Uncertainty analysis method for surface ecological parameter measurement and calculation

Abstract

The invention provides an uncertainty analysis method for surface ecological parameter measurement and calculation. The method comprises the following steps: acquiring an observation value of a target index for a target area; sampling is carried out from prior probability distribution of measurement and calculation factors contained in the target index measurement and calculation model, a plurality of candidate prior parameter combinations are obtained, the measurement and calculation factors comprise model input variables and error parameters, and the model input variables comprise prior surface ecological parameters; the multiple candidate prior parameter combinations are input into a target index measurement and calculation model, respective index measurement and calculation results of the multiple candidate prior parameter combinations are output, and the index measurement and calculation results comprise measurement and calculation values of the target index values; according to the measured value of the target index and the observed value of the target index, a target prior parameter combination is selected from the multiple candidate prior parameter combinations, and the target prior parameter combination comprises target prior surface ecological parameters; and according to the probability distribution determined based on the target prior surface ecological parameters, calculating the uncertainty for the surface ecological parameters.

Description

Technical Field

[0001] This disclosure relates to the field of land surface ecological parameter measurement technology, and in particular to an uncertainty analysis method for land surface ecological parameter measurement. Background Technology

[0002] Surface ecological parameters are key indicators reflecting vegetation growth, assessing ecosystem function and carbon storage, and are widely used in fields such as ecological environment monitoring, natural resource management, and global climate change research. The accuracy of surface ecological parameter measurements can provide reliable data support for assessing the effectiveness of land greening, measuring forest carbon sequestration, and implementing ecological protection and restoration projects.

[0003] Traditional uncertainty assessment methods struggle to handle the nonlinear coupling of multiple sources of error simultaneously, resulting in overly coarse or inaccurate uncertainty estimates that lack reliability. Summary of the Invention

[0004] In view of this, in order to at least partially solve at least one of the aforementioned technical problems, this disclosure provides an uncertainty analysis method for the measurement of surface ecological parameters.

[0005] Specifically, the uncertainty analysis method for land surface ecological parameter measurement includes: obtaining observed values ​​of target indicators for the target area; sampling from the prior probability distribution of the measurement factors included in the target indicator measurement model to obtain multiple candidate prior parameter combinations, wherein the measurement factors include: model input variables and error parameters, and the model input variables include prior land surface ecological parameters; inputting the multiple candidate prior parameter combinations into the target indicator measurement model respectively, and outputting the indicator measurement results of each of the multiple candidate prior parameter combinations, wherein the indicator measurement results include the measured values ​​of the target indicator; selecting the target prior parameter combination from the multiple candidate prior parameter combinations based on the measured values ​​and observed values ​​of the target indicator, wherein the target prior parameter combination includes the target prior land surface ecological parameters; and calculating the uncertainty for the land surface ecological parameters based on the probability distribution determined by the target prior land surface ecological parameters.

[0006] According to embodiments of this disclosure, selecting a target prior parameter combination from multiple candidate prior parameter combinations based on the measured value and observed value of the target indicator includes: constructing a likelihood function based on the measured value and observed value of the target indicator; calculating the joint posterior probability of any candidate prior parameter combination based on the likelihood function and the prior probability distribution; comparing the joint posterior probability with the joint posterior probability calculated in the previous iteration; and determining at least one target prior parameter combination based on the comparison result.

[0007] According to embodiments of this disclosure, the prior probability distribution of the error parameter is determined by the following operations: obtaining an observation dataset of the target index obtained by multiple observations of the target area; determining a super-prior probability distribution based on the probability distribution type of the observation dataset; determining the variance of the prior probability distribution of the error parameter as the value sampled from the super-prior probability distribution; and determining the prior probability distribution of the error parameter based on the variance.

[0008] According to embodiments of this disclosure, determining a prior probability distribution based on the probability distribution type of the observation dataset includes: determining the prior probability distribution type based on the probability distribution type, distribution parameters, and conjugate distribution of the observation dataset; and determining the prior probability distribution based on preset distribution parameters and prior probability distribution type.

[0009] According to embodiments of this disclosure, the probability distribution type of the observed dataset is either a normal distribution or a uniform distribution; the type of the super-prior probability distribution is determined based on the probability distribution type, distribution parameters, and conjugate prior distribution of the observed dataset, including: when the probability distribution of the observed dataset is a normal distribution, determining the inverse gamma distribution of the conjugate distribution as the distribution type of the super-prior probability distribution based on the variance of the distribution parameters; when the probability distribution of the observed dataset is a uniform distribution, determining the Pareto distribution of the conjugate distribution as the distribution type of the super-prior probability distribution based on the upper bound of the distribution parameters.

[0010] According to embodiments of this disclosure, the surface ecological parameters include vegetation cover or forest biomass.

[0011] According to embodiments of this disclosure, when the surface ecological parameter is vegetation cover, the model input variables of the target index calculation model include: vegetation cover, which represents the area proportion of vegetation in a pixel; normalized vegetation index value of pure vegetation pixels; normalized vegetation index value of pure bare soil pixels; and a first error parameter; the target index value output by the target index calculation model includes: normalized vegetation index.

[0012] According to embodiments of this disclosure, when the surface ecological parameter is vegetation cover, multiple candidate prior parameter combinations are input into the target index calculation model, and the index calculation results of each candidate prior parameter combination are output. This includes: calculating the product of the prior vegetation cover and the prior normalized vegetation index value of the prior pure vegetation pixel to obtain the vegetation index contribution value; calculating the product of the prior bare soil cover and the prior normalized vegetation index value of the prior pure bare soil pixel to obtain the bare soil index contribution value, wherein the prior bare soil cover is determined based on the prior vegetation cover; summing the vegetation index contribution value, the bare soil index contribution value, and the first error parameter to output the calculated value of the normalized vegetation index.

[0013] According to embodiments of this disclosure, when the surface ecological parameter is forest biomass, the model input variables of the target index calculation model include: forest biomass; tree height; second error parameter; and the target index value output by the target index calculation model includes diameter at breast height (DBH).

[0014] According to embodiments of this disclosure, when the surface ecological parameter is forest biomass, multiple candidate prior parameter combinations are input into the target index calculation model, and the index calculation results of each candidate prior parameter combination are output. This includes: calculating the difference between the prior forest biomass and the second error parameter to obtain an estimated value of forest biomass; calculating the product of the model prior coefficient, the model prior correction coefficient, and the allometric growth regulation factor of tree height to obtain a combined correction factor, wherein the allometric growth regulation factor of tree height is calculated based on the prior tree height; calculating the ratio of the estimated value of forest biomass to the combined correction factor to obtain a candidate baseline value for diameter at breast height (DBH); and calculating and outputting the measured value of DBH based on the prior influence index of DBH on forest biomass and the candidate baseline value for DBH.

[0015] According to embodiments of this disclosure, by incorporating observation errors, model structure differences, and parameter uncertainties into a unified Bayesian probability model, prior samples of error standard deviations and their probabilities are generated through sampling from the super-prior distribution; prior sample combinations and their probabilities are generated through sampling from the prior distribution; based on the likelihood function constructed from the observation data and the calculation model, and the prior and super-prior, the posterior distributions of each factor are updated according to Bayes' theorem, and sampling is performed from the posterior distribution using the Markov chain Monte Carlo method; finally, a posterior probability distribution is formed based on the posterior sample set, which can simultaneously output parameter estimates and their corresponding standard uncertainties. Therefore, for uncertainty analysis of surface ecological parameter calculations, it does not depend on a specific model, can flexibly adapt to various ecological calculation models, and achieves efficient and stable automated calculations through Markov chain Monte Carlo sampling, thus providing more reliable quantitative results of uncertainty for high-precision applications such as carbon sink measurement and ecological assessment in regional-scale remote sensing inversion and ground estimation. Attached Figure Description

[0016] The above and other objects, features and advantages of this disclosure will become clearer from the following description of embodiments with reference to the accompanying drawings, in which:

[0017] Figure 1 A flowchart is shown for an uncertainty analysis method for calculating surface ecological parameters according to an embodiment of the present disclosure.

[0018] Figures 2(a) to 2(i) show a vegetation cover sampling chain of nine plots according to an embodiment of the present disclosure.

[0019] Figures 3(a) to 3(i) show the posterior probability distribution of vegetation cover in nine plots according to embodiments of the present disclosure.

[0020] Figure 4(a) shows a multispectral image of a target area obtained by using the infrared band at a flight altitude of 30m according to an embodiment of the present disclosure.

[0021] Figure 4(b) shows a multispectral image of a target area obtained by using the near-infrared band at a flight altitude of 100m according to an embodiment of the present disclosure.

[0022] Figure 5(a) shows a normalized vegetation index image obtained by observing a target area using the infrared band at a flight altitude of 30m according to an embodiment of the present disclosure.

[0023] Figure 5(b) shows a normalized vegetation index image obtained by observing a target area in the near-infrared band at a flight altitude of 100m according to an embodiment of the present disclosure.

[0024] Figure 6 A vegetation cover sampling chain according to another embodiment of the present disclosure is shown.

[0025] Figure 7 A visualization of the spatial distribution of vegetation cover according to an embodiment of the present disclosure is shown.

[0026] Figure 8 A posterior probability distribution of vegetation cover according to another embodiment of this disclosure is shown.

[0027] Figure 9A A sequence diagram of posterior mean vegetation cover according to an embodiment of the present disclosure is shown.

[0028] Figure 9B A sequence diagram of the posterior standard deviation of vegetation cover according to an embodiment of the present disclosure is shown.

[0029] Figure 10A A prior probability distribution diagram of tree height in forest biomass measurement according to an embodiment of the present disclosure is shown.

[0030] Figure 10B A prior probability distribution of diameter at breast height (DBH) in forest biomass measurement according to an embodiment of this disclosure is shown.

[0031] Figure 11 A forest biomass sampling chain according to an embodiment of this disclosure is shown.

[0032] Figure 12 The posterior probability distribution of forest biomass according to an embodiment of this disclosure is shown. Detailed Implementation

[0033] The embodiments of the present disclosure will now be described with reference to the accompanying drawings. However, it should be understood that these descriptions are exemplary only and are not intended to limit the scope of the disclosure. In the following detailed description, numerous specific details are set forth to provide a thorough understanding of the embodiments of the present disclosure for ease of explanation. However, it will be apparent that one or more embodiments may be practiced without these specific details. Furthermore, descriptions of well-known structures and techniques are omitted in the following description to avoid unnecessarily obscuring the concepts of the present disclosure.

[0034] The terminology used herein is for the purpose of describing particular embodiments only and is not intended to limit this disclosure. The terms “comprising,” “including,” etc., as used herein indicate the presence of the stated features, steps, operations, and / or components, but do not exclude the presence or addition of one or more other features, steps, operations, or components.

[0035] All terms used herein (including technical and scientific terms) have the meanings commonly understood by those skilled in the art, unless otherwise defined. It should be noted that the terms used herein are to be interpreted in a manner consistent with the context of this specification, and not in an idealized or overly rigid way.

[0036] When using expressions such as "at least one of A, B and C", they should generally be interpreted in accordance with the meaning that is commonly understood by those skilled in the art (e.g., "a system having at least one of A, B and C" should include, but is not limited to, a system having A alone, a system having B alone, a system having C alone, a system having A and B, a system having A and C, a system having B and C, and / or a system having A, B and C, etc.).

[0037] Currently, traditional methods for uncertainty analysis of land surface biological parameters, such as error propagation laws based on Taylor series expansions or Monte Carlo simulations based on the assumption of a single distribution, have significant limitations: First, they typically only handle single or a few sources of uncertainty, making it difficult to achieve collaborative modeling and joint propagation of errors from multiple sources, including observations, models, and parameters. Second, these methods often fail to effectively incorporate valuable prior knowledge such as expert experience and historical data, leading to uncertainty assessments that may deviate from reality. Finally, traditional methods usually output only a general total error or range, making it difficult to clearly analyze the specific contributions of each uncertainty source, which is detrimental to model optimization and improvement. This is especially problematic in operational applications such as carbon sink monitoring and trading, and ecological environment damage assessment, where the accuracy and reliability of results are extremely critical, lacking a quantifiable measure of the reliability of the calculation results.

[0038] In view of this, this disclosure provides an uncertainty analysis method for the calculation of surface ecological parameters, so as to at least solve some of the above-mentioned technical problems existing in related technologies.

[0039] Figure 1 A flowchart is shown for an uncertainty analysis method for calculating surface ecological parameters according to an embodiment of the present disclosure.

[0040] like Figure 1 As shown, the uncertainty analysis method for the calculation of surface ecological parameters includes operations S110 to S150.

[0041] In operation S110, the observation values ​​of the target indicators for the target area are obtained.

[0042] In operation S120, samples are taken from the prior probability distribution of the measurement factors included in the target index measurement model to obtain multiple candidate prior parameter combinations. The measurement factors include model input variables and error parameters. The model input variables include prior surface ecological parameters.

[0043] In operation S130, multiple candidate prior parameter combinations are input into the target index calculation model, and the index calculation results of each candidate prior parameter combination are output. The index calculation results include the calculated value of the target index.

[0044] In operation S140, based on the measured value and the observed value of the target indicator, a target prior parameter combination is selected from multiple candidate prior parameter combinations, wherein the target prior parameter combination includes target prior surface ecological parameters.

[0045] In operation S150, the uncertainty of the surface ecological parameters is calculated based on the probability distribution determined by the prior surface ecological parameters of the target.

[0046] According to embodiments of this disclosure, the target area refers to the geographical area to be analyzed, such as a forest or an ecological reserve. The target index refers to observable data used to indirectly estimate surface ecological parameters. The observed values ​​of the target index can be obtained through remote sensing image processing or ground-based measurements.

[0047] According to embodiments of this disclosure, the target indicator calculation model is a mathematical model that links surface ecological parameters with target indicators. In vegetation cover calculation, it can be a pixel-based bisection model or a needle-based model; in forest biomass calculation, it can be an allometric growth equation model. Calculation factors refer to all unknown parameters in the model, divided into two categories: model input variables, including prior surface ecological parameters, i.e., parameters to be estimated, such as vegetation cover (FVC), forest biomass (AGB), and other structural parameters, such as NDVIsoil, NDVIveg, tree height (H), and diameter at breast height (DBH). Error parameters refer to the deviation between model predictions and observed values. Prior probability distributions refer to the probability distributions set for each calculation factor based on historical data, expert knowledge, or experience before obtaining observational data, used to express the initial understanding of the possible values ​​of that factor.

[0048] According to embodiments of this disclosure, selecting a target prior parameter combination from multiple candidate prior parameter combinations based on the measured value and observed value of the target indicator includes: constructing a likelihood function based on the measured value and observed value of the target indicator; calculating the joint posterior probability of any candidate prior parameter combination based on the likelihood function and the prior probability distribution; comparing the joint posterior probability with the joint posterior probability calculated in the previous iteration; and determining at least one target prior parameter combination based on the comparison result.

[0049] The posterior distribution of each factor is updated based on Bayes' theorem, and sampling is performed from the posterior distribution using the Markov Chain Monte Carlo (MCMC) method. Specifically, a set of specific samples is drawn from the posterior distribution of each factor through sampling. For example, sampling algorithms such as Markov Chain Monte Carlo (MCMC) are used to estimate the posterior probability distribution, obtaining estimated values ​​of surface ecological parameters and their uncertainty characterization. MCMC sampling methods include Gibbs sampling, Metropolis-Hastings sampling, or combinations thereof. The Metropolis-Hastings algorithm is used for sampling below, with the following steps: setting initial values ​​for one set of candidate prior parameter combinations. This will serve as the first sample in the following iterations. Determine a probability distribution g( | ), randomly generate the next set of samples Candidate values. For each iteration i: from g( | Generate a set of samples in ) ; Calculate the acceptance rate α = p( |γ,R(m)) / p( |γ,R(m)); If α ≥ 1, accept the sample group. For a new sample, if α < 1, generate a uniformly random number u from the interval [0, 1]; if u ≤ α, accept the sample. If u > α, then reject the sample group. And set from this group of samples Revert to the previous set of samples .

[0050] Among them, g( | The distribution p(m|γ, R(m)) can be chosen to follow a Gaussian distribution so that the samples of p(m|γ, R(m)) resemble a random walk chain. Furthermore, to reduce the computation time required to obtain a certain number of samples m from p(m|γ, R(m)), a parallelized version of Metropolis-Hastings is typically used to generate the parallel chain. Finally, sampling stops when the iteration termination condition is met, and multiple sets of target posterior parameter combinations are obtained. The iteration termination condition can be, for example, a pre-set number of sampling times, such as 50,000 samplings. Through iterative filtering, the set of parameters that can better explain the observation data gradually converges, forming a posterior understanding of the surface ecological parameters.

[0051] The probability distribution for determining the target posterior surface ecological parameter refers to extracting all possible values ​​of the surface ecological parameter from all accepted parameter combinations; these values ​​constitute the posterior probability distribution of the parameter. Furthermore, based on this posterior probability distribution, its statistical characteristics are calculated to quantify the uncertainty.

[0052] According to embodiments of this disclosure, by incorporating observation errors, model structure differences, and parameter uncertainties into a unified Bayesian probability model, prior samples of error standard deviations and their probabilities are generated through sampling from the super-prior distribution; prior sample combinations and their probabilities are generated through sampling from the prior distribution; based on the likelihood function constructed from the observation data and the calculation model, and the prior and super-prior, the posterior distributions of each factor are updated according to Bayes' theorem, and sampling is performed from the posterior distribution using the Markov chain Monte Carlo method; finally, a posterior probability distribution is formed based on the posterior sample set, which can simultaneously output parameter estimates and their corresponding standard uncertainties. Therefore, for uncertainty analysis of surface ecological parameter calculations, it does not depend on a specific model, can flexibly adapt to various ecological calculation models, and achieves efficient and stable automated calculations through Markov chain Monte Carlo sampling, thus providing more reliable quantitative results of uncertainty for high-precision applications such as carbon sink measurement and ecological assessment in regional-scale remote sensing inversion and ground estimation.

[0053] According to embodiments of this disclosure, the prior probability distribution of the model input variables is determined through the following operations: obtaining historical observation data related to the target area and parameter value ranges recorded by expert experience; determining the probability distribution type and distribution parameters of the model input variables based on historical observation data and expert experience; and determining the prior probability distribution of the model input variables based on the distribution type and distribution parameters. Specifically, the prior probability distribution of the measured factors is constructed based on preprocessed historical data, measured samples, or expert experience of the target area. The measured factors are then preprocessed, including outlier removal, missing value completion, and error correction. Prior information includes key statistics such as the data distribution type, mean, standard deviation, median, maximum, and minimum values. The prior probability model includes the relevant distribution type (e.g., normal distribution, uniform distribution, gamma distribution, beta distribution, etc.) and corresponding parameter estimates.

[0054] According to embodiments of this disclosure, calculating the uncertainty for a surface ecological parameter based on a posterior probability distribution determined based on the target surface ecological parameter includes: calculating the mean of the target posterior surface ecological parameter based on the posterior probability distribution determined based on the target surface ecological parameter, wherein the mean is used to characterize the estimated value of the surface ecological parameter; and calculating the posterior standard deviation of the target surface ecological parameter based on the posterior probability distribution determined based on the target surface ecological parameter, wherein the standard deviation is used to calculate the uncertainty characterizing the estimated value of the surface ecological parameter.

[0055] According to an embodiment of this disclosure, the posterior mean of the target surface ecological parameters is calculated based on the posterior probability distribution determined based on the target surface ecological parameters, referring to formula (1).

[0056] Formula (1)

[0057] in, Let q be the target posterior surface ecological parameter obtained from each sampling in the posterior probability distribution, and q be the number of samples of the target posterior surface ecological parameter.

[0058] According to an embodiment of this disclosure, the prior variance of the target surface ecological parameters is calculated based on the posterior mean of the target surface ecological parameters, referring to formula (2).

[0059] Formula (2)

[0060] According to an embodiment of this disclosure, the posterior standard deviation of a single target surface ecological parameter is calculated based on the posterior mean of the target surface ecological parameter, referring to formula (3).

[0061] Formula (3)

[0062] in, The uncertainty of the posterior surface ecological parameters for a single target.

[0063] According to an embodiment of this disclosure, the posterior uncertainty of all target surface ecological parameters is calculated based on the posterior standard deviation of the target surface ecological parameters, referring to formula (4).

[0064] Formula (4)

[0065] According to embodiments of this disclosure, determining the prior probability distribution of each calculation factor in the target index calculation model based on the observation dataset includes: obtaining the observation dataset of each calculation factor of the target index calculation model obtained by observing the target area; and determining the prior probability distribution of each calculation factor based on the corresponding probability distribution type or distribution space under physical constraints of the observation dataset.

[0066] Specifically, determining the prior probability distribution of each measurement factor based on the corresponding probability distribution type of the observation dataset includes: determining the parameter value of the probability density function calculated from the observation dataset corresponding to one of the measurement factors as the distribution parameter of the prior probability distribution corresponding to that measurement factor; and determining the prior probability distribution of that measurement factor based on the distribution parameter. Determining the prior probability distribution of each measurement factor based on the distribution space under physical constraints includes: determining the physical distribution space boundary corresponding to one of the measurement factors as the distribution parameter of the prior probability distribution corresponding to that measurement factor; and determining the prior probability distribution of that measurement factor based on the distribution parameter.

[0067] According to embodiments of this disclosure, the prior probability distribution of the error parameters of the target index measurement model is determined by the following operations: obtaining an observation dataset of the target index obtained by multiple observations of the target area; determining a super-prior probability distribution based on the probability distribution type of the observation dataset; determining the variance of the prior probability distribution of the error parameters by the values ​​sampled from the super-prior probability distribution; and determining the prior probability distribution of the error parameters based on the variance.

[0068] For example, if we calculate the vegetation cover of a watershed (target area) using 10 remote sensing images taken at different times (multiple observations), we obtain 10 sets of observation data (observation datasets: such as 65%, 68%, 63%, 70%, etc.). First, we analyze the probability distribution of these 10 sets of data. We find that the data follows a normal distribution (probability distribution type). Based on this distribution characteristic, we choose a reasonable prior probability distribution (such as the commonly used gamma distribution, used to describe prior information for non-negative parameters like variance). Then, we randomly select a value from this gamma distribution (e.g., 2.5) and use it as the variance of the prior distribution of the error parameter. That is, in subsequent calculations, the fluctuation range of the error is initially set at a variance of 2.5. Finally, based on this variance, we determine the complete prior probability distribution of the error parameter, such as a normal distribution N~(0,2.5), indicating that the error fluctuates around a mean of 0 with a variance of 2.5, providing an initial hypothesis for the error in subsequent accurate inversion or data correction.

[0069] According to embodiments of this disclosure, the introduction of a super-prior distribution ensures that the prior distribution of the error parameter also carries reasonable uncertainty, rather than a fixed single value. This hierarchical Bayesian modeling approach can more realistically reflect the error characteristics in actual observations. In subsequent index inversion or data correction, the results obtained based on this error parameter will have a more reliable uncertainty range, which not only improves the estimation accuracy of the index but also provides a quantitative basis for verifying the reliability of the results. For example, in long-term monitoring of watershed vegetation cover, it can more accurately identify whether data fluctuations are caused by real ecological changes or observation errors.

[0070] According to embodiments of this disclosure, determining a prior probability distribution based on the probability distribution type of the observation dataset includes: determining the prior probability distribution type based on the probability distribution type, distribution parameters, and conjugate distribution of the observation dataset; and determining the prior probability distribution based on preset distribution parameters and prior probability distribution type.

[0071] Multiple observation datasets of vegetation cover in the target area have been obtained, and testing has shown that they follow a normal distribution (the probability distribution type of the observation dataset). Since the conjugate distribution corresponding to the variance parameter of the normal distribution in Bayesian statistics is a gamma distribution, we first determine the prior probability distribution type as a gamma distribution. Then, combining this with preset distribution parameters for remote sensing observation errors of vegetation cover in the research field—for example, setting the shape parameter of the gamma distribution to 2 and the scale parameter to 1.5—and the selected gamma distribution type, we can completely determine the prior probability distribution, i.e., the inverse gamma distribution with a shape parameter of 2 and a scale parameter of 1.5. This provides a range of values ​​for the variance of the prior distribution of the error parameter extraction.

[0072] According to embodiments of this disclosure, the probability distribution type of the observed dataset is either a normal distribution or a uniform distribution. Determining the type of the super-prior probability distribution based on the probability distribution type, distribution parameters, and conjugate prior distribution of the observed dataset includes: if the probability distribution of the observed dataset is a normal distribution, determining the inverse gamma distribution of the conjugate distribution as the type of the super-prior probability distribution based on the variance of the distribution parameters. If the probability distribution of the observed dataset is a uniform distribution, determining the Pareto distribution of the conjugate distribution as the type of the super-prior probability distribution based on the upper bound of the distribution parameters.

[0073] According to embodiments of this disclosure, after conducting multiple joint field and remote sensing observations of forest biomass (target indicator) in a certain forest area, a corresponding observation dataset was obtained. Statistical testing confirmed that the dataset follows a normal distribution. In the conjugate prior system of Bayesian statistics, when the mean of the normal distribution is known, the conjugate prior distribution of its variance parameter is an inverse gamma distribution. Therefore, combining the normal distribution type of the observation dataset, the known mean of the normal distribution, and the matching rules of the conjugate prior distribution, the inverse gamma distribution can be clearly identified as the specific type of the super-prior probability distribution.

[0074] According to embodiments of this disclosure, by using the normal distribution of the observed dataset as a basis and combining the conjugate prior rule to select the inverse gamma distribution as the super-prior distribution type, and then sampling the variance of the error prior distribution through the super-prior distribution, the initial assumptions of the error parameters are made to perfectly match the statistical characteristics of the observed data, avoiding the bias caused by subjectively setting the error parameters. For example, in the inversion of ecological indicators such as forest biomass and vegetation cover, the prior distribution of the error parameters is no longer an empirical value detached from the data, but a statistic strongly bound to the distribution of the observed data, thus improving the scientific nature of error modeling from the source. Utilizing the conjugate properties of the normal distribution and the inverse gamma distribution, when subsequently deriving the posterior probability distribution of the error parameters, it can be ensured that the posterior probability distribution and the prior distribution both belong to the inverse gamma distribution type, and the analytical solution of the posterior probability distribution can be obtained quickly without complex integral calculations. This significantly reduces the computational complexity in high-dimensional parameter spaces, especially in scenarios involving multiple observations and joint inversion of multiple indicators, which can significantly improve the efficiency of model iteration and parameter calibration and reduce computational consumption.

[0075] According to embodiments of this disclosure, the likelihood function is constructed to reflect a data model that reflects observation errors. Specifically, it combines the measurement model we will use (such as a model for calculating vegetation cover) with actual remote sensing observation data to characterize the distribution patterns of these observation data. This function can be based on statistical patterns obtained from multiple actual measurements, or it can be constructed using empirical formulas commonly used in the field of remote sensing.

[0076] For example, to calculate vegetation cover, one could use the needle-pricking method: mark the rope at specific intervals with a fine needle (2mm in diameter), then insert the needle vertically downwards from above the measurement point at the same interval within the quadrat, counting the number of times the needle hits vegetation (n) and the total number of pricks (N). Dividing n by N gives the vegetation cover. Alternatively, the pixel-based bisection method could be used: calculate the vegetation cover using the Normalized Difference Vegetation Index (NDVI). First, find the NDVI value representing the bare soil area without vegetation, then find the NDVI value representing the purely vegetated area. Using the NDVI value of the current observation area and these two values, the vegetation cover can be calculated. The likelihood function can combine these calculation models with actual observation data to verify the model's robustness. If remote sensing NDVI data (i.e., observation data) is obtained, and the pixel-based bisection method is used to calculate vegetation cover, the likelihood function can measure the statistical consistency between the observed data and the model's predicted values. High consistency indicates strong model robustness and low uncertainty; low consistency requires model adjustment. On the other hand, constructing a likelihood function based on the measurement model and adding prior errors to the model allows for the mathematical representation of observation errors. Actual observations (such as remote sensing imaging and in-situ needle penetration measurements) will always have errors. The likelihood function can express the distribution pattern of these errors (such as whether the errors are concentrated in a certain range or fluctuate randomly) in mathematical form. Subsequent decisions, whether calibrating the model or evaluating the reliability of the results, can be based on this error pattern.

[0077] According to embodiments of this disclosure, introducing hyperparameters to construct a hierarchical Bayesian model provides a hierarchical logical framework for the model's assumptions and the patterns in the data. The prior distribution represents the initial assumptions about the model's input parameters, the likelihood function describes the fit between the observed data and the model, and the hyperparameters add a layer of reasonable constraints to the parameters in the prior distribution, allowing these two to be hierarchically correlated. That is, errors from different sources (biases in the observed data itself, errors in the model's calculation process, and defects in the model's structure) are all brought together into a single analytical framework, no longer treated piecemeal, but constrained by unified rules. The hyper-prior is a tool specifically used to describe the prior distribution of the variance parameter. After introducing hyperparameters, the previous likelihood function becomes one where the mean is known, but the variance is still uncertain. This acknowledges the inherent uncertainty between our estimated parameters and the actual observed results, ultimately concentrating all sources of uncertainty in the measurement and modeling process into a single framework. This random parameter facilitates a unified analysis of the impact of these uncertainties later on.

[0078] According to embodiments of this disclosure, surface ecological parameters include vegetation cover or forest biomass.

[0079] According to the embodiments of this disclosure, when the surface ecological parameter is vegetation cover, and the target index calculation model is the needle acupuncture method, the input variables of the target index calculation model include: vegetation cover, which represents the area ratio of vegetation in the quadrat; comprehensive calculation error parameters; the target index value output by the target index calculation model includes: the number of times the needle contacts the vegetation.

[0080] According to embodiments of this disclosure, when the surface ecological parameter is vegetation cover, and the target index calculation model is a pixel-based method, the input variables of the target index calculation model include: vegetation cover, which represents the area proportion of vegetation in a pixel; the normalized vegetation index value of a pure vegetation pixel; the normalized vegetation index value of a pure bare soil pixel; and a first error parameter; the target index value output by the target index calculation model includes: the normalized vegetation index.

[0081] According to embodiments of this disclosure, when the surface ecological parameter is vegetation cover, multiple candidate prior parameter combinations are input into the target index calculation model, and the index calculation results of each candidate prior parameter combination are output. This includes: calculating the product of the prior vegetation cover and the prior normalized vegetation index value of the prior pure vegetation pixel to obtain the vegetation index contribution value; calculating the product of the prior bare soil cover and the prior normalized vegetation index value of the prior pure bare soil pixel to obtain the bare soil index contribution value, wherein the prior bare soil cover is determined based on the prior vegetation cover; summing the vegetation index contribution value, the bare soil index contribution value, and the first error parameter to output the calculated value of the normalized vegetation index.

[0082] According to embodiments of this disclosure, when the surface ecological parameter is vegetation cover, a prior distribution is set for the model input variables, including: a prior distribution of vegetation cover; a prior distribution of the comprehensive calculation error; wherein, a super-prior distribution is set for the prior distribution parameter of the comprehensive calculation error, where the distribution parameter refers to the probability density function parameter. The target index calculation model maps the factor vector to the observation space, and constructs an observation likelihood function accordingly to measure the statistical consistency between the observed data and the model output.

[0083] When the target index measurement model is the acupuncture method, the observed data is the number of times the needle comes into contact with the vegetation, and the model output is the model prediction value of the number of times the needle comes into contact with the vegetation under the sampling conditions of vegetation coverage.

[0084] Based on the Bayesian model, the posterior sample index calculation results of the model input variables are obtained through the Markov chain Monte Carlo method, including: the posterior distribution of vegetation cover; the posterior distribution of the comprehensive calculation error; and the posterior distribution of the probability density function parameters of the comprehensive calculation error.

[0085] Specifically, the target indicator calculation model is based on formula (5).

[0086] Formula (5)

[0087] Where n represents the number of times the needle came into contact with the vegetation during the actual observation, which is a count-type indicator obtained from the field measurement; N represents the total number of needle insertions, which is the total number of experiments conducted in the field. To comprehensively calculate error parameters, this is used to characterize the deviation of the "number of times the needle comes into contact with vegetation" from the theoretical value in actual observations, and to quantify the uncertainty in the observation process.

[0088] Example 1

[0089] To obtain ground vegetation cover data for model building and validation in the target area, nine representative quadrats were selected for ground surveys. The main vegetation types covered included four species: Calligonum mongolicum, Haloxylon ammodendron, Nitraria tangutorum, and Ephedra sinica. Each quadrat had an area of ​​100m × 100m.

[0090] The vegetation cover of each quadrat was determined using the needle prick method. To ensure the representativeness and spatial uniformity of the measurements, four measurement lines were laid out in each quadrat, specifically including: (1) two measurement lines laid out along the diagonal direction; (2) two center lines passing through the center of the quadrat, parallel to the sides of the quadrat, and with a length equal to the sides of the quadrat. Sampling was carried out at 1m intervals along each measurement line, that is, a point was set every 1m along each line to record whether vegetation was hit. A total of four independent measurements were completed for each quadrat to obtain observational data for estimating the vegetation cover of the quadrat. The average vegetation cover of the surveyed quadrat is shown in Table 1.

[0091] Table 1. Vegetation cover of nine sample plots using the needle acupuncture method.

[0092]

[0093] Based on the four measurement records and the average of the four measurement data, key statistics such as the distribution type, mean, standard deviation, median, maximum and minimum values ​​of the data were obtained, and the prior distributions of the nine sample plots were constructed.

[0094] Figures 2(a) to 2(i) illustrate the vegetation cover sampling chain of nine plots according to an embodiment of this disclosure. As shown in Figures 2(a) to 2(i), the horizontal axis represents the number of samplings, and the vertical axis represents the vegetation cover. The curves are continuous records of vegetation cover data obtained from multiple observations of this plot using the needle-pricking method. For example, within the same plot, needle-pricking and coverage recording are repeated in chronological or spatial order, thus forming a chain of observations arranged sequentially. The nine figures shown in Figures 2(a) to 2(i) correspond to nine different plots, each with its own coverage observation sequence (i.e., a sampling chain). These sampling chains together constitute the coverage observation dataset of the nine plots. The fluctuations in the curves of each figure reflect the observational differences in vegetation cover within the corresponding plot, such as the changes in coverage measured at different needle positions within the same plot. These fluctuation data can be used to subsequently calculate the mean and uncertainty of the coverage, and also reflect the spatial heterogeneity of vegetation distribution within a single plot.

[0095] Figure 2(a) is the pinprick coverage sampling chain for plot number 0831-1; Figure 2(b) is the pinprick coverage sampling chain for plot number S2; Figure 2(c) is the pinprick coverage sampling chain for plot number M1; Figure 2(d) is the pinprick coverage sampling chain for plot number M2; Figure 2(e) is the pinprick coverage sampling chain for plot number Z2; Figure 2(f) is the pinprick coverage sampling chain for plot number B1; Figure 2(g) is the pinprick coverage sampling chain for plot number B2; Figure 2(h) is the pinprick coverage sampling chain for plot number S1; Figure 2(i) is the pinprick coverage sampling chain for plot number S3.

[0096] Taking Figure 2(a) as an example, the horizontal axis represents the number of samplings, that is, the nth record of the needle-based observation of the plot. For example, 0 on the horizontal axis is the 1st sampling, 500 is the 501st sampling, and so on. The vertical axis represents the observed value of vegetation cover, which is the result of the vegetation cover obtained in that sampling. In Figure 2(a), the range is between 0.088 and 0.096, that is, the coverage is approximately 8.8% to 9.6%.

[0097] a priori m=FVC, that is The distribution was modeled as following a Gaussian distribution, where the mean was the arithmetic mean of the four transects and the average comprehensive needle-punch coverage of the sample plots; the standard deviation was the standard deviation of the needle-punch coverage of the sample plots. in, This represents the average vegetation cover. This represents the standard deviation of vegetation cover.

[0098] Furthermore, a likelihood function is constructed, where, It is assumed to follow a uniform distribution. The upper bound is the maximum coverage of the sample plot by needle prick plus the error, and the error is usually selected as 1-3 times the standard deviation. The lower bound is the minimum coverage of the needle prick minus the standard deviation or 0 (which can be adjusted according to the comparison between the sampling results and the true values ​​based on the calculated posterior probability distribution). o represents the observed data.

[0099] By defining a prior distribution for the error term and introducing model structure error parameters, a Bayesian model is constructed. Using the concept of conjugate distributions in probability theory to select... To ensure and Maintain the same family of distributions. One such conjugate distribution is the inverse gamma distribution parameterized by shape parameter α and scale parameter β. Choose α=1 and β=0.1 so that the samples of σ are generally located in the range [0,1], which corresponds to the value range of vegetation cover.

[0100] We use the Markov chain Monte Carlo method to draw samples from the posterior probability distribution. The number of samplings is set to 5000, and the first 1000 samples are discarded to eliminate initial value bias, accelerate chain convergence, and improve the robustness of the posterior estimate.

[0101] Figures 3(a) to 3(i) show the posterior probability distribution of vegetation cover in nine plots according to embodiments of the present disclosure.

[0102] As shown in Figures 3(a) to 3(i), the horizontal axis represents the possible values ​​of vegetation cover; the vertical axis represents the probability density corresponding to that value. Each figure represents the posterior probability distribution of vegetation cover for the corresponding plot. The "peak-shaped pattern" in the figure reflects the probability concentration trend of the vegetation cover in that plot. The horizontal axis corresponding to the peak value is the most likely value of vegetation cover for that plot (similar to the "optimal estimate"). The width of the figure reflects the uncertainty of the vegetation cover. The narrower the figure, the more concentrated the possible values ​​of vegetation cover are across multiple measurements, and the lower the uncertainty. The wider the figure, the greater the fluctuation of the vegetation cover value across multiple measurements, and the higher the uncertainty. Figures 3(a) to 3(i) show the "most likely value" and "uncertain range" of vegetation cover for each of the nine plots. For example, if the peak value of the posterior probability distribution of a certain sample plot is around 0.1 and the graph is relatively narrow, it indicates that the coverage of the sample plot is about 10% and the results are very stable; if the peak value of another sample plot is around 0.09 and the graph is slightly wider, it indicates that its coverage is about 9%, but the results fluctuate slightly more.

[0103] Figure 3(a) is the posterior probability distribution of the coverage of the sample plot 0831-1 by the needle-punch method; Figure 3(b) is the posterior probability distribution of the coverage of the sample plot S3 by the needle-punch method; Figure 3(c) is the posterior probability distribution of the coverage of the sample plot M1 by the needle-punch method; Figure 3(d) is the posterior probability distribution of the coverage of the sample plot M3 by the needle-punch method; Figure 3(e) is the posterior probability distribution of the coverage of the sample plot Z3 by the needle-punch method; Figure 3(f) is the posterior probability distribution of the coverage of the sample plot B1 by the needle-punch method; Figure 3(g) is the posterior probability distribution of the coverage of the sample plot B3 by the needle-punch method; Figure 3(h) is the posterior probability distribution of the coverage of the sample plot S1 by the needle-punch method; Figure 3(i) is the posterior probability distribution of the coverage of the sample plot S3 by the needle-punch method.

[0104] As shown in Figure 3(a), the horizontal axis represents the possible values ​​of vegetation cover; the vertical axis represents the probability density corresponding to that value. The higher the probability density, the more likely the value of the vegetation cover is to be the true value. In Figure 3(a), the height of the bars corresponds to the probability density of different coverage values. The higher the bar, the greater the probability of that coverage value. The dashed line (marked as the mean vegetation cover) is the center of the posterior probability distribution, representing the optimal estimate of the vegetation cover of this plot. In Figure 3(a), the mean corresponds to approximately 0.092, which is about 9.2% of the coverage. The overall narrow peak distribution indicates that the possible values ​​of the coverage of this plot are concentrated around the mean, with low uncertainty and high stability of the observation results.

[0105] Based on the target prior surface ecological parameters of vegetation cover, the mean, standard deviation, and standard uncertainty of the posterior samples are calculated to measure the uncertainty of the parameters. Table 2 shows the quantification of the uncertainty of vegetation cover by the needle-punch method in nine sample plots.

[0106] Table 2. Uncertainty of Coverage by Needle Puncture Method in Nine Plots

[0107]

[0108] Taking plot 0831-1 as an example, the arithmetic mean is 0.092, the variance is 1.301, the standard deviation is 0.001, and the final calculated standard uncertainty of vegetation cover is 1.803.

[0109] When the target index calculation model is the pixel-based binary method, the observed data is the normalized vegetation index, and the model output is the predicted vegetation coverage under the sampling conditions.

[0110] Based on the Bayesian model, the posterior sample index calculation results of the model input variables are obtained through the Markov chain Monte Carlo method, including: posterior probability distribution of vegetation cover; posterior probability distribution of comprehensive calculation error; posterior probability distribution of the probability density function parameter of comprehensive calculation error; posterior probability distribution of normalized vegetation index value of pure vegetation pixels; and posterior probability distribution of normalized vegetation index value of pure bare soil pixels.

[0111] Specifically, the target indicator calculation model can also refer to formula (6).

[0112] Formula (6)

[0113] in, Normalized Difference Vegetation Index; Vegetation coverage is the percentage of the area occupied by vegetation in a pixel. The normalized vegetation index value is the value of a pure vegetation pixel. The normalized vegetation index value is for a pure bare soil pixel. This is the first comprehensive measurement error parameter, and ; The standard deviation of the first comprehensive measurement error parameter describes the source of uncertainty error between the measured value and the observed value of the target indicator when measuring vegetation cover.

[0114] Example 2

[0115] We acquired optical remote sensing data from drones and used the pixel-based binary classification method to calculate vegetation cover. Taking the Ephedra plot among the multiple plots acquired by drones during the survey as an example.

[0116] Figure 4(a) shows a multispectral image of a target area obtained by using the infrared band at a flight altitude of 30m according to an embodiment of the present disclosure. Figure 4(b) shows a multispectral image of a target area obtained by using the near-infrared band at a flight altitude of 100m according to an embodiment of the present disclosure.

[0117] Figure 5(a) shows a normalized vegetation index (NDI) image of a target area obtained by infrared observation at a flight altitude of 30m according to an embodiment of the present disclosure. Figure 5(b) shows a normalized vegetation index (NDI) image of a target area obtained by near-infrared observation at a flight altitude of 100m according to an embodiment of the present disclosure.

[0118] Preprocessing of UAV optical remote sensing data includes sensor noise suppression, radiometric correction of shaded areas using homomorphic filtering, and radiometric calibration; calculation of NDVI values. Vegetation cover of the surveyed plots is calculated. Key statistics such as data distribution type, mean, standard deviation, median, maximum, and minimum values ​​are obtained.

[0119] Figure 6 A vegetation cover sampling chain according to another embodiment of the present disclosure is shown.

[0120] like Figure 6 As shown, the horizontal axis represents the number of samplings (4000 in total); the vertical axis represents the observed values ​​of vegetation cover (FVC) (the fluctuation range includes negative values, which are actually temporary anomalies caused by observation errors). The observed values ​​fluctuate overall, reflecting random errors in the sampling process or the heterogeneity of vegetation spatial distribution within the sample plots.

[0121] Table 3 shows the statistics of the minimum, maximum, mean, and standard deviation of the NDVI data from the sample plots.

[0122] Table 3. Distribution of NDVI data in Ephedra 1 plot

[0123]

[0124] Set priors m=[FVC, ],in, N represents a normal distribution, and [i, j] represents the x and y coordinates of a pixel. This represents the standard deviation of vegetation cover calculated using the pixel binarization method for the entire image. U represents a uniform distribution. This represents the upper bound of a uniform distribution, specifically the NDVI value corresponding to a cumulative percentage of 94% across the entire image. This represents the upper bound of the uniform distribution, i.e., the NDVI value corresponding to a cumulative NDVI percentage of 96%. U represents a uniform distribution. This represents the upper bound of a uniform distribution, here corresponding to the NDVI value when the cumulative percentage of NDVI distribution across the entire image is 4%. This represents the upper bound of the uniform distribution, i.e., the NDVI value corresponding to a cumulative NDVI percentage of 6%. The pixel dichotomy method is introduced into the likelihood function: Wherein, NDVI represents the observations obtained based on remote sensing technology. A priori distribution is defined for the error term, and model structure error parameters are introduced to construct a Bayesian model. Using the concept of conjugate distributions in probability theory to select... To ensure and Maintain the same family of distributions. One such conjugate distribution is the inverse gamma distribution parameterized by shape parameter α and scale parameter β. Choose α = 1 and β = 0.1 so that the samples of σ are generally located in the range [0,1], which corresponds to the value range of vegetation cover.

[0125] We use the Markov chain Monte Carlo method to draw samples from the posterior probability distribution. The number of samplings is set to 5000, and the first 1000 samples are discarded to eliminate initial value bias, accelerate chain convergence, and improve the robustness of the posterior estimate.

[0126] Figure 7 A visualization of the spatial distribution of vegetation cover according to an embodiment of the present disclosure is shown.

[0127] like Figure 7 As shown, the background is a grayscale image of the sample plot. The light-colored areas represent vegetation, and the dark-colored areas represent bare soil. The darker sampling points are random sampling points, and the color bars on the right correspond to the FVC values ​​(0~1). This shows the spatial distribution characteristics of FVC within the sample plot, that is, the light-colored areas (vegetation) correspond to higher FVC values. The random sampling points mark the sampling locations, intuitively reflecting the spatial heterogeneity of vegetation cover.

[0128] Figure 8 A posterior probability distribution of vegetation cover according to another embodiment of this disclosure is shown.

[0129] like Figure 8 As shown, the horizontal axis represents the possible values ​​of FVC, and the vertical axis represents the probability density. The middle dashed line represents the mean of FVC (0.21), and the dashed lines on the left and right represent the mean ± 1 standard deviation (-0.01~0.43). The observed FVC value of one random point is (0.19), with a standard deviation of 0.22. The probability describes the most likely values ​​and uncertainty range of FVC. The mean of 0.21 is the optimal estimate of FVC, and the standard deviation of 0.22 reflects the uncertainty of the estimate. The distribution shape reflects the central tendency of FVC values.

[0130] The uncertainty of the parameters is measured by calculating the mean and standard deviation of the posterior samples. Table 4 shows the quantification of the FVC uncertainty using the pixel-dichotomy method for the sample plots.

[0131] Table 4. Uncertainty of vegetation cover in Ephedra 1 sample plot

[0132]

[0133] The posterior mean is 0.2547, the variance is 0.0479, the standard deviation is 0.2190, and the calculated standard uncertainty is 0.0035.

[0134] Figure 9A A sequence diagram of posterior mean vegetation cover according to an embodiment of the present disclosure is shown. Figure 9B A sequence diagram of the posterior standard deviation of vegetation cover according to an embodiment of the present disclosure is shown.

[0135] like Figure 9A As shown, the horizontal axis represents the cell index (100 cells in total), and the vertical axis represents the posterior mean of vegetation cover. Figure 9A The differences and fluctuations in the optimal estimates of vegetation cover across multiple units are shown. The mean vegetation cover values ​​vary across different units, reflecting the differences in cover across different areas within the sample plots.

[0136] like Figure 9B As shown, the horizontal axis is the cell index, and the vertical axis is the posterior standard deviation of vegetation cover. Figure 9B This section illustrates the differences in uncertainty in vegetation cover estimation results across multiple units. The fluctuations in standard deviation reflect the varying observation / modeling uncertainties across different units; some units have higher standard deviations, indicating slightly lower reliability of their vegetation cover estimation results.

[0137] According to embodiments of this disclosure, when the surface ecological parameter is forest biomass, the model input variables of the target index calculation model include: forest biomass; tree height; second error parameter; and the target index value output by the target index calculation model includes diameter at breast height (DBH).

[0138] According to embodiments of this disclosure, when the surface ecological parameter is forest biomass, multiple candidate prior parameter combinations are input into the target index calculation model, and the index calculation results of each candidate prior parameter combination are output. This includes: calculating the difference between the prior forest biomass and the second error parameter to obtain an estimated value of forest biomass; calculating the product of the model prior coefficient, the model prior correction coefficient, and the allometric growth regulation factor of tree height to obtain a combined correction factor, wherein the allometric growth regulation factor of tree height is calculated based on the prior tree height; calculating the ratio of the estimated value of forest biomass to the combined correction factor to obtain a candidate baseline value for diameter at breast height (DBH); and calculating and outputting the calculated value of DBH based on the prior influence index of DBH on forest biomass and the candidate baseline value for DBH.

[0139] With forest biomass as the surface ecological parameter, prior distributions are set for the model input variables, including: prior distribution of forest biomass; prior distribution of tree height; and prior distribution of the comprehensive measurement error. A super-prior distribution is set for the prior distribution parameter of the comprehensive measurement error, where the distribution parameter refers to the probability density function parameter. The target index measurement model maps the factor vector to the observation space, and constructs an observation likelihood function to measure the statistical consistency between the observed data and the model output. The observed data is diameter at breast height (DBH), and the model output is the model prediction of DBH under the factor vector sampling conditions. Based on the Bayesian model, the posterior sample index measurement results of each model input variable are obtained through the Markov chain Monte Carlo method, including: posterior distribution of forest biomass; posterior distribution of tree height; posterior distribution of the comprehensive measurement error; and posterior distribution of the probability density function parameter of the comprehensive measurement error.

[0140] Specifically, the target indicator calculation model is based on formula (7).

[0141] Formula (7)

[0142] in, Diameter at breast height (DBH) is the diameter of a tree trunk at a height of 1.3 meters above the ground. For forest biomass; a) model coefficients used to calibrate the basic estimation order of biomass; b) index of the influence of diameter at breast height (DBH) on forest biomass; c) index of the influence of tree height on forest biomass; d) model correction coefficients. This is the second comprehensive calculation error parameter, and ; The standard deviation of the second comprehensive measurement error parameter describes the source of uncertainty error between the measured value and the observed value of the target indicator when measuring forest biomass.

[0143] Example 3

[0144] Tree height (H) and diameter at breast height (DBH) were measured through field surveys in forest plots. The collected data underwent preprocessing, including outlier removal, missing value completion, and error correction, to eliminate data anomalies caused by observational errors or recording biases. Based on statistical analysis methods, distribution characteristics of each input factor were learned and descriptive statistics were performed to analyze key statistics such as data distribution type, mean, and standard deviation.

[0145] Figure 10A A prior probability distribution diagram of tree height in forest biomass measurement according to an embodiment of the present disclosure is shown. Figure 10B A prior probability distribution of diameter at breast height (DBH) in forest biomass measurement according to an embodiment of this disclosure is shown.

[0146] according to Figure 10A and Figure 10B It can be seen that tree height and diameter at breast height (DBH) approximately follow a normal distribution. Therefore, we set the prior distribution of tree height as a normal distribution, where the expected value is the arithmetic mean of tree height and the standard deviation is the standard deviation of tree height; the prior distribution of DBH also follows a normal distribution, where the expected value is the arithmetic mean of DBH and the standard deviation is the standard deviation of DBH.

[0147] like Figure 10A As shown, the horizontal axis represents the possible values ​​of tree height, and the vertical axis represents the probability density; the middle dashed line represents the mean height (12.25m), and the dashed lines on the left and right sides represent the mean ± 1 standard deviation (9.00~15.49m), with a standard deviation of 3.25m; the shaded area covered by the dashed lines on the left and right sides represents the range of the mean ± 1 standard deviation, which represents the main distribution range of the height. Figure 10A The study describes the most likely values ​​and uncertain range of tree height. The mean of 12.25m is the optimal estimate of tree height, and the standard deviation of 3.25m reflects the degree of height fluctuation. The distribution pattern reflects the central tendency of height.

[0148] like Figure 10B As shown, the horizontal axis represents the possible values ​​of chest diameter, and the vertical axis represents the probability density; the middle dashed line represents the mean chest diameter (15.45cm), and the dashed lines on the left and right sides represent the mean ± 1 standard deviation (10.38~20.51cm), with a standard deviation of 5.07cm; the shaded area covered by the dashed lines on the left and right sides represents the main distribution range of chest diameter. Figure 10B The most likely values ​​and uncertainties for diameter at breast height (DBH) of trees are described. The mean of 15.45 cm is the best estimate of DBH, the standard deviation of 5.07 cm reflects the heterogeneity of DBH, and the distribution pattern reflects the concentration of DBH.

[0149] The allometric growth equation of the corresponding tree species in pure stands was used as the forest biomass measurement model, according to formula (7). The AGB value per tree was calculated, and key statistics such as distribution type, mean, and standard deviation of AGB per tree in the sample plot were analyzed. The AGB distribution of the stand follows a normal distribution, with its expected value being the arithmetic mean of the calculated AGB and its standard deviation being the calculated standard deviation of AGB for the sample plot. Considering the comprehensive measurement error model, it can be written as: ,in, The second comprehensive measurement error parameter is represented by the inverse gamma distribution, which is used as a hyperparameter. The conjugate priors of the probability density function have shapes controlled by shape parameter α and scale parameter β. The default values ​​are α = 3 and β = 1. This ensures that the variance of the sampling results is typically distributed within [0, AGB]. std The interval is defined. The likelihood function is updated to... .

[0150] Figure 11A forest biomass sampling chain according to an embodiment of this disclosure is shown. For example... Figure 11 As shown, the horizontal axis represents the number of samplings (approximately 17,500 in total), and the vertical axis represents the observed parameter values ​​(range 26–36). After multiple samplings of forest biomass, the observed values ​​are concentrated between 28 and 36, reflecting random errors in the sampling process or parameter differences between individual trees.

[0151] The random sampling number was set to 5000 times, with the first 1000 samples discarded to eliminate initial value bias, accelerate chain convergence, and improve the robustness of the posterior estimate. Each sampling step involved: first generating candidate values ​​from the prior distributions of H and DBH; then calculating AGB candidate values ​​using the allometric growth equation; and finally, evaluating the degree of match between the candidate value and the observed AGB data based on the likelihood function to determine whether to accept the candidate value. Ultimately, the posterior sample chain for AGB consisted of 4000 valid samples.

[0152] The uncertainty of the parameters is measured by calculating the mean, standard error, and uncertainty (standard deviation) of the posterior sample. The uncertainty of the input parameters (the standard deviations of H and DBH) is passed to the AGB estimate through the model and manifested as the uncertainty of the posterior parameters, as shown in Table 5.

[0153] Table 5. Uncertainty regarding forest biomass in twelve sample plots

[0154]

[0155] Figure 12 The posterior probability distribution of forest biomass according to an embodiment of this disclosure is shown.

[0156] like Figure 12 As shown, the horizontal axis represents the height (H) corresponding to AGB, and the vertical axis represents the probability density. The dashed line represents the mean height corresponding to AGB, indicating the height level corresponding to the optimal estimate of AGB. Combined with the distribution of tree height, the concentrated range of AGB values ​​is shown, with the mean position representing the parameter level corresponding to the optimal estimate of AGB. Taking plot GH144 as an example, the arithmetic mean of biomass in plot GH144 is 31.1536, the variance is 2.2425, the standard error is 0.0237, and the uncertainty is 1.4975.

[0157] The embodiments of this disclosure have been described above. However, these embodiments are for illustrative purposes only and are not intended to limit the scope of this disclosure. Although various embodiments have been described above, this does not mean that the measures in the various embodiments cannot be used advantageously in combination. Various substitutions and modifications can be made by those skilled in the art without departing from the scope of this disclosure, and all such substitutions and modifications should fall within the scope of this disclosure.

Claims

1. A method for uncertainty analysis in the measurement of surface ecological parameters, comprising: Obtain observations of target indicators for the target region; Multiple candidate prior parameter combinations are obtained by sampling from the prior probability distribution of the measurement factors included in the target indicator measurement model. The measurement factors include: model input variables and error parameters. The model input variables include prior surface ecological parameters. The multiple candidate prior parameter combinations are respectively input into the target index calculation model, and the index calculation results of each of the multiple candidate prior parameter combinations are output. The index calculation results include the calculated value of the target index. Based on the calculated value of the target indicator and the observed value of the target indicator, a target prior parameter combination is selected from the plurality of candidate prior parameter combinations, wherein the target prior parameter combination includes target prior surface ecological parameters; The uncertainty of the surface ecological parameters is calculated based on the probability distribution determined by the prior surface ecological parameters of the target.

2. The method according to claim 1, wherein, Based on the calculated value and observed value of the target indicator, a target prior parameter combination is selected from the plurality of candidate prior parameter combinations, including: A likelihood function is constructed based on the measured value and the observed value of the target indicator; Calculate the joint posterior probability of any set of candidate prior parameter combinations based on the likelihood function and the prior probability distribution; The joint posterior probability is compared with the joint posterior probability calculated in the previous iteration, and at least one set of target prior parameter combinations is determined based on the comparison result.

3. The method according to claim 1, wherein, The prior probability distribution of the error parameter is determined by the following operation: Obtain the observation dataset of the target index obtained by conducting multiple observations of the target area; Based on the probability distribution type of the observed dataset, determine the prior probability distribution; The value sampled from the prior probability distribution is determined as the variance of the prior probability distribution of the error parameter; Based on the variance, determine the prior probability distribution of the error parameter.

4. The method according to claim 3, wherein, Based on the probability distribution type of the observed dataset, the prior probability distribution is determined, including: Based on the probability distribution type, distribution parameters, and conjugate distribution of the observed dataset, determine the prior probability distribution type; The super-prior probability distribution is determined based on the preset distribution parameters and the prior probability distribution type.

5. The method according to claim 4, wherein, The probability distribution of the observed dataset is either a normal distribution or a uniform distribution; The step of determining the type of the hyperprior probability distribution based on the probability distribution type, distribution parameters, and conjugate prior distribution of the observed dataset includes: When the probability distribution of the observed dataset is a normal distribution, the distribution type of the inverse gamma distribution of the conjugate distribution is determined by the variance of the distribution parameter; When the probability distribution of the observed dataset is uniform, the Pareto distribution of the conjugate distribution is determined to be the distribution type of the super-prior probability distribution by the upper bound of the distribution parameter.

6. The method according to claim 1, wherein, The surface ecological parameters include vegetation cover or forest biomass.

7. The method according to claim 6, wherein, When the surface ecological parameter is vegetation cover, the model input variables of the target index calculation model include: Vegetation coverage, which characterizes the proportion of area occupied by vegetation in a pixel; Normalized vegetation index value of pure vegetation pixels; Normalized Difference Vegetation Index (NDI) values ​​for pure bare soil pixels; First error parameter; The target indicator values ​​output by the target indicator calculation model include: Normalized Difference Vegetation Index (NDVI).

8. The method according to any one of claims 6 or 7, wherein, When the surface ecological parameter is vegetation cover, the multiple candidate prior parameter combinations are respectively input into the target index calculation model, and the index calculation results of each of the multiple candidate prior parameter combinations are output, including: The vegetation index contribution value is obtained by multiplying the prior vegetation cover by the prior normalized vegetation index value of the prior pure vegetation pixel. The bare soil index contribution value is obtained by multiplying the prior bare soil coverage by the prior normalized vegetation index value of the prior bare soil pixel. The prior bare soil coverage is determined based on the prior vegetation coverage. The contribution values ​​of the vegetation index, the contribution value of the bare soil index, and the first error parameter are summed to calculate and output the measured value of the normalized vegetation index.

9. The method according to claim 6, wherein, When the surface ecological parameter is forest biomass, the model input variables of the target index calculation model include: Forest biomass; Tree height; Second error parameter; The target indicator value output by the target indicator calculation model includes chest diameter.

10. The method according to any one of claims 6 or 9, wherein, When the surface ecological parameter is forest biomass, the multiple candidate prior parameter combinations are input into the target index calculation model, and the index calculation results of each of the multiple candidate prior parameter combinations are output, including: The difference between the prior forest biomass and the second error parameter is calculated to obtain an estimated value of forest biomass; The combined correction factor is obtained by multiplying the model prior coefficients, the model prior correction coefficients, and the allometric growth regulation factor of tree height, wherein the allometric growth regulation factor of tree height is calculated based on the prior tree height. The ratio of the estimated forest biomass to the combined correction factor is calculated to obtain the candidate baseline value for diameter at breast height (DBH). The calculated value of the diameter at breast height (DBH) is calculated and output based on the prior influence index of DBH on forest biomass and the candidate baseline value of DBH.

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