, which may lead to an over-fitting phenomenon. The sample size can be increased to verify the model of nitrogen content evaluation in large areas of crops and improve the stability of the model. Meanwhile, this paper only studies the estimation model for wheat nitrogen content. In the future, remote sensing monitoring of other biochemical parameters of wheat will be added to evaluate the nutritional status of wheat more comprehensively through hyperspectral data. Besides, the paper only uses the ground experiment data for crops, which has a large workload and a single data source. Subsequent research can select multi-source remote sensing data, such as unmanned aerial vehicle images or satellite images, to estimate the nitrogen content in crop leaves.