Traveltime computation with the linearized eikonal equation

THE LINEARIZED EIKONAL EQUATION

The eikonal equation, describing the traveltime propagation in an isotropic medium, has the form

$$\left(\nabla \tau\right)^2 = n^2(x,y,z)\;,$$ (1)

where $\tau (x,y,z)$ is the traveltime (eikonal) from the source to the point with the coordinates $(x, y, z)$, and $n$ is the slowness at that point (the velocity $v$ equals $1/n$.) In Appendix A, I review a basic derivation of the eikonal and transport equations. To formulate a well-posed initial-value problem on equation (1), it is sufficient to specify $\tau$ at some closed surface and to choose one of the two branches of the solution (the wave going from or to the source.)

Equation (1) is nonlinear. The nonlinearity is essential for producing multiple branches of the solution. Multi-valued eikonal solutions can include different types of waves (direct, reflected, diffracted, head, etc.) as well as different branches of caustics. To linearize equation (1), we need to assume that an initial estimate $\tau_0$ of the eikonal $\tau$ is available. The traveltime $\tau_0$ corresponds to some slowness $n_0$, which can be computed from equation (1) as

$$n_0 = \left\vert\nabla \tau_0\right\vert\;.$$ (2)

Let us denote the residual traveltime $\tau - \tau_0$ by $\tau_1$ and the residual slowness $n - n_0$ by $n_1$. With these definitions, we can rewrite equation (1) in the form

$$\left(\nabla \tau_0 + \nabla \tau_1 \right)^2 = \left(\nabl... ...au_1\right)^2 = (n_0 + n_1)^2 = n_0^2 + 2 n_0 n_1 + n_1^2\;,$$ (3)

or, taking into account equality (2),

$$2 \nabla \tau_0 \cdot \nabla \tau_1 + \left(\nabla \tau_1\right)^2 = 2 n_0 n_1 + n_1^2\;.$$ (4)

Neglecting the squared terms, we arrive at the equation

$$\nabla \tau_0 \cdot \nabla \tau_1 = n_0 n_1\;,$$ (5)

which is the linearized version of the eikonal equation (1). The accuracy of the linearization depends on the relative ratio of the slowness perturbation $n_1$ and the true slowness model $n$. Though it is difficult to give a quantitative estimate, the ratio of 10% is generally assumed to be a safe upper bound.

The intimate connection of the linearized eikonal equation and traveltime tomography is discussed in Appendix B.

Traveltime computation with the linearized eikonal equation