Boletín de Ciencias de la Tierra

1.1. Study area

The study area is located in southern Peru, spanning from 15.6° to 18.2° south latitude, and from 68.7° to 76.2° west longitude. This region was affected by the 2001 earthquake, which had 134 aftershocks distributed southeastward, suggesting that the rupture propagated in that direction [12]. Fig. 1 illustrates the location of the study area and the distribution of aftershocks according to their magnitude. Blue circles correspond to aftershocks with Mw between 5 and 5.5, green circles from 5.5 to 6.5, and red circles from 6.5 to 7.5.

Figure 1: Geographic distribution of the study area, 2001 Arequipa earthquake epicenter, and spatial pattern of aftershocks by magnitude

Source: Adapted from Tavera, H. and Antayhua, 2022.

The main Arequipa 2001 earthquake is represented by a yellow star, while the rupture zone is marked with a green rectangle.

The 2001 earthquake in southern Peru caused numerous fatalities and severe damage in Tacna, Arequipa, Ayacucho, and Moquegua, as well as in Arica and Iquique (Chile), with an intensity of VII-VIII on the Modified Mercalli scale [13]. The aftershock analysis identified a rupture area of 370 x 150 km², with propagation towards the southeast [14]. More than 80 localities were affected, with 217,400 people impacted and 17,580 houses destroyed [15]. Key seismic source parameters were determined through advanced studies, as reflected in Table 1, while Fig. 2 illustrates its focal mechanism and intensities.

Figure 2: Focal mechanism and tectonic context of the 2001 Arequipa earthquake (Mw 8.4). The beach ball symbol illustrates a reverse fault mechanism. Red arrows indicate compressive forces from the Nazca Plate subducting beneath the South American Plate at ~6.9 cm/year. The rupture zone, aftershock distribution (colored by magnitude), and Modified Mercalli intensities across southern Peru are also depicted, highlighting the regional seismic impact.

Source: Adapted from Tavera et al., 2020.

2.1. Fourier Transform (FFT)

The FFT is essential in seismology for analyzing seismic signals, decomposing them into their frequency components, and facilitating the interpretation of seismic phenomena [9]. The Fourier Series allows the representation of any periodic function as an infinite sum of sine and cosine functions at different frequencies. The coefficients aₖ and bₖ indicate the contribution of each frequency in the function [17].

The Short-Time Fourier Transform (STFT) is essential for analyzing non-stationary signals, as it segments the signal into time windows to track changes in its frequency content [1]. The STFT enables a time-frequency analysis, where each window reveals the spectral distribution at that moment [18]. The result is a time-frequency graph, where the horizontal axis represents time and the vertical axis represents frequency, showing how frequencies evolve (Fig. 3).

Figure 3: Schematic representation of the STFT for time-frequency analysis of seismic signals

Source: The authors

2.2. Wavelet Transform (WT)

Unlike Fourier, the WT uses a mother wavelet, an oscillatory function localized in time and frequency, allowing the decomposition of signals into scaled and translated components [17]. This method enables concurrent analysis in both time and frequency domains, facilitating the identification of trends and oscillations [19]. CWT spectra, amplitudes are represented through a chromatic scale where warm colors (reds/oranges) indicate higher energy magnitudes, while cool colors (blues/greens) correspond to lower amplitudes [2]. Eq. (2) defines a scaled and translated wavelet used in the Continuous Wavelet Transform (CWT) [20].

Fig. 4 shows the process of the WT, a technique for analyzing signals in the time and scale domains [21]. The input signal represents a vibration that varies over time [22]. During the Wavelet Transform process, the signal is decomposed into different scales to identify its frequency content over time [23]. The analysis reveals that large scales correspond to low frequencies, while small scales capture high frequencies, enabling more detailed resolution than the Fourier Transform [2]. In Table 2, a comparison of FFT and WT in terms of seismology is presented.

Figure 4: Conceptual diagram of the WT for multiscale time-frequency analysis of seismic signals

Source: The authors.

3.1. Database download

The data were obtained from the IRIS seismic network, specifically from the Ñaña station (NNA), located at latitude -11.988° and longitude -76.842°, selected for its proximity to the earthquake epicenter.

Fig. 5 shows the flowchart of the process followed in this research. The diagram outlines a detailed and structured approach to analyzing seismic signals using two complementary techniques: FFT and WT. The results from both methods were used to determine fundamental parameters about the seismic source, allowing for a better understanding of the nature of earthquakes and their aftershocks. Once the data (10 aftershocks) were downloaded, two main programs were used to process the data.

Figure 5: Flowchart of the methodological framework for seismic signal analysis using FFT and WT

Source: The authors

The aftershock data were selected between June 23 and July 1, 2001, with magnitudes ranging from 5 to 10, and depths from 0 to 75 km. The selection of aftershocks was guided by the need to ensure waveform quality and consistency in depth and magnitude for reliable waveform analysis. Events with magnitudes ≥5.0 and depths ≤75 km were chosen to maintain a homogeneous dataset suitable for spectral and time-frequency analysis. This threshold also ensures that selected events generate records with a sufficient signal-to-noise ratio, essential for accurate estimation of source parameters. Although other events of lower magnitude exist, they were excluded due to limitations in data quality or lack of clear waveform characteristics. Table 3 shows the database of the 10 aftershocks, with their main characteristics (Date, time, latitude, longitude, depth, magnitude type, and location).

3.2. Determination of seismic source parameters with FFT

The Seismic Analysis Code (SAC) software is a widely used tool in seismology for the management and analysis of seismic data in time series [25]. It allows the application of FFT to calculate the spectrum of a signal and uses high-pass, low-pass, and band-pass filters to remove noise and enhance seismic detection [26].

SAC is essential for determining seismic source parameters such as 𝑀o, source radius, corner frequency 𝑓𝑐, and stress drop (Δ𝜎), allowing the analysis of fault dynamics and potential damage.

For seismic signal processing, high-pass, low-pass, and band-pass filters were applied using SAC software to eliminate noise and enhance spectral detection. Signals were corrected to physical units suitable for calculating parameters such as Mo and Δ𝜎. In the Wavelet analysis, signals were bandpass filtered between 2 and 4 Hz, amplitudes were squared, absolute values were taken, then smoothed, and normalized to unity.

The 𝑓𝑐 is related to the rupture size and earthquake duration, being lower in high-magnitude events and higher in smaller ones [27]. The flat part of the spectrum (Ω₀) is found at low frequencies, where the amplitude is constant, indicating a uniform distribution of energy [28] (Fig. 6).

Figure 6: Fourier spectrum, calculating the corner frequency 𝑓𝑐 and the flat part Ω₀, of the 2001 Arequipa earthquake.

Source: The authors

3.3. Determination of seismic source parameters with WT

MATLAB is a key tool in technical scientific programming, enabling seismic signal analysis through digital processing, including the interpretation of phases, amplitude, and polarity [29]. Additionally, it facilitates the use of the Continuous Wavelet Transform (CWT) to represent non-stationary signals in the time-frequency domain [30].

The use of MATLAB and the Wavelet Transform allows for the analysis of non-stationary seismic signals, decomposing them into different frequency and time scales to characterize the dynamics of the seismic source [31]. The seismic signal shows energy variations over time, with higher energy at low frequencies (surface waves) in the first minutes, which decreases over time. In Fig. 7, high scales correspond to low frequencies, associated with surface waves, while low scales represent high frequencies of P or S waves. High intensity peaks indicate energy release, and the decrease towards the end reflects the reduction in seismic activity.

Figure 7: 3D wavelet spectrogram CWT of the 2001 Arequipa earthquake, generated in MATLAB

Source: The authors.

3.3.1. Calculation of rupture time

The rupture time was estimated following the methodologies of [32]. The vertical BHZ component was filtered between 2-4 Hz, squared, and normalized to enhance signal clarity. Rupture duration was then measured from the P-wave onset to the point where the amplitude decayed to 20% of its maximum, providing reliable results without the need for instrumental response correction.

The seismic record used for this analysis corresponds to a vertical BHZ component, with a velocity equivalent to that of a broadband station, allowing for a detailed study of the waveforms [33] (Fig.8a). To enhance the clarity of the signal, a bandpass filter is applied to the record, restricting frequencies to a range of 2 to 4 Hz, which effectively isolates the relevant seismic frequencies while minimizing noise interference (Fig. 8b).

Figure 8: (a) Vertical component seismic signal of the Arequipa Earthquake of 2001, (b) Normalized seismic signal (P-wave), (c) Calculation of rupture duration, (d) Calculation of maximum radiated energy MaxEa (J)

Source: The authors.

Once the filtered signal is obtained, its amplitude values are squared, making them proportional to the seismic energy radiation. Additionally, to facilitate further processing, the absolute values of the amplitudes are taken, smoothed, and normalized to unity, ensuring a consistent comparison across different signals [34]. This transformation allows a better understanding of the energy distribution and variations within the seismic event.

In terms of signal decay analysis, the duration of the pause is measured from the initial arrival of the P-wave until the amplitude reduces to 20% of its maximum value, providing insights into the dissipation of seismic energy over time. This approach highlights the gradual attenuation of the seismic waves and their energy release patterns [35].

It is also crucial to emphasize that, since the analysis focuses only on the temporal component of the seismic signal, adjusting the instrumental response is not required. This simplifies the workflow while maintaining the integrity of the data, ensuring that the results remain reliable and representative of the actual seismic phenomenon under investigation. Since the analysis relies on the signal’s temporal evolution rather than absolute amplitude, instrumental correction is unnecessary. The waveform shape remains intact, preserving data integrity and ensuring reliable, representative results of the seismic event (Fig. 8c).

4.1. Comparison of the parameters of the Arequipa 2001 earthquake and its aftershocks

Table 4 presents the main source parameters: Mo, R, and Δσ estimated using FFT; and Fc, MaxEa, and Tr obtained using WT. These values provide complementary information on the event's size, energy release, and rupture dynamics.

Table 5 summarizes the seismic source parameters obtained through FFT and WT transforms, comparing the mainshock (Mw 8.4) and its aftershocks (Mw 5.3-6.7), including M₀, fracture radius, stress drop (Δσ), radiated energy (MaxEa), and rupture durations.

4.2. Seismic Moment (Mo)

Fig. 9 illustrates the seismic moment (Mo) of the main shock and aftershocks using a logarithmic scale on the vertical axis to capture the wide range of values. The main shock, represented in red, exhibits a seismic moment of 1.58 × 10²¹ N·m. This reflects the immense amount of energy released during the primary rupture. In contrast, the aftershocks, shown in blue, have significantly lower moments, ranging from 3.55 × 10¹⁶ N·m to 1.41 × 10¹⁸ N·m. These events mainly contributed to the release of residual stresses and had a much smaller impact on the overall deformation. The use of a logarithmic scale emphasizes the contrast in energy between the main event and the subsequent seismic.

Figure 9: Seismic Moment Values for the Arequipa Earthquake (Main Shock) and its Aftershocks.

Source: The authors

4.3. Fracture Radius

Fig. 10 presents the fracture radius (in kilometers) for each seismic event. The main shock, again in red, displays the largest fracture radius-3.15 km-indicating a much greater rupture area compared to the aftershocks, which are shown in green. The aftershocks generally range between 0.6 km and 2 km in radius. This graph clearly highlights the spatial extent of the main rupture compared to the more localized failures of the smaller seismic events. The use of contrasting colors once more helps differentiate the dominant main shock from the supporting seismic sequence.

Figure 10: Fracture Radius for the Arequipa Earthquake (Main Shock) and its Aftershocks

Source: The authors

Fig. 11 displays the log-log relationship between Mo (N·m) and fracture radius (r), with lines representing different values of stress drop (Δ𝜎 = 0.1, 1, 10, and 100 MPa). The red star marks the mainshock, located above the 10 MPa line, while the aftershocks cluster near the same line, suggesting similar stress drop magnitudes. This graphical representation follows the scaling law Δ𝜎 ∝ M₀ / r³, allowing for visual interpretation of rupture mechanics.

Figure 11: Log-Log Plot of Seismic Moment vs Fracture Radius with Stress Drop (Δσ) Iso-Lines

Source: The authors.

4.4. Maximum radiated energy

The main earthquake radiated 40.80 J of energy, while the aftershocks had values ranging from 21.19 J to 26.29 J, according to analysis using the WT.

Fig. 12 shows the relationship between rupture duration (in seconds) and radiated energy in a seismic context, specifically for a Tsunami Generator (TG). The x-axis measures rupture duration, and the y-axis represents radiated energy (blue). Each point in the graph represents a seismic event, differentiated by colors and shapes. The main earthquake is highlighted with a red star, located in the upper left corner, indicating that it had the longest rupture duration and the highest radiated energy. The other points correspond to aftershocks, labeled in the legend as Aftershock 1, Aftershock 2, etc., with different colors and symbols. It is observed that as the rupture duration decreases, the radiated energy also decreases, suggesting a positive correlation between both variables. The aftershocks exhibit lower energy and shorter rupture durations compared to the main event, which is expected in a seismic sequence.

Figure 12: Relationship between rupture duration and radiated energy for the 2001 Arequipa mainshock and its aftershocks

Source: The authors.