Math 2.0

Geophysics Definition

Travel-time tomography

Builds on Seismic tomographyFermat's principleEikonal equationRadon transform

Definition

Let v>0v > 0 be the wave speed on a region Ω\Omega and let u=1/vu = 1/v be the slowness. For a source ss and a receiver rr, the travel time is t(s,r)=∫γu dℓ.t(s, r) = \int_{\gamma} u \, d\ell. Here dℓd\ell is arc length, and γ\gamma is the ray from ss to rr, a path that makes the travel time stationary by Fermat's principle. Travel-time tomography is the seismic tomography that estimates uu from the observed travel times tit_i of many source-receiver pairs. In linearised form, for a reference slowness u0u_0 with rays γi0\gamma_i^0, it solves the following equations for the slowness perturbation δu=u−u0\delta u = u - u_0: ti−ti0=∫γi0δu dℓ,i=1,…,N.t_i - t_i^0 = \int_{\gamma_i^0} \delta u \, d\ell, \qquad i = 1, \dots, N. Here ti0t_i^0 are the travel times in u0u_0.

Remarks

The linearisation keeps the reference rays, because by Fermat's principle the first-order change of the path does not change the time. Example: two unit square cells side by side with slownesses u1,u2u_1, u_2; a horizontal ray through both has time u1+u2u_1 + u_2, and a vertical ray through cell 11 has time u1u_1. Times t1=1.2t_1 = 1.2 and t2=0.5t_2 = 0.5 give u1=0.5u_1 = 0.5 and u2=0.7u_2 = 0.7, so both cells are determined. Non-example of a well-posed set-up: with only the horizontal ray, only the sum u1+u2u_1 + u_2 is known, so the data do not determine the model. For a constant reference speed the rays are straight lines, and the data are samples of the Radon transform (in two dimensions) of δu\delta u. Travel times in a general medium solve the eikonal equation ∣∇t∣=u\abs{\nabla t} = u, which is used to compute rays and times.

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Calculus of variations, concept, no evidence levelCalculus of variationsconceptEikonal equation, concept, no evidence levelEikonal equationconceptEuler–Lagrange equation, concept, no evidence levelEuler–Lagrange equati…conceptGeodesic, concept, no evidence levelGeodesicconceptRadon transform, concept, no evidence levelRadon transformconceptSeismic tomography, concept, no evidence levelSeismic tomographyconceptFermat's principle, statement, no evidence levelFermat's principlestatementTravel-time tomography, definition, no evidence levelTravel-time tomographydefinition
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  1. Calculus of variations concept
  2. Eikonal equation concept
  3. Euler–Lagrange equation concept
  4. Geodesic concept
  5. Radon transform concept
  6. Seismic tomography concept
  7. Fermat's principle statement
  8. Travel-time tomography definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Seismic_tomography
About this page
TagJ9W1CBXYS9
URIm2://J9W1CBXYS9
Kinddefinition
WritingW0 stub
MSC86A22, 86A15, 44A12
DefinesTravel-time tomography
UsesSeismic tomography, Fermat's principle, Eikonal equation, Radon transform
Changed3 November 2026

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