NO. 15 · Seismic Methods

Prepare for Full-Waveform Inversion

The physics, the adjoints, and the judgment FWI actually demands. Build the wave equation and the optimization underneath it, then run the inversion playbook from starting model to a defended result.

You can state the FWI objective, explain where the gradient comes from, choose a starting model and a frequency schedule on purpose, and defend a converged run with honest diagnostics instead of a pretty picture.

23 competencies · 5 interactive widget challenges · 12 to 18 hours of guided study

For geophysicists heading into velocity model building or FWI teams

Draws on: A Mathematician's Survey Physics-Informed Neural Networks for Exploration Seismology Seismic Data Processing

The mathematics FWI stands on

Derivatives and integrals, sharpened

FWI is one long derivative chain, and the fundamental theorem of calculus is the receipt the whole method runs on.

Reads: The Derivative The Fundamental Theorem of Calculus

Gradients and Jacobians

The model update is a gradient and nothing more; the Jacobian is what prices each parameter's influence on the data.

Reads: Vector-Valued and Multivariable Functions The Total Derivative as a Jacobian

Linear systems, rank, and eigenvalues

Null spaces and conditioning decide what your data can resolve and what it never will, before any inversion is run.

Reads: Linear Maps and Matrix Representations Rank-Nullity and Conditions for Invertibility Eigenvalues, Eigenvectors, and Diagonalization

Least squares and gradient descentwidget challenge

Every FWI update is a step downhill on a misfit surface; learn the step, the direction, and the ways both go wrong.

Reads: Linear algebra primer Optimization basics

Fourier, the inversion scheduler

FWI schedules itself in frequency bands from low to high; the Fourier transform is the scheduler you must read fluently.

Reads: The Fourier Transform on the Real Line The Fourier transform, slowly

Wave physics

The wave equation as mathematics

This is the PDE the whole method fits to data; meet it with clean notation before the field units arrive.

Reads: What Is a Differential Equation? The Wave Equation

The wave equation, field edition

The same law in seismic units and seismic language, thirty minutes from derivation to shot gather.

Reads: Wave equation in 30 minutes

Time domain and Helmholtz

Time stepping or frequency domain is the first modeling decision every FWI project makes; know what each choice buys and costs.

Reads: The 1D acoustic wave equation The 2D acoustic wave equation Frequency-domain (Helmholtz) formulation

Boundaries and absorbers

Reflections off the edge of your grid look exactly like geology; absorbing conditions and PMLs are how you keep the lie out of the wavefield.

Reads: Absorbing boundary conditions (ABC) Perfectly matched layers (PML)

The processing context

Velocities before FWI

FWI starts where semblance and tomography stop; know exactly what the conventional flow already bought you.

Reads: Stacking, RMS, and interval velocities Velocity picking on semblance gathers Tomographic velocity inversion

Migration and RTM

RTM is FWI's forward-and-adjoint engine wearing an imaging hat; learn to read its artifacts here, where they are cheap.

Reads: Pre-stack depth migration: Kirchhoff Reverse-Time Migration (RTM) Migration artifacts & QC

Velocity model building

The starting model decides whether FWI converges or cycle skips; building one is a craft with rules, not a download.

Reads: Velocity model building for migration

Classical FWI

The inverse problem, stated

Misfit, gradient, regularization: FWI written as mathematics on one page, which is how you will be asked about it.

Reads: The inverse problem, mathematically

FWI in practicewidget challenge

Low frequencies first, encoded sources when compute bites: the strategy layer that separates a run that converges from a run that burns a cluster.

Reads: FWI in practice: low-frequency strategies Encoded FWI & computational strategies

Elastic and anisotropic FWI

Real rocks shear and real basins are anisotropic; know when the acoustic shortcut stops being a shortcut and starts being wrong.

Reads: Elastic and anisotropic FWI

FWI quality control

Synthetic-versus-recorded overlays are the lie detector; run them before the project review runs them on you.

Reads: FWI QC: synthetic-vs-recorded matching

Learned wavefields

The PINN idea

Put the PDE residual into the loss and a network becomes a solver; the forward-versus-inverse hinge is where FWI walks in.

Reads: The PINN idea: PDE residual as a loss term The forward problem vs the inverse problem

PINN-FWI, first contactwidget challenge

The classical loss and the PINN loss side by side, then a full 1D velocity inversion you can hold in your head end to end.

Reads: The classical FWI loss vs the PINN-FWI loss 1D velocity inversion end-to-end

Marmousi and frequency continuationwidget challenge

Multiscale continuation is the anti-cycle-skip discipline, and Marmousi is where every inversion idea proves it can leave the whiteboard.

Reads: 2D Marmousi-class inversion: setup Multi-scale frequency continuation

Cycle skipping and multiparameter trade-offs

Cycle skipping is FWI's signature failure and multiparameter crosstalk is its quiet one; learn to detect both before trusting any model.

Reads: Cycle skipping: detection and remedies Multi-parameter inversion (vp, vs, rho)

Defending an inversion run

Convergence diagnostics and hybrid PINN-plus-classical options are how you argue a velocity model is real to people paid to doubt it.

Reads: Defending an inversion run: convergence diagnostics PINN-augmented classical FWI

Capstone

Capstone: Marmousi inversionwidget challenge

The field-standard testbed end to end, from starting model to defended result; if it works here you have earned the acronym.

Reads: Marmousi velocity inversion (the field-standard testbed)

The master workflow card

The entire preparation compressed onto one card you can carry into an FWI kickoff meeting.

Reads: Master PINN-for-seismology workflow card