## 3 Method and Results
### Porosity Versus Cosine of the Phase Modeling  
To understand why the cosine of the instantaneous phase is always chosen at the top of the multi-attribute list and why other phase attributes make at least half of the list when using Groups 3 and 4, a synthetic 3D volume was modeled in which the inline variations represent porosity from 5 to 35 porosity units (PU) at the 5-PU interval. The crossline variation represents thickness from 1 to 101 m at the 10-m interval. This is to evaluate the effect of porosity and thickness changes on the seismic phase response.  
Figure 5(a) shows the amplitude and cosine of the instantaneous phase variations with thickness for an inline of the constant porosity of 20 PU. Figure 5(b) shows such variations with porosity for a crossline of the constant thickness of 51 m.  
\begin{table}
\begin{tabular}{c c} \hline
**Group 1** & **Group 2** \\ \hline Cosine Instantaneous Phase & Cosine Instantaneous Phase \\ \hline Integrated Absolute Amplitude & Relative Geologic Age \\ \hline Filter 5 Hz/10 Hz–15 Hz/20 Hz & Filter 5 Hz/10 Hz–15 Hz/20 Hz \\ \hline X-Coordinate & Filter 45 Hz/50 Hz–55 Hz/60 Hz \\ \hline Filter 45 Hz/50 Hz–55 Hz/60 Hz & Average Frequency \\ \hline Average Frequency & X-Coordinate \\ \hline Amplitude Weighted Phase & Filter 55 Hz/60 Hz–65 Hz/70 Hz \\ \hline Amplitude Weighted Cosine Phase & Y-Coordinate \\ \hline Quadrature Trace & Instantaneous Frequency \\ \hline Raw Seismic & Filter 35 Hz/40 Hz–45 Hz/50 Hz \\ \hline
**Group 3** & **Group 4** \\ \hline Cosine Instantaneous Phase & Cosine Instantaneous Phase \\ \hline Magnitude at 50 Hz & Relative Geologic Age \\ \hline Phase at 30 Hz & Cosine of the Phase at 110 Hz \\ \hline Cosine of the Phase at 110 Hz & Phase at 30 Hz \\ \hline Magnitude at 30 Hz & Filter 45 Hz/50 Hz–55 Hz/60 Hz \\ \hline Amplitude at 70 Hz & Semblance \\ \hline Filter 45 Hz/50 Hz–55 Hz/60 Hz & Average Frequency \\ \hline Cosine of the Phase at 70 Hz & Cosine of the Phase at 70 Hz \\ \hline Cosine of the Phase at 30 Hz & Phase at 70 Hz \\ \hline Magnitude at 50 Hz & Cosine of the Phase at 30 Hz \\ \hline \end{tabular}
\end{table}
Table 1: Top ten attributes selected by stepwise regression to predict porosity.  
Figure 6 indicates that amplitude and cosine of the phase vary regularly with changing porosity or thickness.  
To better illustrate such changes, Figure 7 shows a surface plot in which the x-axis represents porosity, the x-axis represents thickness, and the color is the cosine of the instantaneous phase from the previously described synthetic model for a constant time sample at 700 ms from the synthetic cube. The pattern of almost linear and parallel lines of the constant phase indicates systematic cosine of the instantaneous phase sensitivity to porosity and thickness, thereby explaining why the seismic phase can be useful in rock properties prediction. In addition, phase has the advantage of being less affected by overburden attenuation and transmission losses than amplitude, making it possibly more robust in relating the seismic response to the layer impedance variations than amplitude. This a potentially important advantage of phase over amplitude in quantitative analyses of seismic data for rock properties prediction.  
Figure 6: (**a**) Left: inline of synthetic seismic data that represents constant porosity of 20 PU; crossline variations represent thickness changes. Right: cosine of the phase of the inline on the left. (**b**) Left: crossline of synthetic seismic data that represents a constant thickness of 51 m; inline variations represent porosity changes. Right: cosine of the instantaneous phase of the crossline on the left. Polarity: an increase in amplitude corresponds to an increase in acoustic impedance.  
Figure 7 shows a plot that represents cosine instantaneous phase variations with porosity.