Joint inversion method and system for detecting deep leakage channel of landfill

Abstract

The invention provides a joint inversion method and system for landfill deep layer leakage channel detection, and belongs to the technical field of refuse landfill detection, and the method comprises the steps: carrying out the detection of a target region, and obtaining a detection result as the prior information of resistivity joint inversion; constructing a forward modeling equation based on a non-structural finite element method, and solving the forward modeling equation to obtain a data prediction value in an inversion process; constructing an ERT-TEM combined three-dimensional inversion algorithm based on a data prediction value in an inversion process; based on the prior information, an ERT-TEM joint inversion algorithm is adopted to obtain a joint inversion result of the resistivity; and completing positioning of the leakage structure of the target area by using a joint inversion result. According to the method, the result obtained by micro-motion detection is used as a prior information constraint of joint inversion, so that the anti-interference capability of calculation is enhanced, and the leakage structure of the target area can be positioned more accurately.

Description

Technical Field

[0001] This invention relates to the field of landfill detection technology, and in particular to a joint inversion method and system for detecting deep leakage channels in landfills. Background Technology

[0002] With rapid socio-economic development, high population concentration, and rapid urban construction, the rate of waste growth worldwide is three times faster than the rate of economic growth. Given the large number of contaminated sites and the high risk of contamination, the management, remediation, and restoration of contaminated sites are urgently needed.

[0003] Deep seepage channels are the main pathways for leachate transport in landfills, and their precise detection and location are prerequisites for the scientific prevention, control, and remediation of landfill pollution. Drilling sampling is a traditional and direct method for obtaining information on landfill pollution, accurately determining the content of pollutants at certain points. However, it has significant drawbacks, such as: limited sampling quantity, making it difficult to comprehensively obtain the distribution of pollutants; drilling and monitoring well construction can damage underground structures, causing secondary pollution; and long processing times and poor real-time performance, making it unsuitable for long-term monitoring. Compared to drilling methods, geophysical methods can obtain information on pollutants in terms of area or volume. Furthermore, it is a non-destructive detection technology that can be implemented quickly and facilitates dynamic monitoring of pollution diffusion and migration trends, making it an important technical means for the monitoring and remediation of contaminated sites.

[0004] Currently, commonly used geophysical methods for landfill detection mainly include transient electromagnetic methods, induced polarization methods, DC resistivity methods, high-precision gravity and magnetic methods, and seismic wave methods. These methods have achieved good results in distinguishing and defining the extent of landfills and leachate pollution, as well as the migration direction and pathways of heavy metal pollutants. However, they still have limitations in detecting and locating deep seepage channels in complex geological landfills. The main problems are:

[0005] (1) A single geophysical method cannot accurately characterize the complex structure and medium properties of leakage channels;

[0006] (2) The positioning accuracy of deep leakage channels is low and there is a lack of precise positioning technology. Summary of the Invention

[0007] To overcome the shortcomings of existing technologies, the purpose of this invention is to provide a joint inversion method and system for detecting deep leakage channels in landfills.

[0008] To achieve the above objectives, the present invention provides the following solution:

[0009] A joint inversion method for detecting deep leakage channels in landfills includes:

[0010] Step 1: Use the micro-motion detection method to detect the target area and obtain the detection results as prior information for resistivity joint inversion;

[0011] Step 2: Construct forward equations based on the unstructured finite element method and solve the forward equations to obtain the predicted data values ​​in the inversion process;

[0012] Step 3: Based on the data prediction values ​​during the inversion process, establish separate three-dimensional inversion algorithms for the DC resistivity method and the transient electromagnetic method, and use the inversion results of the separate three-dimensional inversion algorithms for the DC resistivity method and the transient electromagnetic method to construct the ERT-TEM joint three-dimensional inversion algorithm;

[0013] Step 4: Based on the prior information, the ERT-TEM joint inversion algorithm is used to obtain the joint inversion result of resistivity;

[0014] Step 5: Use the joint inversion results to locate the leakage structure in the target area.

[0015] Preferably, in step 2, by constitutive relations, neglecting the influence of displacement current, and eliminating the magnetic field strength, the electric field diffusion double curl equation that satisfies the electromagnetic diffusion law is obtained:

[0016] ;

[0017] Where r is the displacement vector, t is time, E(r,t) is the electric field intensity at position r at time t, and J s (r,t) represents the source current density; Dirichlet boundary conditions are applied on the outer boundary Γ of the simulation region, and the electric field satisfies: Let n be the unit outward normal vector of the outer boundary. Based on the vector finite element method, the space is discretized, and the time derivative is discretized using the back Euler method to obtain a system of linear equations for the forward modeling equations:

[0018] KE=b;

[0019] Where K is the global coefficient matrix of all elements, b is the right-hand side term of the equation substituted with boundary conditions, the electric field E in the forward equation is solved using a direct solver, and the corresponding electromagnetic field value is obtained through interpolation, thus obtaining the data prediction value d in the inversion process. pre .

[0020] Preferably, the Tikhonov regularization method is used, and the separate inversion objective function is defined as follows:

[0021] ;

[0022] in, For data error terms, W is a model constraint term. dIt is a data weighting matrix based on observed data errors, d pre It is the predicted value of the data, d obs It is the data observation value, W m It is the model roughness matrix. is the regularization factor, m is the updated model parameters, and m0 is the initial model parameters;

[0023] A transformation function is used to ensure that the inverted resistivity is within the range of actual rock and ore values:

[0024] ;

[0025] Where: m i These are the inversion space parameters; The resistivity value of the inversion unit; , These represent the lower and upper limits of the resistivity of the target region, respectively.

[0026] Preferably, the inversion result is obtained by minimizing the individual inversion objective function using the L-BFGS optimization algorithm.

[0027] Preferably, in step 4, the formula is used:

[0028] ;

[0029] Construct a joint inversion objective function; where, This indicates the error term in the ERT data. This indicates the error term in the TEM data. Represents the model error term. For regularization terms, β represents the weight of the ERT data item, β represents the weight of the TEM data item, and W represents the weight of the TEM data item. z Depth-weighted function:

[0030] ;

[0031] Among them, g d1 and g d2 Let represent the gradient values ​​of the corresponding data terms in the ERT and TEM objective functions, respectively. To ensure that the denominator is not zero, the minimum value is obtained.

[0032] This invention also provides a joint inversion system for detecting deep leakage channels in landfills, comprising:

[0033] The detection module is used to detect the target area using a micro-motion detection method and obtain the detection results as prior information for resistivity joint inversion;

[0034] The forward modeling module is used to construct forward modeling equations based on the unstructured finite element method and solve the forward modeling equations to obtain the predicted data values ​​in the inversion process;

[0035] The inversion module is used to establish separate three-dimensional inversion algorithms for the DC resistivity method and the transient electromagnetic method based on the data prediction values ​​during the inversion process, and to construct the ERT-TEM joint three-dimensional inversion algorithm using the inversion results of the separate three-dimensional inversion algorithms for the DC resistivity method and the transient electromagnetic method.

[0036] The joint inversion module uses the ERT-TEM joint inversion algorithm based on the prior information to obtain the joint inversion result of resistivity;

[0037] The leakage structure location module is used to locate the leakage structure in the target area using the joint inversion results.

[0038] The present invention also provides an electronic device, including a bus, a transceiver, a memory, a processor, and a computer program stored in the memory and executable on the processor. The transceiver, the memory, and the processor are connected via the bus. When the computer program is executed by the processor, it implements the steps in the above-described joint inversion method for detecting deep leakage channels in landfills.

[0039] The present invention also provides a computer-readable storage medium having a computer program stored thereon, which, when executed by a processor, implements the steps in the above-described joint inversion method for detecting deep leakage channels in landfills.

[0040] According to specific embodiments provided by the present invention, the present invention discloses the following technical effects:

[0041] This invention provides a joint inversion method for detecting deep seepage channels in landfills, comprising: using a micro-motion detection method to detect a target area and obtain detection results; obtaining the underground velocity structure based on the detection results, and transforming it into a corresponding resistivity model based on empirical formulas and physical property correlation functions, which serves as prior information for obtaining resistivity joint inversion; constructing separate three-dimensional inversion algorithms for DC resistivity method and transient electromagnetic method, respectively, and an ERT-TEM joint inversion algorithm; using the prior information as the initial inversion model, employing the ERT-TEM joint inversion method to obtain the joint inversion result of resistivity; and using the joint inversion result to locate the seepage structure in the target area. This invention enhances the anti-interference capability of the calculation by using the results obtained from micro-motion detection as prior information constraints for joint inversion, enabling more accurate location of the seepage structure in the target area. Attached Figure Description

[0042] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the embodiments will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0043] Figure 1 A flowchart of a joint inversion method for detecting deep seepage channels in landfills, provided as an embodiment of the present invention.

[0044] Figure 2 This is a comparison chart of the results of the joint inversion method provided by the embodiments of the present invention and the traditional inversion method. Detailed Implementation

[0045] 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.

[0046] To make the above-mentioned objects, features and advantages of the present invention more apparent and understandable, the present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments.

[0047] like Figure 1 As shown, this invention provides a joint inversion method for detecting deep leakage channels in landfills, comprising:

[0048] Step 1: Use the micro-motion detection method to detect the target area and obtain the detection results as prior information for resistivity joint inversion;

[0049] Step 2: Construct forward equations based on the unstructured finite element method and solve the forward equations to obtain the predicted data values ​​in the inversion process;

[0050] Step 3: Based on the data prediction values ​​during the inversion process, establish separate three-dimensional inversion algorithms for the DC resistivity method and the transient electromagnetic method, and use the inversion results of the separate three-dimensional inversion algorithms for the DC resistivity method and the transient electromagnetic method to construct the ERT-TEM joint three-dimensional inversion algorithm;

[0051] Step 4: Based on the prior information, the ERT-TEM joint inversion algorithm is used to obtain the joint inversion result of resistivity;

[0052] Step 5: Use the joint inversion results to locate the leakage structure in the target area.

[0053] The principles of this invention will be further explained below in conjunction with specific application scenarios:

[0054] Step 100: Derivation of the forward modeling formula

[0055] The forward modeling equations are constructed based on Maxwell's equations satisfied by the electric and magnetic fields in the time domain. Required parameters include: electrical conductivity. Dielectric constant; obtained through rock and mineral laboratory measurements or historical geological data. and permeability Typically, vacuum values ​​or empirical values ​​based on lithology are used; the current density Js is calculated from the parameters of the transmitting device; the boundary conditions are simplified by setting the far-field boundary electric field attenuation to 0 (Dirichlet). The expression for Maxwell's equations is:

[0056] , ;

[0057] Where E, B, H, J, and D represent the electric field strength, magnetic induction intensity, electric displacement vector, magnetic field strength, and conduction current density, respectively, and t is time. The above physical quantities exhibit constitutive relations: , , By applying constitutive relations, neglecting the influence of displacement current, and eliminating the magnetic field strength H, the electric field diffusion double curl equation that satisfies the electromagnetic diffusion law can be derived:

[0058] ;

[0059] In the formula, r is the displacement vector, t is time, E(r,t) is the electric field intensity at position r at time t, and J s (r,t) represents the source current density. Dirichlet boundary conditions are applied to the outer boundary Γ of the simulation region (assuming it is sufficiently far from the emission source), and the electric field satisfies: In the formula, n is the unit outward normal vector of the outer boundary. Based on the vector finite element method, the space is discretized, and the time derivative is discretized using the backward Euler method to obtain a system of linear equations concerning the forward modeling equations:

[0060] KE=b;

[0061] Where K is the global coefficient matrix of all elements, and b is the right-hand side term of the equation substituted with the boundary conditions. Next, a direct solver is used to solve for the electric field E in the forward equation, and the corresponding electromagnetic field value is obtained through interpolation, yielding the predicted data value d for the inversion. pre In order to continue the subsequent inversion calculation work.

[0062] Step 200: Inversion calculations using the DC resistivity method (ERT) and transient electromagnetic method (TEM)

[0063] The inversion uses a Tikhonov regularization method, and the objective function for the inversion is defined as follows:

[0064] ;

[0065] in, For data error terms, W is a constraint term in the resistivity model. d It is a data weighting matrix based on observed data errors, d pre It is the predicted value of the data, d obs It is the data observation value, W m It is the roughness matrix of the resistivity model. is the regularization factor, which balances the weights of the data error term and the model constraint term; m is the updated resistivity model parameter; and m0 is the initial resistivity model parameter.

[0066] The variable used in ERT and TEM inversion is the resistivity of the medium. The following transformation function is used to ensure that the inverted resistivity is within the range of actual rock and ore values:

[0067] ;

[0068] Where: m i These are the inversion space parameters; The resistivity value of the inversion unit; , These represent the lower and upper limits of the resistivity of the study area, respectively. The gradients of their objective functions are calculated as follows:

[0069]

[0070]

[0071] Where J1 and J2 are the sensitivity matrices for the ERT and TEM methods, respectively, and W d1 and W d2 These are data weighting matrices based on observed data errors, d 1 pre and d 2 pre The two methods represent the predicted data values, d. 1 obs and d 2 obs It is the data observation value, W m This is the model roughness matrix. Next, we use adjoint forward modeling to calculate the gradient of the objective function; adjoint forward modeling is a mature method for calculating the gradient of the objective function. Then, we use the L-BFGS optimization algorithm to minimize the objective function. Finally, we use the currently mature data fitting difference to determine whether the inversion has terminated, laying the foundation for the next step of joint inversion.

[0072] Step 300: Implementation of joint inversion using the DC resistivity method and the transient electromagnetic method

[0073] To better fit the geological units of underground landfills and achieve efficient and high-precision calculations using both TEM and ERT methods at different exploration depths and resolutions, a second-order smoothing calculation method based on a minimum structure model and regularization is employed to construct a model under unstructured triangular mesh conditions. A quasi-Newton optimization method (L-BFGS) is used to solve the inversion objective function to ensure stability during the inversion process and reduce memory requirements. The joint inversion objective function is constructed as follows:

[0074] ;

[0075] To enhance the advantages of both methods at different inversion depths, the depth weighting function formula is designed as follows:

[0076] ;

[0077] Its function is to enhance the resolution complementarity between shallow ERT and deep TEM, where g d1 and g d2 These represent the corresponding data terms in the gradients of the ERT and TEM objective functions, respectively. and The gradient value is expressed as:

[0078]

[0079]

[0080] To ensure the minimum value where the denominator is not zero, α and β represent the data proportion weights of ERT and TEM, respectively, and are usually determined based on the results of individual inversions. During the inversion process, the gradients of the data fitting terms for ERT and TEM need to be calculated separately, and the parameter model is updated using L-BFGS, iteratively calculating and minimizing the objective function value. The final output inversion result yields the subsurface three-dimensional resistivity distribution structure. The final results show that this method combines the shallow high resolution of ERT with the deep detection capabilities of TEM, enabling more accurate location of seepage channels; it also reduces the ambiguity of geophysical inversion results alone, avoiding false anomalies.

[0081] Step 400: Calculation of Joint Resistivity Inversion

[0082] The inversion of ERT and TEM is significantly affected by the initial model. To further ensure that this method can effectively and accurately locate deep seepage channels, other geophysical exploration methods such as seismic exploration, micromotion detection, and gravity and magnetic methods can be used to first determine the approximate underground geological structure. For the ERT-TEM joint inversion method, this information is called prior information. The prior information mainly used in this invention comes from micromotion detection: 1. Array deployment: Deploy circular or triangular arrays on the surface to collect environmental noise signals. 2. Data processing: Calculate the spatial autocorrelation function of each array. 3. Inversion dispersion curve: Extract the Rayleigh wave phase velocity dispersion curve and invert the underground shear wave velocity (V). s 4. Geological Interpretation: Geological interpretation is performed by dividing the strata according to the differences in the subsurface shear wave velocity structure. The above are all the implementation steps of micromotion detection. At present, this method is very mature and there are no application difficulties. In order to strengthen the constraint of prior information on joint inversion technology, based on the above results, the subsurface shear wave velocity structure profile obtained by micromotion detection is imported into COMSOL modeling software. According to the borehole resistivity logging data, a physical property correlation function that fits the shear wave velocity and resistivity is constructed to convert the velocity structure into a resistivity structure. Finally, an initial resistivity model is generated in COMSOL modeling software, which yields a subsurface structure resistivity model that conforms to the micromotion detection results. For ERT-TEM joint inversion, when there is no prior information, the initial model used for inversion is usually a uniform half-space model. However, by using the resistivity model obtained by micromotion detection through formula conversion as the initial resistivity model for inversion calculation, joint inversion results based on prior information can be obtained, realizing a three-dimensional fine characterization of the subsurface resistivity structure.

[0083] For the theoretical numerical model, the position of the landfill and its surrounding high-resistivity membrane in the model is calibrated to simulate the prior information obtained from the measured data through micro-motion detection, and then the prior information constraint is inverted. Figure 2 The results show that both ERT and TEM methods can roughly invert the extent of leakage zones in underground landfills. However, the ERT inversion results have a larger numerical error in terms of leakage anomalies due to the presence of a high-resistivity membrane; the TEM inversion results are affected by high resistivity, producing more false anomalies and resulting in a more ambiguous picture. ERT-TEM combines the advantages of both methods, improving resolution and reducing false anomalies caused by the high-resistivity membrane. Compared with traditional landfill detection methods, this method has significant advantages.

[0084] Compared with existing landfill detection methods, this invention significantly reduces spatial positioning errors and abnormal resistance value errors when detecting the location of impermeable layers. This method successfully combines the high-resolution characteristics of DC resistivity method in detecting shallow, thin-layer structures with the advantages of transient electromagnetic method in accurately locating deep, low-resistivity contaminants, overcoming the problem of multiple solutions in single inversion methods and providing more reliable technical support for diagnosing landfill leakage problems.

[0085] The advantages of this invention are mainly as follows: 1. Transient electromagnetic and DC resistivity measurements can be performed simultaneously, saving significant manpower and resources. 2. The joint inversion results have higher accuracy than any single result, greatly reducing the ambiguity of a single method. 3. Using the results obtained from micro-motion detection as prior information constraints for the joint inversion enhances the computation's anti-interference capability and enables more accurate localization of leakage structures.

[0086] This invention also provides a joint inversion system for detecting deep leakage channels in landfills, comprising:

[0087] The detection module is used to detect the target area using a micro-motion detection method and obtain the detection results;

[0088] The prior information acquisition module is used to obtain the underground velocity model based on the micro-motion detection results, and then obtain the corresponding resistivity model through empirical formulas and other methods. This is the prior information.

[0089] The forward modeling module implements forward modeling calculations for both the DC resistivity method and the transient electromagnetic method based on the irregular finite element method.

[0090] The inversion module enables separate three-dimensional inversion of the DC resistivity method and transient electromagnetic method based on the L-BFGS optimization algorithm, as well as the ERT-TEM joint three-dimensional inversion algorithm.

[0091] The joint inversion module uses the prior information as the initial inversion model. The ERT-TEM joint inversion method is then employed to obtain the joint inversion results for resistivity.

[0092] The various embodiments in this specification are described in a progressive manner, with each embodiment focusing on its differences from other embodiments. Similar or identical parts between embodiments can be referred to interchangeably. For the systems disclosed in the embodiments, since they correspond to the methods disclosed in the embodiments, the descriptions are relatively simple; relevant parts can be referred to the method section.

[0093] This article illustrates the principle and implementation of a joint inversion method for detecting deep leakage channels in landfills through specific examples. The descriptions of these embodiments are merely illustrative of the method and its core concepts. Furthermore, those skilled in the art will recognize that variations in the specific implementation methods and the scope of the joint inversion method for detecting deep leakage channels in landfills may occur based on the principles of this invention. Therefore, the content of this specification should not be construed as limiting the scope of this invention.

Claims

1. A joint inversion method for detecting deep leakage channels in landfills, characterized in that, include: Step 1: Use the micro-motion detection method to detect the target area and obtain the detection results as prior information for resistivity joint inversion; Step 2: Construct forward equations based on the unstructured finite element method and solve the forward equations to obtain the predicted data values ​​in the inversion process; Step 3: Based on the data prediction values ​​during the inversion process, establish separate three-dimensional inversion algorithms for the DC resistivity method and the transient electromagnetic method, and use the inversion results of the separate three-dimensional inversion algorithms for the DC resistivity method and the transient electromagnetic method to construct the ERT-TEM joint three-dimensional inversion algorithm; Step 4: Based on the prior information, the ERT-TEM joint inversion algorithm is used to obtain the joint inversion result of resistivity; Step 5: Use the joint inversion results to locate the leakage structure in the target area.

2. The joint inversion method for detecting deep leakage channels in landfills according to claim 1, characterized in that, In step 2, by applying the constitutive relation, neglecting the influence of displacement current, and eliminating the magnetic field strength, the electric field diffusion double curl equation that satisfies the electromagnetic diffusion law is obtained: ； Where r is the displacement vector, t is time, E(r,t) is the electric field intensity at position r at time t, and J s (r,t) represents the source current density; Dirichlet boundary conditions are applied on the outer boundary Γ of the simulation region, and the electric field satisfies: Let n be the unit outward normal vector of the outer boundary. Based on the vector finite element method, the space is discretized, and the time derivative is discretized using the back Euler method to obtain a system of linear equations for the forward modeling equations: KE=b; Where K is the global coefficient matrix of all elements, b is the right-hand side term of the equation substituted with boundary conditions, the electric field E in the forward equation is solved using a direct solver, and the corresponding electromagnetic field value is obtained through interpolation, thus obtaining the data prediction value d in the inversion process. pre .

3. The joint inversion method for detecting deep leakage channels in landfills according to claim 2, characterized in that, In step 3, the Tikhonov regularization-based method is used to define the individual inversion objective function as follows: ； in, For data error terms, W is a model constraint term. d It is a data weighting matrix based on observed data errors, d pre It is the predicted value of the data, d obs It is the data observation value, W m It is the model roughness matrix. is the regularization factor, m is the updated model parameters, and m0 is the initial model parameters; A transformation function is used to ensure that the inverted resistivity is within the range of actual rock and ore values: ； Where: m i These are the inversion space parameters; The resistivity value of the inversion unit; , These represent the lower and upper limits of the resistivity of the target region, respectively.

4. The joint inversion method for detecting deep leakage channels in landfills according to claim 3, characterized in that, In step 3, the L-BFGS optimization algorithm is used to minimize the individual inversion objective function to obtain the inversion result.

5. The joint inversion method for detecting deep leakage channels in landfills according to claim 4, characterized in that, In step 4, the formula is used: ； Construct a joint inversion objective function; where, This indicates the error term in the ERT data. This indicates the error term in the TEM data. Represents the model error term. For regularization terms, β represents the weight of the ERT data item, β represents the weight of the TEM data item, and W represents the weight of the TEM data item. z Depth-weighted function: ； Among them, g d1 and g d2 These represent the corresponding data terms in the ERT and TEM objective functions, respectively. and The gradient value, To ensure that the denominator is not zero, the minimum value is obtained.

6. A joint inversion system for detecting deep leakage channels in landfills, characterized in that, include: The detection module is used to detect the target area using a micro-motion detection method and obtain the detection results as prior information for resistivity joint inversion; The forward modeling module is used to construct forward modeling equations based on the unstructured finite element method and solve the forward modeling equations to obtain the predicted data values ​​in the inversion process; The inversion module is used to establish separate three-dimensional inversion algorithms for the DC resistivity method and the transient electromagnetic method based on the data prediction values ​​during the inversion process, and to construct the ERT-TEM joint three-dimensional inversion algorithm using the inversion results of the separate three-dimensional inversion algorithms for the DC resistivity method and the transient electromagnetic method. The joint inversion module uses the ERT-TEM joint inversion algorithm based on the prior information to obtain the joint inversion result of resistivity; The leakage structure location module is used to locate the leakage structure in the target area using the joint inversion results.

7. An electronic device comprising a bus, a transceiver, a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the transceiver, the memory, and the processor are connected via the bus, characterized in that, When the computer program is executed by the processor, it implements the steps in the joint inversion method for detecting deep seepage channels in landfills as described in any one of claims 1-5.

8. A computer-readable storage medium having a computer program stored thereon, characterized in that, When the computer program is executed by the processor, it implements the steps in the joint inversion method for detecting deep seepage channels in landfills as described in any one of claims 1-5.

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