A model structure error correction method coupling data driving and physical constraints

Abstract

This invention discloses a model structure error correction method that couples data-driven approaches with physical constraints. The invention constructs a corrected groundwater model, comprising a groundwater numerical model, a structure error correction model, and observation errors. Markov chain Monte Carlo simulation is used to identify the posterior distributions of the parameters of the groundwater numerical model and the hyperparameters of the structure error correction model. Based on these posterior distributions, the prediction results of the corrected groundwater model are obtained. The prediction performance of the corrected groundwater model is then evaluated. This invention constructs a novel combinatorial likelihood function, explicitly integrating physical mechanism constraints into the construction process of the error correction model. This effectively constrains the structure error model while correcting structural deviations in the groundwater model, thus significantly improving the performance of the corrected prediction results in conforming to physical mechanisms.

Description

Technical Field

[0001] This invention relates to the field of groundwater numerical simulation technology, specifically to a method for correcting model structure errors that couples data-driven and physical constraints. Background Technology

[0002] Numerical simulation of groundwater is a core technical means for groundwater resource management, pollution prevention and control, and related engineering decision-making. It can quantitatively characterize the spatiotemporal distribution characteristics of groundwater flow and pollutants, and is widely used in the scientific management and protection of groundwater resources. However, in actual groundwater numerical simulation work, due to the complexity of groundwater systems and the scarcity of hydrogeological information, the constructed groundwater models are usually simplifications of the real system. This often results in unreasonable model parameters and structures, leading to random and systematic prediction errors. If model structure bias is not considered during model identification, model parameters may be over-identified to compensate for model structure errors, resulting in an underestimation of model prediction bias and uncertainty.

[0003] To address the aforementioned model structure error problem, existing technologies employ a data-driven method (DDM) to correct model structure errors. This method explicitly corrects model structure bias by establishing a statistical model, representing the corrected prediction as the sum of the physical mechanism model, the structure error model, and the observation error.

[0004] Among numerous data-driven methods, Gaussian Process Regression (GPR) models can learn the complex relationships between dependent and independent variables from historical data and are often used to construct error correction models to improve their predictive capabilities. Applying GPR to groundwater model structure error correction is simple in principle and easy to operate, but the following problems exist in practical applications: (1) The constructed structural error model lacks a physical mechanism, which may lead to the obtained prediction results that violate the actual physical mechanism, such as not satisfying the law of conservation of mass, pollutant concentration being less than 0, etc.

[0005] (2) Existing methods for constraining coupled physical mechanisms can usually only constrain the structural error model, but cannot effectively constrain the structural error correction process.

[0006] (3) There is a lack of general methods for the structural error correction process coupled with multiple forms of physical mechanism constraints.

[0007] In summary, existing groundwater model structure error correction techniques based on GPR have many shortcomings and cannot meet the high precision and high reliability requirements of actual groundwater numerical simulation work. Therefore, developing a groundwater model structure error correction method that can solve the above-mentioned technical defects has become a technical problem that urgently needs to be solved by those skilled in the art. Summary of the Invention

[0008] The purpose of this invention is to provide a model structure error correction method that couples data-driven and physical constraints. By constructing a new combinatorial likelihood function, the physical mechanism constraints are explicitly integrated into the construction process of the error correction model, thereby effectively constraining the structural error model while correcting the structural deviation of the groundwater model.

[0009] To solve the above-mentioned technical problems, the present invention provides the following technical solution: A method for correcting model structure errors by coupling data-driven and physical constraints, comprising the following steps: A structural error correction model is constructed based on Gaussian process regression. A modified groundwater model is constructed; the modified groundwater model includes a groundwater numerical model, a structural error correction model, and observation errors; the modified groundwater model : ; In the formula, M represents the groundwater numerical model, θ is the parameter of the groundwater numerical model; b(x, φ) represents the structural error correction model, x represents the input of Gaussian process regression, φ represents the hyperparameter of Gaussian process regression, and ε represents the random measurement error.

[0010] Markov chain Monte Carlo simulation was used to identify the posterior distribution of groundwater numerical model parameters and structural error correction model hyperparameters; The prediction results of the corrected groundwater model are obtained based on the posterior distribution of the parameters of the groundwater numerical model and the hyperparameters of the structural error correction model. The predictive performance of the modified groundwater model can be evaluated based on the Nash coefficient (NSE), mean absolute error (MAE), and root mean square error (RMSE).

[0011] According to the above technical solution, the prior distribution of the structural error correction model b(x, φ) follows a multivariate Gaussian distribution. ,Right now .

[0012] The multivariate Gaussian distribution is determined by both the mean function and the covariance function. The covariance function used is the squared exponential covariance function: ; In the formula, Let represent the element in the i-th row and j-th column of the covariance matrix. Indicates input point and The covariance function between them, where T represents the transpose. It is an index function, if but ,otherwise . , and Together they constitute hyperparameters ,in Indicates the feature length scale. control The marginal likelihood, Describe the measurement error.

[0013] According to the above technical solution, the Markov chain Monte Carlo simulation recognition step includes: Obtain the parameters θ of the groundwater numerical model and the hyperparameters of the structural error correction model. The prior distribution; The physical mechanism constraint is introduced by constructing a combined likelihood function; Based on the combined likelihood function, the groundwater numerical model parameter θ, and the hyperparameters of the structural error correction model. From the prior distribution, construct the posterior distribution of parameters that satisfy the physical mechanism; Based on Markov chain Monte Carlo, iterative sampling of the posterior distribution of parameters satisfying the physical mechanism is performed to generate groundwater numerical model parameters. and structural error correction model hyperparameters The posterior samples.

[0014] Based on Markov chain Monte Carlo (MCMC), iterative sampling of the target posterior distribution satisfying the physical mechanism is performed. In each parameter update step, the acceptance probability is calculated using a combinatorial likelihood function, guiding the Markov chain to converge to a physically reasonable parameter region with good data fit. Stationary samples generated through MCMC enable effective estimation of the posterior mean, variance, and confidence interval of the parameters, ultimately achieving simultaneous identification of the posterior distribution of parameters by fusing physical constraints with Bayesian statistical inference.

[0015] According to the above technical solution, the combined likelihood function is: ; In the formula, D represents the observed data, μ represents the prior mean of the structural error correction model, ∑ represents the error covariance matrix, and F represents the corrected groundwater model. This represents the combined likelihood function. This indicates how well the modified groundwater model's predictions satisfy the physical constraints, where θ represents the parameters of the groundwater numerical model, φ represents the hyperparameters of the Gaussian process regression, and T represents the transpose. This represents the number of physical constraints, and n represents the number of observation data.

[0016] Among them, the combined likelihood function considers both the physical constraints and the likelihood function of the observed data. That is, the combined likelihood function consists of two parts: one part is the traditional likelihood function, which describes the closeness between the prediction result and the observed data, and the other part represents the degree to which the prediction result satisfies the physical constraints. Wherein, represents the likelihood function indicating the degree to which the prediction result satisfies the physical constraints: .

[0017] Based on the above technical solution, the hyperparameters of the structural error correction model Including hyperparameters in the squared exponential covariance function of Gaussian process regression and and the prior mean function of the structural error correction model .

[0018] According to the above technical solution, the modified groundwater model is in Predicted value : ; In the formula, This indicates that the groundwater numerical model is in The predicted value at that location, The structural error correction model is shown in The predicted value at that location, This indicates that the observation error is obtained using multivariate normal sampling.

[0019] Among them, the predicted values ​​of the structural error correction model posterior mean satisfy: ; Predictions from the structural error correction model Posterior covariance satisfy: ; In the formula, the superscript asterisk represents a sign indicating that the symbol is parallel to the symbol. Related variables. express C represents the prior covariance matrix. Calculated as , Calculated as . Represents training data points and predicted data points The covariance matrix between them Represents predicted data points and predicted data points The covariance matrix between them, where k represents the covariance calculation.

[0020] Based on structural error correction model b Posterior distribution at The structural error correction model b is obtained using multivariate normal sampling. Predicted value at location .

[0021] According to another aspect of the present invention, the present invention further provides an electronic device, comprising: one or more processors; and a storage device for storing one or more programs, wherein, when the one or more programs are executed by the one or more processors, the one or more processors cause the one or more processors to perform a model structure error correction method coupled with data-driven and physical constraints as described in any of the above technical solutions.

[0022] According to another aspect of the present invention, the present invention further provides a storage medium storing at least one instruction, which is loaded and executed by a processor to implement a model structure error correction method coupled with data-driven and physical constraints as described in any of the above technical solutions.

[0023] Compared with existing technologies, the beneficial effects achieved by this invention are as follows: By constructing a novel combined likelihood function, this invention explicitly integrates physical mechanism constraints into the construction process of the error correction model, effectively constraining the structural error model while correcting structural deviations in the groundwater model. It can be used for structural error correction of complex groundwater models, effectively constraining the calibration process of model parameters and overcoming the shortcomings of traditional structural error correction methods that may lead to parameter overfitting; it broadens the expression forms of physical mechanism constraints, overcoming the shortcomings of traditional structural error correction methods that cannot effectively couple physical mechanism constraints, and is applicable to coupling multiple forms of physical mechanism constraints; it effectively constrains the performance of the corrected prediction in conforming to physical mechanisms, avoiding the shortcomings of traditional structural error correction methods that may produce unreasonable prediction results, and improving the accuracy and physical consistency of model predictions. Attached Figure Description

[0024] The accompanying drawings are provided to further illustrate the invention and form part of the specification. They are used in conjunction with embodiments of the invention to explain the invention and do not constitute a limitation thereof. In the drawings: Figure 1This is a flowchart of the steps of a model structure error correction method that couples data-driven and physical constraints according to the present invention. Figure 2 This is a schematic diagram of an ideal three-dimensional aquifer pollutant transport model; Figure 3 This is a comparison of the posterior distribution of parameters in an ideal three-dimensional groundwater pollutant transport model under uncoupled and coupled physical mechanism constraints. Figure 4 This is a graph showing the change over time in the error rate of aquifer pollutant mass conservation predicted by an ideal three-dimensional groundwater pollutant transport model. Figure 5 This is a diagram showing the simulated pollutant concentrations in wells 1 and 2 of the aquifer, obtained by a structural error correction method based on coupled and uncoupled physical mechanisms. Detailed Implementation

[0025] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0026] Taking an ideal three-dimensional aquifer pollutant transport model as an example, this paper applies a model structure error correction method that couples data-driven and physical constraints. By applying physical constraints to the process of correcting groundwater model structure errors using data-driven methods, groundwater models with significant structural defects are corrected. Specific steps include (…). Figure 1 )of: Preparation phase: An ideal three-dimensional aquifer pollutant transport model, such as Figure 2 As shown in (a), there is an aquifer that is 1000m long, 200m wide, and 20m thick. It consists of three layers, from top to bottom: Layer 1, Layer 2, and Layer 3, with thicknesses of 9m, 2m, and 9m respectively. Layer 1 is a highly permeable aquifer, and Layer 3 is a moderately permeable aquifer, separated by a poorly permeable layer (Layer 2) of uneven thickness and discontinuous distribution. The porous medium in each layer is homogeneous and horizontally isotropic, with a vertical anisotropy of 3 and a porosity of 0.3. The left and right boundaries are of the first kind, with a hydraulic gradient of 0.1, while all other boundaries are impermeable.

[0027] A pollutant leakage source exists near the left boundary of Layer 1, with coordinates x=150m, y=110m, z=20m, and a leakage rate of 25 kg / d. Two pumping wells, Well 1 and Well 2, are located on the right boundary of Layer 3. Well 1 has coordinates x=750m, y=150m, z=5m and a pumping rate of 5 m³ / d; Well 2 has coordinates x=850m, y=70m, z=1m and a pumping rate of 10 m³ / d. Both wells are capable of monitoring the pollutant concentration within them.

[0028] Revision phase: S1. Construct a structural error correction model based on Gaussian process regression; S2. Construct a modified groundwater model; the modified groundwater model includes a groundwater solute transport model, a structural error correction model, and an observation error model. The solute transport sub-model is established based on MT3DMS.

[0029] Revised groundwater model : ; In the formula, M represents the groundwater numerical model, θ is the parameter of the groundwater solute transport model; b(x, φ) represents the structural error correction model, x represents the input of Gaussian process regression, φ represents the hyperparameter of Gaussian process regression, and ε represents the random measurement error.

[0030] For real-world groundwater simulations, due to the difficulty in obtaining sufficient aquifer structure information, especially for thin and poorly continuous weakly permeable aquifers, simplified aquifer structures are typically used to represent the real groundwater system. In this case, all aquifers are simplified to be of uniform thickness, based on... Figure 2 (b) shows the model structure used to establish a three-dimensional groundwater pollutant transport model. The model is numerically solved using MODFLOW-2005 and MT3D, and Gaussian process regression (GPR) is used to correct the model structure error caused by the simplification of the aquifer structure.

[0031] In addition, with Figure 2 Compared to the real model shown in (a), the two models have the same model settings, except for the aquifer structure. The parameter settings are shown in Table 1. The model runs for 2 years with a time step of 10 days. The pollutant concentration values ​​in the two pumping wells simulated by the real model are used as observation data, and Gaussian white noise with a mean of 0 and a variance of 0.01 is applied to them, resulting in a total of 146 observation data. For each pumping well, the first 45 observation data are used for model identification, and the last 28 observation data are used for model validation.

[0032] The parameter settings for the ideal three-dimensional aquifer pollutant transport model are shown in Table 1: Table 1. Parameter settings for an ideal three-dimensional aquifer pollutant transport model.

[0033] S3. This case study uses a simplified aquifer structure groundwater model to simulate the transport process of pollutants. The hydraulic conductivity coefficients (K1, K2, and K3) and dispersion (L1, L2, and L3) of the three layers are set as random variables, and their prior distribution is defined as uniform. The hyperparameters σ, λ, and σ of the GPR model follow exponential, Gamma, and uniform distributions, respectively. The parameters θ of the groundwater numerical model and the hyperparameters of the structural error correction model are obtained. The prior distribution of the structural error correction model, where the hyperparameters are... Including hyperparameters in the squared exponential covariance function of Gaussian process regression and and the prior mean function of the structural error correction model The prior information on the physical parameters and hyperparameters is shown in Table 2.

[0034] Table 2 Prior settings for parameters of the ideal three-dimensional aquifer pollutant transport model

[0035] S4. Introduce the physical mechanism constraints by constructing a combined likelihood function. Specifically: A contaminant mass conservation constraint is imposed on the process of correcting structural errors using the structural error correction method. At any given time, the total mass of leaked contaminants equals the sum of three parts: the mass of contaminants in the aquifer, the mass of contaminants flowing out through the aquifer boundary, and the mass of contaminants discharged from the aquifer through pumping wells. Furthermore, in this case study, the contaminant mass conservation error should always remain zero during the simulation, thus yielding an equality constraint.

[0036] Therefore, for any time The mass conservation constraint can be expressed as: ; in, Indicates the total mass of the leaked pollutants; This indicates the mass of pollutants trapped in the aquifer; This indicates the mass of pollutants flowing out through the aquifer boundary. This indicates the mass of pollutants discharged from the aquifer through a pumping well.

[0037] The stringency of each physical constraint remains consistent. Therefore, a confidence level for the accuracy of each constraint is defined. With all values ​​being 0.2, the likelihood function of the physical constraints can be obtained: .

[0038] A combined likelihood function is constructed based on the likelihood function of physical constraints and the likelihood function describing the closeness between the predicted results and the observed data.

[0039] S5. Based on the defined prior distribution of parameters and the combined likelihood function, Markov chain Monte Carlo (MCMC) simulation is used to identify the posterior distribution of the parameters. This serves as a structural error correction method coupled with physical constraints.

[0040] Simultaneously, based on the defined prior distribution of parameters and the traditional likelihood function describing the similarity between predicted results and observed data, Markov chain Monte Carlo (MCMC) simulation is used to identify the posterior distribution of the parameters. This serves as a structural error correction method uncoupled from physical constraints.

[0041] For the structural error correction methods with coupled and uncoupled physical constraints, the same MCMC parameter settings are used. Parameter sampling is performed using the DREAMzs algorithm, employing three parallel Markov chains, each with an iteration length of 8000 and a warm-up period of 4000. Based on the identified posterior distribution of the parameters, the performance of pollutant concentration prediction for model structural correction using coupled and uncoupled physical constraints is evaluated respectively.

[0042] In this case study, based on structural error correction methods constrained by coupled and uncoupled physical mechanisms, the posterior distributions of the physical parameters and hyperparameters of the corrected groundwater model were obtained, such as... Figure 3 As shown in the figure, a comparative analysis of the posterior distributions of the parameters identified by the two methods reveals significant differences in the results. For the six physical parameters, compared to the method without coupled physical constraints, after coupling physical constraints, the posterior distributions of the remaining parameters, except for parameter K2 which has a similar posterior distribution, exhibit significantly smaller distribution ranges and larger peak probability densities. The changes in the posterior distributions of parameters K1, K3, and L3 are particularly pronounced. For the hyperparameters of the GPR model, the posterior distributions of σ and λ identified by the method with coupled physical constraints have relatively smaller means, smaller ranges, and higher peak probability densities, while the posterior distribution of parameter σε is more similar to that without coupled physical constraints.

[0043] The pollutant prediction results obtained from the structural error correction method based on coupled and uncoupled physical mechanisms can show the change of the pollutant mass conservation error rate (ER) over time, such as... Figure 4 As shown. ER is defined as: ; Where, m t and m s These represent the actual total mass of pollutants in the system and the predicted total mass of pollutants, respectively.

[0044] Comparative analysis of the ER values ​​obtained by the two methods reveals that the method with coupled physical constraints exhibits a smaller pollutant mass conservation error compared to the prediction results of the structural error correction method without coupled physical constraints. Throughout the simulation, the pollution source leaked 25 kg of pollutants daily. Without coupled physical constraints, the peak ER predicted by the structural error correction method was 26.79%, with a mass conservation error as high as 6.69 kg, and an average ER of 13.37% with an average mass conservation error of 3.34 kg. After coupling physical constraints, the peak ER predicted by the structural error correction method decreased to 1.90%, with a mass conservation error of 0.35 kg, and an average ER of -0.48% with an average mass conservation error of only -0.12 kg. The structural error correction method with coupled mass conservation constraints demonstrates a better physical mechanism, effectively reducing unrealistic and unreasonable predictions and improving the reliability of the model predictions.

[0045] S6. Based on the posterior distribution of the model parameters, the corrected prediction results for groundwater pollutant concentrations can be obtained, such as... Figure 5 As shown. The corrected groundwater model is in Predicted value : ; In the formula, This indicates that the groundwater numerical model is in The predicted value at that location, The structural error correction model is shown in The predicted value at that location, This indicates that the observation error is obtained using multivariate normal sampling.

[0046] Depend on Figure 5 In the table, (a) is the prediction result obtained from well 1 without coupled physical constraints, (c) is the prediction result obtained from well 2 without coupled physical constraints, (b) is the prediction result obtained from well 1 with coupled physical constraints, and (d) is the prediction result obtained from well 2 with coupled physical constraints. Figure 5 It is evident that the simulation results for Well 1 and Well 2 within the aquifer are similar. Without coupled physical constraints, while the model can capture the changing trends of pollutant concentrations, it significantly overestimates the concentrations during the validation period, resulting in a large deviation between the predicted mean and the observed data. After coupling with physical constraints, the prediction performance of pollutant concentrations is improved, especially during the validation period. Similarly, the structural error correction method without coupled physical constraints provides a very wide 95% prediction interval, while the structural error correction method with coupled physical constraints produces a more reasonable 95% prediction interval, significantly reducing prediction uncertainty.

[0047] The predictive performance of the two methods can be quantitatively evaluated by using Nash coefficient (NSE), mean absolute error (MAE), and root mean square error (RMSE) as indicators, as shown in Table 3. After coupling with physical constraints, during the validation period, the NSE of the pollutant concentration prediction results for Well 1 and Well 2 improved from negative values ​​to 0.8562 and 0.8990, respectively; the MAE decreased by 92.13% and 78.97%, respectively; and the RMSE decreased by 90.10% and 74.95%, respectively. The improvement in model predictive performance is very significant.

[0048] Table 3. Prediction and Evaluation of the Ideal Three-Dimensional Aquifer Pollutant Transport Model

[0049] In this case study, the structural error correction method coupled with physical constraints outperformed the method without physical constraints during the validation period, and both methods showed similar predictive performance during the identification period. Furthermore, the structural error correction method with coupled physical constraints exhibited minimal difference in predictive performance between the identification and validation periods, while the method without physical constraints significantly outperformed the method during the identification period. This suggests that coupled physical constraints can effectively mitigate overfitting during parameter identification. Therefore, the structural error correction method that couples data-driven and physical mechanism constraints can effectively correct system prediction biases caused by model structural errors and improve the predictive reliability of the groundwater model.

[0050] It should be noted that, in this document, relational terms such as "first" and "second" are used only to distinguish one entity or operation from another, and do not necessarily require or imply any such actual relationship or order between these entities or operations. Furthermore, the terms "comprising," "including," or any other variations thereof are intended to cover non-exclusive inclusion, such that a process, method, article, or apparatus that comprises a list of elements includes not only those elements but also other elements not expressly listed, or elements inherent to such process, method, article, or apparatus.

[0051] Finally, it should be noted that the above descriptions are merely preferred embodiments of the present invention and are not intended to limit the present invention. Although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art can still modify the technical solutions described in the foregoing embodiments or make equivalent substitutions for some of the technical features. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. A method for correcting model structure errors that couples data-driven and physical constraints, characterized in that, The steps include: A structural error correction model is constructed based on Gaussian process regression. A modified groundwater model is constructed; the modified groundwater model includes a groundwater numerical model, a structural error correction model, and observation errors; Markov chain Monte Carlo simulation was used to identify the posterior distribution of groundwater numerical model parameters and structural error correction model hyperparameters; The prediction results of the corrected groundwater model are obtained based on the posterior distribution of the parameters of the groundwater numerical model and the hyperparameters of the structural error correction model. Evaluate the predictive performance of the modified groundwater model.

2. The method for correcting model structure errors by coupling data-driven and physical constraints according to claim 1, characterized in that, The prior distribution of the structural error correction model follows a multivariate Gaussian distribution.

3. The method for correcting model structure errors by coupling data-driven and physical constraints according to claim 1, characterized in that, The Markov chain Monte Carlo simulation identification steps include: Obtain the parameters θ of the groundwater numerical model and the hyperparameters of the structural error correction model. The prior distribution; The physical mechanism constraint is introduced by constructing a combined likelihood function; Based on the combined likelihood function, the groundwater numerical model parameter θ, and the hyperparameters of the structural error correction model. By combining the prior distributions, a posterior distribution of parameters that satisfies the physical mechanism is constructed; Based on Markov chain Monte Carlo, iterative sampling of the posterior distribution of parameters satisfying the physical mechanism is performed to generate groundwater numerical model parameters. and structural error correction model hyperparameters The posterior samples.

4. The method for correcting model structure errors by coupling data-driven and physical constraints according to claim 3, characterized in that, The combined likelihood function: ； In the formula, D represents the observed data, μ represents the prior mean of the structural error correction model, ∑ represents the error covariance matrix, and F represents the corrected groundwater model. This represents the combined likelihood function. This indicates how well the modified groundwater model's predictions satisfy the physical constraints, where θ represents the parameters of the groundwater numerical model, φ represents the hyperparameters of the Gaussian process regression, and T represents the transpose. This represents the number of physical constraints, and n represents the number of observation data.

5. The method for correcting model structure errors by coupling data-driven and physical constraints according to claim 4, characterized in that, The hyperparameters of the structural error correction model Including hyperparameters in the squared exponential covariance function of Gaussian process regression and and the prior mean function of the structural error correction model .

6. The method for correcting model structure errors by coupling data-driven and physical constraints according to claim 1, characterized in that, The modified groundwater model is in Predicted value : ； In the formula, This indicates that the groundwater numerical model is in The predicted value at that location, The structural error correction model is shown in The predicted value at that location, This indicates that the observation error is obtained using multivariate normal sampling.

7. An electronic device, characterized in that, include: One or more processors; A storage device for storing one or more programs, wherein when the one or more programs are executed by the one or more processors, the one or more processors cause the one or more processors to perform the method according to any one of claims 1-6.

8. A storage medium, characterized in that, The storage medium stores at least one instruction, which is loaded and executed by a processor to implement the method as described in any one of claims 1-6.

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