The Analytical Analysis of the Ricker Wavelet

The Stokes differential equation takes into account the viscoelastic effects when the seismic wave propagates through subsurface stratum and thus is more realistic than the elastic model. Since the Ricker wavelet satisfies this equation, it has been widely used in seismic analysis such as in the seismic modeling. And the seismic characteristic frequency is commonly used in seismic attribute analysis and the quality factor inversion like the central frequency and the instantaneous frequency. So by combining these two aspects, the analytical solutions of the Ricker wavelet is analyzed and some useful conclusions have been obtained that can be used directly in the seismic analysis.

In this paper, the characteristic frequency is first analyzed in the Fourier domain using the Fourier transform. The centre frequency is defined using different equations and so is the corresponding bandwidth. But they generally follows the linear relationship with the dominant frequency of the Ricker wavelet by solving the Gamma function, thus are replaceable by each other. Then the Hilbert transform is also considered to get an analytical solution of the Ricker wavelet, where the hyper-geometric function is induced. Besides, by including the absorption effects the attenuated Ricker wavelet is also analyzed.