# The eikonal equation

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

Consider a plane wave function P(x, y, z; t) with a spatially varying amplitude P0(x, y, z) and spatially varying traveltime T(x, y, z) (Section H.2)

 ${\displaystyle P(x,y,z;t)=P_{0}(x,y,z)\exp {\Big \{}-i\omega {\Big [}t-T(x,y,z){\Big ]}{\Big \}}.}$ (3)

Assume for the moment that the wave amplitude P0(x, y, z) does not vary spatially, but is a constant. Then, the plane-wave solution of the form defined by equation (8-3) satisfies the scalar wave equation (8-1). Substitution of equation (8-3) into equation (8-1) yields

 ${\displaystyle {\frac {\partial ^{2}P}{\partial x^{2}}}+{\frac {\partial ^{2}P}{\partial y^{2}}}+{\frac {\partial ^{2}P}{\partial z^{2}}}={\frac {1}{v^{2}(x,y,z)}}{\frac {\partial ^{2}P}{\partial t^{2}}}}$ (1)

 ${\displaystyle {\left({\frac {\partial T}{\partial x}}\right)}^{2}+{\left({\frac {\partial T}{\partial y}}\right)}^{2}+{\left({\frac {\partial T}{\partial z}}\right)}^{2}={\frac {1}{v^{2}(x,y,z)}},}$ (4)

which is called the eikonal equation. It is a ray-theoretical approximation to the scalar wave equation (8-1). Derivation of equation (8-4) is provided in Section H.2.

While the solution of the scalar wave equation (8-1) represents the wavefield P(x, y, z; t) at a point in space (x, y, z) and at an instant of time t, the solution of the eikonal equation (8-4) represents the traveltime T(x, y, z) for a ray passing through a point (x, y, z) in a medium with velocity v(x, y, z). Specifically, T(x, y, z) = constant represents the wavefront of constant phase at an instant of time. The wave is propagated from one wavefront to the next by way of raypaths, which are perpendicular to the wavefronts.

When the medium velocity is not constant but is an arbitrary function of space variables v(x, y, z), and the wave amplitude P0(x, y, z) is not constant but also varies spatially, then the traveltime function T(x, y, z) of equation (8-3) is not a solution to the eikonal equation (8-4). For a wave function, such as that given by equation (8-3), with spatially varying amplitudes, the eikonal equation is a good approximation to the wave equation only at a high-frequency limit (Section H.2). The high-frequency limit is equivalent to small wavelengths. How small should the wavelength be for a solution of the eikonal equation to be a good approximation to the wave equation? The approximation is valid if the fractional change in the velocity gradient is much less than the wave frequency [1]. In practice, this means that the eikonal equation can be used to compute traveltimes if the velocity-depth model does not contain large velocity gradients.

Just as a solution to the scalar wave equation can be formulated using a finite-difference scheme (migration principles), the eikonal equation (8-4) also can be solved using the finite-difference technique (Section H.3). Consider the 3-D computation mesh sketched in Figure H-2b. We want to compute the traveltime T of the eikonal equation (H-27) at grid point (x + Δx, y, z + Δz) using the known traveltimes at grid points (x, y, z), (x + Δx, y, z), and (x + Δx, y + Δy, z).

Computing the traveltimes at depth z + Δz from those at depth z means extrapolating T in the z-direction. Rewrite equation (8-4) in the form of an extrapolation equation as [2]

 ${\displaystyle v{\frac {\partial T}{\partial z}}={\sqrt {1-\left(v{\frac {\partial T}{\partial x}}\right)^{2}-\left(v{\frac {\partial T}{\partial y}}\right)^{2}}}.}$ (5)

As shown in Section H.3, the 3-D eikonal equation (8-5) can be solved for the traveltime values T(x, y, z) for the propagating wavefront through a velocity field v(x, y, z) in the subsurface using a finite-difference scheme.

Figure 8.5-4 shows the traveltime contours through the same velocity-depth model as in Figure 8.5-3 computed by using the eikonal equation. These may be considered as wavefronts expanding from a source at the surface located at 3000 m from the origin. Note that the shape of the traveltime contours honors the boundaries with velocity contrast. Nevertheless, the traveltime contours do not show the abrupt changes as would be the case at a boundary such as the salt-sediment interface with sharp velocity contrast.

To explain the smooth behavior of the traveltime contours at layer boundaries in Figure 8.5-4, refer to the sketch in Figure 8.5-5 of an expanding wavefront through a flat interface with a large velocity contrast. At the critical angle of refraction, waves travel along the layer boundary with the faster velocity of the underlying layer, Eventually, these waves are refracted back into the overlying layer and are recorded in the form of first arrivals; thus, they often are called head waves. Note that the wavefront associated with the head wave tends to smooth out the sharp change in the expanding wavefront as it crosses over the layer boundary with velocity contrast. It is this effect that we see in Figure 8.5-4 where the traveltime contours make a smooth transition from the sediment to the salt layer.

Ideally, the traveltime contours should pronounce the sharp layer boundaries. While some solutions to the eikonal equation include the head waves [4] [5] as shown in Figure 8.5-4, others exclude the head wave [2] [6]. Figure 8.5-6 shows the traveltime contours derived from the solution of the eikonal equation that has excluded the head wave. Note the sharp changes in the traveltime contours, which at some locations appear to coincide nicely with the salt-sediment layer boundary. Nevertheless, the match is not consistent at all locations and the traveltime contours show rapid fluctuations that are not physically plausable.

Figure 8.5-7a shows traveltime contours that were computed using a finite-difference solution to the eikonal equation. The velocity-depth model consists of layers with velocities that vary from 1500 m/s to 4000 m/s. Note that the eikonal solution, although it is associated only with the fastest arrival, always yields a raypath from a source at the surface to a point in the subsurface through the gridded velocity-depth model. Some of the raypaths are indicated by trajectories that have a solid circle at the end. The smooth behavior of the traveltime contours is associated with the head wave that often is the first arrival.

In contrast with the eikonal solution, the wavefront construction yields multiple arrivals as shown in Figure 8.5-7b. Again, some of the raypaths are indicated by trajectories that have a solid circle at the end. Note the multiple raypaths associated with a single point in the subsurface. Where a head wave or diffraction develops, the wavefront construction leaves a gap in the traveltime contour. With wavefront construction, however, you can be selective in the type of rays that you would like to include in the summation of amplitudes. Figure 8.5-7c shows traveltime contours associated with the reflecting boundary represented by the thick curve. In fact, the solution from wavefront construction in this case includes arrivals associated with reflected waves as well as transmitted waves.

The eikonal solution does not necessarily yield the maximum energy along the single-arrival raypath. In wavefront construction, on the other hand, amplitudes associated with multiple arrivals can be included in the summation. Thus, the chance of capturing the maximum-energy arrival increases, albeit at an increase in computational cost. By including multiple arrivals in the summation, the likelihood of attaining a complete image of a complex structure also is higher. Figure 8.5-8a shows raypaths from a single source at the surface that illuminate portions of the steep flank of a salt dome. By using the arrivals associated with these raypaths, only, in Kirchhoff summation, the resulting partial image shown in Figure 8.5-8b is obtained. By including in the summation the arrivals associated with the reflections off the salt flank as well as the arrivals associated with the reflections from a deeper interface that bounce off the salt flank (Figure 8.5-8c), a more complete image can be obtained (Figure 8.5-8d).

## References

1. Officer, 1958, Officer, C. B., 1958, Introduction to the theory of sound transmission: McGraw-Hill Book Co.
2. Reshef and Kosloff, 1986, Reshef, M. and Kosloff, D., 1986, Migration of common-shot gathers: Geophysics, 51, 324–331.
3. Nichols, 1996, Nichols, D. E., 1996, Maximum-energy traveltimes calculated in the seismic frequency band: Geophysics, 61, 253–263.
4. Vidale, 1988, Vidale, J., 1988, Finite-difference calculation of traveltimes: Bull. Seis. Soc. Am., 78, 2026–2076.
5. Podvin and Lecomte, 1991, Podvin, P. and Lecomte, I., 1991, Finite-difference computation of traveltimes for velocity-depth models with strong velocity contrast across layer boundaries — a massively parallel approach: Geophys. J. Int., 105, 271–284.
6. Reshef, 1991, Reshef, M., 1991, Prestack depth imaging of three-dimensional shot gathers: Geophysics, 56, 1158–1163.
7. Lecomte, 1999, Lecomte, I., 1999, Local and controlled prestack depth migration in complex areas: Geophys. Prosp., 47, 799–818.