Ablation study

The purpose of the ablation experiment was to study the impact of removing specific parts of the composite model on its overall performance. A lack of performance loss after removing some parts indicates that these parts are less important in the composite model. In contrast, if the performance of the model decreases significantly after removal, the design of the parts is considered essential. To conduct ablation research on the proposed deep-learning model, three deep-learning models were implemented: LSTM, UNet, and UNet-LSTM, and their amplitude prediction performances were evaluated and compared. To control the influence of the parameters on the model performance, the composite and single models used the same parameters during the ablation study. The main parameters of the model and training results are listed in Table 2. Although the independent LSTM and UNet models can converge normally, the proposed composite model achieved lower training and validation losses under the same parameter configuration; after fusing the two models, better pulse-amplitude prediction performance was achieved.

Evaluation of parameter estimation performance

Existing pulse amplitude calculation methods are mostly digital shaping methods, which are effective for estimating the amplitude of most nuclear pulses and are therefore widely used in digital energy spectrometers. References [2, 3] indicate that the digital shaping method poses significant challenges in the amplitude estimation of pulses with shape distortion. This study further explains the relationship between the amplitude estimation results of existing methods and the degree of pulse distortion, and it demonstrates the effectiveness of the UNet-LSTM model in addressing the challenges in distorted-pulse height estimation.

Twenty pulses in the test set and the amplitude prediction output of the model were subjected to inverse normalization processing. Twenty pulses with different degrees of pulse distortion are shown in Fig. 5 with the ground truth fixed at 600.

Fig. 5Full-screen View

(Color online) Simulated pulse sequence diagram

Each distortion pulse in the test set contained different effective sampling points representing varying degrees of distortion. The pulse height estimation results obtained using the UNet-LSTM model and the digital trapezoidal shaping method are shown in Fig. 6. As the degree of pulse distortion decreased, the error in the pulse heights obtained using the digital shaping method also gradually decreased. For example, P1 contains 40 valid sampling points representing the pulse with the most severe distortion in the test set. A maximum relative error of 65.67% was obtained when the digital trapezoidal shaping method was used to estimate the amplitude. As the number of sampling points increased, the degree of pulse distortion decreased, and the relative error of the digital trapezoidal shaping gradually decreased. In Figure 5 the rise time of digital trapezoidal shaping includes 80 sampling points, and the flat top time is 0. Therefore, when the number of sampling points for the original negative exponential pulse was 80, the relative error of the corresponding digital shaping result was reduced to 0.02%.

Fig. 6Full-screen View

(Color online) Comparison of pulse-height estimation

To quantify the prediction ability of the UNet-LSTM model for pulse height, ΔA represents the absolute error of the pulse height estimation, δA represents the relative error, A_real represents the true pulse height, and A_NN represents the pulse height predicted by the neural network model, UNet-LSTM. Thus, the calculations for the absolute and relative errors of the UNet-LSTM model for the distorted pulse-height estimation are shown in Eq. (3) and Eq. (4), respectively. ${\text{Δ}}_A=ABS\left(A_{\text{real}}-A_{\text{NN}}\right)$ (3) ${\delta }_A=\frac{{\text{Δ}}_A}{A_{\text{real}}}\times 100\%$ (4) The 20 pulses shown in Fig. 6 were analyzed for pulse height using the traditional trapezoidal shaping algorithm and the neural network model proposed in this study. The results are shown in Table 3. The average relative error of the trapezoidal shaping algorithm in estimating the pulse height of the sequence is 28.01%, while that of the UNet-LSTM model is approximately 0.64%. This proves that the UNet-LSTM model is not affected by pulse distortion when estimating pulse amplitude parameters, addressing the limitations facing existing methods in pulse height estimation.

Each distorted pulse in the test set contains different effective sampling points representing different degrees of distortion. For example, P1 contains 40 sampling points, representing the most severely distorted pulse in the test set, and its corresponding digital shaping results had a maximum relative error of 65.67%. As the number of sampling points increased, the degree of pulse distortion decreased, and the relative error of the digital trapezoidal shaping gradually decreased. In Fig. 5 the rise time of digital trapezoidal shaping includes 80 sampling points, and the flat top time is 0. Therefore, when the number of sampling points for the original negative exponential pulse was 80, the relative error of the corresponding digital shaping results was reduced to 0.02%. Finally, the average relative error of the trapezoidal shaping algorithm for estimating the pulse height in this sequence was 28.01%. Using the UNet-LSTM model to unfold the input pulse sequence and estimate the pulse height, the estimations are not affected by pulse distortion, with an average relative error of approximately 0.64%.

Evaluation of count correction performance

According to the principle of a multichannel analyzer (MCA), each pulse amplitude in the X-ray fluorescence spectrum corresponds to a count in the energy histogram. When the pulse output of the measurement system experiences an amplitude loss during the digital processing stage, the corresponding histogram drifts to the left. When the number of pulses is sufficient, false characteristic peaks exist in the generated energy spectrum. For samples with a single component, the number of characteristic peaks of the elements in the measured full spectrum is limited. Therefore, the pulse amplitude output of the measurement system mostly fluctuates within a small range of fixed values, which determine the energy resolution of the spectrum. Quantification was performed based on the half-widths of the characteristic peaks in the spectrum. The 20 simulated pulses mentioned earlier were analyzed using traditional MCA as an example. Owing to the normalization of the input pulse amplitude in the model, the channel address range of the full spectrum was set to 0∼1, including 1000 channel addresses. When the pulse with an amplitude of 0.6 is increased by one, the count on the channel address of 0.6 increases by one, resulting in an X-ray spectrum, as shown in Fig. 7(a).

Fig. 7Full-screen View

(Color online) Simulated energy spectrum (a) without the model and with a total count of 20; (b) with the model and a total count of 20; (c) without the model and with a total count of 2000; (d) with the model and a total count of 2000

In this figure, the channel address range of the characteristic peak ROI is approximately 0.610–0.628 with a peak area of 19. A false peak formed by distorted pulses appeared near the 0.484th channel address on the left side of the characteristic peak, with a peak area of 1, as indicated by the green shaded area in Fig. 7(a). The CNN-LSTM model proposed in this study was added before the MCA unit to achieve an accurate estimation of the pulse parameters. A histogram of the characteristic peaks obtained using the model is shown in Fig. 7(b), with the ROI of the characteristic peaks reduced to approximately 0.600–0.609; however, the total count within this range increased to 20. Therefore, using the model to estimate the pulse amplitude not only ensures that the counting of characteristic peak areas is not lost but also eliminates false peaks caused by pulse distortion.

Analysis of generalizability experiment results

To conduct a more thorough statistical analysis of the model performance and address potential biases in the simulation dataset used for model training, a ⁵⁵Fe standard source was used as a test object in the experimental testing phase to provide sufficient pulse sequences for the analysis of the experimental results. The experimental conditions were configured as listed in Table 4.

Using the experimental platform listed in Table 4, the pulse sequence during the measurement process was saved offline to obtain the measured pulse dataset. The pulse dataset was processed according to the processing method of Dataset II to obtain a negative exponential pulse dataset with approximately 40–80 effective sampling points. After trapezoidal shaping of the pulses in this dataset, the required validation dataset was obtained, which was defined as Dataset III, and used to validate the trained model and demonstrate its generalizability. Dataset III contained 5000 pulses with an amplitude range of approximately 20–2000 mv. Each pulse contained 256 sampling points and one amplitude parameter.

The proposed model was compared with other state-of-the-art models to perform pulse height estimation tasks on simulated and measured datasets. The results are shown in Table 5. The selected control model includes both lightweight single models such as LeNet and composite models such as CNN-LSTM, which have been demonstrated to be effective for pulse estimation [30]. The comparison results in Table 5 further demonstrate that because the measured pulse contains more uncertainty than the simulated pulse, the validation loss of each model on the measured pulse dataset increases compared to the simulated dataset. However, UNet-LSTM still performed better than the other models on different datasets, demonstrating good robustness and generalizability.

It is worth noting that although UNet-LSTM achieved better performance on two different datasets, its parameter count was also much larger than that of the other models. Models like LeNet with small parameter numbers and short computation times, can also perform well in most simple tasks after training. Therefore, in application scenarios that require small, lightweight models, UNet-LSTM is not the best choice. This composite model with many parameters and complex calculations is more suitable for more complex time-series analyses, such as the nuclear pulse height estimation and fluorescence spectroscopy analysis tasks in this study.

Analysis of spectral experiment results

The commonly used techniques for optimizing fluorescence spectra can be divided into two categories from the perspective of processing objects, as shown in Table 6.

Category I is for nuclear pulses with the main purpose of obtaining more reliable pulse heights. Category II is for the spectrum itself, of which the most representative and widely used is spectral smoothing. In the simulation research section, a comparison was made between the performance of the traditional digital trapezoidal shaping method and the classical neural network model in pulse height estimation tasks, all of which belong to Category I spectral processing techniques.

Here The experimental conditions listed in Table 4 were used to analyze the results of the spectral optimization experiments. The energy spectrum obtained from the digital MCA was used as the reference spectrum, and on this basis, the spectral smoothing and pulse height estimation units were added separately. The spectral processing flow and the results are shown in Fig. 8. The traditional spectrum acquisition process primarily includes a probe section composed of a high-performance silicon drift detector (FAST-SDD) and preamplifier, a CR differential shaping section, and a digital signal processing unit composed of an operational amplifier, high-precision ADC, digital shaper, and MCA, as shown in Fig. 8 (a). The difference in the spectral analysis process between the added spectral smoothing units and traditional methods lies in the digital signal processing. Although traditional spectral analysis also has functions, such as shaping and filtering in digital signal processing, it does not add a dedicated spectral smoothing unit located after the MCA. In this study, a five-point average (FPA) was used to smooth and filter the generated spectrum, as shown in Fig. 8(b). In another control group, the trained UNet-LSTM model was used to estimate the height of the digitized pulse sequence, and the modified energy spectrum was obtained through an MCA. The spectrum processing is shown in Fig. 8(c).

Fig. 8Full-screen View

Spectral analysis process with (a) the traditional method, (b) FPA, and (c) UNet-LSTM. The energy spectrum of (d)the traditional method, (e) FPA, and (f)UNet-LSTM

The distorted pulse obtained during the measurement process was amplified and trapezoidal to obtain a distorted pulse height. This distortion in the pulse height exists in the form of a false peak before the characteristic peak in the spectrum obtained by traditional spectroscopic methods, as shown in Fig. 8(d) and (e). Although the spectral smoothing method using multipoint averaging can optimize the spectrum to a certain extent, it has no effect on the false peaks caused by distorted pulses. When using the UNet-LSTM model to estimate the measured pulse height, the model can accurately output the pulse height with high accuracy, even for distorted pulses with incomplete widths. Therefore, after predicting the pulse height using the model, the energy spectrum obtained by the MCA can effectively correct the false peaks caused by pulse distortion, as shown in Fig. 8(f).

To quantify the corrective effect of the UNet-LSTM model on the measured X-ray spectrum of the ⁵⁵Fe standard source, two indicators, the correction ratio R_correct and the effective ratio R_effect, are introduced. R_correct represents the proportion of the difference in the peak area of the false peak, before and after calling the model to the peak area of the characteristic peak. R_effect represents the proportion of the increase in the characteristic peak area after calling the model to the loss value of the false-peak area. The calculations are given by Eq. (5) and Eq. (6). $R_{\text{correct}}=\frac{S_{\text{false}-\text{MCA}}-S_{\text{false}-\text{UNet}}}{S_{\text{ROI}-\text{MCA}}}\times 100\%$ (5) $R_{\text{effect}}=\frac{S_{\text{ROI}-\text{UNet}}-S_{\text{ROI}-\text{MCA}}}{S_{\text{false}-\text{MCA}}-S_{\text{false}-\text{UNet}}}\times 100\%$ (6) As shown in Fig. 9, the false peak area was located in the channel interval of approximately 1024–1280, and the ROI of the characteristic peak was captured in the channel interval of approximately 1280–1536. The peak area is represented by S, S_false-MCA represents the peak area of the false peak in the traditional spectral analysis results, S_false-UNet represents the peak area of the false peak after calling the UNet-LSTM model for spectral analysis correction, S_ROI-MCA represents the peak area of the characteristic peak ROI in the traditional spectral analysis results, and S_ROI-UNet represents the peak area of the characteristic peak ROI after calling the UNet-LSTM model for spectral correction.

Fig. 9Full-screen View

(Color online) Peak area analysis of (a) a false peak and (b) characteristic peak

Ten measurements of the ⁵⁵Fe standard source were taken for the analysis, as shown in Table 5. The comparison shows that the peak area of the characteristic peak ROI in the energy spectrum obtained using the traditional MCA was similar to that obtained using FPA filtering. However, the filtered energy spectrum had smoother spectral lines in the low count-rate region; therefore, it has a lower standard deviation than traditional MCA methods in multiple measurement processes. Although FPA filtering can reduce the standard deviation of the measurement and stabilize the measurement results, it has no corrective effect on false peaks in the X-ray spectrum. The ⁵⁵Fe standard source X-ray spectrum obtained using the UNet-LSTM model to predict the pulse height has two typical features. First, the peak area of the characteristic-peak ROI was improved, and the standard deviation of multiple measurement results was significantly reduced. Second, the peak area of false peaks was significantly reduced. According to the above two characteristics, combined with the theorem of energy conservation, it can be inferred that the reduced false peak area should be corrected to the characteristic-peak ROI, and the correction effect can be evaluated using the R_valid index defined above. From Table 5, it can be observed that approximately 91% of the peak area loss in the false-peak area can be corrected to the characteristic peak ROI, and the proportion of the corrected peak area to the peak area of the characteristic-peak ROI was approximately 1.32%, which is an indispensable part of high-count-rate applications (Table 7).