Q-factor estimation using bisection algorithm with power spectrum

Abstract

The centroid frequency shift (CFS) method is a widely used [Formula: see text] estimation approach. However, the CFS approach assumes that the amplitude spectrum of a source wavelet has a particular shape, which can cause systematic error in [Formula: see text] estimation. Moreover, the amplitude spectrum at high and low frequencies is susceptible to random noise, which can reduce the robustness of [Formula: see text] estimation using the CFS method. To improve the accuracy and robustness of [Formula: see text] estimation, we have developed a [Formula: see text] extraction method using the bisection algorithm based on the centroid frequency shift of power spectrum (BPCFS). In the BPCFS approach, we first obtain the source and the received wavelet. Then, we calculate the centroid frequency of the attenuated wavelet and that of the received wavelet from the power spectrum. Based on the obtained centroid frequencies, we establish an equation containing only one variable — the [Formula: see text] factor. Introducing the Jeffrey divergence to measure the attenuation of the power spectrum, we prove that this equation has only one root when [Formula: see text] is greater than zero. The root of this equation, which is the desired [Formula: see text] factor, is obtained through the bisection algorithm; we do not make any assumption about the shape of the amplitude spectrum. The noise-free numerical tests indicate that the BPCFS gives more accurate results than the CFS, which demonstrates that the shape of the wavelet spectrum has little effect on the [Formula: see text] estimation accuracy for BPCFS. Gaussian random and blue noise tests also show that the stability of BPCFS is better than that of CFS. The frequency band selection for BPCFS is also more flexible when the amplitude spectrum of the seismic wavelet is band limited. The application to real vertical seismic profile data further demonstrate the effectiveness and feasibility of the new method.