Analytical Tools in Seismic Waveform Inversion

1. What Is Waveform Inversion?

Waveform inversion is a powerful method in seismology used to determine earthquake source parameters and Earth structure by comparing observed seismograms to synthetic waveforms. The goal is to adjust model parameters so that synthetic waveforms generated from a forward model closely match recorded data.

Mathematically:

d(t) = G(m, t) + ε(t)

Where:
- d(t) = observed seismogram
- G(m, t) = synthetic waveform generated from model m
- ε(t) = noise or residual error

This is a nonlinear inverse problem that requires solving the forward wave equation and iteratively minimizing the misfit between data and model.

2. Forward Modeling

The forward model simulates seismic wave propagation through the Earth based on an assumed source and medium. It solves the elastodynamic wave equation:

ρ ∂²u/∂t² = ∇·σ + f

Where:
- u = displacement
- σ = stress tensor
- f = body force (source term)

Numerical methods include: - Finite Difference Time Domain (FDTD) - Spectral Element Methods (SEM) (e.g., SPECFEM3D) - Normal Mode Summation (for long-period waves) - WKBJ Approximation (for layered media)

These methods compute synthetic seismograms from assumed fault and Earth structure models.

3. Inversion Algorithms

The inversion minimizes a misfit function such as:

Φ(m) = ½ Σ || dᵢ − Gᵢ(m) ||²

Common algorithms:

Gradient-Based (Linearized)

  • Least squares
  • Gauss-Newton
  • Conjugate gradient

Useful when the model can be linearized and a good initial guess is available.

Global/Nonlinear

  • Simulated annealing
  • Genetic algorithms
  • Monte Carlo methods
  • Bayesian inversion (e.g., MCMC)

Used when the problem is complex, with many local minima.

4. Types of Models Inferred

Focal Mechanism / Moment Tensor Inversion

  • Estimates fault orientation (strike, dip, rake) and type of slip
  • Uses first-motion polarities and full waveform fits

Finite Fault Inversion

  • Estimates spatial and temporal slip on a fault surface
  • Requires dense azimuthal station coverage

Full-Waveform Tomography

  • Inverts for Earth’s velocity structure (Vp, Vs, density)
  • Requires 3D modeling and significant computational resources

5. Key Tools & Software

Tool Purpose
SPECFEM3D 3D waveform modeling
ISOLA Moment tensor inversion
MTinv Focal mechanism inversion
SES3D Full-waveform forward modeling
ObsPy Python library for waveform processing
PyFK Frequency-wavenumber synthetic modeling

These tools support both forward modeling and inversion workflows.

6. Observable Features Matched

Seismologists fit various features of the waveform: - Amplitude and phase - P- and S-wave arrivals - Surface wave dispersion - Waveform polarity and coda

Matching can be performed in: - Time domain - Frequency domain - Hybrid metrics (e.g., cross-correlation, envelope fitting)

7. Summary

Waveform inversion enables: - Accurate recovery of earthquake source parameters (moment tensor, rupture slip) - High-resolution imaging of Earth’s interior - Quantitative uncertainty estimation

It relies on a combination of: - High-fidelity seismograph data - Physics-based forward models - Numerical optimization and statistical inference