Nuclide identification model based on background comparisons

The concept of monoenergetic decomposition^[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 $Event\left(n;e_m\left(n\right),\Delta {\text{τ}}_m\left(n\right)\right)$, is defined as the nth gamma ray recorded by the detector. This ray is emitted by the mth monoenergetic source with a measured energy $e_m\left(n\right)$ and interarrival time $\Delta {\tau }_m\left(n\right)$ between the current and previous event. The EMS of a target radionuclide ($EM{S}_{\text{rn}}$) is then represented as $EM{S}_{\text{rn}}\left(N;\underset{\_}{e},\text{Δ}\underset{\_}{\tau }\right)$. $EM{S}_{\text{rn}}\left(N;\underset{\_}{e},\Delta \underset{\_}{\tau }\right):={\displaystyle \sum _{m=1}^M{\displaystyle \sum _{n=1}^{N_m}Event\left(n;e_m\left(n\right),\Delta {\text{τ}}_m\left(n\right)\right)}}$ (1)

The index m represents the mth monoenergetic source of a target radionuclide and the total number of monoenergetic sources is M. The detector detects $N_m$ events from the mth monoenergetic source during the detection period. $\underset{\_}{e}$ is defined as the complete set of energies composing $EM{S}_{\text{rn}}$ along with $\text{Δ}\underset{\_}{\tau }$, which is the corresponding set of interarrival times. N is the sum of the set $N_m$.

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]). The corresponding formula is presented in Eq. (2), where ${\overline{e}}_m$ and ${\sigma }_{e_m}$ are the expected value and standard deviation of the Gaussian distribution, respectively. ${\lambda }_m$ is the expected value of the exponential distribution and ${\alpha }_m$ is the emission probability of the mth monoenergetic source. $\begin{array}{l}P\left(e_m\left(n\right)|\text{radionuclide}\right)=P\left(e_m\left(n\right)\text{|}{\overline{e}}_m,{\sigma }_{e_m}^2\right)\\ =\frac{1}{\sqrt{2\pi }{\sigma }_{e_m}}{e}^{-\frac{{\left(e_m\left(n\right)-{\overline{e}}_m\right)}^2}{2{\sigma }_{e_m}^2}}\\ P\left(\Delta {\text{τ}}_m\left(n\right)|\text{radionuclide}\right)=P\left(\Delta {\text{τ}}_m\left(n\right)|{\lambda }_m\right)={\lambda }_m{e}^{-{\lambda }_m\text{Δ}{\tau }_m\left(n\right)}\\ P(Event\left(n;e_m\left(n\right),\Delta {\text{τ}}_m\left(n\right)\text{|radionuclide}\right)={\alpha }_m\times P\left(e_m\left(n\right)|\text{radionuclide}\right)\\ \times P\left(\Delta {\tau }_m\left(n\right)|\text{radionuclide}\right)\end{array}$ (2)

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. $\begin{array}{l}P\left(e_m\left(n\right)|\text{background}\right)=P\left(e_m\left(n\right){\text{|e}}_m^{\text{left}},{\text{e}}_m^{\text{right}}\right)=\frac{1}{{\text{e}}_m^{\text{right}}-{\text{e}}_m^{\text{left}}},\\ P\left(\Delta {\text{τ}}_m\left(n\right)|\text{background}\right)=P\left(\Delta {\text{τ}}_m\left(\text{n}\right)\text{|}{\lambda }_m^{\text{base}}\right)={\lambda }_m^{\text{base}}{e}^{-{\lambda }_m^{\text{base}}\text{Δ}{\tau }_m\left(n\right)},\\ P(Event\left(n;e_m\left(n\right),\Delta {\text{τ}}_m\left(n\right)\text{|background}\right)=\frac{{\lambda }_m^{\text{base}}}{M\left({\text{e}}_m^{\text{right}}-{\text{e}}_m^{\text{left}}\right)}{e}^{-{\lambda }_m^{\text{base}}\text{Δ}{\tau }_m\left(n\right)}.\end{array}$ (3)

In Eq. (3), ${\text{e}}_m^{\text{left}}$ and ${\text{e}}_m^{\text{right}}$ are the left and right energies of the ROI, respectively, and ${\lambda }_m^{\text{Δ}{\tau }_{\text{base}}}$ 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. $\begin{array}{l}{H}_1:EM{S}_{\text{rn}}\left(N;\underset{\_}{e},\text{Δ}\underset{\_}{\tau }\right)~P\left(\underset{\_}{e},\text{Δ}\underset{\_}{\tau }|\text{radionuclide}\right)\\ ［\text{if radionuclide is detected}］,\\ H_0:EM{S}_{\text{rn}}\left(N;\underset{\_}{e},\text{Δ}\underset{\_}{\tau }\right)~P\left(\underset{\_}{e},\text{Δ}\underset{\_}{\tau }|\text{background}\right)\\ ［\text{if radionuclide is not detected}］\end{array}$ (4)

When a radionuclide is detected, $P\left(\underset{\_}{e},\text{ Δ}\underset{\_}{\tau }|\text{radionuclide}\right)$ is considered as the probabilistic model of the EMS. Otherwise, the probabilistic model of the EMS is $P\left(\underset{\_}{e},\text{ Δ}\underset{\_}{\tau }|\text{background}\right)$. The optimal solution to this binary decision problem is provided by the Wald sequential probability ratio test^[21]. It is assumed that the Nth detected photon is emitted from the mth monoenergetic source. According to the definition in Eq. (1), there are a total of $N_m$ photons detected from the mth monoenergetic source and the likelihood ratio $Ratio\left(N\right)$ is obtained by applying the Newman–Pearson theorem as follows: (5)

The decision function $\Lambda \left(N\right)$ is obtained by taking the logarithm of $Ratio\left(N\right)$, as shown in Equation (6). $\begin{array}{l}\Lambda \left(N\right)=\Lambda \left(N-1\right)+\ln P(Event(N_m;e_m\left(N_m\right),\\ \Delta {\text{τ}}_m\left(N_m\right)\text{|}EM{S}_{\text{rn}}\left(N-1;\underset{\_}{e},\text{Δ}\underset{\_}{\tau }\right),H_1)\\ -\text{ln}P\left(Event(N_m;e_m\left(N_m\right),\Delta {\text{τ}}_m\left(N_m\right)\text{|}EM{S}_{\text{rn}}\left(N-1;\underset{\_}{e},\text{Δ}\underset{\_}{\tau }\right),H_0\right).\end{array}$ (6)

With the $N\text{th}$ photon recorded by the detector, the posterior probabilistic distribution parameter of the EMS model under $H_1$ 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 $T_1$ and $T_0$ are the detection thresholds^[22]. $\begin{array}{l}\Lambda \left(N\right)>T_1 ,\text{Accept} H_1,\\ T_0\le \Lambda \left(N\right)\le T_1,\text{ Continue,}\\ \Lambda \left(N\right)<T_0 ,\text{Accept} H_0.\end{array}$ (7)

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 mth monoenergetic source under $H_1$ is defined in Eq. (8), where the probabilistic model is decomposed by applying Bayes’ rule^[32,33] and ${\alpha }_m\left(N_m\right)$ is the emission rate estimated by the $N_m$th photon detected from the mth monoenergetic source. $\begin{array}{l}P\left(Event(N_m;e_m\left(N_m\right),\Delta {\text{τ}}_m\left(N_m\right)\text{|}EM{S}_{\text{rn}}\left(N-1;\underset{\_}{e},\text{Δ}\underset{\_}{\tau }\right),H_1\right)\\ =P(\Delta {\text{τ}}_m\left(N_m\right)|e_m\left(N_m\right),{\alpha }_m\left(N_m\right),EM{S}_{\text{rn}}\left(N-1;\underset{\_}{e},\text{Δ}\underset{\_}{\tau }\right),H_1)\\ \times P(e_m\left(N_m\right)|{\alpha }_m\left(N_m\right),EM{S}_{\text{rn}}\left(N-1;\underset{\_}{e},\text{Δ}\underset{\_}{\tau }\right),H_1)\\ \times P({\alpha }_m\left(N_m\right)|EM{S}_{\text{rn}}\left(N-1;\underset{\_}{e},\text{Δ}\underset{\_}{\tau }\right),H_1).\end{array}$ (8)

$P({\alpha }_m\left(N_m\right)|EM{S}_{\text{rn}}\left(N-1;\underset{\_}{e},\text{Δ}\underset{\_}{\tau }\right),H_1)$ is the probability that the detected photon is emitted by the mth monoenergetic source. Additionally, ${\alpha }_m\left(N_m\right)$ represents the detection ratio for the N_mth photon. The total detected number of photons is N and there are N_m photons emitted by the mth monoenergetic source. Therefore, the following equation can be derived: $P\left({\alpha }_m\left(N_m\right)\text{|}EM{S}_{\text{rn}}\left(N-1;\underset{\_}{e},\text{Δ}\underset{\_}{\tau }\right),H_1\right)=\frac{N_m}{N}.$ (9)

The measured energy of a photon detected from the mth monoenergetic source is assumed to follow a Gaussian distribution under $H_1$, as shown in Eq. (10). $P(e_m\left(N_m\right)|{\alpha }_m\left(N_m\right),EM{S}_{\text{rn}}\left(N-1;\underset{\_}{e},\text{Δ}\underset{\_}{\tau }\right),H_1)=\frac{1}{\sqrt{2\pi }{\delta }_{e_m}\left(N_m\right)}{e}^{-\frac{{\left(e_m\left(N_m\right)-{\overline{e}}_m\left(N_m\right)\right)}^2}{2{\sigma }_{e_m}^2\left(N_m\right)}},$ (10)

where ${\overline{e}}_m\left(N_m\right)$ and ${\delta }_{e_m}\left(N_m\right)$ are the expected value and standard deviation of the posterior distribution, respectively.

The arrival time interval $\text{Δ}\tau$ is exponentially distributed and ${\lambda }_m\left(N_m\right)$ is the expected value. Therefore, the following equation can be derived: $P\left(\Delta {\tau }_m\left(N_m\right){\text{|e}}_m\left(N_m\right),{\alpha }_m\left(N_m\right),EM{S}_{\text{rn}}\left(N-1;\underset{\_}{e},\text{Δ}\underset{\_}{\tau }\right),H_1\right)={\lambda }_m\left(N_m\right)e^{-{\lambda }_{\text{m}}\left(N_m\right)\text{Δ}{\tau }_m\left(N_m\right)}.$ (11)

Then, the probabilistic model of the EMS under hypothesis $H_1$ is given by Eq. (12). $\begin{array}{l}P\left(Event(N_m;e_m\left(N_m\right),\Delta {\text{τ}}_m\left(N_m\right)\text{|}EM{S}_{\text{rn}}\left(N-1;\underset{\_}{e},\text{Δ}\underset{\_}{\tau }\right),H_1\right)\\ =\frac{N_m{\lambda }_m\left(N_m\right)}{N\sqrt{2\pi }{\sigma }_{e_m}\left(N_m\right)}\times e^{-{\lambda }_m\left(N_m\right)\text{Δ}{\tau }_m\left(N_m\right)-\frac{{\left(e_m\left(N_m\right)-{\overline{e}}_m\left(N_m\right)\right)}^2}{2{\sigma }_{e_m}^2\left(N_m\right)}}\end{array}$ (12)

The probabilistic model of the EMS under hypothesis $H_0$ is given by Eq. (13). $\begin{array}{l}P\left(Event(N_m;e_m\left(N_m\right),\Delta {\text{τ}}_m\left(N_m\right)\text{|}EM{S}_{\text{rn}}\left(N-1;\underset{\_}{e},\text{Δ}\underset{\_}{\tau }\right),H_0\right)\\ =\frac{{\lambda }_m^{\text{base}}}{M\left(e_m^{\text{right}}-e_m^{\text{left}}\right)}{e}^{-{\lambda }_m^{\text{base}}\text{Δ}{\tau }_m\left(N_m\right)}\end{array}$ (13)

The decision function is obtained by substituting Eqs. (12) and (13) into Eq. (6), as shown in Eq. (14). $\begin{array}{l}\Lambda \left(N\right)=\Lambda \left(N-1\right)+\text{ln}{N}_m-\text{lnN}+\ln \left(\frac{{\lambda }_m\left(N_m\right)}{\sqrt{2\pi }{\sigma }_{e_m}\left(N_m\right)}\right)-{\lambda }_m\left(N_m\right)\text{Δ}{\tau }_m\left(N_m\right)-\frac{{\left(e_m\left(N_m\right)-{\overline{e}}_m\left(N_m\right)\right)}^2}{2{\sigma }_{e_m}^2\left(N_m\right)}\\ +\text{ln}M+\text{ln}\left(e_m^{\text{right}}-e_m^{\text{left}}\right)-\text{ln}\left({\lambda }_m^{\text{base}}\right)+{\lambda }_m^{\text{base}}\text{Δ}{\tau }_m\left(N_m\right).\end{array}$ (14)

Implementation of the proposed method

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 ${\lambda }_m^{\text{base}}$ of the mth 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), 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 $H_1$, whereas the other evaluates the log-likelihood under hypothesis $H_0$. 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 $H_1$ 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 $H_0$ 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).

Fig. 1Full-screen View

Detection architecture

A two-stage structure was designed to perform photon discrimination, as shown in 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]. The energy discriminator performs a confidence interval test, as shown in Eq. (15), where n is the particle ordinal number, $k_{e_m}$ is the confidence coefficient of the energy, and $e_m^{\text{t}}$ and ${\sigma }_m^{\text{t}}$ are the mean and standard deviation of the energy, respectively, which can be obtained experimentally. $e_m^{\text{t}}- k_{e_m}{\sigma }_m^{\text{t}}\le e_m\left(n\right)\le e_m^{\text{t}}+ k_{e_m}{\sigma }_m^{\text{t}}$ (15)

The interarrival time discriminator also performs a confidence interval test, as shown in Eq. (16). $\text{Δ}{\tau }_m^{\text{base}}- k_{\overline{\gamma }}{\sigma }_{\Delta {\tau }_{\text{base}}}\le \Delta {\overline{\tau }}_m\le \text{ Δ}{\tau }_m^{\text{base}}+ k_{\overline{\gamma }}{\sigma }_{\Delta {\tau }_{\text{base}}}, {\sigma }_{\Delta {\tau }_{\text{base}}}=\Delta {\tau }_{\text{base}}/\sqrt{n},$ (16) where $\Delta {\overline{\tau }}_m$ is the average interarrival time and $k_{\overline{\gamma }}$ is the confidence coefficient. $\text{Δ}{\tau }_m^{\text{base}}$ is the mean interarrival time under conditions with natural background radiation and ${\sigma }_{\Delta {\tau }_{\text{base}}}$ is the width of the confidence interval.

The posterior probability distribution parameter estimation of the EMS under hypothesis $H_1$ 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^[28,29], the Gaussian–Gamma distribution was applied to model the prior distribution of ${\overline{e}}_m$ and ${\left({\sigma }_{e_m}^2\right)}^{-1}$ because the precision parameter $pre{c}_{e_m}$ is defined as ${\left({\sigma }_{e_m}^2\right)}^{-1}$. The prior distributions ${\tau }_{e_m}\left(0\right)$ and ${\overline{e}}_m\left(0\right)$ are defined in Equation (17), where $a_0$, $b_0$, $u_0$, and $v_0$ are hyperparameters^[30]. For a weak prior, $u_0$ is set to $e_m^t$, $a_0$ and $b_0$ are set to one and 100, respectively, and ${\lambda }_0$ is set to one. $\begin{array}{l}pre{c}_{e_m}\left(0\right)~Gamma\left(a_0,b_0\right),\\ {\overline{e}}_m\left(0\right)~N\left(u_0,{\left(v_{0}pre{c}_{e_m}\left(0\right)\right)}^{-1}\right),\\ P\left({\overline{e}}_m\left(0\right),pre{c}_{e_m}\left(0\right)\right)=N({\overline{e}}_m\left(0\right)|u_0,{\left(v_{0}pre{c}_{e_m}\left(0\right)\right)}^{-1})Gamma(pre{c}_{e_m}\left(0\right)|a_0,b_0)\end{array}$ (17)

Equation (18) models the prior distribution ${\lambda }_m\left(0\right)$ of the expected value of the interarrival ${\lambda }_m$ as Gamma. $c_0$ and $d_0$ are also hyperparameters, where $c_0$ is set to one and $d_0$ is set to ${\lambda }_m^{\text{base}}$. $\begin{array}{l}{\lambda }_m\left(0\right)~Gamma\left(c_0,d_0\right)\\ P\left({\lambda }_m\left(0\right)\right)=Gamma({\lambda }_m\left(0\right)|c_0,d_0)\end{array}$ (18)

Up to the nth arrival from the mth monoenergetic source, the posterior distributions of ${\overline{e}}_m\left(n\right)$, ${\tau }_{e_m}\left(n\right)$, and ${\lambda }_m\left(n\right)$ are updated as shown in Equation (19), where $u_n$, $v_n$, $a_n$, $b_n$, $c_n$, and $d_n$ are the hyperparameters of the posterior distributions^[31]. (19)

The hyperparameters can be inferred in analytical form^[31] via conjugacy, as shown in Equation (20). $\begin{array}{l}{e}_m^{\text{mean}}\left(n\right)=\frac{{{\displaystyle \sum }}_{i=1}^n{e}_m\left(i\right)}{n}\\ {\lambda }_m^{\text{mean}}\left(n\right)=\frac{{{\displaystyle \sum }}_{i=1}^n{\lambda }_m\left(i\right)}{n}\\ u_n=\frac{{\lambda }_0{u}_0+{{\displaystyle \sum }}_{i=1}^n{e}_m\left(i\right)}{{\lambda }_0+n}\\ v_n=v_0+n\\ a_n=a_0+n/2\\ b_n=b_0+\frac{1}{2}{\displaystyle \sum }_{i=1}^n{\left(e_m\left(i\right)-e_m^{\text{mean}}\left(n\right)\right)}^2+\frac{{\lambda }_{0}n\left(e_m^{\text{mean}}\left(n\right)-u_0\right)}{2{\lambda }_n}\\ c_n=c_0+n/2\\ d_n=d_0+\frac{1}{2}{\displaystyle \sum }_{i=1}^n{\lambda }_m\left(i\right)\end{array}$ (20)

The expectation of the posterior distribution can be used to obtain the parameters of the EMS model under $H_1$. $\begin{array}{l}{\overline{e}}_m\left(n\right)=u_n\\ {\sigma }_{e_m}\left(n\right)=\sqrt{\frac{1}{{\tau }_{e_m}\left(n\right)}}=\sqrt{\frac{b_n}{a_n}}\\ {\lambda }_m\left(\text{n}\right)=\frac{c_n}{d_n}\end{array}$ (21)

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 $T_1$, a target radionuclide exists in the detection environment. If the decision function value exceeds the threshold $T_0$, then there is no target radionuclide in the detection environment. Otherwise, the detection process continues until one of the thresholds is exceeded.

Background experiment

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]. ¹³⁷Cs and ⁶⁰Co were selected as target radionuclides. The radioactivity of the ¹³⁷Cs source was 8700 Bq and the radioactivity of the ⁶⁰Co source was 1500 Bq. A LaBr3 (Ce) detector of size $\text{Ф}1.5\times 1.5$ 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. 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.

The spectrum and EMS detected in the background are presented in Fig. 2. The ROIs in the spectrum are labeled with red, blue, and green regions. The ROI of ¹³⁷Cs ranges from 651 to 671 keV. The first ROI of ⁶⁰Co ranges from 1154 to 1186 keV and the second ROI of ⁶⁰Co ranges from 1310 to 1350 keV. It is clear that the counts in the ROI approximately follow a uniform distribution.

Fig. 2Full-screen View

(Color online) Detected EMS and spectra in different radiation fields. (a) Description of the test bench; (b) Spectrum obtained under background conditions with a detection interval of 250 s; (c) 137Cs radiation energy spectrum; (d) ⁶⁰Co radiation energy spectrum

Figure 2(a) presents a description of the test bench. Figure 2(b) presents the energy spectrum and EMS time-domain scatter diagram for natural background radiation. Figure 2(c) presents the energy spectrum and EMS time-domain scatter diagram of the target nuclide ¹³⁷Cs and natural background radiation. Figure 2(d) presents the energy spectrum and EMS time-domain scatter diagram of the target nuclide ⁶⁰CO 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]. The relevant formula is presented in Eq. (22), where LeftChannel and RightChannel are the edges of the selected ROI, $Spectru{m}_i$ is the count of the ith channel in the spectrum, and T is obtained by taking the average of the counts in the ROI. $Ch{i}_{\text{square}}={\displaystyle \sum _{i=\text{LeftChannel}}^{\text{RightChannel}}\frac{{\left({\text{Spectrum}}_i-T\right)}^2}{T}.}$ (22)

For the ROI of ¹³⁷Cs, the Chi-square test result was 26.32. For the first ROI of ⁶⁰Co (［1155, 1185］ keV), the result was 35.85 and for the second ROI of ⁶⁰Co (［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 ${\text{λ}}_{{\text{Δτ}}_{\text{base}}}$, was also calculated. The average interarrival time $\Delta {\tau }_{\text{base}}$ 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 ¹³⁷Cs under background conditions are presented in 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 ⁶⁰Co under background conditions are presented in 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 ⁶⁰Co is absent in the background.

Fig 3.Full-screen View

¹³⁷Cs and ⁶⁰Co Detection results under background radiation conditions

Radionuclide detection experiment

The ¹³⁷Cs source was placed 35 cm in front of the LaBr₃ 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). 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). $Ch{i}_{\text{square}}= {\displaystyle \sum _{i=\text{LeftChannel}}^{\text{RightChannel}}\frac{{\left({\text{Spectrum}}_i-{\text{IntegralValue}}_i\right)}^2}{{\text{IntegralValue}}_i}},$ (23) where LeftChannel and RightChannel are the edges of the selected ROI, $Spectru{m}_i$ is the count of the ith channel in the spectrum, and $IntegralValue$

is obtained using Eq. (24). ${\text{IntegralValue}}_i=\text{TotalCount} \times \underset{e_{\text{left}}^i}{\overset{e_{\text{right}}^i}{{\displaystyle \int }}}\frac{1}{\sqrt{2\pi }{\sigma }^t}{e}^{\frac{-\left(e-e^{\text{t}}\right)}{2{\left({\delta }^{\text{t}}\right)}^2}},$ (24) where $TotalCount$ is the total count in the ROI, and $e_{\text{left}}^i$ and $e_{\text{right}}^i$ are the left and right energies of the ith bin in the spectrum, respectively. Additionally, $e^{\text{t}}$ and ${\sigma }^{\text{t}}$ are preset distribution parameters from 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 ¹³⁷Cs radiation field and long-term detection conforms to a normal distribution.

A rapid nuclide identification method was used to detect ¹³⁷Cs in the corresponding radiation field. The detection function is presented in Fig. 4(a), where the decision function gradually increases with events that are processed and eventually exceeds the high threshold, indicating that the ¹³⁷Cs source is detected in 3.8 s, where the count rate of the ROI is 5.05 s⁻¹.

Fig. 4Full-screen View

(Color online) ¹³⁷Cs and ⁶⁰Co decision results in ¹³⁷Cs and ⁶⁰Co spectra. (a) Decision function and spectrum obtained when 137Cs is identified. (b) Decision function and spectrum obtained when ⁶⁰Co is identified.

There were only a few counts in the ROI before ¹³⁷Cs 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 ¹³⁷Cs at low values.

The ⁶⁰Co source was placed 20 cm in front of the LaBr₃ detector and detection was performed for 100 s. The EMS and spectrum detected are presented in 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 ⁶⁰Co 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 ⁶⁰Co in the corresponding radiation field. The detection results are presented in Fig. 4(b), the decision function gradually increased with events that are processed and eventually exceeds the high threshold, indicating that there is a ⁶⁰Cs source in the radiation field. The proposed method can detect ⁶⁰Cs 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 ⁶⁰Co 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 ⁶⁰Co with such a low counts.

Detection performance experiments

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. 5Full-screen View

Relationship between detection time and detection distance. Identification times required to cover the distances to the 137Cs source (left) and 60Co source (right).

As shown in the figure above, when the detection distance is set to 5 cm, the ¹³⁷Cs 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 ¹³⁷Cs and target nuclide ⁶⁰CO reach 7 s and 12 s, respectively. For the measurement of ⁶⁰Co, 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.