A method for estimating leaf area index of forest land with high spatial resolution

Abstract

The present invention discloses a method for estimating the leaf area index of forest land with high spatial resolution, comprising the following steps: using a Gaussian process regression model to model the relationship between forest land reflectance and leaf area index LAI as a multidimensional surface to achieve an initial estimation of LAI; enhancing the model's ability to handle data variability through variational heteroscedastic Gaussian process regression, while outputting an uncertainty assessment of each estimated value; and evaluating the prediction accuracy of the enhanced model and the accuracy of the uncertainty estimation. The present invention adopts the above-mentioned estimation method, and has excellent estimation capabilities for forest land LAI at a resolution of 10m. Relying on Sentinel‑2 satellite data, it quickly and accurately generates a 10m resolution LAI product, and provides the uncertainty of the estimation result while inverting the LAI, reflecting the accuracy and possible deviation of the model prediction result, which is of great significance for vegetation growth monitoring, surface change simulation and forest planning.

Description

Technical Field

[0001] The invention relates to the technical field of ecological remote sensing, and in particular to a method for estimating a leaf area index of a forest with high spatial resolution. Background Art

[0002] With the rapid development of satellite remote sensing technology, it is more convenient to obtain global land surface parameters such as leaf area index. Sentinel-2 satellite has been widely used in the fields of ecological environment and resources. Estimating regional and global scale vegetation LAI based on S2 remote sensing data is an effective means of dynamic monitoring of vegetation.

[0003] There are many methods for inverting LAI from remote sensing data, which can be divided into three categories: (1) parameter regression method; (2) method based on physical model; (3) non-parametric regression method. Among them, parameter regression method is favored in LAI inversion because of its simplicity and fast calculation speed. However, due to the influence of interference factors such as canopy structure, vegetation density, soil background, atmosphere, etc., the relationship between vegetation index and LAI varies with different regions or vegetation types, resulting in poor universality. In terms of physical methods, the radiation transfer model (RTM) inverts the biophysical parameters of vegetation by establishing the relationship between LAI, leaf optical properties and canopy reflectivity. However, due to the complexity of RTM and the limited amount of observations, more prior data are needed in the process of solving vegetation parameters. It is difficult to accurately describe the spatiotemporal changes of the surface in the process of calculating input parameters, resulting in ill-posed problems in the inversion process. Non-parametric algorithms optimize weights or coefficients in the existing data training stage and minimize the estimation error of the prediction results by continuously adjusting parameters.

[0004] The Gaussian process regression (GPR) model is an efficient parameter regression method. It assumes that the variance of the noise process is independent of the input data, which is inconsistent with the actual situation and limits the accuracy of its application. In order to overcome this limitation, scholars have proposed a variety of improvement schemes, among which the Variational Heteroscedastic Gaussian Processes (VHGPR) has shown excellent performance in the application of estimating land surface parameters.

[0005] Existing inversion methods based on Sentinel-2 data generally include its 20m spatial resolution band, but lack of attention to the specific use of Sentinel-2's 10m spatial resolution band data limits its potential in high-resolution application scenarios. The reflectivity of 10m spatial resolution provides more detailed information, which can capture finer features and details of small objects, thereby reducing uncertainties in subsequent applications. This is particularly critical for monitoring heterogeneous surfaces, such as urban vegetation and crops. Summary of the invention

[0006] The purpose of the present invention is to provide a forest land leaf area index estimation method with high spatial resolution, which has excellent estimation capability for forest land LAI at the 10m resolution level. Relying on Sentinel-2 satellite data, it can quickly and accurately generate 10m resolution LAI products, and provide the uncertainty of the estimation results while inverting LAI, reflecting the accuracy and possible deviation of the model prediction results, which is of great significance for vegetation growth monitoring, surface change simulation and forest planning.

[0007] To achieve the above object, the present invention provides a method for estimating leaf area index of forest land with high spatial resolution, comprising the following steps:

[0008] S1. Establishment of Gaussian process-based regression model GPR: The relationship between forest reflectance data and leaf area index is modeled as a multidimensional surface. By collecting forest remote sensing data and measured LAI data, a GPR model is established to make an initial estimate of the leaf area index LAI.

[0009] S2. Introducing Variational Heteroskedastic Gaussian Process Regression (VHGPR): Based on the GPR model, a variational inference framework is established to optimize model parameters, and heteroskedastic components are introduced to simulate data variability in different geographical environments;

[0010] S3. Model validation and uncertainty assessment: The performance of the VHGPR model was evaluated by statistical tests and cross-validation methods, and an independent test dataset was used to verify the prediction accuracy of the model.

[0011] Preferably, based on the Gaussian process regression GPR in step S1, a model of reflectivity and LAI is established as follows:

[0012]

[0013] Among them, r is the surface spectral reflectance, w is the parameter of the regression model, and k θ (r,r') is the covariance function with θ as parameter, ε n The variance is σ 2 Gaussian noise, θ and σ 2 Hyperparameters for tuning model performance.

[0014] Preferably, a covariance matrix is ​​introduced to accurately describe the relationship between different r. Under the condition of formula (1), the samples extracted from the position set obey the mean of zero and the covariance matrix K rr The kernel function form of the joint multivariate Gaussian distribution is the square exponential form, as shown below:

[0015]

[0016] Among them, k θ (r,r') is the kernel function of Gaussian distribution.

[0017] Preferably, when new test data is introduced outside the training data set, the prior equation (1) constructs a joint prior distribution between the training data set and the new test data set, as shown below:

[0018]

[0019] Among them, k r* Represents new test data r * and the covariance between the training data r, I n is the identity matrix.

[0020] Preferably, the posterior distribution of the LAI value of the new data point is predicted based on the above model, and its posterior mean μ is given * and uncertainty assessment indicators, namely variance As shown below:

[0021]

[0022]

[0023] Preferably, in step S2, VHGPR introduces a noise sequence with variable variance, as shown below:

[0024]

[0025] g(r)~GP(μ0,k g (r,r')) (7)

[0026] Among them, ε n The variance is e g(r) Gaussian noise.

[0027] Preferably, in step S3, the main indicator for evaluating the accuracy of each model is the determination coefficient R 2 and root mean square error RMSE;

[0028] Among them, R 2 It indicates the proportion of the change in the response variable y explained by the independent variable x in the linear regression model, and is used to evaluate the goodness of fit of the regression model coefficients. 2 is calculated by dividing the regression sum of squares by the total sum of squares;

[0029] RMSE is the root mean square error, which indicates the degree of dispersion between the model prediction value and the actual value. The formula is as follows:

[0030]

[0031] Among them, x is the predicted value, y is the measured value, and n is the total number of data.

[0032] Therefore, the present invention adopts the above-mentioned forest land high spatial resolution leaf area index estimation method to achieve efficient forest land LAI estimation at the 10m accuracy level. The core advantage is that, using Sentinel-2 remote sensing data, this technology can quickly generate accurate 10-meter resolution LAI images, while being able to carefully display the accuracy and deviation of the model prediction results, bringing significant advantages to the production of LAI products. This technology is of vital importance for accurately monitoring vegetation growth conditions, efficiently simulating surface changes, and formulating scientific forest management and planning strategies. By accurately capturing subtle vegetation characteristics and surface changes, the present invention has broad application potential in many fields such as vegetation growth detection, environmental monitoring, and resource management.

[0033] The technical solution of the present invention is further described in detail below through the accompanying drawings and embodiments. BRIEF DESCRIPTION OF THE DRAWINGS

[0034] Figure 1 It is the land classification map and location of Saihanba Forestry Center (SFC) in the study area of ​​this invention;

[0035] Figure 2 The distribution diagram of the correlation between field LAI and LAI estimated by VHGPR, SAVI and SA-BP; (a) is the field observation verification result of the VHGPR model; (b) is the field observation verification result of the SAVI model; (c) is the field observation verification result of the SA-BP model;

[0036] Figure 3 This is the LAI map generated by VHGPR in Saihanba in 2018. Each LAI map is marked with the corresponding date of year (DOY).

[0037] Figure 4 is the mean value of the uncertainty associated with the LAI map expressed as standard deviation (SD) around the LAI estimate in 2018;

[0038] Figure 5 is the relative uncertainty of the LAI map in 2018 expressed as the coefficient of variation;

[0039] Figure 6 It is a local verification comparison diagram in an embodiment of the present invention. DETAILED DESCRIPTION

[0040] The technical solution of the present invention is further described below through the accompanying drawings and embodiments.

[0041] The present invention provides a method for estimating a forest land leaf area index with high spatial resolution, comprising the following steps:

[0042] S1. Establishment of Gaussian process-based regression model GPR: The relationship between forest reflectance data and leaf area index is modeled as a multidimensional surface. By collecting forest remote sensing data and measured LAI data, a GPR model is established to make an initial estimate of the leaf area index LAI.

[0043] S2. Introducing Variational Heteroskedastic Gaussian Process Regression (VHGPR): Based on the GPR model, a variational inference framework is established to optimize model parameters, and heteroskedastic components are introduced to simulate data variability in different geographical environments;

[0044] S3. Model validation and uncertainty assessment: The performance of the VHGPR model was evaluated by statistical tests and cross-validation methods, and an independent test dataset was used to verify the prediction accuracy of the model.

[0045] Example 1

[0046] The present invention provides a method for estimating the leaf area index of a forest with high spatial resolution, such as Figure 1 As shown, the present invention selects Saihanba Forest Farm (SFC) in Chengde City, Hebei Province as the test area. SFC is located in a typical artificial forest area with rich vegetation species. It is also one of the source areas of Luanhe River and Liaohe River. The vegetation growth season in this area is mainly concentrated in May-September.

[0047] From June 7 to September 12, 2018, a comprehensive field observation experiment was conducted in the Luanhe River Basin to obtain the LAI ground measurement data of SFC. The measurement objects of LAI include woodlands and grasslands. The main measuring instruments are LAI2000, LAIPhoto, TRAC, etc. Since the field measurement is carried out in an area of ​​25m×25m, it is believed that the data of each point matches the area of ​​10m×10m. The present invention uses 58 high-quality field LAI points and high-quality S2 data of the same period. S2 further improves the temporal, spatial and spectral resolution of remote sensing data, and has great potential in vegetation monitoring. The present invention uses 4 10-meter bands (blue, green, red, and near-infrared) for the construction of the LAI inversion model.

[0048] Table 1Sentinel-2MSI band settings

[0049] Band Description Wavelength(μm) Resolution(m) 1 Coastalaerosol 0.433-0.453 60 2 Blue* 0.458-0.523 10 3 Green* 0.543-0.578 10 4 Red* 0.650-0.680 10 5 VegetationRedEdge 0.698-0.713 20 6 VegetationRedEdge 0.733-0.748 20 7 VegetationRedEdge 0.773-0.793 20 8 NIR* 0.785-0.900 10 8A NarrowNIR 0.855-0.875 20 9 Watervapour 0.935-0.955 60 10 SWIR–Cirrus 1.365-1.385 60 11 SWIR-1 1.565-1.655 20 12 SWIR-2 2.100-2.280 20

[0050] Gaussian process regression (GPR) is a non-parametric model that uses Gaussian process as a priori to perform regression analysis on data. Compared with other machine learning methods, GPR has great advantages in modeling, but in the general Gaussian regression process, the noise variance of all signals is consistent, which is inconsistent with the actual situation. As an improved solution of GPR, VHGPR can retain the advantages of GPR and add variable noise in the regression process to achieve better regression effect.

[0051] S1. Establish a Gaussian process-based regression model GPR.

[0052] The complex relationship between forest reflectance data and leaf area index is modeled as a multidimensional surface. An appropriate covariance function is selected to define the prior distribution of the function space, which can capture the nonlinear relationship between LAI and reflectance. By collecting forest remote sensing data and measured LAI data, this relationship is modeled using Gaussian process regression to achieve an initial estimate of LAI.

[0053] The Gaussian process is a random process consisting of an infinite number of random variables that obey the Gaussian distribution and are defined on a continuous area. GPR regards the function space corresponding to the regression model as a Gaussian process and establishes the relationship between reflectivity and LAI as shown below:

[0054]

[0055] Among them, r is the surface spectral reflectance, w is the parameter of the regression model, and k θ (r,r') is the covariance function with θ as parameter, ε n The variance is σ 2 Gaussian noise, θ and σ 2 Hyperparameters for tuning model performance.

[0056] A Gaussian process is a random process that satisfies the condition that any finite subset of a set of random variables follows a multivariate Gaussian distribution. In Gaussian process regression, there is an assumption that if two (i.e. close to each other) are similar, then the corresponding correlations are also similar. Therefore, the covariance matrix is ​​introduced to describe the relationship between different r. Under the conditions of formula (1), the samples extracted from the position set have a mean of zero and a covariance matrix of K. rr The kernel function form of the joint multivariate Gaussian distribution is the square exponential form, as shown below:

[0057]

[0058] Among them, k θ(r, r') is the kernel function of the Gaussian distribution. This method can more carefully capture the subtle differences in geographical locations and characteristics of different training sets, thereby improving the accuracy and reliability of LAI estimation.

[0059] If new test data is introduced outside the training data set, the prior equation (1) will construct a joint prior distribution between the training data set and the new test data set, as shown below:

[0060]

[0061] Among them, k r* Represents new test data r * and the covariance between the training data r, I n is the identity matrix. This step supports the model's ability to generalize to unknown data.

[0062] Based on the above model, the posterior distribution of the LAI value of the new data point is predicted, and its posterior mean μ is given. * and uncertainty assessment indicators, namely variance As shown below:

[0063]

[0064]

[0065] This provides a quantitative uncertainty assessment for LAI estimation, greatly enhancing the practicality and interpretability of the model.

[0066] S2. Introduce variational heteroskedastic Gaussian process regression VHGPR.

[0067] Based on GPR, a VHGPR model that can handle the inherent variability of the data is further developed. This step includes designing a variational inference framework to optimize the model parameters, while introducing heteroscedastic components to simulate data variability under different geographical and environmental conditions. The VHGPR model can output the uncertainty of each LAI estimate, providing important information for subsequent analysis.

[0068] VHGPR introduces a noise sequence with variable variance to more accurately simulate the uncertainty and variability in nature, as shown below:

[0069]

[0070] g(r)~GP(μ0,k g (r,r')) (7)

[0071] Among them, ε n The variance is e g(r)The Gaussian noise further improves the adaptability and accuracy of the model to the complex changes of forest LAI.

[0072] This form of noise distribution makes VHGPR difficult to analyze and a closed-form solution cannot be obtained directly. Therefore, a complex method called marginal variation (MV) approximation is introduced in VHGPR to improve efficiency and prevent overfitting. At the same time, MV ensures the computational efficiency of the model and provides an analytical expression for the Kulbeck-Leibler divergence between the proposed distribution and the true posterior distribution. By minimizing the proposed distribution and hyperparameters, a true posterior estimate is generated while performing model selection simultaneously.

[0073] S3. Model validation and uncertainty assessment.

[0074] The performance of the VHGPR model was evaluated through a series of statistical tests and cross-validation methods. An independent test dataset was used to verify the prediction accuracy of the model, and calculations such as the root mean square error (RMSE) and the coefficient of determination (R 2 ) and other statistical indicators. In addition, the characteristics and effectiveness of the LAI estimation results and uncertainty estimates were analyzed. The ultimate goal is to ensure that the method can robustly estimate the leaf area index in different woodland environments while providing an effective uncertainty evaluation of the estimation results.

[0075] Coefficient of determination R 2 It indicates the proportion of the change in the response variable y explained by the independent variable x in the linear regression model, and is used to evaluate the goodness of fit of the regression model coefficients. 2 is calculated by dividing the regression sum of squares by the total sum of squares.

[0076] RMSE is the root mean square error, which indicates the degree of dispersion between the model prediction value and the actual value. Its formula is:

[0077]

[0078] Among them, x is the predicted value, y is the measured value, and n is the total number of data. These evaluation indicators provide a quantitative measure of the performance and application effect of the model, ensuring its high accuracy and reliability in practical applications.

[0079] Example 2

[0080] 1. Verify with field data.

[0081] like Figure 2 As shown in the figure, (a) is the field observation verification result of the VHGPR model, (b) is the field observation verification result of the SAVI model, and (c) is the field observation verification result of the SA-BP model. In the SFC area, the 10m LAI estimated by VHGPR is in good agreement with the field observation value, R 2The value is 0.8696 and the RMSE is 0.5078, which indicates that the LAI estimated by VHGPR has high accuracy.

[0082] SAVI performed the worst among the three LAI estimation models, with an acceptable R 2 The SA-BP model constructs the relationship between LAI and reflectivity, rather than VI, which avoids the saturation problem. 2 The average value is 0.8095 and the RMSE is 0.568, which is still worse than VHGPR.

[0083] 2. SFC LAI distribution map generated by VHGPR.

[0084] like Figure 3 As shown in Figure 2, the LAI distribution map of SFC in 2018 was generated based on the VHGPR model and S2 reflectivity data. The LAI estimation results of SFC well reflect the growth trend of vegetation in the region, and the LAI distribution map well reflects the spatial distribution characteristics and temporal change trend of vegetation. The forest vegetation is most lush in July, which is consistent with the change of forest LAI in the present invention area.

[0085] Taking the most vigorous LAI map of all the maps on the 208th day as an example, the Larix principis-rupprechtii forest grew the best, with the highest LAI value in the map. There are also some obvious low-value areas in the study area. The LAI values ​​of Betula platyphylla and Pinus tabulaeformis are relatively low compared with the Larix principis-rupprechtii forest in the main area of ​​the study area. In the western region, it is mainly composed of Pinus sylvestris var. mongolica, Larix principis-rupprechtii forest, young forest and a small amount of spruce. The LAI value of this area is lower than that of the central region where Larix principis-rupprechtii forest grows in large numbers. Compared with other tree species, the growth period of Betula platyphylla starts the earliest, with obvious green indications on the 185th day. However, there is no significant difference between Betula platyphylla and other tree species at the end of growth.

[0086] 3. SFC related uncertainty and relative uncertainty.

[0087] Since VHGPR is developed under a Bayesian framework, it provides not only the leaf area index estimate but also the estimate uncertainty. The relative uncertainty CV can be calculated from this uncertainty diagram, as Figure 4 good Figure 5 As shown:

[0088]

[0089] These two errors provide additional information about the estimation results at the pixel scale.

[0090] Table 2 Statistics of average errors at different times

[0091] DOY MeanSD MeanCV(%) 180 0.1212 10.2851 185 0.1360 10.0728 195 0.1176 7.5763 208 0.1104 5.0239 215 0.1059 5.2505 230 0.1170 6.7886 243 0.1167 6.6019 258 0.1269 8.5541

[0092] In the associated uncertainty of the LAI map, such as Figure 4 As shown in Figure 1, lower SD values ​​represent more reliable LAI estimates, while higher light blue and green values ​​reflect higher uncertainty. Combining the error distribution graph and statistical table, such as Figure 4 , Figure 5 As shown in Table 2, the series of related uncertainty diagrams in 2018 have the following main features:

[0093] (1) The overall uncertainty is relatively high in the early and late growing seasons.

[0094] (2) Uncertainty is higher near ridges and water bodies.

[0095] The reasons for these two characteristics are:

[0096] (1) The LAI ground survey data dates are mostly concentrated in the middle of the growing season when vegetation growth is vigorous. Unlike the early and late growing season, the sparse vegetation canopy is easily affected by shrub or grassland vegetation, which increases uncertainty.

[0097] (2) The water bodies near SFC are mostly wetland landforms, which will greatly affect the accuracy of LAI inversion and increase its uncertainty. However, the forest vegetation near the ridge is sparse, and the open-air vegetation is mostly grassland and shrubs, which increases the uncertainty of the inversion results.

[0098] Due to the inherent calculation method, CV values ​​are significantly higher in the early and late growing seasons and in areas with smaller LAI values. In addition, there are also some areas with higher CV at all growth nodes.

[0099] In order to further explore the causes of these anomalies, some typical areas with large coefficients of variation were randomly selected and compared with the corresponding land cover map (LC), false color composite map (FCS), LAI distribution map, SD distribution map, and NDVI distribution map, such as Figure 6 shown.

[0100] LC, FCS, LAI SD, CV and NDVI represent pseudo-color land cover map, composite map, LAI distribution map, SD distribution map and NDVI distribution map, respectively. LC legends 1 to 6 correspond to Larix principis-rupprechtii plantations, Betula platyphylla plantations, Pinus tabulaeformis plantations, Pinus sylvestris var. mongolica plantations, Picea asperata plantations and young forests, respectively; AF represents 6 different typical areas.

[0101] Combining the NDVI distribution map and the LAI distribution map, it can be seen that the vegetation in these areas is relatively sparse. At the same time, these areas are light blue in FCS, so it is considered that these areas are not vegetation. The reason for this inaccuracy is that the result map is masked according to the 30m land cover map. When the resolution is low, the identification of the ground objects is not accurate enough, resulting in the occurrence of misclassification.

[0102] In area C, the relative error in some areas is also significantly higher. The vegetation type in this area is sparse, with low LAI and NDVI values. The reason for this error is that there is only one field data of a spruce plantation when modeling, so the estimation accuracy of spruce is not ideal.

[0103] In area D, the vegetation in this area is relatively sparse, and the corresponding CV value is particularly high, which is due to the inherent computational complexity of CV and the poor estimation ability of the model for sparse vegetation.

[0104] Finally, in areas E and F, especially in area E, the CV values ​​of the VHGPR estimates are significantly higher than those of other vegetation types in the young forest vegetation area. When collecting the measured points, young forests were not given much attention, there were no similar training points in the VHGPR model, and the LAI value of this land cover type was low, so the estimation accuracy of young forests was not particularly reliable, and the CV value was very high.

[0105] This paper focuses on the 10m LAI inversion based on S2 data. High spatial resolution LAI data is of great significance for vegetation growth detection, surface change simulation and forest planning. S2 data has the advantages of high resolution and short recurrence, but most methods of inverting LAI using S2 data are not based solely on the 10m resolution band, and only obtain LAI results with a 20m resolution.

[0106] The present invention considers the applicability of the VHGPR method to invert LAI from S2 data in four bands: blue, green, red, and near infrared, and compares the results with the results of the vegetation index model (SAVI) empirical model and the simulated annealing algorithm (SA-BP) model. Among them, the accuracy of VHGPR is better than that of SAVI and SA-BP, R 2 The value of the estimated LAI is 0.8696, and the RMSE is 0.5078. In order to compare the estimation characteristics of SA-BP and VHGPR with similar verification accuracy, the correlation analysis of the LAI distribution maps estimated by the two methods was performed. The study found that there is an obvious upper limit to the range of SA-BP estimation values, which is affected by the maximum value of the training sample. In comparison, VHGPR is less affected by this factor.

[0107] The advantage of the VHGPR model is that it can determine the importance of input data and eliminate unimportant bands, which has unique advantages in large-scale mapping. In addition, VHGPR can provide uncertainty estimates and relative uncertainty estimates of prediction results, which is very useful for error source analysis and mapping applications. Although the accuracy difference between the two is not particularly obvious, VHGPR is superior in LAI model inversion and product generation due to its error estimation ability.

[0108] Based on the VHGPR method, this paper generates the LAI distribution map, SD distribution map and CV distribution map of the cloud-free S2 image in 2018, and analyzes the error sources and characteristics of VHGPR in combination with the surface cover map and false color composite map. The results show that the VHGPR method can quickly and accurately generate 10m LAI from S2 data, and can reflect the accuracy and deviation of the model estimation results. It is a feasible 10m LAI inversion scheme.

[0109] Therefore, the present invention adopts the above-mentioned forest land high spatial resolution leaf area index estimation method to achieve efficient forest land LAI estimation at 10m accuracy level. The core advantage is that, using Sentinel-2 remote sensing data, this technology can quickly generate accurate 10-meter resolution LAI images, while being able to carefully display the accuracy and deviation of the model prediction results, bringing significant advantages to the production of LAI products. This technology is of vital importance for accurately monitoring vegetation growth conditions, efficiently simulating surface changes, and formulating scientific forest management and planning strategies. By accurately capturing subtle vegetation characteristics and surface changes, the present invention has broad application potential in multiple fields such as vegetation growth detection, environmental monitoring and resource management, especially in areas with diverse forest components.

[0110] Finally, it should be noted that the above embodiments are only used to illustrate the technical solution of the present invention rather than to limit it. Although the present invention has been described in detail with reference to the preferred embodiments, those skilled in the art should understand that they can still modify or replace the technical solution of the present invention with equivalents, and these modifications or equivalent replacements cannot cause the modified technical solution to deviate from the spirit and scope of the technical solution of the present invention.

Claims

1. A method for estimating leaf area index of forest land with high spatial resolution, characterized in that: The following steps are involved: S1. Establishment of Gaussian process-based regression model GPR: The relationship between forest reflectance data and leaf area index LAI is modeled as a multidimensional surface. By collecting forest remote sensing data and measured LAI data, a GPR model is established to make an initial estimate of the leaf area index LAI. Based on the Gaussian process regression GPR in step S1, the model y of reflectivity and LAI is established as follows: Among them, r is the surface spectral reflectance, w is the parameter of the regression model, and k θ (r,r') is the covariance function with θ as parameter, ε n The variance is σ 2 Gaussian noise, θ and σ 2 To adjust the hyperparameters of the model performance; The covariance matrix is ​​introduced to describe the relationship between different r. Under the condition of formula (1), the samples extracted from the position set have a mean of zero and a covariance matrix of K. rr The joint multivariate Gaussian distribution of , with the covariance function of θ as parameter, is in the form of square exponential, as shown below: Among them, k θ (r,r') is the covariance function with θ as parameter; When new test data is introduced outside the training data set, the prior equation (1) is used to construct a joint prior distribution of the training data set and the new test data set, as shown below: in, Represents new test data r * and the covariance between the training data r, I n is the identity matrix; Based on the above model, the posterior distribution of the LAI value of the new data point is predicted, and its posterior mean μ is given. * and uncertainty assessment indicators, namely variance As shown below: S2. Introducing Variational Heteroskedastic Gaussian Process Regression (VHGPR): Based on the GPR model, a variational inference framework is established to optimize model parameters, and heteroskedastic components are introduced to simulate data variability in different geographical environments; VHGPR introduces a noise sequence with variable variance as follows: g(r)~GP(μ0,k g (r,r')) (7); Among them, ε n The variance is e g(r) Gaussian noise; S3. Model validation and uncertainty assessment: The performance of the VHGPR model was evaluated by statistical tests and cross-validation methods, and the prediction accuracy of the model was verified using an independent test dataset. The index for evaluating the accuracy of each model is the determination coefficient R 2 and root mean square error RMSE; Among them, R 2 It indicates the proportion of the change in the response variable y explained by the independent variable x in the linear regression model, and is used to evaluate the goodness of fit of the regression model coefficients. 2 is calculated by dividing the regression sum of squares by the total sum of squares; RMSE is the root mean square error, which indicates the degree of dispersion between the model prediction value and the actual value. The formula is as follows: Among them, x is the predicted value, y is the measured value, and n is the total number of data.

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