A tree compatibility biomass model construction method fusing lagrange constraint and fourier expansion

Abstract

The application discloses a tree compatibility biomass model construction method fusing a Lagrange constraint and a Fourier expansion, and belongs to the technical field of forest resource monitoring.The existing model forms are disordered, are only compatible at a numerical level, are complex to solve, and are difficult to analyze growth rules.The application standardizes independent variables, uniformly adopts a third-order semi-interval Fourier base function to construct a component model, splits the component through a residual error iteration two-division, and realizes coefficient level compatibility by combining a Lagrange constraint least square, so that model verification and growth amount derivation are simultaneously completed.The model structure is unified, the solution is stable, and the application can be widely used in forest investigation, carbon sink measurement, forest growth simulation and other scenes.

Description

Technical Field

[0001] This invention relates to the field of forest resource monitoring technology, specifically to a method for constructing a tree compatibility biomass model that integrates Lagrange constraints and Fourier expansion. Background Technology

[0002] Currently, mainstream biomass compatibility modeling methods in the industry mostly employ power functions or nested proportional equations to build model systems. While these approaches can achieve summation and unification at the biomass estimation level, they suffer from several inherent drawbacks. The first drawback is the inconsistency in the mathematical forms of models corresponding to different tree components at different levels, making it difficult to form a complete and unified analytical framework for the interpretation of various parameters. The second drawback is that compatibility can only be achieved at the final predicted numerical level, failing to establish equivalent constraint relationships at the model expansion coefficient level. The third drawback is that traditional monotonic functions cannot accurately characterize the fluctuation and distribution characteristics of organ biomass at different growth stages of trees. The fourth drawback is the complex structure of model expressions, making it inconvenient to perform growth rate differentiation and incremental distribution analysis. The fifth drawback is the high computational dimensionality when all components are included in the constraint solution at once, leading to unstable parameter estimation and difficulty in convergence. Therefore, the industry urgently needs a new modeling method with a unified structure, natural compatibility at the coefficient level, lower solution difficulty, and ease of growth analysis. Summary of the Invention

[0003] To address the technical problem, this invention provides a method for constructing a tree compatibility biomass model that integrates Lagrange constraints and Fourier expansion, comprising:

[0004] S1. Collect the diameter at breast height (DBH), tree height, tree age, and the product of the square of DBH and tree height of the test sample trees as candidate modeling independent variables. Collect at least two of the following biomass of the test sample trees that have a hierarchical decomposition subordinate relationship as the measured values ​​of component biomass: whole tree biomass, aboveground biomass, underground biomass, trunk biomass, crown biomass, branch biomass, leaf biomass, coarse root biomass, and fine root biomass. Construct a set of components to be modeled. Mark the components without measured biomass data as null values ​​and exclude them from their independent fitting range. Output the sample tree dataset containing the values ​​of the independent variables of the test sample trees and the measured values ​​of component biomass.

[0005] S2. Based on the sample tree dataset, extract the modeling independent variable values ​​for each sample tree, calculate the minimum and maximum values ​​of the modeling independent variable among all sample trees, convert the modeling independent variable values ​​of each sample tree into standardized independent variable values ​​according to the standardization formula, so that the standardized values ​​fall within the closed interval [0,1], and output the standardized independent variable values ​​corresponding to each sample tree.

[0006] S3. Based on standardized independent variable values, a unified third-order half-interval Fourier basis function is constructed, and all component biomass models share the same set of basis functions;

[0007] S4. In the component hierarchy, the upper-level overall component is the parent level, and the lower-level subdivided components obtained by splitting the parent level are the child components. Based on the third-order half-interval Fourier basis function and sample tree dataset, the highest-level parent biomass model is fitted first. When whole tree biomass data exists, the whole tree total biomass model is fitted. When whole tree biomass data is missing, the measured highest-level component model is fitted. The Fourier expansion coefficient vector of the parent model is estimated by ordinary least squares method, and the parent model coefficients are output.

[0008] S5. Based on the parent model coefficients and the set of components to be modeled, a hierarchical binary splitting strategy is adopted to construct a compatible splitting relationship. Taking the current parent biomass as the initial constraint benchmark, the component with the highest biomass ratio among the components to be modeled corresponding to the current parent is selected as the target component. The current parent is split into the target component biomass and the non-target component biomass. The non-target component biomass is used as the parent in the next round. The above splitting process is repeated until all components in the set of components to be modeled are covered. The multi-level binary splitting relationship and coefficient compatibility constraint rules are output.

[0009] S6. Based on the multi-level binary splitting relationship and coefficient compatibility constraint rules, the third-order half-interval Fourier basis function and the measured biomass values ​​of each sub-component, with the objective of minimizing the sum of squared residuals between the predicted and measured values ​​of all sub-components, and with the constraint that the coefficients of the parent model are equal to the sum of the coefficients of the corresponding child model, the Fourier expansion coefficient vectors of each sub-component are solved by the Lagrange constrained least squares method, and the model coefficients of each sub-component are output.

[0010] S7. Based on the model coefficients of each sub-component, calculate the fitting effect evaluation index of the biomass model of each component, and output the model evaluation results;

[0011] S8. Based on the model coefficients of each sub-component and the third-order half-interval Fourier basis functions, the biomass model of each component is differentiated to obtain the rate of change of biomass with the modeling independent variable. The negative rate of change is set to 0 to obtain the non-negative growth function, and the growth analysis results of each component are output.

[0012] S9. Based on the model coefficients of each sub-component, the third-order half-interval Fourier basis functions, and the range of standardized independent variables, the compatibility of each level of the model is numerically verified within the observation range, and the compatibility test results and the final biomass model are output.

[0013] S10. For scenarios where only a parent model exists and some child components lack measured data, the model coefficients and biomass prediction formulas for the unobserved components are derived using the parent model coefficients and the coefficients of the existing measured child components. The measured fitted model and the derived model are distinguished and output synchronously.

[0014] Furthermore, the specific formula in S2 for converting the modeling independent variable values ​​of each sample tree into standardized independent variable values ​​according to the standardization formula is as follows:

[0015]

[0016] in, To represent the standardized independent variable value of the i-th sample tree, Let the values ​​of the independent variables for modeling the i-th sample tree be taken. To represent the minimum value of this modeling independent variable among all sample trees, This represents the maximum value of the modeling independent variable among all sample trees.

[0017] Furthermore, the specific form of the third-and-a-half-interval Fourier basis function in S3 is as follows:

[0018]

[0019] in, This represents a vector of third- and half-interval Fourier basis functions. The standardized independent variable represents the values ​​of π, where π represents the mathematical constant pi. The basis function vector contains 7 terms, which are constant basis terms in order. First-order cosine term First-order sine term Second-order cosine term Second-order sine term Third-order cosine term Third-order sine term T is the transpose symbol;

[0020] The biomass model expression for any component is:

[0021]

[0022] in, This represents the predicted biomass value of component j under the standardized independent variable value z. The Fourier expansion coefficient vector of component j is specifically:

[0023]

[0024] The explicit formula for calculating the Fourier expansion of any component is:

[0025]

[0026] in This represents the coefficient of the constant term of component j. Denotes the coefficients of the first-order cosine term of component j. This represents the coefficient of the first-order sine term of component j. Denotes the coefficients of the second-order cosine term of component j. Denotes the coefficients of the second-order sine term of component j. Denotes the coefficients of the third cosine term of component j. Let represent the coefficients of the third-order sine term of component j.

[0027] Furthermore, the coefficients of the parent model in S4 are solved using the ordinary least squares method, as follows:

[0028]

[0029] The expression for the biomass model of the parent component is:

[0030]

[0031] in, This represents the Fourier expansion coefficient vector of the parent component. This represents the measured sample size of the parent component. This represents the measured biomass value of the parent component in the i-th sample. Let represent the transpose of the Fourier basis function vector corresponding to the i-th sample. arg represents the parent coefficient vector to be estimated. This represents the value of θ that minimizes the subsequent summation terms. This represents the predicted biomass value of the current parent under the standardized independent variable value z.

[0032] Furthermore, the specific implementation of the step-by-step binary search strategy in S5 is as follows:

[0033] The initial parent is the total biomass of the entire tree. During each split, the component with the highest biomass percentage in the set of components to be modeled corresponding to the current parent is selected as the target component. The current parent is then split into the biomass of the target component and the biomass of non-target components. The functional relationship for the split is as follows:

[0034]

[0035] The compatibility constraint relationship at the coefficient level is as follows:

[0036]

[0037] in, This represents the predicted biomass value of the target component under the standardized independent variable value z. This represents the predicted biomass value of a non-target component under the standardized independent variable value z. This represents the Fourier expansion coefficient vector of the current parent element. This represents the Fourier expansion coefficient vector of the target component. Represents the Fourier expansion coefficient vector of the non-target component;

[0038] After the split is completed, As the parent biomass for the next round, the component with the highest biomass percentage is selected from the set of remaining components to be modeled for the next round of splitting. The above process is repeated until all components to be modeled are included in the splitting system.

[0039] Furthermore, the specific implementation of the Lagrange-constrained least squares method in S6 is as follows:

[0040] Based on the binary splitting relationship and coefficient compatibility constraint rules, an objective function is constructed for the set of sub-components corresponding to each round of splitting:

[0041]

[0042] The constraints are:

[0043]

[0044] Construct the Lagrangian function:

[0045]

[0046] Where m represents the number of child components corresponding to the current parent, L is the Lagrangian objective function, j is the child component index, and i is the sample index. This represents the measured biomass value of the j-th sub-component of the i-th sample tree. This is the transpose of the Lagrange multiplier vector;

[0047] By taking the first-order partial derivatives of all sub-component coefficient vectors with respect to the Lagrange multipliers and setting the derivatives to 0, the final Fourier expansion coefficient vectors of each sub-component can be obtained.

[0048] Furthermore, the evaluation indicators for the fit of the biomass models of each component in S7 include the residual sum of squares, total sum of squares, coefficient of determination, and Akaike information criterion.

[0049] Sum of squared residuals:

[0050]

[0051] Total sum of squares:

[0052]

[0053] Coefficient of determination:

[0054]

[0055] Akaike Information Guidelines:

[0056]

[0057] in, Let j be the sum of squared residuals of the j-th component. Let the sum of squares of the j-th component be . Let be the determination coefficient of the j-th component model. Let i be the sample wood number, and let i be the information criterion for the j-th component. The measured biomass of the j-th component of the i-th sample tree is... This represents the predicted biomass value for the j-th component of the i-th sample tree. Let be the mean of the measured biomass of all samples of component j. Let j be the total number of measured samples of the j-th component. For natural logarithm operations, Let be the number of valid parameters to be estimated for the j-th component model.

[0058] Furthermore, in S8, the rate of change of biomass with respect to the modeling independent variable is obtained by differentiating the biomass model for each component. The calculation formula used to obtain the non-negative growth function by setting the negative rate of change to 0 is as follows:

[0059] First derivative of biomass under standardized independent variables:

[0060]

[0061] Rate of change of biomass under the original independent variable:

[0062]

[0063] Non-negative growth function:

[0064]

[0065] in, Let j be the first derivative of biomass of component j with respect to the standardized independent variable z. , , , , , Let J be the Fourier coefficients of component j. Pi Let x be the rate of change of biomass of component j with respect to the original modeling independent variable x. , These represent the maximum and minimum values ​​of the independent variables being modeled, respectively. Let j be the non-negative growth function of the original independent variable x corresponding to component j. This is for calculating the maximum value.

[0066] Furthermore, the difference calculation formula used in S9 to numerically verify the compatibility of models at each level within the observation range is as follows:

[0067]

[0068]

[0069]

[0070]

[0071] in, , , , This represents the biomass compatibility difference between different biomass levels. This is the predicted value for the total tree biomass. This is a predicted value for aboveground biomass. This is a predicted value for the underground biomass. This is a predicted value for trunk biomass. This is a predicted value for canopy biomass. This is a predicted value for tree branch biomass. Predicted values ​​for leaf biomass. This is a predicted value for coarse root biomass. This is a predicted value for fine root biomass.

[0072] Furthermore, in S10, the formula for calculating the model coefficients and biomass prediction of unobserved components using the parent model coefficients and existing measured sub-component coefficients is as follows:

[0073] Missing Component Biomass Prediction Model:

[0074]

[0075] Missing component Fourier coefficient vector:

[0076]

[0077] in, This is a biomass estimation model without actual measured sample components. For sub-component biomass models with available measured data, Calculate the coefficient vector for the missing components. The coefficient vector of the parent component. This is the coefficient vector of the sub-components that have actual measured samples.

[0078] The beneficial effects of this invention are:

[0079] This invention employs a unified third-order half-interval Fourier function to construct a biomass model for all tree components at all levels, eliminating the drawbacks of the chaotic model forms in existing technologies. Relying on Lagrange equality constraints, it ensures hierarchical conservation of all Fourier coefficients, guaranteeing from the model's underlying structure that the total biomass and the biomass of subdivided components are strictly summed and equal under any independent variable. This invention designs a residual iterative binary decomposition method, breaking down the joint solution of high-dimensional multi-component systems into multiple sets of binary low-dimensional programming calculations. This significantly reduces the solution difficulty while improving the stability of parameter estimation. Throughout the modeling process, there is no need for manually pre-setting component allocation ratios; all expanded parameters are automatically fitted based on measured data, reducing bias caused by human intervention. Attached Figure Description

[0080] Figure 1 This is a graph showing the fitting results of the total biomass of this invention;

[0081] Figure 2 This is a compatibility fitting curve diagram of all components of the present invention;

[0082] Figure 3 This is a scatter plot verifying the observed and predicted values ​​of this invention.

[0083] Figure 4 A schematic diagram of the biomass fixation hierarchy and compatibility decomposition relationship of various tree components;

[0084] Figure 5 This is a schematic diagram of a step-by-step binary search strategy. Detailed Implementation

[0085] The technical solution of the present invention will be further described below with reference to embodiments, but it is not limited thereto. Any modifications or equivalent substitutions to the technical solution of the present invention without departing from the spirit and scope of the technical solution of the present invention should be covered within the protection scope of the present invention. In the following embodiments, process equipment or devices not specifically specified are all conventional equipment or devices in the art. Unless specifically specified, the technical means used in the embodiments of the present invention are all conventional means well known to those skilled in the art.

[0086] Example 1: A method for constructing a tree compatibility biomass model that integrates Lagrange constraints and Fourier expansion, comprising:

[0087] S1. Collect the diameter at breast height (DBH), tree height, tree age, and the product of the square of DBH and tree height of the test sample trees as candidate modeling independent variables. Collect at least two of the following biomass of the test sample trees that have a hierarchical decomposition subordinate relationship as the measured values ​​of component biomass: whole tree biomass, aboveground biomass, underground biomass, trunk biomass, crown biomass, branch biomass, leaf biomass, coarse root biomass, and fine root biomass. Construct a set of components to be modeled. Mark the components without measured biomass data as null values ​​and exclude them from their independent fitting range. Output the sample tree dataset containing the values ​​of the independent variables of the test sample trees and the measured values ​​of component biomass.

[0088] S2. Based on the sample tree dataset, extract the modeling independent variable values ​​for each sample tree, calculate the minimum and maximum values ​​of the modeling independent variable among all sample trees, convert the modeling independent variable values ​​of each sample tree into standardized independent variable values ​​according to the standardization formula, so that the standardized values ​​fall within the closed interval [0,1], and output the standardized independent variable values ​​corresponding to each sample tree.

[0089] S3. Based on standardized independent variable values, a unified third-order half-interval Fourier basis function is constructed, and all component biomass models share the same set of basis functions;

[0090] S4. In the component hierarchy, the upper-level overall component is the parent level, and the lower-level subdivided components obtained by splitting the parent level are the child components. Based on the third-order half-interval Fourier basis function and sample tree dataset, the highest-level parent biomass model is fitted first. When whole tree biomass data exists, the whole tree total biomass model is fitted. When whole tree biomass data is missing, the measured highest-level component model is fitted. The Fourier expansion coefficient vector of the parent model is estimated by ordinary least squares method, and the parent model coefficients are output.

[0091] S5. Based on the parent model coefficients and the set of components to be modeled, a hierarchical binary splitting strategy is adopted to construct a compatible splitting relationship. Taking the current parent biomass as the initial constraint benchmark, the component with the highest biomass ratio among the components to be modeled corresponding to the current parent is selected as the target component. The current parent is split into the target component biomass and the non-target component biomass. The non-target component biomass is used as the parent in the next round. The above splitting process is repeated until all components in the set of components to be modeled are covered. The multi-level binary splitting relationship and coefficient compatibility constraint rules are output.

[0092] S6. Based on the multi-level binary splitting relationship and coefficient compatibility constraint rules, the third-order half-interval Fourier basis function and the measured biomass values ​​of each sub-component, with the objective of minimizing the sum of squared residuals between the predicted and measured values ​​of all sub-components, and with the constraint that the coefficients of the parent model are equal to the sum of the coefficients of the corresponding child model, the Fourier expansion coefficient vectors of each sub-component are solved by the Lagrange constrained least squares method, and the model coefficients of each sub-component are output.

[0093] S7. Based on the model coefficients of each sub-component, calculate the fitting effect evaluation index of the biomass model of each component, and output the model evaluation results;

[0094] S8. Based on the model coefficients of each sub-component and the third-order half-interval Fourier basis functions, the biomass model of each component is differentiated to obtain the rate of change of biomass with the modeling independent variable. The negative rate of change is set to 0 to obtain the non-negative growth function, and the growth analysis results of each component are output.

[0095] S9. Based on the model coefficients of each sub-component, the third-order half-interval Fourier basis functions, and the range of standardized independent variables, the compatibility of each level of the model is numerically verified within the observation range, and the compatibility test results and the final biomass model are output.

[0096] S10. For scenarios where only a parent model exists and some child components lack measured data, the model coefficients and biomass prediction formulas for the unobserved components are derived using the parent model coefficients and the coefficients of the existing measured child components. The measured fitted model and the derived model are distinguished and output synchronously.

[0097] Specifically, Figure 1 The fitting effect of the total biomass model was verified by fitting curves, R2, and AIC. Figure 2 The model displays the trends of total biomass and biomass of each sub-component (aboveground, underground, trunk, crown, branches, leaves, thick and thin roots, etc.) with diameter at breast height, reflecting the model's compatibility constraints on the biomass of parent-child components. Figure 3 The model's prediction accuracy was further quantified and verified by scattering the observed and predicted biomass values ​​of each component. The hierarchical decomposition relationship is the correspondence that a parent-level overall component can be broken down into several lower-level sub-components. Figure 4 This visually demonstrates the layer-by-layer binary decomposition of total tree biomass, where the biomass of each parent component equals the sum of the biomass of its child components, representing an instance of component compatibility constraints. Figure 5 The logic shows that the total biomass is recursively split into two branches according to the priority of component biomass, and each split is divided into two branches: component and non-component.

[0098] Furthermore, the specific formula in S2 for converting the modeling independent variable values ​​of each sample tree into standardized independent variable values ​​according to the standardization formula is as follows:

[0099]

[0100] in, To represent the standardized independent variable value of the i-th sample tree, Let the values ​​of the independent variables for modeling the i-th sample tree be taken. To represent the minimum value of this modeling independent variable among all sample trees, This represents the maximum value of the modeling independent variable among all sample trees.

[0101] Specifically, this step maps the original modeling independent variables to the closed interval [0,1] through range normalization, eliminating the differences in dimensions and magnitudes of indicators such as diameter at breast height (DBH) and tree height, so that the values ​​of independent variables of different trees are on a uniform scale, providing a standardized numerical basis for the subsequent construction of Fourier basis functions and the solution of model coefficients.

[0102] Furthermore, the specific form of the third-and-a-half-interval Fourier basis function in S3 is as follows:

[0103]

[0104] in, This represents a vector of third- and half-interval Fourier basis functions. The standardized independent variable represents the values ​​of π, where π represents the mathematical constant pi. The basis function vector contains 7 terms, which are constant basis terms in order. First-order cosine term First-order sine term Second-order cosine term Second-order sine term Third-order cosine term Third-order sine term T is the transpose symbol;

[0105] The biomass model expression for any component is:

[0106]

[0107] in, This represents the predicted biomass value of component j under the standardized independent variable value z. The Fourier expansion coefficient vector of component j is specifically:

[0108]

[0109] The explicit formula for calculating the Fourier expansion of any component is:

[0110]

[0111] in This represents the coefficient of the constant term of component j. Denotes the coefficients of the first-order cosine term of component j. This represents the coefficient of the first-order sine term of component j. Denotes the coefficients of the second-order cosine term of component j. Denotes the coefficients of the second-order sine term of component j. Denotes the coefficients of the third cosine term of component j. Let represent the coefficients of the third-order sine term of component j.

[0112] Specifically, this step constructs a third-order half-interval Fourier basis function that fits the standardized independent variable whose value is located in the closed interval [0,1]. All components share the same set of basis functions to ensure the uniformity of the parameter system. By flexibly fitting the nonlinear variation law of biomass with growth index through the linear combination of 7 basis functions, the model's ability to fit complex growth relationships is improved, and a unified explicit calculation basis is provided for subsequent hierarchical compatibility constraints, growth rate differentiation and compatibility verification.

[0113] Furthermore, the coefficients of the parent model in S4 are solved using the ordinary least squares method, as follows:

[0114]

[0115] The expression for the biomass model of the parent component is:

[0116]

[0117] in, This represents the Fourier expansion coefficient vector of the parent component. This represents the measured sample size of the parent component. This represents the measured biomass value of the parent component in the i-th sample. Let represent the transpose of the Fourier basis function vector corresponding to the i-th sample. arg represents the parent coefficient vector to be estimated. This represents the value of θ that minimizes the subsequent summation terms. This represents the predicted biomass value of the current parent under the standardized independent variable value z.

[0118] Specifically, this step takes the total biomass or the highest-level component that has been measured as the parent. Based on the constructed unified Fourier basis function, the ordinary least squares method is used to minimize the sum of squared residuals between the predicted and measured values, and the optimal coefficient vector of the parent model is obtained. This provides the benchmark parameters and initial splitting starting point for the subsequent iterative bisection of the remaining residuals and the hierarchical compatibility constraints.

[0119] Furthermore, the specific implementation of the step-by-step binary search strategy in S5 is as follows:

[0120] The initial parent is the total biomass of the entire tree. During each split, the component with the highest biomass percentage from the set of components to be modeled corresponding to the current parent is selected as the target component. The current parent is then split into the biomass of the target component and the biomass of non-target components. The functional relationship for the split is as follows:

[0121]

[0122] The compatibility constraint relationship at the coefficient level is as follows:

[0123]

[0124] in, This represents the predicted biomass value of the target component under the standardized independent variable value z. This represents the predicted biomass value of a non-target component under the standardized independent variable value z. This represents the Fourier expansion coefficient vector of the current parent element. This represents the Fourier expansion coefficient vector of the target component. Represents the Fourier expansion coefficient vector of the non-target component;

[0125] After the split is completed, As the parent biomass for the next round, the component with the highest biomass percentage is selected from the set of remaining components to be modeled for the next round of splitting. The above process is repeated until all components to be modeled are included in the splitting system.

[0126] Specifically, this step constructs a biomass compatibility relationship based on the inherent organ hierarchy of trees to address the inconsistency in the summation of predicted values ​​for each component in the traditional independent modeling model. The preferred natural hierarchical decomposition function relationship is as follows:

[0127]

[0128]

[0129]

[0130]

[0131] Since all components use third-order half-interval Fourier basis functions, the above-mentioned summation compatibility constraints at the function level can be directly and equivalently transformed into linear constraints at the coefficient level:

[0132]

[0133]

[0134]

[0135]

[0136] This design ensures, from the mathematical model's underlying level, that the sum of the predicted biomass values ​​of the parent level and the corresponding predicted values ​​of the child level is strictly equal, fundamentally satisfying the compatibility requirements of tree biomass and providing a basic rule for subsequent Lagrange-constrained least squares solutions.

[0137] Furthermore, the specific implementation of the Lagrange-constrained least squares method in S6 is as follows:

[0138] Based on the binary splitting relationship and coefficient compatibility constraint rules, an objective function is constructed for the set of sub-components corresponding to each round of splitting:

[0139]

[0140] The constraints are:

[0141]

[0142] Construct the Lagrangian function:

[0143]

[0144] Where m represents the number of child components corresponding to the current parent, L is the Lagrangian objective function, j is the child component index, and i is the sample index. This represents the measured biomass value of the j-th sub-component of the i-th sample tree. This is the transpose of the Lagrange multiplier vector;

[0145] By taking the first-order partial derivatives of the coefficient vectors of all sub-components with the Lagrange multipliers and setting the derivatives to 0, the final Fourier expansion coefficient vectors of each sub-component can be obtained.

[0146] Specifically, this step aims to minimize the sum of squared residuals between the predicted and measured biomass values ​​of all sub-components. It uses the constraint that the parent coefficient equals the sum of the coefficients of all sub-components as a forced constraint. The Lagrange multiplier method transforms the optimization problem with equality constraints into an unconstrained extremum problem. In the iterative bisection strategy of the remaining residuals, by taking the first-order partial derivatives of the sub-component coefficient vector and the Lagrange multipliers and setting them to zero, the optimal coefficients of the two sub-components satisfying the compatibility constraint can be obtained simultaneously. This method avoids the compatibility problem of inconsistency between the sum of the sub-component predicted values ​​and the parent predicted values ​​during independent fitting, and reduces the computational complexity of simultaneous multi-component modeling through round-by-round constraint solving.

[0147] Furthermore, the evaluation indicators for the fit of the biomass models of each component in S7 include the residual sum of squares, total sum of squares, coefficient of determination, and Akaike information criterion.

[0148] Sum of squared residuals:

[0149]

[0150] Total sum of squares:

[0151]

[0152] Coefficient of determination:

[0153]

[0154] Akaike Information Guidelines:

[0155]

[0156] in, Let j be the sum of squared residuals of the j-th component. Let the sum of squares of the j-th component be , Let be the determination coefficient of the j-th component model. Let i be the sample wood number, and let i be the information criterion for the j-th component. The measured biomass of the j-th component of the i-th sample tree is... This represents the predicted biomass value for the j-th component of the i-th sample tree. Let be the mean of the measured biomass of all samples of component j. Let j be the total number of measured samples of the j-th component. For natural logarithm operations, Let be the number of valid parameters to be estimated for the j-th component model.

[0157] Specifically, the residual sum of squares and the total sum of squares characterize the model fitting bias from two dimensions: the degree of error dispersion and the degree of data variation, respectively; the coefficient of determination, by comparing the residuals with the total variation, intuitively reflects the model's ability to interpret measured biomass data, and the closer the value is to 1, the better the fit; the Akaike Information Criterion introduces a parameter penalty term on the basis of goodness of fit, and avoids model overfitting through a balancing mechanism, which is especially suitable for cross-sectional comparisons when there are differences in sample size or number of parameters between different components.

[0158] Furthermore, in S8, the rate of change of biomass with respect to the modeling independent variable is obtained by differentiating the biomass model for each component. The calculation formula used to obtain the non-negative growth function by setting the negative rate of change to 0 is as follows:

[0159] First derivative of biomass under standardized independent variables:

[0160]

[0161] Rate of change of biomass under the original independent variable:

[0162]

[0163] Non-negative growth function:

[0164]

[0165] in, Let j be the first derivative of biomass of component j with respect to the standardized independent variable z. , , , , , Let J be the Fourier coefficients of component j. Pi Let x be the rate of change of biomass of component j with respect to the original modeling independent variable x. , These represent the maximum and minimum values ​​of the independent variables being modeled, respectively. Let j be the non-negative growth function of the original independent variable x corresponding to component j. This is for calculating the maximum value.

[0166] Furthermore, the difference calculation formula used in S9 to numerically verify the compatibility of models at each level within the observation range is as follows:

[0167]

[0168]

[0169]

[0170]

[0171] in, , , , This represents the biomass compatibility difference between different biomass levels. This is the predicted value for the total tree biomass. This is a predicted value for aboveground biomass. This is a predicted value for the underground biomass. This is a predicted value for trunk biomass. This is a predicted value for canopy biomass. This is a predicted value for tree branch biomass. Predicted values ​​for leaf biomass. This is a predicted value for coarse root biomass. This is a predicted value for fine root biomass.

[0172] Furthermore, in S10, the formula for calculating the model coefficients and biomass prediction of unobserved components using the parent model coefficients and existing measured sub-component coefficients is as follows:

[0173] Missing Component Biomass Prediction Model:

[0174]

[0175] Missing component Fourier coefficient vector:

[0176]

[0177] in, This is a biomass estimation model without actual measured sample components. For sub-component biomass models with available measured data, Calculate the coefficient vector for the missing components. The coefficient vector of the parent component. This is the coefficient vector of the sub-components that have actual measured samples.

Claims

1. A method for constructing a tree compatibility biomass model that integrates Lagrange constraints and Fourier expansion, characterized in that, include: S1. Collect the diameter at breast height (DBH), tree height, tree age, and the product of the square of DBH and tree height of the test sample trees as candidate modeling independent variables. Collect at least two of the following biomass of the test sample trees that have a hierarchical decomposition subordinate relationship as the measured values ​​of component biomass: whole tree biomass, aboveground biomass, underground biomass, trunk biomass, crown biomass, branch biomass, leaf biomass, coarse root biomass, and fine root biomass. Construct a set of components to be modeled. Mark the components without measured biomass data as null values ​​and exclude them from their independent fitting range. Output the sample tree dataset containing the values ​​of the independent variables of the test sample trees and the measured values ​​of component biomass. S2. Based on the sample tree dataset, extract the modeling independent variable values ​​for each sample tree, calculate the minimum and maximum values ​​of the modeling independent variable among all sample trees, convert the modeling independent variable values ​​of each sample tree into standardized independent variable values ​​according to the standardization formula, so that the standardized values ​​fall within the closed interval [0,1], and output the standardized independent variable values ​​corresponding to each sample tree. S3. Based on standardized independent variable values, a unified third-order half-interval Fourier basis function is constructed, and all component biomass models share the same set of basis functions; S4. In the component hierarchy, the upper-level overall component is the parent level, and the lower-level subdivided components obtained by splitting the parent level are the child components. Based on the third-order half-interval Fourier basis function and sample tree dataset, the highest-level parent biomass model is fitted first. When whole tree biomass data exists, the whole tree total biomass model is fitted. When whole tree biomass data is missing, the measured highest-level component model is fitted. The Fourier expansion coefficient vector of the parent model is estimated by ordinary least squares method, and the parent model coefficients are output. S5. Based on the parent model coefficients and the set of components to be modeled, a hierarchical binary splitting strategy is adopted to construct a compatible splitting relationship. Taking the current parent biomass as the initial constraint benchmark, the component with the highest biomass ratio among the components to be modeled corresponding to the current parent is selected as the target component. The current parent is split into the target component biomass and the non-target component biomass. The non-target component biomass is used as the parent in the next round. The above splitting process is repeated until all components in the set of components to be modeled are covered. The multi-level binary splitting relationship and coefficient compatibility constraint rules are output. S6. Based on the multi-level binary splitting relationship and coefficient compatibility constraint rules, the third-order half-interval Fourier basis function and the measured biomass values ​​of each sub-component, with the objective of minimizing the sum of squared residuals between the predicted and measured values ​​of all sub-components, and with the constraint that the coefficients of the parent model are equal to the sum of the coefficients of the corresponding child model, the Fourier expansion coefficient vectors of each sub-component are solved by the Lagrange constrained least squares method, and the model coefficients of each sub-component are output. S7. Based on the model coefficients of each sub-component, calculate the fitting effect evaluation index of the biomass model of each component, and output the model evaluation results; S8. Based on the model coefficients of each sub-component and the third-order half-interval Fourier basis functions, the biomass model of each component is differentiated to obtain the rate of change of biomass with the modeling independent variable. The negative rate of change is set to 0 to obtain the non-negative growth function, and the growth analysis results of each component are output. S9. Based on the model coefficients of each sub-component, the third-order half-interval Fourier basis functions, and the range of standardized independent variables, the compatibility of each level of the model is numerically verified within the observation range, and the compatibility test results and the final biomass model are output. S10. For scenarios where only a parent model exists and some child components lack measured data, the model coefficients and biomass prediction formulas for the unobserved components are derived using the parent model coefficients and the coefficients of the existing measured child components. The measured fitted model and the derived model are distinguished and output synchronously.

2. The method for constructing a tree compatibility biomass model integrating Lagrange constraints and Fourier expansion as described in claim 1, characterized in that, In S2, the specific formula for converting the modeling independent variable values ​​of each sample tree into standardized independent variable values ​​according to the standardization formula is as follows: in, To represent the standardized independent variable value of the i-th sample tree, Let the values ​​of the independent variables for modeling the i-th sample tree be taken. To represent the minimum value of this modeling independent variable among all sample trees, This represents the maximum value of the modeling independent variable among all sample trees.

3. The method for constructing a tree compatibility biomass model integrating Lagrange constraints and Fourier expansion as described in claim 2, characterized in that, The specific form of the third-order half-interval Fourier basis function in S3 is as follows: in, This represents a vector of third- and half-interval Fourier basis functions. The standardized independent variable represents the values ​​of π, where π represents the mathematical constant pi. The basis function vector contains 7 terms, which are constant basis terms in order. First-order cosine term First-order sine term Second-order cosine term Second-order sine term Third-order cosine term Third-order sine term T is the transpose symbol; The biomass model expression for any component is: in, This represents the predicted biomass value of component j under the standardized independent variable value z. The Fourier expansion coefficient vector of component j is specifically: The explicit formula for calculating the Fourier expansion of any component is: in This represents the coefficient of the constant term of component j. Denotes the coefficients of the first-order cosine term of component j. This represents the coefficient of the first-order sine term of component j. Denotes the coefficients of the second-order cosine term of component j. Denotes the coefficients of the second-order sine term of component j. Denotes the coefficients of the third cosine term of component j. Let represent the coefficients of the third-order sine term of component j.

4. The method for constructing a tree compatibility biomass model integrating Lagrange constraints and Fourier expansion as described in claim 3, characterized in that, The coefficients of the parent model in S4 are solved using the ordinary least squares method. The expression for the biomass model of the parent component is: in, This represents the Fourier expansion coefficient vector of the parent component. This represents the measured sample size of the parent component. This represents the measured biomass value of the parent component in the i-th sample. Let represent the transpose of the Fourier basis function vector corresponding to the i-th sample. arg represents the parent coefficient vector to be estimated. This represents the value of θ that minimizes the subsequent summation terms. This represents the predicted biomass value of the current parent under the standardized independent variable value z.

5. In the method for constructing a tree compatibility biomass model integrating Lagrange constraints and Fourier expansion as described in claim 4, the specific implementation of the stepwise binary search strategy in S5 is as follows: The initial parent is the total biomass of the entire tree. During each split, the component with the highest biomass percentage in the set of components to be modeled corresponding to the current parent is selected as the target component. The current parent is then split into the biomass of the target component and the biomass of non-target components. The functional relationship for the split is as follows: The compatibility constraint relationship at the coefficient level is as follows: in, This represents the predicted biomass value of the target component under the standardized independent variable value z. This represents the predicted biomass value of a non-target component under the standardized independent variable value z. This represents the Fourier expansion coefficient vector of the current parent element. This represents the Fourier expansion coefficient vector of the target component. Represents the Fourier expansion coefficient vector of the non-target component; After the split is completed, As the parent biomass for the next round, the component with the highest biomass percentage is selected from the set of remaining components to be modeled for the next round of splitting. The above process is repeated until all components to be modeled are included in the splitting system.

6. The method for constructing a tree compatibility biomass model integrating Lagrange constraints and Fourier expansion as described in claim 5, characterized in that, The specific implementation of the Lagrange-constrained least squares method in S6 is as follows: Based on the binary splitting relationship and coefficient compatibility constraint rules, an objective function is constructed for the set of sub-components corresponding to each round of splitting: The constraints are: Construct the Lagrangian function: Where m represents the number of child components corresponding to the current parent, L is the Lagrangian objective function, j is the child component index, and i is the sample index. This represents the measured biomass value of the j-th sub-component of the i-th sample tree. This is the transpose of the Lagrange multiplier vector; By taking the first-order partial derivatives of all sub-component coefficient vectors with respect to the Lagrange multipliers and setting the derivatives to 0, the final Fourier expansion coefficient vectors of each sub-component can be obtained.

7. The method for constructing a tree compatibility biomass model integrating Lagrange constraints and Fourier expansion as described in claim 6, characterized in that, The evaluation indices for the fit of the biomass models for each component in S7 include the residual sum of squares, total sum of squares, coefficient of determination, and Akaike information criterion. Sum of squared residuals: Total sum of squares: Coefficient of determination: Akaike Information Guidelines: in, Let j be the sum of squared residuals of the j-th component. Let the sum of squares of the j-th component be . Let be the determination coefficient of the j-th component model. Let i be the sample wood number, and let i be the information criterion for the j-th component. The measured biomass of the j-th component of the i-th sample tree is... This represents the predicted biomass value for the j-th component of the i-th sample tree. Let be the mean of the measured biomass of all samples of component j. Let j be the total number of measured samples of the j-th component. For natural logarithm operations, Let be the number of valid parameters to be estimated for the j-th component model.

8. The method for constructing a tree compatibility biomass model integrating Lagrange constraints and Fourier expansion as described in claim 7, characterized in that, In S8, the rate of change of biomass with respect to the modeling independent variable is obtained by differentiating the biomass model for each component. The calculation formula used to obtain the non-negative growth function by setting the negative rate of change to 0 is as follows: First derivative of biomass under standardized independent variables: Rate of change of biomass under the original independent variable: Non-negative growth function: in, Let j be the first derivative of biomass of component j with respect to the standardized independent variable z. , , , , , Let J be the Fourier coefficients of component j. Pi Let x be the rate of change of biomass of component j with respect to the original modeling independent variable x. , These represent the maximum and minimum values ​​of the independent variables being modeled, respectively. Let j be the non-negative growth function of the original independent variable x corresponding to component j. This is for calculating the maximum value.

9. The method for constructing a tree compatibility biomass model integrating Lagrange constraints and Fourier expansion as described in claim 8, characterized in that, The difference calculation formula used in S9 to numerically verify the compatibility of models at each level within the observation range is as follows: in, , , , This represents the biomass compatibility difference between different biomass levels. This is the predicted value for the total tree biomass. This is a predicted value for aboveground biomass. This is a predicted value for the underground biomass. This is a predicted value for trunk biomass. This is a predicted value for canopy biomass. This is a predicted value for tree branch biomass. Predicted values ​​for leaf biomass. This is a predicted value for coarse root biomass. This is a predicted value for fine root biomass.

10. The method for constructing a tree compatibility biomass model integrating Lagrange constraints and Fourier expansion as described in claim 9, characterized in that, In S10, the formula for calculating the model coefficients and biomass prediction of unobserved components using the parent model coefficients and existing measured child component coefficients is as follows: Missing Component Biomass Prediction Model: Missing component Fourier coefficient vector: in, This is a biomass estimation model without actual measured sample components. For sub-component biomass models with available measured data, Calculate the coefficient vector for the missing components. The coefficient vector of the parent component. This is the coefficient vector of the sub-components that have actual measured samples.

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