# Difference between revisions of "Fermat’s principle"

m (removed internal link) |
(added image) |
||

Line 16: | Line 16: | ||

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 <ref name=ch08r33>Officer, 1958, Officer, C. B., 1958, Introduction to the theory of sound transmission: McGraw-Hill Book Co.</ref>. 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]] <ref name=ch08r28>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.</ref> <ref name=ch08r41>Vesnaver, 1996, Vesnaver, A., 1996, Ray tracing based on Fermat’s principle in irregular grids: Geophys. Prosp., 44, 741–760.</ref>. | 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 <ref name=ch08r33>Officer, 1958, Officer, C. B., 1958, Introduction to the theory of sound transmission: McGraw-Hill Book Co.</ref>. 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]] <ref name=ch08r28>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.</ref> <ref name=ch08r41>Vesnaver, 1996, Vesnaver, A., 1996, Ray tracing based on Fermat’s principle in irregular grids: Geophys. Prosp., 44, 741–760.</ref>. | ||

+ | |||

+ | [[file:ch08_fig5-9.png|thumb|left|{{figure number|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'' + Δ''z'' within a specified aperture, say grid points 1 to 11. An average velocity between the grid points at depth ''z'' and ''z'' + Δ''z'' 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'' + Δ''z'', choose that which corresponds to the minimum traveltime. | 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'' + Δ''z'' within a specified aperture, say grid points 1 to 11. An average velocity between the grid points at depth ''z'' and ''z'' + Δ''z'' 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'' + Δ''z'', choose that which corresponds to the minimum traveltime. |

## Latest revision as of 07:05, 2 October 2014

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]}.

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* + Δ*z* within a specified aperture, say grid points 1 to 11. An average velocity between the grid points at depth *z* and *z* + Δ*z* 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* + Δ*z*, 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.

## References

- ↑ Officer, 1958, Officer, C. B., 1958, Introduction to the theory of sound transmission: McGraw-Hill Book Co.
- ↑ 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.
- ↑ Vesnaver, 1996, Vesnaver, A., 1996, Ray tracing based on Fermat’s principle in irregular grids: Geophys. Prosp., 44, 741–760.

## See also

- 3-D prestack depth migration
- Kirchhoff summation
- Calculation of traveltimes
- The eikonal equation
- Summation strategies
- Migration aperture
- Operator antialiasing
- 3-D common-offset depth migration

## External links