Fast nuclide identification based on a sequential bayesian method

Fast nuclide identification based on a sequential bayesian method 1 first-author https://orcid.org/0000-0001-6243-0354 1 corresp *shinecore@163.com *shinecore@163.com shinecore@163.com https://orcid.org/0000-0001-8840-7793 1 1 https://orcid.org/0000-0002-7968-0402 1 https://orcid.org/0000-0002-4273-6062 1 1 Chengdu University of Technology, The College of Nuclear Technology and Automation Engineering, ChengDu 610059, China Chengdu University of Technology, The College of Nuclear Technology and Automation Engineering, ChengDu 610059, China The rapid identification of radioactive substances in public areas is crucial. However, traditional nuclide identification methods only consider information regarding the full energy peaks of the gamma-ray spectrum and require long recording times, which lead to long response times. In this paper, a novel identification method using the event mode sequence (EMS) information of target radionuclides is proposed. The EMS of a target radionuclide and natural background radiation were established as two different probabilistic models and a decision function based on Bayesian inference and sequential testing was constructed. The proposed detection scheme individually processes each photon. When a photon is detected and accepted, the corresponding posterior probability distribution parameters are estimated using Bayesian inference and the decision function is updated. Then, value of the decision function is compared to preset detection thresholds to obtain a detection result. Experiments on different target radionuclides ( 137 Cs and 60 Co) were performed. The count rates of the regions of interest (ROI) in the backgrounds between [651, 671], [1154, 1186] , and [1310, 1350] keV were 5.05, 3.83, and 3.61 CPS, respectively. The experimental results demonstrate that the proposed method can identify 137 Cs in 3.8 s with a full energy peak count rate of 5.05 s −1 and can identify 60 Co in 4.1 s with a full energy peak count rate of 7.44 s −1 . The results demonstrate that the proposed method can detect radioactive substances with low activity. Natural radiation Nuclide identification Sequential testing Nuclear safety Introduction 1 Radioactive substances are a threat to public safety when they are present in public areas. In such cases, nuclide identification is crucial for inspecting radioactive substances. Traditional radionuclide identification methods [ 1 1 - 2 2 1-2 ] are based on the full energy peak information of the gamma-ray spectrum and require that the particle events generated by the decay of nuclides are treated as statistics in the spectrum. Peak-finding algorithms such as the derivative peak searching method, second derivative peak identification method [ 3 3 ] , covariance peak-search method [ 4 4 ] , and symmetric zero-area peak searching method [ 5 5 ] use full energy peaks information to identify nuclides based on the energy of the gamma rays from nuclides. To obtain a low-statistic spectrum, a long spectrum recording time is required, resulting in a slow response time. To compensate for the shortcomings of traditional radionuclide identification methods, novel detection methods have been proposed, including the adaptive filtering method [ 6 6 ] , chaos oscillator method, blind-source separation method, and stochastic resonance method [ 4 4 ] . The adaptive filtering method is a weak signal detection method with an optimal parameter adjustment function that was developed based on the Kalman filter. This method can automatically optimize a nuclear signal. For the Chaos oscillator method, a nuclear signal in the form of a periodic signal is superimposed with a fixed-frequency chaotic oscillator. The mixed signal is then detected and processed using phase differences. The blind-source separation method uses an analysis algorithm to obtain the best estimate of a nuclear signal. The stochastic resonance method uses the synergistic effects of noise and signals to analyze nuclear signals. Sullivan adopted wavelet transforms and the modulus maxima method [ 7 7 – 8 8 7–8 ] to achieve accurate nuclide identification using low-resolution spectra. Several artificial neural network (ANN) methods have also been introduced in previous studies [ 9 9 – 12 12 9–12 ] based on their significant success in other research fields. Liang used the K-L transform to extract gamma-ray spectrum features and trained an ANN to perform radionuclide identification. Experiments and tests demonstrated that the ANN method is effective for rapid radionuclide identification [ 12 12 ] . Furthermore, Yang et al. proposed a radionuclide identification method based on machine learning and pattern recognition [ 13 13 – 18 18 13–18 ] , and Chong Jie and Yi Ming used a fuzzy-logic-based algorithm to discriminate characteristic peaks and identify radionuclides [ 19 19 , 23 23 , 24 24 ] . Although the aforementioned gamma ray spectrum analysis methods have proven to be reliable and practical, a long recording time is required to obtain a spectrum with low statistical error. Therefore, these techniques cannot meet the requirements for real-time radionuclide inspection in public areas. In 1945, Wald proposed the sequential probability ratio testing theory [ 22 22 ] , which can be used to perform real-time analysis and judgment regarding whether a hypothesis is true based on recorded events. The results satisfied statistical requirements. The sequential probability ratio test combined with the Bayesian law for posterior probability distributions has been applied in many fields [ 23 23 , 24 24 ] . Candy proposed a sequential Bayesian radionuclide identification method in 2008. This method combines the sequential probability ratio test and Bayes rule for radiation monitoring and radionuclide identification, and establishes a radiation detection model. In 2010, Candy et al. developed SRaDS, which is a radiation monitoring system using a sequential-Bayesian-based radionuclide identification method [ 25 25 ] . SRaDS automatically eliminates irrelevant and unexpected photons during the detection process and improves detection efficiency. This system abandons the concept of the energy spectrum statistics used for traditional nuclide recognition. It uses a Bayesian algorithm to group each target photon, compares preset parameters for each photoelectric event sequence, and calculates the probability of the target radionuclide corresponding to the photoelectric event sequence. Finally, the statistical results of multiple single-energy ray groups are used to determine whether the target radionuclides are detected. This method can successfully detect radionuclides when the number of recorded photons is small and the measurement time is short [ 25 25 , 26 26 ] . This method makes full use of event mode sequence (EMS) information. An EMS of the target radionuclide can be treated as a composition of EMSs emitted from several monoenergetic source components. A well-defined probability model can then be used to describe the EMS of the target radionuclide. Therefore, a sequential Bayesian detection scheme can be applied to identify radioactive nuclides in real time by incorporating energy, count rate, and emission ratio information [ 24 24 , 27 27 ] . Unlike the traditional full-spectrum-analysis-based method, this method can detect radionuclides under low-count conditions. However, the results of this method are affected by the background count and some simulation experiments by Xiang Qingpei verified that it may be inaccurate in cases with high background counts [ 25 25 , 26 26 ] . In this study, two different probabilistic models were established based on the EMSs of natural background radiation and target nuclide radiation fields. The posterior distribution parameters of the probabilistic models were then estimated using Bayes’ law. Finally, nuclide identification was performed via sequential detection. Once a photon has been detected and processed, the posterior probability distribution parameters can be estimated using Bayesian inference and a decision function can be updated and compared to preset detection thresholds to determine whether the target radionuclide is detected. In this paper, a rapid nuclide identification model and sequential detection scheme are introduced, followed by the presentation of validation experiments on target radionuclides ( 137 Cs and 60 Co). Methods 1 Nuclide identification model based on background comparisons 2 The concept of monoenergetic decomposition [ 13 13 ] has been proposed to leverage EMS information. Suppose that the sequence of photons emitted by a radionuclide is a set of photons emitted by several separate monoenergetic sources. An <math id="M1"><mrow><mi>E</mi><mi>v</mi><mi>e</mi><mi>n</mi><mi>t</mi><mrow><mo>(</mo><mrow><mi>n</mi><mo>;</mo><msub><mi>e</mi><mi>m</mi></msub><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mo>,</mo><mi>Δ</mi><msub><mtext>τ</mtext><mi>m</mi></msub><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow><mo>)</mo></mrow></mrow></math> , is defined as the n th gamma ray recorded by the detector. This ray is emitted by the m th monoenergetic source with a measured energy <math id="M2"><mrow><msub><mi>e</mi><mi>m</mi></msub><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></math> and interarrival time <math id="M3"><mrow><mi>Δ</mi><msub><mi>τ</mi><mi>m</mi></msub><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></math> between the current and previous event. The EMS of a target radionuclide ( <math id="M4"><mrow><mi>E</mi><mi>M</mi><msub><mi>S</mi><mrow><mtext>rn</mtext></mrow></msub></mrow></math> ) is then represented as <math id="M5"><mrow><mi>E</mi><mi>M</mi><msub><mi>S</mi><mrow><mtext>rn</mtext></mrow></msub><mrow><mo>(</mo><mrow><mi>N</mi><mo>;</mo><munder accentunder="true"><mi>e</mi><mo>_</mo></munder><mo>,</mo><mtext>Δ</mtext><munder accentunder="true"><mi>τ</mi><mo>_</mo></munder></mrow><mo>)</mo></mrow></mrow></math> . <math display="block" id="M6"><mrow><mi>E</mi><mi>M</mi><msub><mi>S</mi><mrow><mtext>rn</mtext></mrow></msub><mrow><mo>(</mo><mrow><mi>N</mi><mo>;</mo><munder accentunder="true"><mi>e</mi><mo>_</mo></munder><mo>,</mo><mi>Δ</mi><munder accentunder="true"><mi>τ</mi><mo>_</mo></munder></mrow><mo>)</mo></mrow><mo>:</mo><mo>=</mo><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>m</mi><mo>=</mo><mn>1</mn></mrow><mi>M</mi></munderover><mrow><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mrow><msub><mi>N</mi><mi>m</mi></msub></mrow></munderover><mrow><mi>E</mi><mi>v</mi><mi>e</mi><mi>n</mi><mi>t</mi><mrow><mo>(</mo><mrow><mi>n</mi><mo>;</mo><msub><mi>e</mi><mi>m</mi></msub><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mo>,</mo><mi>Δ</mi><msub><mtext>τ</mtext><mi>m</mi></msub><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow><mo>)</mo></mrow></mrow></mstyle></mrow></mstyle></mrow></math> The index m represents the m th monoenergetic source of a target radionuclide and the total number of monoenergetic sources is M . The detector detects <math id="M7"><mrow><msub><mi>N</mi><mi>m</mi></msub></mrow></math> events from the m th monoenergetic source during the detection period. <math id="M8"><munder accentunder="true"><mi>e</mi><mo>_</mo></munder></math> is defined as the complete set of energies composing <math id="M9"><mrow><mi>E</mi><mi>M</mi><msub><mi>S</mi><mrow><mtext>rn</mtext></mrow></msub></mrow></math> along with <math id="M10"><mrow><mtext>Δ</mtext><munder accentunder="true"><mi>τ</mi><mo>_</mo></munder></mrow></math> , which is the corresponding set of interarrival times. N is the sum of the set <math id="M11"><mrow><msub><mi>N</mi><mi>m</mi></msub></mrow></math> . Considering instrument noise and experimental uncertainty, the measured energy of the gamma rays emitted by a monoenergetic source in a radiation field follows a Gaussian distribution and the interarrival times follow an exponential distribution (based on the Poisson statistics of nuclear decay[ 28 28 ]). The corresponding formula is presented in Eq. (2), where <math id="M12"><mrow><msub><mover accent="true"><mi>e</mi><mo>¯</mo></mover><mi>m</mi></msub></mrow></math> and <math id="M13"><mrow><msub><mi>σ</mi><mrow><msub><mi>e</mi><mi>m</mi></msub></mrow></msub></mrow></math> are the expected value and standard deviation of the Gaussian distribution, respectively. <math id="M14"><mrow><msub><mi>λ</mi><mi>m</mi></msub></mrow></math> is the expected value of the exponential distribution and <math id="M15"><mrow><msub><mi>α</mi><mi>m</mi></msub></mrow></math> is the emission probability of the m th monoenergetic source. <math display="block" id="M16"><mtable columnalign="left"><mtr><mtd><mi>P</mi><mrow><mo>(</mo><mrow><msub><mi>e</mi><mi>m</mi></msub><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mo>|</mo><mtext>radionuclide</mtext></mrow><mo>)</mo></mrow><mo>=</mo><mi>P</mi><mrow><mo>(</mo><mrow><msub><mi>e</mi><mi>m</mi></msub><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mtext>|</mtext><msub><mover accent="true"><mi>e</mi><mo>¯</mo></mover><mi>m</mi></msub><mo>,</mo><msubsup><mi>σ</mi><mrow><msub><mi>e</mi><mi>m</mi></msub></mrow><mn>2</mn></msubsup></mrow><mo>)</mo></mrow></mtd></mtr><mtr><mtd><mo>=</mo><mfrac><mn>1</mn><mrow><msqrt><mrow><mn>2</mn><mi>π</mi></mrow></msqrt><msub><mi>σ</mi><mrow><msub><mi>e</mi><mi>m</mi></msub></mrow></msub></mrow></mfrac><msup><mi>e</mi><mrow><mo>−</mo><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><msub><mi>e</mi><mi>m</mi></msub><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mo>−</mo><msub><mover accent="true"><mi>e</mi><mo>¯</mo></mover><mi>m</mi></msub></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mrow><mn>2</mn><msubsup><mi>σ</mi><mrow><msub><mi>e</mi><mi>m</mi></msub></mrow><mn>2</mn></msubsup></mrow></mfrac></mrow></msup></mtd></mtr><mtr><mtd><mi>P</mi><mrow><mo>(</mo><mrow><mi>Δ</mi><msub><mtext>τ</mtext><mi>m</mi></msub><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mo>|</mo><mtext>radionuclide</mtext></mrow><mo>)</mo></mrow><mo>=</mo><mi>P</mi><mrow><mo>(</mo><mrow><mi>Δ</mi><msub><mtext>τ</mtext><mi>m</mi></msub><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mo>|</mo><msub><mi>λ</mi><mi>m</mi></msub></mrow><mo>)</mo></mrow><mo>=</mo><msub><mi>λ</mi><mi>m</mi></msub><msup><mi>e</mi><mrow><mo>−</mo><msub><mi>λ</mi><mi>m</mi></msub><mtext>Δ</mtext><msub><mi>τ</mi><mi>m</mi></msub><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></msup></mtd></mtr><mtr><mtd><mi>P</mi><mo stretchy="false">(</mo><mi>E</mi><mi>v</mi><mi>e</mi><mi>n</mi><mi>t</mi><mrow><mo>(</mo><mrow><mi>n</mi><mo>;</mo><msub><mi>e</mi><mi>m</mi></msub><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mo>,</mo><mi>Δ</mi><msub><mtext>τ</mtext><mi>m</mi></msub><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mtext>|radionuclide</mtext></mrow><mo>)</mo></mrow><mo>=</mo><msub><mi>α</mi><mi>m</mi></msub><mo>×</mo><mi>P</mi><mrow><mo>(</mo><mrow><msub><mi>e</mi><mi>m</mi></msub><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mo>|</mo><mtext>radionuclide</mtext></mrow><mo>)</mo></mrow></mtd></mtr><mtr><mtd><mo>×</mo><mi>P</mi><mrow><mo>(</mo><mrow><mi>Δ</mi><msub><mi>τ</mi><mi>m</mi></msub><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mo>|</mo><mtext>radionuclide</mtext></mrow><mo>)</mo></mrow></mtd></mtr></mtable></math> When a radionuclide identification instrument is placed in a public area, the background radiation is constant. For an EMS measured in the presence of natural background radiation, the region of interest (ROI) in the spectrum consists of the Compton platform and other types of background radiation. Therefore, we modeled the measured energy of a gamma ray emitted by a monoenergetic source in an ROI uniformly and the interarrival times exponentially. <math display="block" id="M17"><mtable columnalign="left"><mtr><mtd><mi>P</mi><mrow><mo>(</mo><mrow><msub><mi>e</mi><mi>m</mi></msub><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mo>|</mo><mtext>background</mtext></mrow><mo>)</mo></mrow><mo>=</mo><mi>P</mi><mrow><mo>(</mo><mrow><msub><mi>e</mi><mi>m</mi></msub><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow><msubsup><mrow><mtext>|e</mtext></mrow><mi>m</mi><mrow><mtext>left</mtext></mrow></msubsup><mo>,</mo><msubsup><mtext>e</mtext><mi>m</mi><mrow><mtext>right</mtext></mrow></msubsup></mrow><mo>)</mo></mrow><mo>=</mo><mfrac><mn>1</mn><mrow><msubsup><mtext>e</mtext><mi>m</mi><mrow><mtext>right</mtext></mrow></msubsup><mo>−</mo><msubsup><mtext>e</mtext><mi>m</mi><mrow><mtext>left</mtext></mrow></msubsup></mrow></mfrac><mo>,</mo></mtd></mtr><mtr><mtd><mi>P</mi><mrow><mo>(</mo><mrow><mi>Δ</mi><msub><mtext>τ</mtext><mi>m</mi></msub><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mo>|</mo><mtext>background</mtext></mrow><mo>)</mo></mrow><mo>=</mo><mi>P</mi><mrow><mo>(</mo><mrow><mi>Δ</mi><msub><mtext>τ</mtext><mi>m</mi></msub><mrow><mo>(</mo><mtext>n</mtext><mo>)</mo></mrow><mtext>|</mtext><msubsup><mi>λ</mi><mi>m</mi><mrow><mtext>base</mtext></mrow></msubsup></mrow><mo>)</mo></mrow><mo>=</mo><msubsup><mi>λ</mi><mi>m</mi><mrow><mtext>base</mtext></mrow></msubsup><msup><mi>e</mi><mrow><mo>−</mo><msubsup><mi>λ</mi><mi>m</mi><mrow><mtext>base</mtext></mrow></msubsup><mtext>Δ</mtext><msub><mi>τ</mi><mi>m</mi></msub><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></msup><mo>,</mo></mtd></mtr><mtr><mtd><mi>P</mi><mo stretchy="false">(</mo><mi>E</mi><mi>v</mi><mi>e</mi><mi>n</mi><mi>t</mi><mrow><mo>(</mo><mrow><mi>n</mi><mo>;</mo><msub><mi>e</mi><mi>m</mi></msub><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mo>,</mo><mi>Δ</mi><msub><mtext>τ</mtext><mi>m</mi></msub><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mtext>|background</mtext></mrow><mo>)</mo></mrow><mo>=</mo><mfrac><mrow><msubsup><mi>λ</mi><mi>m</mi><mrow><mtext>base</mtext></mrow></msubsup></mrow><mrow><mi>M</mi><mrow><mo>(</mo><mrow><msubsup><mtext>e</mtext><mi>m</mi><mrow><mtext>right</mtext></mrow></msubsup><mo>−</mo><msubsup><mtext>e</mtext><mi>m</mi><mrow><mtext>left</mtext></mrow></msubsup></mrow><mo>)</mo></mrow></mrow></mfrac><msup><mi>e</mi><mrow><mo>−</mo><msubsup><mi>λ</mi><mi>m</mi><mrow><mtext>base</mtext></mrow></msubsup><mtext>Δ</mtext><msub><mi>τ</mi><mi>m</mi></msub><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></msup><mo>.</mo></mtd></mtr></mtable></math> In Eq. (3), <math id="M18"><mrow><msubsup><mtext>e</mtext><mi>m</mi><mrow><mtext>left</mtext></mrow></msubsup></mrow></math> and <math id="M19"><mrow><msubsup><mtext>e</mtext><mi>m</mi><mrow><mtext>right</mtext></mrow></msubsup></mrow></math> are the left and right energies of the ROI, respectively, and <math id="M20"><mrow><msubsup><mi>λ</mi><mi>m</mi><mrow><mtext>Δ</mtext><msub><mi>τ</mi><mrow><mtext>base</mtext></mrow></msub></mrow></msubsup></mrow></math> is the background count rate of the ROI. The hypothesis test defined in Eq. (4) is applied to identify the nuclide through sequential detection based on the two hypotheses. <math display="block" id="M21"><mtable columnalign="left"><mtr><mtd><msub><mi>H</mi><mn>1</mn></msub><mo>:</mo><mi>E</mi><mi>M</mi><msub><mi>S</mi><mrow><mtext>rn</mtext></mrow></msub><mrow><mo>(</mo><mrow><mi>N</mi><mo>;</mo><munder accentunder="true"><mi>e</mi><mo>_</mo></munder><mo>,</mo><mtext>Δ</mtext><munder accentunder="true"><mi>τ</mi><mo>_</mo></munder></mrow><mo>)</mo></mrow><mo>~</mo><mi>P</mi><mrow><mo>(</mo><mrow><munder accentunder="true"><mi>e</mi><mo>_</mo></munder><mo>,</mo><mtext>Δ</mtext><munder accentunder="true"><mi>τ</mi><mo>_</mo></munder><mo>|</mo><mtext>radionuclide</mtext></mrow><mo>)</mo></mrow></mtd></mtr><mtr><mtd><mi>［</mi><mtext>if radionuclide is detected</mtext><mi>］</mi><mo>,</mo></mtd></mtr><mtr><mtd><mo> </mo><mo> </mo><msub><mi>H</mi><mn>0</mn></msub><mo>:</mo><mi>E</mi><mi>M</mi><msub><mi>S</mi><mrow><mtext>rn</mtext></mrow></msub><mrow><mo>(</mo><mrow><mi>N</mi><mo>;</mo><munder accentunder="true"><mi>e</mi><mo>_</mo></munder><mo>,</mo><mtext>Δ</mtext><munder accentunder="true"><mi>τ</mi><mo>_</mo></munder></mrow><mo>)</mo></mrow><mo>~</mo><mi>P</mi><mrow><mo>(</mo><mrow><munder accentunder="true"><mi>e</mi><mo>_</mo></munder><mo>,</mo><mtext>Δ</mtext><munder accentunder="true"><mi>τ</mi><mo>_</mo></munder><mo>|</mo><mtext>background</mtext></mrow><mo>)</mo></mrow></mtd></mtr><mtr><mtd><mi>［</mi><mtext>if radionuclide is not detected</mtext><mi>］</mi></mtd></mtr></mtable></math> When a radionuclide is detected, <math id="M22"><mrow><mi>P</mi><mrow><mo>(</mo><mrow><munder accentunder="true"><mi>e</mi><mo>_</mo></munder><mo>,</mo><mtext> Δ</mtext><munder accentunder="true"><mi>τ</mi><mo>_</mo></munder><mo>|</mo><mtext>radionuclide</mtext></mrow><mo>)</mo></mrow></mrow></math> is considered as the probabilistic model of the EMS. Otherwise, the probabilistic model of the EMS is <math id="M23"><mrow><mi>P</mi><mrow><mo>(</mo><mrow><munder accentunder="true"><mi>e</mi><mo>_</mo></munder><mo>,</mo><mtext> Δ</mtext><munder accentunder="true"><mi>τ</mi><mo>_</mo></munder><mo>|</mo><mtext>background</mtext></mrow><mo>)</mo></mrow></mrow></math> . The optimal solution to this binary decision problem is provided by the Wald sequential probability ratio test [ 21 21 ] . It is assumed that the N th detected photon is emitted from the m th monoenergetic source. According to the definition in Eq. (1), there are a total of <math id="M24"><mrow><msub><mi>N</mi><mi>m</mi></msub></mrow></math> photons detected from the m th monoenergetic source and the likelihood ratio <math id="M25"><mrow><mi>R</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mrow><mo>(</mo><mi>N</mi><mo>)</mo></mrow></mrow></math> is obtained by applying the Newman–Pearson theorem as follows: http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23770191&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23770191&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23770191&type=middle http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23770196&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23770196&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23770196&type=middle http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23770201&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23770201&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23770201&type=middle The decision function <math id="M26"><mrow><mi>Λ</mi><mrow><mo>(</mo><mi>N</mi><mo>)</mo></mrow></mrow></math> is obtained by taking the logarithm of <math id="M27"><mrow><mi>R</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mrow><mo>(</mo><mi>N</mi><mo>)</mo></mrow></mrow></math> , as shown in Equation (6). <math display="block" id="M28"><mtable columnalign="left"><mtr><mtd><mi>Λ</mi><mrow><mo>(</mo><mi>N</mi><mo>)</mo></mrow><mo>=</mo><mi>Λ</mi><mrow><mo>(</mo><mrow><mi>N</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow><mo>+</mo><mi>ln</mi><mi>P</mi><mo stretchy="false">(</mo><mi>E</mi><mi>v</mi><mi>e</mi><mi>n</mi><mi>t</mi><mo stretchy="false">(</mo><msub><mi>N</mi><mi>m</mi></msub><mo>;</mo><msub><mi>e</mi><mi>m</mi></msub><mrow><mo>(</mo><mrow><msub><mi>N</mi><mi>m</mi></msub></mrow><mo>)</mo></mrow><mo>,</mo></mtd></mtr><mtr><mtd><mi>Δ</mi><msub><mtext>τ</mtext><mi>m</mi></msub><mrow><mo>(</mo><mrow><msub><mi>N</mi><mi>m</mi></msub></mrow><mo>)</mo></mrow><mtext>|</mtext><mi>E</mi><mi>M</mi><msub><mi>S</mi><mrow><mtext>rn</mtext></mrow></msub><mrow><mo>(</mo><mrow><mi>N</mi><mo>−</mo><mn>1</mn><mo>;</mo><munder accentunder="true"><mi>e</mi><mo>_</mo></munder><mo>,</mo><mtext>Δ</mtext><munder accentunder="true"><mi>τ</mi><mo>_</mo></munder></mrow><mo>)</mo></mrow><mo>,</mo><msub><mi>H</mi><mn>1</mn></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mo>−</mo><mtext>ln</mtext><mi>P</mi><mrow><mo>(</mo><mrow><mi>E</mi><mi>v</mi><mi>e</mi><mi>n</mi><mi>t</mi><mo stretchy="false">(</mo><msub><mi>N</mi><mi>m</mi></msub><mo>;</mo><msub><mi>e</mi><mi>m</mi></msub><mrow><mo>(</mo><mrow><msub><mi>N</mi><mi>m</mi></msub></mrow><mo>)</mo></mrow><mo>,</mo><mi>Δ</mi><msub><mtext>τ</mtext><mi>m</mi></msub><mrow><mo>(</mo><mrow><msub><mi>N</mi><mi>m</mi></msub></mrow><mo>)</mo></mrow><mtext>|</mtext><mi>E</mi><mi>M</mi><msub><mi>S</mi><mrow><mtext>rn</mtext></mrow></msub><mrow><mo>(</mo><mrow><mi>N</mi><mo>−</mo><mn>1</mn><mo>;</mo><munder accentunder="true"><mi>e</mi><mo>_</mo></munder><mo>,</mo><mtext>Δ</mtext><munder accentunder="true"><mi>τ</mi><mo>_</mo></munder></mrow><mo>)</mo></mrow><mo>,</mo><msub><mi>H</mi><mn>0</mn></msub></mrow><mo>)</mo></mrow><mo>.</mo></mtd></mtr></mtable></math> With the <math id="M29"><mrow><mi>N</mi><mtext>th</mtext></mrow></math> photon recorded by the detector, the posterior probabilistic distribution parameter of the EMS model under <math id="M30"><mrow><msub><mi>H</mi><mn>1</mn></msub></mrow></math> is estimated using Bayesian inference. The decision function is then updated according to the resulting parameters and compared to preset detection thresholds to obtain detection results. The Wald sequential probability ratio test result is defined in Eq. (7), where <math id="M31"><mrow><msub><mi>T</mi><mn>1</mn></msub></mrow></math> and <math id="M32"><mrow><msub><mi>T</mi><mn>0</mn></msub></mrow></math> are the detection thresholds [ 22 22 ] . <math display="block" id="M33"><mtable columnalign="left"><mtr><mtd><mi>Λ</mi><mrow><mo>(</mo><mi>N</mi><mo>)</mo></mrow><mo>></mo><msub><mi>T</mi><mn>1</mn></msub><mo> </mo><mo>,</mo><mtext>Accept</mtext><mo> </mo><msub><mi>H</mi><mn>1</mn></msub><mo>,</mo></mtd></mtr><mtr><mtd><msub><mi>T</mi><mn>0</mn></msub><mo>≤</mo><mi>Λ</mi><mrow><mo>(</mo><mi>N</mi><mo>)</mo></mrow><mo>≤</mo><msub><mi>T</mi><mn>1</mn></msub><mo>,</mo><mtext> Continue,</mtext></mtd></mtr><mtr><mtd><mi>Λ</mi><mrow><mo>(</mo><mi>N</mi><mo>)</mo></mrow><mo><</mo><msub><mi>T</mi><mn>0</mn></msub><mo> </mo><mo>,</mo><mtext>Accept</mtext><mo> </mo><msub><mi>H</mi><mn>0</mn></msub><mo>.</mo></mtd></mtr></mtable></math> After defining a detection method, it is necessary to infer parameters and finalize a decision function. Based on the specified probability distribution form and observed data, the posterior distribution of the m th monoenergetic source under <math id="M34"><mrow><msub><mi>H</mi><mn>1</mn></msub></mrow></math> is defined in Eq. (8), where the probabilistic model is decomposed by applying Bayes’ rule [ 32 32 , 33 33 ] and <math id="M35"><mrow><msub><mi>α</mi><mi>m</mi></msub><mrow><mo>(</mo><mrow><msub><mi>N</mi><mi>m</mi></msub></mrow><mo>)</mo></mrow></mrow></math> is the emission rate estimated by the <math id="M36"><mrow><msub><mi>N</mi><mi>m</mi></msub></mrow></math> th photon detected from the m th monoenergetic source. <math display="block" id="M37"><mtable columnalign="left"><mtr><mtd><mi>P</mi><mrow><mo>(</mo><mrow><mi>E</mi><mi>v</mi><mi>e</mi><mi>n</mi><mi>t</mi><mo stretchy="false">(</mo><msub><mi>N</mi><mi>m</mi></msub><mo>;</mo><msub><mi>e</mi><mi>m</mi></msub><mrow><mo>(</mo><mrow><msub><mi>N</mi><mi>m</mi></msub></mrow><mo>)</mo></mrow><mo>,</mo><mi>Δ</mi><msub><mtext>τ</mtext><mi>m</mi></msub><mrow><mo>(</mo><mrow><msub><mi>N</mi><mi>m</mi></msub></mrow><mo>)</mo></mrow><mtext>|</mtext><mi>E</mi><mi>M</mi><msub><mi>S</mi><mrow><mtext>rn</mtext></mrow></msub><mrow><mo>(</mo><mrow><mi>N</mi><mo>−</mo><mn>1</mn><mo>;</mo><munder accentunder="true"><mi>e</mi><mo>_</mo></munder><mo>,</mo><mtext>Δ</mtext><munder accentunder="true"><mi>τ</mi><mo>_</mo></munder></mrow><mo>)</mo></mrow><mo>,</mo><msub><mi>H</mi><mn>1</mn></msub></mrow><mo>)</mo></mrow></mtd></mtr><mtr><mtd><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>Δ</mi><msub><mtext>τ</mtext><mi>m</mi></msub><mrow><mo>(</mo><mrow><msub><mi>N</mi><mi>m</mi></msub></mrow><mo>)</mo></mrow><mo>|</mo><msub><mi>e</mi><mi>m</mi></msub><mrow><mo>(</mo><mrow><msub><mi>N</mi><mi>m</mi></msub></mrow><mo>)</mo></mrow><mo>,</mo><msub><mi>α</mi><mi>m</mi></msub><mrow><mo>(</mo><mrow><msub><mi>N</mi><mi>m</mi></msub></mrow><mo>)</mo></mrow><mo>,</mo><mi>E</mi><mi>M</mi><msub><mi>S</mi><mrow><mtext>rn</mtext></mrow></msub><mrow><mo>(</mo><mrow><mi>N</mi><mo>−</mo><mn>1</mn><mo>;</mo><munder accentunder="true"><mi>e</mi><mo>_</mo></munder><mo>,</mo><mtext>Δ</mtext><munder accentunder="true"><mi>τ</mi><mo>_</mo></munder></mrow><mo>)</mo></mrow><mo>,</mo><msub><mi>H</mi><mn>1</mn></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mo>×</mo><mi>P</mi><mo stretchy="false">(</mo><msub><mi>e</mi><mi>m</mi></msub><mrow><mo>(</mo><mrow><msub><mi>N</mi><mi>m</mi></msub></mrow><mo>)</mo></mrow><mo>|</mo><msub><mi>α</mi><mi>m</mi></msub><mrow><mo>(</mo><mrow><msub><mi>N</mi><mi>m</mi></msub></mrow><mo>)</mo></mrow><mo>,</mo><mi>E</mi><mi>M</mi><msub><mi>S</mi><mrow><mtext>rn</mtext></mrow></msub><mrow><mo>(</mo><mrow><mi>N</mi><mo>−</mo><mn>1</mn><mo>;</mo><munder accentunder="true"><mi>e</mi><mo>_</mo></munder><mo>,</mo><mtext>Δ</mtext><munder accentunder="true"><mi>τ</mi><mo>_</mo></munder></mrow><mo>)</mo></mrow><mo>,</mo><msub><mi>H</mi><mn>1</mn></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mo>×</mo><mi>P</mi><mo stretchy="false">(</mo><msub><mi>α</mi><mi>m</mi></msub><mrow><mo>(</mo><mrow><msub><mi>N</mi><mi>m</mi></msub></mrow><mo>)</mo></mrow><mo>|</mo><mi>E</mi><mi>M</mi><msub><mi>S</mi><mrow><mtext>rn</mtext></mrow></msub><mrow><mo>(</mo><mrow><mi>N</mi><mo>−</mo><mn>1</mn><mo>;</mo><munder accentunder="true"><mi>e</mi><mo>_</mo></munder><mo>,</mo><mtext>Δ</mtext><munder accentunder="true"><mi>τ</mi><mo>_</mo></munder></mrow><mo>)</mo></mrow><mo>,</mo><msub><mi>H</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mo>.</mo></mtd></mtr></mtable></math> <math id="M38"><mrow><mi>P</mi><mo stretchy="false">(</mo><msub><mi>α</mi><mi>m</mi></msub><mrow><mo>(</mo><mrow><msub><mi>N</mi><mi>m</mi></msub></mrow><mo>)</mo></mrow><mo>|</mo><mi>E</mi><mi>M</mi><msub><mi>S</mi><mrow><mtext>rn</mtext></mrow></msub><mrow><mo>(</mo><mrow><mi>N</mi><mo>−</mo><mn>1</mn><mo>;</mo><munder accentunder="true"><mi>e</mi><mo>_</mo></munder><mo>,</mo><mtext>Δ</mtext><munder accentunder="true"><mi>τ</mi><mo>_</mo></munder></mrow><mo>)</mo></mrow><mo>,</mo><msub><mi>H</mi><mn>1</mn></msub><mo stretchy="false">)</mo></mrow></math> is the probability that the detected photon is emitted by the m th monoenergetic source. Additionally, <math id="M39"><mrow><msub><mi>α</mi><mi>m</mi></msub><mrow><mo>(</mo><mrow><msub><mi>N</mi><mi>m</mi></msub></mrow><mo>)</mo></mrow></mrow></math> represents the detection ratio for the N m th photon. The total detected number of photons is N and there are N m photons emitted by the m th monoenergetic source. Therefore, the following equation can be derived: <math display="block" id="M40"><mrow><mi>P</mi><mrow><mo>(</mo><mrow><msub><mi>α</mi><mi>m</mi></msub><mrow><mo>(</mo><mrow><msub><mi>N</mi><mi>m</mi></msub></mrow><mo>)</mo></mrow><mtext>|</mtext><mi>E</mi><mi>M</mi><msub><mi>S</mi><mrow><mtext>rn</mtext></mrow></msub><mrow><mo>(</mo><mrow><mi>N</mi><mo>−</mo><mn>1</mn><mo>;</mo><munder accentunder="true"><mi>e</mi><mo>_</mo></munder><mo>,</mo><mtext>Δ</mtext><munder accentunder="true"><mi>τ</mi><mo>_</mo></munder></mrow><mo>)</mo></mrow><mo>,</mo><msub><mi>H</mi><mn>1</mn></msub></mrow><mo>)</mo></mrow><mo>=</mo><mfrac><mrow><msub><mi>N</mi><mi>m</mi></msub></mrow><mi>N</mi></mfrac><mo>.</mo></mrow></math> The measured energy of a photon detected from the m th monoenergetic source is assumed to follow a Gaussian distribution under <math id="M41"><mrow><mtext> </mtext><msub><mi>H</mi><mn>1</mn></msub></mrow></math> , as shown in Eq. (10). <math display="block" id="M42"><mrow><mi>P</mi><mo stretchy="false">(</mo><msub><mi>e</mi><mi>m</mi></msub><mrow><mo>(</mo><mrow><msub><mi>N</mi><mi>m</mi></msub></mrow><mo>)</mo></mrow><mo>|</mo><msub><mi>α</mi><mi>m</mi></msub><mrow><mo>(</mo><mrow><msub><mi>N</mi><mi>m</mi></msub></mrow><mo>)</mo></mrow><mo>,</mo><mi>E</mi><mi>M</mi><msub><mi>S</mi><mrow><mtext>rn</mtext></mrow></msub><mrow><mo>(</mo><mrow><mi>N</mi><mo>−</mo><mn>1</mn><mo>;</mo><munder accentunder="true"><mi>e</mi><mo>_</mo></munder><mo>,</mo><mtext>Δ</mtext><munder accentunder="true"><mi>τ</mi><mo>_</mo></munder></mrow><mo>)</mo></mrow><mo>,</mo><msub><mi>H</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mrow><msqrt><mrow><mn>2</mn><mi>π</mi></mrow></msqrt><msub><mi>δ</mi><mrow><msub><mi>e</mi><mi>m</mi></msub></mrow></msub><mrow><mo>(</mo><mrow><msub><mi>N</mi><mi>m</mi></msub></mrow><mo>)</mo></mrow></mrow></mfrac><msup><mi>e</mi><mrow><mo>−</mo><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><msub><mi>e</mi><mi>m</mi></msub><mrow><mo>(</mo><mrow><msub><mi>N</mi><mi>m</mi></msub></mrow><mo>)</mo></mrow><mo>−</mo><msub><mover accent="true"><mi>e</mi><mo>¯</mo></mover><mi>m</mi></msub><mrow><mo>(</mo><mrow><msub><mi>N</mi><mi>m</mi></msub></mrow><mo>)</mo></mrow></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mrow><mn>2</mn><msubsup><mi>σ</mi><mrow><msub><mi>e</mi><mi>m</mi></msub></mrow><mn>2</mn></msubsup><mrow><mo>(</mo><mrow><msub><mi>N</mi><mi>m</mi></msub></mrow><mo>)</mo></mrow></mrow></mfrac></mrow></msup><mo>,</mo></mrow></math> where <math id="M43"><mrow><msub><mover accent="true"><mi>e</mi><mo>¯</mo></mover><mi>m</mi></msub><mrow><mo>(</mo><mrow><msub><mi>N</mi><mi>m</mi></msub></mrow><mo>)</mo></mrow></mrow></math> and <math id="M44"><mrow><msub><mi>δ</mi><mrow><msub><mi>e</mi><mi>m</mi></msub></mrow></msub><mrow><mo>(</mo><mrow><msub><mi>N</mi><mi>m</mi></msub></mrow><mo>)</mo></mrow></mrow></math> are the expected value and standard deviation of the posterior distribution, respectively. The arrival time interval <math id="M45"><mrow><mtext>Δ</mtext><mi>τ</mi></mrow></math> is exponentially distributed and <math id="M46"><mrow><msub><mi>λ</mi><mi>m</mi></msub><mrow><mo>(</mo><mrow><msub><mi>N</mi><mi>m</mi></msub></mrow><mo>)</mo></mrow></mrow></math> is the expected value. Therefore, the following equation can be derived: <math display="block" id="M47"><mrow><mi>P</mi><mrow><mo>(</mo><mrow><mi>Δ</mi><msub><mi>τ</mi><mi>m</mi></msub><mrow><mo>(</mo><mrow><msub><mi>N</mi><mi>m</mi></msub></mrow><mo>)</mo></mrow><msub><mrow><mtext>|e</mtext></mrow><mi>m</mi></msub><mrow><mo>(</mo><mrow><msub><mi>N</mi><mi>m</mi></msub></mrow><mo>)</mo></mrow><mo>,</mo><msub><mi>α</mi><mi>m</mi></msub><mrow><mo>(</mo><mrow><msub><mi>N</mi><mi>m</mi></msub></mrow><mo>)</mo></mrow><mo>,</mo><mi>E</mi><mi>M</mi><msub><mi>S</mi><mrow><mtext>rn</mtext></mrow></msub><mrow><mo>(</mo><mrow><mi>N</mi><mo>−</mo><mn>1</mn><mo>;</mo><munder accentunder="true"><mi>e</mi><mo>_</mo></munder><mo>,</mo><mtext>Δ</mtext><munder accentunder="true"><mi>τ</mi><mo>_</mo></munder></mrow><mo>)</mo></mrow><mo>,</mo><msub><mi>H</mi><mn>1</mn></msub></mrow><mo>)</mo></mrow><mo>=</mo><msub><mi>λ</mi><mi>m</mi></msub><mrow><mo>(</mo><mrow><msub><mi>N</mi><mi>m</mi></msub></mrow><mo>)</mo></mrow><msup><mi>e</mi><mrow><mo>−</mo><msub><mi>λ</mi><mtext>m</mtext></msub><mrow><mo>(</mo><mrow><msub><mi>N</mi><mi>m</mi></msub></mrow><mo>)</mo></mrow><mtext>Δ</mtext><msub><mi>τ</mi><mi>m</mi></msub><mrow><mo>(</mo><mrow><msub><mi>N</mi><mi>m</mi></msub></mrow><mo>)</mo></mrow></mrow></msup><mo>.</mo></mrow></math> Then, the probabilistic model of the EMS under hypothesis <math id="M48"><mrow><msub><mi>H</mi><mn>1</mn></msub></mrow></math> is given by Eq. (12). <math display="block" id="M49"><mtable columnalign="left"><mtr><mtd><mi>P</mi><mrow><mo>(</mo><mrow><mi>E</mi><mi>v</mi><mi>e</mi><mi>n</mi><mi>t</mi><mo stretchy="false">(</mo><msub><mi>N</mi><mi>m</mi></msub><mo>;</mo><msub><mi>e</mi><mi>m</mi></msub><mrow><mo>(</mo><mrow><msub><mi>N</mi><mi>m</mi></msub></mrow><mo>)</mo></mrow><mo>,</mo><mi>Δ</mi><msub><mtext>τ</mtext><mi>m</mi></msub><mrow><mo>(</mo><mrow><msub><mi>N</mi><mi>m</mi></msub></mrow><mo>)</mo></mrow><mtext>|</mtext><mi>E</mi><mi>M</mi><msub><mi>S</mi><mrow><mtext>rn</mtext></mrow></msub><mrow><mo>(</mo><mrow><mi>N</mi><mo>−</mo><mn>1</mn><mo>;</mo><munder accentunder="true"><mi>e</mi><mo>_</mo></munder><mo>,</mo><mtext>Δ</mtext><munder accentunder="true"><mi>τ</mi><mo>_</mo></munder></mrow><mo>)</mo></mrow><mo>,</mo><msub><mi>H</mi><mn>1</mn></msub></mrow><mo>)</mo></mrow></mtd></mtr><mtr><mtd><mo>=</mo><mfrac><mrow><msub><mi>N</mi><mi>m</mi></msub><msub><mi>λ</mi><mi>m</mi></msub><mrow><mo>(</mo><mrow><msub><mi>N</mi><mi>m</mi></msub></mrow><mo>)</mo></mrow></mrow><mrow><mi>N</mi><msqrt><mrow><mn>2</mn><mi>π</mi></mrow></msqrt><msub><mi>σ</mi><mrow><msub><mi>e</mi><mi>m</mi></msub></mrow></msub><mrow><mo>(</mo><mrow><msub><mi>N</mi><mi>m</mi></msub></mrow><mo>)</mo></mrow></mrow></mfrac><mo>×</mo><msup><mi>e</mi><mrow><mo>−</mo><msub><mi>λ</mi><mi>m</mi></msub><mrow><mo>(</mo><mrow><msub><mi>N</mi><mi>m</mi></msub></mrow><mo>)</mo></mrow><mtext>Δ</mtext><msub><mi>τ</mi><mi>m</mi></msub><mrow><mo>(</mo><mrow><msub><mi>N</mi><mi>m</mi></msub></mrow><mo>)</mo></mrow><mo>−</mo><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><msub><mi>e</mi><mi>m</mi></msub><mrow><mo>(</mo><mrow><msub><mi>N</mi><mi>m</mi></msub></mrow><mo>)</mo></mrow><mo>−</mo><msub><mover accent="true"><mi>e</mi><mo>¯</mo></mover><mi>m</mi></msub><mrow><mo>(</mo><mrow><msub><mi>N</mi><mi>m</mi></msub></mrow><mo>)</mo></mrow></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mrow><mn>2</mn><msubsup><mi>σ</mi><mrow><msub><mi>e</mi><mi>m</mi></msub></mrow><mn>2</mn></msubsup><mrow><mo>(</mo><mrow><msub><mi>N</mi><mi>m</mi></msub></mrow><mo>)</mo></mrow></mrow></mfrac></mrow></msup></mtd></mtr></mtable></math> The probabilistic model of the EMS under hypothesis <math id="M50"><mrow><msub><mi>H</mi><mn>0</mn></msub></mrow></math> is given by Eq. (13). <math display="block" id="M51"><mtable columnalign="left"><mtr><mtd><mi>P</mi><mrow><mo>(</mo><mrow><mi>E</mi><mi>v</mi><mi>e</mi><mi>n</mi><mi>t</mi><mo stretchy="false">(</mo><msub><mi>N</mi><mi>m</mi></msub><mo>;</mo><msub><mi>e</mi><mi>m</mi></msub><mrow><mo>(</mo><mrow><msub><mi>N</mi><mi>m</mi></msub></mrow><mo>)</mo></mrow><mo>,</mo><mi>Δ</mi><msub><mtext>τ</mtext><mi>m</mi></msub><mrow><mo>(</mo><mrow><msub><mi>N</mi><mi>m</mi></msub></mrow><mo>)</mo></mrow><mtext>|</mtext><mi>E</mi><mi>M</mi><msub><mi>S</mi><mrow><mtext>rn</mtext></mrow></msub><mrow><mo>(</mo><mrow><mi>N</mi><mo>−</mo><mn>1</mn><mo>;</mo><munder accentunder="true"><mi>e</mi><mo>_</mo></munder><mo>,</mo><mtext>Δ</mtext><munder accentunder="true"><mi>τ</mi><mo>_</mo></munder></mrow><mo>)</mo></mrow><mo>,</mo><msub><mi>H</mi><mn>0</mn></msub></mrow><mo>)</mo></mrow></mtd></mtr><mtr><mtd><mo>=</mo><mfrac><mrow><msubsup><mi>λ</mi><mi>m</mi><mrow><mtext>base</mtext></mrow></msubsup></mrow><mrow><mi>M</mi><mrow><mo>(</mo><mrow><msubsup><mi>e</mi><mi>m</mi><mrow><mtext>right</mtext></mrow></msubsup><mo>−</mo><msubsup><mi>e</mi><mi>m</mi><mrow><mtext>left</mtext></mrow></msubsup></mrow><mo>)</mo></mrow></mrow></mfrac><msup><mi>e</mi><mrow><mo>−</mo><msubsup><mi>λ</mi><mi>m</mi><mrow><mtext>base</mtext></mrow></msubsup><mtext>Δ</mtext><msub><mi>τ</mi><mi>m</mi></msub><mrow><mo>(</mo><mrow><msub><mi>N</mi><mi>m</mi></msub></mrow><mo>)</mo></mrow></mrow></msup></mtd></mtr></mtable></math> The decision function is obtained by substituting Eqs. (12) and (13) into Eq. (6), as shown in Eq. (14). <math display="block" id="M52"><mtable columnalign="left"><mtr><mtd><mi>Λ</mi><mrow><mo>(</mo><mi>N</mi><mo>)</mo></mrow><mo>=</mo><mi>Λ</mi><mrow><mo>(</mo><mrow><mi>N</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow><mo>+</mo><mtext>ln</mtext><msub><mi>N</mi><mi>m</mi></msub><mo>−</mo><mtext>lnN</mtext><mo>+</mo><mi>ln</mi><mrow><mo>(</mo><mrow><mfrac><mrow><msub><mi>λ</mi><mi>m</mi></msub><mrow><mo>(</mo><mrow><msub><mi>N</mi><mi>m</mi></msub></mrow><mo>)</mo></mrow></mrow><mrow><msqrt><mrow><mn>2</mn><mi>π</mi></mrow></msqrt><msub><mi>σ</mi><mrow><msub><mi>e</mi><mi>m</mi></msub></mrow></msub><mrow><mo>(</mo><mrow><msub><mi>N</mi><mi>m</mi></msub></mrow><mo>)</mo></mrow></mrow></mfrac></mrow><mo>)</mo></mrow><mo>−</mo><msub><mi>λ</mi><mi>m</mi></msub><mrow><mo>(</mo><mrow><msub><mi>N</mi><mi>m</mi></msub></mrow><mo>)</mo></mrow><mtext>Δ</mtext><msub><mi>τ</mi><mi>m</mi></msub><mrow><mo>(</mo><mrow><msub><mi>N</mi><mi>m</mi></msub></mrow><mo>)</mo></mrow><mo>−</mo><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><msub><mi>e</mi><mi>m</mi></msub><mrow><mo>(</mo><mrow><msub><mi>N</mi><mi>m</mi></msub></mrow><mo>)</mo></mrow><mo>−</mo><msub><mover accent="true"><mi>e</mi><mo>¯</mo></mover><mi>m</mi></msub><mrow><mo>(</mo><mrow><msub><mi>N</mi><mi>m</mi></msub></mrow><mo>)</mo></mrow></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mrow><mn>2</mn><msubsup><mi>σ</mi><mrow><msub><mi>e</mi><mi>m</mi></msub></mrow><mn>2</mn></msubsup><mrow><mo>(</mo><mrow><msub><mi>N</mi><mi>m</mi></msub></mrow><mo>)</mo></mrow></mrow></mfrac></mtd></mtr><mtr><mtd><mo>+</mo><mtext>ln</mtext><mi>M</mi><mo>+</mo><mtext>ln</mtext><mrow><mo>(</mo><mrow><msubsup><mi>e</mi><mi>m</mi><mrow><mtext>right</mtext></mrow></msubsup><mo>−</mo><msubsup><mi>e</mi><mi>m</mi><mrow><mtext>left</mtext></mrow></msubsup></mrow><mo>)</mo></mrow><mo>−</mo><mtext>ln</mtext><mrow><mo>(</mo><mrow><msubsup><mi>λ</mi><mi>m</mi><mrow><mtext>base</mtext></mrow></msubsup></mrow><mo>)</mo></mrow><mo>+</mo><msubsup><mi>λ</mi><mi>m</mi><mrow><mtext>base</mtext></mrow></msubsup><mtext>Δ</mtext><msub><mi>τ</mi><mi>m</mi></msub><mrow><mo>(</mo><mrow><msub><mi>N</mi><mi>m</mi></msub></mrow><mo>)</mo></mrow><mo>.</mo></mtd></mtr></mtable></math> Implementation of the proposed method 2 First, a radionuclide identification instrument is installed in a public area to detect the EMS of the target radionuclide under conditions with natural background radiation for a long period (at least hundreds of seconds). The count rate <math id="M53"><mrow><msubsup><mi>λ</mi><mi>m</mi><mrow><mtext>base</mtext></mrow></msubsup></mrow></math> of the m th monoenergetic source in the natural background radiation is then obtained by taking average counts from the ROI. Next, a parallel detection architecture is proposed to implement the sequential detection scheme. As shown in Fig. 1(a) Fig. 1(a) , there are two types of detection channels that process the detected photons emitted by a monoenergetic source. The first type is designed to infer the parameters of the probabilistic model and evaluate the log-likelihood under hypothesis <math id="M54"><mrow><msub><mi>H</mi><mn>1</mn></msub></mrow></math> , whereas the other evaluates the log-likelihood under hypothesis <math id="M55"><mrow><msub><mi>H</mi><mn>0</mn></msub></mrow></math> . When a photon is detected in the first type of channel, photon discrimination is performed. If an event is accepted by one of the channels, then posterior distribution parameter estimation is performed. Finally, the log-likelihood is calculated based on the probabilistic model under hypothesis <math id="M56"><mrow><msub><mi>H</mi><mn>1</mn></msub></mrow></math> with updated parameters. .Event discrimination is first performed using the second type of channel. Finally, the proposed method performs a log-likelihood evaluation based on the probabilistic model under hypothesis <math id="M57"><mrow><msub><mi>H</mi><mn>0</mn></msub></mrow></math> with constant parameters. After an event is processed using the detection channels, the current decision function value is updated according to Eq. (14) and compared to the preset detection thresholds to obtain detection results according to Eq. (7). A two-stage structure was designed to perform photon discrimination, as shown in Fig. 1(b) Fig. 1(b) . The energy discriminator is used to determine which channel a photon should be processed in, after which a detection rate discriminator uses the interarrival time for verification [ 13 13 ] . The energy discriminator performs a confidence interval test, as shown in Eq. (15), where n is the particle ordinal number, <math id="M58"><mrow><msub><mi>k</mi><mrow><msub><mi>e</mi><mi>m</mi></msub></mrow></msub></mrow></math> is the confidence coefficient of the energy, and <math id="M59"><mrow><msubsup><mi>e</mi><mi>m</mi><mtext>t</mtext></msubsup></mrow></math> and <math id="M60"><mrow><msubsup><mi>σ</mi><mi>m</mi><mtext>t</mtext></msubsup></mrow></math> are the mean and standard deviation of the energy, respectively, which can be obtained experimentally. <math display="block" id="M61"><mrow><msubsup><mi>e</mi><mi>m</mi><mtext>t</mtext></msubsup><mo>−</mo><mo> </mo><msub><mi>k</mi><mrow><msub><mi>e</mi><mi>m</mi></msub></mrow></msub><msubsup><mi>σ</mi><mi>m</mi><mtext>t</mtext></msubsup><mo>≤</mo><mo> </mo><msub><mi>e</mi><mi>m</mi></msub><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mo>≤</mo><mo> </mo><msubsup><mi>e</mi><mi>m</mi><mtext>t</mtext></msubsup><mo>+</mo><mo> </mo><msub><mi>k</mi><mrow><msub><mi>e</mi><mi>m</mi></msub></mrow></msub><msubsup><mi>σ</mi><mi>m</mi><mtext>t</mtext></msubsup></mrow></math> The interarrival time discriminator also performs a confidence interval test, as shown in Eq. (16). <math display="block" id="M62"><mrow><mtext>Δ</mtext><msubsup><mi>τ</mi><mi>m</mi><mrow><mtext>base</mtext></mrow></msubsup><mo>−</mo><mo> </mo><msub><mi>k</mi><mover accent="true"><mi>γ</mi><mo>¯</mo></mover></msub><msub><mi>σ</mi><mrow><mi>Δ</mi><msub><mi>τ</mi><mrow><mtext>base</mtext></mrow></msub></mrow></msub><mo>≤</mo><mi>Δ</mi><msub><mover accent="true"><mi>τ</mi><mo>¯</mo></mover><mi>m</mi></msub><mtext> </mtext><mo>≤</mo><mtext> Δ</mtext><msubsup><mi>τ</mi><mi>m</mi><mrow><mtext>base</mtext></mrow></msubsup><mo>+</mo><mo> </mo><msub><mi>k</mi><mover accent="true"><mi>γ</mi><mo>¯</mo></mover></msub><msub><mi>σ</mi><mrow><mi>Δ</mi><msub><mi>τ</mi><mrow><mtext>base</mtext></mrow></msub></mrow></msub><mo>,</mo><mo> </mo><msub><mi>σ</mi><mrow><mi>Δ</mi><msub><mi>τ</mi><mrow><mtext>base</mtext></mrow></msub></mrow></msub><mo>=</mo><mtext> </mtext><mi>Δ</mi><msub><mi>τ</mi><mrow><mtext>base</mtext></mrow></msub><mo>/</mo><msqrt><mi>n</mi></msqrt><mo>,</mo></mrow></math> where <math id="M63"><mrow><mi>Δ</mi><msub><mover accent="true"><mi>τ</mi><mo>¯</mo></mover><mi>m</mi></msub></mrow></math> is the average interarrival time and <math id="M64"><mrow><msub><mi>k</mi><mover accent="true"><mi>γ</mi><mo>¯</mo></mover></msub></mrow></math> is the confidence coefficient. <math id="M65"><mrow><mtext>Δ</mtext><msubsup><mi>τ</mi><mi>m</mi><mrow><mtext>base</mtext></mrow></msubsup></mrow></math> is the mean interarrival time under conditions with natural background radiation and <math id="M66"><mrow><mo> </mo><msub><mi>σ</mi><mrow><mi>Δ</mi><msub><mi>τ</mi><mrow><mtext>base</mtext></mrow></msub></mrow></msub></mrow></math> is the width of the confidence interval. The posterior probability distribution parameter estimation of the EMS under hypothesis <math id="M67"><mrow><msub><mi>H</mi><mn>1</mn></msub></mrow></math> is achieved using Bayesian inference. Because energy is modeled as a Gaussian distribution and the interarrival time is modeled as an exponential distribution (both belonging to the exponential family of distributions), it is convenient to find conjugate prior distributions and infer the posterior distribution in analytical form. Following research on Bayesian inferencing [ 28 28 , 29 29 ] , the Gaussian–Gamma distribution was applied to model the prior distribution of <math id="M68"><mrow><msub><mover accent="true"><mi>e</mi><mo>¯</mo></mover><mi>m</mi></msub></mrow></math> and <math id="M69"><mrow><msup><mrow><mrow><mo>(</mo><mrow><msubsup><mi>σ</mi><mrow><msub><mi>e</mi><mi>m</mi></msub></mrow><mn>2</mn></msubsup></mrow><mo>)</mo></mrow></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></mrow></math> because the precision parameter <math id="M70"><mrow><mi>p</mi><mi>r</mi><mi>e</mi><msub><mi>c</mi><mrow><msub><mi>e</mi><mi>m</mi></msub></mrow></msub></mrow></math> is defined as <math id="M71"><mrow><msup><mrow><mrow><mo>(</mo><mrow><msubsup><mi>σ</mi><mrow><msub><mi>e</mi><mi>m</mi></msub></mrow><mn>2</mn></msubsup></mrow><mo>)</mo></mrow></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></mrow></math> . The prior distributions <math id="M72"><mrow><msub><mi>τ</mi><mrow><msub><mi>e</mi><mi>m</mi></msub></mrow></msub><mrow><mo>(</mo><mn>0</mn><mo>)</mo></mrow></mrow></math> and <math id="M73"><mrow><msub><mover accent="true"><mi>e</mi><mo>¯</mo></mover><mi>m</mi></msub><mrow><mo>(</mo><mn>0</mn><mo>)</mo></mrow></mrow></math> are defined in Equation (17), where <math id="M74"><mrow><msub><mi>a</mi><mn>0</mn></msub></mrow></math> , <math id="M75"><mrow><msub><mi>b</mi><mn>0</mn></msub></mrow></math> , <math id="M76"><mrow><msub><mi>u</mi><mn>0</mn></msub></mrow></math> , and <math id="M77"><mrow><msub><mi>v</mi><mn>0</mn></msub></mrow></math> are hyperparameters [ 30 30 ] . For a weak prior, <math id="M78"><mrow><msub><mi>u</mi><mn>0</mn></msub></mrow></math> is set to <math id="M79"><mrow><msubsup><mi>e</mi><mi>m</mi><mi>t</mi></msubsup></mrow></math> , <math id="M80"><mrow><msub><mi>a</mi><mn>0</mn></msub></mrow></math> and <math id="M81"><mrow><msub><mi>b</mi><mn>0</mn></msub></mrow></math> are set to one and 100, respectively, and <math id="M82"><mrow><msub><mi>λ</mi><mn>0</mn></msub></mrow></math> is set to one. <math display="block" id="M83"><mtable columnalign="left"><mtr><mtd><mi>p</mi><mi>r</mi><mi>e</mi><msub><mi>c</mi><mrow><msub><mi>e</mi><mi>m</mi></msub></mrow></msub><mrow><mo>(</mo><mn>0</mn><mo>)</mo></mrow><mo>~</mo><mi>G</mi><mi>a</mi><mi>m</mi><mi>m</mi><mi>a</mi><mrow><mo>(</mo><mrow><msub><mi>a</mi><mn>0</mn></msub><mo>,</mo><msub><mi>b</mi><mn>0</mn></msub></mrow><mo>)</mo></mrow><mo>,</mo></mtd></mtr><mtr><mtd><msub><mover accent="true"><mi>e</mi><mo>¯</mo></mover><mi>m</mi></msub><mrow><mo>(</mo><mn>0</mn><mo>)</mo></mrow><mo>~</mo><mi>N</mi><mrow><mo>(</mo><mrow><msub><mi>u</mi><mn>0</mn></msub><mo>,</mo><msup><mrow><mrow><mo>(</mo><mrow><msub><mi>v</mi><mn>0</mn></msub><mi>p</mi><mi>r</mi><mi>e</mi><msub><mi>c</mi><mrow><msub><mi>e</mi><mi>m</mi></msub></mrow></msub><mrow><mo>(</mo><mn>0</mn><mo>)</mo></mrow></mrow><mo>)</mo></mrow></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></mrow><mo>)</mo></mrow><mo>,</mo></mtd></mtr><mtr><mtd><mi>P</mi><mrow><mo>(</mo><mrow><msub><mover accent="true"><mi>e</mi><mo>¯</mo></mover><mi>m</mi></msub><mrow><mo>(</mo><mn>0</mn><mo>)</mo></mrow><mo>,</mo><mi>p</mi><mi>r</mi><mi>e</mi><msub><mi>c</mi><mrow><msub><mi>e</mi><mi>m</mi></msub></mrow></msub><mrow><mo>(</mo><mn>0</mn><mo>)</mo></mrow></mrow><mo>)</mo></mrow><mo>=</mo><mi>N</mi><mo stretchy="false">(</mo><msub><mover accent="true"><mi>e</mi><mo>¯</mo></mover><mi>m</mi></msub><mrow><mo>(</mo><mn>0</mn><mo>)</mo></mrow><mo>|</mo><msub><mi>u</mi><mn>0</mn></msub><mo>,</mo><msup><mrow><mo>(</mo><mrow><msub><mi>v</mi><mn>0</mn></msub><mi>p</mi><mi>r</mi><mi>e</mi><msub><mi>c</mi><mrow><msub><mi>e</mi><mi>m</mi></msub></mrow></msub><mrow><mo>(</mo><mn>0</mn><mo>)</mo></mrow></mrow><mo>)</mo></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo><mi>G</mi><mi>a</mi><mi>m</mi><mi>m</mi><mi>a</mi><mo stretchy="false">(</mo><mi>p</mi><mi>r</mi><mi>e</mi><msub><mi>c</mi><mrow><msub><mi>e</mi><mi>m</mi></msub></mrow></msub><mrow><mo>(</mo><mn>0</mn><mo>)</mo></mrow><mo>|</mo><msub><mi>a</mi><mn>0</mn></msub><mo>,</mo><msub><mi>b</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mtd></mtr></mtable></math> Equation (18) models the prior distribution <math id="M84"><mrow><msub><mi>λ</mi><mi>m</mi></msub><mrow><mo>(</mo><mn>0</mn><mo>)</mo></mrow></mrow></math> of the expected value of the interarrival <math id="M85"><mrow><msub><mi>λ</mi><mi>m</mi></msub></mrow></math> as Gamma. <math id="M86"><mrow><msub><mi>c</mi><mn>0</mn></msub></mrow></math> and <math id="M87"><mrow><msub><mi>d</mi><mn>0</mn></msub></mrow></math> are also hyperparameters, where <math id="M88"><mrow><msub><mi>c</mi><mn>0</mn></msub></mrow></math> is set to one and <math id="M89"><mrow><msub><mi>d</mi><mn>0</mn></msub></mrow></math> is set to <math id="M90"><mrow><msubsup><mi>λ</mi><mi>m</mi><mrow><mtext>base</mtext></mrow></msubsup></mrow></math> . <math display="block" id="M91"><mtable columnalign="left"><mtr><mtd><msub><mi>λ</mi><mi>m</mi></msub><mrow><mo>(</mo><mn>0</mn><mo>)</mo></mrow><mo>~</mo><mi>G</mi><mi>a</mi><mi>m</mi><mi>m</mi><mi>a</mi><mrow><mo>(</mo><mrow><msub><mi>c</mi><mn>0</mn></msub><mo>,</mo><msub><mi>d</mi><mn>0</mn></msub></mrow><mo>)</mo></mrow></mtd></mtr><mtr><mtd><mi>P</mi><mrow><mo>(</mo><mrow><msub><mi>λ</mi><mi>m</mi></msub><mrow><mo>(</mo><mn>0</mn><mo>)</mo></mrow></mrow><mo>)</mo></mrow><mo>=</mo><mi>G</mi><mi>a</mi><mi>m</mi><mi>m</mi><mi>a</mi><mo stretchy="false">(</mo><msub><mi>λ</mi><mi>m</mi></msub><mrow><mo>(</mo><mn>0</mn><mo>)</mo></mrow><mo>|</mo><msub><mi>c</mi><mn>0</mn></msub><mo>,</mo><msub><mi>d</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mtd></mtr></mtable></math> Up to the n th arrival from the m th monoenergetic source, the posterior distributions of <math id="M92"><mrow><msub><mover accent="true"><mi>e</mi><mo>¯</mo></mover><mi>m</mi></msub><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></math> , <math id="M93"><mrow><msub><mi>τ</mi><mrow><msub><mi>e</mi><mi>m</mi></msub></mrow></msub><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></math> , and <math id="M94"><mrow><msub><mi>λ</mi><mi>m</mi></msub><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></math> are updated as shown in Equation (19), where <math id="M95"><mrow><msub><mi>u</mi><mi>n</mi></msub></mrow></math> , <math id="M96"><mrow><msub><mi>v</mi><mi>n</mi></msub></mrow></math> , <math id="M97"><mrow><msub><mi>a</mi><mi>n</mi></msub></mrow></math> , <math id="M98"><mrow><msub><mi>b</mi><mi>n</mi></msub></mrow></math> , <math id="M99"><mrow><msub><mi>c</mi><mi>n</mi></msub></mrow></math> , and <math id="M100"><mrow><msub><mi>d</mi><mi>n</mi></msub></mrow></math> are the hyperparameters of the posterior distributions [ 31 31 ] . http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23770221&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23770223&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23770229&type= The hyperparameters can be inferred in analytical form [ 31 31 ] via conjugacy, as shown in Equation (20). <math display="block" id="M101"><mtable columnalign="left"><mtr><mtd><msubsup><mi>e</mi><mi>m</mi><mrow><mtext>mean</mtext></mrow></msubsup><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mrow><msubsup><mstyle mathsize="140%" displaystyle="true"><mo>∑</mo></mstyle><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></msubsup><msub><mi>e</mi><mi>m</mi></msub><mrow><mo>(</mo><mi>i</mi><mo>)</mo></mrow></mrow><mi>n</mi></mfrac></mtd></mtr><mtr><mtd><msubsup><mi>λ</mi><mi>m</mi><mrow><mtext>mean</mtext></mrow></msubsup><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mrow><msubsup><mstyle mathsize="140%" displaystyle="true"><mo>∑</mo></mstyle><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></msubsup><msub><mi>λ</mi><mi>m</mi></msub><mrow><mo>(</mo><mi>i</mi><mo>)</mo></mrow></mrow><mi>n</mi></mfrac></mtd></mtr><mtr><mtd><msub><mi>u</mi><mi>n</mi></msub><mo>=</mo><mfrac><mrow><msub><mi>λ</mi><mn>0</mn></msub><msub><mi>u</mi><mn>0</mn></msub><mo>+</mo><msubsup><mstyle mathsize="140%" displaystyle="true"><mo>∑</mo></mstyle><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></msubsup><msub><mi>e</mi><mi>m</mi></msub><mrow><mo>(</mo><mi>i</mi><mo>)</mo></mrow></mrow><mrow><msub><mi>λ</mi><mn>0</mn></msub><mo>+</mo><mi>n</mi></mrow></mfrac></mtd></mtr><mtr><mtd><msub><mi>v</mi><mi>n</mi></msub><mo>=</mo><msub><mi>v</mi><mn>0</mn></msub><mo>+</mo><mi>n</mi></mtd></mtr><mtr><mtd><msub><mi>a</mi><mi>n</mi></msub><mo>=</mo><msub><mi>a</mi><mn>0</mn></msub><mo>+</mo><mi>n</mi><mo>/</mo><mn>2</mn></mtd></mtr><mtr><mtd><msub><mi>b</mi><mi>n</mi></msub><mo>=</mo><msub><mi>b</mi><mn>0</mn></msub><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><munderover><mstyle mathsize="140%" displaystyle="true"><mo>∑</mo></mstyle><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msup><mrow><mo>(</mo><mrow><msub><mi>e</mi><mi>m</mi></msub><mrow><mo>(</mo><mi>i</mi><mo>)</mo></mrow><mo>−</mo><msubsup><mi>e</mi><mi>m</mi><mrow><mtext>mean</mtext></mrow></msubsup><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow><mo>)</mo></mrow><mn>2</mn></msup><mo>+</mo><mfrac><mrow><msub><mi>λ</mi><mn>0</mn></msub><mi>n</mi><mrow><mo>(</mo><mrow><msubsup><mi>e</mi><mi>m</mi><mrow><mtext>mean</mtext></mrow></msubsup><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mo>−</mo><msub><mi>u</mi><mn>0</mn></msub></mrow><mo>)</mo></mrow></mrow><mrow><mn>2</mn><msub><mi>λ</mi><mi>n</mi></msub></mrow></mfrac></mtd></mtr><mtr><mtd><msub><mi>c</mi><mi>n</mi></msub><mo>=</mo><msub><mi>c</mi><mn>0</mn></msub><mo>+</mo><mi>n</mi><mo>/</mo><mn>2</mn></mtd></mtr><mtr><mtd><msub><mi>d</mi><mi>n</mi></msub><mo>=</mo><msub><mi>d</mi><mn>0</mn></msub><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><munderover><mstyle mathsize="140%" displaystyle="true"><mo>∑</mo></mstyle><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msub><mi>λ</mi><mi>m</mi></msub><mrow><mo>(</mo><mi>i</mi><mo>)</mo></mrow></mtd></mtr></mtable></math> The expectation of the posterior distribution can be used to obtain the parameters of the EMS model under <math id="M102"><mrow><msub><mi>H</mi><mn>1</mn></msub></mrow></math> . <math display="block" id="M103"><mtable columnalign="left"><mtr><mtd><msub><mover accent="true"><mi>e</mi><mo>¯</mo></mover><mi>m</mi></msub><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mo>=</mo><msub><mi>u</mi><mi>n</mi></msub></mtd></mtr><mtr><mtd><msub><mi>σ</mi><mrow><msub><mi>e</mi><mi>m</mi></msub></mrow></msub><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mo>=</mo><msqrt><mrow><mfrac><mn>1</mn><mrow><msub><mi>τ</mi><mrow><msub><mi>e</mi><mi>m</mi></msub></mrow></msub><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></mfrac></mrow></msqrt><mo>=</mo><msqrt><mrow><mfrac><mrow><msub><mi>b</mi><mi>n</mi></msub></mrow><mrow><msub><mi>a</mi><mi>n</mi></msub></mrow></mfrac></mrow></msqrt></mtd></mtr><mtr><mtd><msub><mi>λ</mi><mi>m</mi></msub><mrow><mo>(</mo><mtext>n</mtext><mo>)</mo></mrow><mo>=</mo><mfrac><mrow><msub><mi>c</mi><mi>n</mi></msub></mrow><mrow><msub><mi>d</mi><mi>n</mi></msub></mrow></mfrac></mtd></mtr></mtable></math> Then, the decision function is updated according to the parameters and the detection result is obtained by comparing the decision function to the preset detection thresholds. If the decision function value exceeds the threshold <math id="M104"><mrow><msub><mi>T</mi><mn>1</mn></msub></mrow></math> , a target radionuclide exists in the detection environment. If the decision function value exceeds the threshold <math id="M105"><mrow><msub><mi>T</mi><mn>0</mn></msub></mrow></math> , then there is no target radionuclide in the detection environment. Otherwise, the detection process continues until one of the thresholds is exceeded. Experiments 1 Background experiment 2 In this experiment, the spectrum of the background was recorded for a long duration. The count rate of the ROI was stable and the relative statistical error was small. The target radioactive sources were added and the spectrum was collected to form a stable spectrum. The energy distribution model of the characteristic peaks in the ROI was then analyzed. For a target radionuclide, the count rate of the ROI in the background can be obtained through a long-term detection experiment [ 34 34 ] . 137 Cs and 60 Co were selected as target radionuclides. The radioactivity of the 137 Cs source was 8700 Bq and the radioactivity of the 60 Co source was 1500 Bq. A LaBr3 (Ce) detector of size <math id="M106"><mrow><mtext>Ф</mtext><mn>1.5</mn><mo>×</mo><mn>1.5</mn></mrow></math> inches was used in this experiment. The EX-03 multi-channel analyzer was developed by the Chengdu University of Technology to obtain the EMS. The parameters for Eqs. (15) and (16) are listed in Table 1 Table 1 . The experiment was performed in a laboratory environment to obtain the count rate of the ROI in the background and the detection time was set to 250 s. The threshold used in this study was obtained experimentally and the results of the experiments demonstrated that a threshold of 3.98 is acceptable. 1 1 Detection parameter 1 1 Value 1 1 <math id="M107"><mrow><msubsup><mi>e</mi><mn>1</mn><mtext>t</mtext></msubsup></mrow></math> of the 1st component of 137 Cs 1 1 661 keV 1 1 <math id="M108"><mrow><msubsup><mi>σ</mi><mn>1</mn><mtext>t</mtext></msubsup></mrow></math> of the 1st component of 137 Cs 1 1 10 keV 1 1 <math id="M109"><mrow><msubsup><mi>e</mi><mn>1</mn><mtext>t</mtext></msubsup></mrow></math> of the 1st component of 60 Co 1 1 1170 keV 1 1 <math id="M110"><mrow><msubsup><mi>σ</mi><mn>1</mn><mtext>t</mtext></msubsup></mrow></math> of the 1st component of 60 Co 1 1 16 keV 1 1 <math id="M111"><mrow><msubsup><mi>e</mi><mn>2</mn><mtext>t</mtext></msubsup></mrow></math> of the 2nd component of 60 Co 1 1 1330 keV 1 1 <math id="M112"><mrow><msubsup><mi>σ</mi><mn>2</mn><mtext>t</mtext></msubsup></mrow></math> of the 2nd component of 60 Co 1 1 20 keV 1 1 <math id="M113"><mrow><msub><mi>k</mi><mrow><msub><mi>e</mi><mi>m</mi></msub></mrow></msub></mrow></math> , <math id="M114"><mrow><msub><mi>k</mi><mover accent="true"><mi>γ</mi><mo>¯</mo></mover></msub></mrow></math>   1 1 1.0 1 1 <math id="M115"><mrow><msub><mi>T</mi><mn>0</mn></msub></mrow></math>   1 1 −3.98 1 1 <math id="M116"><mrow><msub><mi>T</mi><mn>1</mn></msub></mrow></math>   1 1 +3.98 The spectrum and EMS detected in the background are presented in Fig. 2 Fig. 2 . The ROIs in the spectrum are labeled with red, blue, and green regions. The ROI of 137 Cs ranges from 651 to 671 keV. The first ROI of 60 Co ranges from 1154 to 1186 keV and the second ROI of 60 Co ranges from 1310 to 1350 keV. It is clear that the counts in the ROI approximately follow a uniform distribution. Figure 2(a) Figure 2(a) presents a description of the test bench. Figure 2(b) Figure 2(b) presents the energy spectrum and EMS time-domain scatter diagram for natural background radiation. Figure 2(c) Figure 2(c) presents the energy spectrum and EMS time-domain scatter diagram of the target nuclide 137 Cs and natural background radiation. Figure 2(d) Figure 2(d) presents the energy spectrum and EMS time-domain scatter diagram of the target nuclide 60 CO and natural background radiation. Mathematically, the Chi-square goodness-of-fit test was performed to validate that the energy detected in the ROI under natural background radiation conforms to a uniform distribution [ 31 31 ] . The relevant formula is presented in Eq. (22), where LeftChannel and RightChannel are the edges of the selected ROI, <math id="M117"><mrow><mi>S</mi><mi>p</mi><mi>e</mi><mi>c</mi><mi>t</mi><mi>r</mi><mi>u</mi><msub><mi>m</mi><mi>i</mi></msub></mrow></math> is the count of the i th channel in the spectrum, and T is obtained by taking the average of the counts in the ROI. <math display="block" id="M118"><mrow><mi>C</mi><mi>h</mi><msub><mi>i</mi><mrow><mtext>square</mtext></mrow></msub><mo>=</mo><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mtext>LeftChannel</mtext></mrow><mrow><mtext>RightChannel</mtext></mrow></munderover><mrow><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><msub><mrow><mtext>Spectrum</mtext></mrow><mi>i</mi></msub><mo>−</mo><mi>T</mi></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mi>T</mi></mfrac><mo>.</mo></mrow></mstyle></mrow></math> For the ROI of 137 Cs, the Chi-square test result was 26.32. For the first ROI of 60 Co (［1155, 1185］ keV), the result was 35.85 and for the second ROI of 60 Co (［1310, 1350］ keV), the result was 47.10. The calculated chi-square values were compared to the 95% confidence threshold values, which were 31.41, 43.77, and 55.76, respectively. All of the calculated values are less than the corresponding threshold values, indicating that the energy in the ROI measured under the background conditions was uniformly distributed. The count rate of the ROI under natural radiation background, denoted as <math id="M119"><mrow><msub><mtext>λ</mtext><mrow><msub><mrow><mtext>Δτ</mtext></mrow><mrow><mtext>base</mtext></mrow></msub></mrow></msub></mrow></math> , was also calculated. The average interarrival time <math id="M120"><mrow><mi>Δ</mi><msub><mi>τ</mi><mrow><mtext>base</mtext></mrow></msub></mrow></math> was obtained by taking the reciprocal of the count rate. The average count rates in the background for ［651, 671］ keV, ［1154, 1186］ keV and ［1310, 1350］ keV were 5.05, 3.83 and 3.61 per second, respectively. The corresponding average interarrival times were 0.1980 s, 0.2611 s, and 0.2770 s, respectively. The rapid nuclide identification method was used to detect target radionuclides under background conditions. The detection results for 137 Cs under background conditions are presented in Fig. 3(a) Fig. 3(a) . The detection function gradually decreases along with events that are processed until the detection function exceeds the low threshold, indicating that the background is detected in 5 s. The detection results for 60 Co under background conditions are presented in Fig. 3(b) Fig. 3(b) . This detection function also gradually decreases with the events that are processed until the detection function exceeds the low threshold, indicating that 60 Co is absent in the background. Radionuclide detection experiment 2 The 137 Cs source was placed 35 cm in front of the LaBr 3 detector to verify the proposed method under a radiation field and detection was performed for 100 s. The detected EMS and spectrum are presented in Fig. 2 (b) Fig. 2 (b) . A fitted Gaussian function (red line) is placed above the ROI, and it is clear that the energy detected in the ROI under a radiation field approximately follows a normal distribution. In addition to this graphical depiction, the Chi-square test was performed to validate that the energy detected in the ROI under a radiation field approximately follows a normal distribution. The relevant formula is presented in Eq. (23). <math display="block" id="M121"><mrow><mi>C</mi><mi>h</mi><msub><mi>i</mi><mrow><mtext>square</mtext></mrow></msub><mo>=</mo><mo> </mo><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mtext>LeftChannel</mtext></mrow><mrow><mtext>RightChannel</mtext></mrow></munderover><mrow><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><msub><mrow><mtext>Spectrum</mtext></mrow><mi>i</mi></msub><mo>−</mo><msub><mrow><mtext>IntegralValue</mtext></mrow><mi>i</mi></msub></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mrow><msub><mrow><mtext>IntegralValue</mtext></mrow><mi>i</mi></msub></mrow></mfrac></mrow></mstyle><mo>,</mo></mrow></math> where LeftChannel and RightChannel are the edges of the selected ROI, <math id="M122"><mrow><mi>S</mi><mi>p</mi><mi>e</mi><mi>c</mi><mi>t</mi><mi>r</mi><mi>u</mi><msub><mi>m</mi><mi>i</mi></msub></mrow></math> is the count of the i th channel in the spectrum, and <math id="M123"><mrow><mi>I</mi><mi>n</mi><mi>t</mi><mi>e</mi><mi>g</mi><mi>r</mi><mi>a</mi><mi>l</mi><mi>V</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi><mo> </mo></mrow></math> is obtained using Eq. (24). <math display="block" id="M124"><mrow><msub><mrow><mtext>IntegralValue</mtext></mrow><mi>i</mi></msub><mo>=</mo><mtext>TotalCount</mtext><mo> </mo><mo>×</mo><munderover><mstyle mathsize="140%" displaystyle="true"><mo>∫</mo></mstyle><mrow><msubsup><mi>e</mi><mrow><mtext>left</mtext></mrow><mi>i</mi></msubsup></mrow><mrow><msubsup><mi>e</mi><mrow><mtext>right</mtext></mrow><mi>i</mi></msubsup></mrow></munderover><mfrac><mn>1</mn><mrow><msqrt><mrow><mn>2</mn><mi>π</mi></mrow></msqrt><msup><mi>σ</mi><mi>t</mi></msup></mrow></mfrac><msup><mi>e</mi><mrow><mfrac><mrow><mo>−</mo><mrow><mo>(</mo><mrow><mi>e</mi><mo>−</mo><msup><mi>e</mi><mtext>t</mtext></msup></mrow><mo>)</mo></mrow></mrow><mrow><mn>2</mn><msup><mrow><mrow><mo>(</mo><mrow><msup><mi>δ</mi><mtext>t</mtext></msup></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></mfrac></mrow></msup><mo>,</mo></mrow></math> where <math id="M125"><mrow><mi>T</mi><mi>o</mi><mi>t</mi><mi>a</mi><mi>l</mi><mi>C</mi><mi>o</mi><mi>u</mi><mi>n</mi><mi>t</mi></mrow></math> is the total count in the ROI, and <math id="M126"><mrow><msubsup><mi>e</mi><mrow><mtext>left</mtext></mrow><mi>i</mi></msubsup></mrow></math> and <math id="M127"><mrow><msubsup><mi>e</mi><mrow><mtext>right</mtext></mrow><mi>i</mi></msubsup></mrow></math> are the left and right energies of the i th bin in the spectrum, respectively. Additionally, <math id="M128"><mrow><msup><mi>e</mi><mtext>t</mtext></msup></mrow></math> and <math id="M129"><mrow><msup><mi>σ</mi><mtext>t</mtext></msup></mrow></math> are preset distribution parameters from Table 1 Table 1 . The chi-square test result for the ROI ［651,671］ was 26.32. When comparing the calculated Chi-square value to the value of the 95% confidence threshold of 31.41, the calculated value is less than the threshold, indicating that the energy spectrum in the ROI under the 137 Cs radiation field and long-term detection conforms to a normal distribution. A rapid nuclide identification method was used to detect 137 Cs in the corresponding radiation field. The detection function is presented in Fig. 4(a) Fig. 4(a) , where the decision function gradually increases with events that are processed and eventually exceeds the high threshold, indicating that the 137 Cs source is detected in 3.8 s, where the count rate of the ROI is 5.05 s −1 . There were only a few counts in the ROI before 137 Cs that were identified and the counts in the ROI were not sufficient to form a peak. According to Eqs. (23) and (24), the Chi-square test was performed to determine whether the spectrum in the ROI follows a Gaussian distribution. The test value was 41.76, which is greater than the threshold value of 31.41, indicating that traditional full-spectrum-analysis-based radionuclide identification methods cannot identify 137 Cs at low values. The 60 Co source was placed 20 cm in front of the LaBr 3 detector and detection was performed for 100 s. The EMS and spectrum detected are presented in Fig. 2(c) Fig. 2(c) , where two fitted Gaussian functions (red and blue lines) are placed in the ROI. The energy detected in the ROI under the 60 Co radiation field approximately follows a Gaussian distribution. By using Eqs. (23) and (24), the Chi-square test was also performed for validation. The Chi-square test results for ROI ［1155, 1185］ and ［1310, 1350］ keV were 35.85, and 47.10, respectively. The calculated Chi-square values were compared to the values of the 95% confidence threshold, which were 43.77 and 55.76, respectively. All calculated values were less than the threshold values, indicating that the energy in the ROI detected in the spectrum under radiation and long-term detection conforms to a Gaussian distribution. The rapid nuclide identification method was used to detect 60 Co in the corresponding radiation field. The detection results are presented in Fig. 4(b) Fig. 4(b) , the decision function gradually increased with events that are processed and eventually exceeds the high threshold, indicating that there is a 60 Cs source in the radiation field. The proposed method can detect 60 Cs in 4.1 s, where the count rate of the corresponding ROI is 7.44 s −1 . Only a few counts in the ROI before 60 Co were detected. To determine if the counts in the ROI were insufficient to form a peak, according to Eqs. (23) and (24), a Chi-square test was performed to demonstrate that the spectrum in the ROI did not follow a Gaussian distribution. The test values of the ROIs of ［1155, 1185］ keV and ［1310, 1350］ keV were 48.59 and 60.34, respectively, and the 95% confidence threshold values were 43.77 and 55.76, respectively. It is clear that the calculated values are greater than the detection thresholds. Therefore, traditional full-spectrum-analysis-based radionuclide identification methods cannot identify 60 Co with such a low counts. Detection performance experiments 2 Experiments were conducted while varying the distance between the source and detector to evaluate the performance of the proposed method. The step length was 5 cm and the distance ranged from 5 to 100 cm. For each distance, the detection experiments were repeated 20 times to analyze the statistics of the identification time. The average detection time and variance for each test group were calculated and plotted. The results for the identification time required to cover the distance between the source and detector are presented in Fig. 5 Fig. 5 . As shown in the figure above, when the detection distance is set to 5 cm, the 137 Cs detection time is 0.2 s. As the distance increases, the detection time also increases. When the distance reaches 95 cm, the average detection time increases to 6.19 s. Regarding the two types of sources with low activity in the experiment, when the detection distance reaches 90 cm, the detection times of the target nuclide 137 Cs and target nuclide 60 CO reach 7 s and 12 s, respectively. For the measurement of 60 Co, the average detection time is within 2 s when the detection distance is within 25 cm. As the detection distance increases, the detection time also increases and the relative distance reaches 60 cm. The average detection time is 6 s, which meets the requirements for rapid identification. Conclusions and prospects 1 Based on the Bayesian sequential detection method, this paper introduced a rapid nuclide identification method for radionuclide inspection in public areas. Unlike other traditional full-spectrum-analysis-based methods, in this study, based on the EMS measurement of background radiation and the target radiation field, two EMS sequences were established, a priori models were estimated based on effective particle events, and an a posteriori model was obtained. The target nuclides were identified via sequential detection of the posteriori models. To prove that the efficiency of the proposed background-comparison-based radionuclide identification method can meet the requirements of rapid identification under different measurement conditions, several groups of experiments were conducted. 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Tech.  28 , 150 ( 2017 ). doi: 10.1007/s41365-017-0293-9 http://doi.org/10.1007/s41365-017-0293-9 Evaluation of terrestrial radiation in the Chengdu Plain using 1/250,000-scale geochemical prospecting data All authors contributed to the study conception and design. Material preparation, data collection and analysis were performed by Xiao-Zhe Li, Qing-Xian Zhang, He-Yi Tan, Zhi-Qiang Cheng. The first draft of the manuscript was written by Xiao-Zhe Li and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript. 10.1007/s41365-021-00982-z This work was supported by the National Natural science Foundation of China (No. 41774190), the Special-funded program on national key scientific instruments and equipment development (No. 2017YFC0602105), and the Department of Science and Technology of Sichuan Province (No. 2020JDRC0109) . 2021-06-06 2021-08-12 2021-08-16 2021-12-20 2022-01-14 © China Science Publishing & Media Ltd. (Science Press), Shanghai Institute of Applied Physics, the Chinese Academy of Sciences, ChineseNuclear Society and Springer Nature Singapore Pte Ltd. 2021 This is a post-peer-review, pre-copyedit version of an article published in Nuclear Science and Techniques. The final authenticated version is available online at: http://dx.doi.org/10.1007/s41365-021-00982-z http://dx.doi.org/10.1007/s41365-021-00982-z . 2021 Nuclear Science and Techniques 12 32