The acquisition of seismic exploration data in remote locations presents several logistical and economic criticalities. The irregular distribution of sources and/or receivers facilitates seismic acquisition operations in these areas. A convenient approach is to deploy nodal receivers on a regular grid and to use sources only in accessible locations, creating an irregular source–receiver layout. It is essential to evaluate, adapt, and verify processing workflows, specifically for near-surface velocity model estimation using surface-wave analysis, when working with these types of datasets. In this study, we applied three surface-wave techniques (i.e., wavelength–depth (W/D) method, laterally constrained inversion (LCI), and surface-wave tomography (SWT)) to a large-scale 3D dataset obtained from a hard-rock site using the irregular source–receiver acquisition method. The methods were fine-tuned for the data obtained from hard-rock sites, which typically exhibit a low signal-to-noise ratio. The wavelength–depth method is a data transformation method that is based on a relationship between skin depth and surface-wave wavelength and provides both S- and P-wave velocity (

In order to overcome the difficulties of collecting seismic data in remote areas such as foothills and forests, a new acquisition method has been recently introduced, in which the nodal receivers are deployed in a regular grid, while the source locations are restricted to reachable areas such as the access roads (Lys et al., 2018). This approach creates an irregular source–receiver outline that raises the necessity to evaluate, verify, and test the seismic processing workflow. Here, we focus on the application of surface-wave methods to the data recorded through an irregular source–receiver scheme with the purpose of near-surface velocity model estimations. These velocity models can be used for engineering purposes or as input in the exploration processing workflow to improve static corrections and ground roll removal.

Surface-wave methods are powerful tools for subsurface characterization. Most of these methods process the data to extract the surface-wave phase velocity dispersion curve (DC) from seismic records and invert these DCs individually to estimate the velocity models. Since the energy decay of surface-wave wavefields in depth depends on their wavelength, the investigation depth of surface-wave methods is related to the maximum recovered wavelength and can be considerably variable, ranging from a few meters (e.g., Xia et al., 2002; Feng et al., 2005; Comina et al., 2011; Pan et al., 2018) to several tens of meters (e.g., Mordret et al., 2014; Da Col et al., 2020) or even to a few kilometers (e.g., Ritzwoller and Levshin, 1998; Kennett and Yoshizawa, 2002; Fang et al., 2015). The estimated models from surface-wave techniques can be used in many applications, such as near-surface site characterization (Lai, 1998; Xia, 2014; Foti et al., 2015), static corrections (Mari, 1984; Roy et al., 2010), and ground roll prediction and damping (Blonk and Herman, 1994; Ernst et al., 2002; Halliday et al., 2010).

Different methods can be adopted for extracting DCs from the seismic records and inverting them (see, for instance, Papadopoulou (2021) for a thorough review of the different processing techniques and characteristics of the estimated DCs). The retrieval of a velocity model from the DC can be based on simple data transformations or on model optimization approaches with different inversion strategies. According to the chosen workflow, the computational cost and model resolution may vary, and identifying the optimal approach for the analysis is an important task. Here, we compare three different methods (i.e., wavelength–depth data transform, laterally constrained inversion, and surface-wave tomography) that are rarely used for near-surface 3D model estimation. We apply these methods to a large-scale test dataset acquired from irregular deployed source–receiver layout. The data were collected from a hard-rock site, which is typically characterized by a lower signal-to-noise ratio compared to data from loose granular material (Papadopoulou, 2021).

Regardless of the type of surface-wave technique, since phase velocity DCs are known to have lower sensitivity to P-wave velocity (

The earliest applications of LCI were on resistivity data (Auken and Christiansen, 2004; Wisén et al., 2005; Auken et al., 2005). The first successful application of LCI to surface waves was performed by Wisén and Christiansen (2005). Despite LCI's capability as an effective tool for estimating near-surface models, its full potential in practical applications has not been fully exploited. In this technique, several multi-channel DCs available along a line or over an area are associated with local relevant 1D models and inverted simultaneously. The parameters of the 1D models are connected laterally and vertically through a set of constraints, whose strength controls the variations between model parameters at adjacent model points (Boiero and Socco 2010). As a result, consistent and smooth estimated pseudo-2D or 3D models are usually obtained from the LCI applications.

In the context of earthquake seismology, SWT is a well-established method for

In the literature, DC estimation methods are usually categorized into multi-channel and two-station methods, even though there are no theoretical or significant technical differences between the two approaches (Papadopoulou, 2021). The multi-channel technique is the most common approach, in which the recordings from an array of receivers (in a 2D scheme) go through a wavefield transform (e.g.,

Here, we show the application of the three surface-wave methods (W/D, LCI, and SWT) to estimate both

In this paper, we first introduce the site and describe the acquired data. Then, we explain the multi-channel and two-station DC estimation processing techniques. Then, we briefly describe the W/D, LCI, and SWT velocity model estimation methods and show their application to the dataset. We use the W/D method to estimate the a priori Poisson's ratio required by the LCI and SWT methods, which we then employ to transform their

The location is a limestone quarry in Aurignac in the south of France (Fig. 1a). In Fig. 1b, we show the satellite view of the site superimposed with the elevation map of the area. From north-west to south-east, a significant natural (outside the pits) and human-made (inside the pits) elevation contrast is present, which can cause highly scattered surface waves. In Fig. 1c, we show the geological map of the area from the website of the French Geological Survey (BRGM). The central, eastern, and northern parts of the site are characterized by stiff formations belonging to Thanetian and Sparnacian stages, primarily composed of stiff limestone and marl. In the western zone, recent loose deposits are present (Ypresian), creating a significant lateral variation between the east and west portions of the site. The very dense limestone with dolomite layers from Danian stage is outcropping in the north, outside of the investigated area, and is expected to be reached in shallow subsurface in the investigated zone.

Acquisition parameters of the dataset for outside the mining pits.

The seismic campaign was conducted inside and outside the two open mining pits to test the irregular source–receiver layout acquisition technique at a hard-rock site and provide an exploration dataset to be used for testing different processing approaches. Altogether, 918 receivers were deployed on a regular grid (area of 1.7

To minimize the effect of elevation contrasts (Fig. 1b), we split the data into two sub-datasets (north and south), each corresponding to an area with relatively flat topography. In Fig. 2, we show the acquisition layout, where different colors are used for each sub-dataset.

The satellite view of the Aurignac site (© Google Earth) superimposed with the acquisition layout. The data are divided into two sub-datasets shown with different colors, each within a relatively flat area. The recordings from the highlighted shot (green circle) are plotted in Fig. 3.

An example seismogram from the northern zone of the Aurignac site. The shot location is highlighted with a green circle in Fig. 2.

In Fig. 3, we show the first 2

Two example DC estimations for the same location using two different receiver arrays.

Multi-channel dispersion analysis is usually performed by selecting recordings from multiple inline receivers with a source and performing domain transform (

To minimize the impact of lateral variability on the DC estimation of the 3D data, we consider the recordings from receivers spread over an area (Wang et al., 2015; Xia et al., 2009; Park, 2019). For each DC estimation, we select the receivers inside a square area (window) and consider the sources within a certain distance from the center of the square. We use the phase shift method (Park et al., 1998) to estimate the

Multi-channel DC estimation from the field data.

For the field data, we considered a window of 100

The two-station DCs are estimated applying interferometry to the recordings from receiver couples aligned with a source, assuming straight ray approximation. We use the algorithm developed by Da Col et al. (2020) and modified by Khosro Anjom et al. (2021). First, an automatic search is performed to find the receiver couples aligned with the source at each azimuth angle, considering 1° of tolerance for the deviation from a straight path. Given the scale of the site, we consider the propagation path as occurring over a plain area, and we neglect the great-circle approximation. Then, the traces are narrow-band filtered at various frequencies, using zero-phase Gaussian filters, and the filtered traces of the receiver couples are cross-correlated and assembled to form the cross-multiplication matrix. We use a third-order spline interpolator to convert the cross-multiplication matrix to the frequency–velocity domain. We stack the cross-multiplication matrices computed from the records of the same two stations, but different sources, to increase the signal-to-noise ratio. Finally, at each frequency, the phase velocity is picked as the maximum of the cross-multiplication matrix. To avoid cycle skipping, we use the closest multi-channel DC as a reference, and we automatically pick maxima closest to the reference DCs. To minimize the contamination of the fundamental mode by higher surface-wave modes, we damp the higher-mode data using the muting strategy of Khosro Anjom et al. (2019).

We applied the two-station DC estimation method to the data from the north of the site only (blue markers in Fig. 2). We performed an automatic search of the receiver couples aligned with sources within a 250

Pseudo-slices of the estimated path-averaged DCs from the north of the site shown within wavelength ranges of

In Fig. 7a–f, we show the data coverage within different wavelength ranges, where the color scale shows the path-averaged phase velocity. The data exhibit very high coverage for wavelengths between 40 to 220

The only inputs of the W/D method are the estimated multi-channel DCs. The method, as described in Khosro Anjom et al. (2021), is composed of four main steps: (i) the clustering of DCs, (ii) the selection of a reference DC for each cluster and the estimation of the corresponding time-average

Since the same W/D relationship is applied to different DCs to transform them into velocity models, to apply the method at sites with significant lateral variations, the DCs must be clustered into more homogenous sets, and one W/D relationship should be estimated and applied separately to each cluster of DCs. We use the hierarchical clustering algorithm developed by Khosro Anjom et al. (2017) to cluster the DCs.

For each cluster, a reference DC and its corresponding time-average

We use the estimated experimental W/D relationship to directly transform all DCs of the cluster into time-average

Clustering of the multi-channel DCs.

The method's inputs are the multi-channel DCs and the initial models at the location of the local DCs. An initial model defined as the thickness, density, Poisson's ratio, and

The inversion method is a deterministic least-squares inversion based on Auken and Christiansen (2004), which was developed by Boiero (2009) and modified by Khosro Anjom (2021) to support parallel computing. At each iteration the

The inputs of the tomographic inversion are the path-averaged DCs from the two-station method and the initial model. The parameters of the initial model are the thickness,

We use the tomographic inversion algorithm developed by Boiero (2009) and modified by Khosro Anjom et al. (2021). An essential part of the tomographic inversion is the computation of synthetic path-averaged DCs corresponding to the observed ones. We compute the path-averaged DCs, assuming a straight ray path approximation between the two receivers and as reciprocal of the average slowness along the paths discretized over the model grid. The phase velocities at the location of the discretized paths are computed by bi-linear interpolation of the phase velocities from local DCs corresponding to the adjacent model points (Boiero, 2009).

Similar to the LCI algorithm, a damped least-squares method (Marquart, 1963) with lateral constraints is used to iteratively update the model until the minimum misfit between synthetic and observed DCs is reached. The only parameter that updates in the inversion is

The clustering of all the estimated DCs generated two clusters. In Fig. 8a, we show the estimated DCs with the color scale based on the clustering of the DCs in Fig. 8b. The DCs of the western cluster (cluster A, shown in blue in Fig. 8a) present lower phase velocities compared to the eastern DCs (cluster B, shown in green in Fig. 8a).

The steps of estimating the reference W/D relationship and apparent Poisson's ratio for cluster A.

The steps of estimating the reference W/D relationship and apparent Poisson's ratio for cluster B.

The estimated

In Figs. 9 and 10, we show the steps of estimating the reference W/D relationship and apparent Poisson's ratio for the reference DCs of clusters A and B. We considered variable Poisson's ratios between 0.1 and 0.45 for the Monte Carlo inversion. Based on the information from the site, we considered density of 2000

For both clusters, the W/D relationship and apparent Poisson's ratio were not available for the first 20

The estimated

The estimated DCs from the two clusters were transformed to interval

Poisson's ratio estimation for clusters A and B.

We defined an initial model composed of nine layers overlying a half-space with constant thicknesses of 15

The estimated

We performed an unconstrained and several laterally constrained inversions to find the optimal level of constraints according to the strategy described in Boiero and Socco (2010). We chose a lateral constraint on

Given the high data coverage from the estimated path-averaged DCs (Fig. 7), we defined a dense model grid on the considered northern zone, composed of 300 1D models, aiming at obtaining a high-resolution model. We used the same initial models defined for LCI (Sect. 5.2).

The estimated

Comparison between the wavelength distributions of the multi-channel and two-station dispersion data.

In Fig. 15, we show the estimated

We showed the application of three surface-wave methods for

In Fig. 16, we show the wavelength distribution of the estimated multi-channel DCs in blue, which shows dense data sampling up to wavelengths of 300

Checkerboard test.

In Fig. 16, in gray, we also show the wavelength distribution of the estimated two-station DCs. Even though the total number of DCs from the two-station analysis (1301) is far more than from the multi-channel analysis (545), the large wavelength data points (

Isosurfaces of the estimated

The application of the W/D method to the dataset provided both

The box plot showing the difference between the

To compare the estimated models from each method quantitatively, we compute the difference between the estimated

The difference between the estimated

The geological map of the site, obtained from the French Geological Survey (©BRGM;

We compute the total differences between the estimated models of every two methods as

The total difference between the estimated

The approximated computational costs for each method.

In Table 3, we provide the approximated computational costs for each part of the three methods. The most time-consuming step of all methods is the DC estimation, which also involves expert user intervention. Compared to W/D and LCI, SWT usually requires more DCs to reach adequate data coverage for the tomographic inversion. We estimated 1301 DCs for SWT applied to the north of the site, whereas only 174 DCs were estimated for the application of the W/D and LCI methods to the same zone. The W/D relationship and Poisson's ratio estimation are a common stage for all three methods. The inversion running times (for LCI and SWT) given in Table 3 are for a single inversion trial using 10 CPU cores. Usually, in addition to an unconstrained inversion, several constrained inversions are performed to reach a satisfactory model in schemes of SWT and LCI methods, whereas the W/D method can be applied faster and is efficient for processing large-scale datasets. It is noteworthy to mention that SWT was limited to the northern zone due to computational limitations. The tomographic inversion with 1301 DCs and 300 model points was performed by a workstation equipped with 128

In Fig. 20, we show the geological map superimposed with the satellite view of the area and with the horizontal slice of the estimated

The application of the calibrated multi-channel (W/D and LCI) and two-station (SWT) methods showed promising results for the processing of the data with irregular source–receiver layout. The W/D method is cost effective and also provides

We showed the application of three surface-wave methods, W/D, LCI, and SWT, to estimate

The data are licensed to Politecnico di Torino, which allows research activities conducted only by Politecnico di Torino. As a result, the data cannot be made publicly available. The codes for the three surface-wave methods were developed in other projects. Nevertheless, the code for the W/D data transformation method may be made available by contacting the corresponding author.

FKA worked on the application of the methods to the dataset with the supervision of FA and LVS. FKA wrote the original paper draft, with the contributions and revisions of FA and LVS.

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 made in the text, published maps, institutional affiliations, or any other geographical representation in this paper. While Copernicus Publications makes every effort to include appropriate place names, the final responsibility lies with the authors.

We would like to thank GALLEGO TECHNIC Geophysics for licensing the field data. Farbod Khosro Anjom would like to thank TotalEnergies for supporting his PhD, during which this study was carried out. We also thank the topical editor, Caroline Beghein, and the anonymous reviewers for their thorough reviews and useful suggestions.

This paper was edited by Caroline Beghein and reviewed by three anonymous referees.