Quantification method of unfrozen water content in frozen soil based on hierarchical bayes

Abstract

This invention discloses a method for quantifying the unfrozen water content in permafrost based on hierarchical Bayesian methods, comprising the following steps: Step 1, establishing a hierarchical SFCC framework; Step 2, establishing a data layer and its corresponding likelihood function; Step 3, establishing a parameter layer and its corresponding prior distribution; Step 4, establishing a hyperparameter layer and its corresponding hyperprior distribution; Step 5, correcting the parameters in the heat conduction equation based on the freezing process; Step 6, establishing an SFCC model by coupling the improved heat conduction equation; Step 7, obtaining the posterior distribution using the Markov chain Monte Carlo method and taking a 95% confidence interval. This invention, based on a hierarchical Bayesian framework, combines corrected specific heat and thermal conductivity with the introduction of phase change heat and convective heat into the heat conduction equation, constructing a freezing characteristic curve model capable of quantifying the unfrozen water content with a very small number of measured points. This invention has a mature mathematical foundation and can provide a new method for obtaining the unfrozen water content during the freezing process of permafrost in cold regions.

Description

Technical Field

[0001] This invention relates to a method for quantifying the unfrozen water content in permafrost, specifically a method for quantifying the unfrozen water content in permafrost based on hierarchical Bayesian methods. Background Technology

[0002] With global warming and the advancement of engineering construction in cold regions, a large amount of infrastructure is being built in permafrost areas. These regions are widely covered by permafrost and seasonally frozen soil, and the freeze-thaw cycle caused by low temperatures within the permafrost has become a global scientific problem. Engineering projects in cold regions, such as building foundations, bridges, tunnels, highways, and railways, are directly affected by the freeze-thaw cycle of the soil. Under this effect, the main cause of engineering defects is the frost heave and thaw settlement phenomenon triggered by the phase change of water driven by temperature gradients. Therefore, accurately determining the unfrozen water content within the permafrost is crucial in practical engineering projects.

[0003] However, existing inventions for obtaining unfrozen water content mostly focus on fitting SFCC (Freezing Temperature Coefficient of Combustion) after measuring a large number of experimental points, or on calculating based on complex expressions established by a certain theory. These methods fail to reduce the cost and time required for experimental measurements or the application threshold of theoretical formulas from an engineering application perspective. Specifically, existing technologies have the following shortcomings: First, laboratory methods for measuring unfrozen water content, such as nuclear magnetic resonance (NMR), are costly or time-consuming; second, in-situ methods for measuring unfrozen water content, such as time domain reflectance (TDR), have significant systematic errors; third, the SFCC expressions established by theory contain complex parameters that are difficult to obtain and have a high barrier to entry for use and understanding. Due to these shortcomings, there is currently a lack of a rapid method for obtaining unfrozen water content in permafrost in cold regions that can reduce costs and technical barriers.

[0004] This invention aims to solve the aforementioned problems. Specifically, based on a layered Bayesian framework, it fully utilizes empirical information from similar sites with similar geological features and introduces the heat conduction equation during the freezing process as a physical constraint. Ultimately, it establishes a freezing characteristic curve model capable of quantifying the unfrozen water content under conditions with very few measured points. This invention will facilitate the rapid determination of the unfrozen water content in permafrost in cold regions for engineering applications, providing convenience for practical engineering projects. Summary of the Invention

[0005] To address the aforementioned issues, this invention discloses a method for quantifying the unfrozen water content in permafrost based on layered Bayesian methods. Compared to other experimental and theoretical methods, the method described in this invention is low in cost and easy to apply.

[0006] The objective of this invention is achieved through the following technical solution:

[0007] A method for quantifying the unfrozen water content in permafrost based on hierarchical Bayesian methods includes the following steps:

[0008] Step 1: Establish the SFCC layered framework;

[0009] Step 1 includes the following sub-steps:

[0010] a. In order to introduce the Bayesian framework into the freezing process of the freeze-thaw cycle of permafrost in cold regions, the Freezing Characteristic Curve (SFCC) is stratified, and considering the three different data types in the SFCC, it is divided as follows.

[0011] (1)

[0012] In the formula, Here, T represents the unfrozen water content, and T represents the temperature, indicating the data layer of the model; z represents all the parameters in the model, indicating the parameter layer. It is used to characterize the errors in experimental measurements and theoretical derivations, and is one of the hyperparameters. Other hyperparameters are derived from the distribution function of the parameters.

[0013] b. To integrate prior information from similar sites, SFCC is estimated based on a Bayesian framework, quantifying the uncertainty in SFCC. Representing all parameters to be updated, this function embeds all data based on measurements and priors into a posterior probability density function (PDF), which can be represented as follows.

[0014] (2)

[0015] in, is the likelihood function, which characterizes the degree of fit between the measured data and a given set of parameters, corresponding to the data layer; It includes prior distribution and hyperprior distribution, corresponding to the parametric layer and hyperparametric layer, respectively; c is a normalization constant, ensuring... The integral is 1.

[0016] Step 2: Establish the data layer and its corresponding likelihood function;

[0017] Step 2 includes the following sub-steps:

[0018] a. For a set of measurements Unfrozen water content If the value is positive, its distribution follows a normal distribution.

[0019] (3)

[0020] in, For a given value with a mean of zero and a standard deviation of , Gaussian random variables.

[0021] b. Consider all measurements from the target site and similar sites, with the overall likelihood function being...

[0022] (4)

[0023] In the formula, For the number of similar sites, the first The venue represents the target venue; This represents the number of available measurements in the i-th site.

[0024] c. Due to and All parameters are unknown and awaiting update. It is represented as.

[0025] (5)

[0026] Step 3: Establish the parameter layer and its corresponding prior distribution;

[0027] Step 3 includes the following sub-steps:

[0028] a. Treat parameters from different sites as realizations of the same prior distribution. Taking the absolute values ​​of all parameters, their prior distribution follows a log-normal distribution:

[0029] (6)

[0030] In the formula, and These are the mean and standard deviation of the parameter, respectively. Characterizing the similarity between different sites, while This reflects the variability of z across different locations.

[0031] b. The parameters of a site are independent of each other, and the joint prior PDF of different parameters in site i is:

[0032] (7)

[0033] Where n is the number of model parameters.

[0034] c. and These are new parameters, or hyperparameters, derived from the prior PDF. These hyperparameters are also unknown and awaiting updating. Represented as:

[0035] (8)

[0036] Based on all the parameters currently pending updates, the joint prior PDF is:

[0037] (9)

[0038] in, , and This represents the hyperprior distribution corresponding to the hyperparameter layer.

[0039] Step 4: Establish the hyperparameter layer and its corresponding hyperprior distribution;

[0040] Step 4 includes the following sub-steps:

[0041] a. Hyperparameters Set as an interval A uniform distribution on the surface is represented as .

[0042] (10)

[0043] b. Hyperparameters Suppose it is a seminormal distribution with a mean of zero:

[0044] (11)

[0045] In the formula, It is the reciprocal of the variance. The smaller, The less prior information there is.

[0046] c. Hyperparameters Assuming an exponential distribution:

[0047] (12)

[0048] Step 5: Establish the extended likelihood function of the coupled improved heat conduction equation;

[0049] Step 5 includes the following sub-steps:

[0050] a. During the process of water transforming into ice, the thermal properties of the soil change as the unfrozen water content decreases due to the difference in thermal properties between water and ice. Simultaneously, the water-to-ice phase transition releases heat of phase transition, and the pressure changes caused by the change in water and ice content induce the flow of water and air, generating convective heat during this flow.

[0051] b. Determine the amount of heat conducted through the soil unit:

[0052] (13)

[0053] In the formula, Enthalpy change represents the difference in heat transferred into and out of the soil unit; and Passing through a plane on the unit soil respectively ( and The heat of soil; C is the specific heat of the soil. This refers to the soil density. and These are phase change heat and convection heat, respectively.

[0054] c. Determine the heat difference between the soil element and the element that is transferred in and out according to Fourier's law of cooling. (Axially through the plane) and plane The calories are as follows:

[0055] (14)

[0056] (15)

[0057] in, is the thermal conductivity of the soil.

[0058] d. Determine the variation of specific heat with unfrozen water content. Since specific heat has the property of being a weighted average of the soil's components, the corrected specific heat is expressed as:

[0059] (14)

[0060] in, These are the volume fractions of water, gas, and ice, respectively. These are the specific heats of soil particles, water, air, and ice, respectively, and are generally taken as... , kJ / (kg·°C).

[0061] e. Determine the variation of thermal conductivity with unfrozen water content. The thermal conductivity of soil is controlled by the thermal conductivity and volume fraction of its components. The corrected thermal conductivity is expressed as:

[0062] (15)

[0063] in, These are the thermal conductivity coefficients of soil particles, water, air, and ice, respectively. , W / (m·°C).

[0064] f. The heat of phase transition released by the water-ice phase transition is determined by the magnitude and content change of the latent heat of phase transition.

[0065] (16)

[0066] Where L is the latent heat of phase change, taken as 334.56 kJ / kg; Let the density of water be 1 kg / m³. 3 .

[0067] g. Convective heat is the heat generated by the migration of water and air through the pores of soil.

[0068] (17)

[0069] Where h is the heat transfer coefficient, the magnitude of which depends on the medium in which the flow occurs, W / (m²). 2 • °C); A is the heat exchange surface area, which in soil is the surface area of ​​soil pores, in m³. 2 .

[0070] h. During the freezing process, the unknown properties of the soil particles remain unchanged:

[0071] (18)

[0072] i. Determine the heat conduction equation during the freezing process. Substituting the corrected specific heat and thermal conductivity, and introducing phase change heat and convection heat, we obtain the expression for the heat conduction equation during the freezing process:

[0073] (19)

[0074] Step 6: Couple the improved heat conduction equations to establish the SFCC model;

[0075] a. Couple the improved heat conduction equation and establish the extended likelihood function:

[0076] (20)

[0077] (twenty one)

[0078] The equation is implicit, characterized by: estimating the unfrozen water content through a mathematical framework. The values ​​represent all possible values ​​in the probability space, but unfrozen water contents above the initial moisture content and below zero do not conform to the actual freezing process. Therefore, the layered Bayesian framework needs to be constrained by the improved heat conduction equation to obtain the solution in the physical space. .

[0079] b. By embedding the extended likelihood function into a hierarchical Bayesian framework, a hierarchical model of soil freezing characteristic curves based on hierarchical Bayes is obtained:

[0080] (twenty two)

[0081] Step 7: Obtain the posterior distribution using the Markov chain Monte Carlo method and take the 95% confidence interval.

[0082] Step 7 includes the following sub-steps:

[0083] a. Obtain the posterior distribution using the Markov Chain Monte Carlo (MCMC) method. A large number of numerical solutions were obtained, and values ​​other than 2.5% and 97.5% of all solutions were removed, and a 95% confidence interval was taken.

[0084] The beneficial effects of this invention are:

[0085] This invention, based on a layered Bayesian framework and combined with the heat conduction equation during the freezing process, constructs an SFCC model capable of quantifying the unfrozen water content under conditions of minimal on-site measurement points. This method fully considers the limitations of experimental cost and time in practical engineering, and various types of data and parameters can be quickly obtained through simple geotechnical experiments. The quantitative framework provided by this invention can accurately predict the unfrozen water content of permafrost in cold regions during the freezing process, demonstrating significant application value. In addition to the aforementioned objectives, features, and advantages, this invention also has other objectives, features, and advantages. The invention will now be described in further detail with reference to the accompanying drawings. Attached Figure Description

[0086] Figure 1 This is a schematic diagram of the freezing characteristic curve in this invention.

[0087] Figure 2 This is a schematic diagram of the frozen feature curve layering framework and the Bayesian framework in this invention.

[0088] Figure 3 This is a schematic diagram of heat conduction in the soil during the freezing process in this invention.

[0089] Figure 4 This is a distribution diagram of the unfrozen water content in the soil sample used in this invention.

[0090] Figure 5 This is a graph showing the prediction results of the model in this invention.

[0091] Figure 6 This is a comparison chart of the measured and predicted unfrozen water content of soil samples in this invention.

[0092] Detailed Implementation Instructions

[0093] A method for quantifying the unfrozen water content in permafrost based on hierarchical Bayesian methods includes the following steps:

[0094] Step 1: Establish the SFCC layered framework;

[0095] Step 1 includes the following sub-steps:

[0096] a. To introduce a Bayesian framework into the freezing process of freeze-thaw cycles in cold-region permafrost, the freezing characteristic curve (SFCC) is layered, such as... Figure 1As shown, an SFCC curve has a data layer consisting of temperature and unfrozen water content, a parameter layer consisting of curve parameters, and a hyperparameter layer derived from the distribution function of the parameters. Considering the three levels of data in SFCC, we can divide them as follows.

[0097] (1)

[0098] In the formula, Here, T represents the unfrozen water content, and T represents the temperature, indicating the data layer of the model; z represents all the parameters in the model, indicating the parameter layer. It is used to characterize the errors in experimental measurements and theoretical derivations, and is one of the hyperparameters. Other hyperparameters are derived from the distribution function of the parameters.

[0099] b. To integrate prior information from similar sites, SFCC is estimated based on a Bayesian framework, quantifying the uncertainty in SFCC. Representing all parameters to be updated, this function embeds all data based on measurements and priors into a posterior probability density function (PDF), which can be represented as follows.

[0100] (2)

[0101] in, is the likelihood function, which characterizes the degree of fit between the measured data and a given set of parameters, corresponding to the data layer; It includes prior distribution and hyperprior distribution, corresponding to the parametric layer and hyperparametric layer, respectively; c is a normalization constant, ensuring... The integral is 1. The correspondence between different levels in SFCC and the Bayesian framework is as follows: Figure 2 As shown.

[0102] Step 2: Establish the data layer and its corresponding likelihood function;

[0103] Step 2 includes the following sub-steps:

[0104] a. For a set of measurements Unfrozen water content If the value is positive, its distribution follows a normal distribution.

[0105] (3)

[0106] in, For a given value with a mean of zero and a standard deviation of , Gaussian random variables.

[0107] b. Consider all measurements from the target site and similar sites, with the overall likelihood function being...

[0108] (4)

[0109] In the formula, For the number of similar sites, the first The venue represents the target venue; This represents the number of available measurements in the i-th site.

[0110] c. Due to and All parameters are unknown and awaiting update. It is represented as.

[0111] (5)

[0112] Step 3: Establish the parameter layer and its corresponding prior distribution;

[0113] Step 3 includes the following sub-steps:

[0114] a. Treat parameters from different sites as realizations of the same prior distribution. Taking the absolute values ​​of all parameters, their prior distribution follows a log-normal distribution:

[0115] (6)

[0116] In the formula, and These are the mean and standard deviation of the parameter, respectively. Characterizing the similarity between different sites, while This reflects the variability of z across different locations.

[0117] b. The parameters of a site are independent of each other, and the joint prior PDF of different parameters in site i is:

[0118] (7)

[0119] Where n is the number of model parameters.

[0120] c. and These are new parameters, or hyperparameters, derived from the prior PDF. These hyperparameters are also unknown and awaiting updating. Represented as:

[0121] (8)

[0122] Based on all the parameters currently pending updates, the joint prior PDF is:

[0123] (9)

[0124] in, , and This represents the hyperprior distribution corresponding to the hyperparameter layer.

[0125] Step 4: Establish the hyperparameter layer and its corresponding hyperprior distribution;

[0126] Step 4 includes the following sub-steps:

[0127] a. Hyperparameters Set as an interval A uniform distribution on the surface is represented as .

[0128] (10)

[0129] b. Hyperparameters Suppose it is a seminormal distribution with a mean of zero:

[0130] (11)

[0131] In the formula, It is the reciprocal of the variance. The smaller, The less prior information there is.

[0132] c. Hyperparameters Assuming an exponential distribution:

[0133] (12)

[0134] Step 5: Establish the extended likelihood function of the coupled improved heat conduction equation;

[0135] Step 5 includes the following sub-steps:

[0136] a. During the process of water transforming into ice, the thermal properties of the soil change as the unfrozen water content decreases due to the difference in thermal properties between water and ice. Simultaneously, the water-to-ice phase transition releases heat of phase transition, and the pressure changes caused by the change in water and ice content induce the flow of water and air, generating convective heat during this flow.

[0137] b. Determine the amount of heat conducted through the soil unit:

[0138] In equation (13), Enthalpy change represents the difference in heat transferred into and out of the soil unit; and Passing through a plane on the unit soil respectively ( and The heat of soil; C is the specific heat of the soil. This refers to the soil density. and These are phase change heat and convection heat, respectively.

[0139] c. Determine the heat difference between the soil elements entering and leaving the soil element according to Fourier's law of cooling. For example... Figure 3 As shown in (a), it passes through the plane along the axial direction. and plane The calories are as follows:

[0140] (14)

[0141] (15)

[0142] in, is the thermal conductivity of the soil.

[0143] d. Determine the variation of specific heat with unfrozen water content. Since specific heat has the property of being a weighted average of the soil's components, the corrected specific heat is expressed as:

[0144] (14)

[0145] in, These are the volume fractions of water, gas, and ice, respectively. These are the specific heats of soil particles, water, air, and ice, respectively, and are generally taken as... , kJ / (kg·°C).

[0146] e. Determine the variation of thermal conductivity with unfrozen water content. The thermal conductivity of soil is controlled by the thermal conductivity and volume fraction of its components. The corrected thermal conductivity is expressed as:

[0147] (15)

[0148] in, These are the thermal conductivity coefficients of soil particles, water, air, and ice, respectively, and are generally taken as... , W / (m·°C).

[0149] f. such as Figure 3 As shown in (b), water transforms into ice upon freezing. The heat of phase transition released during the water-ice phase transition is determined by both the magnitude of the latent heat of phase transition and the change in its content.

[0150] (16)

[0151] Where L is the latent heat of phase change, taken as 334.56 kJ / kg; Let the density of water be 1 kg / m³. 3 .

[0152] g. Changes in soil pore pressure due to variations in water and ice content lead to the migration of water and ice, such as... Figure 3 (b) Convection heat is the heat generated by the migration of water and air in the pores of soil.

[0153] (17)

[0154] Where h is the heat transfer coefficient, the magnitude of which depends on the medium in which the flow occurs, W / (m²). 2 • °C); A is the heat exchange surface area, which in soil is the surface area of ​​soil pores, in m³. 2 .

[0155] h. During the freezing process, the unknown properties of the soil particles remain unchanged:

[0156] (18)

[0157] i. Determine the heat conduction equation during the freezing process. Substituting the corrected specific heat and thermal conductivity, and introducing phase change heat and convection heat, we obtain the expression for the heat conduction equation during the freezing process:

[0158] (19)

[0159] Step 6: Couple the improved heat conduction equations to establish the SFCC model;

[0160] Step 6 includes the following sub-steps:

[0161] a. Couple the improved heat conduction equation and establish the extended likelihood function:

[0162] (20)

[0163] (twenty one)

[0164] The equation is implicit, characterized by: estimating the unfrozen water content through a mathematical framework. The values ​​represent all possible values ​​in the probability space, but unfrozen water contents above the initial moisture content and below zero do not conform to the actual freezing process. Therefore, the layered Bayesian framework needs to be constrained by the improved heat conduction equation to obtain the solution in the physical space. .

[0165] b. By embedding the extended likelihood function into a hierarchical Bayesian framework, a hierarchical model of soil freezing characteristic curves based on hierarchical Bayes is obtained:

[0166] (twenty two)

[0167] Step 7: Obtain the posterior distribution using the Markov chain Monte Carlo method and take the 95% confidence interval.

[0168] Step 7 includes the following sub-steps:

[0169] a. Obtain the posterior distribution using the Markov Chain Monte Carlo (MCMC) method. A large number of numerical solutions were obtained, and values ​​other than 2.5% and 97.5% of all solutions were removed, and a 95% confidence interval was taken.

[0170] Verification of unfrozen water content in frozen soil:

[0171] This invention evaluates the accuracy of the predicted unfrozen water content by combining theoretical derivation with experimental verification: first, the posterior distribution is calculated based on the stratified model of the freezing characteristic curve; then, the 95% confidence interval and its mean are taken; and finally, the data are compared and verified with the measured data.

[0172] This invention utilizes freezing test data from three sets of published literature on different types of soil samples, covering three typical soil types: clay, silty clay, and sand. The initial moisture content of the soil samples ranged from 10% to 30%, and the dry density ranged from 1.5 to 3 g / cm³. 3 The aforementioned data has undergone complete preliminary analysis and grouping in previous work, providing a clear and well-defined verification basis for the reliability assessment of this invention. The selected data covers typical working conditions across multiple dimensions, including soil type, initial moisture content, and dry density, and is highly representative. Detailed soil sample physical properties are systematically compiled as shown in Table 1, and detailed freezing test data are as follows: Figure 4 As shown.

[0173] This invention uses the coefficient of determination R 2 The root mean square error (RMSE) verifies the actual predictive performance of the theoretical model:

[0174] (twenty three)

[0175] In the formula, These are the measured value and the predicted value, respectively. This is the average of the measured values.

[0176] (twenty four)

[0177] Table 1. Basic physical properties of soil samples and site division.

[0178]

[0179] The coefficient of determination R of the model of this invention 2 All values ​​are above 0.98, and the root mean square error (RMSE) is less than 0.003, demonstrating excellent predictive performance.

[0180] like Figure 2 As shown, Figure 2 This diagram illustrates the hierarchical division of the frozen feature curve and the corresponding Bayesian framework for each level in this invention. It also describes the basic data layer in SFCC. Its corresponding likelihood function; the parameter layer used to depict the curve. The corresponding prior distribution; the hyperparameters derived from the distribution function of the parameter layer and their corresponding hyperprior distributions.

[0181] like Figure 3 (a)-(b) are schematic diagrams illustrating the freezing process of the soil described in this invention. They depict a) heat conduction within a soil unit during the freezing process; and b) the water-ice phase transition and water and gas migration phenomena occurring within the soil unit during freezing.

[0182] like Figure 4 The figure shows the soil sample freezing test data used in this invention. It describes the similar but different initial moisture content across different sites. During freezing (temperature decrease), the unfrozen water content decreases sharply within a narrow temperature range, and then gradually levels off after reaching a certain level. Throughout the process, the SFCC (soil condensation concentration) shows a similar pattern, but the specific values ​​differ, fully reflecting the similarities and variability between different sites.

[0183] like Figure 5 As shown, the 95% confidence intervals calculated based on this invention closely wrap around the measured values. The coefficient of determination (R²) of the mean of the 95% confidence interval with respect to the measured values ​​is... 2 The values ​​were all above 0.98, and the root mean square error (RMSE) was all less than 0.003, which statistically verified the reliability of the model.

[0184] like Figure 6 As shown, the vast majority of data points are closely distributed on both sides of the 1:1 line, intuitively demonstrating a high degree of agreement between the model's predicted values ​​and the measured values. This proves that the model of this invention can be applied to different types of permafrost and exhibits good predictive performance.

[0185] In summary, the method for quantifying the unfrozen water content of frozen soil in cold regions obtained by this invention can effectively predict the unfrozen water content at different temperatures for soil samples with different initial moisture contents and different soil types. Applying the method for quantifying the unfrozen water content of this invention to obtain the unfrozen water content of actual soil samples is feasible.

Claims

1. A method for quantifying the unfrozen water content in frozen soil based on hierarchical Bayesian methods, characterized in that, It consists of the following steps: Step 1: Establish the SFCC layered framework; Step 2: Establish the data layer and its corresponding likelihood function; Step 3: Establish the parameter layer and its corresponding prior distribution; Step 4: Establish the hyperparameter layer and its corresponding hyperprior distribution; Step 5: Correct the parameters in the heat conduction equation based on the freezing process; Step 6: Couple the improved heat conduction equations to establish the SFCC model; Step 7: Obtain the posterior distribution using the Markov chain Monte Carlo method and take the 95% confidence interval.

2. The method for quantifying the unfrozen water content in frozen soil based on layered Bayesian methods according to claim 1, characterized in that: Step 1 includes: a. To introduce a Bayesian framework into the freezing process of freeze-thaw cycles in cold-region permafrost, the Freezing Characteristic Curve (SFCC) is stratified, considering three different data types within the SFCC, and is divided as follows: (1) In the formula, Here, T represents the unfrozen water content, and T represents the temperature, indicating the data layer of the model; z represents all the parameters in the model, indicating the parameter layer. It is used to characterize the errors in experimental measurements and theoretical derivations, and is one of the hyperparameters. Other hyperparameters are derived from the distribution function of the parameters. b. To integrate prior information from similar sites, SFCC is estimated based on a Bayesian framework, quantifying the uncertainty in SFCC using... Representing all parameters to be updated, this function embeds all data based on measurements and priors into a posterior probability density function (PDF), which can be represented as follows: (2) in, is the likelihood function, which characterizes the degree of fit between the measured data and a given set of parameters, corresponding to the data layer; It includes prior distribution and hyperprior distribution, corresponding to the parametric layer and hyperparametric layer, respectively; c is a normalization constant, ensuring... The integral is 1.

3. The method for quantifying the unfrozen water content in frozen soil based on hierarchical Bayesian methods according to claim 1, characterized in that: Step 2 includes: a. For a set of measurements Unfrozen water content For positive values, their distribution follows a normal distribution: (3) in, For a given value with a mean of zero and a standard deviation of , Gaussian random variables; b. Considering all measurements from the target site and similar sites, the overall likelihood function is: (4) In the formula, For the number of similar sites, the first The venue represents the target venue; This represents the number of available measurements in the i-th site; c. Due to and All parameters are unknown and awaiting update. Represented as: (5)。 4. The method for quantifying the unfrozen water content of frozen soil based on hierarchical Bayesian methods according to claim 1, characterized in that: Step 3 includes: a. Treating parameters from different sites as realizations of the same prior distribution, the prior distribution after taking the absolute value of all parameters is a log-normal distribution: (6) In the formula, and These are the mean and standard deviation of the parameter, respectively. Characterizing the similarity between different sites, while This reflects the variability of z across different locations; b. The parameters of a site are independent of each other, and the joint prior PDF of different parameters in site i is: (7) Where n is the number of model parameters; c. and These are new parameters, or hyperparameters, derived from the prior PDF. These hyperparameters are also unknown and are parameters to be updated. Represented as: (8) Based on all the parameters currently pending updates, the joint prior PDF is: (9) in, , and This represents the hyperprior distribution corresponding to the hyperparameter layer.

5. The method for quantifying the unfrozen water content in frozen soil based on hierarchical Bayesian methods according to claim 1, characterized in that: Step 4 includes: a. Hyperparameters Set as an interval A uniform distribution on the surface is represented as: (10) b. Hyperparameters Suppose it is a seminormal distribution with a mean of zero: (11) In the formula, It is the reciprocal of the variance. The smaller, The less prior information; c. Hyperparameters Assuming an exponential distribution: 。 (12)。 6. The method for quantifying the unfrozen water content in frozen soil based on hierarchical Bayesian methods according to claim 1, characterized in that: Step 5 includes: a. During the process of water phase turning into ice, due to the difference in thermal properties between ice and water, the thermal properties of the soil will change as the unfrozen water content decreases. At the same time, the phase transition heat is released during the water-ice phase transition, and the pressure change caused by the change in water and ice content will cause the flow of water and air, which will generate convective heat during the flow of water and air. b. Determine the amount of heat conducted through the soil unit: (13) In the formula, Enthalpy change represents the difference in heat transferred into and out of the soil unit; and Passing through a plane on the unit soil ( and The heat of soil; C is the specific heat of the soil. This refers to the soil density. and These are phase change heat and convection heat, respectively. c. Determine the heat difference between the incoming and outgoing soil elements according to Fourier's law of cooling, and pass through the plane axially. and plane The calories are as follows: (14) (15) in, The thermal conductivity of the soil; d. Determine the variation of specific heat with unfrozen water content. Since specific heat has the property of being a weighted average of the various components of the soil, the corrected specific heat is expressed as: (14) in, These are the volume fractions of water, gas, and ice, respectively. These are the specific heats of soil particles, water, air, and ice, respectively, and are generally taken as... , kJ / (kg·°C); e. Determine the variation of thermal conductivity with unfrozen water content. The thermal conductivity of soil is controlled by the thermal conductivity and volume fraction of its components. The corrected thermal conductivity is expressed as: (15) in, These are the thermal conductivity coefficients of soil particles, water, air, and ice, respectively, and are generally taken as... W / (m·°C); f. The heat of phase transition released during the water-ice phase transition is determined by both the magnitude of the latent heat of phase transition and the change in its content: (16) Where L is the latent heat of phase change, taken as 334.56 kJ / kg; Let the density of water be 1 kg / m³. 3 ; g. Convective heat is the heat generated by the migration of water and air through the pores of soil: (17) Where h is the heat transfer coefficient, the magnitude of which depends on the medium in which the flow occurs, W / (m²). 2 • °C); A is the heat exchange surface area, which in soil is the surface area of ​​soil pores, in m³. 2 ; h. During the freezing process, the unknown properties of the soil particles remain unchanged: (18) i. Determine the heat conduction equation under the freezing process, substitute the corrected specific heat and thermal conductivity, and introduce phase change heat and convection heat to obtain the expression of the heat conduction equation under the freezing process: 。 (19)。 7. The method for quantifying the unfrozen water content in frozen soil based on hierarchical Bayesian methods according to claim 1, characterized in that: Step 6 includes: a. Couple the improved heat conduction equation and establish the extended likelihood function: (20) (21) The equation is implicit, characterized by: estimating the unfrozen water content through a mathematical framework. The values ​​represent all possible values ​​in the probability space. However, unfrozen water contents above the initial moisture content and below zero do not conform to the actual freezing process. Therefore, the layered Bayesian framework needs to be constrained by the improved heat conduction equation to obtain the solution in the physical space. ; b. By embedding the extended likelihood function into a hierarchical Bayesian framework, a hierarchical model of soil freezing characteristic curves based on hierarchical Bayes is obtained: 。 (22)。 8. The method for quantifying the unfrozen water content in frozen soil based on hierarchical Bayesian methods according to claim 1, characterized in that: Step 7 includes: a. Obtain the posterior distribution using the Markov Chain Monte Carlo (MCMC) method. A large number of numerical solutions were obtained, and values ​​other than 2.5% and 97.5% of all solutions were removed, and a 95% confidence interval was taken.

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