Temporal windowing and inverse transform of the wavefield in the Laplace-Fourier domain

Temporal windowing is a valuable process, which can help us to focus on a specific event in a seismogram. However, applying the time window is difficult outside the time domain. We suggest a windowing method which is applicable in the Laplace-Fourier domain. The window function we adopt is defined as a product of a gain function and an exponential damping function. The Fourier transform of a seismogram windowed by this function is equivalent to the partial derivative of the Laplace-Fourier domain wavefield with respect to the complex damping constant. Therefore, we can obtain a windowed seismogram using the partial derivatives of the Laplace-Fourier domain wavefield. We exploit the time-windowed wavefield, which is modeled directly in the Laplace-Fourier domain, to reconstruct subsurface velocity model by waveform inversion in the Laplace-Fourier domain. We present the windowed seismograms by introducing an inverse Laplace-Fourier transform technique and demonstrate the effect of temporal windowing in a synthetic Laplace-Fourier domain waveform inversion example.