CN109521456B - Gamma radiation source item inversion method and system based on regularization least square method - Google Patents
Gamma radiation source item inversion method and system based on regularization least square method Download PDFInfo
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Abstract
The invention discloses a regularization least square method-based gamma radiation source item inversion method and a system thereof, wherein the method comprises the following steps: s1, carrying out regional division on gamma radiation source items, wherein the radioactivity in each source item division region is uniformly distributed in space; s2, determining an objective function for solving the radioactivity value in each source item division region through an optimization problem according to dose rate measurement data and uncertainty thereof at a plurality of external measurement positions and the contribution of the unit radioactivity of each source item division region to the dose rate at each measurement position; and S3, solving the objective function to obtain the radioactivity value in each source item division region. According to the method and the system provided by the invention, activity data of gamma radiation source items which are relatively stable and are matched with the precision of measurement data are inverted on the basis of a given source item division mode and nuclide species according to the measurement data and the dosage rate contribution matrix and on the basis of a regularization least square principle.
Description
Technical Field
The invention relates to the technical field of radioactive source item characterization in the nuclear facility overhaul and decommissioning processes, in particular to a gamma radiation source item inversion method and system based on a regularization least square method.
Background
The radioactive source item characterisation runs through all stages of the design construction, operation and decommissioning of the nuclear facility, with source item investigations having different goals and effects at different stages. The radiation source item monitoring in the operation stage is an important means for evaluating the operation state and the pollution level of nuclear facilities, and provides basic data for occupational irradiation evaluation, source item and dosage control; meanwhile, in the later operation stage, the development of targeted source item measurement provides important basis for formulating a source item investigation scheme and radioactive calculation program verification in the retirement stage, even prolonging the service life of the unit and the like. The source item investigation of the transition stage is taken as an important work of the stage, and provides a basis for formulating a specific decommissioning scheme. Among different radiation source items, the gamma radiation source item is generally used as an easily-detectable nuclide, provides a basis for source item characterization of other difficultly-detectable nuclides (pure beta and alpha), and generally determines the activity of the difficultly-detectable nuclide by adopting the ratio of the easily-detectable nuclide to the activity of the difficultly-detectable nuclide determined by a sampling analysis result.
In the above radioactive source item characterization of the nuclear facility operation and transition phase, the measurement means adopted for the gamma radiation source item includes source item scanning, radiation imaging, energy spectrum measurement, dose rate measurement, and the like. Under the conditions that nuclide information in a gamma radiation source item is determined, a place with high dose rate, a measurement space is relatively narrow and the like, a dose rate measurement mode is adopted to carry out inversion of the gamma radiation source item.
Because the dose rate measurement result generally contains measurement uncertainty, the uncertainty is influenced by factors such as relative inherent error of measurement equipment, direction correspondence, energy response, field measurement conditions, statistical fluctuation and the like, and the normal matrix condition number of a dose rate response matrix of the system under the condition of given gamma radiation source item division is larger, the phenomenon of overfitting can be caused by directly carrying out the inversion of the gamma radiation source item according to the dose rate measurement result; moreover, due to system characteristics, for example, a large negative correlation exists between adjacent regions of the inversion source term, which may cause a solution oscillation phenomenon, and if the inversion algorithm is not used properly, a negative value may occur.
Disclosure of Invention
Aiming at the defects in the prior art, the invention aims to provide a gamma radiation source item inversion method and system based on a regularized least square method, and gamma radiation source item data which are relatively stable and are matched with the measurement data precision can be inverted through the method and the system.
In order to achieve the purpose, the technical scheme adopted by the invention is as follows:
a gamma radiation source item inversion method based on a regularized least square method comprises the following steps:
s1, carrying out regional division on gamma radiation source items, wherein the radioactivity in each source item division region is uniformly distributed in space;
s2, determining an objective function for solving the radioactivity value in each source item division region through an optimization problem according to dose rate measurement data and uncertainty thereof at a plurality of external measurement positions and the contribution of the unit radioactivity of each source item division region to the dose rate at each measurement position;
and S3, solving the target function to obtain a corresponding formal solution, adding a regularization term on the basis of the formal solution to obtain a general solution, and solving according to a deviation criterion posterior strategy to obtain the radioactivity value in each source term division region.
Further, in the above-mentioned method for inverting a gamma radiation source term based on the regularized least square method, in step S2, the objective function includes:
wherein, A = (A) 1 ，A 2 ，…，A M ) T Dividing regions for M source itemsA vector of radioactivity values of (c), D = (D) 1 ，D 2 ，…，D N ) T A vector of dose rate measurement data at N measurement positions, E = (ε) ij )，i＝<1，N>，j＝<1，M>N is more than or equal to M, and is a dosage rate contribution matrix formed by the contribution of the unit radioactivity of the divided regions of M source items to the dosage rates at N measurement positions, S D ＝diag(σ 1 2 ,σ 2 2 ,…,σ N 2 ) An uncertainty matrix is formed of the uncertainties of the measured dose rate data at the N measurement locations.
Further, according to the gamma radiation source term inversion method based on the regularized least square method, in step S3, the objective functions (1) and (2) are derived from a and then solved for a, so as to obtain corresponding formal solutions respectively;
A＝(E T E) -1 E T D (3)
wherein,
the covariance matrix of A has diagonal elements as the variance of each component of A, and other off-diagonal elements as the correlation between each component.
Further, according to the gamma radiation source term inversion method based on the regularized least square method, in step S3, regularization terms are respectively added on the basis of the formal solutions (3) and (4) to obtain general solutions;
wherein α is regularizationParameter, α Q T Q is a regularization term, Q is a regularization matrix, and W is a weight matrix.
Further, according to the gamma radiation source item inversion method based on the regularized least square method, in step S3, a posterior strategy is solved according to a deviation criterion to obtain a radioactivity value in each source item division region, and in the strategy, regularization parameters meet the following deviation equation;
wherein,
for the regularization solution when the regularization parameter is alpha, D δ Measuring data for a dose rate with a measurement error level δ;
solving the deviation equation (7) according to a Newton method to obtain a regularization parameter alpha matched with the error level of the measured dose rate data opt According to the regularization parameter α opt Finally, a regularization solution, namely a radioactivity value in each source item division region is obtained.
The embodiment of the invention also provides a gamma radiation source item inversion system based on a regularization least square method, which comprises the following steps:
the system comprises a region division module, a radiation source detection module and a radiation detection module, wherein the region division module is used for performing region division on gamma radiation source items, and the radioactivity in each source item division region is uniformly distributed in space;
the determining module is used for determining an objective function for solving the radioactivity value in each source item division region through an optimization problem according to dose rate measurement data and uncertainty thereof at a plurality of external measurement positions and dose rate contribution of unit radioactivity of each source item division region to each measurement position;
and the solving module is used for solving the target function to obtain a corresponding formal solution, adding a regularization term on the basis of the formal solution to obtain a general solution, and solving according to a deviation criterion posterior strategy to obtain the radioactivity value in each source item division region.
Further, as mentioned above, the objective function includes:
wherein, A = (A) 1 ，A 2 ，…，A M ) T A vector of radioactivity values in the M source-term-divided regions, D = (D) 1 ，D 2 ，…，D N ) T A vector of dose rate measurement data at N measurement positions, E = (ε) ij )，i＝<1，N>，j＝<1，M>N is more than or equal to M, and is a dosage rate contribution matrix formed by the contribution of the unit radioactivity of the divided regions of M source items to the dosage rates at N measurement positions, S D ＝diag(σ 1 2 ,σ 2 2 ,…,σ N 2 ) An uncertainty matrix is formed of the uncertainties of the measured dose rate data at the N measurement locations.
Further, as to the gamma radiation source item inversion system based on the regularized least square method, the solving module is specifically configured to solve a after derivation of the objective functions (1) and (2) on a, so as to obtain corresponding formal solutions respectively;
A＝(E T E) -1 E T D (3)
wherein,
covariance matrix of A whose diagonal elements are each of AThe variance of each component, and the other off-diagonal elements represent the correlation between the components.
Further, as to the gamma radiation source term inversion system based on the regularized least square method, the solving module is specifically configured to add regularization terms respectively on the basis of the formal solutions (3) and (4) to obtain general solutions;
where α is a regularization parameter, α Q T Q is a regularization term, Q is a regularization matrix, and W is a weight matrix.
Further, as to the above-mentioned gamma radiation source item inversion system based on the regularization least square method, the solving module is specifically configured to obtain a radioactivity value in each source item partition region according to a deviation criterion posterior strategy, where a regularization parameter satisfies the following deviation equation;
wherein,
for the regularization solution when the regularization parameter is α, D δ Dose rate measurement data for a measurement error level of δ;
solving the deviation equation (7) according to the Newton method to obtain a regularization parameter alpha matched with the error level of the measured dose rate data opt According to the regularization parameter α opt Finally, a regularization solution, namely a radioactivity value in each source item division region is obtained.
The invention has the beneficial effects that: according to the method and the system provided by the invention, activity data of gamma radiation source items which are relatively stable and matched with the precision of measurement data are inverted on the basis of a given source item division mode and nuclide species according to the measurement data and the dosage rate contribution matrix and on the basis of a regularized least square principle. The method can be applied to the fields of radioactive source item characterization of complex objects in the decommissioning transition stage of nuclear facilities, radiation protection optimization in the nuclear facility overhaul and decommissioning processes and the like.
Drawings
Fig. 1 is a schematic flowchart of a gamma radiation source term inversion method based on a regularized least square method according to an embodiment of the present invention;
fig. 2 is a schematic structural diagram of a gamma radiation source term inversion system based on a regularized least square method according to an embodiment of the present invention.
Detailed Description
The invention is described in further detail below with reference to the drawings and the detailed description.
As shown in fig. 1, a method for inverting gamma radiation source term based on regularized least square method includes:
s1, carrying out regional division on gamma radiation source items, wherein the radioactivity in each source item division region is uniformly distributed in space;
s2, determining an objective function for solving the radioactivity value in each source item division region through an optimization problem according to dose rate measurement data and uncertainty thereof at a plurality of external measurement positions and dose rate contribution of unit radioactivity of each source item division region to each measurement position;
and S3, solving the target function to obtain a corresponding formal solution, adding a regularization term on the basis of the formal solution to obtain a general solution, and solving according to a deviation criterion posterior strategy to obtain the radioactivity value in each source term division region.
In step S2, the objective function includes:
wherein, A = (A) 1 ，A 2 ，…，A M ) T A vector of radioactivity values in the M source-term partition region, D = (D) 1 ，D 2 ，…，D N ) T A vector of dose rate measurement data at N measurement positions, E = (ε) ij )，i＝<1，N>，j＝<1，M>N is more than or equal to M, and is a dosage rate contribution matrix formed by the contribution of the unit radioactivity of the divided regions of M source items to the dosage rates at N measurement positions, S D ＝diag(σ 1 2 ,σ 2 2 ,…,σ N 2 ) An uncertainty matrix is formed of the uncertainties of the measured dose rate data at the N measurement locations.
In the step S3, the objective functions (1) and (2) are differentiated for A and then solved for A, and corresponding formal solutions are obtained respectively;
A＝(E T E) -1 E T D (3)
wherein,
the covariance matrix of A has diagonal elements as the variance of each component of A, and other off-diagonal elements as the correlation between each component.
In the step S3, regularization terms are respectively added on the basis of the formal solutions (3) and (4) to obtain general solutions;
where α is a regularization parameter, α Q T Q is a regularization term, Q is a regularization matrix, and W is a weight matrix.
In the step S3, solving according to a deviation criterion posterior strategy to obtain the radioactivity value in each source item division region, wherein in the strategy, the regularization parameter meets the following deviation equation;
wherein,
for the regularization solution when the regularization parameter is α, D δ Measuring data for a dose rate with a measurement error level δ;
solving the deviation equation (7) according to a Newton method to obtain a regularization parameter alpha matched with the error level of the measured dose rate data opt According to the regularization parameter α opt Finally, a regularization solution, namely a radioactivity value in each source item division region is obtained.
The basic principle of the invention is as follows:
assuming that the activity of each source item inside the waste liquid tank is uniformly distributed, the radioactivity distribution area is V, and the distribution density of the radioactive nuclide in the tank is V
Unit activity of site to measurement location
The contribution of the measured quantity (dose) is
The total radioactivity inside the waste liquid tankThe activity was:
at the measuring position
The measured data of (a) are:
the problem that needs to be solved now is: based on N measured data
Calculate the Total Activity A eval Is determined based on the estimated values of the measured values,
wherein, the ith source item in the tank divides the activity of the regional radionuclide
The following optimization problem is satisfied:
in fact, it is exactly the inverse problem of the above two measurement problems (1), (2), that is, the dose rate measurement data (or sampling measurement data) from outside the waste liquid tank is inverted to the source item inside the waste liquid tank. Due to calculation
In time, it is assumed that the radioactivity distribution within each source term partition is spatially uniform, and thus the inverted source terms are also source terms corresponding to a uniform distribution.
If the uncertainty of the measured data is considered and the weight w is added, then the following results are obtained:
writing the formulas (4) and (5) in a matrix form, wherein the matrix form comprises the following steps:
wherein D = (D) 1 ，D 2 ，…，D N ) T Is a vector formed by measurement data at different spatial points outside the waste liquid tank;
A＝(A 1 ，A 2 ，…，A M ) T the vector is composed of the radioactivity activity of each source item division region, namely the quantity to be solved;
E＝(ε ij )，i＝<1，N>，j＝<1，M>n ≧ M is a dose rate contribution matrix, i.e., the dose rate contribution of the unit activity of the jth partial source entry dividing region (assuming the radioactivity distribution is spatially uniform within that region) to the ith measurement position;
S D ＝diag(σ 1 2 ,σ 2 2 ,…,σ N 2 ) The uncertainty matrix is formed by uncertainties of the measured dose rates at N measurement positions; the form of the matrix actually contains the assumption that the measurement results at the individual measurement locations are statistically independent of each other.
The above two objective functions are derived for a and then solved for a, yielding the following formal solutions for the optimization problems (6) and (7):
A＝(E T E) -1 E T D (8)
result A = (A) 1 ，A 2 ，…，A M ) T Has a covariance matrix of
The diagonal elements of the components are the variances of the components of A, and other off-diagonal elements represent the correlation between the components.
The inversion method based on the regularization least square principle comprises the following steps:
in the specific source item inversion process, in order to take account of the numerical stability and the model fitting degree of the inversion result, a regularized least square solution is adopted. The regularized least square method is based on a formal solution and is solved after a regularization term is additionally added. The general solution is in the form of:
in the above formula, α is a regularization parameter, Q is a regularization matrix, W is a weight matrix, S D Is part of the weight. There are many methods for selecting regularization parameters, which are generally classified into a priori strategy and a posterior strategy. Among them, the prior strategy has a value of theoretical analysis, but it is often difficult to verify the conditions under which it is applied in practice, so the posterior strategy (determining regularization parameters matching with the error level of the measurement data according to a certain principle in the process of calculating the regularization solution) and the method thereof are practical. The invention adopts a posterior strategy according to a deviation criterion (Discrenancy principal). In this method, the regularization parameters satisfy the following deviation equation:
in the above formula
Is the regularization solution when the regularization parameter is α (i.e., the expression in (10) or (11), but a function with α as a variable), is the dose rate measurement data with a measurement error level of δ. Solving the (12) according to the Newton method to obtain the regularization parameter alpha matched with the error level of the measured dosage rate data opt And then eventually a regularization solution may be obtained.
Example one
The evaluation of the complex gamma radiation source term is taken as an example for explanation. Wherein the source item is divided into 15 regions, the detector measurement point data is 20, and the dose rate contribution matrix E is calculated by the monte carlo method. According to the performance characteristics (relative inherent error, direction correspondence, energy response, statistical fluctuation and other factors) of a detector for measuring the dosage rate and field measurement conditions, the error level of dosage rate measurement data is determined, and the optimal regularization parameter is determined according to a deviation principle. The source term inversion results (relative shares) without and with regularization according to the ML-EM algorithm are then shown in table 1.
TABLE 1
As can be seen from table 1, when no regularization is used, the relative shares between some neighboring blocks may differ by more than 10 magnitude (14 and 15 regions) due to model overfitting (physically unreasonable solution), and the inversion result using regularization shows a relatively regular spatial distribution of source terms.
According to the method, gamma radiation source item data which are relatively stable and are matched with the precision of measurement data are inverted on the basis of a regularized least square principle according to the measurement data (gamma radiation dose rate) and a system dose rate response matrix; the method can be applied to the fields of radioactive source item representation of complex objects in the decommissioning Transition Phase (Transition Phase) of the nuclear facility, nuclear facility overhaul, radiation protection optimization in the decommissioning process and the like.
As shown in fig. 2, an embodiment of the present invention further provides a system for inverting a gamma radiation source term based on regularized least square method, including:
the region dividing module 1 is used for performing region division on gamma radiation source items, wherein the radioactivity in each source item division region is uniformly distributed in space;
the determining module 2 is used for determining an objective function for solving the radioactivity value in each source item division region through an optimization problem according to the dose rate measurement data and the uncertainty thereof at the external multiple measurement positions and the dose rate contribution of the unit radioactivity of each source item division region to each measurement position;
and the solving module 3 is used for solving the target function to obtain a corresponding formal solution, adding a regularization term on the basis of the formal solution to obtain a general solution, and solving according to a deviation criterion posterior strategy to obtain the radioactivity value in each source term division region.
The objective function includes:
wherein, A = (A) 1 ，A 2 ，…，A M ) T A vector of radioactivity values in the M source-term-divided regions, D = (D) 1 ，D 2 ，…，D N ) T A vector of rate measurement data at the N measurement locations, E = (ε) ij )，i＝<1，N>，j＝<1，M>N is more than or equal to M, and is a dosage rate contribution matrix formed by the contribution of the unit radioactivity of the region divided by M source items to the dosage rates at N measurement positions，S D ＝diag(σ 1 2 ,σ 2 2 ,…,σ N 2 ) An uncertainty matrix is formed of the uncertainties of the measured dose rate data at the N measurement positions.
The solving module 3 is specifically configured to solve a after the objective functions (1) and (2) are derived from a, to obtain corresponding formal solutions respectively;
A＝(E T E) -1 E T D (3)
wherein,
the covariance matrix of A has diagonal elements as the variance of each component of A, and other off-diagonal elements as the correlation between each component.
The solving module 3 is specifically configured to add regularization terms respectively on the basis of the formal solutions (3) and (4) to obtain general solutions;
where α is a regularization parameter, α Q T Q is a regularization term, Q is a regularization matrix, and W is a weight matrix.
The solving module 3 is specifically configured to solve to obtain a radioactivity value in each source partition according to a deviation criterion posterior strategy, where in the strategy, the regularization parameter satisfies the following deviation equation;
wherein，
For the regularization solution when the regularization parameter is α, D δ Measuring data for a dose rate with a measurement error level δ;
solving the deviation equation (7) according to a Newton method to obtain a regularization parameter alpha matched with the error level of the measured dose rate data opt According to the regularization parameter α opt Finally, a regularization solution, namely a radioactivity value in each source item division region is obtained.
It will be apparent to those skilled in the art that various changes and modifications may be made in the present invention without departing from the spirit and scope of the invention. Thus, if such modifications and variations of the present invention fall within the scope of the claims of the present invention and their equivalents, the present invention is intended to include such modifications and variations.
Claims (2)
1. A gamma radiation source item inversion method based on regularized least square method is characterized by comprising the following steps:
s1, carrying out regional division on gamma radiation source items, wherein the radioactivity in each source item division region is uniformly distributed in space;
s2, determining an objective function for solving the radioactivity value in each source item division region through an optimization problem according to dose rate measurement data and uncertainty thereof at a plurality of external measurement positions and dose rate contribution of unit radioactivity of each source item division region to each measurement position;
s3, solving the target function to obtain a corresponding formal solution, adding a regularization term on the basis of the formal solution to obtain a general solution, and solving according to a deviation criterion posterior strategy to obtain a radioactivity value in each source term partition region;
in step S2, the objective function includes:
wherein, A = (A) 1 ，A 2 ，…，A M ) T A vector of radioactivity values in the M source-term-divided regions, D = (D) 1 ，D 2 ，…，D N ) T A vector of rate measurement data at the N measurement locations, E = (ε) ij )，i＝<1，N>，j＝<1，M>N is more than or equal to M, and is a dosage rate contribution matrix formed by the contribution of the unit radioactivity of the divided regions of M source items to the dosage rates at N measurement positions, S D ＝diag(σ 1 2 ,σ 2 2 ,…,σ N 2 ) An uncertainty matrix consisting of the uncertainties of the measured dose rate data at the N measurement positions;
in the step S3, the objective functions (1) and (2) are differentiated for A and then solved for A, and corresponding formal solutions are obtained respectively;
A＝(Ε T Ε) -1 Ε T D (3)
wherein,
the covariance matrix is A, the diagonal elements of the covariance matrix are the variances of all the components of A, and other off-diagonal elements represent the correlation among all the components;
in the step S3, regularization terms are respectively added on the basis of the formal solutions (3) and (4) to obtain general solutions;
where α is a regularization parameter, α Q T Q is a regularization term, Q is a regular matrix, and W is a weight matrix;
in the step S3, solving according to a deviation criterion posterior strategy to obtain the radioactivity value in each source item division region, wherein in the strategy, the regularization parameter meets the following deviation equation;
wherein,
for the regularization solution when the regularization parameter is alpha, D δ Measuring data for a dose rate with a measurement error level δ;
solving the deviation equation (7) according to a Newton method to obtain a regularization parameter alpha matched with the error level of the measured dose rate data opt According to the regularization parameter α opt Finally, a regularization solution, namely a radioactivity value in each source item partition region is obtained.
2. A gamma radiation source item inversion system based on regularized least squares, comprising:
the region dividing module is used for performing region division on the gamma radiation source items, wherein the radioactivity in each source item division region is uniformly distributed in space;
the determining module is used for determining an objective function for solving the radioactivity value in each source item division region through an optimization problem according to dose rate measurement data and uncertainty thereof at a plurality of external measurement positions and the contribution of the unit radioactivity of each source item division region to the dose rate at each measurement position;
the solving module is used for solving the target function to obtain a corresponding formal solution, adding a regularization term on the basis of the formal solution to obtain a general solution, and solving according to a deviation criterion posterior strategy to obtain a radioactivity value in each source item division region;
the objective function includes:
wherein, A = (A) 1 ，A 2 ，…，A M ) T A vector of radioactivity values in the M source-term-divided regions, D = (D) 1 ，D 2 ，…，D N ) T A vector of dose rate measurement data at N measurement positions, E = (ε) ij )，i＝<1，N>，j＝<1，M>N is more than or equal to M, and is a dosage rate contribution matrix formed by the contribution of the unit radioactivity of the divided regions of M source items to the dosage rates at N measurement positions, S D ＝diag(σ 1 2 ,σ 2 2 ,…,σ N 2 ) An uncertainty matrix composed of uncertainties of the dose rate data measured at the N measurement positions;
the solving module is specifically used for solving the A after derivation of the objective functions (1) and (2) to the A, and obtaining corresponding formal solutions respectively;
A＝(Ε T Ε) -1 Ε T D (3)
wherein,
the covariance matrix is A, the diagonal elements of the covariance matrix are the variances of all the components of A, and other off-diagonal elements represent the correlation among all the components;
the solving module is specifically used for respectively adding regularization terms on the basis of the formal solutions (3) and (4) to obtain general solutions;
where α is a regularization parameter, α Q T Q is a regularization term, Q is a regular matrix, and W is a weight matrix;
the solving module is specifically used for solving according to a deviation criterion posterior strategy to obtain the radioactivity value in each source item division region, wherein in the strategy, the regularization parameter meets the following deviation equation;
wherein,
for the regularization solution when the regularization parameter is alpha, D δ Measuring data for a dose rate with a measurement error level δ;
solving the deviation equation (7) according to a Newton method to obtain a regularization parameter alpha matched with the error level of the measured dose rate data opt According to the regularization parameter α opt Finally, a regularization solution, namely a radioactivity value in each source item partition region is obtained.
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