SeidarT is designed to enhance usability and performance in geophysical modeling, particularly for wave propagation in snow and ice environments. One of its key features is that it is open-source and hosted on GitHub, accompanied by robust documentation to support users in its implementation and development. Its architecture is designed to be open and extensible, making it easy to integrate with additional Python packages or other scripting languages as needed. The design philosophy follows a modular and flexible approach to cater to a wide range of geophysical applications and user skill sets. The core of the software is built around the staggered grid finite difference time domain (FDTD) method, which provides robust and accurate solutions for 4th order seismic and 2nd-order electromagnetic wave propagation.

Model development in SeidarT leverages illustration software used to generate PDF and PNG files. There are open source tools such as GIMP and Inkscape, but the actual choice is the user's preference. This approach provides users with significant control over defining complex geometries, such as the varying sloped sidewalls and changes in ice facies found in glaciers. Programmatically defining such intricate geometries can be cumbersome and limiting, whereas using RGB values in raster based illustrations to identify different materials streamlines the process, allowing for more precise and versatile model creation.

Models are organized in a JSON (JavaScript Object Notation) file where users can define domain parameters, source locations, and the properties associated with the materials present within the domain. The use of JSON provides a readable platform, and this approach streamlines the setup and management of simulations, making it straightforward to configure and run their models.

The software supports the calculation of bulk properties from Euler angle inputs for material fabrics, but material tensors can be manually input when available. Users can input text files from Electron Backscatter Diffraction (EBSD) directly into the program for material homogenization, employing three Euler angles for Bunge rotation (z–x–z) and fourth-order tensors. This allows for detailed incorporation of natural sample (i.e. ice core) observations, which can then be compared with Ground Penetrating Radar (GPR) and seismic observations, in order to provide correlation between observed and modeled data.

Building and creating a wave propagation model follows the simple image plus JSON workflow (Fig. ). A rectilinear Cartesian grid with uniform spacing along each coordinate direction, represented by a PNG image, establishes the simulation domain, where each pixel corresponds to an FDTD node and each color is automatically assigned an integer ID. This ID serves as a relational value that is later referenced in the model configuration to link material properties and wave parameters to regions in the grid. With the PNG defined, the user creates a JSON project file specifying domain attributes (dimensions, resolution), material settings (e.g., stiffness or permittivity and conductivity tensors), and model parameters (seismic or electromagnetic mode). Tensors can be supplied explicitly or generated, and for 2.5D simulations the x–z plane properties are extruded in the y-direction with corresponding CPML settings.

Only two files are strictly required to run a simulation: the JSON project file and the PNG image. At build time, the user specifies at least the number of time steps and source center frequency, and may optionally define initial conditions. SeidarT automatically checks that the grid spacing and velocity model satisfy the Courant–Friedrichs–Lewy (CFL) condition and chooses a stable time step . Default CPML parameters are also initiated using in ranges provided by , but these values can be tuned individually to improve boundary performance. While running the simulation, outputs are written either as Fortran unformatted binary files (.dat) at each time step or chunked block outputs compressed using Zstandard formatting . This increases storage requirements with the tradeoff of preserving the full wavefield, allowing post hoc construction of arbitrary receiver arrays and extraction of particle-velocity or electric-field time series at any node, including dense virtual-sensor networks.

SeidarT supports common-offset profiling by extending internal class variables to generate series of model runs. For common-offset surveys, a CSV file specifies paired source and receiver locations, with each row containing a source–receiver pair. Results from common shot gathers and common-offset surveys can be exported as pickle-formatted files and/or CSV files. Pickle files retain model parameters and metadata, time series, and derived products within a single object, while CSV output facilitates import into external signal-processing workflows. Receiver geometry for common-offset surveys is fixed for a given run, so changing array layout requires rerunning the survey, but the modular design makes it straightforward to set up, rerun, and analyze different acquisition scenarios.

Wave propagation refers to the manner in which elastic (seismic) or electromagnetic perturbations travel through often heterogeneous materials. Accurately capturing those waves – especially in anisotropic or attenuating media – requires a discretization strategy that resolves both fine‐scale interfaces and broad‐scale structures. While finite‐element (FEM) and spectral‐element (SEM) methods offer high accuracy per degree of freedom, they introduce substantial implementation complexity and often rely on external libraries. By contrast, a finite‐difference time‐domain (FDTD) approach, though computationally heavier, remains remarkably minimalist: it directly discretizes Maxwell's equations (for EM) or the elastodynamic equations (for seismic) on a simple Cartesian grid. This simplicity makes the codebase easier to inspect, modify, and extend.

The FDTD approach has undergone continuous refinement to handle increasingly complex media and boundary conditions. Maxwell's equations or the equations of motion can be discretized using a grid where the electric and magnetic field (or stress and velocity) components are staggered in both space and time. The early work of laid a strong foundation with the development of velocity-stress finite-difference methods for horizontal shear wave propagation. By introducing a staggered‐grid formulation, was able to discretize both velocity and stress without artificially smoothing internal boundaries, computing each component so that it is centered between neighboring components. During each time step, the velocity from the previous step is first used to update the current stress, and this updated stress is then used to compute the current velocity before advancing to the next time step. His method was based on the first‐order hyperbolic system for SH‐wave propagation and became the prototype for many later FDTD schemes.

Implementation of Maxwell's equations and the elastic wave equation follow similar approaches and we will lay out both sequentially. To derive the staggered‐grid formulation, we start from the continuous constitutive relations that couple stress to particle velocity in an elastic medium. The generalized form for the constitutive equations of motion is 1σij,t+γijσij=Cijkl(uk,lt+ul,kt)2+Sij(x,t)2ρui,tt=σji,j+Fi(s), where σ is the Cauchy stress tensor, C is the 4th-order stiffness tensor, ρ is the density, u is displacement, and γ describes viscoelastic attenuation. Sources are prescribed either as stress-type moment-tensor terms (e.g., double couple, CLVD, explosive) 3Sij(x,t)=Mijw(t)δ(x-xs), where Mij is the moment tensor and δ(x-xs) denotes a point source at location xs, or as equivalent body forces 4Fi(s)(x,t)=ρe^iw(t)δ(x-xs), with e^i the ith unit basis vector. The source wavelet w(t) may be chosen from a range of user-defined functions (e.g., Gaussian pulse, Ricker, higher-order Gaussian derivatives, multi-modal sweeps, chirps), allowing the incident spectrum and time–frequency content to be tailored to a given application. Finite-difference expansions of Eqs. ()–() follow standard staggered-grid formulations , and in SeidarT we employ a 2nd-order stencil on the staggered grids to obtain an overall 4th-order scheme in space.

The finite difference expansion of Eqs. () and () are relatively straightforward at this point and can be referred to in more detail in . This approach ensured that both the stress and velocity fields are modeled in heterogeneous media with a minimum of 2nd-order accuracy in both time and space. We have applied a 2nd-order finite difference stencil to both grids to provide a 4th-order modeling scheme.

Attenuation is modeled through the amplitude decay of a wave traveling distance d from a directional source with initial amplitude A0 5A=A0e-αd, where α is a 2nd-order tensor with components applied component-wise in a chosen coordinate frame, 6αij=πfcQij=γijc. Here f is frequency, c is phase velocity, and Q is the quality factor, which is often treated as frequency independent over limited bandwidths . In the most general case, the attenuation tensor γ is 4th-order and couples relaxation of all stress components. For computational efficiency and stability, we assume component-wise relaxation and reduce γ to a 2nd-order tensor while retaining the ability to represent direction- and mode-dependent attenuation in the materials of interest.

Electromagnetic wave propagation is described by a parallel structure governed by Maxwell's equations . Faraday's law, 7ϵijkEk,j=-μilHl,t, relates spatial gradients of the electric field E to temporal changes in the magnetic field H, where ϵijk is the Levi-Civita symbol and μ is the magnetic permeability tensor. Ampère's law for a lossy medium, 8ϵijkHk,j-σijEj+εijEj,t=Ji0, introduces the conductivity tensor σ, permittivity tensor ε, and source term J0, which is treated analogously to the body-force source in Eq. (). For many non-magnetic materials and in air–ice–water mixtures relevant to cryospheric studies, the magnetic permeability tensor is isotropic and close to that of free space, allowing us to set μil=μ0δil without loss of generality. In this lossy, anisotropic medium, wave attenuation is fundamentally captured by the conductivity tensor σ. For a given frequency, σ maps directly to the amplitude attenuation vector α in Eq. (), where an increase in conductivity fundamentally dictates a higher spatial attenuation constant.

The numerical implementation of electromagnetic wave propagation and seismic wave propagation follows very similar approaches. While the equations of motion and Maxwell's equations describe different physical phenomena, they share structural similarities since they are both systems of partial differential equations; thus, they can be decomposed similarly into their respective finite-difference approximations. The primary difference between them is the material properties (i.e. stiffness and permittivity) that relate the stress and velocity fields or the electric and magnetic fields.

To suppress artificial reflections from the outer model boundaries, SeidarT uses a Convolutional Perfectly Matched Layer (CPML) formulation . Without an absorbing layer, outgoing waves reflect at the domain limits and contaminate the interior solution . CPML extends the classical PML by introducing complex-frequency stretching and recursive convolution terms to improve damping of grazing and evanescent waves, especially in anisotropic media.

Following , the CPML parameters along each Cartesian direction i∈{x,y,z} are defined as 9αi(x)=αmax⁡1-xidm,10si(x)=smax⁡xidn,11κi(x)=1+κmax⁡-1xidn, where d is the CPML thickness, xi≤d is the distance from the inner CPML interface into the layer, m and n are grading exponents, and αmax⁡, smax⁡, and κmax⁡ control the strength and distribution of damping. We can then compute the auxiliary coefficients 12ai(x)=dκid+κiαibi-1,13bi(x)=exp⁡-dκi+αiΔt, to be implemented into recursive convolution memory variable. Large m and n push most attenuation toward the outer edge of the CPML, which is desirable for long wavelengths, whereas smaller exponents concentrate damping closer to the physical domain.

For electromagnetic simulations, relate smax⁡ to the desired reflection coefficient via 14sopt=m+1min⁡(Δxi)μ0ε0,smax⁡=βsopt, where μ0 and ε0 are the permeability and permittivity of free space, respectively, and β is an O(1) tuning factor. Values in the ranges 5≤κmax⁡≤10 and 0.9≤β≤2 typically achieve reflection levels on the order of -60 to -70 dB . further recommend choosing 15αmax⁡=πωc, with ωc=2πfc the dominant angular frequency of the source. In SeidarT, we introduce an additional scaling factor γ and, for the EM case, rescale by ε0 to remain consistent with the use of relative permittivity, giving 16αmax⁡=2πfcγε0.

For seismic simulations, the same functional forms are used for αi and κi, while smax⁡ is adapted to the elastic case by incorporating the reflection coefficient Γ and a quasi–pressure-wave speed ũp : 17smax⁡=-eΓ(m+1)ũp2d. In anisotropic media, ũp is evaluated from the largest diagonal stiffness components (i.e., the dominant “P-like” speed) because the true pressure mode no longer propagates with a single isotropic velocity.

Although SeidarT supplies reasonable default values for αmax⁡, κmax⁡, m, and n, performance can be further tuned. Roughly:

κi≈1 reduces CPML to a classical PML, degrading damping for grazing and low-frequency waves, whereas large κi effectively thickens the layer but may introduce dispersion if chosen too high.

The explicit staggered-grid schemes in SeidarT must satisfy both a Courant–Friedrichs–Lewy (CFL) condition and a wavenumber-bandlimit criterion to ensure stability and accuracy. For a maximum wave speed umax⁡ on a grid with minimum spacing Δxmin⁡ in D dimensions, the CFL condition is 18CFL≤umaxΔtΔx≤1, A standard von Neumann stability analysis of the 2nd-order explicit scheme for the scalar wave equation yields a quadratic constraint on the directional Courant numbers as ∑i=1Dλi≤1 where λi=umaxΔt/Δxi. For a uniform grid with λi=CFL this reduces to λi≤1/D, which is the default choice when initiating a model. The time step is computed from Eq. (), and when more stability is needed or non-uniform grid spacing across axes, the default CFL can be redefined to an appropriate value.

In addition to the CFL limit, the wavenumber-bandlimit criterion 19Δxmax≤umin4fc, is checked to avoid spatial aliasing of the shortest resolved wavelengths where umin is the minimum relevant phase velocity and fc is the central frequency of the source. This condition ensures at least four grid points per smallest propagating wavelength. Violating the CFL condition typically produces severe numerical instability and exponential amplitude growth, whereas violating the bandlimit criterion leads to phase errors, dispersion, and spurious modes. SeidarT checks Eqs. () and () suggests a maximum allowable spatial step. If numerical dispersion or artifacts appear near the high-frequency end of the modeled band, users can either refine the grid (reducing Δx) or decrease the CFL to introduce additional temporal safety margin.

Discretization of the model domain on a rectilinear grid produces staircased artifacts at non-axis-aligned material boundaries, introducing a numerical error that is distinct from, and independent of, a poorly tuned CFL (Eq. ) or wavenumber-bandlimit (Eq. ) criterion. Staircase discretization introduces an error of O(1) concentrated at the boundary that decays exponentially with the number of grid cells away from the interface, in addition to an O(h) phase error in the effective reflection coefficient . This error is most severe for interfaces oriented near 45° to the grid axes and vanishes for grid-aligned boundaries . Consequently, achieving accuracy comparable to a smoothly represented boundary can require grid spacing 2–4 times finer than the wavenumber-bandlimit criterion alone would dictate . A stable solution does not guarantee a geometrically or physically well-posed one, and results should be checked against independent resolution or convergence tests before quantitative interpretation.

Given a specific crystal system (i.e cubic, hexagonal, etc.) many of the terms in Eq. () are zero due to coefficients in C being zero. However, anisotropy, heterogeneities, and preferred crystal orientation fabrics (PCOF) have been observed in glacier systems and are shown to have significant impacts on the flow dynamics of glaciers , therefore we consider the bulk effective medium of snow and ice in our numerical implementation which, as a circumstance, produces stiffness tensors that exhibit lower symmetry.

For reasons of being computationally fast and easy to implement while maintaining an acceptable degree of accuracy , we use the Hill average which is the arithmetic mean from and upper and lower bounds, respectively, in order to calculate the bulk effective medium . For 2nd-order tensor, as seen in the electromagnetic constitutive equations (Eqs.  and ), we can compute the bulk permittivity and conductivity tensors from rotation matrices derived from the Euler angles – ϕ, θ, ψ.

The orientation of a crystal relative to a reference coordinate system can be described by a rotation matrix. A rotation matrix can be formed in a variety of ways (e.g. roll, pitch, yaw), but for simplicity and commonality we can consider the Bunge (z–x–z) Euler angle rotation . For the rotation about each of the three axes for a given angle θ we have 20Rx(θ)=1000cos⁡θ-sin⁡θ0sin⁡θcos⁡θ,Ry(θ)=cos⁡θ0sin⁡θ010-sin⁡θ0cos⁡θRz(θ)=cos⁡θ-sin⁡θ0sin⁡θcos⁡θ0001, therefore, the z–x–z rotator is matrix product of the three rotation matrices of 3 independent angles 21R(ϕ,θ,ψ)=Rz(ϕ)Rx(θ)Rz(ψ). and rotation of a 3×3 tensor follows as 22T′=RTR⊤. Thus we can compute the Voigt and Reuss tensors, TVoigt and TReuss approximations from 23TVoigt=1n∑i=1nνiTi24TReuss-1=1n∑i=1nνiTi-125THill=TVoigt+TReuss2 where ν is the fractional volume of the crystal .

For a 2nd-order tensor such as the permittivity tensor, rotation and bulk effective estimation, Eqs. ()–() can easily be applied. However, rotation of the stiffness tensor requires additional steps to compute bulk effective estimates. For a 6×6 matrix rotation such as those required for the stiffness and compliance tensor, it can be shown that 26Cijkl=RipRjqRkrRlsCpqrs although, for brevity, we will revert to the simplification from who provided a more computationally efficient and concise description of the rotation matrix needed for a 6×6 rotation. The Bond matrix is thus written as 27B=R112R122R1322R12R132R13R112R11R12R212R222R2322R22R232R23R212R21R22R312R322R3322R32R332R33R312R31R32R21R31R22R32R23R33R22R33+R23R32R21R33+R23R31R22R31+R21R32R31R11R32R12R33R13R12R33+R13R32R13R31+R11R33R11R32+R12R31R11R21R12R22R13R23R12R23+R13R22R13R21+R11R23R11R22+R12R21. Likewise, for the rotation of the compliance matrix, as a consequence of the properties of orthogonal matrices, we can avoid computing the inverse and simply apply the transpose inverse bond matrix, N, such that 28S′=NSN⊤ where S=C-1 is the compliance matrix.

Estimating the isotropic tensor from a single anisotropic crystal can follow two approaches. One method involves applying rotations from Euler angles to generate a large population of crystal orientations, which must reach extremely high numbers to effectively converge to the isotropic case. However, this approach is computationally intensive and can be approximated more efficiently. For a transversely isotropic material, the shear and bulk moduli can be extracted directly from the stiffness coefficients. Under the Voigt approximation, the effective shear modulus is simply the arithmetic mean of the shear components c44 and c66, while the effective bulk modulus is given by 29KVoigt=c11+c12+2c13+2c33+2c44/6. Similarly, the Reuss effective shear and bulk moduli can be calculated from the compliance tensor.

For seismic specific modeling, numerical instabilities can occur in regions where there exists high density gradients most commonly observed in the air-surface interface and is due to division by small values. As seen in Eq. () the density moves to the right hand side of the equation and becomes part of the denominator in all of the terms. In purely elastic‐wave simulations, that boundary is usually handled as a traction‐free (free‐surface) condition, i.e. 62σzz=σxz=σyz=0 However, when modeling fluid‐ or air‐filled cavities deeper within the snowpack (for example englacial air pockets, crevasse voids, or infrasound–seismic coupling problems), the density can be sharply reduced locally in places other than the surface.

Whenever the physical wavelength λ is comparable to – or smaller than – the characteristic size of a cavity, homogenization breaks down and an effective medium approximation is no longer valid. To mitigate this, we replace the sharp discontinuity at the solid/fluid (or solid/air) interface with a smoothed transition zone. Specifically, we apply one of three discrete averaging schemes across the grid cells straddling the boundary. These being: the geometric mean 63ρ¯geo=ρ1⋅ρ2, the harmonic mean 64ρ¯harm=2ρ1ρ2ρ1+ρ2, and the arithmetic mean 65ρ¯arithmetic=ρ1+ρ22 which represents the midpoint between the 2 values.

The geometric mean remains closest to the smaller of the two values, helping preserve impedance contrast and energy flux across the boundary, and thus better conserving energy. Likewise, the harmonic mean is biased toward smaller values of modulus or density, effectively weighting low values more heavily. However, unlike the geometric mean, it does not preserve properties such as multiplicative consistency and self-reciprocity, which can lead to violations of energy conservation in this context. It produces a transition that is slightly smoother than the geometric mean while still retaining much of the contrast. In contrast, the arithmetic mean yields the smoothest variation between solid and fluid or air and is the most numerically stable, but it can significantly smear sharp features and does not conserve energy.

These differences reflect a tradeoff between numerical stability and physical accuracy. The choice of averaging scheme should therefore be guided by the modeling objective, particularly whether preserving sharp impedance contrasts is more important than maximizing numerical stability.

To accelerate numerical wave propagation across large spatial domains, OpenMP multi-threading and CUDA GPU offloading are integrated directly into the core time-stepping loops. The subroutines support four compile-time execution modes: single-threaded serial CPU, multi-threaded CPU processing, GPU offloading, and a hybrid CPU–GPU pipeline.

Performance was evaluated using three transversely isotropic ice models with grid resolutions of 250×250 (small), 500×500 (medium), and 1000×1000 (large) nodes. Benchmarks were conducted on a workstation equipped with an AMD Ryzen 7 7700 8-core CPU (8 threads allocated), an NVIDIA GeForce RTX 5060 GPU, and 64 GB of DDR5 RAM running Fedora 44, compiled using gfortran/gcc.

Computational complexity varies significantly by dimensionality and physics kernel. Analytical floating-point operation (FLOP) counts per time step are 51⋅nx⋅nz and 342⋅nx⋅nz for 2D and 2.5D electromagnetic subroutines, and 178⋅nx⋅nz and 1252⋅nx⋅nz for 2D and 2.5D seismic subroutines, respectively. Because grid-based finite-difference updates are primarily memory-bandwidth limited, theoretical FLOP counts do not fully reflect real-world runtime. We therefore quantify domain-independent performance using Million Grid Cell Updates per Second (Mcellss-1) : 66Mcellss-1=nx⋅nz⋅nttcompute×106 where nx and nz are grid spatial dimensions, nt is total time steps, and tcompute is wall-clock execution time spent within the finite-difference solver kernel.

Figure  summarizes execution throughput across backends, grid sizes, and I/O strategies with block outputs of 50 time steps. CPU multi-threading consistently yields peak performance across nearly all grid scales and physics domains, delivering up to 35.7 Mcellss-1 in 2D and 27.5 Mcellss-1 in 2.5D configurations. Serial CPU performance scales competently on smaller domains (250×250) but degrades relative to thread scaling on larger grids. Conversely, GPU offloading exhibits significant latency overhead on small domain sizes, achieving parity or speedups only as grid resolution increases toward 1000×1000. Comparing I/O implementations reveals that Block I/O consistently outperforms Legacy per-channel file writing, reducing write penalties by up to 40 % on bandwidth-constrained runs.

We begin with an electromagnetic zero-offset profile (ZOP) that replicates the ZOP surveys conducted by , verifying that the synthetic arrivals and the signal attenuation in Fig.  are in agreement with the experimental field results.

Gusmeroli et al. (2010) investigated the vertical distribution of water in the polythermal Storglacieren, Sweden, by combining borehole radar measurements with nearby temperature observations. They collected 100 MHz zero-offset profiles between 2 pairs of ∼ 80 m deep hot-water drilled boreholes in the upper ablation area to obtain high-resolution radio-wave speed and amplitude profiles across the ice column. These radar data were corrected for borehole geometry, electronic timing errors, and the influence of air bubbles, then converted to volumetric water content using a dielectric mixing model, yielding nearly uniform temperate-ice water contents of about 0.6 % with localized enhancements at the cold–temperate transition surface (CTS). Thermistor-based estimates at the CTS supported the radar-derived water contents and demonstrated that the temperate layer is several times softer than the overlying cold ice, providing a quantitative hydrothermal structure that is well suited for numerical experiments.

The vertical dielectric permittivity, temperature, and water content profiles provided by are used to build a layered model in which we can simulate the 2D electromagnetic response. Our simulation shows a clear reduction in wave velocity and an increase in attenuation below the CTS, consistent with the observed increase in water content within the temperate layer. The SeidarT derived conductivity estimates were slightly tuned to provide better energy loss behavior in facies with higher water content. We contribute the mismatch in our modeled conductivity values to chemical differences and acidity that is not constrained well enough to be modeled with SeidarT.

The ZOP survey of was conducted in water filled boreholes and their effects were corrected for in their observed travel times. Our initial setup included these water filled boreholes but we observed high levels of ringing within them, so we chose to leave them out. This assumption limits our source accuracy and energy coupling, but our simulated waveform amplitudes and travel times exhibit similar behavior with the field-based measurements.

Our results show very similar results to those provided by , but there is a slight difference in their interpretation and what we see. They attributed the hyperbola seen propagating from the surface (left side) of the image to be due to reflections from a nearby ablation stake. While we do not dispute this, we see a similar hyperbola forming from the reflection off of the air-ice interface even though our model does not contain an ablation stake.

Next, we illustrate a synthetic 2.5D electromagnetic simulation that incorporates three transversely isotropic media with different degrees of anisotropy: strong (66 % – ε∥=10 and ε⟂=2), moderate (33 % – ε∥=4 and ε⟂=2), and near-isotropic (<1%-ε∥=3.205 and ε⟂=3.179) with the latter being SeidarT's intrinsic values for ice. In all cases, the symmetry axis is confined to the x–y plane and is oriented 45° from the x-axis. The x-direction is defined as zero degrees with positive rotation in the counterclockwise direction. The model is energized by an x-polarized point source, and the plane wave energy is measured as a function of the propagation direction. The source to receiver distance is 21.375 m to allow for the development of a polarized plane wave.

The results presented in Fig. () show distinct anisotropy-induced polarization effects because the source energy aligns with the symmetry axes. In the high-anisotropy models, electric field energy is preferentially aligned along the fast and slow axes of the material, demonstrating birefringent splitting and directionally dependent energy maxima. In contrast, the near-isotropic (ice) model exhibits minimal directional dependence, as expected. Some z-polarized leakage is observed across models, particularly in high anisotropy scenarios. This leakage is attributable to the three-dimensional nature of the wavefield and confirms that the solver correctly captures mode conversion and polarization rotation due to anisotropic material properties. The observed numerical dispersion in the high anisotropy model is due to the high contrast in velocities and the difficulty in accommodating the directional dependence into the absorbing boundary.

We validated our seismic wave propagation model using three distinct tests: body-wave imaging, surface wave dispersion analysis, and anisotropy/attenuation assessment. Each test was designed to benchmark the solver against expected physical phenomena in layered or anisotropic elastic media. Table () provides the isotropic velocity model for the surface wave and body wave focused simulations. Each of these models was tuned to present clear modes.