Modelling and Simulation of the Propagation of P-SV Seismic Waves from Earthquakes: Application to Deep Earthquakes in Acre, Brazil

Abstract: Brazil is located in the central-eastern portion of the South American Plate, meaning that the country mostly experiences low-intensity seismic activity within its territory. However, some geological faults in this region have generated intense earthquakes. In this context, we intend to describe a recent earthquake of magnitude around 6.5 M_b that occurred at a depth of approximately 600 km in the state of Acre, Brazil. In this work, we modeled the propagation of P-SV seismic waves using a two-dimensional system of partial differential equations (PDEs) in a two-dimensional vertical rectangular domain. The source is modeled by a Gaussian pulse function. The initial quiescence condition and Neumann boundary conditions are used. The PDE system is discretized by the finite difference method (FDM) and solved by the Gauss-Seidel method (GSM). The numerical simulations obtained describe the propagation of attenuated seismic waves in multiple geological layers, simulating intense and deep earthquakes in Acre. We used the propagation of perfect seismic waves to validate the model. The results include images of the simulations and theoretical seismograms simulating the vertical and horizontal displacement in the epicenter region and 200 km east and west of the epicenter.

• The paper introduces a numerical framework to simulate coupled P-SV seismic waves from deep-focus earthquakes in Acre, Brazil.
• It uses a second-order finite difference method to discretize elastodynamic PDEs, validated against analytical solutions for correct wave propagation.
• Synthetic seismograms demonstrate strong attenuation in deep earthquakes, highlighting crucial implications for seismic risk assessment.

Numerical Modeling and Simulation of P-SV Seismic Wave Propagation in Acre, Brazil

Introduction and Motivation

This paper presents a comprehensive numerical framework for modeling and simulating the propagation of coupled P (primary) and SV (vertically polarized shear) seismic waves from deep-focus earthquakes, with application to events in Acre, Brazil—one of the most consequential yet least studied seismic regions in the country. Although Brazil is typically associated with low seismicity, deep earthquakes of magnitude exceeding 6.0 have been repeatedly observed in Acre, especially along the Tarauacá fault. The study addresses the need for quantitative seismic risk assessment in a region where recent seismic events have reached depths of 600 km and magnitudes greater than 6.5.

Mathematical and Computational Framework

The propagation of P-SV waves is formulated as a system of coupled elastodynamic partial differential equations (PDEs) in a two-dimensional vertical (x, z) domain. The wave source is represented using a Gaussian pulse, facilitating a realistic point-source approximation, and wave attenuation is incorporated via depth- and material-dependent loss terms. Boundary conditions are specified as Neumann-type for the entire domain, with custom boundary attenuation layers devised to prevent spurious numerical reflections.

The governing PDEs are discretized using a second-order finite difference method (FDM) in both space and time. The computational implementation further exploits regressive approximations at boundaries to preserve stability and accuracy. The resulting large, sparse linear system is solved at each time step using the Gauss-Seidel iterative method. The chosen domain size and discretization parameters are configured to resolve high-fidelity wave dynamics for the spatial and temporal scales relevant to deep Acre earthquakes.

Validation and Physical Parameters

Model validation is conducted by simulating idealized, non-attenuated (perfect) wave propagation with homogeneous media. In this limiting case, the system admits analytical solutions, thus enabling direct verification of the finite difference scheme and computational pipeline. The authors report correct propagation velocities and energy conservation, with boundary damping successfully suppressing artificial reflections. These initial tests establish both the correctness and robustness of the computational approach before transitioning to the heterogeneous, layered Earth model required for Acre-specific simulations.

For the full simulation, heterogeneous elastic parameters corresponding to distinct Earth layers (crust, upper mantle, transition zones) are incorporated. Empirical P- and S-wave velocities ($V_p$, $V_s$) and densities ($\rho$) as functions of depth are adopted from geological studies, with attenuation factors assigned based on lithological estimates for the Acre crust and mantle materials. The model domain is designed to span both the epicentral region and lateral distances up to several hundred kilometers, accommodating displacement tracking at the epicenter and at selected sites east and west of the fault.

Results: Propagation Dynamics and Theoretical Seismograms

Simulations for recent Acre deep-focus earthquakes produce detailed space-time maps of coupled P and SV wavefronts propagating from hypocentral depths near 600 km upwards toward the surface and laterally within the computational domain. Key features include wavefront bifurcation due to distinct $V_p(z)$ and $V_s(z)$, and reflection/refraction at geophysical layer boundaries, particularly at the mantle transitions. Notably, simulated results demonstrate strong attenuation with increasing distance and depth—quantitatively confirming why even high-magnitude events in Acre result in limited surface damage.

Synthetic seismograms are extracted at the epicenter and at points 200 km east and west. These records capture both vertical (Uz) and horizontal (Ux) displacement components, with amplitude and envelope characteristics reflecting both source dynamics and medium heterogeneity. Estimated magnitudes derived from wave amplitudes and periods match reported earthquake strengths (Mb ≈ 6.5–6.8), validating the model’s realism in reproducing observed seismic phenomena.

Implications and Future Prospects

The paper provides the first numerical simulation evidence that deep, high-magnitude Acre earthquakes produce strongly attenuated ground motions at the surface, consistent with the observed lack of major damage despite significant seismic energy release. This finding underscores the critical role of both event depth and geologic structure in regional hazard assessment and points to the limitations of using only magnitude and epicentral distance as risk proxies.

Practical implications include guiding seismic monitoring infrastructure placement, as regional variations in crust–mantle properties and wave attenuation should inform site-specific risk evaluations. The theoretical insights gained also serve to motivate further investigations into multi-source scenarios that align with the complex structure of the Tarauacá fault. The authors suggest the importance of acquiring higher-resolution geophysical data on the upper crust in Acre, which would enhance parameterization and accuracy in future numerical models.

For computational geophysics, the demonstrated coupling of PDE-based modeling, high-order FDM discretization, and robust boundary attenuation treatments sets a technical benchmark for future regional seismic studies, especially in intraplate settings with deep-focus event regimes.

Conclusion

This study delivers a rigorous numerical modeling framework for the simulation of P-SV seismic wave propagation from deep earthquakes in the Acre region, capturing both complex wavefield dynamics and the resultant theoretical seismograms relevant to recent high-magnitude events (2601.03177). The results quantitatively demonstrate strong wave attenuation at great depth and explain the limited surface effects observed for Acre’s significant seismicity. The approach advocates for continued, data-informed refinement of regional Earth models and boundary treatments, outlining a path for predictive simulation-based seismic risk assessment across understudied stable continental regions.