Vertical seismic profiling (VSP) provides a direct observation of seismic waveforms propagating to various depths within the earth’s subsurface. The Q analysis or attenuation (1/Q) analysis based on direct comparison between individual waveforms at different depths, however, suffers from the problem of instability commonly due to fluctuations inherent in the frequency spectrum of each waveform. To improve the stability, we considered frequency and time variations and conducted Q analysis on an integrated observation. First, we transformed the time- (or depth-) frequency-domain spectrum to a 1D attenuation measurement with respect to a single variable, the product of time and frequency. Although this 1D measurement has a higher signal-to-noise ratio than the 2D spectrum in the time-frequency domain, it can also be used to further generate a stabilized compensation function. Then, we implemented two Q-analysis methods by data fitting (in a least-squares sense) to either the attenuation measurement or the data-driven gain function. These two methods are theoretically consistent and practically robust for conducting Q analysis on field VSP data.

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