Traveltime computation with the linearized eikonal equation

Bibliography

Alkhalifah, T., 1997, Acoustic approximations for processing in transversely isotropic media: submitted to Geophysics.

Biondi, B., 1992, Solving the frequency-dependent eikonal equation: 62nd Ann. Internat. Mtg, Soc. of Expl. Geophys., 1315-1319.

Fowler, P. J., 1994, Finite-difference solutions of the 3-D eikonal equation in spherical coordinates: 64th Ann. Internat. Mtg, Soc. of Expl. Geophys., 1394-1397.

Gray, S. H., and W. P. May, 1994, Kirchhoff migration using eikonal equation traveltimes: Geophysics, 59, 810-817.

Lavrentiev, M. M., V. G. Romanov, and V. G. Vasiliev, 1970, Multidimensional inverse problems for differential equations: Springer-Verlag, volume  167 of Lecture Notes in Mathematics.

Nichols, D. E., 1994, Imaging complex structures using band-limited Green's functions: PhD thesis, Stanford University.

Popovici, M., 1991, Finite difference travel time maps, in SEP-70: Stanford Exploration Project, 245-256.

Schneider, W. A., 1995, Robust and efficient upwind finite-difference traveltime calculations in three dimensions: Geophysics, 60, 1108-1117.

van Trier, J., and W. W. Symes, 1991, Upwind finite-difference calculation of traveltimes: Geophysics, 56, 812-821.

Cerveny, V., I. A. Molotkov, and I. Pšencik., 1977, Ray method in seismology: Univerzita Karlova.

Vidale, J. E., 1990, Finite-difference calculation of traveltimes in three dimensions: Geophysics, 55, 521-526.

Appendix A

A SIMPLE derivation of the eikonal and transport equations

In this Appendix, I remind the reader how the eikonal equation is derived from the wave equation. The derivation is classic and can be found in many popular textbooks. See, for example, (Cerveny et al., 1977).

Starting from the wave equation,

$${\frac{\partial^2 P}{\partial x^2}} + {\frac{\partial^2 P}... ...z^2}} = {n^2 (x,y,z) \frac{\partial^2 P}{\partial t^2}}\;,$$ (8)

we introduce a trial solution of the form

$$P (x,y,t) = A (x,y,z) f (t - \tau (x,y,z))\;,$$ (9)

where $\tau$ is the eikonal, and $A$ is the wave amplitude. The waveform function $f$ is assumed to be a high frequency (discontinuous) signal. Substituting solution (F-2) into equation (F-1), we arrive at the constraint

$$\Delta A f - 2 \nabla A \cdot \nabla \tau f' - A \Delta \tau f ' + A \left(\nabla \tau\right)^2 f'' = n^2 A f''\;.$$ (10)

Here $\Delta \equiv \nabla^2$ denotes the Laplacian operator. Equation (F-3) is as exact as the initial wave equation (F-1) and generally difficult to satisfy. However, we can try to satisfy it asymptotically, considering each of the high-frequency asymptotic components separately. The leading-order component corresponds to the second derivative of the wavelet $f''$. Isolating this component, we find that it is satisfied if and only if the traveltime function $\tau (x,y,z)$ satisfies the eikonal equation (1).

The next asymptotic order corresponds to the first derivative $f'$. It leads to the amplitude transport equation

$$2 \nabla A \cdot \nabla \tau + A \Delta \tau = 0\;.$$ (11)

The amplitude, defined by equation (F-4), is often referred to as the amplitude of the zero-order term in the ray series. A series expansion of the function $f$ in high-frequency asymptotic components produces recursive differential equations for the terms of higher order. In practice, equation (F-4) is sufficiently accurate for describing the major amplitude trends in most of the cases. It fails, however, in some special cases, such as caustics and diffraction.

Appendix B

CONNECTION OF THE LINEARIZED EIKONAL EQUATION AND TRAVELTIME TOMOGRAPHY

The eikonal equation (1) can be rewritten in the form

$$\mathbf{n} \cdot \nabla \tau = n \;,$$ (12)

where $\mathbf{n}$ is the unit vector, pointing in the traveltime gradient direction. The integral solution of equation (G-1) takes the form

$$\tau = \int_{\Gamma (\mathbf{n})} n dl\;,$$ (13)

which states that the traveltime $\tau$ can be computed by integrating the slowness $n$ along the ray $\Gamma (\mathbf{n})$, tangent at every point to the gradient direction $\mathbf{n}$.

Similarly, we can rewrite the linearized eikonal equation (5) in the form

$$\mathbf{n_0} \cdot \nabla \tau_1 = n_1 \;,$$ (14)

where $\mathbf{n_0}$ is the unit vector, pointing in gradient direction for the initial traveltime $\tau_0$. The integral solution of equation (G-3) takes the form

$$\tau_1 = \int_{\Gamma (\mathbf{n_0})} n_1 dl\;,$$ (15)

which states that the traveltime perturbation $\tau_1$ can be computed by integrating the slowness perturbation $n_1$ along the ray $\Gamma (\mathbf{n_0})$, defined by the initial slowness model $n_0$. This is exactly the basic principle of traveltime tomography.

I have borrowed this proof from Lavrentiev et al. (1970), who used linearization of the eikonal equation as the theoretical basis for traveltime inversion.

Traveltime computation with the linearized eikonal equation