Born Forward Modeling

The trace $d({\bf {g}}\vert{\bf {s}})$ excited by a harmonic point source at ${\bf {s}}$ and recorded by a geophone at ${\bf {g}}$ is given by the Born modeling equation:

$$\delta d({\bf {g}}\vert{\bf {s}})= \omega^2 \int_{\Omega} G({\bf {g}}\vert{\bf {x}})\delta m({\bf {x}})G({\bf {x}}\vert{\bf {s}}) dx^2$$ (4.3)

$$\rightarrow {\bf\delta} {\bf {d}}= {\bf L}{\bf\delta} {\bf {m}},$$

where $G({\bf {g}}\vert{\bf {x}})$ is the Helmholtz Green's function for the background velocity model, the model function perturbed from the background model is given by $\delta m({\bf {x}})=2 \delta s({\bf {x}}) s({\bf {x}}) \rightarrow {\bf\delta} {\bf {m}}$ , $s({\bf {x}})$ is the background slowness model, $\delta s({\bf {x}})$ is the perturbation of the slowness field, and $\omega$ is the angular frequency. For notational economy, this equation can be represented in operator notation by ${\bf\delta} {\bf {d}}={\bf L}{\bf\delta} {\bf {m}}$ , where ${\bf\delta} {\bf {d}}$ represents the scattered seismic field $\delta d({\bf {g}}\vert{\bf {s}})$ under the weak scattering approximation, ${\bf {L}}$ represents the integral operator, and $\Omega$ defines the integration points in the model region.

The integration in equation 4.3 is over the entire model space, but if the trace is windowed about a specific event then the integration can be approximated by that over the event's first Fresnel zone associated with the specific source-receiver pair. For example, if the trace only contains the transmitted arrival, then $\Omega=\Omega_{trans.}$ defines the points in the yellow colored wavepath in Figure 4.4a of the diving wave's first Fresnel zone; only velocity perturbations in this zone will significantly affect the character of the diving wave arrival in the trace.