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

Appendix A: Higher-order expansions

Traveltime behavior due to source perturbations can be estimated more accurately using higher-order formulations. Considering that $l_x$, $l_y$, and $l_z$ represent source perturbations in the $x$, $y$, and $z$ directions, respectively, a full representation of the second derivative behavior is given by the following symmetric matrix

$$\left( \begin{array}{ccc} D_{xx} & D_{xy} & D_{xz} \\ D_{xy... ...} & \frac{\partial^2 \tau}{\partial l_z^2} \end{array}\right).$$ (25)

$D_{xx}$ is evaluated using the first order linear differential equation 20, where $\tau$ is obtained from solving the eikonal equation and $D_x$ is evaluated from equation 4.

Similarly, higher order approximations in $l_y$ and $l_z$ are given by

$$\nabla D_y \cdot \nabla D_y \, + \,\nabla \tau \cdot \nabla D_{yy} \, = \, \frac{1}{2} \frac{\partial^2 w}{\partial y^2}$$ (26)

and

$$\nabla D_z \cdot \nabla D_z \, + \,\nabla \tau \cdot \nabla D_{zz} \, = \, \frac{1}{2} \frac{\partial^2 w}{\partial z^2},$$ (27)

respectively, which represent the diagonal terms of the matrix in equation E-1.

To obtain the non-diagonal components of the matrix we differentiate equation 3 with respect to $l_y$, instead of $l_x$, yielding:

$$2 \left(\frac{\partial^2 \tau}{\partial x \partial l_x}\right) \... ...u}{\partial y} \, \frac{\partial^3 \tau}{\partial y \partial l_x \partial l_y}+$$

$$2 \left(\frac{\partial^2 \tau}{\partial z \partial l_x}\right) \l... ...al z \partial l_x \partial l_y} = \frac{\partial^2 w}{\partial x \partial y}.$$ (28)

Substituting the second derivative of traveltime with respect to source location $D_{xy}=\frac{\partial^2 \tau}{\partial l_x \partial l_y}$ into equation E-4 provides us with a first order linear equation in $D_{xy}$ given by:

$$2 \left(\frac{\partial D_x}{\partial x}\right) \left(\frac{\parti... ... \, + 2 \frac{\partial \tau}{\partial y} \, \frac{\partial D_{xy}}{\partial y}+$$

$$2 \left(\frac{\partial D_x}{\partial z}\right) \left(\frac{\parti... ...rac{\partial D_{xy}}{\partial z} = \frac{\partial^2 w}{\partial x \partial y}.$$ (29)

or

$$\nabla D_x \cdot \nabla D_y \, + \,\nabla \tau \cdot \nabla ... ..., = \, \frac{1}{2} \frac{\partial^2 w}{\partial x \partial y}.$$ (30)

Similar equations for the rest of the matrix components are given by

$$\nabla D_x \cdot \nabla D_z \, + \,\nabla \tau \cdot \nabla ... ..., = \, \frac{1}{2} \frac{\partial^2 w}{\partial x \partial z},$$ (31)

and

$$\nabla D_y \cdot \nabla D_z \, + \,\nabla \tau \cdot \nabla ... ..., = \, \frac{1}{2} \frac{\partial^2 w}{\partial y \partial z},$$ (32)

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