## 4 Discussion  
Spectral broadening outside the original band of the seismic signal is controversial [6]. In the present study, the spectrally broadened seismic data is considered as an attribute: an arbitrary mathematical transformation of a seismic trace. Although [PERSON] and [PERSON] [6] discussed under what circumstances such broadening can be viewed as physically accurate, [PERSON] et al. [2] showed--for the same area in the Daqing field--that multi-attribute analysis of spectrally broadened data resulted in better predictions of well log porosity by all measures compared with a similar analysis with the original conventional seismic data. Based on the results, the spectrally broadened data is a significant input for multi-attribute analysis as indicated by cross-validation and blind well tests. The significance of the attribute was demonstrated by ties to well data and the performance in predicting porosity in both the training and validation [2], rather than by theoretical argument. The attributes were derived from spectrally broadened seismic data and classified into four groups: conventional attributes (Group 1), conventional attributes and relative geologic age (Group 2), conventional attributes and components of spectral decomposition (Group 3), and all attributes combined (Group 4). Then, stepwise regression was performed for the attributes in each group to create a multi-attribute list that included the ten attributes within each group that yielded the best prediction error. From the attributes included in the stepwise regressions, the cosine of the instantaneous phase was selected as the most significant attribute in all the cases, followed by relative geologic age when available. For Groups 3 and 4, more than half of the selections by stepwise regression were phase attributes. The high ranking of relative geologic age on the attribute lists is related to its ability to pinpoint where in a stratigraphic sequence  
Figure 17: From left to right: porosity and gamma-ray log. Group 4 predicted porosity volume, with a red line indicating the well path, corresponding to the spectrally broadened seismic section. The blue rectangle highlights the channel-like feature. Polarity: an increase in amplitude corresponds to an increase in acoustic impedance. a particular layer occurs, thereby taking advantage of any geologic succession in rock properties that may be learned in training. Conversely, the selection of the cosine of the instantaneous phase first in all the cases was an unexpected outcome, as it was the stepwise regression process that selected many phase attributes among the top 10 attributes. The relationship between phase and porosity is explained with a synthetic wedge model (Figures 6 and 7), showing that phase is sensitive to porosity and thickness in seismically thin layers. The isophase trends in the cross-plots of phase versus porosity and thickness suggest that the seismic phase response is related to the phase of the series of reflection coefficients for a given thickness. In addition, this is directly related to reservoir porosity through layer impedance. In synthetic data, the cosine of the instantaneous phase varies systematically with the porosity at a given time thickness. This relationship is particularly relevant in the present study that focuses on porous, thin, and discontinuous reservoirs. The cosine of the instantaneous phase removes all amplitude information that could be ambiguous in predicting porosity in thin layers and with overburden effects. As a result, the seismic phase potentially is more readily correlated to porosity than amplitude alone in thin layers. For this reason, it is understandable from a physical perspective that the attribute is selected first by stepwise regression and that the other phase attributes are also selected. The agreement between the predicted and actual porosity increases from Group 1 to Group 4.