Single Frequency Response Modeling for a Shot

Following Nihei and Li (2006) and Sirgue et al. (2008), the single frequency response of a velocity model $v(\textbf{x})$ for a shot at $\textbf{s}$ can be modelled with the time-domain finite-difference method by solving the equation

$$(\bigtriangledown^{2}-\frac{1}{v^2}\frac{\partial^2}{\partial t^2... ...x},\textbf{s})=-Re[W(\omega_{s})e^{i\omega_{s}t}]\delta(\textbf{x}-\textbf{s}),$$ (35)

where the source wavelet is a harmonic wave with its amplitude and phase specified by the source $W(\omega_{s})$ . Assuming that the trace length $T$ is long enough to include all the dominant arrivals, the recorded wavefield at the receiver location ${P}(t,\textbf{g},\textbf{s})$ reaches steady state after propagation time $T$ . Therefore, the single frequency response can be extracted with the following formula

$$\tilde{P}(\omega_s,\textbf{g},\textbf{s})=\frac{1}{T} \int_{T}^{2T}{P}(t,\textbf{g},\textbf{s})e^{-i\omega_s t}dt.$$ (36)

Note that the simulation time is increased from $T$ to $2T$ . For a single shot, the above integration can be computed over a single period ( $\frac{2\pi}{\omega_s}$ ) instead of $T$ . Repeating the above solution for different frequencies and digitizing the records give the frequency domain data $\tilde{\textbf{d}}_{i\omega,ig,is}$ .