Surface waves are widely used to model shear-wave velocity of the subsurface. Surface wave tomography (SWT) has recently gained popularity for near-surface studies. Some researchers have used straight-ray SWT in which it is assumed that surface waves propagate along the straight line between receiver pairs. Alternatively, curved-ray SWT can be employed by computing the paths between the receiver pairs using a ray-tracing algorithm. The SWT is a well-established method in seismology and has been employed in numerous seismological studies. However, it is important to make a comparison between these two SWT approaches for near-surface applications since the amount of information and the level of complexity in near-surface applications are different from seismological studies. We apply straight-ray and curved-ray SWT to four near-surface examples and compare the results in terms of the quality of the final model and the computational cost. In three examples we optimise the shot positions to obtain an acquisition layout which can produce high coverage of dispersion curves. In the other example, the data have been acquired using a typical seismic exploration 3D acquisition scheme. We show that if the source positions are optimised, the straight-ray can produce S-wave velocity models similar to the curved-ray SWT but with lower computational cost than the curved-ray approach. Otherwise, the improvement of inversion results from curved-ray SWT can be significant.
Surface waves are commonly analysed to build shear-wave velocity (VS) models. Surface wave tomography (SWT) is a well-established method in seismological studies, where numerous researchers have used it to construct subsurface velocity models at global and regional scales by inverting earthquake signals (Woodhouse and Dziewonski, 1984; Ekstrom et al., 1997; Ritzwoller and Levshin, 1998; Boschi and Dziewonski, 1999; Simons et al., 1999; Boschi and Ekstrom, 2002; Yao et al., 2010). Some authors have applied SWT using ambient noise cross-correlation to retrieve regional crustal structures (Shapiro et al., 2004; Lin et al., 2008; Kästle et al., 2018).
The SWT usually consists of three steps (Yoshizawa and Kennett, 2004; Yao et
al., 2008). First, different path-averaged dispersion curves (DCs) are computed for different receiver pairs aligned with a source. Then, the DCs are inverted to build phase velocity maps at different periods (frequency). Finally, the obtained phase velocity maps are inverted to produce 1D VS models at different locations. However, the efficiency of SWT can be increased by the direct inversion of the path-averaged DCs, i.e. skipping the intermediate step of building phase velocity maps (Boschi and Ekstrom, 2002; Boiero, 2009; Fang et al., 2015).
Traditionally, in seismology SWT has been employed assuming great-circle propagation of surface waves (Trampert and Woodhouse, 1995; Ekstrom et al., 1997; Passier et al., 1997; Ritzwoller and Levshin, 1998; Boschi and Dziewonski, 1999; van Heijst and Woodhouse, 1999; Simons et al., 1999; Boschi and Ekstrom, 2002; Lin et al., 2008; Yao et al., 2010; Bussat and Kugler, 2011; Kästle et al., 2018). However, some researchers have employed SWT not assuming the great-circle propagation of surface waves (Spetzler et al., 2002; Yoshizawa and Kennet, 2004; Lin et al., 2009). SWT has been used in seismological studies for decades and different SWT approaches have been compared by seismologists. For instance, Laske (1995) studied deviations from a straight line in the propagation of long-period surface waves and concluded that they usually have small effects on the propagation phase. Spetzler et al. (2001) applied both straight-ray and curved-ray SWT methods. They computed the maximum deviations of ray paths from straight lines and pointed out that this maximum is typically below the estimated resolution, except for long paths at short periods. Some studies showed that a more complex forward modelling in SWT did not improve the results (Sieminski et al., 2004; Levshin et al., 2005), while other studies reported obtaining better results (Ritzwoller et al., 2002; Yoshizawa and Kennett, 2004; Zhou et al., 2005). Trampert and Spetzler (2006) pointed out that the choice of regularisation has a major impact on SWT results. They studied SWT methods based on ray theory (straight ray and curved ray) and scattering theory in which the integral along the ray path is replaced by the integral over an influence zone. They showed that these methods are statistically alike and any model from one method can be obtained by the other one by changing the value of the regularisation. They concluded that the only option to increase the resolution of the model is to increase and homogenise the data coverage. Bozdag and Trampert (2008) compared straight-ray and curved-ray SWT methods in their study and mentioned that performing ray tracing could be so time-consuming that the potential gain in crustal corrections on a global scale might not be worth the additional computational effort imposed by ray tracing. Despite seismological studies, a comparison between the performance of straight-ray and curved-ray SWT at the near-surface scale is missing. In near-surface studies, the shot locations can be optimised to ensure that a high coverage of DCs can be achieved. This abundance of information facilitates shallow 2D or 3D characterisation with great details. Due to its ability to provide high lateral resolution, SWT has recently attracted the attention of researchers for near-surface studies, where high lateral heterogeneity is expected. Few researchers have used SWT for near-surface characterisation assuming straight-ray propagation of surface waves. Kugler et al. (2007) characterised shallow water marine sediments using Scholte waves dispersion data, Picozzi et al. (2009) applied SWT on high-frequency seismic noise data to construct a VS model up to 25 m in depth, Rector et al. (2015) employed SWT to obtain a VS model in a mining site, Ikeda and Tsuji (2020) successfully applied SWT in laterally heterogeneous media, Papadopoulou (2021) showed the applicability of SWT in near-surface characterisation in a mining site consisting of hard rocks and Khosro Anjom (2021) constructed a 3D VS model applying SWT on a large 3D dataset acquired for testing purposes in a mining area.
Since the level of complexity and lateral heterogeneity in the near surface is expected to be higher than in most seismological studies, the straight-ray approximation of surface waves may not be valid and curved-ray tomography should be used by means of ray tracing at each frequency. Fang et al. (2015) applied SWT on a shallow crustal study considering the effect of heterogeneity on wave propagation. They performed surface wave ray tracing at each frequency using a fast-marching method (Rawlinson and Sambridge, 2004). Wu et al. (2018) applied curved-ray SWT to obtain a shallow VS model at a mining site. Barone et al. (2021) applied different tomography methods, including eikonal tomography, to 3D active seismic data.
In curved-ray SWT, at each iteration a ray tracing method is applied at each frequency component to compute the ray path between the source and receivers. Even though this can increase the accuracy of the final model, it will lead to higher computational cost compared with straight-ray SWT. The computational efficiency is of great importance in seismic near-surface as, compared to seismological studies, the abundance of data at active seismic near-surface projects can significantly increase the computational cost. Therefore, it is necessary to investigate the gained improvement together with the associated additional computational cost from the curved-ray SWT over the straight-ray approach.
We apply straight-ray and curved-ray SWT on four datasets. Two examples include 3D synthetic models containing lateral velocity heterogeneity. We then apply SWT on two field datasets to evaluate the method on real data. For each dataset, 3D VS models from straight-ray and curved-ray SWT are obtained by direct inversion of DCs, and the accuracy and computational efficiency of the two approaches are compared.
In this section the applied methodology is described. Besides explaining the straight-ray and curved-ray SWT approaches and the differences between them, we also describe the employed procedure to optimise source positions and the process to estimate the DCs from the raw data.
For a given (random or regular) array configuration, we can optimise the locations of shots to ensure having high coverage DCs with minimum number of shots based on the guidelines by Da Col et al. (2020). In this approach, many shot positions are defined as the potential shot candidates. For each shot, we find all receiver pairs aligned with the shot. After computing all the possible receiver pairs for all the defined shot positions, the shots are sorted based on the number of in-line receiver pairs. Then we pick the shots which could provide the greatest number of unique pairs (i.e. potential DCs). If the data coverage is satisfactory also from the azimuthal point of view, we consider the selected shots as the final ones. Otherwise, more shots are added to increase the data coverage.
From the presented four examples in this study, the shot positions have been optimised for three examples (case studies 1–3). We also use a dataset (case study 4) where the acquisition layout mimics at a smaller scale the classical seismic exploration 3D cross-spread acquisition scheme with orthogonal lines of sources and receivers. This dataset, not being optimised (for a SWT study) will help in analysing the criticalities introduced by a non-optimal acquisition scheme.
Once the acquisition layout is finalised, the DCs are estimated from the acquired data. We use a MATLAB code (Papadopoulou, 2021) that automatically retrieves the DCs between each receiver pair that are collinear with a source.
Here, we provide the general concepts based on which the code estimated the
DCs from the raw data (for detail see Papadopoulou, 2021). For each receiver
pair, a frequency domain narrow band-pass Gaussian filter, which was
originally proposed by Dziewonski and Hales (1972), is used to analyse the
traces into monochromatic components. The traces are then cross-correlated
frequency by frequency to produce the cross-correlation matrix. The phase
velocities of surface waves correspond to the maxima on the
cross-correlation matrix, but there are many maxima because in the
two-station method the observed phase delay is invariant under
For the frequency band of the generic
We carry out our experiments in a Cartesian coordinate system. The
subsurface is discretised into a set of 3D grid blocks where it is assumed
that the only unknown parameter of each grid block is the VS value while the Poisson ratio
To obtain the forward response in the curved-ray SWT, first the ray path
between the generic receiver pair
The ray path between receivers
The path-average phase slowness along the discretised path for each
frequency (
We obtain the vector of the simulated DC for the receiver pair (
The employed inversion algorithm is based on the method proposed by Boiero (2009). We solve the inverse problem aiming at minimising the misfit
function (
To estimate the DCs from raw data, we have used the auto-picking code
(Papadopoulou, 2021) in which the DCs are sampled uniformly in frequency.
This means that each DC is non-uniformly sampled in terms of wavelength
which can drive the inversion algorithms to the shallowest part of the
subsurface without any significant updates in the deeper portion of the
initial velocity model (Khosro Anjom and Socco, 2019a). To address this
issue, a wavelength-based weighting scheme was applied in the inversion
process to compensate for this non-uniformity (see Khosro Anjom et al.,
2021, for details). Hence, the
In this section, we apply straight-ray and curved-ray SWT approaches on four
datasets and compare the results. For each example, the straight-ray and
curved-ray SWT inversions start from the same initial model. Other inversion
parameters (
The Blocky model consists of a sequence of layers with vertically increasing
velocity values, surrounding two blocks of velocity anomalies which extend 4 m in the horizontal and vertical directions (Fig. 2). The receivers are located
in a regular grid with 1 m spacing in an area of 20 m
True VS model.
Geophysical parameters of the Blocky model.
The estimated 971 DCs are shown in Fig. 3a. The initial model for the
inversion is defined as a 5-layer 3D model where the thickness (
The VS models from SWT inversion. Straight-ray SWT results are
shown at:
Figure 4 shows that straight-ray and curved-ray SWT have modelled the location and the value of the high-velocity anomaly relatively accurately. The model from the curved-ray method is slightly superior at the grid blocks surrounding the high-velocity box (the black arrows in Fig. 4). In the case of a low-velocity anomaly, curved-ray SWT provided better results, since the bottom half of the low-velocity block is better resolved by the curved-ray approach (the white arrows in Fig. 4b and e). As shown in Table 4, the curved-ray approach produced slightly lower model misfit than straight-ray SWT.
The sand bar model is designed to simulate a saturated environment where a
sand layer is buried in unconsolidated clay. The model contains a
curve-shaped high-velocity anomaly (the sand layer) embedded between two
low-velocity clay layers (Fig. 6). The geophysical parameters of the sand
bar model are shown in Table 2. The receivers are distributed at the surface
as a regular grid with 2 m spacing (Fig. 5a). We defined sources at the same
location of receivers and for each source we computed the aligned receiver
pairs. Then, we picked 13 shots (Fig. 5a) which provided the highest data
coverage to generate the synthetic data. The same finite difference code
used for the Blocky model was used to obtain the sand bar synthetic dataset
and no error was added to the synthetic data. The source is a function of
Ricker wavelets with a dominant frequency of 40 Hz, and the minimum element
size of the mesh grid was set to 0.1 m to prevent numerical dispersion. The
time stepping was equal to
Geophysical parameters of the sand bar model.
True VS model.
The retrieved 1207 DCs are depicted in Fig. 6a. The defined initial model is
a 3D model with 10 layers of constant 1 m thickness where the VS values are
fixed at 80 m s
SWT inversion results. Straight-ray SWT results are shown at:
We can see in Fig. 7 that the VS models from both approaches are similar to each
other. Figure 7 shows that both methods have successfully located the
high-velocity anomaly. Not only are the velocity values close to the true
model (Fig. 5b to d) but the shape of the anomaly has also been clearly retrieved. The vertical slices at
The data were acquired in a field near Pijnacker, South Holland, the Netherlands
(Fig. 8a). An area of 27 m
Each inversion block extends 3 m horizontally. In this case the initial
model contains 6 layers where the layers get thicker with depth. The first
2 layers are 1 m thick, followed by 2 layers of 2 m and 2 of 3 m. The
initial model is defined regardless of the well information. The well data
are used later to assess the inversion results. The inversion started from
an initial VS value of 60 m s
The misfit function values at different iterations of SWT inversions.
SWT inversion results. Straight-ray SWT results are shown at
We can see in Fig. 10 that also in this case the VS models from the
straight-ray and curved-ray SWT are similar. The difference between the
horizontal slices (Fig. 10a and d) are clearer than the vertical ones. As
shown in the black dashes, the cells around the high velocity
portion have lower VS values in the curved-ray (Fig. 10d) than straight-ray SWT
(Fig. 10a). A previous 2D full waveform study (Bharadwaj et al., 2015) on a
clay field, which was not very far from the field location of our study,
obtained a VS model in the range of 40–80 m s
Parameters of the initial model for the inversion in the Pijnacker field.
The field data were acquired at the National Research Council (CNR) headquarters
in Turin, Italy (Fig. 11a and b). The site consists of compacted sand and
gravel formations surrounding an artificial loose sand body. The sand body
occupies an area of 5 m
The inversion started from an 8-layer 3D model where the horizontal and
vertical sizes of each grid are 0.5 m, VS is equal to 200 m s
The values of misfit function at different iterations of SWT inversions.
The VS models from SWT inversion. The boundaries of the sand body
are superimposed in black dashes. The results of the straight-ray SWT inversion are shown
as
Figure 13 shows that the differences between straight-ray and curved-ray models are more pronounced in this example. There are some cells with relatively high-velocity values inside the sand body in the model obtained from the straight-ray SWT (Fig. 13a). The boundaries of the loose sand body at the surface are better retrieved by the curved-ray SWT (Fig. 10d). The areas shown in white dashes in Fig. 13b and e show that the gradual increase of the VS values with depth inside the sand body is clearer in the model from the curved-ray SWT(Fig. 13e). The black arrow in Fig. 13c shows the high-velocity cells inside the loose sand body in the retrieved model from the straight-ray SWT. The reason is that in this area (close to the interface of the sand body and the background medium) the assumed paths in the straight-ray approach are much shorter than the true paths and therefore the obtained VS from the inversion becomes unrealistically high. This artefact does not exist in the corresponding slice from the curved-ray SWT (Fig. 13f).
We have applied straight-ray and curved-ray SWT to four datasets and compared the results. In this section, we investigate the results in more detail considering ray paths, data weighting, models and data misfits, and computational cost.
The improvement of the model obtained by the curved-ray SWT compared to straight-ray SWT, particularly at the boundaries of velocity anomalies, has been shown in the synthetic and real world examples. Some selected examples of the computed ray paths at the last iteration of the curved-ray SWT are depicted in Fig. 14.
Examples of the computed ray paths at the last iteration of the
curved-ray SWT inversion for
In all the three models in Fig. 14 the receivers A and B are located outside
the velocity anomalies, and we see that the computed ray paths between them
do not cross the anomalies. Therefore, the obtained paths do not deviate
considerably from a straight line. In Fig. 14a, the high-velocity anomaly
exists at the depth range of 3–6 m. Hence, we can see that the
high-frequency components of the DCs, which correspond to the shallow parts
of the model, do not deviate from a straight line; however, the lower frequencies
(i.e. higher wavelengths) for the C-D and G-H pairs have deviated from a
straight line and travelled through the high-velocity parts. In Fig. 14b,
the depth of velocity anomalies is in the range of 2–6 m. We see that also in
this case the ray paths for higher frequencies have almost no deviations
from a straight line as they do not cross the anomalies. However, we can
see for the obtained paths between B-C and D-E pairs that the lower
frequencies have bypassed the low-velocity anomaly. Similarly,
lower frequencies in the case of the G-F pair have deviated from a straight path and travelled through the high-velocity anomaly. In Fig. 14c, the sand body
(low-velocity anomaly) starts at the surface and reaches a maximum depth of 2.5 m. Its area shrinks from 5 m
Even though the exact boundaries of the anomaly (sand layer) are unknown for the Pijnacker field, the computed ray paths can provide helpful insights. For instance, the computed ray paths from straight-ray and curved-ray SWT, for the DCs data with the wavelengths in the range of 6–9 m are displayed in Fig. 15.
The computed ray paths for the data points with wavelengths in the
range of 6–9 m for the Pijnacker field dataset from
As the initial VS model is vertically and horizontally homogeneous, the initial ray paths for both straight-ray and curved-ray SWT are straight lines. As shown in Fig. 15a, the ray paths do not change during the inversion in the straight-ray approach. However, the paths are updated at every iteration of the curved-ray SWT inversion. We see that some areas in Fig. 15b (shown in dashed red) are bypassed by almost all the rays even though the data coverage of these areas in the straight-ray approach (Fig. 15a) is considerably higher. Therefore, these portions correspond to the low-velocity materials, i.e. clay and peat. The area between these low-velocity portions has both a higher concentration of ray paths and higher average phase velocity values. Therefore, they probably show the sand layer. These locations agree with the obtained VS model from the curved-ray SWT inversion (Fig. 10d).
The ray paths for the data points of CNR example with wavelengths
in the range of 0–1 m from
We have shown the inversion results from the straight-ray and curved-ray
SWT. In this part, we compare the results quantitatively. We carried out the
inversions on 40 cores on a cluster with the processor type of
Intel® Xeon® E5-2650 v3. In Table 4, we report
the number of iterations (
The quantitative comparison of straight-ray (SR) and curved-ray (CR) SWT of the various case studies.
We define the relative data misfit (
We can see in Table 4 that in all examples except for the Blocky model, the
curved-ray SWT has converged in less iterations than the straight-ray SWT.
However, the curved-ray SWT has increased
In all the examples presented, the computed VS models from the straight-ray and curved-ray SWT do not differ significantly except for the CNR field. Since the source positions had not been optimised in this example, the DCs coverage is low, particularly in the shallower portion of the medium. Figure 16 depicts the ray paths of the DCs data with the wavelengths in the range of 0–1 m.
We can see in Fig. 16a that some areas of the medium are not covered with straight rays, especially outside the sand body. It should be noted that for both cases, the ray paths at the first iteration are straight lines as the initial model has a constant VS value for all cells. However, in the curved-ray approach (Fig. 16b) the ray paths are flexible and can adjust to the updated subsurface velocity during the inversion process. Figure 16b also shows that the ray paths responded properly to the edges of the loose sand body, travelling through the faster part of the model.
As mentioned previously, uneven sampling of DC data in terms of wavelength can be problematic in SWT inversion. For instance, most of the extracted DC data (81 %) of the Pijnacker field have wavelengths less than 3 m while the available well data from the area suggest that the depth of the target is expected to vary in the range of 2–7 m. This can be a serious problem as the inversion might reach the defined stopping criteria without any significant updates in the deeper portion of the initial velocity model. Figure 17 shows the obtained VS models with (Fig. 17a and b) and without (Fig. 17c and d) the wavelength-based weighting at 5.5 m depth.
In Fig. 17, we can see the improvement of the model after applying the wavelength-based weighting method (Fig. 17a and b) compared with the non-weighted model (Fig. 17c and d) where the non-weighted inversions have barely retrieved any pattern to model the target (sand layer).
We have shown that optimisation of source positions can provide higher data coverage than a typical 3D cross-spread acquisition scheme. We have evaluated straight-ray and curved-ray SWT at the near-surface scale and a comparison of our results might not necessarily agree with previous (global or regional scale) seismological studies. For instance, Spetzler et al. (2001) pointed out that the maximum deviations of ray paths from a straight line is mostly below the estimated resolution. However, we have shown that the deviation from a straight line can be resolved at the near-surface scale (Fig. 14). We also showed that in the case of low data coverage, using the curved-ray approach significantly improves the obtained VS model. This result might not agree completely with the result of the (seismological) study by Trampert and Spetzler (2006), where they pointed out that increasing the data coverage is the main factor to increase the resolution of the model. Our results show that in the case of high data coverage, the model improvement achieved in the curved-ray approach may not be worth the additional computational effort, which agrees with the results of the study by Bozdag and Trampert (2008).
The impact of weighting on the SWT inversion results for the
Pijnacker dataset. The computed VS model at the depth of 5.5 m from
We applied SWT to four datasets and built near-surface VS models. We compared the obtained results from the straight-ray and curved-ray SWT in terms of data misfit, model misfit, and computational cost. We showed that compared to the straight-ray approach, using curved-ray SWT improves the accuracy of the computed VS model. We illustrated that the acquisition layout can play an important role in the data coverage obtained and consequently in the inversion results. We showed that the classical cross-spread acquisition layout (which was used in the CNR example) may not provide high DC coverage. In this case, the improvement of inversion results from the curved-ray SWT can be significant. We also showed that in the case of high data coverage, which can be achieved by optimisation of source positions, the difference between the obtained VS models from the straight-ray and curved-ray approaches can be very small even in the presence of high lateral variation of the velocity.
The data can be made available by contacting the corresponding author. The code can also be made available by contacting the corresponding author after carrying out some additional work to make it user-friendly.
MK worked on the processing and inversion of all the datasets, with coordination and supervision of ES and LVS. MK and ES contributed to seismic data acquisition in Pijancker field and LVS coordinated the seismic data acquisition in the CNR site. MK wrote the original paper draft, with contributions from all the authors.
The contact author has declared that none of the authors has any competing interests.
Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
We would like to thank Seismic Mechatronics for providing the vibrator source, CNR group for giving access to data acquisition, Compagnia di San Paolo for funding the PhD of Mohammadkarim Karimpour, and all the people involved in data acquisition.
This paper was edited by Ulrike Werban and reviewed by Emanuel Kästle, Fabrizio Magrini, and one anonymous referee.