SPECFEM: Spectral-Finite-Element Method

Introduction

The Spectral Element Method (SPECFEM) is a numerical modeling technique used for simulating wave propagation in complex media. It combines the accuracy of spectral methods with the geometric flexibility of finite elements, making it ideal for geophysical applications such as seismic wave simulations, gravity and magnetic modeling, and inversion processes.

This blog provides an overview of forward and inverse modeling using SPECFEM and highlights its application in solving Poisson’s and Laplace equations for subsurface imaging.

1. Forward Modeling with SPECFEM

Forward modeling calculates how waves propagate through a given medium based on its physical properties.

Wave Equation in Elastic Media

The fundamental wave equation is expressed as:

[ \rho \frac{\partial^2 u}{\partial t^2} = \nabla \cdot \sigma + f ]

where:

• ( \rho ): Density.
• ( u ): Displacement vector.
• ( \sigma ): Stress tensor.
• ( f ): External force or source term.

Discretization and Element Assembly

SPECFEM divides the domain into spectral elements, allowing accurate approximation of wavefields within each element. The displacement field is expanded in terms of Legendre polynomials, enabling high-order accuracy.

The final system of equations is solved using explicit time integration:

[ M \frac{\partial^2 U}{\partial t^2} + K U = F ]

where:

• ( M ): Mass matrix.
• ( K ): Stiffness matrix.
• ( F ): Force vector.

2. Inverse Modeling with SPECFEM

Inverse modeling estimates subsurface parameters (e.g., density, velocity) by minimizing the difference between observed and simulated data.

Formulation of the Inverse Problem

The inverse problem minimizes the objective function:

[ \Phi = \Phi_d + \alpha \Phi_m ]

where:

• ( \Phi_d ): Data misfit (observed vs. predicted).
• ( \Phi_m ): Model regularization to enforce smoothness or compactness.
• ( \alpha ): Weighting parameter controlling trade-off.

The misfit is calculated as:

[ \Phi_d = \frac{1}{2} \sum_{i=1}^N (d_i^{obs} - d_i^{calc})^2 ]

Adjoint Method for Gradient Calculation

SPECFEM uses the adjoint method to efficiently compute gradients for updating model parameters.

1. Forward Simulation: Compute the wavefield using the initial model.
2. Adjoint Simulation: Propagate residuals backward to determine sensitivity.
3. Model Update: Adjust parameters using gradients:

[ m^{k+1} = m^k - \gamma \nabla \Phi ]

where ( \gamma ) is the step size.

3. Applications of SPECFEM

• Seismic Wave Propagation: Models earthquake waveforms and subsurface responses.
• Gravity and Magnetic Fields: Solves Poisson’s and Laplace equations to model subsurface density and magnetization.
• Structural Imaging: Detects faults, fractures, and ore bodies.
• Reservoir Characterization: Evaluates properties of hydrocarbon reservoirs.

4. Key Features of SPECFEM

• High Accuracy: Uses spectral interpolation for smooth approximations.
• Geometric Flexibility: Models complex boundaries and topographies.
• Parallel Processing: Supports MPI and GPU acceleration for large-scale simulations.
• Adaptive Mesh Refinement: Focuses computational effort on areas of interest.

5. Example: Gravity Inversion Using SPECFEM

Poisson’s equation models gravitational potential:

[ \nabla^2 \phi = -4 \pi G \rho ]

where:

• ( \phi ): Gravitational potential.
• ( G ): Gravitational constant.
• ( \rho ): Density distribution.

Forward Modeling computes the gravity response for a given density model.

Inverse Modeling minimizes the data misfit to recover the density distribution:

[ \Phi = |d_{obs} - d_{calc}|^2 + \alpha |m| ]

The solution is refined iteratively using the adjoint method, improving model accuracy while reducing computational costs.

6. Why Use SPECFEM?

SPECFEM offers a versatile framework for solving geophysical problems, particularly in:

• Simulating seismic wave propagation through heterogeneous media.
• Modeling magnetic and gravity anomalies for subsurface exploration.
• Combining forward and inverse modeling to refine geological interpretations.

Conclusion

SPECFEM’s combination of spectral accuracy and finite element flexibility makes it a powerful tool for geophysical simulations. Whether modeling seismic waves or analyzing potential field data, its ability to handle complex geometries and large datasets ensures reliable results.

For more information, download example scripts and data processing tools here.