1 Introduction

Shear wave velocity is a valuable indicator of the dynamic behaviour of the soil due to its relationship with the shear modulus (i.e. μ = ρ · Vs2), which depends on strain level. Therefore, it is the important parameter in construction purposes and engineering studies. Shear waves are the type of elastic waves, where particle motion is perpendicular to the direction of wave propagation. These waves are also called S-wave or Secondary wave because of its appearance on the seismogram immediate after the first arrival P-wave. Shear waves travel slowly in the rock as compared to the P-wave. The P-wave velocity is 1.37 times the S-wave velocity. Shear wave velocity is used to classify the soil types based on Vs30 (S-wave velocity over top 30 m of depth) by characterizing the effect of sediment stiffness on the ground motion. There are many soil classifications based on Vs30 like NEHRP, EUROCODE8, Uniform Building Code (UBC), etc. which is used for earthquake engineering design with the expectation that sites in the same class will respond similarly to a given earthquake. The shear wave velocity can be estimated by many geophysical methods. Out of which, Horizontal to Vertical Spectral Ratio (HVSR) technique is one of the geophysical method to estimate the shear wave velocity and for the study of ground motion amplification. This technique is non-destructive and cheapest. However, there is no evidence of the particular composition of the noise field at HV curve. The spectral ratio curve has been assumed to be dominated by the body waves around the peak of the H/V [6, 10, 15]. The HVSR ratio curve from ambient noise on other hand, has been related to the Rayleigh wave ellipticity by Lermo and Chávez-García [11], Malischewsky and Scherbaum [13]. When Rayleigh and Love waves come from various directions an ellipticity analysis becomes very complicated. However, successful inversion schemes based on surface wave ellipticity have been used [2] . The description of HV curve by taking into account the combined effect of complete seismic ambient-vibration wave field [21]. This is called diffuse field assumption. The purpose of this paper is to get the best shear wave velocity model by modelling Horizontal to Vertical Spectral Ratio curve by using Simulated Annealing Method based on software developed by García-Jerez et al. [9]. The input to this software require HV spectral ratio data in very limited samples. In the present work, a program has been developed to model the spectral ratio curve from ambient noise data and obtain a reliable limited sampled spectral ratio data.

2 Methodology

It has been verified by many workers that earthquake ground motion is strongly effected by local site conditions [1, 19]. Site effects is characterized by the means of spectral ratios of recorded motions with respect to reference rock site (e.g. Cadet et al. [7]). On the other hand, for more quiescent areas seismic noise severe as a major tool to provide useful information about spectral ratio contents of ground motion. Nakamura [14] was among initial workers who have related spectral ratio of horizontal to vertical component of recorded ground motion with the site effects. Lermo and Chávez-García [11] has obtained reliable estimates of the soil resonance frequencies from the spectral ratio of horizontal with vertical motions of microtremors. The HV curve has been studied to explain its strengths and limitations by different workers viz. [4, 6, 12, 17]. The HV ratio curve reveals the site dominant frequency f0 but the amplitude of this HV curve is not well understood [17]. The recent work of Lunedei and Albarello [12] is a significant step towards the understanding of the amplitude as well as frequency part of HV spectrum.

In the present work, noise is assumed to be diffuse wavefield containing all types of elastic waves. Underlying this assumption is the connection between diffuse fields and the Green’s function [8, 20, 23,24,25], which implies the proportionality between the average energy densities of a diffuse field and the imaginary part of Green’s Function at the source [16, 20]. This can be summarised as:

$$\langle {u}_{i}({x}_{A}, \omega ){u}_{j}^{*}({x}_{B}, \omega )\rangle \propto \mathrm{Im}[{G}_{ij}({x}_{A}, {x}_{B}; \omega )]$$
(1)

where \({G}_{ij}({x}_{A}, {x}_{B}; \omega )\) is the Green’s function defined as the displacement in the ith direction at point \({x}_{B}\) due to the application of a unit harmonic point force in the jth direction applied at point \({x}_{A}\), \({u}_{i}\left({x}_{A}, \omega \right)\) is the displacement field component in i direction at point \({x}_{A}\), \(\omega =2\pi f\) is the angular frequency. For coinciding source and receiver, the autocorrelation of both the components gives the directional energy density \({E}_{i} \left(x, \omega \right)\) at point x as:

$${E}_{i} \left(x, \omega \right)=\rho {\omega }^{2}\langle {\Vert {u}_{i}\left(x, \omega \right)\Vert }^{2}\rangle \propto \mathrm{Im} \left[{G}_{ii}\left(x, x; \omega \right)\right]$$
(2)

In this expression \(\rho\) is the mass density. Therefore, the HV can be expressed as the square root of the ratio of the corresponding imaginary parts of Green’s tensor components [2, 22]:

$$HV \left(x,\omega \right)= \sqrt{\frac{\mathrm{Im }\left[{G}_{11}\left(x, x; \omega \right)\right]+\mathrm{Im }\left[{G}_{22}\left(x, x; \omega \right)\right]}{\mathrm{Im }\left[{G}_{33}\left(x, x; \omega \right)\right]}}$$
(3)

In above expression the subscripts 1 and 2 represent the horizontal directions while the subscript 3 corresponds to the vertical one, respectively.

HV-Inv is a computer code for forward calculation and inversion of H/V spectral ratios of ambient noise (HVSRN) based on the diffuse field assumption (DFA) developed by García-Jerez et al. [9]. The software supports joint inversion of HVSRN and dispersion curves by using several local and global algorithms which include simulated annealing algorithm.

The definition of an optimal solution for any optimisation problems can be an extremely challenging task or some time it is practically difficult. This is because when a problem gets very large, we need to search a vast number of the possible solution to find the optimal solution. This type of problem can be solved using Simulated Annealing Technique (SAT). This is a probabilistic technique for approximating the global optimum of a given task. It is efficacious and commonly used form of optimization problems, which is used for solving the unconstrained and bound-constrained optimization problems. Simulated annealing is not affected by the initial guess, as it is a global optimization technique and gives the global optima instead of local optima, which is an optimal solution [5]. This algorithm was originally motivated by the process of annealing of metal. “Annealing” discusses the process of thermodynamics, which involves the cooling and heating of the material to modify its physical properties due to change in its internal structure [18]. The algorithm is a hill-climbing except instead of picking the best move, it picks a random move. At each iteration of the simulated annealing technique, a new point is randomly generated. If the selected iteration improves the solution, then it is accepted. Otherwise, the algorithm makes the iteration with some probability less than one. The difficulty with this approach is that while it rapidly finds a local minimum, it cannot get from there to the global minimum. There are so many uses of Simulated Annealing (SA technique in different domains. It has a broad range of application that is still being explored. In the present work the spectral content of HVSR curve is mainly dependent on the P and S wave velocity profile of region and density of the subsurface layers. Initial range of these parameters is selected from the heuristic guess based on geophysical and geological observations. The iterative computation of HVSR curve is based on Simulated Annealing algorithm. In the present work, HV-Inv has been used to study the site specific case study of the Bayasi region in the Garhwal Himalaya.

The ground vibration created by any disturbance at the surface of the earth can be recorded by three-component recorders. These recorders record ground vibrations in two horizontally and one vertically oriented sensor. Nakamura [14] has shown that the ratio of the amplitude spectrum of the horizontal component to the vertical component is directly related to the shear wave velocity profile of the subsurface of the earth. Normally in this technique, the ground disturbance created by passing vehicle is recorded for a specific duration of time in three-component sensors i.e. two horizontal components along north–south and east–west direction (\({a}_{ns}(t)\), \({a}_{ew}(t)\)) and one vertical component (\({a}_{v}(t)\)) as shown in Fig. 1. Fourier transform of these components are as follows:

$${\text{a}}_{{{\text{ns}}}} \left( {\text{t}} \right) \mathop{\longrightarrow}\limits^{{}}{\text{A}}_{{{\text{ns}}}} \left( {\text{f}} \right)$$
(4)
$${\text{a}}_{{{\text{ew}}}} \left( {\text{t}} \right)\mathop{\longrightarrow}\limits^{{}}{\text{A}}_{{{\text{ew}}}} \left( {\text{f}} \right)$$
(5)
$${\text{a}}_{{\text{v}}} \left( {\text{t}} \right)\mathop{\longrightarrow}\limits^{{}}{\text{A}}_{{\text{v}}} \left( {\text{f}} \right)$$
(6)
Fig. 1
figure 1

Time series record of the a North South ‘ans(t)’, b East West ‘aew(t)’ and c vertical ‘av(t)’ component of ground vibrations created by passing vehicle

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Amplitude spectrum of each component is calculated as:

$${\mathrm{Amp}}_{\mathrm{ns}}\left(\mathrm{f}\right)={[\mathrm{Re}{\left({\mathrm{A}}_{\mathrm{ns}}\left(\mathrm{f}\right)\right)}^{2}+\mathrm{Im}{\left({\mathrm{A}}_{\mathrm{ns}}\left(\mathrm{f}\right)\right)}^{2}]}^{1/2}$$
(7)
$${\mathrm{Amp}}_{\mathrm{ew}}\left(\mathrm{f}\right)={[\mathrm{Re}{\left({\mathrm{A}}_{\mathrm{ew}}\left(\mathrm{f}\right)\right)}^{2}+\mathrm{Im}{\left({\mathrm{A}}_{\mathrm{ew}}\left(\mathrm{f}\right)\right)}^{2}]}^{1/2}$$
(8)
$${\mathrm{Amp}}_{\mathrm{v}}\left(\mathrm{f}\right)={[\mathrm{Re}{\left({\mathrm{A}}_{\mathrm{v}}\left(\mathrm{f}\right)\right)}^{2}+\mathrm{Im}{\left({\mathrm{A}}_{\mathrm{v}}\left(\mathrm{f}\right)\right)}^{2}]}^{1/2}$$
(9)

As we have two horizontal and one vertical component, so there are two amplitude spectrums of NS and EW component. The amplitude spectrum of \({ a}_{ns}(t)\), \({a}_{ew}(t)\) and \({a}_{v}(t)\) is shown in Fig. 2.

Fig. 2
figure 2

Amplitude spectrum of time series shown in Fig. 1 along North South, East west and Vertical component of ground vibration

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We can calculate the average of horizontal components either by geometrical or root mean square amplitude of horizontal components by the following formula:

$${\text{Root}}\,{\text{mean}}\,{\text{square}} = \left[ {\left\{ {\left( {{\text{Amp}}_{{{\text{NS}}}} \left( {\text{f}} \right)} \right)^{2} + \left( {{\text{Amp}}_{{{\text{EW}}}} \left( {\text{f}} \right)} \right)^{2} } \right\}/2} \right]^{1/2}$$
(10)

or

$${\text{Geometrical}}\,{\text{mean}} = \left[ {{\text{Amp}}_{{{\text{NS}}}} \left( {\text{f}} \right) \times {\text{Amp}}_{{{\text{EW}}}} \left( {\text{f}} \right)} \right]{ }^{1/2}$$
(11)

In the present work, Geometrical mean has been used for obtaining the H/V curve at any site. H/V curve can be obtained by the following expression and is shown in Fig. 3.

Fig. 3
figure 3

Ratio of amplitude spectrum of horizontal by vertical component of record shown in Fig. 1

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$$\mathrm{HV}\left(\mathrm{f}\right)={\left[{\mathrm{Amp}}_{\mathrm{NS}}\left(\mathrm{f}\right)\times {\mathrm{Amp}}_{\mathrm{EW}}\left(\mathrm{f}\right)\right]}^\frac{1}{2}/{\mathrm{Amp}}_{\mathrm{V}}(\mathrm{f})$$
(12)

According to system theory, the output ‘y(t)’ is obtained when any input ‘x(t)’ is passed through a system as shown in Fig. 4.

Fig. 4
figure 4

Flow graph of forward modelling approach

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In any mathematical approach, when input is known and system response is known and we are calculating the output, then such system is termed as the forward approach. Similarly, when the output of the system is known and we are interested to calculate input from the system, then this approach is termed as inverse approach. In both approaches, we know the system response. Inverse approach is shown in Fig. 5.

Fig. 5
figure 5

Flow graph of inverse modelling approach

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The obtained HVSR curve at any site is theoretically represented by the formulation given which is mainly dependent on parameters of the earth model. In the formulation, system response mainly depends on P wave velocity (\(\alpha\)), Shear wave velocity (\(\beta\)), the density of the medium (\(\rho\)) and Poisson ratio (\(\sigma\)). The purpose of present work is to select final earth parameters using Simulated Annealing algorithm developed by García-Jerez et al. [9]. Different authors have attempted to explain the H/V phenomenology in terms of SH waves [10, 14], of Rayleigh waves [3, 11, 13], and by adding the effects of Love. Recent studies have considered the role of all waves, known as a total field [21, 22]. It was found that results differ not only as a function of the type of waves but also as a function of the temporal and spatial distribution of the sources and their strength. The inversion of the H/V curve requires not only the knowledge of the specific sources acting at the site but also of several other soil parameters (e.g., Poisson’s ratio, damping of each layer, 2D effects, etc.) which cannot be easily determined. Under these condition various algorithms helps in selecting most preferred model out of various possible range of modelling parameters.

Modelling of HV curve can be made by Simulated Annealing algorithm in HV-Inv software. This software requires HV curve as an input function which is modelled by using appropriate parameter space that can be estimated using several algorithms supported in the software. In the present work the acquired data consist of a time series of 12 min length that has sampling frequency 128 Hz. Therefore, the data of ground vibration consist of 92,160 samples along three different mutually perpendicular directions. This gives a similar number of samples in HV curve in frequency domain. However, software HV-Inv needs very less number of data point in the HV curve as an input to the algorithm. Therefore, the challenging task before us is to re discretize 92,160 samples into very less samples retaining the spectral property of HV curve. We need to discretize it so that all peaks should remain preserved in an HV curve of fewer samples. For this purpose, a software DISHV is prepared in FORTRAN that prepares an HV curve of small length i.e. of nearly 30 samples in frequency range 0.1–100. Input to this program is a time series in NS, EW and Vertical component of ground vibrations. The program calculates FFT of these input ground vibration in various small windows which can be decided by the user. Presently 38,000 samples at sampling frequency 128 Hz has been used as input of ground vibration along with NS, EW and Vertical direction. Figure 6 shows the time series along with NS, EW and vertical direction. This entire ground vibration is divided into 10 windows and HV curve in each window is calculated. Mean and Standard deviation of all HV curve is calculated which is used as input to the algorithm. Figure 7a shows the HV curve when window length consists of 3800 samples. The software DISHV further check the number of points in each logarithmic window and discretize uniformly 10 samples in each window. The discretize HV curve having 39 points is shown in Fig. 7b. Comparison of HV curve that is having 3800 samples and 39 points clearly shows that peak are preserved and both curves has characteristic similarity shown in Fig. 7c. This reduces approximately 100 times the sample present in actual data, thereby retaining the main spectral contents of the HV curve. The output from DISHV program is further used as an input to HV-Inv program for modelling of HV curve using Simulated Annealing algorithm.

Fig. 6
figure 6

a NS, b EW and c vertical component of ground vibration

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Fig. 7
figure 7

a HV curve obtained from time series containing 3800 samples, b recreating HV curve from obtained curve using 39 samples and c comparison of HV curve having 3800 and 39 samples respectively

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3 Case Study for the Bayasi Site, Uttarakhand

The ambient noise data has been recorded at the Bayasi site, Uttarakhand of the Garhwal Himalaya in two horizontal and one vertical direction shown in Fig. 8. The amplitude spectrum of time series of recorded noise data is shown in Fig. 9a along North–South, East–west and vertical component of ground vibration. The H/V curve obtained by the ratio of averaged horizontal amplitude spectrum to that of the vertical spectrum is shown in Fig. 9b. The obtained H/V curve is then used as input to the DISHV program to discretize the HV curve for less number of samples. The discretize HV curve having 36 samples is shown in Fig. 9c.

Fig. 8
figure 8

Time series record of the a North South ‘ans(t)’, b East west ‘aew(t)’ and c vertical ‘av(t)’ component at Bayasi site

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Fig. 9
figure 9

a Amplitude spectrum of time series, b ratio of amplitude spectrum of horizontal by vertical component of record, c HV curve from 36 samples

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The parameter ranges of P-wave velocity, S-wave velocity and density are chosen for the inversion of HV curve using simulated annealing algorithm is shown in Fig. 10. The average velocity profile and best fit model of VP and VS obtained after the inversion process is shown in Fig. 10 with black colour line and red colour line respectively. Figure 11 is showing the best-modelled HV curve in red colour after inversion and the yellow and blue lines showing the set of forward HV curves for the different test model. The output obtained from Hv-Inv software by using input spectral ratio at 36 points using Simulated Annealing algorithm is shown in Fig. 12.

Fig. 10
figure 10

Set of tested models (blue and yellow lines), average and best-fit models showing with black and red colour line

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Fig. 11
figure 11

Best modelled HV curve for the best model after inversion shown in red colour line

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Fig. 12
figure 12

HVInv program window of the inversion of HV curve using HV-Inv

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4 Results and Discussion

The best-fit S-wave velocity model obtained from the simulated annealing algorithm is further refined by iterative modelling of HV curve using forward modelling approach. The minute change in the thickness and velocity according to the user gives the best model after the refinement. The modelled HV curve after the refinement is shown in Fig. 13. The one dimensional Shear wave velocity structure obtained after the refinement of model obtained from Simulated Annealing algorithm is shown in Fig. 14.

Fig. 13
figure 13

Final HV curve obtained after the refinement

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Fig. 14
figure 14

Final S-wave velocity model after the further refinement in the model

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The final model obtained after refinement shows that the first layer is of thickness 0.45 m with S-wave velocity 100.0 m/s and the second layer of thickness 9 m with S-wave velocity about 320.1 m/s and the half-space velocity is about 645.2 m/s. The final model obtained after refinement in the model through iterative forward modelling of the model selected from Simulated Aneling algorithm is shown in Table 1. The work done in this paper shows that the HV-Inv program for the inversion of HV curve effectively model the HV curve by a global Simulated Annealing technique and allow the user to further refine the model obtained from the inversion process. The geological input of the area obtained from the field investigation of the region also support this observation.

Table 1 Final model parameter after the refinement
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5 Conclusion

The ratio of the amplitude of horizontal to the vertical component of ground vibration created by an ambient noise is dependent on the subsurface velocity structure of the investigating area. Ground vibration data from ambient noise recording system has been recorded at the Bayasi site in the Garhwal Himalaya. A computer code DISHV in FORTRAN has been developed in this work has been used for obtaining HV curve with a limited number of data points that can be effectively used as an input to the HV-Inv software developed by García-Jerez et al. [9]. The HV curve obtained from DISHV is further used to obtained theoretical HV curve corresponding to a model of earth by using simulated annealing algorithm. The obtained velocity model from simulated annealing is utilized for the forward modelling approach that further refines the obtained shear wave velocity model by comparing the simulated and the observed HV curves.