Continuous time-varying Q-factor estimation method in the time-frequency domain

Introduction

The viscoelasticity and heterogeneity of subsurface media result in amplitude attenuation and phase distortion of seismic waves propagating in the media, considerably reducing the seismic data resolution. The study of the absorption and attenuation characteristics of underground media is of great significance to the inversion of geophysical properties and the distribution analysis of oil and gas reservoirs. The Q-factor is an important parameter for describing the above characteristics that reflect the formation structure and the comprehensive effect of fluid properties such as saturation, porosity , and permeability (Winkler and Nur, 1982). The Q-estimation methods are implemented in the time domain, frequency domain, or time-frequency domain.

The time-domain methods mainly include the amplitude decay method, the rise-time method (Kjartansson, 1979), the wavelet modeling method (Jannsen et al., 1985), and the analytic signal method (Engelhard, 1996). The signal-to-noise ratio of acquired seismic data is low and the amplitude information is unreliable because of the influence of several environmental factors and complex medium structure. Thus, the accuracy and stability of Q values estimated in the time domain often cannot meet actual production requirements. Frequency information is relatively less affected by noise and data amplitude, so its stability is generally better than that of the time-domain method (Yan et al., 2001; Gong et al., 2009). The frequency domain methods include the spectral ratio (SR) method (Bath, 1974), the matching method (Raikes and White, 1984), the spectral modeling method (Jannsen et al., 1985), etc. The SR method is a widely used method with high accuracy in estimating the Q values without noise; however, it is greatly affected by the selected spectral bandwidth and noise (Tonn, 1991). Considering these factors, some scholars improved the SR method. Wang et al. (2015a) calculated the Q values using the logarithmic spectral area difference (LSAD) method, and Liu et al. (2018) introduced frequency-based linear fitting in the SR method. Quan and Harris (1997) proposed the centroid frequency shift (CFS) method based on the fact that the centroid frequency experiences a downshift when seismic waves propagate in viscoelastic media. This method deduces the relationship between the Q-factor and the centroid frequency when the amplitude spectrum shape is Gaussian, rectangular, or triangular. However, these are different from the shape of the actual seismic amplitude spectrum, so the CFS method may produce systematic errors. Some scholars analyzed and improved the CFS method, such as the Q-estimation method based on the combinations of statistical frequency attributes (Zhao et al., 2013) and the Q-value inversion using the centroid frequency of the energy spectrum (Wang et al., 2015b). Both techniques avoid the assumption of the amplitude spectrum of seismic wavelets. Zhang and Ulrych (2002) proposed the peak frequency shift (PFS) method to estimate the Q values of common midpoint (CMP) gathers, and Gao and Yang (2007) used this method to estimate the Q values of vertical seismic profiling (VSP) data. Li and Liu (2015) fitted the amplitude spectrum using the frequency-weighted-exponential function and derived a formula for estimating the Q value. Hu et al. (2013) established the relationship between the centroid and peak frequencies under the assumption of the Ricker wavelet and proposed an improved frequency shift (IFS) method. Li et al. (2015) estimated the Q values using the dominant and centroid frequencies. These frequency-domain methods must use the Fourier transform to calculate the frequency spectrum of the seismic signal in the selected time window. However, the actual seismic reflection data interfere with each other, causing some difficulties in selecting the appropriate type and length of the time window function. Furthermore, the frequency spectrum obtained using the Fourier transform reflects the average eff ect of a certain frequency component, reducing calculation accuracy of the Q-factors.

Several scholars calculate the Q values in the timefrequency domain to avoid selecting time windows and reduce the influence of spectral interference between adjacent seismic refl ection waves. Reine et al. (2009) compared the Q-factors estimated using the SR method in four different time-frequency transform domains and concluded that the time-frequency transform with the variable window could achieve more accurate and stable calculation results. Wang (2004) transformed poststack seismic data from the time domain to the time-frequency domain using the Gabor transform and then calculated the Q values according to the one-dimensional function of the product of frequency and time. Lupinacci and Oliveira (2015) estimated the Q-factors of the poststack seismic section in the Gabor transform domain using three different methods. An (2015) used the energy density ration method to estimate the Q values of CMP gathers based on an improved S-transform. Hao et al. (2016) derived the Q-estimation formula of the SR method in the generalized S-transform domain. Liu et al. (2011) improved the division operation of the SR method using shaping regularization and proposed a stable Q-estimation method based on the S-transform. Both Liu et al. (2016a) and Liu et al. (2016b) estimated the Q-factors based on the local time-frequency transform (LTFT). Wu et al. (2018) proposed a continuous spectral ratio slope method based on the generalized S-transform to avoid the extraction of target layers. Because the time-frequency transform can characterize the local attributes of the seismic data, this paper develops a continuous time-varying Q-estimation method in the time-frequency domain. The LTFT is an adaptive time-frequency transform proposed by Liu and Fomel (2013), based on the Fourier transform and solves the underdetermined problem of the least-squares solution of the adaptive Fourier series through shaping regularization (Fomel, 2007b). LTFT can adjust the frequency range and frequency sampling interval while providing variable resolution in the time direction.

When estimating the Q value in the frequency domain, accurately extracting the instantaneous spectrum of the target layer’s top and bottom interfaces is necessary. However, this is often difficult to achieve in actual data processing. This paper proposes a continuous time-varying Q-estimation method in the time-frequency domain based on the local centroid frequency to avoid picking up the target layer. The local centroid frequency is obtained by solving the inversion problem of the centroid frequency in the time-frequency domain within the constraint of the shaping regularization condition (Fomel, 2007b). This frequency is not calculated according to the instantaneous spectrum information at a specific moment but by adjusting the smoothing parameter to use the spectrum information around the moment. Therefore, we can get a relatively stable local centroid frequency at times without effective spectrum information. In this paper, the local centroid frequency and CFS method are combined to calculate the timevarying equivalent Q-factors and the time-varying interval Q-factors. Furthermore, we implement inverse Q filtering processing (Wang, 2006,2002) using the continuous time-varying equivalent Q-factors.

Continuous time-varying Q-factor estimation method in the time-frequency domain