Numerical analysis of the geophysical parameter sensitivity on the multimode Love wave dispersion curves

The characteristics of multimode Love wave dispersion curves are dependent on various geophysical parameters, namely shear wave velocity (SWV), layer thickness and density. This paper presents a novel numerical analysis of these parameters to determine which ones have a significant effect on the dispersion curves of multimode Love waves. Identifying parameter sensitivity is a crucial step in the inversion procedure of geophysical modeling. Our numerical analysis focuses on perturbing three geophysical parameters that influence the dispersion curves. The results confirm that SWV parameter is the most sensitive. By employing the same perturbation procedure, our numerical analysis reveals that the SWV parameter has a highly significant impact on the multimode Love wave dispersion curves. The average relative error (RE) values are found to be 27.33% for alpha = −0.2 and 25.51% for alpha = +0.2. Conversely, perturbing the layer thickness parameter demonstrates no significant influence on the dispersion curves, resulting in average RE values of 5.69% for alpha = −0.2 and 7.96% for alpha = +0.2. Furthermore, the perturbation of the density parameter exhibits an extremely negligible impact on the multimode Love wave dispersion curves, with average RE values on the order of 10−14%.


Introduction
In recent years, along with the advancement of computer technology, numerical analysis has been widely conducted in various disciplines, including in physics and its applications.A computational program employing convolutional neural networks (CNNs) have conducted to perform a fast approximation of effective hydrodynamic parameters in porous media research and applications.This paper presents a novel methodology for permeability prediction from micro-CT scans of geological rock samples [1].Mathematical and numerical modeling system have also tested to investigate transport mechanism between the growth media and the cells [2].A program of data-driven dispersion curve inversion network (DispINet) has presented to perform directly maps the multimode dispersion curves to the S-wave velocity model.The well-trained DispINet could be used to predict the S-wave velocity model.At same time, a MATLAB-based packed FDwave3D is presented for synthetic wavefield and seismogram modeling in 3D anisotropic media.The code in this packed has been tested on the numerical examples of layered and overthrust models [3].Other Matlab package is also developed for noisy dispersion curve picking (NDCP), which allows full control over several parameters to properly identify group velocity dispersion curves [4].In addition, a novel depiction of Love wave dispersion and inversion has published for inversely dispersive medium by full SH-wavefield reflectivity method.Their numerical example denoted that inversion of the effective dispersion curve can excellently estimate the shear-wave velocity (SWV) [5].
Analysis of dispersion curves in the surface wave method is the most important step for estimating SWV profiles.The SWV profiles is one of geophysical parameters that can be carried out employing a set of postulated earth model from observation data.In this case, the inversion modeling delivers the estimated SWV from dispersion curves of surface wave.An automatic inversion procedures can be successfully applied only in the case in which branch of modal curves are selected within a proper frequency range [6].
Inversion of the surface wave dispersion curve was originally developed for determining the SWV profile of soil deposits and pavement systems.It has recently been used to determine stiffness, one of the key earth parameters in geophysical, environmental, and geotechnical engineering applications [7][8][9].Several authors have shown that the inverse analysis of dispersion curves including higher modes of the Rayleigh wave can enhance the accuracy and reduce the non-uniqueness of estimated SWV [10][11][12].In such applications, the Love wave method can be a very interesting alternative method for deriving a SWV reversal profile [13,14].
Joint inversion codes of the effective Rayleigh wave and multimode Love wave dispersion curves have been developed and can be utilized for geotechnical site investigation [15].However, in this inversion approach and almost inversion procedures in the previously study, the analysis of geophysical parameter sensitivity encompassing SWV, layer thickness, and density parameters has not been comprehensively conducted.In addition, the initial input of parameter modeling in this inversion procedure has not incorporated the geophysical parameter sensitivity approaches as a priori information or constraints.In this paper for the primary time we carried out a numerical analysis to identify the parameters should be sensitively affected the multimode Love wave dispersion curves.
To study the sensitivity of Love wave dispersion curves to changes in geophysical parameters, we introduce perturbations to these parameters.Perturbing parameters involves making small adjustments or adding random noise to the values of shear wave velocity, density, or layer thickness.The purpose of perturbation is to understand how variation in specific geophysical parameters impact the Love wave dispersion curves.Parameter perturbation helps quantify the uncertainty in Love wave dispersion curve predictions.In general, for Love wave dispersion curves, parameter perturbation involves making small adjustments to geophysical parameters to examine the sensitivity, uncertainty, and robustness of the dispersion curve predictions.Here, we implement the MATLAB programming language to carry out the computation of multimode Love dispersion curve and numerical analysis of geophysical parameter sensitivity.

Literature review
In Geophysics, Love waves represents a specific type of surface waves.Numerical analysis of the sensitivity of geophysical parameters, based on the characteristics of multimode Love wave dispersion curves, can be carried out by initially understanding the basic properties of surface waves.This involves knowing the general concept of how surface waves propagate in layered media, realizing the fundamental equation that describes Love waves propagating in such layered media, and following a specific procedure for calculating the dispersion curves of Love waves.

Basic properties of surface waves
In addition to body waves (primary waves or P-waves and shear wave or S-waves) penetrating deeply into the earth, a seismic source such as an explosion or earthquake can also generates waves that travel along the surface of the earth and often called as surface waves (in seismic exploration known as ground roll).The most observed surface waves are Rayleigh waves, after Lord Rayleigh who predicted their existence in 1887.A Rayleigh wave is frequently dominant events in seismograms, because of their greater energy and because of a geometrical spreading that is lower than that of body waves spreading energy in all directions (spherical spreading in a homogeneous medium).This wave travels along the surface of the earth and involves an interference of the P-wave and SV-wave.The particle motion is confined to the vertical plane that includes the direction of propagation of the wave.The amplitude of the Rayleigh wave motion decreases exponentially with depth.
The second type of surface waves is called Love waves, after Augustus Edward Hough Love, who developed a theory for their existence early in the twentieth century.A Love wave is formed through the constructive interference of multiply reflected S-waves.The particle motion is horizontal (SH) and in the direction of SHwave.The amplitude of a Love wave motion decreases also exponentially with depth.
The surface wave method (SWM) is a seismic characterization method based on the analysis of the surface wave dispersion curves.The dispersive nature of surface waves in a layered medium is utilized to estimate a SWV (i.e., stiffness) profile of the test site.The complete testing procedure can be divided into three basic steps [16]: (1) generation and measurement of surface wave in the field (acquisition step); (2) data processing and extraction of an experimental dispersion curves (processing step); (3) inversion modeling of the experimental dispersion curves to obtain a model of the estimated shear wave velocity with depth profile (inversion step).
In each case, the depth to which the waves penetrate is a function of their frequency.Higher frequencies have shorter wavelengths, because of the relationship f = v/l (where f is frequency, v is velocity, and l is wavelength), and penetrate shallower into the earth.Due to velocity commonly increases with depth in the layered earth, the higher frequencies pass through material with lower average velocity.The existence of vertical medium-velocity gradients in the real earth caused velocity of surface waves varies with frequency (i.e., velocity is a function of frequency), which is that they are dispersed.The propagation of surface waves in a vertically heterogeneous medium shows a dispersive phase velocity (dispersion curve).Dispersion means that different frequencies have different phase velocities.
On seismograms, surface waves appear as long wave trains in which the frequency slowly increases, and they are not compact wavelets like body waves.Surface wave is employed to estimate a general picture of subsurface earth structure through the analysis of their phase velocity.Due to different zones of the earth have different distributions of phase velocity with depth each zone is characterized by different dispersion curves (plots of the variation of phase velocity with frequency).

Surface waves in the layered medium
For finite motions, the exact equation for motion in the continuum can be derived from the theory of elastic wave that provides mathematical relationships between stresses and strains in the medium [17,18].Applying Newton's second law to the medium gives: In the case in which sources or body forces such as gravity are not being considered ( f i = 0), u i which is the displacement of the particle can be written as: where x j ( j = 1, 2, 3) are the Cartesian coordinate axes.
Basically, surface waves propagating in elastic medium can be written as wave equations and their solutions of the equation of motion have to satisfy boundary conditions everywhere (no body force considered).In the Following there are four boundary conditions (BC) that must be satisfied by equation of motion (equation ( 2)): • BC1 (radiation condition): The displacement in infinite depth is zero (no displacements at infinity or at the bottom half-space), i.e., u/z → ∞ = 0 • BC2 (radiation continuity condition): Displacement must be continuous at any interfaces or across any layer boundaries (where the elastic moduli have a discontinuity), i.e., u z d u z d • BC3 (traction condition): Traction must be zero at the free surface (no traction at the free surface), i.e., σ/z = 0 = 0 • BC4 (traction continuity condition): Traction must be continuous at any interfaces or across any layer boundaries (where the elastic moduli have a discontinuity), i.e., z d z d i i1 Corresponding to shear horizontal wave (SH wave) or Love wave, it is necessary to introduce an important topic supporting description of this wave.In this section, a description of Love waves propagating in the 1D layered medium will be discussed in briefly.

Love waves propagating in the layered medium
The description of Love waves propagating in the layered medium is based on the equation of motion in the elastic medium that can be generally rewritten as follows.
This equation shows an x component of displacement (u) and other components can be written as similarly.The σ xx , σ xy , and σ xz are stresses and can be written as follow using Lame constants λ and μ.
where ν and w are y component and z component of displacement respectively.
Type of solution for SH waves that satisfies equations (3)-( 6) is generally assumed as follows. where, Stresses in x-y plane can be written as follows in terms of separation of variables.
Here, y 1 (z) and y 2 (z) are simultaneous solution for layered medium.Substituting set solutions from equation (7) into equation (6) yields, Comparison between (9) and (10) indicates in explicit form that, Substituting z = z m+1 into the equations (15) and (16) gives the solution on the boundary of (m + 1) as follows Substituting these A m ¢ and B m ¢ into the equations ( 15) and (16) yields These equations can be written as a matrix form y y y y cosh sinh where θ = ν βm d m and d m = z m −z m+1 is the thickness of layer mth.Using matrix notation, equation (23) can be rewritten as follow The matrix B m (d m ) is so called a layer matrix for layer m.This matrix is essentially equal to the matrix so called Haskell layer matrix [19].Advantage of the matrix (23) is that the elements of the matrix are always real.Solution on the layer boundary can then be calculated using the layer matrix iteratively from the initial condition Y n .Initial condition in the layer boundary n is where μ n ν βn must be larger than 0 so that y 1 and y 2 go to 0 where z goes to infinite.This condition implies that phase velocity or dispersion curves of Love wave must be slower that SWV of bottom layer (β n ).At the free surface, y 2 must be 0 so that stress goes to 0. Therefore, characteristic equations for Love wave can be written as follows.
F c y or F c y y , 0 , 0 2 6 where c is dispersion curve.The dispersion curve of Love waves is based on the integrating the equation (11) and its actual calculation procedure uses the equation (26).Other calculation and simulation based on technology computer aided design (TCAD) are implemented on the study of optoelectronic characteristics and related switching behavior of one monolithically integrated silicon light-emitting device (LED) and verification of almost-linear modulation curve [20].

Methodology/approach
This paper introduces a research procedure based on the quantitative method by implementing a numerical approach.In general, the numerical approach can be certainly carried out if the algorithm of the actual calculation from an empirical system characteristic is obviously established.Based on the lists of equation presented in the literature review section, the actual calculation of the multimode Love wave dispersion curves using Haskell layer matrix method can be succinctly outlined as follow [21]: • Define the model parameters of surface wave including number of layer (n), SWV values for each layer (β), the value of density values for each layer (ρ), thickness values of each layer (H).
• Read input of global parameters of surface waves including minimum, maximum and shift/increments of frequency desired, minimum, maximum and shift/increments of S-wave velocity, and number of mode (N) desired.
• Process the looping for the frequency.
• Process the looping for the S-wave velocity.
• Calculate the initial condition based on the equation (25).
• Process the looping for the layer iteration.
• Calculate variable y 1 (z) and y 2 (z) from bottom to top layer using equation (23).
• End the looping for the layer iteration.
• End the looping for the S-wave velocity.
• End the looping for the frequency.
• Display the dispersion curves data (pairs of phase velocity versus frequency).
According to main equation listed in this algorithm, there are three geophysical parameters should be sensitively affected the characteristics of the multimode Love wave dispersion curve, that is [22]: 1. the SWV values for each layer (β), 2. the layer thickness values of each layer (H), and

the density values of each layer (ρ)
Several stages should be carried out to investigate which geophysical parameter most sensitive are: • Stage 1: Generating a numerical computation of the multimode Love wave dispersion curves on the specified profile as the referenced dispersion curves.
• Stage 2: Analyzing the sensitivity of the SWV values for each layer by presenting the perturbation coefficients upon the referenced profile.In this stage, the geophysical parameter of the SWV values is variative changed using perturbation coefficients and other geophysical parameters and input global parameters should be assumed as constant parameters.To compute how many percentages of the relative error of each perturbation coefficient, it is quantitatively employed the equation as follow: where RE i is a relative error of ith mode, NF is number of frequency points for all finite phase velocity values, PV k rp is a phase velocity value at reference profile, and PV k pc is a phase velocity value obtained from a perturbation coefficient.
• Stage 3: Analyzing the sensitivity of the thickness values for each layer by presenting the perturbation coefficients upon the referenced profile.In this stage, the geophysical parameter of the layer thickness values is variative changed using perturbation coefficients.Other geophysical parameters and input global parameters should be assumed as constant parameters.The equation (27) is also used to compute the relative error.
• Stage 4: Analyzing the sensitivity of the density values for each layer by presenting the perturbation coefficients upon the referenced profile.In this stage, the geophysical parameter of the density values is variative changed using perturbation coefficients and other geophysical parameters and input global parameters should be assumed as constant parameters.It is alike to the Stage 3, the equation ( 27) is also used to compute the relative error.
• Stage 5: Comparing the sensitivities of physical parameters obtained from the Stage 2, Stage 3, and Stage 4 by assuming all stages should be applied the same perturbation coefficients.In this stage, it can be furthermore verified, which the geophysical parameter can most sensitively impact the characteristic of the multimode Love wave dispersion curves.

Generating dispersion curves from the referenced profile
In this section, the actual calculation of the multimode Love wave dispersion curves is implemented on the synthetic site model called the referenced profile.This profile represents the regular velocity structure where the value of the SWV in each layer increases monotonically with increasing layers.It consists of four layers over a half-space.The geophysical parameters of the referenced profile are listed in table 1 and their visualization is shown on figure 1. Computation results of multimode Love wave dispersion curves obtained from the referenced profile can be seen in figure 2.
In general, the legends displayed in figure 2 use text of Mode1, Mode 2, Mode 3, etc.Here, Mode 1 is correlated by the first mode or fundamental mode, Mode 2 for the second mode, Mode 3 for the third mode, etc It is an essential assumption that the first curve closest to the vertical axis will always be associated with the first mode, followed by the second mode, the third mode and so on.The validation process for multimode Love wave dispersion is being conducted, and this process is adaptable or adjustable concerning earlier research.The representations of dispersion curves in this study closely resemble the results characterized in previous work, suggesting a degree of consistency or agreement between the findings of the current study and those of prior research [15,21,22].
However, multimode offer several advantages in geophysics and seismic studies compared to their singlemode counterparts.Multimode dispersion curves provide a more comprehensive view of the relationship between wave characteristics across different modes.This increased sensitivity allows for a more detailed analysis of subsurface structures and material properties.The existence of multiple modes increases the   resolution of the dispersion curves, allowing for a finer analysis of subtle variations in the subsurface.This heightened resolution is crucial in applications such as environmental monitoring, where precise delineation of subsurface structures is essential.In general, multimode dispersion curves carry more information about the subsurface than single-mode curves [22].
Trend of the multimode Love wave dispersion curves produced from a SWV profile can be analyzed employing the cut-off frequency approach.The quantity of the cut-off frequency (CF) is defined as a value at which the SWV in a certain mode reaches its peak velocity value.The structure of the SWV illustrated by the referenced profile has a cut-off frequency as presented in table 2.
The cut-off frequency average obtained from the total summation of differences between adjacent elements of cut-off frequencies of the dispersion curves divided by the number of modes minus one (N−1) is defined as: Where CF ̅ is the cut-off frequency average , N is number of modes, h = 1,2, K, N. As illustration, the cut-off frequency (CF) listed in table 2 have the cut-off frequency average CF ̅ of 9.5 Hz.

Analyzing the geophysical parameter sensitivity of the SWV
In this section, it is important to introduce firstly the alpha (α) parameter perturbating the values of SWV from the referenced profile.The alpha parameter is set so that the SWV values generated will be smaller and/or greater than the SWV values at the referenced profile.In an effort to analyze the impact of varying SWV values on the dispersion curve, the alpha parameter is numerically defined in the row matrix elements as follows: 0.20, 0.18, 0.16, 0.14, 0.12, 0.10, 0.08, 0.06, 0.04, 0.02 0.00, 0.02, 0.04, 0.06, 0.08, 0.10, 0.12, 0.14, 0.16, 0.18, 0.20 30 As an illustration, if it is known that the SWV value in the top layer of the referenced profile is 200 m s −1 , then by applying the α = −0.2, the SWV value changes to (1−α) × 200 = 240 m s −1 and for the alpha parameter α = 0.2, the SWV value changes to (1−α) × 200 = 160 m s −1 .Perturbation of the SWV values in each layer of the reference profile due to the application of the alpha parameter can be in detail seen in table 3.
Furthermore, the impact of the geophysical parameters from the perturbation of SWV values on the multimode Love wave dispersion curves can be graphically seen in figure 3.This figure shows realistically the deviation of the dispersion curve due to the perturbation of the SWV value for the four modes which is jointly plotted all together with the dispersion curve obtained from the referenced profile.
To perform the impact of perturbation of SWV values based on the alpha parameter, the relative error values for each mode are then plotted all together.Figure 4 illustrates the relative error (%) versus the perturbation of alpha parameter.It is observed that the greater the absolute value of alpha (perturbation parameter), the larger the error becomes.This confirms the previous results on water waves [23,24], and also wave-like solutions of Lotka-Volterra model near its equilibrium point [25].
In figure 4, Model V provides insights into the impact of perturbation coefficients on the system.Specifically, the perturbation coefficients of α-negative (from equation 29) and α-positive (from equation 30) do not have a  significant influence on the relative error.The results show that these coefficients have almost similar values of absolute mean slope, which are recorded as |−2.75| and |2.55| consecutively.Numerically, when dealing with the symmetrical Model V, it becomes evident that introducing perturbations to the reference model valuewhether through α-negative or α-positive perturbation coefficients-leads to dispersion curve characteristics that are almost identical.
Another investigation in order to evaluate the impact of SWV perturbation on the dispersion curve is an analysis based on the values of the cut-off frequency average (CF ̅ ) and its relative error percentage (PCF ̅ ) that usually defined as: Where PCF p is the percentage of relative error of (CF) at p-perturbation based on the α parameter, CF p ̅ is the value of the cut-off frequency average at p-perturbation, and CF o is the value of the cut-off frequency average for the original or reference profile.Perturbation of the SWV values based on the αparameter as shown in table 3 should numerically produce the values of CF and PCF p listed in table 4.
Based on the listed numerical values on table 4 and data visualization on figure 4, it is verified that the greater the value of the alpha parameter, the higher the RE value.

Analyzing the geophysical parameter sensitivity of the layer thickness values of each layer (H)
By using a similar procedure to section 4.2, in this section equations (29) and (30) will be consistently used to perturbate the layer thickness values for each layer.The perturbation of the layer thickness values in each layer of the reference profile due to the application of the alpha parameter can be in detail seen in table 5.
Graphically, the impact of the geophysical parameters from the perturbation of layer thickness values on the multimode Love wave dispersion curves can be seen in figure 5.This figure realistically shows the deviation of  the dispersion curve due to the perturbation of the layer thickness value for the first four modes which is jointly plotted all together with the dispersion curve obtained from the referenced profile.
To appraise the impact of the layer thickness perturbation on the dispersion curve, the values of the cut-off frequency average (CF) and the percentages of the relative error of CF (PCF) are incorporated in the analysis.These values can be respectively calculated using equation (28) and equation (31).Perturbation of the layer thickness values based on the alpha parameter as shown in table 5 produce the values of CF and PCF listed in table 6.

Conclusion
The characteristics of multimode Love wave dispersion curves depend on three geophysical parameters, namely the shear wave velocity (SWV) values for each layer (β), the layer thickness values for each layer (H), and the density values for each layer (ρ).These parameters are expected to sensitively affect Love wave dispersion curves.Following a thorough numerical analysis, we found that that the shear wave velocity (SWV) values for each layer exhibit greatest sensitivity compared to both the layer thickness values and density values.In terms of relative error (RE) values, the perturbation of the SWV parameter manifests a highly significant impacton multimode Love wave dispersion curves, with average relative error (RE) values highest than those associated with other parameters.When we examine the average cut-off frequency values, the relative error percentages (PCF) due to perturbation of the SWV parameter in multimode Love wave dispersion curves are also significantly greatest than those of other parameters.

Equation ( 11 ) 4 .
Writing σ xx and σ xy using y 1 and substituting them into the equation of motion (3) gives The equations (9) and (10) are basic equation for SH waves.These equations can be rewritten as are simultaneous equations for y 1 and y 2 , where β is shear wave velocity (SWV) defined as Calculation of love wave dispersion curve Solution for equation(11) can easily be obtained in the layered medium, in which each layer has constant density (ρ = ρ m ) and SWM values (β = β m ) in the layer of index m, as follow y z A e B e 15

Figure 1 .
Figure 1.Visualization the geophysical parameter model of the referenced profile.

Figure 2 .
Figure 2. Dispersion curves computation resulted from the referenced profile.

Figure 3 .
Figure 3. Deviation of the dispersion curve due to the perturbation of SWV values for the first four mode.

Figure 4 .
Figure 4.The relative error (RE) obtained from the perturbation of SWV based on the alpha parameter.

Figure 5 .
Figure 5. Deviation of the dispersion curve due to the perturbation of layer thickness values for the first four mode.To evaluate the perturbation impact of layer thickness values based on the alpha parameter, the relative error values for each mode are then plotted all together.Figure 6 presents the relative error (%) versus the perturbation of layer thickness values based on the alpha parameter.

Figure 6 .
Figure 6.The relative error (RE) obtained from the perturbation of the layer thickness value based on the alpha parameter.

Figure 7 .
Figure 7. Deviation of the dispersion curve due to the perturbation of density values for the first four mode.The impact of the perturbation of density values based on the alpha parameter performs very small relative error values in the order of 10 −14 % for each mode.Figure 8 presents the relative error (%) versus the perturbation of density values based on the alpha parameter.

4. 4 .
Analyzing the geophysical parameter sensitivity of the density It is a very much different result from the two previous geophysical parameters, the deviation of the dispersion curve due to the perturbation of density value based on the utilization of same alpha parameter presents the similar dispersion curve among them to dispersion curve of the referenced profile (see figure7).

Table 1 .
Model Parameters of the referenced profile.

Table 2 .
The cut-off frequency of the referenced profile.

Table 3 .
Variation of alpha parameter corresponding to the perturbation of SWV values (m/s) for all layer.

Table 4 .
Perturbation of SWV values based on the α parameter corresponding to the CF values and RE of the CF ̅ .

Table 5 .
Variation of alpha parameter corresponding to the perturbation of the layer thickness values.

Table 6 .
Perturbation of layer thickness values based on the α parameter corresponding to the CF and RE of the CF values.