A non-stationary random field modeling method for cross-correlation of rock and soil parameters

Abstract

The invention discloses a non-stationary random field modeling method of mutual correlation of rock and soil parameters, which is realized by the following steps: determining the statistical characteristics of the rock and soil parameters to be simulated and the model scale, determining the autocorrelation distance and the correlation function, determining the mutual correlation between the parameters, performing finite element calculation grid division, determining the covariance matrix and the mutual correlation matrix of the surface parameters and the trend component, performing Cholesky decomposition of the covariance matrix and the mutual correlation matrix, forming a stationary mutual correlation random field through matrix operation, and combining and generating a non-stationary mutual correlation random field through a detrending method according to the stationary mutual correlation random field of the surface parameters and the trend component, thereby realizing the modeling of multiple random fields that simultaneously consider the non-stationary characteristics of the rock and soil parameters developing with depth and the mutual correlation characteristics with obvious correlation between the parameters.

Description

Technical Field

[0001] The invention belongs to the field of geotechnical engineering, and more specifically, relates to a cross-correlation non-stationary random field modeling method for geotechnical body parameters. Background Art

[0002] As a natural geological product, rock and soil bodies will produce obvious spatial variability due to the influence of stress history, physical and chemical weathering and various geological tectonic actions. This variability can be divided into two categories. The first category is the variability caused by different soil types, which is mainly distributed vertically. When encountering geological actions such as stratum uplift and uplift, corresponding rotation effects often occur. The current geological survey report also describes this aspect of variability. On the other hand, there is the variability of the same type of soil itself, which is manifested in the difference in parameters of the same type of soil at different locations.

[0003] According to existing research, soil parameters are often not independent of each other. For example, there is a non-negligible correlation between friction angle and cohesion, porosity and water content, and gravity and water content. This correlation often has a great impact on the calculation results of engineering safety analysis, so its correlation is often considered through orthogonal transformation, Copula function and other means. It should be noted that the random field that considers the correlation of soil parameters currently often uses cross-correlation parameter analysis of soil sample parameters at multiple depths to generate a stationary random field in the entire space. However, studies have shown that due to the influence of stress levels, soil parameters such as internal friction angle, undrained shear strength and porosity often change with depth, showing non-stationary characteristics as a whole. Therefore, it is necessary to propose a method for generating non-stationary random fields that consider the cross-correlation between parameters. Summary of the invention

[0004] The present invention provides a non-stationary random field modeling method for cross-correlation of rock and soil parameters in order to solve the technical problems existing in the known technology.

[0005] The technical solution adopted by the present invention to solve the technical problems existing in the known technology is: a non-stationary random field modeling method of the cross-correlation of rock and soil parameters, the method is carried out according to the following steps:

[0006] Step 1: Determine the statistical characteristics and model scale of the rock and soil surface parameters and trend components to be simulated;

[0007] Step 2: Determine the autocorrelation distance and autocorrelation function;

[0008] Step 3: Determine the correlation between parameters;

[0009] Step 4: Perform finite element calculation grid division, and determine the covariance matrix and cross-correlation matrix of surface parameters and trend components according to steps 1, 2, and 3;

[0010] Step 5: Perform Cholesky decomposition on the covariance matrix and the cross-correlation matrix in step 4 to form a stationary cross-correlation random field, including a decomposition method and a generation method. The decomposition method is: perform Cholesky decomposition on the autocorrelation matrix and the cross-correlation matrix respectively:

[0011] A i =L i L i T (1)

[0012] R j =P j P j T (2)

[0013] Where: A i is the autocorrelation matrix, R j is the cross-correlation matrix, L i and P j Decompose the lower triangular matrix for Cholesky;

[0014] The generation method is:

[0015] H i =L 2i-1 ζ k P j T (3)

[0016] K i =L 2i ζ m P j T (4)

[0017] Where: H i is the relevant standard Gaussian random field of surface soil parameters, take the corresponding column; K i is the standard Gaussian random field of the trend component parameters of soil parameters, and the corresponding column is taken; ζ k ζ m are two independent normal distribution vector matrices corresponding to each soil parameter, where ζ k is the surface soil parameter, ζ m is the trend component parameter.

[0018] Step 6: Generate a non-stationary cross-correlation random field from the stationary cross-correlation random field obtained in step 5 by detrending the method. Assume that the surface soil parameters belong to the log-normal distribution random field and the trend component parameters belong to the normal distribution random field. Then:

[0019] G i =H i ′+K i′γz (5)

[0020] H i ′=exp(μ lni +σ lni H i ) (6)

[0021] K i ′=μ+σK i (7)

[0022] Where: H i' is the log-normal random field of surface soil parameters, K i' is the normal random field of trend component coefficient, G i is the non-stationary random field of soil parameters, γ is the soil mass, z is the depth from the ground surface, μ lni and σ lni is the mean and standard deviation of the normally distributed variable corresponding to the lognormal distribution, μ and σ are the mean and standard deviation of the normal distribution.

[0023] In step 1, the statistical characteristics of geotechnical parameters include the mean, variance, probability distribution type of at least two related parameters in the surface layer and the mean variance of the trend component parameters calculated by the two related parameters within the modeling range, and the calculation method is as follows:

[0024]

[0025]

[0026] Where: n is the number of statistical samples, x i is the i-th sample value, μ is the mean, σ 2 is the variance.

[0027] In step 2, the autocorrelation distance is calculated by using the recursive space method through the on-site statistical parameter values, and the Γ 2 (Δz), draw a graph of the relationship between the two, and take Γ 2 For points with stable changes, the relevant distance is obtained by the following formula:

[0028] δ=Γ 2 (Δz)·Δz (10)

[0029] Where: Γ 2 is the variance reduction function, Δz is the distance, and δ is the correlation distance.

[0030] In the step 2, the autocorrelation function selects an exponential correlation function:

[0031]

[0032] Where: τ x and τz are the relative distances between any two points in space in the horizontal and vertical directions respectively; θ x and θ z are the relevant distances in the corresponding directions respectively.

[0033] In step 3, the correlation between parameters includes: the correlation between top parameters and the correlation between trend components, which is determined using the Pearson correlation coefficient:

[0034]

[0035] Where: X and Y are the surface sample point values ​​or corresponding trend component values ​​of the two soil parameters respectively; μ X and μ Y is the sample mean, σ X and σ Y is the standard deviation; the correlation coefficient directly obtained from the measured data when calculating the correlation coefficient will undergo the following changes when it is transformed from the standard normal space to the lognormal space, namely:

[0036]

[0037] Where: 0X1X2 is the cross-correlation coefficient in the standard normal space, ρ X1X2 is the cross-correlation coefficient in lognormal space, COV c and are the coefficients of variation of cohesion and friction angle.

[0038] In the fourth step, when performing finite element mesh division, the site to be simulated is decomposed into small units, and the centroid coordinates are obtained to determine the autocorrelation coefficient matrix:

[0039]

[0040] Where: lm is the centroid of the mth unit and the lth unit after mesh division (x m , z m ), (x l , z l ) corresponds to the autocorrelation coefficient, which is calculated by the above formula (11); A i is the autocorrelation matrix of soil surface parameters and trend component parameters. When there are two related soil parameters, there are 4 autocorrelation matrices, i = 1-4;

[0041] The cross-correlation matrix is ​​constructed in the following form:

[0042]

[0043] Where: ρx1x2 is the correlation coefficient between two related soil surface parameters or two trend component coefficients X1 and X2, which is calculated using formula (12); R i It is the correlation coefficient matrix of soil surface parameters or soil parameters or soil trend components. When there are two related soil parameters, there are 2 cross-correlation matrices, j = 1-2.

[0044] The advantages and positive effects of the present invention are:

[0045] (i) The present invention provides a random field modeling method that simultaneously considers the cross-correlation and non-stationary characteristics of rock and soil parameters, which is more practical.

[0046] (ii) The random field construction method provided by the present invention can obtain a traditional stationary random field that takes into account the correlation of soil parameters by adjusting relevant parameters.

[0047] (III) In addition to being used for random field construction, the random field construction method provided by the present invention can also be used to study the magnitude relationship between the influence of the correlation of trend component parameters and the influence of the correlation of random fluctuation components on the cross-correlation of soil parameters. BRIEF DESCRIPTION OF THE DRAWINGS

[0048] Figure 1 A non-stationary random field about the friction angle considering the cross-correlation obtained by applying the present invention;

[0049] Figure 2 A non-stationary random field of cohesion considering cross-correlation is obtained for the application of the present invention. DETAILED DESCRIPTION

[0050] In order to further understand the content, features and effects of the present invention, the following embodiments are given as examples and described in detail with reference to the accompanying drawings:

[0051] Taking the two-dimensional random field considering cohesion and friction angle as an example, a random soil mass of 20m*20m is generated:

[0052] The first step is to calculate the mean, variance, probability distribution type of the two related parameters of cohesion and friction angle in the surface layer, as well as the mean variance of the trend component parameters of the two related parameters within the modeling range. The calculation method is as follows:

[0053]

[0054]

[0055] Where: n is the number of statistical samples, x i is the i-th sample value, μ is the mean, σ 2 Variance

[0056] Taking silty clay as an example, the average surface cohesion is μ c =8kPa, standard deviation is σ c =1kPa; the average friction angle is The standard deviation is Cohesion trend function parameter μ K1 =0.2, standard deviation is σ K1 =0.04; friction angle trend function parameter μ K2 =0.3, standard deviation is σ K2 =0.02;

[0057] In the second step, the autocorrelation distance is calculated by using the recursive space method through the on-site statistical parameter values, and the Γ 2 (Δz), draw a graph of the relationship between the two, and take Γ 2 For points with stable changes, the relevant distance is obtained by the following formula:

[0058] δ=Γ 2 (Δz)·Δz (10)

[0059] Where: Γ 2 is the variance reduction function, Δz is the distance, and δ is the correlation distance.

[0060] For simplicity, the horizontal correlation distance of the surface parameters and trend component coefficients of friction angle and cohesion is taken as 20m, and the vertical correlation distance is 2m.

[0061] The autocorrelation function selects the exponential correlation function:

[0062]

[0063] Where: τ x and τ z are the relative distances between any two points in space in the horizontal and vertical directions respectively; θ x and θ z are the relevant distances in the corresponding directions respectively.

[0064] The third step is to determine the correlation between parameters, which includes two parts: the correlation between top parameters and the correlation between trend components, which is determined using the Pearson correlation coefficient:

[0065]

[0066] Where: X and Y are the surface sample point values ​​or corresponding trend component values ​​of the two soil parameters respectively; μ X and μ Y is the sample mean, σ X and σ Y is the standard deviation.

[0067] It should be noted that the cross-correlation coefficient obtained by directly using the measured data when calculating the correlation coefficient will not change when converting from the standard normal space to the general normal space, but will change when converting to the logarithmic normal space, that is:

[0068]

[0069] Where: 0X1X2 is the cross-correlation coefficient in the standard normal space, ρ X1X2 is the cross-correlation coefficient in lognormal space, COV c and are the coefficients of variation of cohesion and friction angle.

[0070] According to the values ​​in existing papers, the correlation coefficient of the surface parameters of friction angle and cohesion is taken as -0.6, and for the trend component coefficient, the correlation coefficient is temporarily taken as 0.9.

[0071] The fourth step is to divide the finite element mesh into small units and obtain their centroid coordinates to determine the autocorrelation coefficient matrix:

[0072]

[0073] Where: lm is the centroid of the mth unit and the lth unit after mesh division (x m , z m ), (x l , z l ) corresponds to the autocorrelation coefficient, which is calculated by the above formula (4); A i is the autocorrelation matrix of soil surface parameters and trend component parameters. When there are two related soil parameters, there are 4 autocorrelation matrices, i=1-4.

[0074] The cross-correlation matrix is ​​constructed in the following form:

[0075]

[0076] Where: ρx1x2 is the correlation coefficient between two related soil surface parameters or two trend component coefficients X1 and X2, which is calculated using formula (5); R i It is the correlation coefficient matrix of soil surface parameters or soil parameters or soil trend components. When there are two related soil parameters, there are 2 cross-correlation matrices, i=1-2.

[0077] The fifth step is to perform Cholesky decomposition on the autocorrelation matrix and the cross-correlation matrix respectively:

[0078] A i =L i Li T (1)

[0079] R i =P i P i T (2)

[0080] Where: A i is the autocorrelation matrix, R i is the cross-correlation matrix, L i and P i Cholesky factorization of the lower triangular matrix.

[0081] Taking double parameters as an example, two independent normal distribution vector matrices ζ1ζ2 are generated, where ζ1 is the surface soil parameter and ζ2 is the trend component parameter.

[0082] H1=L1(ζ1P1 T (n, 1)) (16)

[0083] H2=L3ζ1P1 T (17)

[0084] K1=L2ζ2P2 T (18)

[0085] K2=L4ζ2P2 T (19)

[0086] Where: H1 and H2 are the related standard Gaussian random fields of surface soil parameters 1 and 2; K1 and K2 are the related standard Gaussian random fields of trend component parameters of soil parameters 1 and 2.

[0087] In the sixth step, when constructing the non-stationary random field by the detrending method, it is assumed that the surface soil parameters belong to the log-normal distribution random field, and the trend component parameters belong to the normal distribution random field, then:

[0088] G1=H1′+K1′γz (20)

[0089] G2=H2′+K2′γz (21)

[0090] H i ′=exp(μ lni +σ lni H i )i=1,2 (6)

[0091] K i ′=μ+σK i i=1,2 (7)

[0092] Where: H i'is the log-normal random field of surface soil parameters, K i' is the normal random field of the trend component coefficient, G1 and G2 are the non-stationary random fields of soil parameters 1 and 2, γ is the soil mass, z is the depth from the surface, μ lni and σ lni is the mean and standard deviation of the normally distributed variable corresponding to the lognormal distribution, μ and σ are the mean and standard deviation of the normal distribution.

[0093] The results of the non-stationary random fields of friction angle and cohesion considering cross-correlation are given in Figure 1 and Figure 2 ,

[0094] Although the preferred embodiments of the present invention have been described above in conjunction with the accompanying drawings, the present invention is not limited to the above-mentioned specific embodiments, which are merely illustrative and not restrictive. Under the guidance of the present invention, ordinary technicians in this field can make many forms without departing from the scope of protection of the present invention and the claims, all of which fall within the scope of protection of the present invention.

Claims

1. A non-stationary random field modeling method for cross-correlation of rock and soil parameters, characterized in that: The method proceeds as follows: Step 1: Determine the statistical characteristics and model scale of the rock and soil surface parameters and trend components to be simulated; Step 2: Determine the autocorrelation distance and autocorrelation function; Step 3: Determine the correlation between parameters; Step 4: Perform finite element calculation grid division, and determine the covariance matrix and cross-correlation matrix of surface parameters and trend components according to steps 1, 2, and 3; Step 5: Perform Cholesky decomposition on the covariance matrix and the cross-correlation matrix in step 4 to form a stationary cross-correlation random field, including a decomposition method and a generation method. The decomposition method is: perform Cholesky decomposition on the autocorrelation matrix and the cross-correlation matrix respectively: HAS i =L i L i T (1) R j =P j P j T (2) Where: A i is the autocorrelation matrix, R j is the cross-correlation matrix, L i and P j Decompose the lower triangular matrix for Cholesky; The generation method is: H i =L 2i-1 g k P j T (3) K i =L 2i g m P j T (4) Where: H i is the relevant standard Gaussian random field of surface soil parameters, take the corresponding column; K i is the standard Gaussian random field of the trend component parameters of soil parameters, and the corresponding column is taken; ζ k ζ m are two independent normal distribution vector matrices corresponding to each soil parameter, where ζ k is the surface soil parameter, ζ m is the trend component parameter; Step 6: Generate a non-stationary cross-correlation random field from the stationary cross-correlation random field obtained in step 5 by detrending the method. Assume that the surface soil parameters belong to the log-normal distribution random field and the trend component parameters belong to the normal distribution random field. Then: G i =H i ′+K i ′γz (5) H i ′=exp(μ lni +σ lni H i ) (6) K i ′=μ+σK i (7) Where: H i ‘ is the logarithmic normal random field of surface soil parameters, K i ′ is the normal random field of trend component coefficient, G i is the non-stationary random field of soil parameters, γ is the soil mass, z is the depth from the ground surface, μ lni and σ lni is the mean and standard deviation of the normally distributed variable corresponding to the lognormal distribution, μ and σ are the mean and standard deviation of the normal distribution.

2. The cross-correlation non-stationary random field modeling method of rock and soil mass parameters according to claim 1 is characterized in that: In step 1, the statistical characteristics of geotechnical parameters include the mean, variance, probability distribution type of at least two related parameters in the surface layer and the mean variance of the trend component parameters calculated by the two related parameters within the modeling range, and the calculation method is as follows: Where: n is the number of statistical samples, x i is the i-th sample value, μ is the mean, σ 2 is the variance.

3. The cross-correlation non-stationary random field modeling method of rock and soil mass parameters according to claim 1 is characterized in that: In step 2, the autocorrelation distance is calculated by using the recursive space method through the on-site statistical parameter values, and the Γ 2 (Δz), draw a graph of the relationship between the two, and take Γ 2 For points with stable changes, the relevant distance is obtained as follows: δ=T 2 (Δz)·Δz (10) Where: 2 is the variance reduction function, Δz is the distance, and δ is the correlation distance.

4. The cross-correlation non-stationary random field modeling method of rock and soil mass parameters according to claim 1 is characterized in that: In the step 2, the autocorrelation function selects an exponential correlation function: Where: τ x and τ z are the relative distances between any two points in space in the horizontal and vertical directions respectively; θ x and θ z are the relevant distances in the corresponding directions respectively.

5. The cross-correlation non-stationary random field modeling method of rock and soil mass parameters according to claim 1 is characterized in that: In step 3, the correlation between parameters includes: the correlation between top parameters and the correlation between trend components, which is determined using the Pearson correlation coefficient: Where: X and Y are the surface sample point values ​​or corresponding trend component values ​​of the two soil parameters respectively; μ X and μ Y is the sample mean, σ X and σ Y is the standard deviation; the correlation coefficient directly obtained from the measured data when calculating the correlation coefficient will undergo the following changes when transformed from the standard normal space to the lognormal space, namely: Where: 0X1X2 is the cross-correlation coefficient in the standard normal space, ρ X1X2 is the cross-correlation coefficient in lognormal space, COV c and are the coefficients of variation of cohesion and friction angle.

6. The cross-correlation non-stationary random field modeling method of rock and soil mass parameters according to claim 1 is characterized in that: In the fourth step, when performing finite element mesh division, the site to be simulated is decomposed into small units, and the centroid coordinates are obtained to determine the autocorrelation coefficient matrix: Where: lm is the centroid of the mth unit and the lth unit after mesh division (x m , z m ), (x l , z l ) corresponds to the autocorrelation coefficient, which is calculated by the above formula (11); A i is the autocorrelation matrix of soil surface parameters and trend component parameters. When there are two related soil parameters, there are 4 autocorrelation matrices, i = 1-4; The cross-correlation matrix is ​​constructed in the following form: Where: ρx1x2 is the correlation coefficient between two related soil surface parameters or two trend component coefficients X1 and X2, which is calculated using formula (12); R i It is the correlation coefficient matrix of soil surface parameters or soil parameters or soil trend components. When there are two related soil parameters, there are 2 cross-correlation matrices, j = 1-2.

Images