Determinant of Jacobian Matrix

The transformation between the data coordinates $(x_g,0)$ , $(x_s,0)$ and $(k_x,k_z)=(k_{sx_o}+k_{gx_o},k_{sz_o}+k_{gz_o})$ is given by

$$\left( \begin{array}{c} dk_x \\ dk_z \end{array} \right) = \left[... ...} \end{array} \right] \left( \begin{array}{c} dx_g \\ dx_s \end{array} \right),$$ (12.1)

where the 2x2 matrix is the Jacobian matrix. The scaled determinant $J$ of the Jacobian matrix is given by

$$J =\omega^4 \left\vert\frac{\partial k_x}{dx_g} \frac{\partial k_z}... ...x_s} - \frac{\partial k_x}{dx_s} \frac{ \partial k_z}{\partial x_g}\right\vert,$$ (12.2)

so that $dk_x dk_z = J dx_g dx_s$ . In the case of a homogeneous medium with velocity $c$ and a scatterer at ${\bf {r}}_o=(x_o,z_o)$ , the model wavenumbers are

$$k_x= \frac{\omega(x_o-x_g)}{c\sqrt{(x_o-x_g)^2+z_o^2}} +\frac{\omega (x_o-x_s)}{c\sqrt{(x_o-x_s)^2+z_o^2}},$$

$$k_z= \frac{\omega z_o}{c\sqrt{(x_o-x_g)^2+z_o^2}} +\frac{\omega z_o}{c\sqrt{(x_o-x_s)^2+z_o^2}},$$ (12.3)

so that the partial derivatives of the wavenumbers can be easily determined. For a heterogeneous medium, the derivatives can be approximated by finite-difference approximations to the first-order derivatives and the wavenumbers can be computed by a ray tracing method. Under the farfield approximation $z\gg L$ , where $L$ is the aperture width of the source-geophone array, so equation G.3 becomes

$$k_x\approx \,\frac{\omega(x_o-x_g)}{cz_o} +\frac{\omega (x_o-x_s)}{cz_o},$$

$$k_z\approx \,\frac{2\omega }{c},$$ (12.4)

where the horizontal wavenumbers are now linear functions of the data variables $x_g$ and $x_s$ . This means that equation F.7 represents the inverse Fourier transform of the model spectrum.