CN111950140A - Dam seepage behavior analysis method considering multiple uncertainties - Google Patents

Dam seepage behavior analysis method considering multiple uncertainties Download PDF

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CN111950140A
CN111950140A CN202010746295.2A CN202010746295A CN111950140A CN 111950140 A CN111950140 A CN 111950140A CN 202010746295 A CN202010746295 A CN 202010746295A CN 111950140 A CN111950140 A CN 111950140A
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王晓玲
余红玲
佟大威
关涛
王佳俊
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Tianjin University
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Abstract

The invention discloses a dam seepage behavior analysis method considering various uncertainties, which comprises the following steps: step 1, constructing a sample pair by using a Latin hypercube sampling method and a dam seepage numerical simulation model; step 2, establishing a proxy model by using the sample pair constructed in the step 1; and 3, quantifying various uncertainties in the inversion process by adopting a random method, calculating seepage parameters to be inverted based on Bayesian rules, and using the calculated seepage parameters to be inverted to correct the dam seepage numerical simulation model, so that the dam seepage behavior is accurately analyzed. According to the invention, various uncertainties are comprehensively considered in the seepage parameter random inversion method, so that dam seepage parameters are accurately and reliably inverted and seepage behavior is reasonably analyzed and evaluated.

Description

Dam seepage behavior analysis method considering multiple uncertainties
Technical Field
The invention relates to a dam seepage behavior analysis technology in water conservancy and hydropower engineering, in particular to a dam seepage behavior analysis method considering various uncertainties.
Background
Seepage parameters (such as permeability coefficients of dam bodies and dam foundations) are key parameters influencing the accuracy of a dam seepage numerical simulation model. However, seepage parameters obtained by laboratory tests or field in-situ tests have large deviations from actual conditions. The parameter inversion analysis method based on the field observation data can obtain more accurate seepage parameters and provide a data base for correcting the dam seepage numerical simulation model, so that the method plays an important role in dam seepage performance analysis.
Many parameter inversion methods are commonly used at present, including least square method[1]Gauss-Newton method[2]Levenberg-Marquardt method[3](ii) a Artificial neural network[4]And genetic algorithm[5]And the like. However, these parametric inversion methods can only obtain deterministic point estimation values of the seepage parameters, but do not consider various uncertainty problems existing in the inversion process, and the obtained inversion result has low reliability.
Various uncertainties such as model uncertainty, parameter uncertainty and measurement uncertainty exist in the seepage parameter inversion process. Model uncertainty generally comes from approximate expression of a concept model to a real seepage physical phenomenon, simplification of a complex hydrogeological model, errors in grid discretization and the like[6]. A large number of parameters to be specified are usually involved in the inverse calculation model, and the parameter uncertainty is caused by the inherent characteristics of the percolation parameters (e.g., permeability coefficient, spatial variability of porosity, etc.) and the human cognitive uncertainty (e.g., ambiguity in the values of the parameters, grayness, uncertainty, etc.)[7]. In addition, during the process of collecting, measuring, recording and processing the monitoring data, errors of the monitoring data, namely measurement uncertainty, can be brought about inevitably due to human or system reasons[8]. Various uncertainties existing in the seepage parameter inversion process inevitably affect the reliability of the parameter inversion result, so that an effective method is necessary for considering and analyzing the uncertainties.
The Bayesian inversion method is a constantThe method is used for parameter inversion and uncertainty evaluation, and the posterior distribution of the parameters to be inverted can be deduced through Bayesian rules under the condition of providing the prior distribution and the measured data of the parameters to be inverted. At present, Bayesian inversion methods are applied to seepage parameter inversion research by many scholars. Lv Peng et al[9]Introducing entropy-blind number theory in a Bayesian inversion method, and performing inversion research on the permeability coefficient of a dam foundation of a certain roller compacted concrete dam; lule, etc[10]In the Bayesian inversion method, a single-component self-adaptive metropolis (SCAM) sampling algorithm is adopted and a weight factor is added, so that the solving efficiency of high-dimensional parameter identification in the complex underground water problem is greatly improved; zhang Shuangsheng, etc[11]Based on a Bayesian formula and a progressive well-adding optimization method with minimum information entropy, a differential evolution self-adaptive Metropolis algorithm is adopted to synchronously invert pollution source parameters and aquifer parameters of a heterogeneous underground aquifer; zhang Jiangjiang river[12]In order to improve the calculation efficiency of parameter inversion, a Bayesian uncertainty analysis method based on a substitution system is provided, so that the key parameters of the underground water solute transport model are efficiently and accurately inverted; li Peng Tao[13]Carrying out inversion research on hydrogeological parameters of an ideal model and a Tribo hydrogeological test field by a Bayesian method based on a nested sampling algorithm; donghai powerful food[14]And carrying out inversion research on the parameters of the underground water pollution source based on a Kriging substitution model and an improved Bayesian inversion method.
In the research, the Bayesian inversion method can accurately obtain the parameter inversion result. However, most of the existing Bayesian inversion methods only focus on the research on parameter uncertainty, and ignore other uncertainties such as model uncertainty and measurement uncertainty. In addition, because the Bayesian inversion method needs to repeatedly call forward models in large quantity to make the posterior distribution of parameters converge, in order to improve the calculation efficiency, most of the Bayesian inversion method researches adopt a proxy model to replace the forward model which consumes time in calculation. However, the proxy model inevitably brings extra uncertainty to the parametric inversion, and few studies are currently made to quantify the uncertainty of the proxy model in the parametric inversion study. Therefore, how to effectively consider model uncertainty, parameter uncertainty, measurement uncertainty and proxy model uncertainty existing in the parameter inversion process is of great significance for obtaining accurate and reliable seepage parameters and reasonably analyzing and evaluating seepage performance.
Disclosure of Invention
The invention aims to provide a dam seepage behavior analysis method aiming at the fact that uncertainty in an inversion process is ignored in most of current seepage parameter inversion methods.
The technical scheme adopted by the invention is as follows: a dam seepage behavior analysis method considering multiple uncertainties comprises the following steps:
step 1, constructing a sample pair by using a Latin hypercube sampling method and a dam seepage numerical simulation model;
step 2, establishing a proxy model by using the sample pair constructed in the step 1;
and 3, quantifying various uncertainties in the inversion process by adopting a random method, calculating seepage parameters to be inverted based on Bayesian rules, and using the calculated seepage parameters to be inverted to correct the dam seepage numerical simulation model, so that the dam seepage behavior is accurately analyzed.
Further, in step 1, the constructing of the sample pair by using the latin hypercube sampling method and the dam seepage numerical simulation model comprises:
determining the value range of the seepage parameter to be inverted according to engineering experience and indoor test and in-situ test data, extracting sample points from the value range of the seepage parameter to be inverted by adopting a Latin hypercube sampling method, inputting the sample points into a dam seepage numerical simulation model one by one for simulation calculation to obtain a response value corresponding to the sample point, and forming the sample pair by the sample point and the response value corresponding to the sample point.
Further, in step 2, the establishing of the proxy model by using the sample pair constructed in step 1 includes:
taking a sample pair with a set proportion as a training sample, and taking the rest sample pair as a test sample;
training a Kriging model as an agent model, wherein the mathematical expression of the Kriging model is shown as formula (1):
Figure BDA0002608456890000031
in the formula, theta is input data and represents seepage parameters to be inverted; f (theta) is an output value of the Kriging model and represents a water head value of a measuring point; beta is a regression coefficient; p is the number of polynomial functions and is determined by the order of the polynomial functions and the number of initial sample points; g (θ) is a known polynomial function about the variable θ, using a zero order polynomial; z (theta) is a Gaussian random term of a Kriging model, and the covariance function of Z (theta) is expressed as shown in formula (2):
Cov[Z(θjj),Z(θkk)]＝σ2R(θjj,θkk) (2)
in the formula, σ2Is the variance of Z (θ); r (theta)jj,θkk) For any two samples thetajjAnd thetakkOf a spatial correlation function of R (theta)jj,θkk) Using a gaussian correlation function, as shown in equation (3):
Figure BDA0002608456890000032
in the formula (I), the compound is shown in the specification,
Figure BDA0002608456890000033
and
Figure BDA0002608456890000034
are each thetajjAnd thetakkThe ii component of (a); n is the number of water head monitoring points; lambda [ alpha ]iiAnd determining the undetermined coefficient by a maximum likelihood estimation method.
Further, in step 3, the quantifying a plurality of uncertainties in the inversion process by using a random method, and the calculating of the seepage parameters to be inverted based on the bayesian rule includes:
replacing the dam seepage numerical simulation model with the proxy model constructed in the step 2, and calculating to obtain posterior distribution of seepage parameters to be inverted based on a Bayesian inversion method, wherein the calculation is as shown in a formula (4):
p(θ|y)∝p(y|θ)p(θ) (4)
in the formula, theta is a seepage parameter to be inverted; p (theta) is a prior distribution function of seepage parameters to be inverted; y is an actual monitoring value; p (y | θ) is a likelihood function; p (theta | y) is a posterior distribution function of seepage parameters to be inverted;
in the process of obtaining the actual monitoring value y, the monitoring data has measurement uncertainty, the measurement uncertainty is recorded as e, the actual monitoring value y can be represented by a true value R and a measurement uncertainty e of a calculation result of a dam seepage numerical simulation model, and the calculation is shown as a formula (5):
y＝R+e (5)
where the measurement uncertainty e follows an independent and identical Gaussian distribution
Figure BDA0002608456890000046
Wherein the content of the first and second substances,
Figure BDA0002608456890000045
determining according to the error of the monitoring instrument;
if the model uncertainty exists in the calculation result of the dam seepage numerical simulation model, and the model uncertainty is recorded as b, the actual monitoring value y in the formula (5) is expanded as shown in the formula (6):
y＝M+b+e (6)
wherein M is a calculated value of a dam seepage numerical simulation model;
in the process of replacing the dam seepage numerical simulation model by the proxy model, proxy model uncertainty exists between the proxy model prediction value and the calculated value of the dam seepage numerical simulation model, if the proxy model uncertainty is recorded as follows, the actual monitoring value y in the formula (6) is further developed as shown in a formula (7):
y＝f++b+e (7)
the model uncertainty b is expressed as b ═ η (y-f), with each component η of ηiSubject to independent and identical Gaussian distributions
Figure BDA0002608456890000041
Wherein the content of the first and second substances,
Figure BDA0002608456890000042
the method is unknown, and the method needs to be obtained by performing joint inversion with seepage parameters to be inverted; each component of the proxy model uncertaintyiSubject to independent and identical Gaussian distributions
Figure BDA0002608456890000043
Wherein the content of the first and second substances,
Figure BDA0002608456890000044
the calculation is expressed by the mean square error of the proxy model of the ith measurement point as shown in equation (8):
Figure BDA0002608456890000051
wherein i represents the ith measuring point; j represents the jth test sample of the proxy model; n is the total number of test samples; mijRepresenting the model simulation value of the jth test sample of the ith test point; f. ofijRepresenting the predicted value of the jth test sample of the proxy model of the ith test point;
Figure BDA0002608456890000052
representing the mean square error of the proxy model of the ith measuring point;
as derived from equations (4) to (8), the distribution of compliance of the actual measurement value y is shown in equation 9:
y|θ,σb～N(μ,∑) (9)
wherein μ ═ f (θ),
Figure BDA0002608456890000053
i is a unitA matrix;
assuming the seepage parameters θ and σ to be invertedbAre independent of each other, formula (4) is rewritten as formula (10):
p(θ,σb|y)∝p(y|θ,σb)p(θ)p(σb) (10)
wherein the likelihood function p (y | θ, σ)b) Written as equation (11):
Figure BDA0002608456890000054
in the formula, the | sigma | is a determinant of the sigma; n is the dimension of the water head monitoring point;
and finally, carrying out Markov Monte Carlo method sampling on the formula (10) to obtain the posterior distribution of the seepage parameter to be inverted, and taking the maximum posterior probability estimation value of the posterior distribution as the inversion value of the seepage parameter to be inverted.
The invention has the beneficial effects that: the dam seepage behavior analysis method considering various uncertainties can consider and quantify various uncertainties such as parameter uncertainty, model uncertainty, measurement uncertainty and proxy model uncertainty in the seepage parameter inversion process, and accurately and reliably obtain dam seepage parameters, so that more reliable support and basis are provided for correction of a dam seepage numerical simulation model and dam seepage behavior analysis. The dam seepage behavior analysis method considering various uncertainties, which is disclosed by the invention, can be used in the field of dam seepage parameter inversion, can also be widely applied to parameter estimation and uncertainty evaluation research in other research fields, and has strong popularization.
Drawings
FIG. 1 is a flow chart of a method for analyzing seepage behavior of a dam in consideration of various uncertainties according to the present invention;
fig. 2 is a schematic view of a dam seepage structure in embodiment 1 of the present invention.
Detailed Description
In order to further understand the contents, features and effects of the present invention, the following embodiments are illustrated and described in detail with reference to the accompanying drawings:
the seepage parameter inversion is an effective means for correcting the dam seepage numerical simulation model. The invention provides a dam seepage performance state analysis method considering multiple uncertainties, and aims to solve the problem that most existing seepage parameter inversion methods ignore uncertainty in the inversion process, and Bayesian inversion methods can consider uncertainty problems but mainly focus on parameter uncertainty research at present and ignore various uncertainties such as model uncertainty, measurement uncertainty and proxy model uncertainty.
As shown in fig. 1, a method for analyzing seepage behavior of a dam considering various uncertainties comprises the following steps:
step 1, constructing a sample pair by using a Latin hypercube sampling method and a dam seepage numerical simulation model.
Determining the value range of the seepage parameter to be inverted according to engineering experience and indoor test and in-situ test data, extracting sample points from the value range of the seepage parameter to be inverted by adopting a Latin Hypercube Sampling (LHS) method, inputting the sample points into a dam seepage numerical simulation model one by one for simulation calculation to obtain a response value corresponding to the sample point, and forming the sample pair by the sample points and the response value corresponding to the sample point.
And 2, establishing a proxy model by using the sample pair constructed in the step 1.
And taking the sample pairs with set proportion as training samples, and taking the rest sample pairs as test samples. The training samples are used to train the agent model, and the testing samples are used to test whether the agent model meets the precision requirement. The invention trains the Kriging model as an agent model. The mathematical expression of the Kriging model is shown in formula (1):
Figure BDA0002608456890000061
in the formula, theta is input data, and represents seepage parameters to be inverted in the invention; f (theta) is an output value of a Kriging model, and represents a water head value of a measuring point in the invention; beta is a regression coefficient; p is the number of polynomial functions and is determined by the order of the polynomial functions and the number of initial sample points; g (theta) is a known polynomial function related to the variable theta, and generally has three forms, namely a zero-order polynomial, a first-order polynomial and a second-order polynomial, and the zero-order polynomial is adopted in the invention; z (theta) is a Gaussian random term of a Kriging model, and the covariance function of Z (theta) is expressed as shown in formula (2):
Cov[Z(θjj),Z(θkk)]＝σ2R(θjj,θkk) (2)
in the formula, σ2Is the variance of Z (θ); r (theta)jj,θkk) For any two samples thetajjAnd thetakkIs determined. The spatial correlation function has various forms such as an exponential form, a gaussian form, a linear form, a spherical form, a cubic polynomial, a spline curve, and the like. The spatial correlation function R (theta) of the present inventionjj,θkk) Using a gaussian correlation function, as shown in equation (3):
Figure BDA0002608456890000071
in the formula (I), the compound is shown in the specification,
Figure BDA0002608456890000072
and
Figure BDA0002608456890000073
are each thetajjAnd thetakkThe ii component of (a); n is the number of water head monitoring points; lambda [ alpha ]iiThe undetermined coefficient can be determined by a maximum likelihood estimation method.
And 3, quantifying multiple uncertainties in the inversion process by adopting a random method, calculating seepage parameters to be inverted based on Bayesian rules, and using the seepage parameters to be inverted to correct the dam seepage numerical simulation model, thereby more accurately analyzing the dam seepage behavior.
The method comprises the following steps of quantifying various uncertainties in the inversion process by adopting a random method, and calculating seepage parameters to be inverted based on Bayesian rules, wherein the seepage parameters to be inverted specifically comprise the following steps: replacing the time-consuming dam seepage numerical simulation model with the proxy model constructed in the step 2, and calculating to obtain posterior distribution of seepage parameters to be inverted based on a Bayesian inversion method, wherein the calculation is shown as a formula (4):
p(θ|y)∝p(y|θ)p(θ) (4)
in the formula, theta is a seepage parameter to be inverted; p (theta) is a prior distribution function of the parameter to be inverted; y is an actual monitoring value; p (y | θ) is a likelihood function; p (θ | y) is the posterior distribution function of the parameter to be inverted.
In the process of obtaining the actual monitoring value y, due to human or system reasons, the monitoring data inevitably has errors, namely measurement uncertainty, and the measurement uncertainty is recorded as e, so that the actual monitoring value y can be represented by a true value R and a measurement uncertainty e of a calculation result of a dam seepage numerical simulation model, and the calculation is as shown in a formula (5):
y＝R+e (5)
where the measurement uncertainty e is a random vector, it is assumed in the present invention that it follows an independent and identical Gaussian distribution
Figure BDA0002608456890000074
Wherein the content of the first and second substances,
Figure BDA0002608456890000075
can be determined according to the error of the monitoring instrument.
Due to the simplification of a complex hydrogeological model, grid discretization errors and the like, model uncertainty inevitably exists in the calculation result of the dam seepage numerical simulation model, and the model uncertainty is recorded as b. Then, the actual monitor value y in equation (5) is expanded as shown in equation (6):
y＝M+b+e (6)
wherein M is a calculated value of a dam seepage numerical simulation model;
in addition, in order to improve Bayesian inversion efficiency, in the process of replacing the dam seepage numerical simulation model with the proxy model, errors inevitably exist between the proxy model predicted value and the calculated value of the dam seepage numerical simulation model, namely the uncertainty of the proxy model is recorded as follows. Then, the actual monitor value y in equation (6) is further expanded as shown in equation (7):
y＝f++b+e (7)
in the present invention, it is proposed that the model uncertainty b is expressed as b ═ η (y-f), η is a random vector, and it can be assumed that each of its components ηiSubject to independent and identical Gaussian distributions
Figure BDA0002608456890000081
Wherein the content of the first and second substances,
Figure BDA0002608456890000082
is unknown and needs to be obtained by joint inversion with seepage parameters to be inverted. In the present invention, it is proposed that the proxy model uncertainty is also a random vector, each of its componentsiSubject to independent and identical Gaussian distributions
Figure BDA0002608456890000083
Wherein the content of the first and second substances,
Figure BDA0002608456890000084
expressed by the Mean Square Error (MSE) of the proxy model at the ith station, the calculation is shown in equation (8):
Figure BDA0002608456890000085
wherein i represents the ith measuring point; j represents the jth test sample of the proxy model; n is the total number of test samples; mijRepresenting the model simulation value of the jth test sample of the ith test point; f. ofijRepresenting the predicted value of the jth test sample of the proxy model of the ith test point;
Figure BDA0002608456890000086
representing the mean square error of the proxy model of the ith measuring point;
as derived from equations (4) to (8), the distribution of compliance of the actual measurement value y is shown in equation 9:
y|θ,σb～N(μ,∑) (9)
wherein μ ═ f (θ),
Figure BDA0002608456890000087
and I is an identity matrix.
The invention assumes the seepage parameters theta and sigma to be invertedbAre independent of each other, the bayesian formula (formula 4) can be rewritten as formula (10):
p(θ,σb|y)∝p(y|θ,σb)p(θ)p(σb) (10)
wherein the likelihood function p (y | θ, σ)b) Written as equation (11):
Figure BDA0002608456890000091
in the formula, the | sigma | is a determinant of the sigma; and N is the dimension of the water head monitoring point.
Finally, Markov Monte Carlo (MCMC) sampling is carried out on a Bayesian formula (formula 10) to obtain the posterior distribution of the seepage parameters to be inverted, and the maximum posterior probability estimation value of the posterior distribution is taken as the inversion value of the seepage parameters to be inverted.
Example 1
As shown in the attached figure 2, the effectiveness and the superiority of the method provided by the invention are proved by inversion of the permeability coefficient of the dam foundation of the earth-rock dam in the embodiment of the invention. The permeability coefficients of the dam body and the impervious wall of the earth-rock dam are respectively set to be 1 multiplied by 10-5m/s and 1X 10- 6m/s. The dam foundation comprises three strata, namely a stratum A, a stratum B and a stratum C, wherein each stratum is provided with an isotropic uniform porous medium. Setting the permeability coefficient of each stratum as a parameter to be inverted, and setting the value range of the permeability coefficient as KA∈[0,2]×10-3m/s、KB∈[0.2,1.8]×10-4m/s、KC∈[0.5,1.5]×10-5m/s, and the prior distribution is set to be uniform. The real values of the permeability coefficients of the stratum A, the stratum B and the stratum C are respectively 1 × 10-3m/s、1×10-4m/s、1×10-5m/s, upstream water level H 115m, downstream water level H2And (5 m), simulating by a dam seepage numerical value to obtain the simulated values of 3 water head monitoring points, and taking the simulated values as corresponding 'actual monitoring values'.
According to the method of the step 1, 50 groups of permeability coefficient sample points are extracted from the value ranges of the three permeability coefficients to be inverted by adopting a Latin hypercube sampling method. Inputting the 50 groups of permeability coefficient sample points into a dam seepage numerical simulation model for simulation calculation to obtain each monitoring point response value corresponding to each group of permeability coefficient sample points, and forming 50 groups of sample pairs by the permeability coefficient sample points and the monitoring point response values.
According to the method of step 2 in the invention, 40 sample pairs in 50 sample pairs are taken as training samples, a Kriging agent model is trained, and the remaining 10 sample pairs are taken as test samples for testing the predictive performance of the agent model.
According to the method of the step 3 in the invention, a Kriging agent model is used for replacing a dam seepage numerical simulation model consuming time in calculation, the posterior distribution of the permeability coefficient to be inverted is obtained by a seepage parameter inversion method considering various uncertainties, the maximum posterior probability estimation value of the posterior distribution is taken as the value of the permeability coefficient to be inverted, namely the permeability coefficient inversion values of the stratum A, the stratum B and the stratum C are respectively 1.046x10-3m/s、1.078x10-4m/s、1.022x10-5m/s. The inversion value is very close to the true value, so the method provided by the invention is effective.
In order to further verify the superiority of the dam seepage behavior analysis method considering various uncertainties, the three methods of comparative analysis are shown in table 1. As can be seen from table 1, compared with the other two methods, the average percentage error of the inversion value of the permeability coefficient obtained by the dam seepage behavior analysis method considering multiple uncertainties provided by the present invention is the lowest and is closer to the "true value", so the method provided by the present invention has superior performance.
Table 1 shows comparison of dam foundation permeability coefficient inversion results by using a method (provided by the invention) of considering parameters and measurement uncertainty, a method (provided by the invention) of considering parameters, measurement uncertainty and model uncertainty, and a method (provided by the invention) of considering parameters, measurement uncertainty, model uncertainty and proxy model uncertainty
Figure BDA0002608456890000101
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Although the preferred embodiments of the present invention have been described above with reference to the accompanying drawings, the present invention is not limited to the above-described embodiments, which are merely illustrative and not restrictive, and those skilled in the art can make many modifications without departing from the spirit and scope of the present invention as defined in the appended claims.

Claims (4)

1. A dam seepage behavior analysis method considering multiple uncertainties is characterized by comprising the following steps:
step 1, constructing a sample pair by using a Latin hypercube sampling method and a dam seepage numerical simulation model;
step 2, establishing a proxy model by using the sample pair constructed in the step 1;
and 3, quantifying various uncertainties in the inversion process by adopting a random method, calculating seepage parameters to be inverted based on Bayesian rules, and using the calculated seepage parameters to be inverted to correct the dam seepage numerical simulation model, so that the dam seepage behavior is accurately analyzed.
2. The method for analyzing seepage behavior of dam considering multiple uncertainties as claimed in claim 1, wherein said constructing sample pairs using Latin hypercube sampling method and dam seepage numerical simulation model in step 1 comprises:
determining the value range of the seepage parameter to be inverted according to engineering experience and indoor test and in-situ test data, extracting sample points from the value range of the seepage parameter to be inverted by adopting a Latin hypercube sampling method, inputting the sample points into a dam seepage numerical simulation model one by one for simulation calculation to obtain a response value corresponding to the sample point, and forming the sample pair by the sample point and the response value corresponding to the sample point.
3. The method for analyzing seepage behavior of dam considering multiple uncertainties as claimed in claim 1, wherein said step 2 of establishing a proxy model using the sample pairs constructed in step 1 comprises:
taking a sample pair with a set proportion as a training sample, and taking the rest sample pair as a test sample;
training a Kriging model as an agent model, wherein the mathematical expression of the Kriging model is shown as formula (1):
Figure FDA0002608456880000011
in the formula, theta is input data and represents seepage parameters to be inverted; f (theta) is an output value of the Kriging model and represents a water head value of a measuring point; beta is a regression coefficient; p is the number of polynomial functions and is determined by the order of the polynomial functions and the number of initial sample points; g (θ) is a known polynomial function about the variable θ, using a zero order polynomial; z (theta) is a Gaussian random term of a Kriging model, and the covariance function of Z (theta) is expressed as shown in formula (2):
Cov[Z(θjj),Z(θkk)]＝σ2R(θjj,θkk) (2)
in the formula, σ2Is the variance of Z (θ); r (theta)jj,θkk) For any two samples thetajjAnd thetakkOf a spatial correlation function of R (theta)jj,θkk) Using a gaussian correlation function, as shown in equation (3):
Figure FDA0002608456880000021
in the formula (I), the compound is shown in the specification,
Figure FDA0002608456880000022
and
Figure FDA0002608456880000023
are each thetajjAnd thetakkThe ii component of (a); n is the number of water head monitoring points; lambda [ alpha ]iiAnd determining the undetermined coefficient by a maximum likelihood estimation method.
4. The method for analyzing the seepage behavior of the dam considering various uncertainties according to claim 1, wherein in step 3, the step of quantifying various uncertainties in the inversion process by using a random method and calculating seepage parameters to be inverted based on Bayesian rules comprises the steps of:
replacing the dam seepage numerical simulation model with the proxy model constructed in the step 2, and calculating to obtain posterior distribution of seepage parameters to be inverted based on a Bayesian inversion method, wherein the calculation is as shown in a formula (4):
p(θ|y)∝p(y|θ)p(θ) (4)
in the formula, theta is a seepage parameter to be inverted; p (theta) is a prior distribution function of seepage parameters to be inverted; y is an actual monitoring value; p (y | θ) is a likelihood function; p (theta | y) is a posterior distribution function of seepage parameters to be inverted;
in the process of obtaining the actual monitoring value y, the monitoring data has measurement uncertainty, the measurement uncertainty is recorded as e, the actual monitoring value y can be represented by a true value R and a measurement uncertainty e of a calculation result of a dam seepage numerical simulation model, and the calculation is shown as a formula (5):
y＝R+e (5)
where the measurement uncertainty e follows an independent and identical Gaussian distribution
Figure FDA0002608456880000024
Wherein the content of the first and second substances,
Figure FDA0002608456880000025
determining according to the error of the monitoring instrument;
if the model uncertainty exists in the calculation result of the dam seepage numerical simulation model, and the model uncertainty is recorded as b, the actual monitoring value y in the formula (5) is expanded as shown in the formula (6):
y＝M+b+e (6)
wherein M is a calculated value of a dam seepage numerical simulation model;
in the process of replacing the dam seepage numerical simulation model by the proxy model, proxy model uncertainty exists between the proxy model prediction value and the calculated value of the dam seepage numerical simulation model, if the proxy model uncertainty is recorded as follows, the actual monitoring value y in the formula (6) is further developed as shown in a formula (7):
y＝f++b+e (7)
the model uncertainty b is expressed as b ═ η (y-f), with each component η of ηiSubject to independent and identical Gaussian distributions
Figure FDA0002608456880000031
Wherein the content of the first and second substances,
Figure FDA0002608456880000032
the method is unknown, and the method needs to be obtained by performing joint inversion with seepage parameters to be inverted; each component of the proxy model uncertaintyiSubject to independent and identical Gaussian distributions
Figure FDA0002608456880000033
Wherein the content of the first and second substances,
Figure FDA0002608456880000034
the calculation is expressed by the mean square error of the proxy model of the ith measurement point as shown in equation (8):
Figure FDA0002608456880000035
wherein i represents the ith measuring point; j represents the jth test sample of the proxy model; n is the total number of test samples; mijRepresenting the model simulation value of the jth test sample of the ith test point; f. ofijRepresenting the predicted value of the jth test sample of the proxy model of the ith test point;
Figure FDA0002608456880000036
representing the mean square error of the proxy model of the ith measuring point;
as derived from equations (4) to (8), the distribution of compliance of the actual measurement value y is shown in equation 9:
y|θ,σb～N(μ,∑) (9)
wherein μ ═ f (θ),
Figure FDA0002608456880000037
i is an identity matrix;
assuming the seepage parameters θ and σ to be invertedbAre independent of each other, formula (4) is rewritten as formula (10):
p(θ,σb|y)∝p(y|θ,σb)p(θ)p(σb) (10)
wherein the likelihood function p (y | θ, σ)b) Written as equation (11):
Figure FDA0002608456880000038
in the formula, the | sigma | is a determinant of the sigma; n is the dimension of the water head monitoring point;
and finally, carrying out Markov Monte Carlo method sampling on the formula (10) to obtain the posterior distribution of the seepage parameter to be inverted, and taking the maximum posterior probability estimation value of the posterior distribution as the inversion value of the seepage parameter to be inverted.
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