An underwater gamma-ray spectrum inversion method based on double regularization and parameter bootstrap
By constructing a system response matrix and introducing sparsity and smoothness regularization constraints, combined with a parameter bootstrapping mechanism, the problem of gamma spectrum overlap caused by scattering effects in underwater NaI detectors was solved, achieving quantitative evaluation and reliability improvement of inversion results, which is suitable for nuclear safety monitoring.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- SOUTHWEAT UNIV OF SCI & TECH
- Filing Date
- 2026-06-15
- Publication Date
- 2026-07-14
Smart Images
Figure CN122386362A_ABST
Abstract
Description
Technical Field
[0001] This invention relates to the field of nuclear radiation detection and data processing technology, specifically to an underwater gamma spectrum inversion method based on dual regularization and parameter bootstrapping. Background Technology
[0002] The statements in this section are provided only as background information in connection with this disclosure and may not constitute prior art.
[0003] In underwater in-situ detection, NaI(Tl) scintillator detectors are widely used due to their strong environmental adaptability and low cost. However, due to the inherently low energy resolution of NaI detectors, coupled with the strong Compton scattering and energy attenuation effect of water on gamma rays, the measured gamma spectrum often exhibits characteristics of a continuous spectrum, severely broadened and overlapping characteristic peaks, and being submerged by a strong scattering background.
[0004] Unfolding analysis aims to recover the true incident information of nuclides through deconvolution. However, in practical applications, underwater energy spectrum inversion is a typical ill-posed inverse problem due to the extremely large condition number of the system response matrix. Existing standard iterative inversion algorithms, such as Maximum Likelihood Expectation Maximization (MLEM), are prone to overfitting to Poisson statistical noise when processing low signal-to-noise ratio data, thus introducing non-physical oscillations and spurious peaks into the reconstructed energy spectrum. Furthermore, traditional algorithms typically provide only a single mathematical point estimate and lack a rigorous uncertainty quantification mechanism, making it impossible to assess the reliability of the spectral results in complex underwater environments.
[0005] Therefore, there is an urgent need for a novel inversion method that can effectively suppress pathological oscillations in a strong scattering background while providing a quantitative uncertainty assessment. Summary of the Invention
[0006] The purpose of this invention is: To address the problems of traditional inversion algorithms in low signal-to-noise ratio underwater environments, which are prone to non-physical oscillations and spurious peaks due to ill-conditioning and lack of confidence quantification mechanisms for inversion results, this paper proposes an underwater gamma spectrum inversion method based on dual regularization and parameter bootstrapping. The objective function, which combines Poisson negative log-likelihood with sparsity and smoothing dual regularization, suppresses ill-conditioning oscillations, and a parameterized bootstrapping resampling mechanism enables a rigorous quantitative assessment of the uncertainty of the inversion results.
[0007] The technical solution of the present invention is as follows: An underwater gamma spectrum inversion method based on dual regularization and parameter bootstrapping includes: Step S1: Acquire underwater measured data and construct the system response matrix; including: Obtain measured energy spectrum data from a NaI detector in a simulated underwater environment in the laboratory; Monte Carlo simulation was used to establish a complete system physical model including the water medium, waterproof packaging and NaI detector, to simulate the energy deposition process of gamma rays in the underwater environment and obtain the simulation results of pulse amplitude spectrum. The measured energy spectrum data was used to extract the full width at half maximum (FWHM) of the NaI detector at different characteristic energies, and a Gaussian energy broadening function was fitted. The simulated pulse amplitude spectrum results were corrected by Gaussian energy broadening and then normalized to construct a system response matrix that combines underwater scattering characteristics with the energy resolution properties of a real detector. ; Step S2: Construct the maximum likelihood estimation objective function that incorporates dual regularization constraints; including: Using the measured energy spectrum data and the system response matrix As input, the Poisson negative log-likelihood function is used as the data fidelity term. At the same time, a sparsity regularization term based on the L1 norm and a smoothness regularization term based on the second difference are introduced to construct the regularized maximum likelihood estimation objective function. Step S3: Iteratively solve for the preliminary inversion energy spectrum; including: An initial estimated spectrum is set, and the regularized maximum likelihood estimation objective function is minimized using a gradient descent iterative algorithm. A non-negativity truncation constraint is applied after each update of the current estimated spectrum along the gradient direction until the iterative convergence condition is met, at which point the preliminary inverted energy spectrum is output. ; Step S4: Uncertainty assessment based on parameterized bootstrapping; including: Based on the preliminary inversion energy spectrum and the system response matrix Forward calculation of the theoretically expected observation spectrum; Using each element of the theoretically expected observation spectrum as the mean parameter of the Poisson distribution, independent random resampling is performed to generate multiple sets of simulated observation spectra; For each set of simulated observation spectra, perform the iterative solution described in step S3 independently, summarize all corresponding inversion results, statistically calculate the confidence interval of each energy channel, and complete the quantitative assessment of the uncertainty of the inversion results.
[0008] Furthermore, in step S1, the Gaussian energy broadening function is fitted using the following formula:
[0009] In the formula: The energy of a single-energy gamma ray is Half the height and width at the location; Energy of a single-energy gamma ray, measured in keV; , , These are the detector characteristic coefficients obtained by least-squares fitting of the measured energy spectrum data; The system response matrix for A dimensional matrix, whose dimensional matrix is the first dimensional matrix. The column is composed of energy. The pulse amplitude spectrum of monoenergetic gamma rays obtained through Monte Carlo simulation is constructed by Gaussian energy broadening correction and normalization; the system response matrix is... matrix elements Indicates unit injection energy as Gamma rays in the detector The probability of generating a response count.
[0010] Furthermore, in step S2, the specific form of the regularized maximum likelihood estimation objective function is as follows:
[0011] In the formula: Let the regularized maximum likelihood estimation objective function be denoted as . The measured energy spectrum data constitutes the 3D data vector; To determine the true incident energy spectrum Dimensional estimation of the energy spectrum vector; For the data fidelity item; For the sparse regularization term; For the smoothness regularization term; , These are the sparse regularization hyperparameter and the smoothness regularization hyperparameter, respectively.
[0012] Furthermore, the data fidelity item The formula for the Poisson negative log-likelihood function used is as follows:
[0013] In the formula: For the data vector In the Actual counts of the Dao; Based on the estimated energy spectrum vector and the system response matrix The first matrix multiplication is calculated as follows Expected count.
[0014] Furthermore, the sparse regularization term With the smoothness regularization term The formulas are as follows:
[0015]
[0016] In the formula: The estimated energy spectrum vector The Each element.
[0017] Furthermore, the specific iterative solution process of step S3 includes: Initial estimated spectrum Set to a flat spectrum where all values are constants; In the In each iteration, the regularized maximum likelihood estimation objective function is calculated. For the current estimated spectrum gradient The formula for calculating the gradient is:
[0018] In the formula: The system response matrix Transpose of; This represents an element-wise division operation between vectors; It is a symbolic function; It is a second-order difference matrix; Adaptive learning rate The estimated spectrum is updated along the gradient direction according to the following formula, and the nonnegativity truncation constraint is applied:
[0019] The iteration stops when the change in the estimated spectrum between two consecutive iterations is less than the convergence threshold or the number of iterations reaches the preset upper limit, and the final result is output. As the preliminary inversion energy spectrum .
[0020] Further, in step S4, the process of generating the simulated observation spectrum includes: Based on the preliminary inversion energy spectrum With the system response matrix Forward calculation of theoretically expected observation spectrum ; Based on the theoretically predicted observation spectrum Each element is used as the mean parameter of the Poisson distribution, and independent random sampling is performed to generate... The simulated observation spectrum described in the group , ,in The value of is not less than 200.
[0021] Furthermore, in step S4, the process of statistically calculating the confidence intervals for each energy channel includes: For each group of simulated observation spectra The iterative solution described in step S3 is performed independently using the same regularization parameters as the main inversion to obtain the corresponding inversion energy spectrum. ; Summary The inversion results are used to calculate each energy channel using the following formula. Statistical mean with standard deviation :
[0022]
[0023] The [2.5%, 97.5%] quantile intervals of each energy channel count were extracted as 95% confidence intervals to complete the quantitative assessment.
[0024] Furthermore, the method also includes step S5 of quantitatively evaluating the inversion results: Calculate the center energy of each characteristic peak in the inverted energy spectrum Standard characteristic energy of the corresponding nuclide relative deviation between ; The peak-Compton ratio is obtained by calculating the ratio of the maximum count value of the full-energy peak to the count value of the Compton edge in the inverted energy spectrum, which is used to quantitatively assess the ability to suppress strong underwater scattering background.
[0025] Furthermore, the method is applied to various typical underwater nuclear safety monitoring scenarios. , or The identification and activity inversion are performed, and the sparse regularization hyperparameters are adaptively selected using the L-curve criterion or generalized cross-validation method. With the smoothness regularization hyperparameter .
[0026] Compared with existing technologies, the advantages of this invention are: 1. This invention constructs the system response matrix by combining Monte Carlo simulation and measured data. It incorporates the Compton scattering effect of the water medium and the true energy resolution characteristics of the NaI detector into the physical model, which solves the defects of simple simulation ignoring the true resolution and simple measurement failing to cover the full energy range. This provides a mathematical basis with extremely high physical accuracy for subsequent accurate inversion.
[0027] 2. This invention innovatively introduces a dual regularization constraint that integrates the L1 norm and second-order difference. Specifically, sparsity regularization effectively suppresses the generation of spurious peaks and sharpens the characteristic full-energy peaks of radionuclides in the context of strong underwater Compton scattering; smoothness regularization effectively suppresses the ill-conditioned nature caused by the high condition number of the system response matrix and eliminates non-physical high-frequency oscillations in the inversion results. These two mechanisms work together to address the technical bias of traditional MLEM algorithms, which overfit statistical noise due to a lack of physical constraints.
[0028] 3. This invention is the first to introduce the parametric bootstrap mechanism into the field of underwater gamma-ray spectrum inversion. By utilizing the physical property that the energy spectrum count follows a Poisson statistical distribution for resampling, it overcomes the limitation of traditional algorithms that can only output a single mathematical point estimate. This enables a rigorous quantitative assessment of the uncertainty of the inversion results and constructs a 95% confidence interval, greatly improving the credibility of nuclide identification results and the reliability of nuclear safety analysis.
[0029] 4. The regularized maximum likelihood estimation algorithm architecture described in this invention is seamlessly adapted to parallel computing and can be efficiently ported and deployed on high-performance multi-core embedded processor platforms, meeting the stringent engineering requirements of underwater in-situ monitoring systems for real-time data processing. Attached Figure Description
[0030] To more clearly illustrate the technical solutions in the embodiments of the present invention, the accompanying drawings used in the description of the embodiments will be briefly introduced below. Obviously, the accompanying drawings described below are only some embodiments recorded in the embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings.
[0031] Figure 1 This is a flowchart illustrating the overall method of the present invention; Figure 2 This is a flowchart of the RMLE inversion algorithm of the present invention. Figure 3 This is a flowchart of the parameterized bootstrap uncertainty evaluation process of the present invention. Detailed Implementation
[0032] It should be noted that relational terms such as "first" and "second" are used merely 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 a process, method, article, or apparatus. Without further limitations, an element defined by the phrase "comprising one..." does not exclude the presence of other identical elements in the process, method, article, or apparatus that includes said element.
[0033] The features and performance of the present invention will be further described in detail below with reference to embodiments.
[0034] Example 1 In underwater in-situ radiation monitoring scenarios, the measured energy spectrum of the NaI detector was obtained. Compared with the true incident energy spectrum The two satisfy linear discrete equations: ( (Statistical noise). Due to the strong Compton scattering and energy attenuation effect of water on gamma rays, the system response matrix... The condition number is extremely large. This system of equations is a typical ill-posed inverse problem. Directly inverting the equations or using the traditional maximum likelihood expectation maximization (MLEM) iterative solution will lead to a severe amplification of statistical noise under low signal-to-noise ratio conditions, producing non-physical oscillation peaks and spurious peaks. At the same time, traditional peak-finding algorithms lack a rigorous uncertainty quantification mechanism.
[0035] Based on the above principles, such as Figure 1 As shown, this embodiment provides an underwater gamma-ray spectrum inversion method based on dual regularization and parameter bootstrapping, specifically including the following steps: Step S1: Acquire underwater measurement data and construct the system response matrix; specifically including: Obtain measured energy spectrum data from a NaI detector in a simulated underwater environment in the laboratory; A full-system physical model, including the water medium, waterproof packaging, and the NaI detector, was established using Monte Carlo simulation to simulate the energy deposition process of gamma rays in the underwater environment and obtain the pulse amplitude spectrum simulation results, i.e., the simulated energy spectrum. The measured energy spectrum data is used to extract the full width at half maximum (FWHM) of the NaI detector at different characteristic energies, and a Gaussian energy broadening function is fitted, which means data preprocessing and energy spectrum technology redistribution are performed on the measured energy spectrum data. The simulated pulse amplitude spectrum results were corrected by Gaussian energy broadening and then normalized to construct a system response matrix that combines underwater scattering characteristics with the energy resolution properties of a real detector. That is, the simulated pulse amplitude spectrum results are subjected to Gaussian energy broadening. The unbroadened response matrix is then broadened to obtain a Gaussian broadened matrix. Finally, a system response matrix that combines underwater scattering characteristics with the energy resolution characteristics of a real detector is constructed. .
[0036] In the specific implementation of the project, the detailed operation process and working principle of step S1 are as follows: First, using an underwater detection platform built in the laboratory, a 200cm×200cm×200cm polyethylene water tank was filled with simulated seawater and a standard gamma radiation source was placed inside. Measured energy spectrum data were acquired by the NaI detector in the underwater environment. The detector's... (81keV, 356keV) (662keV) The full width at half maximum (FWHM) at different characteristic energies of standard radioactive sources such as (1173 keV, 1332 keV) is calculated. The Gaussian energy broadening function is obtained by fitting the data using the least squares method.
[0037] In the formula: The energy of a single-energy gamma ray is Half the height and width at the location; Energy of a single-energy gamma ray, measured in keV; , , These are the detector characteristic coefficients obtained by least-squares fitting of the measured energy spectrum data.
[0038] Secondly, a full-system simulation model was established using Monte Carlo simulation software. The entire energy range (e.g., 50 keV to 3000 keV) was set at uniform logarithmic intervals. One single-energy gamma-ray energy point ( Monte Carlo simulations were performed one by one to obtain the simulation results of the pulse amplitude spectrum of the detector in the underwater environment.
[0039] Subsequently, the Compton scattering effect of the water medium and the true energy resolution characteristics of the NaI detector were simultaneously incorporated into the physical model: a Gaussian broadening correction was applied to each simulated spectrum, that is, for each channel in the simulated spectrum... The count at each address is replaced by a Gaussian-weighted sum centered at that address, followed by normalization. The resulting system response matrix... for A dimensional matrix, whose dimensional matrix is the first dimensional matrix. The corresponding energy of the column is The broadened correction normalization result of monoenergetic gamma rays; its matrix elements It has clear physical and statistical significance, representing the energy per unit injection. Gamma rays in the detector The probability of generating a response count.
[0040] This joint construction method solves the problems of ignoring the resolution of real detectors in simple simulation and the inability of simple field measurement to cover the entire energy range, and significantly improves the physical accuracy of the response matrix.
[0041] Step S2: Construct the maximum likelihood estimation objective function that incorporates dual regularization constraints; specifically including: Using the measured energy spectrum data and the system response matrix As input, the Poisson negative log-likelihood function is used as the data fidelity term. Simultaneously, a sparsity regularization term based on the L1 norm and a smoothness regularization term based on the second-order difference are introduced to construct the regularized maximum likelihood estimation (RMLE) objective function; its specific form is as follows:
[0042] In the formula: Let the regularized maximum likelihood estimation objective function be denoted as . The measured energy spectrum data constitutes the 3D data vector; To determine the true incident energy spectrum Dimensional estimation of the energy spectrum vector; , These are the sparse regularization hyperparameter and the smoothness regularization hyperparameter, respectively.
[0043] The technical principle and the technical problem solved by this step are as follows: (1) Data fidelity item Because nuclear radiation counts physically follow a Poisson distribution, the Poisson negative log-likelihood function has more accurate statistical properties when handling low count rate data compared to the conventional least squares method. Its formula is: In the formula, For actual counts, This is the expected count.
[0044] (2) Sparsity regularization term The formula is: This constraint causes the true incident spectrum to tend to be sparse, which is consistent with the physical prior of the linear gamma spectrum of the nuclide. It can effectively suppress the generation of spurious peaks in the background of strong underwater scattering, sharpen the characteristic full-energy peaks of radionuclides, and improve the energy spectrum resolution.
[0045] (3) Smoothness regularization term The formula is: This constraint restricts the second-order difference of adjacent channel counts, effectively suppresses the ill-conditioned nature caused by the high condition number of the system response matrix, eliminates non-physical high-frequency oscillations in the inversion results, and ensures the continuous smoothness of the inversion energy spectrum.
[0046] It is worth noting that the choice of regularization hyperparameters is crucial to the inversion quality: Excessive suppression of weak peaks at the conference Excessive hyperparameter values can lead to overly smoothed energy spectra, resulting in a loss of resolution. In practical engineering, the L-curve criterion or generalized cross-validation (GCV) method can be used to adaptively select the optimal combination of hyperparameters.
[0047] Step S3: Iteratively solve for the preliminary inversion energy spectrum; such as Figure 2 As shown, it specifically includes: Set the initial estimated spectrum The spectrum is a flat spectrum with all constants (such as constant 1) to avoid introducing artificial prior bias; the regularized maximum likelihood estimation objective function is minimized by the gradient descent iterative algorithm.
[0048] In the In the next iteration, based on the current estimated spectrum and system response matrix Forward calculation of expected observation spectrum and with measured energy spectrum data vector With the expected observation spectrum The element-wise ratios are used to construct the Poisson likelihood gradient, and the objective function is calculated with respect to the current estimated spectrum. gradient :
[0049] In the formula: This is the matrix transpose. This is element-wise division. It is a symbolic function; It is a second-order difference matrix.
[0050] Then an adaptive learning rate was adopted. The estimated spectrum is updated along the gradient direction, and a non-negativity truncation constraint must be applied to conform to the objective physical law that "the energy spectrum count cannot be negative":
[0051] When the estimated spectral change between two consecutive iterations (e.g., the 2-norm distance) (less than the convergence threshold) (usually taken) If the number of iterations reaches a preset limit, stop iterating and output the final result. As the preliminary inversion energy spectrum .
[0052] Step S4: Uncertainty assessment based on parameterized bootstrapping; such as Figure 3 As shown, it specifically includes: Based on the preliminary inversion energy spectrum and the system response matrix Forward calculation of theoretically expected observation spectrum ; by Each element is used as the mean parameter of the Poisson distribution, and independent random sampling is performed to generate... The simulated observation spectrum described in the group , To ensure statistical stability, The value is usually not less than 200 (e.g., B=200 or 500).
[0053] For each set of simulated observation spectra, perform the RMLE iterative solution described in step S3 independently, and then summarize the results. Group inversion results Calculate each energy channel Statistical mean with standard deviation The [2.5%, 97.5%] quantile intervals of each energy channel count were extracted as 95% confidence intervals.
[0054] The significant technical effect of this step is that it introduces a parameterized bootstrapping mechanism into underwater gamma spectrum inversion for the first time, and uses the characteristic that the energy spectrum count follows a Poisson distribution to perform Monte Carlo resampling. This overcomes the shortcomings of traditional algorithms that only provide a single mathematical point estimate solution, and achieves a rigorous quantitative assessment of the uncertainty of the inversion results.
[0055] Step S5: Quantitatively evaluate the inversion results.
[0056] Calculate the center energy of each characteristic peak in the inverted energy spectrum Standard characteristic energy of the corresponding nuclide relative deviation between ; and the ratio of the maximum count value of the full-energy peak to the count value of the Compton edge (peak-Compton ratio).
[0057] To further verify the effectiveness of this embodiment, underwater measurements were conducted. Taking the (661.6keV) gamma spectrum as an example: In the unprocessed measured spectrum, the full-energy peak is severely submerged by the Compton continuum, with a peak-to-Compton ratio of only about 1.2, resulting in severe oscillations in the traditional MLEM algorithm inversion. The method used in this embodiment (setting...) , After 500 iterations, the full-energy peak was clearly restored, the relative deviation of the characteristic energy was less than 0.5%, and the peak-to-candle ratio was significantly improved to over 8.0. The 95% confidence interval obtained by parameterized bootstrapping (B=500) accurately covered the main features of the real spectrum, proving that the algorithm has a strong ability to suppress strong underwater scattering background and is robust.
[0058] Furthermore, the architecture described in this embodiment has high parallel computing potential, can be compiled and ported to a high-performance multi-core embedded processor platform, and achieves real-time processing of in-situ data through multi-threaded parallel optimization, meeting the real-time requirements of underwater nuclear monitoring engineering applications.
[0059] The embodiments described above merely illustrate specific implementation methods of this application, and while the descriptions are detailed and specific, they should not be construed as limiting the scope of protection of this application. It should be noted that those skilled in the art can make various modifications and improvements without departing from the concept of the technical solution of this application, and these modifications and improvements all fall within the scope of protection of this application.
Claims
1. A method for underwater gamma-ray spectrum inversion based on dual regularization and parameter bootstrapping, characterized in that, include: Step S1: Acquire underwater measured data and construct the system response matrix; including: Obtain measured energy spectrum data from a NaI detector in a simulated underwater environment in the laboratory; Monte Carlo simulation was used to establish a complete system physical model including the water medium, waterproof packaging and NaI detector, to simulate the energy deposition process of gamma rays in the underwater environment and obtain the simulation results of pulse amplitude spectrum. The measured energy spectrum data was used to extract the full width at half maximum (FWHM) of the NaI detector at different characteristic energies, and a Gaussian energy broadening function was fitted. The simulated pulse amplitude spectrum results were corrected by Gaussian energy broadening and then normalized to construct a system response matrix that combines underwater scattering characteristics with the energy resolution properties of a real detector. ; Step S2: Construct the maximum likelihood estimation objective function that incorporates dual regularization constraints; including: Using the measured energy spectrum data and the system response matrix As input, the Poisson negative log-likelihood function is used as the data fidelity term. At the same time, a sparsity regularization term based on the L1 norm and a smoothness regularization term based on the second difference are introduced to construct the regularized maximum likelihood estimation objective function. Step S3: Iteratively solve for the preliminary inversion energy spectrum; including: An initial estimated spectrum is set, and the regularized maximum likelihood estimation objective function is minimized using a gradient descent iterative algorithm. A non-negativity truncation constraint is applied after each update of the current estimated spectrum along the gradient direction until the iterative convergence condition is met, at which point the preliminary inverted energy spectrum is output. ; Step S4: Uncertainty assessment based on parameterized bootstrapping; including: Based on the preliminary inversion energy spectrum and the system response matrix Forward calculation of the theoretically expected observation spectrum; Using each element of the theoretically expected observation spectrum as the mean parameter of the Poisson distribution, independent random resampling is performed to generate multiple sets of simulated observation spectra; For each set of simulated observation spectra, perform the iterative solution described in step S3 independently, summarize all corresponding inversion results, statistically calculate the confidence interval of each energy channel, and complete the quantitative assessment of the uncertainty of the inversion results.
2. The underwater gamma-ray spectrum inversion method based on dual regularization and parameter bootstrapping as described in claim 1, characterized in that, In step S1, the Gaussian energy broadening function is fitted using the following formula: In the formula: The energy of a single-energy gamma ray is Half the height and width at the location; Energy of a single-energy gamma ray, measured in keV; , , These are the detector characteristic coefficients obtained by least-squares fitting of the measured energy spectrum data; The system response matrix for A dimensional matrix, whose dimensional matrix is the first dimensional matrix. The column is composed of energy. The pulse amplitude spectrum of monoenergetic gamma rays obtained through Monte Carlo simulation is constructed by Gaussian energy broadening correction and normalization; the system response matrix is... matrix elements Indicates unit injection energy as Gamma rays in the detector The probability of generating a response count.
3. The underwater gamma-ray spectrum inversion method based on dual regularization and parameter bootstrapping as described in claim 2, characterized in that, In step S2, the specific form of the regularized maximum likelihood estimation objective function is as follows: In the formula: Let the regularized maximum likelihood estimation objective function be denoted as . The measured energy spectrum data constitutes the 3D data vector; To determine the true incident energy spectrum Dimensional estimation of the energy spectrum vector; For the data fidelity item; For the sparse regularization term; For the smoothness regularization term; , These are the sparse regularization hyperparameter and the smoothness regularization hyperparameter, respectively.
4. The underwater gamma-ray spectrum inversion method based on dual regularization and parameter bootstrapping as described in claim 3, characterized in that, The data fidelity item The formula for the Poisson negative log-likelihood function used is as follows: In the formula: For the data vector In the Actual counts of the Dao; Based on the estimated energy spectrum vector and the system response matrix The first matrix multiplication is calculated as follows Expected count.
5. The underwater gamma-ray spectrum inversion method based on dual regularization and parameter bootstrapping as described in claim 3, characterized in that, The sparsity regularization term With the smoothness regularization term The formulas are as follows: In the formula: The estimated energy spectrum vector The Each element.
6. The underwater gamma-ray spectrum inversion method based on dual regularization and parameter bootstrapping as described in claim 3, characterized in that, The specific iterative solution process of step S3 includes: Initial estimated spectrum Set to a flat spectrum where all values are constants; In the In each iteration, the regularized maximum likelihood estimation objective function is calculated. For the current estimated spectrum gradient The formula for calculating the gradient is: In the formula: The system response matrix Transpose of; This represents an element-wise division operation between vectors; It is a symbolic function; It is a second-order difference matrix; Adaptive learning rate The estimated spectrum is updated along the gradient direction according to the following formula, and the nonnegativity truncation constraint is applied: The iteration stops when the change in the estimated spectrum between two consecutive iterations is less than the convergence threshold or the number of iterations reaches the preset upper limit, and the final result is output. As the preliminary inversion energy spectrum .
7. The underwater gamma-ray spectrum inversion method based on dual regularization and parameter bootstrapping as described in claim 1, characterized in that, In step S4, the process of generating the simulated observation spectrum includes: Based on the preliminary inversion energy spectrum With the system response matrix Forward calculation of theoretically expected observation spectrum ; Based on the theoretically predicted observation spectrum Each element is used as the mean parameter of the Poisson distribution, and independent random sampling is performed to generate... The simulated observation spectrum described in the group , ,in The value of is not less than 200.
8. The underwater gamma-ray spectrum inversion method based on dual regularization and parameter bootstrapping as described in claim 7, characterized in that, In step S4, the process of statistically calculating the confidence intervals for each energy channel includes: For each group of simulated observation spectra The iterative solution described in step S3 is performed independently using the same regularization parameters as the main inversion to obtain the corresponding inversion energy spectrum. ; Summary The inversion results are used to calculate each energy channel using the following formula. Statistical mean with standard deviation : The [2.5%, 97.5%] quantile intervals of each energy channel count were extracted as 95% confidence intervals to complete the quantitative assessment.
9. The underwater gamma-ray spectrum inversion method based on dual regularization and parameter bootstrapping according to claim 1, characterized in that, The method further includes step S5, which involves quantitatively evaluating the inversion results: Calculate the center energy of each characteristic peak in the inverted energy spectrum Standard characteristic energy of the corresponding nuclide relative deviation between ; The peak-Compton ratio is obtained by calculating the ratio of the maximum count value of the full-energy peak to the count value of the Compton edge in the inverted energy spectrum, which is used to quantitatively assess the ability to suppress strong underwater scattering background.
10. The underwater gamma-ray spectrum inversion method based on dual regularization and parameter bootstrapping according to claim 3, characterized in that, The method has been applied to various typical underwater nuclear safety monitoring scenarios. , or The identification and activity inversion are performed, and the sparse regularization hyperparameters are adaptively selected using the L-curve criterion or generalized cross-validation method. With the smoothness regularization hyperparameter .