Introduction

Microvibration that exists ubiquitously on earth surface is called microtremor. Microtremor is a passive-source vibration signal which comes from nature or human activities and carries abundant information about subsurface geological structures. In seismic exploration, microtremor is generally a kind of interference signal which needs to be removed. However, this background noise can also be used for passive-source exploration. The advantages of microtremor exploration include good adaptability to site conditions, easy to measure and noninvasive. Compared with seismic exploration using explosive source or spark source, microtremor survey is inexpensive and safe. In survey region where severe electromagnetic interference exists, electromagnetic exploration cannot obtain accurate data, but microtremor method will not be affected because it measures ground vibration.

Microtremor exploration has a wide range of application, including conducting seismic microzonation in earthquake research (Büyüksaraç et al. 2013), analyzing site characteristics and site amplification (Ansary and Rahman 2013; Farahani and Zaré 2014) and detecting hydrocarbon reservoir (Saenger et al. 2007; Steiner et al. 2008). Microtremor exploration is more often used in detecting underground shear wave velocity structure and estimating sediment thickness. Common microtremor data acquisition methods contain array acquisition and single-station acquisition. The data acquired by array can be processed by SPAC (spatial autocorrelation) method (Aki 1957) or F–K (frequency wavenumber) method (Capon 1970) to extract phase velocity dispersion curve of Rayleigh wave, which can then be inversed to obtain 1-D underground shear wave velocity (Wang 1986; Ye 2004; He et al. 2007; Xu et al. 2013). SPAC method is more popular than F–K method because it needs less stations.

When using SPAC method to process microtremor array data, only vertical component of each acquisition station is needed. However, for single-station data, not only vertical component but also two horizontal components are needed for horizontal-to-vertical spectral ratio (HVSR or H/V spectral ratio, Nakamura 1989) method.

Research has shown that peak frequency of HVSR curve is consistent with fundamental resonant frequency of site sediment (Sylvette et al. 2006). Delgado et al. (2000b) illustrated the effectiveness of using HVSR method to detect sediment thickness and pointed out that this method is not valid if shear wave velocity of sediment varies irregular with depth. Guéguen et al. (2007) pointed out that in valleys where width-to-depth ratio is small, 2-D or 3-D basin geometry could induce significant error to this method. Several researchers have derived frequency-to-sediment thickness (f–Z) fitting equations in different regions (Ibs-von Seht and Wohlenberg 1999; Delgado et al. 2000a; Parolai et al. 2002; Motamed et al. 2007; D’Amico et al. 2008; Gosar and Lenart 2010; Paudyal et al. 2013) through fitting peak frequency of HVSR with corresponding sediment thickness from boreholes, geological or geophysical data.

This paper applied HVSR technique in a karst collapse area in Guangdong to detect site sediment thickness. Firstly, the relationship of peak frequency and sediment thickness (i.e., the f–Z fitting equation) was obtained based on microtremor data measured beside 52 boreholes in southwest of this collapse area. Survey grid with a spacing of 10 m was arranged in this collapse area to acquire microtremor data; then, peak frequency of HVSR for each measuring point was extracted. Finally, sediment thickness of the collapse area was estimated according to the established f–Z fitting equation.

Site condition

The study site is karst collapse area A, which is located in Conghua district of Guangzhou city, Guangdong province. This site is overlain by Quaternary alluvial deposit without rock outcrop. The location and nearby geology information of collapse area A are shown in Fig. 1.

Fig. 1
figure 1

Regional geological map nearby area A. 1. alluvium in the first terrace; 2. alluvium in the second terrace; 3. monzonitic granite; 4. Lower Carboniferous Datangian Stage Shidengzi Formation; 5. Lower Carboniferous Datangian Stage Ceshui Formation; 6. upper member of Maozifeng Formation, Upper Devonian; 7. fault; 8. river; 9. study site

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Underlying stratum of this site is limestone of Lower Carboniferous Shidengzi Formation. The terrain here is flat, and the elevation ranges from 27 to 31 m. The top soil in area A is mostly cultivated soil except the southwest corner where it is covered by sand. There are 25 sinkholes located in area A, and most of them tend to be circular with a diameter ranges from 0.5 to 5 m.

Methods

Microtremor seismometer records the vibrations of three orthogonal directions, namely two horizontal components (north–south and east–west directions) and one vertical component. Horizontal-to-vertical spectral ratio (HVSR) is computed as geometric mean (Gosar and Lenart 2010) of both horizontal spectra divided by the vertical spectrum. The formula can be written as:

$${\text{HVSR}} = \sqrt {\frac{{S_{\text{NS}} \cdot S_{\text{EW}} }}{{S_{Z}^{2} }}}$$
(1)

where \(S_{\text{NS}}\), \(S_{\text{EW}}\) and \(S_{Z}\) are the amplitude spectrums of south–north, east–west and vertical components, respectively. HVSR can also be computed as arithmetic mean (Ibs-von Seht and Wohlenberg 1999) or root-mean-square (Delgado et al. 2000a) of \(S_{\text{NS}} /S_{Z}\) and \(S_{\text{EW}} /S_{Z}\). Although the formulas for computing HVSR are different, the extraction result of peak frequency will not be influenced.

When processing actual microtremor data, the data were firstly divided into N segments of equal length. HVSR of all the N segments is calculated, and then, M segments with large transient noise caused by vehicles or people are removed. The average HVSR of remaining N–M segments is HVSR of this measuring point.

Theoretical relationship of sediment thickness Z, average shear wave velocity \(\overline{{V_{\text{s}} }}\) and fundamental resonant frequency \(f_{\text{r}}\) is (Ibs-von Seht and Wohlenberg 1999)

$$Z = \frac{{\overline{{V_{\text{S}} }} }}{{4f_{\text{r}} }}$$
(2)

According to Eq. (2), if \(\overline{{V_{\text{s}} }}\) and \(f_{\text{r}}\) are known, the sediment thickness can be calculated. The peak frequency of HVSR coincides well with the fundamental resonance frequency of sediments. However, the average shear velocity is usually unknown. According to the empirical formula, \(\overline{{V_{\text{s}} }}\) has positive correlation with depth \(Z\) due to the increasing confining pressure, and the expression can be written as (Delgado et al. 2000b)

$$\overline{V}_{\text{S}} { = }PZ^{Q}$$
(3)

According to Eqs. (2) and (3), the sediment thickness and peak frequency have the following power–function relationship:

$$Z = af^{b}$$
(4)

In formula (4), \(f\) means the peak frequency of microtremor HVSR. Research results show that the fitting parameters a and b vary with regions. This phenomenon mainly results from the different relationships that the average shear velocity varies with depth due to the diverse components and unlike compaction degree of sediments in different regions. If the f–Z fitting equation of Pearl River Delta is established based on data from known sites, then sediment thickness of karst collapse area A can be estimated.

Results

Fitting equation in Pearl River Delta

In this paper, we acquired microtremor data beside 52 boreholes in area B of Pearl River Delta, China. Sediment thicknesses revealed by these boreholes range from 7.9 to 39.6 m. Overburden of these sites is complex and generally consists of several different components including plain fill, silty clay, different kinds of sands, mucky soil, etc., and the underlying rock basements are limestone of Devonian–Carboniferous. The locations of 52 boreholes are shown as red circles in Fig. 2, where some of them are too close to be distinguished. Despite the fact that these boreholes are located in different geomorphic units, the sediment components are similar.

Fig. 2
figure 2

Location of boreholes (read circles) used for f–Z fitting

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For microtremor data of area B, the sampling frequency was 128 Hz and the sampling time length was 20 min. When processing microtremor data, dividing the 20-min data into 60 segments means each segment has a length of 20 s. Figure 3 shows a 20-s microtremor record beside one borehole in area B. The top, middle and bottom curves represent the vibration of north–south, east–west and up–down directions, respectively. Calculate HVSR of each 20-s segment using formula (1) and then eliminate those with strong disturbance. The HVSR of this measuring point was obtained by calculating the average HVSR of the remaining segments. HVSR beside one borehole is shown in Fig. 4; the red line represents HVSR curve and two black lines represent HVSR plus or minus standard error. The maximum HVSR amplitude is at 3.31 Hz, i.e., the peak frequency is 3.31 Hz.

Fig. 3
figure 3

A 20-s microtremor record beside one borehole

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Fig. 4
figure 4

HVSR beside one borehole

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The peak frequencies, which are sorted from low to high, and corresponding sediment thicknesses of the 52 boreholes are listed in Table 1. The lowest and highest peak frequencies are 1.44 and 6.88 Hz, respectively. Based on these data, the f–Z fitting equation can be established using nonlinear least-square regression. The obtained f–Z fitting equation of Pearl River Delta is:

Table 1 Peak frequencies and corresponding sediment thicknesses in Pearl River Delta
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$$Z = 55f^{ - 1.02}$$
(5)

Several researchers applied HVSR method to study how to detect sediment thickness in different regions and had established corresponding f–Z fitting equations which are applicable to those regions. Table 2 lists four sets of fitting parameters obtained from Germany, Spain and Slovenia, as well as parameters obtained from China’s Pearl River Delta in this paper.

Table 2 Frequency–thickness fitting parameters from different regions
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Ibs-von Seht and Wohlenberg (1999) carried out microtremor measurements in two boundaried areas of the western Lower Rhine Embayment (Germany) to the north of Aachen. Subsurface of the study area is Tertiary and Quaternary sediments cover a basement of Paleozoic. Delgado et al. (2000a) mapped the sediment thickness in Segura River Valley (Spain). The valley is comprised of Late Pleistocene–Holocene sediments (gray silt and sand), Late Miocene–Late Pleistocene sedimentary rocks (conglomerates, marls and sandstones) and Triassic–Cretaceous carbonate rocks. Parolai et al. (2002) derived the f–Z relationship for Cologne area (Germany) where Tertiary and Quaternary sediments (mainly gravel, sand and clays) cover Devonian bedrock. Gosar and Lenart (2010) applied HVSR method to map sediment thickness in Ljubljana Moor basin (Slovenia) as a complementary tool to seismic refraction survey. The overburdens consist of Quaternary lacustrine and fluvial sediments, while bedrocks consist of Carboniferous–Permian sandstones, conglomerates and shales in the northern basin and Triassic dolomite and Jurassic limestone in the central southern basin.

The variation of parameters a and b is caused by the diverse relationship of sediment shear wave velocity and depth in different regions. In essence, this variation results from the difference of sediment composition and compaction degree of the sedimentary layer. For each pair of f–Z fitting parameters in Table 2, calculate sediment thickness when peak frequency ranges from 1 to 10 Hz and then plot the results in double logarithmic coordinate as shown in Fig. 5. The black circles in Fig. 5 represent the measured data in Pearl River Delta, and the black line means the corresponding f–Z fitting equation established in this paper. The peak frequency and sediment thickness relationship (formula 5) derived in this research have a good fitting effect with the determination coefficient \(R^{2} = 0.86\). In double logarithmic coordinate, power function (4) is displayed as a straight line, in which parameter a is the y-intercept at 1 Hz, while parameter b is the slope. Parameter b in Pearl River Delta is relatively small compared with those given in other studies. This indicates that peak frequency decreases slower while sediment thickness increases in Pearl River Delta.

Fig. 5
figure 5

Different fitting curves and obtained data in Pearl River Delta

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The sediment components in Pearl River Delta can be roughly divided into three classifications in terms of shear wave velocity, namely normal layers, soft layers and hard layers. The normal layers contain clay, silty clay, backfill, cultivated soil, etc. The soft layers contain muck, mucky soil, loose sand, etc. And the hard layers contain coarse sand, gravelly sand, pebble, etc.

If the sediment contains soft layers, average shear wave velocity \(\overline{{V_{\text{S}} }}\) would be reduced. According to formula (2), resonant frequency \(f_{\text{r}}\) and \(\overline{{V_{\text{S}} }}\) have a positive relationship, so \(f_{\text{r}}\) would be lower. Therefore, the HVSR peak frequency \(f\) of this site is relatively low in contrast to normal sediment. When applying fitting Eq. (5) to this site, the sediment thickness would be overestimated. Similarly, if the sediment contains hard layers, the sediment thickness would be underestimated. However, when both soft layers and hard layers exist, the estimation error depends on which component dominates the sediment. According to the borehole data used in this paper, when thickness of soft or hard layers is larger than 5 m, a relatively large estimation error would likely to occur.

Detected results in karst collapse area A

The measuring line layout in karst collapse area A is shown in Fig. 6. The measuring line strike is 30°, while the surveying line spacing and measuring point spacing are both 10 m. There are 26 measuring points in each line except lines 1–4 which are short of two points due to site limitation in the west corner. Single-station microtremor data were acquired at each measuring point using the same acquiring parameters as in area B.

Fig. 6
figure 6

Thickness of sediments in collapse area A. 1. river; 2. sinkhole; 3. power line; 4. building; 5. measuring point; 6. borehole

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Analyzing microtremor data acquired in area A, the peak frequency of HVSR for each measuring point was obtained. For areas A and B, the sediments are both alluvial deposits and the underlying strata are limestone. Therefore, the f–Z relationship derived in area B can be applied here. According to Eq. (5), the sediment thickness of each measuring point was calculated. The detected result is shown in Fig. 6.

The interpreted sediment thickness in karst collapse area A ranges from 8 to 27.6 m. The north and south parts have relatively deep sediment thickness, while the west part is shallow. Twenty out of 25 sinkholes in area A are located in regions where the sediment thickness is less than 16 m, while the other five sinkholes are located in regions where the sediment thickness is between 16 and 24 m.

After the microtremor-detecting work, five boreholes numbered CT1, CT2, CT3, CT4 and SK1 were arranged in area A. Columnar diagrams from 0 to 30 m of the five boreholes are shown in Fig. 7. According to the drilling data, the sediments are composed of interbedded sand and silty clay, while some regions contain muck or mucky soil. The topsoil of CT1, CT2, CT3, CT4 is cultivated soil with a thickness of 0.5–0.8 m.

Fig. 7
figure 7

Columnar diagrams of five boreholes in karst collapse area A (the interpreted sediment thickness is, in order 16.0, 16.3, 12.5, 10.5 and 10.2 m): 1. cultivated soil; 2. silty clay; 3. coarse sand; 4. medium sand; 5. fine sand; 6. muck; 7. mucky soil 8. gravelly sand; 9. limestone; 10. karst cavity

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Table 3 lists real sediment thickness, estimated sediment thickness according to microtremor data and the corresponding estimation error of the five boreholes. Estimated sediment thickness for CT2 has a relatively larger error of 4.1 m, while for the other four boreholes, detecting errors are less than 2 m.

Table 3 Detected sediment thickness of the borehole locations in karst collapse area A
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The detecting errors are caused by three reasons:

  1. 1.

    The fitting equation is derived by fitting all the data acquired in a relatively large measuring area using least-square criterion. However, the sediment components, proportion and compaction degree will not be the same in different sites. Therefore, fitting Eq. (5) only gives an estimated value. When applied to a particular measuring point, the fitting equation would inevitably over- or underestimate sediment thickness.

  2. 2.

    Microtremor measurement was carried out in a 10-m grid. Therefore, sediment thickness for sites not located at the measuring point was obtained by interpolation.

  3. 3.

    Microtremor HVSR reflects sediment properties not only directly underneath the measuring point, but also in a certain range around it. As the bedrock surface in karst terrain is undulated, when applying HVSR method to a measuring site, the sediment thickness result estimated by fitting equation is equivalent to smoothing the real site thickness data.

  4. 4.

    The relatively large error for CT2 results from the low shear wave velocity of muck (2.7–3.4 m in depth) and loose–medium sand (3.4–4.3 m in depth). Soft layers cause low HVSR peak frequency according to formula (2). Therefore, the estimated depth is relatively high when applying fitting Eq. (5) to this site.

When combining data of 52 boreholes in area B with data of five boreholes in area A to derive a new f–Z fitting relationship using least-square criterion, however, the same parameters as Eq. (5) were obtained. This result means that the obtained f–Z equation in this paper is reasonable.

Conclusions

This paper established the f–Z fitting equation based on microtremor data acquired beside 52 boreholes in Pearl River Delta. And then, the equation was applied in karst collapse area A.

The error between sediment thicknesses revealed by five boreholes in this collapse area and corresponding values calculated by the f–Z fitting equation is small. Therefore, this fitting equation can be applied to estimate sediment thickness of site in Pearl River Delta or other alluvial fans with similar geological setting.

According to the detected result, 80% sinkholes in area A are located in places where the sediment thickness is less than 16 m. This phenomenon indicates that collapses are more likely to occur in regions with small sediment thickness. Therefore, detecting sediment thickness can help delineating potential karst collapse areas.

When conducting karst collapse investigation, the f–Z fitting equation for this area can be established through analyzing microtremor data beside boreholes nearby with similar geological settings using HVSR method. Then, the sediment thickness of some specific sites in this survey area can be estimated quickly.

Discussion

The theoretical basis of formula (1) is that the subsurface is composed of two horizontally stratified elastic media, namely a soft sedimentary layer with low shear wave velocity lying over a rigid bedrock layer with fast shear wave velocity. Both the sediment and bedrock are horizontally isotropic, which means the model is 1-D. In addition, there must exist a large impedance contrast between sediment and underlying bedrock so a clear HVSR peak can be identified. In consequence, this technic may become ineffective in the presence of gradational weathering or cementation (Lane et al. 2008). It is proved to be very robust to estimate sediment thickness using formula (4) as long as the subsurface abides by the basic assumptions (Bignardi 2017).

The estimation error of sediment thickness mainly results from two aspects. Firstly, the sedimentary layer may consist of several different components so shear wave velocity does not vary according to formula (3). Secondly, horizontal shear wave velocity change makes the subsurface do not satisfy 1-D mode assumption.

Parameters of f–Z fitting equation vary a lot due to diverse geological settings around the world, so it is better to establish a local f–Z fitting equation for site sediment thickness estimation. In order to obtain a suitable fitting equation, the amount of available borehole data should be sufficient and the sediment thicknesses revealed by these boreholes should have a certain span. Besides, it is best that the locations of those boreholes are uniformly distributed in the study area so the overall geology condition can be reflected. If it is not possible to establish an applicable f–Z fitting equation due to lack of borehole data in the study area, researchers can refer to one fitting equation in area with similar geology condition from literatures.