By utilizing the proposed frequency interval analysis method, one can divide the total frequency range into different sub-bands, and analyze the structural behaviors delineated from the sub-bands. In this paper, I use the synchrosqueezing wavelet transform (SSWT) (Xie et al., 2015a; Chen et al., 2014; Daubechies et al., 2011) for the time-frequency delineation of the karsts. I let $\omega_{\min}=1$Hz, $\omega_{\max}=80$Hz, and $\Delta \omega=20$Hz, which is the optimal parameter combination for this dataset. I choose $\omega_{\min}=1$ Hz and $\omega_{\max}=80$ Hz by looking at the averaged frequency spectrum of the seismic data and finding out that amplitude spectrum beyond 80 Hz or below 1 Hz is almost negligible. $\Delta \omega=20$Hz is decided by comparing the delineation results (as shown in Figure 4b) with $\Delta \omega=5,10,20$ Hz, respectively, and founding out that $\Delta \omega=20$Hz gives the best performance. The four 3D cubes that correspond to four different frequency intervals ($<1,20>$ Hz, $<21,40>$ Hz, $<41,60>$ Hz, and $<61,80>$ Hz) are shown in Figures 4a to 4d, respectively. Focusing on the target layer from 1.0 s to 1.4 s, it is obvious that the result of the frequency interval $<21,40>$ clearly shows the distribution of the karsts. Please note that the closer the color is to blue, the higher possibility that area is karst. The collapsed karsts are not formed during continuous sedimentation and thus is usually composed of complicated components. There are no layered structures inside the karsts, which will make seismic wave go through a strong scattering process when traveling through the karst media. The strong scattering will be revealed in seismic image as discontinuities, which will be further revealed in time-frequency maps as low amplitude phenomenon in the dominant frequency range. Thus, the blue (close to zero) is of higher likehood of karst. I then select the best 3D karst characterization result of the frequency interval $<21,40>$ Hz and show it in detail in Figure 9 and will discuss about the details later.

old-z lowf-st-z lowf-sswt-z lowf-stft-z lowf-ltft-z
Figure 6.
Zoomed 2D seismic section (crossline=75) and its zoomed karst mapping results. (a) Amplitude section. (b) Low frequency slice of ST. (c) Low frequency slice of SSWT. (d) Low frequency slice of STFT. (e) Low frequency slice of LTFT.
[pdf] [pdf] [pdf] [pdf] [pdf] [png] [png] [png] [png] [png] [scons]

tf-s tf-ss tf-stft tf-ltft
Figure 7.
Time-frequency maps of the 80th trace in the 2D seismic section as shown in Figure 5a, using (a) ST and using (b) SSWT. (c) STFT. (d) LTFT. The selected trace is highlighted by the blue line in Figure 5a.
[pdf] [pdf] [pdf] [pdf] [png] [png] [png] [png] [scons]

sswt-0 sswt-1 sswt-2 sswt-3 sswt-4 sswt-5 sswt-6 sswt-7 sswt-8 sswt-9 sswt-10 sswt-11 sswt-12 sswt-13 sswt-14 sswt-15
Figure 8.
Constant frequency slices using SSWT. The frequency range is from 21 to 81 Hz. Frequency interval is 4 Hz.
[pdf] [pdf] [pdf] [pdf] [pdf] [pdf] [pdf] [pdf] [pdf] [pdf] [pdf] [pdf] [pdf] [pdf] [pdf] [pdf] [png] [png] [png] [png] [png] [png] [png] [png] [png] [png] [png] [png] [png] [png] [png] [png] [scons]

tfsswt2-2d3d tfsswt2-2d3d-2 tfsswt2-2d3d-3 tfsswt2-2d3d-4
Figure 9.
3D karst features mapping results (flat view) using SSWT with different time (top), crossline (left) and inline (right) slices. (a) Crossline=75, inline=80, time=1.2s. (b) Crossline=45, inline=80, time=1.2s. (c) Crossline=75, inline=50, time=1.2s. (d) Crossline=75, inline=80, time=1.1s.
[pdf] [pdf] [pdf] [pdf] [png] [png] [png] [png] [scons]

stra-int stra-tf
Figure 10.
Different strata slices from the time-frequency delineation result, shown in Figure 10b. (a) Top of Caddo. (b) Strata slice between Caddo and Vineyard. (c) Top of Vineyard. (d) Strata slice between Vineyard and the base.
[pdf] [pdf] [png] [png] [scons]

stra1 stra2 stra3 stra4
Figure 11.
Different strata slices from the time-frequency delineation result, shown in Figure 10b. (a) Top of Caddo. (b) Strata slice between Caddo and Vineyard. (c) Top of Vineyard. (d) Strata slice between Vineyard and the base.
[pdf] [pdf] [pdf] [pdf] [png] [png] [png] [png] [scons]

I then take the selected 2D section for further analysis. I compare the performance using four different time-frequency decomposition approaches: the S transform, the SSWT transform, the short time Fourier transform (STFT), and the local attribute based time-frequency transform (LTFT) (Liu et al., 2011b) . The delineation results using ST, SSWT, STFT, and LTFT are shown in Figures 5b, 5c , 5d , and 5e , respectively. The result from SSWT is very appealing because those karst collapses are all depicted by the time-frequency decomposition (see those pink frame boxes), and the collapse areas are all corresponding to small amplitude in the low frequency slice. However, all other time-frequency transforms cannot decompose the seismic data in such way. In order to make the comparison clearer, I zoom a part from each figure in Figure 5 and show them in Figure 6. The zoomed area is highlighted by the black frame boxes in Figure 5. In Figure 6, the blue color in Figure 6c correlates well with the collapsed area in Figure 6a. The best mapping of karsts using SSWT is due to its high resolution property when analyzing 1D non-stationary seismic signals. Figure 7 shows time-frequency decomposition results of the 80th trace of the 2D seismic section as shown in Figure 5a, using four different approaches (ST, SSWT, STFT, and LTFT). The selected trace is highlighted by blue in Figure 5a. From the comparison, one can see that SSWT has a much higher frequency resolution than ST, STFT, and LTFT. From the single-trace time-frequency slice one can judge roughly in which depth the data show abnormal frequency, and what their corresponding dominant frequency components are. In order to show superior performance of the frequency interval analysis approach as introduced in this paper, over the traditional single frequency slice analysis approach, I plot 16 single frequency slices in Figure 8. The frequency range is from 21 Hz to 81 Hz, and I plot the frequency slices every 4 Hz. It can be observed that the time-frequency decomposition results of subsurface properties are either of too low resolution, as shown in the low frequency slices, or of too high resolution as shown in high frequency slices.

Figure 9 shows 3D karst delineation results (flat view) using SSWT with different time (top), crossline (left) and inline (right) slices. Figure 9a differs from 9b by different crossline sections. From the comparison between Figures 9a and 9b, one can see clearly how the karst varies along the crossline direction: the karst becomes weaker from crossline 75 to crossline 50. In the same way, one can study the karst variation along the inline direction by comparing Figures 9a and 9c. It is also possible to evaluate the volume of the karst controlled area by looping along different directions in the 3D volume (Figure 4b), which is closely related with the karst controlled reserve. Figure 10a shows an interpreted 3D seismic section with three key stratas highlighted. The top pink line, middle green line, and bottom blue line denote the Caddo strata, the Vineyard strata, and the Ellenburger strata, respectively. Figure 10b shows the time-frequency delineation result of the karsts overlaid by the picked three horizons. The Upper Caddo is the top boundary of the productive Bend Conglomerate section, and the Upper Vineyard is a sequence near the base of the Bend Conglomerate section. The Bend Conglomerate section is of Middle Pennsylvanian age (also known as Atokan) and is targeted throughout the 1980s and 1990s as it contains several oil&gas bearing reservoirs. The thickness is around 300-360 meters in this area, with a depth from 1370 to 1830 meters (Hardage et al., 1996b,a). It can be observed from the karsts mapping result shown in Figure 10b that the karsts spreads a deep range from the Upper Caddo to the base of Ellenburger. I also selected different strata slices from Figure 10b and show them in Figure 11. Figure 11a shows a structure surface of the top of Caddo. Figure 11b shows a structure surface between Caddo and Vineyard. Figure 11c shows a structure surface on the top of Vineyard. Figure 11d shows a structure surface between Vineyard and the base of Ellenburger carbonate section. It can be observed that from up to down, the main color turns from cool (blue) to warm (red), indicates that the karstification weakens as the depth increases, which is consistent with the fact that the consolidation becomes stronger as the depth increases and make the deeper strata has stronger collapses and less karsts.

The Boonsville 3D dataset consolidate the proposed frequency interval analysis method in delineating subsurface karst collapses. When I use the term karst collapses, I intend to refer to all those commonly seen karst features, such as incised valleys, eroded caves, sinkholes, and so on. While the reasons causing these karst features are different, they will all cause more or less depressions or collapses to the overburden strata and have a similar situation as the one presented in the Boonsville example. There is a large potential that the proposed framework can be successfully used in any type of karst features, and future validation of the framework on more field datasets are worth being done. The sediment or mud-filled collapse features may not cause significant differences of anomalies in amplitude spectrum and they will both show irregularities on the 3D seismic images since seismic wave will experience strong scattering effects when traveling through the collapsed area. But future investigation should be done to validate this inference.