Method

Time-frequency decomposition maps a 1-D signal in the time domain into a 2-D image in time-frequency space, which can describe the local time-frequency variations of geophysical properties. There have been many different approaches for such decomposition mission, such as the continuous wavelet transform (Daubechies, 1992), S transform (Stockwell et al., 1996), and synchrosqueezing wavelet transform (Daubechies et al., 2011). I will not promote a new algorithmic advance here for better time-frequency decomposition, but instead, I will propose a novel and effective way for mapping the subsurface karst features based on a simple frequency interval analysis approach using 3D seismic data recorded at the surface of the targeted area.

The algorithm can be summarized in three simple steps:

• Step 1: Input the 3D seismic data, and define the minimum and maximum analyzing frequency $\omega_{\min}$ and $\omega_{\max}$ and frequency interval distance $\Delta \omega$.
• Step 2: Apply time-frequency decomposition to each trace of the 3D seismic data and sum the frequency components in each frequency interval $<\omega_i,\omega_{i}+\Delta \omega>$, $i=1,2,\cdots,N$, where $N$ denotes the number of frequency intervals, then reform the summed frequency components of different intervals into several 3D cubes that correspond to different frequency intervals.
• Step 3: Output the different frequency cubes and analyze the results to pick the best frequency interval that can best depicts the karst features.

$\omega_{\min}$ and $\omega_{\max}$ are chosen based on the bandwidth of seismic data. A reference value can be $\omega_{\min}=1$Hz and $\omega_{\max}=100$Hz. The $\Delta \omega$ is the only parameter that may need several trials. One can test the $\Delta \omega$ from 5Hz to 30 Hz, and updating $\Delta \omega$ every 5Hz. The choice of $\Delta \omega$ is a compromise between analysis complexity and the characterization resolution.

It is worth mentioning that time-frequency based karst delineation approach is not new in geophysical study. A typical and standard way to extract subsurface karst features based on time-frequency decomposition is to extract different constant frequency slices, and then to select a certain frequency slice to best represent the karst features based on a prior knowledge and good correlation between the frequency slice and the amplitude map. However, the single frequency slice depiction is sometimes not reliable since higher frequency may provide fine-scale delineation with higher resolution, while lower-frequency may provide large-scale result with lower resolution. The result of the contradiction in traditional approaches is that there might be spurious bright spots or unclear karsts representation. I will just mention the problem here and will show it in detail in the data example. The frequency interval analysis approach provides us a convenient way to control the resolution and reliability of finally depicted karsts by simply adjusting the frequency interval. Although determining the frequency interval still requires some efforts in its current version, my experience shows that frequency interval is usually chosen as 5, 10, or 20Hz.