**Estimation of Q Factor in Seismic Evaluations**

**Sanuja Senanayake**

Geology Undergraduate Student: Winter 2015, University of Calgary.

### Summary

The quality of seismic images varies with the several parameters. Fundamentally, the signal strength plays a major role in the clarity of seismic images. By analyzing the signal quality quantitatively as opposed to qualitatively, we can correct the loss of signal strength over a distance and time. The Q factor is a mathematical representation of signal degeneration. It can be used to evaluate the original seismic wavelet from a distorted wavelet. There are several methods to derive the value of Q. But currently there is no consensus among the geophysicists on which method is more accurate. In this particular study (Lupiancci, Andriano, Oliveira, 2015), researchers evaluated three methods; amplitude decay versus time decay, spectral ratio-based and Wang’s method. After several iterations of the data, they found the Wang method to be more accurate and provided the most consistent dataset. However, it should be highlighted the spectral ratio-based method also provided very accurate results.

### Introduction

The seismic interpretations depend on both the knowledge and experience of the interpreter and the quality of seismic dataset itself. The seismic images are derived by collecting signal data from either seismic surveys or by measuring natural earthquakes. However, the measured data is almost always not the original seismic signal. As waves travel through a medium, they interact with the grains and fluids causing the waves to lose energy. Hence, the attributes of the original seismic signal is not detected by the geophones.

To obtain the original seismic signal from a distorted dataset, the rate or amount of energy loss should be quantified. This can be achieved by assigning a mathematical value to the quality of seismic images. The Q factor is one such mathematical parameter that can be used to quantify the energy loss of the seismic signal.

In physics, attenuation is described as the reduction in signal strength or the energy loss of signals over a distance or time. As the signal travels across a medium, it interacts with the particles within the medium. These interactions result in transfer of energy out of the signal. The Q factor measures how much of the original energy remains at the time of signal detection. Hence the Q factor is inversely proportional to the attenuation.

In this particular study, researchers focused on different methods of deriving the value of Q (Lupiancci, Andriano, Oliveira, 2015). They evaluated the errors in the Q factor obtained through amplitude decay versus time decay, spectral ratio-based and newly suggested Wang’s method.

### Theory and Methodology

In order to analyze the seismic data, the signals must be decomposed using appropriate algorithms. Spectral decomposition is a commonly used method in which the change in frequency is analyzed. However, it is a non-unique process where depending on the algorithm used, the value of Q may vary widely for same dataset. Some possible methods for spectral decomposition includes, but not limited to, continuous wavelet transform, Gabor transform and S transform.

The seismic theory suggests that a wave propagating through an inelastic medium will result in exponential decay of amplitude. This relationship can be used to derive estimations for Q factor by regression applied to the decay function itself. However, the decay function itself is not clearly detected in both real world and in synthetic data. Decay function detection and analysis is especially difficult to obtain in short time windows due to less data values. In this study, researchers calculated Q factors in various time windows and verified the results with inverse Q filtering of a seismic section (Lupiancci, Andriano, Oliveira, 2015).

There are two parts to methodology: the modeling of seismic trace in a dissipative medium and the Q factor estimation approaches. The modelling was done using fundamental principles of wave propagation (1). The U_{0}(ω) is the Fourier transform of the seismic pulse, ω is the angular frequency. V(ω) is the complex phase velocity and x and τ are the travel distance and time, respectively.

For weekly dispersive media, the Q factor is much higher than one (Q >> 1), hence the approximation can be reached using the following (Cravenly, 2001):

in which, V^{R}(ω) is the real phase velocity and the Q(ω) is the medium Q factor. Hence V(ω) term in the first equation (1) can be replaced with the second equation (2). This new equation (3) is used to model the synthetic seismic trace for Q factor estimations.

In the above equation (3), the first exponent deals with the propagation and dispersion while the second exponent is responsible for the amplitude decay. Next, the equation (3) can be rewritten as:

In equation (4), τ = x/V^{R}(ω)

The synthetic seismic data is modeled using frequency domain using the following relationship.

The equation (5) represents the Fourier transform of the trace and the wavelet with respect to the frequency domain. This function was derived using previous studies by numerous geophysics and physicists (Lupiancci, Andriano, Oliveira, 2015). Finally, by assuming transmission effect of the wave is negligible, finally the equation (6) was derived, which was used for calculating the Q factor for the nth layer.

The next part of this study is the analysis of Q factor itself. To obtain the values for Q, the researchers (Lupiancci, Andriano, Oliveira, 2015) relied on Gabor transform, a type of Fourier transform that uses a Gaussian window function to generate a time-frequency amplitude spectrum of a seismic trace.

### Alternative methods of Q Estimation

Three alternative methods were briefly considered in the study. They all involve the amplitude spectrum of the time-frequency transform of the seismic trace. The first method was the amplitude decay versus time method. It is based on the amplitude curve of the time-frequency amplitude spectrum picked for a constant frequency value. For example, in this particular study it was 30 Hz (Lupiancci, Andriano, Oliveira, 2015).

The second method was the spectral ratio-based method. It is based on the measurement of the exponential decay along the frequency axes for a constant time.

The third method was coined as the “Wang’s method” by the authors because it was first introduced by Yanghua Wang in 2004. This involves measurement of the amplitude decay along the compound variable x = ωτ, where the variable x depends on the signal-to-noise ratio of the data. In this particular study researchers picked amplitudes along the curves defined by the equation, x = ωτ in the time-frequency domain. Then the average of the amplitudes was taken to define the exponential decay value s(x) for the following function (equation 7). (Lupiancci, Andriano, Oliveira, 2015)

Using the above three methods, the modelled seismic trace was used to evaluate the Q factor estimation approaches considering a medium formed by random spikes Lupiancci, Andriano, Oliveira, 2015). The estimations were measured at set time intervals; 0.5, 1.0, 2.0 and 3.0 seconds for the all three methods (Table 1-REMOVED due to Copyright).

### Synthetic Data Results

The synthetic data were obtained through the use of equation (5) and (6). Three above mentioned Q estimation methods were then tested for the time windows of 0.5, 1.0, 2.0 and 3.0 seconds. For the spectral ratio-based method, the frequency band 5 to 70 Hz was used. For the Wang’s method, Xa = 23 and Xb = 180 were used for all time windows; equation (7).

The Q factor estimation reading were repeated for Q = 60 and Q = 90 and the results are analyzed (Table 1). The researchers observed that the Q factor estimation become more accurate and precise as the length of the analysis time window increase. Hence the time frame of the analysis directly related to the accuracy and precision of the Q factor. Longer the analysis time window, higher the accuracy of the Q factor. Another conclusion was that the exponential decay function also becomes more coherent as for longer time windows than shorter ones. This was expected by the researchers (Lupiancci, Andriano, Oliveira, 2015) because as the wave propagates through a medium, over time and distance the attenuation also increases. Attenuation modifies the signals through the loss of energy during the interaction between the propagating wave and the particles or atoms of the medium. The amplitude spectrum of the time-frequency transform of the signal is controlled by peaks and valleys due to interference from the reflected pluses. These resulted in oscillations that mask the exponential decay trend. This was highly reflected in analysis performed on short time windows. It should be highlighted that this issue occurred in all methods. However, the Wang’s method was the least affected by this problem. The amplitude decay versus time method showed fluctuations in the estimated Q value based on the chosen frequency. Therefore researchers have used a range of frequencies to reduce errors in estimating the value of Q.

### Real World Data Results

The real world data were obtained using a seismic section from deep water Pelotas Basin, Brazil. The samples were processed using basic seismic processing methods. The processing flow can be summarized as: geometrical spread correction, deconvolution, velocity analysis, parabolic Radon demultiple, dip-moveout correction (DMO), common offset prestack migration (Stolt) and bandpass filter (5 – 55 Hz).

Once the data was processed, the Q factor estimations were obtained through all three (above mention) methods. The amplitude decay versus time method shows a progressive loss of energy through the degradation of the amplitude over a period of time. Compared to the synthetic data, it was found to have higher impedance. The researchers ( Lupiancci, Andriano, Oliveira, 2015) suggest that this is most likely caused by shallow gas within the formation. As the density of the material decreases, the attenuation also increases. The spectral ratio-based method on real world data made it almost impossible to detect the linear relationships between variables. The amplitude decay versus time method on the real world dataset also resulted in data without clear linear relationships. However, the Wang’s method produced a very clear data output for the real world dataset with very strong linear trend. This was predicted in both this study (Lupiancci, Andriano, Oliveira, 2015) as well as previous studies by Wang (2004).

### Outcome of the Research

It was found that the amplitude versus time and spectral ratio-based methods did not work well in real world data. To obtained reasonable results, the spectral ratio-based method requires a careful choice of frequency intervals where the logarithm spectral amplitude ratio versus frequency curve is near liner. Additionally, this method is not robust for the estimation of Q factor due to its sensitivity to the size of the analysis window.

The amplitude decay versus time method performed well in the noise-free synthetic data, but the performance was poor in the real world dataset. This was most likely caused by the fact that this method requires a very good time-varying amplitude gain control in seismic processing. The processing should only correct the geometrical spread effect while preserving the exponential decay caused by attenuation. Hence it is difficult to achieve. Another problem is the presence of stronger reflections with anomalous amplitudes causing erroneous regression.

The Wang’s method was robust for real world datasets. The trace by trace analysis of Q estimation suggested by Wang (2005) provided the most consistent and statistically accurate Q factor estimations. In fact, the standard deviation was small in the two analysis window sizes (6.55 for the 0 – 2.0s window and 9.25 for the 1.0 – 3.0 window). The main reason for the effectiveness of the Wang’s method is that it effectively explores the time-frequency domain. This method shows that Q factor increased as the depth of the real world dataset increases. This is consistent with the petrophysics because once the wave propagates into deeper rock layers, which are more compact, the wave favors less seismic attenuation. Hence a higher Q factor in deeper and more compacted layers.

### Other Studies

There are several different studies have been done on estimation of Q factor. In one particular study by Vassil Davidov (2012) on the earthquake seismology found that the Q factor estimation is very difficult to achieve in the real world scenarios. This is because the geologic materials in which the seismic waves propagate are almost always antistrophic, inelastic and heterogeneous. This results in variation in Q factor in almost every direction as the wave propagates through the medium. The data obtained through the geophones ground stations could not often properly calculate the Q factor due to these complications in physical geology itself.

### Conclusions

While there are no set international standards on obtaining the value of Q factor, the estimations can be derived from variety of methods. In this particular study, researchers found that the Wang’s method proposed in 2004 to be the most accurate out of the three methods studied. Other studies on real world data such as the earthquake study by Davido (2012) also highlighted the need to alternative methods for obtaining Q factor estimations for real world datasets.

### References

Lupinacci, Wagner Moreira, and Sérgio Adriano Moura Oliveira. “Q factor estimation from the amplitude spectrum of the time-frequency transform of stacked reflection seismic data.” Journal of Applied Geophysics 114 (2015): 202-209.

Wang, Y., 2004. Q analysis on reflection seismic data. Geophys. Res. Lett. 31, L17606.

Davidov, Vassil. Seismic quality factor (Q) of the mid-continental crust from regional earthquake seismograms. Diss. Northern Illinois University, 2012.

The research is done by respective authors of the listed papers under refrences. The resarch is not belong to Sanuja Senanayake. This is an online publication of the final Term Paper written by Sanuja Senanayake at the University of Calgary for Geophysics 559: Geophysical Interpretation in Winter 2015. The text is copyrighted to Sanuja Senanayake. You may follow the guidelines posted under the general Site Copyright Notice. Figures used in the original paper have been removed due to copyright laws.

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