Let Ω ⊂ R 2 be a domain and

M = { x 1 , ⋯ , x N } ⊂ Ω

a discretization of the domain Ω with N points. Each point of the discretization M is denoted as a node.

For each one of the nodes of the domain, where the value of U is unknown, E_s-star is defined as a set of selected points E s = { x 0 , x 1 , ⋯ , x s } ⊂ M with the central node x₀ Î M and x i ( i = 1 , ⋯ , s ) ∈ M is a set of points located in the neighborhood of x₀. In order to select the points, different criteria as the four quadrants or distance can be used (Benito et al., 2001) .

We consider an E_s-star with the central node x₀, where (x₀, y₀) are the coordinates of the central node, (x_i, y_i) are the coordinates of the i^th node in the E_s-star, and h_i = x_i − x₀ and k_i = y_i − y₀. If U₀ = U(x₀) is the value of the function at the central node of the star and U_i = U(x_i) are the function values at the rest of the nodes, with i = 1 , ⋯ , s , then, according to the Taylor series expansion it is known that:

U i = U 0 + h i ∂ U 0 ∂ x + k i ∂ U 0 ∂ y               + 1 2 ( h i 2 ∂ 2 U 0 ∂ x 2 + k i 2 ∂ 2 U 0 ∂ y 2 + 2 h i k i ∂ 2 U 0 ∂ x ∂ y ) + ⋯ ,     i = 1 , ⋯ , s (1)

If in Equation (1) the higher than second order terms are ignored, a second order approximation for the U_i function is obtained. This is indicated as u_i. It is then possible to define the function B(u) as:

B ( u ) = ∑ i = 1 s [ ( u 0 − u i ) + h i ∂ u 0 ∂ x + k i ∂ u 0 ∂ y     + 1 2 ( h i 2 ∂ 2 u 0 ∂ x 2 + k i 2 ∂ 2 u 0 ∂ y 2 + 2 h i k i ∂ 2 u 0 ∂ x ∂ y ) ] 2 ω i 2 (2)

where w i = w ( h i , k i ) are positive symmetrical weighting functions, having the property as defined in Lancaster and Salkauskas (1986) that are functions decreasing monotonically in magnitude as the distance to the center of the weighting function increases. See also Levin (1998) for more detail.

Some weighting functions as potential 1 d i s t n or exponential e − n ⋅ d i s t 2 can be

used (Benito et al., 2001) , where n ∈ ℕ . If the norm given by Equation (2) is minimized with respect to the partial derivatives, the following system of linear equations is obtained (for each node of the discretization)

A D = b (3)

The matrix A is a positive definite matrix and the approximation is of second order, θ ( h i 2 , k i 2 ) (Gavete et al., 2017) .

The solution of system Equation (3) is unique and it is a linear combination of the function values obtained at the nodes. Then

D = A − 1 b = − m 0 u 0 + ∑ i = 1 s m i u i (4)

with

m 0 = ∑ i = 1 s m i (5)

The explicit expressions of matrices A, D, b and coefficients m⁰, mⁱ may be seen in (Benito et al., 2001; Benito et al., 2015; Benito et al., 2007) .

The equations of motion for a perfectly elastic, homogeneous and isotropic medium in the domain Ω ⊂ R 2 are

ρ U i , t t = ( λ + μ ) U k , i k + μ U i , k k (6)

The U_i are the components of the displacements, ρ is the density and λ and µ are Lame’s elastic coefficients.

In this paper two types of boundary conditions are considered: Dirichlet boundary conditions and free surface. On the free surface, the following conditions we impose

σ i j ⋅ n j = ( λ ϵ k k δ i j + 2 μ ϵ i j ) ⋅ n j = g i ( t ) (7)

where g_i(t) is equal to zero when there are not forces on the surface.

PML x-dir can be conceptually assumed up by a single transformation of the original equation. Then, wherever and x derivative appears in the wave equations, it is replaced in the form

∂ ∂ x → 1 1 + i δ x ( x ) ω ∂ ∂ x (8)

where i = − 1 in Equation (8).

Substituting the expression Equation (8) in Equation (6) the equations for PML part (Benito et al., 2013) , are:

ρ [ U i , t t + 2 δ U i , t + δ 2 U i ] = ( λ + μ ) U j , j i + μ U i , k k (9)

The explicit scheme formulae for the second spatial derivatives with second order approximation (see Equation (4)), are

[ u i , j k 0 ] t = n Δ t = − m j k 0 u n i 0 + ∑ l = 1 s m j k l u n i l + Θ [ h i 2 , k i 2 ] (10)

where h_l = x_l − x₀ and k_l = y_l − y₀ the superscript n denotes time step, the superscripts 0 and l refer to the central node and the rest of nodes of the star respectively and s is the number of nodes in the star (in this work s = 8 and the star nodes are selected by using the distance criteria) (Benito et al., 2001) .

m j k 0 are the coefficients that multiply the approximate values of the functions U_i at the central node ( u n i 0 ) for the time n∆t in the generalized finite difference explicit expressions for the space derivatives, and m j k l are the coefficients that multiply the approximate values of the functions U_i at the other nodes of the star ( u n i l ). In all these expressions the cross-terms are equal.

The replacement in Equation (1) of the explicit expressions obtained for the spatial derivatives (Equation (10)) and the central-difference formula for the time derivative lead to the explicit differences scheme:

u n + 1 i 0 = 2 u n i 0 − u n − 1 i 0 + ( Δ t ) 2 ρ [ ( λ + μ ) ( − m j i 0 u n i 0 + ∑ l = 1 s m j i l u n j l )     + μ ( − m k k 0 u n i 0 + ∑ l = 1 s m k k l u n j l ) ] + Θ [ ( Δ t ) 2 , h i 2 , k i 2 ] (11)

and the scheme in GFDM for the PML part can be obtained in a similar way:

u n + 1 i 0 = 1 1 + δ Δ t [ ( 2 − δ 2 ( Δ t ) 2 ) u n i 0 + ( Δ t ) 2 ρ [ ( λ + μ ) ( − m j i 0 u n i 0 + ∑ l = 1 s m j i l u n j l )                           + μ ( − m k k 0 u n i 0 + ∑ l = 1 s m k k l u n j l ) ] − ( 1 − δ Δ t ) u n − 1 i 0 ] + Θ [ ( Δ t ) 2 , h i 2 , k i 2 ] (12)

where represents the truncation error of the scheme.

In order to implement the free surface conditions, Equation (7), we need to add the same number of nodes outside of our domain as there exists on the free boundary and place them near it. Substituting the first order derivatives that appear in the free surface conditions Equation (7) by the explicit expressions (Equation (10)) the system of 2p equations is obtained:

{ λ ( − m k p u n k p + ∑ l = 1 s m k l u n k l ) + μ [ ( − m j p u n i p + ∑ l = 1 N m j l u n i l ) + ( − m i p u n j p + ∑ l = 1 N m i l u n j l ) ] } ⋅ n j = g i ( t ) (13)

where the 2p unknowns are the displacements in the p added nodes. These unknowns appear in the summation.

The stability of the schemes has already been studied in (Ureña et al., 2011) and its formulation leads to

Δ t < 4 ( α 2 + β 2 ) [ ( | m 11 0 | + | m 22 0 | ) + ( m 11 0 + m 22 0 ) 2 + ( m 12 0 ) 2 ] (14)

where α and β are the velocities of the P-waves and S-waves respectively.

The stability analysis in the continuum of the PML has already been studied in (Salete et al., 2011) .

The dispersion for the phase and group velocities have also been studied in depth in (Gavete et al., 2017) .

If the density and Lamé coefficients λ and µ are functions of spatial variables, in agreement with Moczo (1998) , in seismological problems the heterogeneous approach can be complicated and, then, so-called heterogeneous formulation is preferred. Therefore, the same formulae are used for all points in the domain and material discontinuities are accounted for spatial variation of the material parameters (Benito et al., 2013) .

A geotechnical investigation has been carried out to elaborate a geotechnical profile on a construction site. Three boreholes with core recovery and an additional cross-hole test were performed in order to obtain a seismic velocity vs. depth profile. Based on the geotechnical investigation aforementioned, the subsoil comprises an upper layer of medium dense to dense sand up to 6-meter depth, overlying a very dense silty sand layer. The λ parameter, the dynamic Shear Modulus and the compression and shear wave velocity vs. depth are shown in Figure 1.

The dynamic parameters shown in Figure 1 have been assigned to the subsoil as a function of soil depth.

The cross-hole test has been analyzed by in a GFDM model, applying a vertical pulse (to generate S-waves) and a horizontal pulse (to generate P-waves) at different depths. The step pulse is applied for a time increment of t = 1e−5 sec. The model is 16 m (vertical) × 6 m (horizontal), with a regular node spacing of 0.05 m. The number of nodes is 38,400. The star is made up of eight nodes plus the central node. The selection of nodes associated to each central node is based

on the distance criterion. The weighting function adopted is the potential function 1/(dist)³. A free surface boundary condition has been imposed to the upper and right limits of the model. An absorbing boundary condition (Perfectly Matched Layer) has been imposed to the lower and left limits of the model.

Figure 2 shows the moment at which the front S-wave reaches the geophone located at 9-meter depth in the GFDM model. The shear, head or Von-Schmidt and surface or Rayleigh waves can clearly be seen in Figure 2.

The vertical displacement over time for the geophone located at 9-meter depth is shown in Figure 3. The shear wave velocity is obtained from the time of wave arrival, taking into account the distance between the source and the geophone (3 meters).

By applying a horizontal pulse at the source location, the P-waves propagation has also been analyzed. The pressure or dilatational, shear, head and surface waves can be seen in Figure 4.

The curve that appears in Figure 5 shows the horizontal displacement u_x over time for the geophone located at 9-meter depth. The P-waves velocity is obtained from the time of wave arrival, similarly to the S-waves case.

The seismic refraction method is usually performed in quarries or open-pit mines to make an estimate of the rock volume that may be extracted from the site and the most appropriate excavation method. For example, the seismic refraction method carried out in a quarry in Valparaiso (Chile) to quantify the weathered and fresh rock volume. The layout of the seismic refraction profiles is shown in Figure 6. As a result of the geophysical study, the longitudinal profile shown in (Figure 7) was obtained.

Basically, the subsoil is composed of three layers (sandy oil, weathered rock and fresh rock).

The purpose of the model is to simulate the real geophysical investigation, measuring the travel time for the wave to reach the geophones located on the

ground surface along the profile. The real data obtained from the geophysical investigation are shown in Figure 8. The different dromochrones (distance vs. travel time curves) have been drawn for the different pulse locations. Three slopes may be distinguished in most of these curves, corresponding to the three soil layers aforementioned: sandy soil, weathered rock and fresh rock. Three representative dromochrones have been selected (Figure 9) to enhance these three slopes.

The slope of the first section of the three curves is very similar, leading to a seismic velocity of 350 - 500 m/s for the sandy soil layer. Nevertheless, the point where the curve changes its slope depends on the thickness of the soil layer. On curve 2, the time when the refracted wave reaches the direct wave is shorter than the one for curves 1 and 3. This difference leads to a smaller thickness of layer 1 in the neighborhood of x = 30 m. Regarding the second section of the curves, the

seismic velocity obtained for the weathered rock layer ranges from 1000 to 2000 m/s. Finally, the seismic velocity for the third layer (fresh rock) ranges from 5000 to 11,000 m/s.

A GFDM model has been used to model the seismic refraction investigation (Figure 10). The horizontal and vertical dimensions of the model are 200 meters and 60 meters, respectively. A regular mesh of nodes has been adopted with a node spacing of 1 meter, leading to a total number of nodes of 12,000. Every star

is made up of eight nodes plus the central node. The selection of nodes associated to each central node is based on the distance criterion. The weighting function adopted is the potential function 1/(dist)³. A free surface boundary condition has been imposed to the upper limit of the model. An absorbing boundary condition (Perfectly Matched Layer) has been imposed to the lower and side limits of the model. The vertical pulse applied at the surface at x = 50 m. (50 meters from the left border) corresponds to the central lobe of a Ricker wavelet. The parameters for the Ricker wavelet adopted are the following ones: Ricker wavelet:

A = [ 1 − 2 π 2 f M 2 ( t − t s ) 2 ] e ( − 2 f M 2 ( t − t s ) 2 )

where: t_s = 0.00015 s and f_M = 1418 s⁻¹.

The time increment adopted is t = 0.00005s. Therefore, the central lobe for the Ricker wavelet can be defined by 7 points, from t = 0 s. until t = 0.0003 s.

The main goal of the model is to reproduce the distance vs. travel time curves obtained in the geophysical investigation. The curves depend not only on the elastic parameters of the soil layers (Youngs Modulus, Shear Modulus), but on their thickness. By varying the elastic parameters and the thickness of the soil layers, a distance vs. travel time curve has been obtained whose seismic velocities lie within the real velocities range. Figure 11 shows the dromochrone obtained from the model.

The seismic velocities calculated from the model data curve in Figure 11 are the following ones:

Layer 1 (sandy soil): vp = 460 m/s.

Layer 2 (weathered rock): vp = 1590 m/s.

Layer 3 (fresh rock): vp = 5000 m/s.

In Figure 12 three curves showing the wave amplitude vs. time are shown. The red one corresponds to the point A (50.0, 0.0) where the pulse has been applied. The green one corresponds to the node B (50.0, 7.0), located 7 meters below the pulse node; and the blue one shows the curve for node C (50.0, 9.0), located 9 meters below the pulse node, that is, 1 meter in the second soil layer (weathered rock).

To illustrate the capability of the model to reproduce the refraction of waves, several screenshots of the model have been shown in Figures 13-15. The refraction is clearly modelled between layers 1 and 2, and between layers 2 and 3.

The graphic scale has been adjusted to enhance the wavefront, which results in a higher color saturation for the zones with higher amplitude. The P waves wavefront has been indicated in the figures, showing the higher wave velocity in the layer 2 compared to layer 1. Besides, the Rayleigh waves can be identified at the surface close to the point where the pulse has been applied. In addition, the refracted waves wavefront reaching the surface can be clearly distinguished in Figure 15.

Figure 16 shows the comparison between the actual seismic velocities recorded in the cross-hole test and those obtained with the model.

As it may be observed in Figure 16, the model-calculated seismic velocities match satisfactorily to those obtained in the cross-hole test, with a 5% maximum deviation. It is therefore concluded that the model is capable to reproduce accurately a cross-hole test, with minimum error in P-wave and S-wave velocities.

Figure 17 shows the real dromochrones along with that one obtained from the model.

The seismic velocities calculated from the GFDM fall within the range for the seismic velocities derived from the field investigation. Therefore, it can be concluded that the GFDM model may successfully simulate the real seismic refraction investigation to obtain the thicknesses and the seismic velocities of the different layers of a geological profile.

It is also necessary to clarify that the discretization of a geological profile in a finite number of different layers is a simplified view of the geology. Frequently, the edge between two layers is not a well-defined line, but a zone where the transition from one layer to another takes place. It happens, for example, in the transition from weathered and fresh rock. This explains the abrupt change in the

slope of the obtained curves, characteristic of these kinds of simulations.