An eikonal based formulation for traveltime perturbation with respect to the source location

Shift in the source location

The eikonal equation appears in the zeroth-order asymptotic expansion of the solution of the wave equation given by the Wentzel, Kramers, and Brillouin (WKB) approximation. It represents the geometrical optics term that contains the most rapidly varying component of the leading behavior of the expansion. In a medium with sloth (slowness squared), $w$, the traveltime $\tau$ for a wavefield emanating from a source satisfies the following formula:

$$\left(\frac{\partial \tau}{\partial x}\right)^2 + \left(\fr... ...^2+ \left(\frac{\partial \tau}{\partial z}\right)^2 =w(x,y,z),$$ (1)

where $(x,y,z)$ are the components of the 3-D medium. At the location of the source $(x_s,y_s,z_s)$, the initial value of time $\tau(x_s,y_s,z_s)=0$ is needed for numerically solving the eikonal equation 1. Moving the source along the $x$-axis a distance $l$ is equivalent to solving the following eikonal equation:

$$\left(\frac{\partial \tau}{\partial x}\right)^2 + \left(\fr... ...+ \left(\frac{\partial \tau}{\partial z}\right)^2 =w(x-l,y,z),$$ (2)

for the same source location. In other words, we are replacing a shift in the source location with an equal distance shift in the velocity field in the opposite direction. Figure 1 shows the operation for a single source and image point combination taking into account the reciprocity principle between sources and receivers.

timeds Figure 1. Illustration of the relation between the initial source location and a perturbed version given by a single source and image point locations. This is equivalent to a shift in the velocity field laterally by $dl$.

Assuming that the sloth (or velocity) field is continuous in the $x$ direction, we differentiate equation 2 with respect to $l$ and get:

$$2 \frac{\partial \tau}{\partial x} \, \frac{\partial^2 \tau}... ...\tau}{\partial z \partial l} = -\frac{\partial w}{\partial x}.$$ (3)

Substituting the change in traveltime field shape due to source perturbation, $D_x=\frac{\partial \tau}{\partial l}$, into equation 3 provides a first order linear equation in $D_x$ given by:

$$2 \frac{\partial \tau}{\partial x} \, \frac{\partial D_x}{\p... ...c{\partial D_x}{\partial z} = - \frac{\partial w}{\partial x}.$$ (4)

Solving for $D_x$ requires the velocity (sloth) field as well as the traveltime field $\tau$ for a source located at the surface at $l_0$. Thus,the traveltime field for a source at $l$ can be approximated by

$$t(x,y,z) \approx \tau(x,y,z) + D_x(x,y,z) (l-l_0).$$ (5)

Equation 4 is velocity dependent, which limits its use for inversion purposes. However, a differentiation of Equation 2 with respect to $x$ produces

$$2 \frac{\partial \tau}{\partial x} \, \frac{\partial^2 \tau}... ... \tau}{\partial x \partial z} = \frac{\partial w}{\partial x}.$$ (6)

Adding equations 3 and 6 yields equation

$$\frac{\partial \tau}{\partial x} \, \frac{\partial^2 \tau}{\... ...{\partial z} \, \frac{\partial^2 \tau}{\partial x \partial z},$$ (7)

which is velocity independent. Substituting again the change in traveltime with source location $D_x=\frac{\partial \tau}{\partial l}$ into equation 7 yields

$$\frac{\partial \tau}{\partial x} \, \frac{\partial D_x}{\par... ...{\partial z} \, \frac{\partial^2 \tau}{\partial x \partial z},$$ (8)

which is a first order linear partial differential equation in $D_x$ with $D_x$=0 at the source. The traveltime derivatives are computed for a given traveltime field $\tau$ corresponding to a source location $l_0$. Equation 8 can be represented in a vector notation as follows:

$$\nabla \tau \cdot \nabla D_x \, = \, \nabla \tau \cdot \nabla \frac{\partial \tau}{\partial x}.$$ (9)

A similar treatment for a change of the source location in $y$ or $z$ yields the following equations, respectively:

$$\frac{\partial \tau}{\partial x} \, \frac{\partial D_y}{\par... ...{\partial z} \, \frac{\partial^2 \tau}{\partial y \partial z},$$ (10)

or

$$\nabla \tau \cdot \nabla D_y \, = \, \nabla \tau \cdot \nabla \frac{\partial \tau}{\partial y}.$$ (11)

and

$$\frac{\partial \tau}{\partial x} \, \frac{\partial D_z}{\par... ...ial \tau}{\partial z} \, \frac{\partial^2 \tau}{\partial z^2},$$ (12)

or

$$\nabla \tau \cdot \nabla D_z \, = \, \nabla \tau \cdot \nabla \frac{\partial \tau}{\partial z}.$$ (13)

The above set of equations provides a tool for calculating first-order traveltime derivatives with respect to the source location. However, a condition for stability is that the velocity field must be continuous. This condition is analogous to conditions used in ray tracing methods and can be enforced using smoothing techniques. Another approach to handle this limitation is discussed later.

An eikonal based formulation for traveltime perturbation with respect to the source location