A multi-element gamma-ray spectrum quantitative inversion algorithm based on polynomial forward and Newton iteration

Abstract

The present application relates to the technical field of prompt gamma neutron activation analysis and non-destructive testing of industrial materials, and particularly relates to a multi-element gamma spectrum quantitative inversion algorithm based on polynomial forward and Newton iteration. The technical scheme comprises the following steps: obtaining a gamma spectrum obtained by neutron activation measurement, and extracting a plurality of gamma detection characteristic values corresponding to a plurality of elements; based on the plurality of gamma detection characteristic values and the true mass fraction of the corresponding sample elements, a polynomial forward model is established, and based on the gamma detection characteristic values and the true mass fraction of the elements, a direct polynomial inverse model is established. The present application effectively improves the accuracy of multi-element gamma spectrum quantitative inversion, enhances the robustness and convergence stability of the algorithm under complex materials, improves the generalization ability of the model to different matrix environments, and has good engineering practicability.

Description

Technical Field

[0001] This invention relates to the field of transient gamma neutron activation analysis and non-destructive testing technology for industrial materials, and particularly to a multi-element gamma energy spectrum quantitative inversion algorithm based on polynomial forward modeling and Newton iteration. Background Technology

[0002] Prompt Gamma Neutron Activation Analysis (PGNAA) is a non-contact, rapid, multi-element simultaneous nuclear analysis technique. Its basic principle is to irradiate the material with a neutron source (such as a DT neutron generator). The atomic nuclei of each element in the material undergo capture or inelastic scattering reactions with the neutrons, transiently emitting gamma rays with characteristic energies. By measuring the gamma energy spectrum and analyzing the characteristic peak information using a detector, the mass fraction of each element in the material can be determined. Currently, PGNAA technology is widely used in the online composition analysis of lumpy and bulk industrial materials such as iron ore, coal, and cement raw materials.

[0003] In practical PGNAA detection of multi-element mixtures, the inversion of elemental mass fractions from gamma-detector eigenvalues ​​is a typical strongly nonlinear inverse problem. The main sources of this nonlinearity include:

[0004] (1) Neutron self-shielding effect: Some elements in the material (such as iron, chlorine, boron, etc.) have a large absorption or scattering cross section for neutrons, which will significantly reduce the effective thermal neutron flux inside the material, causing the characteristic gamma yield to deviate from the ideal linear relationship with the element content.

[0005] (2) Gamma self-absorption effect: The characteristic gamma rays emitted by nuclear reactions are absorbed and attenuated by the material itself during outward transmission. The attenuation intensity is closely related to the gamma energy, the effective atomic number, thickness and density of the material, which further exacerbates the nonlinearity between the detection signal and the element content.

[0006] The two physical effects mentioned above are superimposed on each other, and combined with matrix interference between multiple elements, resulting in a significant nonlinearity, coupling and matrix correlation between the gamma characteristic peak count and the element mass fraction.

[0007] To address the aforementioned nonlinear inversion problem, existing quantitative analysis methods typically employ a single-element independent fitting strategy, establishing an empirical correction curve or polynomial fitting model for each element's "eigenvalue-content". The core assumption of this approach is that the mutual influence between elements is negligible, thus failing to consider the nonlinear coupling effects caused by neutron self-shielding, gamma self-absorption, and matrix interference between multiple elements. Under ideal conditions where the material composition is relatively fixed and the matrix variation is minimal, the single-element independent fitting method can achieve certain results. However, in complex industrial materials such as iron ore, copper slag, and coal, where the content of each element fluctuates greatly and the matrix changes drastically, the quantitative error of this method increases significantly, its generalization ability is insufficient, and it is difficult to meet the requirements of online high-precision detection. Therefore, this application proposes a multi-element gamma-ray energy spectrum quantitative inversion algorithm based on polynomial forward modeling and Newton's iteration. Summary of the Invention

[0008] The purpose of this invention is to address the shortcomings of existing quantitative analysis methods, which typically employ a single-element independent fitting strategy and fail to consider the nonlinear coupling effects caused by neutron self-shielding, gamma self-absorption, and matrix interference between multiple elements. This invention proposes a multi-element gamma spectrum quantitative inversion algorithm based on polynomial forward modeling and Newton iteration.

[0009] Firstly, this application provides a quantitative inversion algorithm for multi-element gamma spectrum based on polynomial forward modeling and Newton iteration, comprising the following steps:

[0010] S1. Obtain the gamma spectrum obtained from neutron activation measurement, and extract the relevant components. One-to-one correspondence of elements One gamma detection feature value, ;

[0011] S2, based on Establish a system based on the gamma detection feature values ​​and the corresponding elemental true quality scores of the samples. Metapolynomial forward model:

[0012]

[0013] in, for Mass fraction of the elements For the first The gamma-ray detection eigenvalues ​​of each element, Indicates the first An orthogonal mapping function with elements;

[0014] S3. Based on the gamma detection feature value and the true mass fraction of the element, establish... Direct polynomial inverse model:

[0015]

[0016] in, Indicates the first A reverse mapping function for elements. For the first Mass fraction of the elements express The gamma detection feature value is used to calculate the initial iterative value of the element mass fraction by substituting the gamma detection feature value of the sample to be tested into the inverse model. ;

[0017] S4. Based on the aforementioned forward model, construct the residual function:

[0018]

[0019] in, Indicates the first A residual function, No. Measured values ​​of gamma eigenvalues This is represented as the solution vector of the current element's quality score. The solution obtained by substituting into the forward model is the first... Theoretical calculated values ​​of elemental gamma detection eigenvalues;

[0020] With the initial iteration value Starting with Newton's iteration formula:

[0021]

[0022] Perform inversion to solve; where, Indicates the first The solution vector after the second Newton iteration. Indicates the first The solution vector after the second Newton iteration. The inverse of the Jacobian matrix, the Jacobian matrix Calculation using central difference numerical differentiation, Represented as the first The residual vector, composed of residual functions of each order, is applied to the solution after each iteration, and a physical constraint projection is applied to the solution to ensure that the solution satisfies... and .

[0023] Optionally, in step S4, the Newton iteration process also employs an adaptive step-size backoff strategy, whereby the step size is reduced until the residual norm decreases after iteration when the residual norm increases.

[0024] Optionally, in step S2, the coefficients of the n-variable polynomial forward model are obtained by least squares fitting.

[0025] Optionally, in step S1, the method for extracting gamma detection feature values ​​is either the peak area method or the full spectrum analysis method.

[0026] Optionally, the physical constraint projection specifically includes: truncating the mass fraction of each element in the iteration result to the [0,1] interval, and according to... Perform a projection operation with normalized scaling.

[0027] Optionally, step S5 is also included: using the mean absolute error (MAE) and / or root mean square error (RMSE) to evaluate the accuracy of the inversion results.

[0028] Optionally, when applied to neutron activation analysis of copper slag materials, n=2, and the two elements are copper and iron, respectively; the polynomial forward model is used to characterize the matrix interference effect of iron mass fraction on the gamma detection characteristic value of copper, and the direct polynomial inverse model is used to calculate the initial iterative values ​​of copper and iron mass fractions.

[0029] Secondly, this application provides a multi-element gamma-ray spectral quantitative inversion system based on polynomial forward modeling and Newton iteration, including:

[0030] The feature extraction module is used to obtain the gamma energy spectrum obtained from neutron activation measurement and extract the gamma detection feature values ​​of each element;

[0031] The model building module is used to build an n-variable polynomial forward model and an n-variable direct polynomial inverse model based on the sample dataset.

[0032] The initial value calculation module is used to input the gamma detection feature value of the sample to be tested into the direct polynomial inverse model to obtain the initial iterative value of the element mass fraction;

[0033] The Newton iteration solution module is used to construct a residual function based on the forward model, perform Newton iteration starting from the initial iteration value, wherein the Jacobian matrix is ​​calculated using central difference numerical differentiation, and physical constraint projection is applied after each iteration to output the final element mass fraction inversion result.

[0034] Thirdly, this application provides an electronic device including a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the processor executes the program to implement the method described in the first aspect.

[0035] Fourthly, this application provides a computer-readable storage medium having a computer program stored thereon, which, when executed by a processor, implements the method described in the first aspect.

[0036] Compared with the prior art, this application includes at least one of the following beneficial technical effects:

[0037] By constructing a high-order polynomial forward model, the nonlinear coupling relationship between element mass fraction and gamma eigenvalue is characterized, compensating for the biases caused by neutron self-shielding, gamma self-absorption and multi-element matrix interference, and the inversion accuracy is higher than that of the single-element independent fitting method.

[0038] A direct polynomial inverse model is used to provide initial values ​​for Newton iteration. The Jacobian matrix is ​​calculated by central difference numerical differentiation and an adaptive step-size backoff strategy is combined to ensure stable convergence of the iteration. After each iteration, a physical constraint projection is applied to ensure that the inversion results meet the mass fraction constraint conditions and maintain reliable performance under complex working conditions.

[0039] The model considers the interactions between multiple elements and matrix effects, and is applicable to different materials such as iron ore, coal, and cement, as well as varying matrix environments, without relying on fixed operating conditions.

[0040] Inversion based on forward physical modeling yields results that conform to the objective laws of material composition, requiring no complex parameter adjustments and can be directly integrated into the PGNAA online analysis system.

[0041] In summary, this invention effectively improves the accuracy of quantitative inversion of multi-element gamma spectroscopy, enhances the robustness and convergence stability of the algorithm under complex materials, improves the model's generalization ability to different matrix environments, and has good engineering applicability. Attached Figure Description

[0042] Figure 1 This is a flowchart of a multi-element gamma-ray spectroscopy quantitative inversion algorithm based on polynomial forward modeling and Newton iteration.

[0043] Figure 2 The results show a comparison between the algorithm of this invention (iterative inversion) and the traditional algorithm (single-element quadratic inversion). Detailed Implementation

[0044] The following specific examples illustrate the implementation of the present invention. Those skilled in the art can easily understand other advantages and effects of the present invention from the content disclosed in this specification. The present invention can also be implemented or applied through other different specific embodiments, and various details in this specification can also be modified or changed based on different viewpoints and applications without departing from the spirit of the present invention. It should be noted that, unless otherwise specified, the following embodiments and features described therein can be combined with each other.

[0045] Example

[0046] like Figure 1As shown, this invention proposes a multi-element gamma-ray energy spectrum quantitative inversion method based on high-order polynomial forward modeling and Newton iteration. This method can fully couple neutron self-shielding, gamma self-absorption, and inter-element matrix interference to achieve high-precision and robust inversion of element mass fractions in complex industrial materials. The specific steps are as follows:

[0047] Symbol definition

[0048] Suppose the elements to be inverted total Species, definition:

[0049] True quality score of elements:

[0050] Gamma-ray spectral elemental characteristic values:

[0051] Physical constraints:

[0052]

[0053]

[0054] For the first Mass fraction of the element.

[0055] S1. Gamma spectrum reading and feature extraction.

[0056] Read the gamma energy spectrum dataset obtained from neutron activation measurements, and extract features from each raw energy spectrum using either the peak area method or full spectrum analysis to obtain the corresponding data. One-to-one correspondence of elements Each gamma detection feature value; simultaneously acquire the corresponding gamma detection feature value for each sample. The true quality scores of each element are used to construct a standardized modeling sample set.

[0057] S2, Construction A polynomial forward model.

[0058] For each element's eigenvalue ,Establish The metapolynomial forward model characterizes the nonlinear coupling mapping relationship between elemental mass fraction and gamma eigenvalues ​​under the combined effects of neutron self-shielding, gamma self-absorption, and matrix effects.

[0059]

[0060] in, for Mass fraction of the elements For the first The mass fraction of each element For the first The gamma-ray detection eigenvalues ​​of each element, Indicates the first The forward mapping function of each element is derived from the mass fraction of multiple elements to calculate the gamma spectrum eigenvalues ​​of the corresponding elements, characterizing the nonlinear positive correlation between element content and gamma detection eigenvalues ​​during neutron activation, and the least squares method is used to fit the polynomial coefficients.

[0061] S3. Construct a direct polynomial inverse model.

[0062] mass fraction of each element ,Establish Direct polynomial inverse model:

[0063]

[0064] in, For the first Mass fraction of the elements express Gamma detection eigenvalues, Indicates the first The inverse mapping function of each element is directly derived from the measured gamma-ray spectrum eigenvalues ​​to obtain the mass fraction of the corresponding element, thus characterizing the inverse mapping relationship from gamma-ray detection eigenvalues ​​to element mass fractions. Let... The solution vector is composed of the element mass fractions, and the output of the inverse model is used as the initial value for Newton's iteration. This is to improve the speed and stability of iterative convergence.

[0065] S4, Newton's iterative optimization inversion.

[0066] Based on the forward model, construct the residual function:

[0067]

[0068] in, Indicates the first A residual function, No. Measured values ​​of gamma eigenvalues This is represented as the solution vector of the current element's quality score. Substitution The first polynomial obtained by solving the polynomial forward model Theoretical calculated values ​​of elemental gamma detection eigenvalues;

[0069] Solve using Newton's iterative formula:

[0070]

[0071] Among them, the Jacobian matrix The solution is obtained using central difference numerical differentiation. Indicates the first The solution vector after the second Newton iteration. Indicates the first The solution vector after the second Newton iteration. Describes the inverse of the Jacobian matrix. Represented as the first The residual vector, composed of residual functions of each order, is generated in each iteration and does not require analytical differentiation. Physical constraint projection is performed after each iteration to ensure that the solution is valid. An adaptive step-size backoff strategy is adopted to ensure stable convergence of the iteration.

[0072] S5. Accuracy assessment and result output.

[0073] The inversion results are quantitatively evaluated using the mean absolute error (MAE) and root mean square error (RMSE).

[0074]

[0075]

[0076] in, This represents the total number of inversion samples used in the accuracy assessment, i.e., the number of data points used for error calculation. This represents the predicted value of the inverted sample calculated using the algorithm model. This represents the true value of the inverted sample.

[0077] Application examples

[0078] This invention uses the PGNAA detection of copper slag as an application scenario to verify the best practice. Based on the field conditions, the material mainly contains two elements with high content and high activation cross-sections: Fe and Cu. Changes in Fe content significantly interfere with the spectral accuracy of Cu. Elements such as Ca, Si, and Al have low content or very small activation cross-sections, and their impact on the energy dispersive spectral characteristics is negligible. Therefore, this embodiment only considers the interaction between Cu and Fe, constructing a binary inversion model with n=2.

[0079] The range of element content is determined based on the on-site working conditions:

[0080] 1) Cu element mass fraction: 0~9 wt%

[0081] 2) Fe element mass fraction: 45~54 wt%

[0082] 3) The mass fraction of O element is determined by... Calculations show that its impact is negligible.

[0083] The gamma spectrum response was simulated using the MCNP program:

[0084] 1) Construct 100 sets of simulation samples according to the above content range using grid difference as calibration samples for training the forward and inverse models;

[0085] 2) Twenty test samples were randomly generated within the range of Cu and Fe content;

[0086] 3) The method of this invention and the traditional spectral decomposition algorithm were used to invert the 20 sets of samples respectively;

[0087] 4) Using the MCNP simulation true value as a reference, calculate the MAE and RMSE to complete the comparison.

[0088] like Figure 2 Experimental results show that the traditional algorithm, i.e., single-element quadratic inversion, achieves a Cu elemental spectrum resolution accuracy of 0.081% MAE and 0.109%, while the method of this invention, i.e., iterative inversion, achieves a Cu elemental spectrum resolution accuracy of 0.043% MAE and 0.053%. This invention effectively suppresses Fe's matrix interference on Cu, significantly reduces nonlinear deviation, and exhibits significantly better accuracy and robustness, meeting the requirements for on-site online detection of copper slag.

[0089] It is worth noting that this invention, by constructing a high-order polynomial forward model, fully characterizes the nonlinear coupling relationship caused by neutron self-shielding, gamma self-absorption, and multi-element matrix interference, thereby significantly improving the accuracy of element mass fraction inversion. Simultaneously, by utilizing a direct polynomial inverse model to provide reliable initial values ​​for Newton iterations, coupled with a numerical differential Jacobian matrix and an adaptive step-size backoff strategy, stable convergence of the iteration process is ensured, avoiding divergence due to improper initial values ​​or excessively large step sizes. Furthermore, the physical constraint projection applied after each iteration ensures that the inversion results always conform to the objective laws of material composition, enhancing the algorithm's generalization ability in different materials and varying matrix environments, such as iron ore, coal, and cement. Overall, this invention requires no complex parameter adjustments and can be directly integrated into the PGNAA online analysis system, demonstrating good engineering practicality.

[0090] The above specific embodiments are merely several optional embodiments of the present invention. Based on the technical solutions of the present invention and the relevant teachings of the above embodiments, those skilled in the art can make various alternative improvements and combinations to the above specific embodiments.

Claims

1. A multi-element gamma-ray energy spectrum quantitative inversion algorithm based on polynomial forward modeling and Newton iteration, characterized in that, Includes the following steps: S1. Obtain the gamma spectrum obtained from neutron activation measurement, and extract the relevant components. One-to-one correspondence of elements One gamma detection feature value, ; S2, based on Establish a system based on the gamma detection feature values ​​and the corresponding true quality scores of the elements in the sample. Multivariate polynomial forward model; S3. Based on the gamma detection feature value and the true mass fraction of the element, establish... The direct polynomial inverse model is used to calculate the initial iterative value of the element mass fraction by substituting the gamma detection eigenvalues ​​of the sample into the inverse model. ; S4. Based on the aforementioned forward model, construct the residual function; With the initial iteration value Starting with Newton's iterative formula, an inversion solution is performed. After each iteration, a physical constraint projection is applied to the solution to ensure that the solution satisfies... and .

2. The multi-element gamma-ray energy spectrum quantitative inversion algorithm based on polynomial forward modeling and Newton iteration as described in claim 1, characterized in that, In step S4, the Newton iteration process also adopts an adaptive step size back-off strategy. When the residual norm increases after iteration, the step size is reduced until the residual decreases.

3. The multi-element gamma-ray energy spectrum quantitative inversion algorithm based on polynomial forward modeling and Newton iteration as described in claim 1, characterized in that, In step S2, establish The polynomial forward model is as follows: in, for Mass fraction of the elements For the first The gamma-ray detection eigenvalues ​​of each element, Indicates the first The forward mapping function of n elements, and the coefficients of the n-variable polynomial forward model are obtained by least squares fitting.

4. The multi-element gamma-ray energy spectrum quantitative inversion algorithm based on polynomial forward modeling and Newton iteration as described in claim 1, characterized in that, In step S1, the method for extracting gamma detection feature values ​​is either the peak area method or the full spectrum analysis method; in step S3... The direct polynomial inverse model is: in, Indicates the first Inverse mapping function for elements, For the first Mass fraction of the elements express Gamma detection feature values.

5. The multi-element gamma-ray energy spectrum quantitative inversion algorithm based on polynomial forward modeling and Newton iteration as described in claim 1, characterized in that, The residual function is: in, Indicates the first A residual function, No. Measured values ​​of gamma eigenvalues This is represented as the solution vector of the current element's quality score. The solution obtained by substituting into the forward model is the first... Theoretical calculated values ​​of elemental gamma detection eigenvalues; Newton's iterative formula is: in, Indicates the first The solution vector after the second Newton iteration. Indicates the first The solution vector after the second Newton iteration. The inverse of the Jacobian matrix, the Jacobian matrix Calculation using central difference numerical differentiation, Represented as the first The residual vector formed by the residual functions of each order in each iteration; The physical constraint projection specifically includes: truncating the mass fraction of each element in the iteration result to the [0,1] interval, and according to... Perform a projection operation with normalized scaling.

6. The multi-element gamma-ray energy spectrum quantitative inversion algorithm based on polynomial forward modeling and Newton iteration as described in claim 1, characterized in that, It also includes step S5: using the mean absolute error (MAE) and / or root mean square error (RMSE) to evaluate the accuracy of the inversion results.

7. The multi-element gamma-ray energy spectrum quantitative inversion algorithm based on polynomial forward modeling and Newton iteration as described in claim 1, characterized in that, When applied to neutron activation analysis of copper slag materials, n=2, and the two elements are copper and iron, respectively; the polynomial forward model is used to characterize the matrix interference effect of iron mass fraction on the gamma detection characteristic value of copper, and the direct polynomial inverse model is used to calculate the initial iterative values ​​of copper and iron mass fractions.

8. A multi-element gamma-ray energy spectrum quantitative inversion system based on polynomial forward modeling and Newton iteration, characterized in that, include: The feature extraction module is used to obtain the gamma energy spectrum obtained from neutron activation measurement and extract the gamma detection feature values ​​of each element; The model building module is used to build an n-variable polynomial forward model and an n-variable direct polynomial inverse model based on the sample dataset. The initial value calculation module is used to input the gamma detection feature value of the sample to be tested into the direct polynomial inverse model to obtain the initial iterative value of the element mass fraction; The Newton iteration solution module is used to construct a residual function based on the forward model, perform Newton iteration starting from the initial iteration value, wherein the Jacobian matrix is ​​calculated using central difference numerical differentiation, and physical constraint projection is applied after each iteration to output the final element mass fraction inversion result.

9. An electronic device comprising a memory, a processor, and a computer program stored in the memory and executable on the processor, characterized in that, When the processor executes the program, it implements the algorithm according to any one of claims 1 to 7.

10. A computer-readable storage medium having a computer program stored thereon, characterized in that, When the program is executed by the processor, it implements the algorithm according to any one of claims 1 to 7.

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