Detecting Karst Voids Based on Dominant Frequencies of Seismic Profiles

Abstract

The presence of near-surface karst voids is an extremely difficult issue in the construction of a high-speed rail (HSR) foundation. Seismic constant-offset profile (COP) method is one of the shallow geophysics technologies which may be used for the detection of karst voids. Although a COP image does not directly reveal the characters related to the anomalies in a karst terrain, the dominant frequency of the COP image in a karst terrain is significantly lower than the dominant frequency over the background without karstification or voids. This dominant-frequency anomaly is due to the strong attenuation effect when seismic waves propagate through any karst voids. Thus, we propose using the dominant-frequency anomalies of the COP image to directly detect near-surface karst voids in a karst terrain. First, we generate a high-resolution time–frequency spectrum for each COP trace, using the modified Wigner-Ville distribution (WVD) algorithm which combines WVD with a multichannel maximum entropy method. Second, we estimate a high-precision dominant-frequency function which varies along the reflection time, based on the corresponding high-resolution time–frequency spectrum. Finally, we detect the geological anomalies by analyzing low-frequency distributions in the dominant-frequency image for all traces. We demonstrate this procedure with a case study for the detection of karst voids within the high-speed rail foundation in a karst terrain, and verify the interpretation of hidden voids, cavities, clays and peats directly with drilling cores.

1 Introduction

Near-surface geological anomalies such as cavities, voids, clays, and peats present an extremely difficult issue in the construction engineering for a high-speed rail foundation (Mirzanejad et al., 2020). Seismic constant-offset profile (COP) survey is a near-surface seismic exploration technology which can be used for detecting shallow anomalies in an engineering project. In this article, we propose a method to exploit the low-frequency anomalies in a COP seismic data set for detecting near-surface karstification.

Hidden karstification presents a serious threat to high-speed rail foundations. If there is a closed karst cavity within the near-surface media, then the karst cavity will lead to wave diffractions, following Huygen's principle of seismic wave theory. Therefore, the characteristics of wave diffractions can be used as evidence for predicting a karst cavity (Dobecki & Upchurch, 2006; Gritto et al., 2014; Ivanov et al. 2005a, b). However, when the cavity shape is complex and irregular, the anomalous characteristics of wave diffractions are not so obvious, and this method will not work effectively (Carpenter et al., 1998).

The presence of geological anomalies such as karst voids within the subsurface media will cause dynamic changes in seismic reflections. Thus, the dynamic information of seismic reflections can be used to identify these anomalies comprehensively. This type of detection method conventionally requires advance knowledge of the specified seismic trace associated with the karst and the frequency band of seismic reflection waves, and then uses various transformations to form a time–frequency spectrum which reveals the karst location. However, the time–frequency spectrum needs to have a sufficiently high resolution in the time direction.

A seismic trace in the COP seismic data set is a non-stationary signal, because the frequency property of the seismic signal varies along the travel time. Time–frequency spectral methods are often used for analyzing a non-stationary signal (Cunningham & Williams, 1993; Gabor, 1946; Kumar & Foufoula-Georgiou, 1997; Mallat & Zhang, 1993; Sejdić et al., 2009; Stockwell et al., 1996; Wang, 2007, 2010, 2021). One of such methods is the Wigner-Ville distribution (WVD) method (Ville, 1948; Wigner, 1932), which satisfies many ideal mathematical characteristics and is able to generate the time–frequency spectrum with relatively concentrated spectral energy distribution. However, one problem with the conventional WVD method is the cross-term interferences among different components in a multicomponent signal (Pachori & Nishad, 2016). Smoothing along either the time or frequency direction can effectively suppress this interference (Franz et al., 1995) but also alters the spectral characteristics of each component of the signal and thus reduces the accuracy of spectral decomposition.

Wang et al. (2020) proposed modifying the WVD method by combining the conventional WVD calculation with a multichannel maximum entropy method (MC-MEM). The MC-MEM method designs a prediction-error filter (PEF) based on the part of the WVD kernel that does not contain the cross-term interference and predicts the kernel sequence beyond that part (Appendix A). Thus, the modified WVD method, “WVD + MC-MEM”, effectively eliminates the cross-term interference in the spectrum. Moreover, the PEF in the MC-MEM is a minimum-phase filter and can also be used to extrapolate the WVD kernel sequence (Ulrych, 1972; Ulrych et al., 1973). The extrapolation extends the length of the WVD kernel and results in a higher resolution than the spectrum generated by the conventional WVD method.

In this article, we exploit the high-resolution spectral feature of the WVD + MC-MEM method to estimate the dominant-frequency function with high precision. This dominant-frequency function varies along seismic trace and is comparable to the instantaneous frequency. Because the instantaneous frequency is a phase differential function (Boashash, 1992; Taner et al., 1979), the conventional calculation is numerically unstable, and the accuracy is strongly influenced by data noise. We calculate the dominant frequency at any given time by an integral of the first moment of the time–frequency spectrum. Because such an integral effectively suppresses the influence of data noise, the dominant-frequency calculation is not only robust, but also has high precision.

We then use the high-precision dominant frequency to identify the hidden karstification underneath a tunnel where a high-speed railway crosses through. Although the original seismic COP image does not show the trivial characteristics of the near-surface anomalies, the dominant-frequency profile clearly indicates low-frequency features related to the anomalies. The drilling cores directly reveal the existence of soft clay, karst cavities and voids underneath the target construction.

In summary, the proposed procedure for the characterization of near-surface seismic anomalies consists of three steps in sequence: (1) generating the time–frequency spectrum for each seismic trace using the modified WVD method, (2) estimating the time-varying dominant frequency by the integral of the first moment of the frequency spectrum, and (3) identifying karst voids based on the anomalous distribution in the dominant-frequency profile.

2 Time–Frequency Spectra with High Resolution

For a nonstationary seismic trace, the spectral characteristics vary along seismic time. We use the WVD + MC-MEM method to generate a high-resolution time–frequency spectrum.

The conventional WVD is defined as (Ville, 1948; Wigner, 1932)

$$W(t,f) = \int\limits_{{ - \infty }}^{\infty } {z(t + \tfrac{\tau }{2})z^{*} (t - \tfrac{\tau }{2})e^{{ - {\text{i}}2\pi f\tau }} {\text{d}}\tau } ,$$

(1)

where $t$ is time, $z(t)$ is the analytic signal of a seismic trace, $z^{*} (t)$ is its complex conjugate, and $f$ is the frequency.

The WVD spectrum at any given time $t$ is a Fourier transform of the kernel $k = z(t + \tfrac{\tau }{2})z^{*} (t - \tfrac{\tau }{2})$ with respect to time $\tau$. Because the WVD kernel is a quadratic function, there exist cross-term interferences among different signal components. The cross-term interferences will degrade the quality of the resultant time–frequency spectrum. To tackle this issue, Wang et al. (2020) proposed a MC-MEM algorithm to modify the WVD kernel, by predicting from the interference-free part of the WVD kernel to the part beyond. This modified WVD method has two features in the time–frequency spectrum. First, it generates the spectrum with effective suppression of the cross-term interferences. Second, it generates the spectrum with a high resolution. The second feature is possible simply because the minimum-phase MC-MEM filter can extrapolate the WVD kernel to a much longer sequence than the length of the data observation sequence.

In this article, we compare this modified WVD method with the wavelet transform (WT) and the generalized S transform (ST) method. The continuous WT method is expressed as

$$X(t,\alpha ) = \frac{1}{{\sqrt \alpha }}\int\limits_{{ - \infty }}^{\infty } {x(\tau )\psi ^{*} \left( {\frac{{\tau - t}}{\alpha }} \right){\text{d}}\tau } ,$$

(2)

where $x(t)$ is the seismic signal, $\psi (t)$ is a mother wavelet, $\psi ^{*} (t)$ is its complex conjugate, and $\alpha$ is a scaling factor. When the scaling factor $\alpha$ increases, the time-domain wavelet function is expanded, and its frequency-domain duration is correspondingly compressed. When the scaling factor $\alpha$ decreases, the time-domain breadth of the wavelet function decreases, and the frequency-domain bandwidth increases. Therefore, the WT spectral resolution changes with the scaling factor $\alpha$. For analyzing high-frequency features, WT will use a high-frequency wavelet to observe the signal, and the scaling factor $\alpha$ will be a smaller value. For analyzing low-frequency features, WT will use a low-frequency wavelet to observe the signal, and the scaling factor will be a larger value.

One widely used mother wavelet in seismic analysis is the generalized Morse wavelet, because of its good localization characteristics (Lilly & Olhede, 2009; Olhede & Walden, 2002). The generalized Morse wavelet (of zeroth order) may be defined in the frequency domain as

$$\hat{\psi }(f) = a_{{\beta ,\gamma }} (2\pi f)^{\beta } e^{{ - (2\pi f)^{\gamma } }} ,\;{\text{for}}\;f \ge {\text{0,}}$$

(3)

where $\beta$ is the frequency-domain compactness parameter, $\gamma$ describes the frequency-domain asymmetry of the Morse wavelet, and $a_{{\beta ,\gamma }}$ is a normalizing constant. We derive the normalizing constant as

$$a_{{\beta ,\gamma }} = \left( {\frac{{e\gamma }}{\beta }} \right)^{{\beta /\gamma }} ,$$

(4)

where e is Euler’s number. Equation (4) makes the peak amplitude of $\hat{\psi }(f)$ unity at the peak frequency $(\beta /\gamma )^{{1/\gamma }} /(2\pi )$, although there are different normalizing constants used in the literature (Kocahan et al., 2017; Lilly & Olhede, 2009). In the example shown in this article, the two adjustable parameters are taken as $\beta = 10$ and $\gamma = 2$. According to Eq. (3), if $\gamma = 2$, the Morse wavelet $\hat{\psi }(f)$ becomes a specific wavelet defined by the analytic derivative of a Gaussian function.

The generalized version of the ST method is expressed as

$$S(t,f) = \int\limits_{{ - \infty }}^{{ + \infty }} {x(\tau )} \frac{{\lambda \left| f \right|^{p} }}{{\sqrt {2\pi } }}e^{{ - \lambda ^{2} f^{{2p}} (t - \tau )^{2} /2}} e^{{ - {\text{i}}2\pi f\tau }} {\text{d}}\tau ,$$

(5)

where $(\lambda ,{\text{ }}p)$ are two adjustable parameters. If compared directly to the continuous WT, the wavelet in  the generalized ST used here is a generalized Gaussian time window function:

$$\psi (t) = \frac{1}{{\sqrt {2\pi } c(f)}}{\text{e}}^{{ - \frac{{t^{2} }}{{2c^{2} (f)}}}} ,$$

(6)

with

$$c(f) = \frac{1}{{\lambda \left| f \right|^{p} }},\;\lambda > 0,\;p > 0.$$

(7)

As shown in Eq. (5), ST adjusts the breadth of the time window function adaptively according to the frequency variation, because of the frequency-dependent factor $c(f)$. The generalized Gaussian window function also contains two adjustable parameters $(\lambda ,{\text{ }}p)$, similar to the generalized Morse wavelet. In the example shown in this article, the generalized ST parameters are taken as $\lambda = 2$ and $p = 0.5$, selected based on visual observation of spectral localization.

Figure 1a is a near-surface seismic trace, arbitrarily selected from a seismic COP data set. The time sampling interval is $\Delta t = 0.125$ ms, and the number of samples per trace is $N = 800$. Figure 1b–d are the time–frequency spectra generated by WT, ST and the modified WVD method, respectively. Figure 1 clearly demonstrates that the spectral resolution of the modified WVD method is much higher than that of WT and ST. The spectral energy is more concentrated in the WVD spectrum.

Fig. 1

a Near-surface seismic trace. b The time–frequency spectrum generated by the continuous WT method. c The spectrum generated by the generalized ST method. d The high-resolution spectrum generated by the modified WVD method

3 Dominant Frequency with High Precision

In this section, we exploit the high-resolution time–frequency spectrum generated by the modified WVD method to estimate the dominant frequency, which will have high precision.

We estimate the dominant frequency using the first moment integral method, expressed as follows:

$$f_{{\text{d}}} (\tau ) = \frac{{\int_{0}^{\infty } {f|W(\tau ,f)|{\text{d}}f} }}{{\int_{0}^{\infty } {|W(\tau ,f)|{\text{d}}f} }},$$

(8)

where $f_{{\text{d}}} (\tau )$ is the dominant frequency at a given time $\tau$, and $|W(\tau ,f)|$ represents the amplitude spectrum at this time. The integral over the frequency f is helpful in suppressing the data noise along the frequency axis.

This continuous form has been used in analytical derivations for the dominant frequency and the frequency bandwidth of the Ricker wavelet (Wang, 2015a) or Wang’s generalized wavelet (Wang, 2015b). In the discrete calculation with a sum over the frequency axis, one practically needs to set the starting frequency and the ending frequency of the summation according to the noise level in the field seismic spectrum. However, because of the high-resolution feature of the modified WVD spectrum, it can simply be summed over the Nyquist frequency range.

The time-varying dominant frequency is comparable to the instantaneous frequency, calculated by the differential of the instantaneous phase (Boashash, 1992; Taner et al., 1979). Figure 2b shows a direct comparison between the estimated dominant-frequency function and the conventional instantaneous frequency function. Two frequency curves are generally consistent, but a significant difference appears between them due to the form of mathematical methods; one is the integral and the other is the differential. The integral method through the first moment produces a stable estimate of the dominant frequency (in red color). In contrast, the calculation of the instantaneous frequency is numerically unstable (in black), and its precision is severely influenced by data noise.

Fig. 2

a Near-surface seismic trace. b Comparison between the dominant-frequency function (in red color) and the instantaneous frequency function (in black color). c Comparison of dominant frequencies estimated from the WT spectrum (in blue color), the ST spectrum (in black) and the spectrum of the WVD + MC-MEM method (in red color)

Figure 2c compares the precision of the dominant-frequency functions estimated from three time–frequency spectra. Time–frequency spectral resolution directly affects the precision of the dominant-frequency estimation. The first WT spectrum produces the lowest dominant frequency, the second ST spectrum leads to the highest dominant frequency, and the third WVD spectrum with the WVD + MC-MEM method has a dominant frequency in between. The resolution of the third dominant-frequency function in the time axis is higher than that estimated from the other two time–frequency spectra.

Although the first moment calculation method suppresses the data noise, the integral summation in theory also reduces the resolution to a certain extent. But because the time–frequency spectral resolution of the WVD + MC-MEM method is far higher than the WT and ST spectra, the estimated dominant frequency still has high resolution when compared to the other two methods.

4 Detecting Hidden Karst Voids

The dominant frequency estimated from the modified WVD spectrum is used to identify the karst voids underneath a tunnel where a high-speed railway crosses through. Prior to the near-surface seismic survey, the roadbed construction of the whole section was completed, and the surface of the foundation was reinforced by concrete. The subsurface lithology of this section is limestone.

Figure 3 displays the seismic COP data set. The common-depth-point (CDP) numbers of the imaging section are between 110 and 330, and the constant shot-receiver offset is 12 m. The profile has been preprocessed, including steps of band-pass filtering, amplitude equalization, frequency–wavenumber (F–K) filtering, and random noise attenuation. The parameters of the band-pass filter are 30–50–600–650 Hz. The method for the amplitude equalization is automatic gain control (with a time window of 60 ms) and cross-channel equalization. The F–K filter is an interactive polygonal filter. The random noise attenuation method is the F–X forward–backward prediction filter (Wang, 1999).

Fig. 3

The constant-offset seismic profile. The constant shot-receiver offset is 12 m

Seismic COP image in Fig. 3 has relatively balanced energy and does not have obvious noise interference. But the image makes for nontrivial solution for detecting of karstification, even if pre-whitening is applied to the image (Doll et al., 1998). First, the seismic image does not show any obvious refraction waves related to karst anomalies. Second, any analysis procedure based on reflection waves would also be tedious and not easy to implement. However, the dominant-frequency distribution is effective in the detection.

In the high-precision dominant-frequency profile, as displayed in Fig. 4a, evaluated from the time–frequency spectrum generated by the modified WVD, there are two obvious low-frequency anomalies between CDP 260 and 320. In the geological interpretation of these two low-frequency anomalies shown in Fig. 4b, the upper anomaly is clay and the lower anomaly is voids or cavities in limestone. When seismic waves propagate through the shallow clay and the lower karst voids, their high-frequency energy is heavily decayed, and thus their dominant frequency is reduced (Rao & Wang, 2009, 2015). The lateral extent of these anomalies in the interpretation section (Fig. 4b) is basically the same as that in the low-frequency imaging section (Fig. 4a). Although the specific depth and thickness of these anomalies cannot be determined on the seismic image profiling section due to the lack of accurate velocity information, the lateral location of these karst anomalies is clearly revealed in the dominant-frequency profile. Figure 4b also indicates two drilling positions for verifying the geological interpretation. Other low-frequency anomalies could be small heterogeneities within the limestone, in contrast to the two significant karst anomalies revealed by drilling cores.

Fig. 4

a Dominant-frequency profile showing low-frequency anomalies. b Geological interpretation based on the dominant-frequency profile. The two vertical columns are drilling wells

Figure 5 displays the actual drilling cores. The cores are arranged from top to bottom according to the drilling depth, and each row in the image is equal to a 1 m length of core samples. In the two drilling positions (CDP 278 and 298), concrete, filled gravel, soft clay and limestone are encountered sequentially. These core samples clearly reveal voids and cavities within the limestone. The dominant-frequency anomalies are in good agreement with the soft clay and the karst voids revealed by the core samples. Mud filling is also seen in the lower part of the karst voids, in which the lithology is clearly different from the limestone background.

Fig. 5

a Drilling cores at the lateral position CDP 278. b Drilling cores at the lateral position CDP 298. These cores reveal concrete, filled gravel, clay and limestone in sequence, and voids and cavities within the limestone

5 Conclusions

In the near-surface seismic COP survey, when seismic waves propagate through a karst terrain, their dominant frequencies vary significantly because of the absorption and attenuation of the high-frequency energy. We have demonstrated that a reliable dominant-frequency estimate may be used directly to detect the near-surface geological anomalies. In contrast to conventional COP analysis, with this procedure there is no need to preselect any possible karst location from the seismic profile. The estimation procedure is robust, and the estimated dominant frequency is accurate.

We have applied the dominant-frequency image to the construction engineering project for the high-speed rail foundation. The low-frequency anomalies in the dominant-frequency image straightforwardly reveal geological anomalies such as voids and cavities in a karst terrain. Actual drilling cores further verify the lithology of these interpreted anomalies. Therefore, this dominant frequency-based method is a cost-effective method for construction engineering projects.

Appendix A: The Multichannel MEM Algorithm

This appendix summarizes the multichannel maximum-entropy method (MC-MEM), proposed by Wang et al. (2020), for the sake of completeness.

The Wigner-Ville kernel $k = z(t + \tfrac{\tau }{2})z^{*} (t - \tfrac{\tau }{2})$ can be discretized into

$$k_{n} (\ell ) = z(n + \ell )z^{*} (n - \ell ),$$

(9)

where $n$ refers to the time sample index. The kernel sequence is conjugate symmetrical, $k_{n} ( - \ell ) = k_{n}^{*} (\ell )$. Thus, for modifying the kernel from $k_{n} (\ell )$ to $\hat{k}_{n} (\ell )$, where the circumflex in $\hat{k}_{n} (\ell )$ indicates modification, we need to consider only half of the kernel sequence $k_{n} (\ell )$, for $\ell = \{ 0,{\text{ 1, }} \ldots ,{\text{ }}L\}$.

The multichannel linear prediction can be made by

$$\begin{gathered} k_{{n + j}} (\ell ) = - \sum\limits_{{i = 1}}^{\beta } {a_{\beta } (i)k_{{n + j}} (\ell - i)}, \hfill \\ k_{{n + j}} (\ell ) = - \sum\limits_{{i = 1}}^{\beta } {a_{\beta }^{*} (i)k_{{n + j}} (\ell + i)}, \hfill \\ \end{gathered}$$

(10)

where $\beta$ is the prediction order, $\{ a_{\beta } (i)\}$ and $\{ a_{\beta }^{*} (i)\}$ are the forward and backward prediction coefficients, respectively (Wang, 1999), and $- J \le j \le J$ is the index of the multiple channels. At a given time sample $n$, the prediction coefficients $\{ a_{\beta } (i)\}$ are estimated by exploiting multiple kernel sequences within a time window $[n - J,{\text{ }}n + J]$.

Exploiting the linearity, the current kernel sequence $k_{n} (\ell )$ may be replaced with an average of multiple channels:

$$\tilde{k}_{n} (\ell ) = \frac{1}{{2J + 1}}\sum\limits_{{j = - J}}^{J} {k_{{n + j}} (\ell )},$$

(11)

where $\tilde{k}_{n} (\ell )$ with a tilde represents an average. Then, the forward and backward prediction errors are measured by

$$\begin{aligned} \tilde{f}_{\beta } (\ell ) &= \tilde{k}_{n} (\ell ) - \left( { - \sum\limits_{{i = 1}}^{\beta } {a_{\beta } (i)\tilde{k}_{n} (\ell - i)} } \right) \\ &= \sum\limits_{{i = 0}}^{\beta } {a_{\beta } (i)\tilde{k}_{n} (\ell - i)} , \\ \tilde{g}_{\beta } (\ell ) &= \tilde{k}_{n} (\ell ) - \left( { - \sum\limits_{{i = 1}}^{\beta } {a_{\beta }^{*} (i)\tilde{k}_{n} (\ell + i)} } \right) \\ &= \sum\limits_{{i = 0}}^{\beta } {a_{\beta }^{*} (i)\tilde{k}_{n} (\ell + i)} , \\ \end{aligned}$$

(12)

where $\{ a_{\beta } (0),{\text{ }}a_{\beta } (1),{\text{ }} \ldots ,{\text{ }}a_{\beta } (\beta )\}$ is the prediction-error filter, and $a_{\beta } (0) = 1$. Because of the averaging, Eq. (12) is a minimum-phase prediction-error filter (PEF). The PEF $\{ a_{\beta } (i)\}$ is determined for $\beta = \{ 1,{\text{ }}2, \ldots ,{\text{ }}M\}$ recursively, using the maximum-entropy method (MEM).

Once the prediction coefficients $a_{{\beta = M}} (i)$ for $i = \{ 1,{\text{ }} \cdots ,{\text{ }}M\}$ are obtained, the linear prediction is preformed to both the original kernel $k_{n} (\ell )$ and the multichannel average $\tilde{k}_{n} (\ell )$. Initially, the prediction of the kernel sequence is set to be

$$\hat{k}_{n} (\ell ) = \frac{1}{{1 + w}}[\tilde{k}_{n} (\ell ) + wk_{n} (\ell )],$$

(13)

where $w$ is a weighting factor, to balance the magnitude of $\tilde{k}_{n} (\ell )$ and $k_{n} (\ell )$. Then, the forward prediction is made by

$$\hat{k}_{n} (\ell ) = - \sum\limits_{{i = 1}}^{M} {a_{{\beta = M}} (i)\hat{k}_{n} (\ell - i)} ,$$

(14)

for $\ell = \{ L_{S} + 1,{\text{ }} \ldots ,{\text{ }}L\}$, where $L_{S}$ is the number of elements in the front part of the kernel sequence, which is supposed to have no cross-term interference and is kept unchanged. The sequence outside this interference-free part is predicted by the minimum-phase filter, and thus the cross-term interference is eliminated from the sequence.