In this work, we use the geomagnetic horizontal “H” component, which exhibits a maximum amplitude and is more susceptible to magnetosphere–ionosphere modulation effects. Consequently, it is more suitable for studying pulsations in conjugate points . The data were sampled at a rate of 1 s for all stations, allowing us to effectively analyze Pc3 pulsations.

We focus on a case study covering the period from 01:00 to 04:00 UT on 25 October 2016, which corresponds to a quiet to moderately disturbed geomagnetic environment (3.5 < Kp < 5; Dst ∼ -30 nT). More information about the space environment conditions for the selected data period is presented and discussed in . Previous studies have shown that some Pc3 pulsations occur in the range of 1<Kp<3.5 during quiet to disturbed periods . A storm sudden commencement (SSC) was recorded at 09:22 UT on the same day, after the selected period. This time period was chosen for the analysis of Pc3 pulsations because it is the only period with data from all four stations, where the amplitudes exceed 0.5 nT and the wave packet contains at least three wave cycles.

Fortunately, the chosen period occurred near the September equinox, ensuring nearly equal solar illumination of both hemispheres. This condition results in interhemispheric symmetry, with similarities in conductivity and atmospheric electric current systems at conjugate points . Such atmospheric characteristics simplify the physical system, as conjugate ionospheres during winter or summer would exhibit significantly different conductivities . Additionally, the moderately disturbed period, characterized by variations in solar wind pressure, offers a unique opportunity to investigate the electrodynamical coupling between the solar wind and the magnetosphere–ionosphere system at low latitudes.

The selected stations are located at low latitudes (L-shell < 2) and are positioned near conjugate points, theoretically placing them close to the feet of the same magnetic field line. Specifically, the Kakioka (KAK) station in Japan is roughly conjugate to Alice Springs (ASP) station in Australia, while San Juan (SJG) station in the United States is roughly conjugate to São Martinho da Serra (SMS) station in Brazil. Figure  illustrates the locations of the conjugate stations. Additionally, the surface area of the SAMA is highlighted on the map by the closed blue line, which corresponds to the 32 000 nT magnitude isoline at the Earth's surface, as calculated using the IGRF-13 model.

The conjugate coordinates were determined using the altitude-adjusted corrected geomagnetic (AACGM) model, which utilizes the full-resolution IGRF-13 magnetic field model, including its non-dipolar components, to trace magnetic field lines between the hemispheres. The tracing process begins by numerically following the magnetic field line from the geographic starting position to the dipole magnetic equatorial plane, defined by the best-fit Earth-centered dipole. From that point, a dipole field line is traced back to the Earth's surface . These steps provide a more physically meaningful mapping between conjugate locations.

We utilized a Python-based wrapper tool

https://pypi.org/project/aacgmv2/ (last access: February 2025)

In this work, we applied two techniques for spectral analysis: time–frequency analysis, which utilizes the Fourier transform, and timescale analysis, based on the continuous wavelet transform.

Coherence analysis can quantify the correlation between signals or similarity between them in the frequency domain , helping to determine whether pulsations come from the same source region in the frequency domain . Differences in observations may indicate disturbances from local processes or asymmetrical effects from external sources. The cross-wavelet transform is a tool used to examine timescale dependencies between two time series, defined as follows: 3Wf,gψ(a,τ)=Wfψ(a,τ)Wgψ(a,τ).

Based on this cross-transform, we can compute the wavelet coherence and a phase measure. They provide linear-quantitative estimators of the degree to which the relationship scales between the two signals . Here, the wavelet coherence is computed using the following expression: 4CW2(a,τ)=Sa-1Wf,gψ(a,τ)2Sa-1Wfψ(a,τ)2Sa-1Wgψ(a,τ)2.

Wavelet coherence (CW2(a,τ)) values, ranging from 0 to 1, indicate the degree of linear relationship between two signals. Values close to 1 suggest a strong correlation, implying the signals likely come from the same source. Values near 0 indicate weak or no correlation, suggesting the signals originate from different sources. The operator “S” is a smoothing function applied in the timescale domain, implemented via convolution of transform coefficients with a Gaussian moving average in both time and scale directions. This smoothing reduces noise introduced by amplification effects . The wavelet coherence phase, denoted as θx,y, is derived from the real and imaginary components of the coefficients. This measurement yields information on the delay of one signal considering the other as a function of time and scale. Implementation of the CWT and wavelet coherence were based on a library of MATLAB functions provided by . A detailed explanation of the methodology can be found in .

Figure  shows the Pc3 pulsations for each conjugate pair (with each pair indicated by red and blue). We initially selected two time intervals representing simultaneous wave packets of Pc3 pulsations. Both intervals have the same duration of 10 min. The time intervals were chosen based on the presence of significant Pc3 pulsation amplitudes in both regions and the significant wavelet coefficients (as seen in Figs.  and , which are described later).

Through visual inspection, the plots in Fig.  show consistent propagation patterns of Pc3 wave packets that are simultaneous and similar between the conjugate stations, particularly between KAK and ASP in the Asia-Pacific region. That is, the occurrence of the Pc3 pulsation was essentially identical, and relative amplitudes were similar. This observation is consistent with some previous investigations at conjugate stations . Additionally, Fig. b shows that Pc3 wave packets in the Southern Hemisphere generally exhibit higher-amplitude intensities compared to those in the Northern Hemisphere for conjugate station pairs. At the SMS station, an enhancement in Pc3 pulsation amplitudes is observed near the center of the South Atlantic Magnetic Anomaly (SAMA). During quiet to moderately disturbed geomagnetic conditions, the SAMA region experiences an increase in Pc3 pulsation amplitudes, which may be associated with electron precipitation in the ionospheric D and E layers. The SMS station, located near the center of the SAMA, recorded the highest amplitudes of these pulsations.

We employ a power spectral method (explained in Sect. ) to analyze the significant peaks and evaluate the similarity between the conjugate station pairs. Figure  displays Welch's power spectral density (PSD) of the filtered time series, calculated using the FFT, for the selected intervals. It also shows the frequency contributions to the Pc3 pulsations across all intervals for the conjugate pairs. In interval 1, two significant discrete spectral peaks are observed for each conjugate pair at 23.4 and 29.2 mHz in both regions. In interval 2, a significant peak is observed at 27.3 mHz in all stations. Furthermore, the amplitudes of the peaks in both intervals are similar in the Asia-Pacific region. In contrast, the amplitudes of the peaks at the SMS station are considerably higher than those at its conjugate station. This result may indicate that the SMS station is influenced by an altered upper atmosphere in the SAMA region, primarily due to the ionosphere. It is important to note that some authors have observed spectra similar to Pc3 pulsations at low latitudes and/or over a wide range of latitudes on the Earth's surface . However, this would be the first time that similar spectra have been observed at conjugate points and at low latitudes, particularly when considering the SAMA region.

The dominant frequencies (23.4, 29.2, and 27.3 mHz) are consistent with some of the discrete frequencies observed in the 20–30 mHz range, as reported by , , , , and . Several studies have reported the manifestation of pulsations and fluctuations at discrete frequencies, often interpreted in terms of field line resonances (FLRs) associated with large-scale waveguide/cavity modes in the Earth's magnetosphere . However, this topic is not the primary focus of the current work, though it could be explored in future studies.

In this section, we analyze the signal energy across different scales over time. Scalograms are used to visualize the location of maximum values and their temporal occurrence, offering an effective way to examine pulsation details and highlight similarities.

For the Asia-Pacific region, Fig.  shows the Pc3 pulsation signal (Fig. a and b), the corresponding scalograms (Fig. c and d), and plots of the global wavelet spectrum (Fig. e and f), each corresponding to the conjugate stations. Panels in the same vertical column correspond to the KAK station in the Northern Hemisphere (on the left) and the ASP station in the Southern Hemisphere (on the right), while panels at the same horizontal position refer to the conjugate station pair. In wavelet scalograms, the horizontal axis represents the time in hours, while the vertical axis represents the signal intensity (in nT) and wavelet scale (in mHz), respectively. The magnitudes of the wavelet coefficients are indicated by the color bar, which has the same range (nT² mHz⁻¹) for the signal analysis of each station.

Figure c and d highlight the characteristics of the Pc3 pulsations. The maximum values of the wavelet coefficients (indicated in yellow) coincide with the most intense Pc3 wave packets observed at each station (Fig. a and b). In the two selected time intervals (1 and 2), the scalograms show higher-intensity spectrum values in the frequency range of 20 to 40 mHz. An inspection of the scalograms reveals similar features in energy amplitude, duration, and scale (converted to central frequency) for the conjugate pairs. The global wavelet spectrum plots highlight two peaks that are similar between the conjugate stations, coinciding with the Pc3 pulsation wave packets observed at each station. However, it can be noted that the peak powers at the ASP station are slightly more intense. Based on the observed similarity in energy patterns, we conclude that the conjugate pairs in the Asia-Pacific region are subject to nearly identical conditions in the magnetosphere–ionosphere system.

Figure  presents the Pc3 pulsation data (top panels), scalogram (middle panels), and plots of the global wavelet spectrum (bottom panels) for the conjugate station, following a format similar to the previous figure but focusing on the America–SAMA region.

Initially, the scalograms show similar and simultaneous energy patterns, with the more intense wavelet coefficients corresponding to the Pc3 pulsation wave packets. However, the coefficients at the SMS station show a slight increase in intensity, particularly during time interval 2. Additionally, the more intense wavelet coefficient values in the scalograms are observed between 20 and 40 mHz. The global wavelet spectrum plots (Fig. e and f) reveal two peaks at the conjugate stations, which align with the Pc3 pulsation wave packets detected at each location. However, it is noted that the peak powers at the SMS station are more intense than at its conjugate. Finally, the results suggest that the SAMA influences the energy of Pc3 pulsations due to ionization in the SAMA region. The use of the CWT technique to investigate timescale characteristics has significantly contributed to this study.

To complete the investigation, we employ coherence analysis to address the following question: for a pair of signals from conjugate stations, how similar are their power spectra?

The wavelet coherence for each conjugate pair in both global regions, calculated using Eq. (), is used to create color map plots (Fig. ). These plots represent the coherence values of Pc3 pulsations between the conjugate pairs, with yellow indicating maximum coherence (close to 1). Arrows indicate the phase relationship between signals: right arrows for in-phase, left arrows for anti-phase, and inclined arrows for intermediate values. Arrows are shown only where the coherence is 0.8 or higher for better visualization.

Figure  shows the wavelet coherence results for the conjugate stations in both the Asia-Pacific and America–SAMA regions. As seen in Fig. a, the time intervals for the KAK–ASP stations exhibit very high coherence (CW>0.7) and in-phase signals for pulsations at the conjugate stations. Coherence in the 20 to 100 mHz band is consistently high between this conjugate pair. and also observed high-coherence Pc3–5 pulsations (using FFT) between conjugate stations, recorded in the Asia-Pacific region. Finally, the high coherence in this region suggests that the Pc3 pulsations observed at the conjugate stations likely originated from the same source in the magnetosphere. These results may indicate that the stations are subject to similar effects and undergo similar processes of energy transfer between the magnetosphere and ionosphere.

In contrast, when examining the time intervals in Fig. b, there are generally moderate to high signal coherence (0.4<CW<0.7) and in-phase signals between the conjugate stations. The SJG–SMS pair shows lower coherence than the KAK–ASP pair, with moderate to high coherence observed in the 20–60 mHz band. This reduced coherence for SJG–SMS may be linked to the station's proximity to the SAMA center, where enhanced electrical conductivity, due to particle precipitation, alters pulsations and impacts the plasmasphere–ionosphere system, thereby modifying the structure of Pc3 pulsations along the magnetic field lines connecting conjugate stations.

In summary, the observations from our case study demonstrate a striking similarity and simultaneity of Pc3 pulsations recorded at conjugate stations in low-latitude regions and across two distinct geographic areas. These pulsations occurred near the time of interhemispheric illumination symmetry, coinciding with the September equinox.

In the Asia-Pacific region, nearly symmetric conditions at the magnetopause, plasmasphere, and ionosphere result in similar and simultaneous Pc3 pulsation amplitudes. Spectral analyses (PSD) and scalograms show significant similarities between conjugate stations, with high coherence and in-phase signals in this region. In contrast, in the America–SAMA region, an enhancement in the amplitude of Pc3 pulsations was observed at the SMS station.

Previous studies have reported similar observations. For instance, , , and observed that magnetic pulsations at conjugate stations outside the South Atlantic Magnetic Anomaly (SAMA) typically exhibit comparable amplitude and phase characteristics. In contrast, studies focused on the SAMA region (though not necessarily at conjugate stations) have shown that Pc3 pulsation amplitudes are often enhanced in this area . Furthermore, research on Pc5 pulsations has also reported amplitude enhancement within the SAMA region . Thus, our case study aligns with previous findings and supports the interpretation that the enhancement of pulsation amplitudes in the SAMA region is a persistent feature.

While spectral analyses and scalograms showed similarities, the SMS station displayed more intense power spectral density and wavelet coefficients. Notably, the discrete spectral frequencies (PSD) were identical in both regions. Moderate to high coherence and in-phase signals were also observed in this region. These differences arise from the asymmetric conditions between the America–SAMA stations. Although both stations experience similar magnetopause conditions, they differ in particle precipitation and plasmasphere–ionosphere dynamics, with the SMS station exhibiting an ionization anomaly. This suggests that the SAMA region influences the characteristics and structure of Pc3 pulsations.

The high coherence observed in both regions suggests that the Pc3 pulsations likely originate from the same external source in the magnetosphere, propagating along magnetic flux lines linking the conjugate stations. However, at the SMS station, the Pc3 pulsations may also be influenced by factors such as the ionization in the SAMA, particle precipitation, and variations in electrical conductivity during slightly perturbed periods. These asymmetries in plasmasphere–ionosphere conditions alter the Pc3 pulsations along the magnetic flux lines in the America–SAMA region, contrasting with the Asia-Pacific region, which is under more symmetric conditions.

The observations from this case study support the conclusion that the SAMA region influences the characteristics of Pc3 pulsations. This study does not address the sources of the pulsations or the internal mechanisms driving the modifications in the magnetic pulsations. Exploring these aspects would necessitate the use of additional methodologies and comprehensive data from ground-based instruments to examine the ionosphere and from satellites to probe the plasmasphere.