Math 2.0

Geophysics Definition

Full waveform inversion

Builds on Seismic tomographyWave equationAdjoint state methodLeast squares

Definition

Let Ω⊂ℝn\Omega \subset \R^n be a region and let mm be a model of the medium on Ω\Omega, for example m=1/c2m = 1/c^2 for a sound speed c>0c > 0. For each source ss, let us(x,t;m)u_s(x, t; m) solve the wave equation m(x) ∂t2us−Δus=fs,us=∂tus=0 at t=0,m(x)\, \partial_t^2 u_s - \Delta u_s = f_s, \qquad u_s = \partial_t u_s = 0 \text{ at } t = 0, where fsf_s is the source term and Δ\Delta is the Laplacian. Let ds,r(t)d_{s,r}(t) be the seismogram recorded at receiver xrx_r for t∈[0,T]t \in [0, T]. Full waveform inversion is the seismic tomography that estimates mm by minimising the least squares waveform misfit J(m)=12∑s,r∫0T(us(xr,t;m)−ds,r(t))2 dt.J(m) = \frac{1}{2} \sum_{s, r} \int_0^T \bigl( u_s(x_r, t; m) - d_{s,r}(t) \bigr)^2 \, dt. The minimisation, usually with a regularisation term, uses local iterative methods from an initial model m0m_0.

Remarks

The gradient of JJ is computed by the adjoint state method. For each source, an adjoint wavefield is driven by the data residuals at the receivers and propagated backward in time. The gradient at xx is a time integral of the product of the adjoint field with ∂t2us\partial_t^2 u_s at xx. So one gradient costs about two wave simulations per source, whatever the number of model parameters. Example of the main difficulty: take a single arrival w(t−τ)w(t - \tau) with w(t)=cos⁡(ωt)w(t) = \cos(\omega t) on a long window. The misfit against the true arrival at τ=0\tau = 0 behaves like 1−cos⁡(ωτ)1 - \cos(\omega \tau), with minima at every τ=2πk/ω\tau = 2\pi k/\omega. A starting model with a time error larger than half a period converges to a wrong minimum; this is called cycle skipping. Non-example: travel-time tomography uses only the arrival times extracted from the records, not the full traces, so it is not full waveform inversion. Elastic, viscoelastic and anisotropic versions replace the acoustic wave equation by the corresponding systems.

Reading path

To understand this page from the ground up, read these 23 pages in order. The foundations come first.

Show the path as a graph
Adjoint state method, concept, no evidence levelAdjoint state methodconceptMatrix, definition, no evidence levelMatrixdefinitionSeismic tomography, concept, no evidence levelSeismic tomographyconceptVector space, definition, no evidence levelVector spacedefinitionWave equation, concept, no evidence levelWave equationconceptTranspose, definition, no evidence levelTransposedefinitionInner product, definition, no evidence levelInner productdefinitionLinear map, definition, no evidence levelLinear mapdefinitionLinear subspace, definition, no evidence levelLinear subspacedefinitionNorm, definition, no evidence levelNormdefinitionInner product space, definition, no evidence levelInner product spacedefinitionOrthogonal vectors, definition, no evidence levelOrthogonal vectorsdefinitionEndomorphism, definition, no evidence levelEndomorphismdefinitionSum of subspaces, definition, no evidence levelSum of subspacesdefinitionNorm induced by an inner product, definition, no evidence levelNorm induced by an in…definitionOrthogonal complement, definition, no evidence levelOrthogonal complementdefinitionDirect sum of subspaces, definition, no evidence levelDirect sum of subspac…definitionA space is the direct sum of a subspace and its orthogonal complement, statement, no evidence levelA space is the direct…statementProjection, definition, no evidence levelProjectiondefinitionOrthogonal projection, definition, no evidence levelOrthogonal projectiondefinitionOrthogonal projection is the closest point, statement, no evidence levelOrthogonal projection…statementLeast squares, concept, no evidence levelLeast squaresconceptFull waveform inversion, definition, no evidence levelFull waveform inversi…definition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Adjoint state method concept
  2. Matrix definition
  3. Seismic tomography concept
  4. Vector space definition
  5. Wave equation concept
  6. Transpose definition
  7. Inner product definition
  8. Linear map definition
  9. Linear subspace definition
  10. Norm definition
  11. Inner product space definition
  12. Orthogonal vectors definition
  13. Endomorphism definition
  14. Sum of subspaces definition
  15. Norm induced by an inner product definition
  16. Orthogonal complement definition
  17. Direct sum of subspaces definition
  18. A space is the direct sum of a subspace and its orthogonal complement statement
  19. Projection definition
  20. Orthogonal projection definition
  21. Orthogonal projection is the closest point statement
  22. Least squares concept
  23. Full waveform inversion definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Seismic_inversion
About this page
Tag26QCWECTG1
URIm2://26QCWECTG1
Kinddefinition
WritingW0 stub
MSC86A22, 35R30, 35L05
DefinesFull waveform inversion
UsesSeismic tomography, Wave equation, Adjoint state method, Least squares
Changed3 November 2026

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