Detection of P-Wave Arrival as a Structural Transition in Seismic Signals: An Approach Based on SVD Entropy

Highlights

What are the main findings?

• P-wave arrival is identified as a structural transition in seismic signals using SVD entropy rather than energy growth.
• Reliable detection accuracy of 93–98% is achieved at low signal-to-noise ratios ($SNR\ge 2$) without any model training or labeled data.
• Low-frequency signal drift can degrade entropy-based detection at the JNKS station, but polynomial detrending restores sensitivity, enabling accurate P-wave detection comparable to other stations. This underscores the importance of proper preprocessing in entropy-based seismic analysis.

What are the implications of the main findings?

• The method is suitable for deployment on distributed IoT/edge seismic sensors within smart city earthquake early warning networks.
• Station-dependent parameter tuning and simple preprocessing (e.g., polynomial detrending) enable robust operation in real urban noise conditions.
• Station-dependent parameter sensitivity highlights the need for adaptive detection strategies. While universal parameters can work, station-specific calibration ensures highest reliability, suggesting future early warning systems should include automatic tuning and preprocessing diagnostics.

Abstract

Early and reliable detection of P-wave arrivals is critical for seismic monitoring and earthquake early warning, particularly under low signal-to-noise ratio (SNR) and non-stationary noise conditions. This study presents an automatic detection method based on singular value decomposition (SVD) entropy computed in sliding time windows with local signal filtering. Within this framework, the P-wave onset is interpreted as a local structural change in the signal rather than a simple energy increase. SVD entropy captures the redistribution of energy among dominant signal components, providing high sensitivity to the initial P-wave arrival even at moderate and low noise levels ($SNR\ge 2$). The method was validated using real seismic data from four regional stations operating under different noise conditions. Analysis of detection parameters revealed strong station dependence. For stations affected by low-frequency drift, polynomial detrending was identified as a necessary preprocessing step to ensure a stable entropy response and reliable detection. The proposed approach achieves detection accuracies of up to 93–98% at $SNR\ge 2$, significantly outperforming the classical STA/LTA algorithm and demonstrating performance comparable to modern deep learning methods. Since the method does not require model training or labeled datasets, it provides an interpretable and computationally efficient solution for automatic seismic monitoring. These properties make the proposed approach particularly suitable for real-time seismic monitoring systems and distributed sensor networks operating under limited computational resources. All computational stages were performed at the Farabi Supercomputer Centre of Al-Farabi Kazakh National University. The method requires no model training or labeled data, making it an interpretable, robust, and computationally efficient solution for automatic seismic monitoring and early warning systems.

1. Introduction

Seismic data analysis is a complex task due to non-stationarity, non-linearity, and high noise levels [1,2,3]. Processing methods based on amplitude, spectral and energy characteristics are of limited adequacy in describing the structural complexity of wave processes and do not allow for the detection of weak but informative changes in seismicity dynamics [4,5,6]. In this regard, the development and application of alternative analysis methods that are capable of considering the information content and degree of orderliness of seismic time series is a pressing issue.

Despite the intensive development of early earthquake warning systems, accurate and timely prediction of the arrival time of P-waves under conditions of low signal-to-noise ratio remains one of the key problems in seismology [7,8,9]. This is because even modern and rapidly developing methods face limitations in reliably detecting subtle changes in the structure of complex signals.

Seismic monitoring plays a crucial role in early earthquake warning, magnitude estimation and epicenter location, as well as in monitoring induced and volcanic seismic activity [10,11,12,13,14]. Among these parameters, the detection of the arrival of the P-wave is of fundamental importance, as it is the first wave recorded by sensors and allows a few seconds to be gained for activation of early warning systems to minimize damage from subsequent destructive waves [15,16,17,18]. In complex scenarios typical of real seismic recordings, stations located in high-noise environments (urban areas, ocean floor), as well as when recording weak seismic events, accurate detection is significantly hampered by the non-stationary nature of the low signal-to-noise ratio and phase interference [19,20,21,22,23,24].

The STA/LTA method and its modifications, as well as statistical approaches such as AIC and CUSUM, have become the interpretable basis for early warning systems due to their simplicity [25,26,27]. Modern methods of the next evolutionary stage, based on machine learning, demonstrate high precision and a reduction in the level of false alarms on labeled data [28,29,30,31]. However, in complex conditions characterized by phase transitions of chaotic noise, nonlinear interference, and non-stationary signals, these approaches may experience reduced robustness in highly noisy and non-stationary environments [32,33,34]. ML models often learn these features implicitly, which limits generalization across regions with different geology, as shown in recent studies [35]. In contrast, the method proposed in this study does not depend on training data from any specific network. By focusing on intrinsic changes in the signal structure rather than learned representations, it is less sensitive to noise variability and propagation differences, enabling broader applicability across diverse seismic environments.

Entropy-based measures have recently attracted attention for the analysis of complex dynamical systems because they quantify the degree of order and structural organization of signals. SVD entropy focuses on the rank structure of the signal, which ensures high sensitivity to the appearance of a new dominant component in the time series (as when a P-wave arrives) [36,37,38,39,40]. This approach to analysing signals with a high proportion of additive noise, based on the study of the spectrum of singular values, can be more informative than Shannon entropy, which is based on the distribution of amplitudes. SVD entropy is derived from the singular value spectrum of the embedding matrix and therefore reflects the redistribution of energy among dominant dynamical components of the time series [41]. The next study has reported that SVD-based entropy reveals detailed characteristics of Shannon entropy [42]. In comparison with machine learning approaches based on labelled datasets and requiring model training, the proposed SVD entropy method operates unsupervised and does not rely on pre-trained models. This eliminates the need for large annotated datasets and reduces computational loads. This independence makes the method particularly suitable for real-time monitoring scenarios and for deployment in environments with limited computational resources, including edge and IoT seismic systems [35]. In addition to seismology, when analysing complex signals, SVD entropy, compared to approximate entropy, did not require additional parameters to be set and allowed for a more stable assessment of the structural complexity of the signal [33,34,35]. In the context of detecting structural transitions—such as the onset of a P-wave under noisy conditions—this property may provide increased sensitivity to the emergence of a new ordered component in the signal. Beyond seismology, SVD entropy has also demonstrated advantages in the analysis of complex signals. In comparison with approximate entropy, it does not require additional parameter tuning and enables a more stable evaluation of structural complexity [43,44,45]. Compared to permutation entropy, SVD entropy offers greater interpretability through direct analysis of singular values, allowing a quantitative assessment of signal orderliness.

The aim of this work is to develop and investigate a method for early detection of P-wave arrival based on the analysis of SVD entropy of seismic signals in sliding time windows with local filtering. Unlike traditional approaches, which focus primarily on energy characteristics, the proposed method considers the arrival of the P-wave as a structural transition in the time series and uses entropy measures to detect it.

Particular attention is paid to analyzing the stability of the method under conditions of moderate and low signal-to-noise ratios characteristic of real seismic recordings. To this end, a wide parametric search of the sliding window length and filter cutoff frequency is used, as well as a cascade detection scheme combining the analysis of the absolute value of SVD entropy and its derivative.

The work is based on real data from the regional seismic network and includes both a quantitative assessment of detection accuracy and a qualitative analysis of algorithm behavior scenarios under various noise conditions. This allows one not only to evaluate the effectiveness of the method, but also to identify its limits of applicability and sensitivity to the characteristics of individual stations. This characteristic also makes the method a source of reliable structural descriptors for machine learning algorithms, which are increasingly being used in IoT-oriented seismic monitoring systems and early warning architectures for smart cities.

The article is organised as follows. Section 2 describes the data set used, the pre-processing scheme and the method for calculating SVD entropy, as well as the criteria for detecting the arrival of a P-wave. Section 3 presents the results of parametric analysis, detection accuracy assessment and analysis of typical scenarios for the algorithm’s operation. Section 4 discusses the results obtained, the limitations of the proposed approach, and possible directions for its further development.

2. Materials and Methods

The proposed method for early detection of P-wave arrival is based on calculating singular spectral entropy (SVD entropy) in sliding windows of the seismic signal. The general scheme of the algorithm is shown in Figure 1. The initial records are loaded from the vertical HHZ channel of several stations, after which they are divided into overlapping time segments of various lengths, for each of which SVD entropy is calculated. They then undergo preliminary processing: time scale synchronisation, normalisation and low-frequency filtering. Next, analysis of the temporal evolution of entropy and its derivative allows the moment of a sharp change in the signal structure to be localised, which is interpreted as the arrival of a P-wave.

This approach combines the advantages of classical detectors based on signal energy changes and modern methods of time series complexity analysis. In addition, parametric scanning of filter cutoff frequencies and window lengths makes it possible to evaluate the stability of the method in various noise conditions and for different stations.

2.1. Dataset Description

The study is based on a set of real seismic data collected over two years—from 1 January 2023 to 31 December 2024—at several stations located in the foothills of the Northern Tien Shan. This region is seismically active, making it a convenient site for testing P-wave early detection algorithms. Seven seismic stations are publicly available within a 3° radius of Almaty city, but four stations (TLG, SHLS, JNKS, and TARG) were selected for analysis based on the criterion of having more than 200 recorded events during 2023–2024. These stations belong to different seismic networks (QZ, KR, and KC). Their relative locations, as well as the distribution of earthquake epicentres, are shown in Figure 2.

As shown in Figure 2, each station is highlighted with a separate colour scheme, with a radius of 333 km (3 degrees). Common events detected by two or three stations simultaneously are highlighted with black dots. All records were obtained through IRIS services using the PyWEED tool (Version 1.0.12), which automatically collects only earthquakes and excludes quarry blasts and other non-tectonic events. This allowed us to automatically collect events for the entire period and ensure uniformity of data loading. Event selection criteria were as follows:

-

Magnitude: min −2.0, max 10.0

-

Depth (km): min 0.0, max 6800.0

-

Distance to station: radius 333 km (3°)

-

Signal-to-noise requirements: no explicit SNR-based selection; all recorded events in different ranges were included

For each event, a 240 s section was downloaded, including 120 s of background noise before the P-wave arrival and 120 s after the arrival. This choice provides a sufficiently long context necessary for calculating entropy characteristics and analysing signal behaviour against natural noise. The 120 s pre-P-wave window was selected to ensure a stable estimate of the background noise, which is essential for accurate entropy computation. The 120 s post-P-wave interval allows full capture of the initial P-wave arrival and its early evolution, thereby providing adequate temporal context for subsequent analysis. Only the vertical HHZ channel with a sampling frequency (fs) of 100 Hz was used.

The initial list of events is shown in

Table 1. Number of events per station.

Network	Station	Instrument	Elevation (m)	Sensitivity	Number of Events	Events After Compilation
QZ	TLG	Trillium Horizon 120V2Slim, S/N 2222 (Nanometrics, Ottawa, ON, Canada)	1214	$4.792\times {10}^8$$1\mathrm{Hz}$	218	208
QZ	SHLS	Trillium Compact Horizon, S/N 8449 (Nanometrics, Ottawa, ON, Canada)	2061	$2.995\times {10}^8$$1\mathrm{Hz}$	253	242
KC	TARG	STS-2, 120 s, 1500 V/m/s, generation 3 electronics (Streckeisen, Pfungen, Switz erland)	3530	$1.492\times {10}^9$ 0.05 Hz	222	215
KR	JNKS	Trillium Compact, 120 s, 754 V/m/s-Centaur, 40 vpp (Nanometrics, Ottawa, ON, Canada)	2248	$2.995\times {10}^8$$1\mathrm{Hz}$	236	226

, which summarizes both the number of events per station and the main station metadata, including instrument type, elevation, and sensitivity. Several restrictions were applied subsequently. First, the recording had to be continuous and contain no gaps. Second, if two events had arrival times that were too close together (less than 120 s apart), only the first event was considered. Finally, for each event, it was necessary to ensure that the arrival of the P-wave could be reliably identified after signal filtering. Recordings with an uncharacteristic or poorly distinguishable signal structure were therefore excluded.

Table 1 presents detailed metadata for each station, including the sensor model, installation elevation, and nominal sensitivity at the reference frequency. Providing this information allows a more accurate interpretation of station-specific performance differences and facilitates reproducibility of the dataset configuration.

Over the two-year period, seismic activity was uneven: some months were characterised by isolated events, while others exhibited sharp increases associated with regional earthquakes. This diverse dataset proved useful for assessing the robustness of the entropy-based approach, as it contains records with varying noise levels, distances to the epicentre, and energy characteristics.

The resulting dataset enables evaluation of the proposed method under conditions that are close to real-world scenarios—including both weak and moderately strong earthquakes, across different stations, instrumentation types, and environmental conditions.

Note: Preprocessing steps such as filtering, time alignment, and local window processing are described in Section 2.3. Instrument correction, detrending, or resampling were not applied, as the analysis was performed on raw amplitude values.

2.2. Sliding-Window Segmentation

Before pre-processing the signals, each recording was divided into a series of overlapping time segments, enabling local analysis of signal evolution over time. This approach allows early changes in waveform structure to be detected before the arrival of the P-wave and is a key element of the entropy method.

The window length was set within a wide range, from 1 to 60 s. This range was deliberately chosen to be wide: short windows allow fast signal fluctuations and sharp changes in its structure to be recorded, while longer windows smooth out noise and provide a stable entropy estimate in low signal-to-noise conditions. The result is a multi-scale representation of the recording, which increases the reliability of subsequent detection.

Segmentation was performed using the sliding window method with a fixed step size. The step size was one second, which at a sampling frequency fs = 100 Hz corresponded to a shift of 100 counts. This choice ensured sufficiently high temporal resolution with reasonable computational load.

Figure 3 shows a diagram of how sliding windows are formed: the original recording is divided into a series of overlapping segments, each of which has its own time-bound interval. Further analysis is performed independently for each window, which allows tracking the dynamics of changes in the signal structure.

The coloured rectangles in Figure 3 illustrate the principle of segmenting the recording into overlapping windows of fixed length. The window length was set to an interval of 1–60 s. Each window is shifted forward by one second relative to the previous one, which ensures the continuity of the time analysis. For each such segment, preliminary processing is then performed, forming a time series of entropy estimates used to identify the moment of arrival of the P-wave.

2.3. Signal Preprocessing

Before calculating entropy characteristics, each seismic record underwent basic pre-processing aimed at bringing the data to a uniform time format and preparing the signal for sliding window analysis. The general processing scheme is shown in Figure 4, and Figure 5 illustrates the sequence of operations using the example of a single time window.

The first stage involved synchronising the time axis. The original ASCII files contained time stamps with microsecond precision, which meant that the start of the recording could fall at any point within a second. To eliminate this offset, the analysis began with the first reading, whose timestamp coincided with a whole second. This approach made it possible to align the timelines of all recordings without changing the source data and ensured the correct formation of the sequence of sliding windows in subsequent stages.

After temporary alignment, the signal was used in its original amplitude scale. Time series normalization was used solely for visualization purposes and did not participate in either recording segmentation or entropy characteristic calculation. All calculations were performed on raw amplitude values, which preserved the physical interpretability of the signal and eliminated the influence of scaling on entropy estimates.

Signal filtering was a key element of pre-processing. Unlike traditional schemes, where the filter is applied to the entire recording, in this study, low-frequency filtering was performed locally, separately for each sliding window. For this purpose, a fourth-order Butterworth low-pass filter was used in zero phase delay mode. This approach made it possible to avoid phase distortions and simultaneously investigate the effect of the cutoff frequency on entropy behaviour at different time scales of analysis.

The cut-off frequency range varied from 1 to 25 Hz, covering the spectrum of low-frequency longitudinal oscillations characteristic of the early stage of the P-wave arrival. Filtering was performed after each time window was formed, which ensured consistency in processing when searching through parameters and eliminated the influence of neighbouring sections of the recording.

Figure 5 illustrates this process using one time segment as an example: it shows the original signal, the selected sliding window, the corresponding fragment of the recording, and the result of its local filtering. It was precisely this filtered window signal that was then used to calculate the SVD entropy.

Thus, pre-processing was limited to two main tasks: coordinating the time grid and preparing the signal for local window analysis with parameterised filtering. This scheme ensured reproducibility of results, correct comparison between events, and stable conditions for subsequent calculation of entropy characteristics.

2.4. SVD-Entropy Computation

SVD entropy was used in this study to quantitatively assess local changes in the seismic signal structure. This measure allows for a compact description of the degree of orderliness of a time series and is particularly suitable for analysing short time segments in noisy conditions, which is important for early detection of P-wave arrival.

Entropy was calculated separately for each sliding time window formed in the previous stage. Thus, SVD entropy was considered a local characteristic of the signal, reflecting its structure at a specific point in time, rather than a global measure for the entire recording.

For each window, a one-dimensional time series was transformed into a matrix of embeddings with minimal dimension. This embedding procedure is analogous to phase space reconstruction, where the embedding dimension can be estimated using the mutual information function and the nearest neighbors method. In the present study, the embedding dimension was chosen as m = 3. This choice is based on practical considerations: for short sliding windows, m = 3 provides sufficient sensitivity of SVD entropy to structural changes at the initial stage of P-wave arrival, while maintaining stability and reproducibility of the entropy estimates.

It should be noted that a sensitivity analysis for m > 3 or m < 3 was not performed. The main objective of this study was to detect the first longitudinal oscillations of the signal, which indicate the onset of P-wave arrival. Within this context, the chosen embedding dimension m = 3 was considered adequate.

Singular value decomposition (SVD) was applied to the resulting embedding matrix, producing three singular values that reflect the energy distribution among the dominant components of the time series. These singular values were normalized so that their sum equaled one, allowing them to be interpreted as relative contributions to the overall signal structure.

Based on the normalized singular values, entropy was calculated in standard information form and then normalized to the range [0, 1]. In this study, SVD entropy was computed using the AntroPy library with parameters order = 3, delay = 1, and normalize = True (AntroPy SVD entropy documentation https://raphaelvallat.com/entropy/build/html/generated/entropy.svd_entropy.html, accessed on 10 December 2025). This normalization facilitates comparisons between different windows, events, and parameter configurations.

Low SVD entropy values correspond to a stable and slightly varying signal, characteristic of background noise. As the first longitudinal oscillations appear, the time series structure becomes more complex, energy is redistributed among singular components, and the entropy value increases. This makes SVD entropy a sensitive indicator of the initial stage of P-wave arrival, even under low signal-to-noise ratio conditions.

The sequence of entropy values calculated for all sliding windows forms a time series H(t) describing the evolution of the local complexity of the signal. This series is further used in Section 2.5 to implement criteria for automatic detection of the arrival of a longitudinal wave.

2.5. Threshold and Derivative Based Detection Criteria

After calculating the time series of SVD entropy for all sliding windows, the next step was to determine the moment of arrival of the P-wave. For this purpose, two complementary criteria based on the behaviour of entropy over time were used: the criterion of absolute entropy level and the criterion of its rate of change. Both approaches are illustrated in Figure 6.

The first criterion is based on the absolute value of SVD entropy. In conditions of background noise, the signal structure remains relatively stable, which is reflected in low and slightly changing entropy values. As the first longitudinal oscillations appear, the structure of the time series becomes more complex, energy is redistributed between singular components, and entropy begins to increase. The moment when the entropy value exceeds a specified threshold is interpreted as the beginning of a significant signal restructuring. An example of such behaviour is shown in Figure 6a.

The threshold value for absolute entropy was selected empirically based on the analysis of a large number of recordings and amounted to approximately 0.05. Practice has shown that this level reliably separates background noise sections from the interval in which P-wave formation begins, regardless of the station and the overall noise level. The main advantage of this criterion is its stability and low sensitivity to random signal fluctuations.

The second criterion is based on the analysis of the first difference of SVD entropy, i.e., on the rate of its change over time. Unlike the absolute value, the entropy derivative is sensitive to sharp structural transitions, even if the entropy values themselves remain relatively low. This approach allows us to fix the initial moment of signal restructuring preceding a steady increase in entropy. An example of how this criterion works is shown in Figure 6b.

A separate threshold of approximately 0.04 was used for derivative entropy. Exceeding this value indicated a sharp change in the structure of the time series and was considered a potential moment of P-wave arrival. This criterion proved to be particularly useful for recordings with a low signal-to-noise ratio, where the increase in absolute entropy can be smoothed and delayed.

The combined use of two criteria allows their advantages to be combined. The derivative criterion provides high sensitivity to initial structural changes, while the threshold based on the absolute entropy value serves as a stabilising factor and reduces the likelihood of false alarms. In practice, the moment of arrival of the P-wave was recorded when one or both criteria were triggered within the expected time interval of the recording.

Thus, analysis of both the level and dynamics of SVD entropy provides reliable and physically interpretable identification of the initial stage of longitudinal wave arrival, which makes the proposed approach robust to variations in noise conditions and seismic event characteristics.

2.6. Parameter-Space Exploration

Before proceeding to the analysis of detection results, the stability of the proposed entropy approach to the selection of processing parameters was evaluated. The aim of this stage was not to optimise the method for individual events, but to investigate how its behaviour is preserved when key parameters vary over a wide range of values.

As part of this analysis, the fixed processing scheme described in Section 2.3, Section 2.4 and Section 2.5 was applied repeatedly with various combinations of parameters. Two parameters that have the greatest impact on the local signal structure were varied: the cutoff frequency of the low-pass filter and the length of the sliding window. All other elements of the pipeline, including the segmentation method, SVD entropy calculation, and detection criteria, remained unchanged.

The filter cutoff frequency varied from 1 to 25 Hz in 1 Hz increments, covering both the low-frequency components of the background signal and the range in which early P-wave energy manifests itself. The length of the sliding window varied from 1 to 60 s. This range allowed us to investigate both short windows, sensitive to rapid structural changes, and longer windows, providing stable entropy estimates in conditions of increased noise.

A complete search of parameters generated 1500 combinations for each record (25 cutoff frequency values and 60 window length values). For each configuration, sliding windows were formed, local filtering was performed, the SVD entropy time series was calculated, and the detection criteria described above were applied.

The results of each run were compared with the reference P-wave arrival marks and classified according to the type of criteria triggered. This approach made it possible not only to record successful or unsuccessful detection, but also to analyse the nature of errors and the dependence of the algorithm’s behaviour on processing parameters.

This parametric analysis allows us to consider the proposed method not as a set of rigidly defined settings, but as a stable procedure that remains effective even with significant variations in filter parameters and window length. The results obtained serve as the basis for a quantitative assessment of detection accuracy, analysis of time shifts, and discussion of the method’s applicability in various seismic conditions, which is presented in the Results section.

3. Results

3.1. Evaluation Framework

To quantitatively assess the detection quality, reference P-wave arrival times provided by the PyWEED service when uploading waveforms were used. Since each recording began 120 s before the reference arrival, the true moment of the P-wave was determined as $t_{P,\mathrm{true}}=t_0+120$ s and used as ground truth in the subsequent comparison.

Only those recordings were included in the analysis for which the signal-to-noise ratio satisfied the condition $SNR\ge 2$ on a linear scale, which corresponds to approximately 3 dB on a logarithmic scale. This threshold is significantly lower than the values used in previous studies ($SNR\ge 8$) and was chosen to assess the robustness of the method under moderate and low signal-to-noise ratios typical of real automatic detection systems.

For each record and each combination of parameters, the algorithm generated an estimate of the arrival time $t_{\det }$. The accuracy of detection was assessed based on the time error

$$\Delta t=t_{\det }-t_{P,\mathrm{true}}$$

(1)

Given that the entropy measure used is local and depends on the length of the sliding window, a systematic bias may occur between the moment of physical arrival of the P-wave and the moment of detection. For this reason, detection was considered correct if the estimate $t_{\det }$ fell within the acceptable time interval of 120 ± 10 s.

Based on this criterion, all results were classified into three categories: correct detection, false alarm, and no detection. These categories reflect different scenarios of algorithm behaviour at different noise levels and signal structures, which are discussed in detail in Section 3.3.

For correct detection, temporal errors were aggregated by station, window length, and filtering frequency, which made it possible to analyse both the average accuracy of the method and its dependence on processing parameters and recording characteristics.

3.2. Sensitivity and Parameter-Space Analysis

Before proceeding with the analysis, it should be noted that the JNKS station is not included in the heat maps provided below. During data processing, it became apparent that the records from this station behaved differently from the others: in some cases, a slow signal drift and unusually pronounced low-frequency fluctuations were observed. Because of this, the SVD entropy reacted in a non-standard way, and the results were highly dependent on the filtering parameters.

Since these characteristics do not allow for a fair comparison of JNKS with other stations in the general parametric grid, its analysis is presented separately. Therefore, this subsection only considers the TLG, SHLS, and TARG stations, while details on JNKS will be discussed in the discussion section.

To assess how the choice of pre-processing parameters affects detection stability, a complete search of combinations of window length (1–60 s) and filter cutoff frequency (1–25 Hz) was performed. Accuracy heat maps were constructed for each station, allowing visual identification of areas where the method works most reliably.

Figure 7 shows the heat map for the TLG station. The figure shows that windows that are too short (1–5 s) lead to a sharp decrease in accuracy, since entropy in this range is highly susceptible to noise. As the window length increases, the behaviour becomes more stable, and a confident zone forms in the range of approximately 15–30 s. The cutoff frequency also has a significant effect: a cutoff frequency of around 4–8 Hz provides the best noise suppression while maintaining the signal shape.

A similar heat map for the SHLS station is shown in Figure 8. Unlike TLG, there is no single stable zone here; instead, there are several scattered clusters of increased accuracy. This structure reflects the more complex noise background of the station. Nevertheless, it is possible to identify a range where the method demonstrates the best results: 10–20 s windows and 6–12 Hz cutoff frequencies.

The results for the TARG station are shown in Figure 9. This station demonstrates the highest accuracy values, but only for relatively long windows—around 30–45 s. This indicates that the signal at TARG has a smoother initial phase, and short segments are not sufficiently informative. The most confident zone is also associated with cutoff frequencies of 4–10 Hz.

Despite the marked differences between stations, it is interesting to assess the possibility of using a single set of parameters for all stations. To this end, their joint parametric space was analysed. The minimum value achieved by the stations, 0.86, corresponding to the SHLS station, was used as the general accuracy threshold. The parameter range in which all three stations simultaneously demonstrate an accuracy of at least 86% is shown in Figure 10.

As can be seen in Figure 10, the overlapping parameter zone is narrow and corresponds to a window length of approximately 27–28 s and a cutoff frequency of approximately 10 Hz. These parameters can be considered a ‘universal’ configuration suitable for use in network automatic detection algorithms when station-specific calibration is not possible. However, it is important to emphasise that when using such universal settings, the detection accuracy for each station is lower than its individual maximum.

To quantitatively assess the effectiveness of the method in the selected universal configuration, the detection results for all events that passed the final data selection were analysed. The number of such events was 177 for the TLG station, 220 for the SHLS station, and 213 for the TARG station. These values correspond to the number of records that met all quality criteria and were used in the subsequent analysis.

Table 2. Detection outcomes for selected parameter configurations.

Window (s)	Cutoff (Hz)	Correct Detections	False Detections	No Detection	Accuracy (%)
Station TLG (177 events)
27	10	166	0	11	93.8
28	10	165	1	11	93.2
Station SHLS (220 events)
27	10	181	21	18	91.8
28	10	180	22	18	93.8
Station TARG (213 events)
27	10	184	1	28	86.9
28	10	188	1	24	88.7

shows the final detection rates for each station at fixed parameters: window size 27–28 s and cutoff = 10 Hz. For each station, the number of correct detections, false alarms, and missed detections is given, as well as the corresponding accuracy calculated relative to the total number of events after the final selection.

As shown in Table 2, the maximum accuracy of wave P determination for the three stations is TLG 93.8%, SHLS 93.8% and TARG 88.7%. Further comparison of the three independent thermal maps made it possible to identify areas of high accuracy for each station. These areas are shown in Figure 11 as coloured clusters. To construct a combined map, individual accuracy thresholds were used, reflecting the data quality and noise conditions of each station by given accuracy: for TARG $\ge 0.93$, for TLG $\ge 0.89$, for SHLS $\ge 0.86$. This approach allows us to objectively identify the ranges of parameters that ensure stable detection for a specific station.

The lack of complete overlap between these areas confirms that each station has its own optimal set of parameters, determined by the local noise background, sensor installation conditions, and characteristics of the recorded waveforms. The resulting zones of maximum accuracy differ significantly: for TLG, the best results are achieved with window lengths of 36–40 s and cutoff frequencies of 4–5 Hz; for SHLS, with window lengths of 11–28 s and cutoff frequencies of 8–13 Hz; for TARG, with window lengths of 29–44 s and a cutoff frequency of 4–9 Hz. The fact that these areas do not coincide completely highlights an important feature of the method: the most reliable detection is achieved by selecting parameters separately for each station, rather than using a single universal setting. Figure 11 shows the parameter ranges where the method works most reliably, but it does not show specific accuracy values. Therefore,

Table 3. Representative station-specific high-accuracy parameter configurations.

Window Size (s)	Cutoff Frequency (Hz)	Accuracy (%)
TLG
36	5	97.2
37	4	98.3
37	5	97.2
38	4	98.3
38	5	97.2
39	4	98.3
39	5	97.2
40	4	98.3
SHLS
11	9	90.5
11	10	90.5
11	11	90
12	8	90.5
12	10	90
13	10	90
14	11	90.5
17	11	90.9
17	12	90.9
18	9	90.9
18	14	90.9
19	8	91.4
20	8	91.4
22	8	91.4
22	14	90.9
24	10	91.4
25	10	91.4
27	10	91.8
28	10	91.8
TARG
29	4	93.9
30	4	93.9
31	4	93.9
32	4	94.4
34	4	93.9
34	6	93.4
35	4	93.9
39	6	93.9
40	6	93.9
41	6	93.9
41	8	93.9
42	6	93.9
42	8	93.9
43	6	93.9
43	8	94.4
44	6	93.9
44	8	94.4
44	9	93.9
45	8	94.4
45	9	93.9
46	8	93.9

provides individual examples of parameters from these ranges, indicating the accuracy achieved for each station.

Based on the results of parametric analysis, configurations were identified for each station at which the highest detection accuracy was achieved. In particular, for the TLG station, the maximum accuracy of 98.3% was obtained with a window length of 37 s and a cutoff frequency of 4 Hz; for the SHLS station, 91.8% with parameters of 27 s and 10 Hz; and for the TARG station, 94.4% with a window of 32 s and a cutoff frequency of 4 Hz.

It should be noted that for each station there are several combinations of parameters that provide a comparable level of accuracy. However, when selecting the final configurations, preference was given to options with the lowest possible cut-off frequency, provided that high detection accuracy was maintained.

3.3. Detection Scenarios Under Different SNR Conditions

After analysing the parametric space and selecting both universal and station-specific parameter configurations, let us consider how the algorithm behaves on real seismic records.

As noted in Section 3.1, only recordings with a signal-to-noise ratio of at least $SNR\ge 2$ on a linear scale (approximately 3 dB) were included in the study. With this restriction, the dataset remains quite heterogeneous and includes both recordings with a clearly defined P-wave structure and cases where the useful signal is only weakly distinguishable from the noise.

In the first two scenarios, P-wave detection is considered correct, since the entropy characteristics demonstrate consistent and physically interpretable behaviour near the reference arrival time. The third scenario corresponds to erroneous cases and is used to analyse the limits of applicability of the method.

In the first scenario, P-wave detection is based on the derivative of SVD entropy. Background noise is characterised by almost stationary entropy behaviour, whereas the appearance of a longitudinal wave is accompanied by a sharp change in the signal structure. This leads to a noticeable jump in the first entropy difference.

If the value of the SVD entropy derivative exceeds a specified threshold (0.04), the corresponding time point is interpreted as the arrival of a P-wave. This scenario is typically implemented for recordings with high or moderately high SNR, where structural changes are sufficiently clear.

Figure 12 illustrates a typical example of this scenario for the TARG station at a high SNR value. It can be seen that the detection time, determined by the entropy derivative, agrees well with the reference P-wave arrival time obtained using PyWEED.

In the second scenario, the derivative of SVD entropy does not show a pronounced peak sufficient for reliable triggering. However, the absolute value of SVD entropy reliably exceeds the specified threshold.

This behaviour is typical for signals with a smoother structure growth or a lower signal-to-noise ratio, when the dynamic criterion is not sensitive enough. In this case, detection is performed based on exceeding the entropy threshold without using its derivative.

In practice, the first and second scenarios are used sequentially, in a cascade scheme: first, the entropy derivative is checked, and if there is no trigger, the absolute value of entropy is checked. This approach increases the overall stability of the method compared to using only one criterion.

An example of the second scenario is shown in Figure 13, where detection is performed based on the SVD entropy threshold at the average SNR value.

The third scenario corresponds to cases where the P wave is either not detected at all or is detected prematurely, significantly earlier than the expected interval. All such cases are classified as erroneous.

Such errors are usually associated with low SNR values or high signal complexity, where entropy characteristics exhibit unstable behaviour. Depending on the nature of the error, two subtypes can be distinguished: lack of detection and premature triggering.

Figure 14 shows an example of a recording with low SNR, where neither the derivative nor the absolute value of the SVD entropy allows the arrival time of the P-wave to be reliably determined.

Figure 15 shows a case of premature triggering, where entropy criteria respond to local signal fluctuations before the actual arrival of the P-wave.

Taken together, the results show that the proposed entropy approach to P-wave detection works reliably across a wide range of parameters and noise conditions. Parametric analysis has identified both universal configurations suitable for use without station-specific calibration and individual parameter ranges that provide maximum accuracy for individual stations.

A qualitative analysis of detection scenarios complements these conclusions, demonstrating that the cascaded use of the derivative and absolute values of SVD entropy allows the arrival time of the P-wave to be reliably recorded even at moderate and low signal-to-noise ratios. At the same time, the nature of detection errors remains reproducible and amenable to physical interpretation.

At the same time, the results highlight the influence of local noise conditions and station characteristics on the behaviour of entropy characteristics. In particular, data from the JNKS station, which demonstrates non-standard entropy behaviour, requires separate consideration and is discussed in the following section. The same section provides an interpretation of the results obtained, an analysis of the limitations of the method, and possible directions for its further development.

To evaluate the computational efficiency of the proposed method, a comparison of processing time was performed with the classical STA/LTA algorithm and a neural-network-based method. All computational experiments were conducted at the Farabi Supercomputing Centre of Al-Farabi Kazakh National University. However, only one server with the following processor configuration was used to perform the calculations: Intel Core i9-14900KF (3.20 GHz, 24 cores, 32 logical processors).

It should be noted that the use of a supercomputer is not a prerequisite for the proposed algorithm to work. Parallel computing was used only to speed up the processing of large numbers of records from different stations, while the algorithm itself is designed to run on standard multiprocessor CPUs.

To compare the computational complexity, the processing time for a single seismic signal lasting 240 s was measured for three methods: the classical STA/LTA algorithm, the proposed SVD entropy method, and the neural network-based method. The results are presented in the

Table 4. Comparison of methods.

Method	Signal Duration	Processing Time	Note
STA/LTA	240 s	0.8 ms	accuracy = 75%
SVD-entropy	240 s	5 ms	comparable in accuracy to NN
Neural network [29]	240 s	2 s	high precision

.

Thus, the proposed method provides a favourable balance between detection accuracy and computational efficiency, making it promising for use in automatic seismic monitoring systems.

4. Discussion

Even with moderate and low signal-to-noise ratios ($SNR\ge 2$), the amplitude of the P-wave may be weak and difficult to distinguish visually. However, the very appearance of the wave leads to a redistribution of the contribution between the main components of the signal, which causes a noticeable change in the SVD entropy values. Thanks to this, the method allows you to record the initial moment of the P-wave arrival in conditions where the energy criteria are unstable.

The use of two criteria—the derivative of SVD entropy and the threshold value of its absolute level—increases the reliability of detection. The derivative is sensitive to sudden changes in the signal structure, while the threshold based on the absolute entropy value reduces the impact of random noise fluctuations. This combination makes the algorithm stable and applicable for analysing real seismic data with varying noise levels.

Although the dataset used records from four stations (TLG, SHLS, TARG, and JNKS), the initial analysis of the parametric space showed that the JNKS station exhibited fundamentally different behaviour compared to the others. In the case of the JNKS station, the observed increasing linear trend is associated with the drift component of the signal, rather than with an increase in its structural complexity.

When using the same method as with TLG, SHLS, and TARG stations, the maximum detection accuracy for JNKS was only 45.6%, which is significantly lower than the values obtained for other stations. This is confirmed by the accuracy heat map in the ‘window size–cutoff frequency’ parameter space shown in Figure 16.

A detailed analysis of events at the JNKS station showed that even at relatively high SNR values (e.g., SNR = 12.2), many recordings exhibit pronounced low-frequency drift and slow amplitude trends. An example of such an event is shown in Figure 17.

Under these conditions, SVD entropy responds primarily to the global trend of the signal rather than to local structural changes associated with the arrival of the P-wave. As a result, detection either occurs prematurely or does not occur at all, despite the visually discernible arrival of the wave.

As can be seen in Figure 17, this is an example of a JNKS station recording for an event with a signal-to-noise ratio (SNR) of 12.2, processed without polynomial detrendering. It is clear that even before the arrival of the P-wave, the signal contains a pronounced smooth increase in amplitude and low-frequency drift.

The vertical blue line corresponds to the reference time of arrival of the P-wave, determined by the PyWEED method. Despite the fairly high SNR, the moment of arrival of the wave is poorly distinguished against the background of the general signal trend: the amplitude increases gradually, without a sharp change in the shape of the oscillations.

The purple line shows the SVD entropy, and the red line shows its first derivative. Before the arrival of the P-wave, the entropy values and their derivatives show a slow change caused not by the appearance of the wave, but by the global drift of the signal. As a result, neither the derivative of the SVD entropy nor its absolute value forms a distinct response that could be unambiguously associated with the moment of arrival of the P-wave.

Thus, this example clearly shows that in the presence of low-frequency drift, SVD entropy responds primarily to the non-stationary trend of the signal, rather than to local structural changes associated with the arrival of a longitudinal wave. This leads to incorrect or delayed detection, even at relatively high signal-to-noise ratios.

Polynomial Detrending. Pre-Processing Stage for JNKS Stations

To eliminate the influence of slow trends in the JNKS station records, polynomial detrending of the signal was applied before calculating the SVD entropy. This procedure effectively removes the low-frequency component while preserving the local structure of the wave process. As shown in Figure 18, after detrendering, the smooth amplitude drift disappears, and the entropy characteristics begin to respond clearly to the moment of arrival of the P-wave. This leads to a fundamental change in the behaviour of the algorithm in the parametric space.

Figure 18 shows the same event at station JNKS (SNR = 12.2), but after applying polynomial detrending to the signal. Unlike the original recording, the overall low-frequency amplitude drift has been eliminated, and the signal remains flat until the arrival of the P-wave.

The black line, corresponding to the normalised signal, shows a stable level until the arrival of the longitudinal wave. The green line shows the detection point, determined based on SVD entropy. The method determined using SVD entropy visually shows more accurate detection.

The purple SVD entropy curve shows a sharp and localised change directly at the moment of the P-wave arrival. Unlike in the case without detrending, entropy no longer responds to slow signal trends and reflects precisely the structural change associated with the appearance of an ordered wave component.

The red curve, corresponding to the first derivative of SVD entropy, forms a distinct peak near the moment of arrival of the P-wave, which allows the beginning of the wave process to be reliably recorded. After detection, the entropy value and its derivative gradually stabilise, reflecting the transition of the signal to a more stable wave state.

Thus, this example demonstrates that preliminary polynomial detrendering is an important processing step for the JNKS station. Eliminating low-frequency drift allows for the restoration of a physically correct SVD entropy response and ensures accurate and stable detection of P-wave arrival even at moderate signal-to-noise ratios. After eliminating low-frequency drift and reanalysing the parametric space, a significantly different picture of detection accuracy was obtained for the JNKS station (Figure 19). In particular, at a signal-to-noise ratio $\left(SNR\right)\ge 2$, the maximum detection accuracy was 93.2% and was achieved with a sliding window size of 35 s and a cutoff frequency of 5 Hz.

The accuracy heat map shown in Figure 19 illustrates the distribution of detection efficiency in the ‘window size–cutoff frequency’ parameter space for the JNKS station after polynomial detrending of the signal. In contrast to the case without preliminary trend correction, the map shows a wide and stable area of high accuracy values, covering medium and long window durations at low and moderate cutoff frequencies.

The most stable detection zone is observed at window sizes of around 30–40 s and cutoff frequencies of 4–6 Hz, which corresponds to the physical characteristics of longitudinal waves and ensures sufficient stability of entropy estimates against noise. With windows that are too short, accuracy decreases sharply due to the high sensitivity of SVD entropy to random fluctuations, while with excessively high cutoff frequencies, the influence of high-frequency noise increases.

The results obtained underscore the fundamental conclusion of the study: optimal detection parameters depend on the specific station and recording conditions. Despite the possibility of selecting a universal configuration of parameters, the highest accuracy is achieved by individually adjusting the window size and cutoff frequency for each station, taking into account the local noise background and signal characteristics. It is this station-dependent adjustment that is the key factor in the stable and accurate detection of P-wave arrival in real seismic data.

The results obtained in this work should be considered in the context of existing approaches to automatic P-wave detection, including both classical algorithms and modern deep learning methods. In particular, ref. [29] proposed a method for detecting P-waves based on continuous wavelet transformation and spectrogram classification using the YOLO model. This approach demonstrated high accuracy (F1-score 91–95%) at independent stations and global events, but requires preliminary data labelling, model training, and significant computational resources. Another study [28] showed that the classic STA/LTA algorithm provides an accuracy of about 0.75, which reflects its limited capabilities in conditions of low signal-to-noise ratio and complex noise background. These results are consistent with the well-known limitations of STA/LTA related to its sensitivity to parameter selection and its focus exclusively on the energy characteristics of the signal.

Against this background, the entropy approach proposed in this paper occupies an intermediate but fundamentally important position. On the one hand, it does not require training and labelled data, unlike deep learning methods. On the other hand, it demonstrates significantly higher accuracy compared to classical energy detectors. In particular, for the TLG, SHLS, TARG, and JNKS stations, detection accuracy of up to 93–98% was achieved at $SNR\ge 2$, which significantly exceeds the STA/LTA indicators and is comparable to the results obtained using neural network models.

The key difference between the entropy approach is its focus on structural changes in the signal rather than on increases in amplitude or energy. This makes the method particularly effective in conditions of weak and noisy events, where classical algorithms are unstable and the application of deep learning may be difficult due to the lack of sufficient training data or the need to transfer the model between stations.

5. Conclusions

This paper proposes and investigates a method for early detection of P-wave arrival based on SVD entropy analysis in sliding time windows. Unlike traditional energy detectors, which consider wave arrival as an increase in signal amplitude or energy, the proposed approach interprets it as a local structural change in the time series, which provides increased resistance to noise and non-stationarity.

Unlike modern deep-learning-based solutions, the proposed method does not rely on labeled training datasets, iterative optimization procedures, or GPU-based infrastructure. This eliminates one of the main practical barriers associated with neural network deployment in seismic monitoring systems.

As a result, the method provides a favorable balance between accuracy and operational simplicity: it achieves performance comparable to data-driven models while remaining transparent, interpretable, and computationally efficient. This makes it particularly attractive for large-scale distributed monitoring networks, stations with limited computational capacity, and regions where labeled seismic datasets are scarce or unavailable

Overall, the results indicate that SVD entropy provides an effective and interpretable tool for detecting structural transitions in seismic signals and can serve as a promising component of automatic seismic monitoring systems. Future work may focus on optimizing parameter selection, improving computational efficiency for real-time applications, and extending the approach to multi-station seismic analysis.