Fermat’s principle

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Seismic Data Analysis
Series Investigations in Geophysics
Author Öz Yilmaz
DOI http://dx.doi.org/10.1190/1.9781560801580
ISBN ISBN 978-1-56080-094-1
Store SEG Online Store

The traveltime along a raypath from one point to another has an extremum value which, for most physical problems, is a minimum. This is the formal statement of Fermat’s principle [1]. The raypath also defines the direction of energy flow. Among a bundle of rays from one point to another, Fermat’s principle can be applied to discard all but one raypath that corresponds to a minimum time of travel from one point to the other. This practical concept can be used to perform traveltime computations for prestack depth migration [2] [3].

Figure 8.5-9  A sketch that illustrates traveltime calculation using Fermat’s principle. See text for details.

Consider the raypath geometry depicted in Figure 8.5-9. The velocity-depth model may be split into a set of horizontal slabs with a specified thickness, say 50 to 300 m. Assume that traveltimes from a source at the surface z = 0 to a depth z through a velocity-depth model already have been computed. Now compute the traveltime from the grid point 6 at depth z to each of the grid points at depth z + Δz within a specified aperture, say grid points 1 to 11. An average velocity between the grid points at depth z and z + Δz along each of the raypaths may be used in the computation. Among the 11 raypaths from the grid point 6 at depth z to grid points 1 to 11 at depth z + Δz, choose that which corresponds to the minimum traveltime.

The process may be continued for all grid points at depth z, and then from one depth to the next, and the traveltimes associated with the minimum-traveltime raypaths through each of the horizontal slabs may be added to compute the total traveltime from a source or receiver point at the surface z = 0 to a reflection point at some depth in the subsurface.


  1. Officer, 1958, Officer, C. B., 1958, Introduction to the theory of sound transmission: McGraw-Hill Book Co.
  2. Meshbey et al., 1993, Meshbey, V., Kosloff, D., Ragoza, Y., Meshbey, O., Egozy, U., and Cozzens, J., 1993, A method for computing traveltimes for an arbitrary velocity model: Presented at the 45th Ann. EAGE Mtg.
  3. Vesnaver, 1996, Vesnaver, A., 1996, Ray tracing based on Fermat’s principle in irregular grids: Geophys. Prosp., 44, 741–760.

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