<figure><img src="image_28.png" /><figcaption>**Fig. 25** Active measurements: example of near field effect and rejection of closest receivers. “×1” indicates the shortest source-receiver distance taken into account in the processing. On these figures, the two black lines indicate the wavelength limit corresponding respectively to the length of the array and to the O’Neil (2003) criterion (length of the array multiplied by 0.4). One can see that the DC curve is rather stable above this limit, whatever the “×1” distance. At high frequency the increase of the source—first geophone distance may alter the DC curve due to lack of energy, so making several test is useful</figcaption></figure>

be obtained with the f-k analysis of passive data that give a representation of energy
distribution (Fig. 26).

**3.4 Estimation of uncertainties**

It is recommended that the observed variability on the experimental dispersion curve
always be reported with the results. This may be represented in the form of condensed
statistical parameters (e.g. mean value and standard deviation). Data variability is a clear
indicator of the reliability of the results and can be used for further assessments during
inversion.

In principle, aleatory and epistemic (i.e. model-based) uncertainties should be clearly separated when constructing the dispersion curve. Aleatory uncertainty is that associated with the primitive measurements for a given site and a given source-receiver array and analyst. Uncertainty of the “derived” data (including the uncertainty of the dispersion curve) can be obtained from that of the primitive data using standard algorithms and approaches (i.e. FOSM, etc).

Epistemic uncertainty on the other hand refers for instance to different methods used to
construct the dispersion curve. Therefore the same uncertainty of the “primitive” data may
actually map into different uncertainty of “derived” data depending on the specific
algorithm used to process the “primitive” data. This uncertainty should thus be kept
separated and treated for instance using logic-tree approaches.

<figure><img src="image_29.png" /><figcaption>**Fig. 26** Checking the azimuthal distribution of sources, example from the characterization of the OGIM station of the RAP (French permanent accelerometric network) (30 m aperture and 900 m aperture arrays respectively). Left example of uniform distribution of sources in the high frequency range, right example of dominant source direction (N280) in the 1–3 Hz frequency band</figcaption></figure>
<figure><img src="image_28.png" /><figcaption>**Fig. 25** Active measurements: example of near field effect and rejection of closest receivers. “×1” indicates the shortest source-receiver distance taken into account in the processing. On these figures, the two black lines indicate the wavelength limit corresponding respectively to the length of the array and to the O’Neil (2003) criterion (length of the array multiplied by 0.4). One can see that the DC curve is rather stable above this limit, whatever the “×1” distance. At high frequency the increase of the source—first geophone distance may alter the DC curve due to lack of energy, so making several test is useful</figcaption></figure>

be obtained with the f-k analysis of passive data that give a representation of energy
distribution (Fig. 26).

**3.4 Estimation of uncertainties**

It is recommended that the observed variability on the experimental dispersion curve
always be reported with the results. This may be represented in the form of condensed
statistical parameters (e.g. mean value and standard deviation). Data variability is a clear
indicator of the reliability of the results and can be used for further assessments during
inversion.

In principle, aleatory and epistemic (i.e. model-based) uncertainties should be clearly separated when constructing the dispersion curve. Aleatory uncertainty is that associated with the primitive measurements for a given site and a given source-receiver array and analyst. Uncertainty of the “derived” data (including the uncertainty of the dispersion curve) can be obtained from that of the primitive data using standard algorithms and approaches (i.e. FOSM, etc).

Epistemic uncertainty on the other hand refers for instance to different methods used to
construct the dispersion curve. Therefore the same uncertainty of the “primitive” data may
actually map into different uncertainty of “derived” data depending on the specific
algorithm used to process the “primitive” data. This uncertainty should thus be kept
separated and treated for instance using logic-tree approaches.

<figure><img src="image_29.png" /><figcaption>**Fig. 26** Checking the azimuthal distribution of sources, example from the characterization of the OGIM station of the RAP (French permanent accelerometric network) (30 m aperture and 900 m aperture arrays respectively). Left example of uniform distribution of sources in the high frequency range, right example of dominant source direction (N280) in the 1–3 Hz frequency band</figcaption></figure>
Variability of measurements due to different source-receiver configurations is a com-
bination of aleatory and epistemic uncertainty however epistemic uncertainty would be
prevailing in case of lateral heterogeneity of soil deposits. Since standard surface wave
testing intrinsically assumes 1D modeling, the uncertainty associated to different source-
receiver configurations should be assumed aleatory since sites with pronounced 2D effects
should be disregarded.

3.4.1 Active measurements

Although uncertainty estimation is necessary, there is not yet any protocol shared inter-
nationally on the best way to measure it from active surface wave analysis.

Independent processing of several recordings acquired with the same layout (shot and receivers) can be used to obtain different estimates of the experimental dispersion curve. These values can then be used to estimate mean and standard deviation of the phase velocity at each frequency. In this case the observed variability is only associated to repeatability of the test and observed discrepancies are mainly due to the effect of ambient vibrations. It is often much lower than the variability taking into account the natural variability of the ground and the uncertainties linked to the choice of the processing technique.

The variability of the results on the experimental dispersion curve can be estimated by considering different source locations for the same array, as well as using subsets of receivers instead of all receivers. With this approach, the uncertainties due to different factors are lumped together (e.g. lateral variations; effect of higher modes; near-field effects; signal-to-noise ratio). This way of uncertainty estimation is probably the one that is most similar to the uncertainty estimation for passive measurement (see below).

**Feedback from InterPacific project** Although no international consensus have yet been reached on the best way to estimate uncertainties for active surface wave analysis, the feedback from several international blind analysis projects (among them the InterPacific one, see Garofalo et al. 2016a) shows that, within the reliable wavelength band, inter-analyst uncertainties, estimated in terms of Coefficient of Variarion (COV), on the average phase velocity range from 5 to 10% depending on the sites (Fig. 27). The uncertainty on dispersion estimates provided by each analysist range between 5 and 20% depending on the site and the analyst’s (Fig. 28).

3.4.2 Passive surveys

For passive experiments, dispersion curves are most often obtained by averaging the
dispersion estimates from the various time blocks. Such uncertainties are related to time
variation of ambient vibration wavefield properties, in relation with the array capability to
separate simultaneous propagating waves across the array, with the lateral variation of the
subsurface, and with near-field effects.

This can be achieved by analyzing, frequency by frequency, the histogram of the
velocity estimates and removing the outliers to get a Gaussian distribution. Mean and
standard deviation can then be estimated at each frequency. Alternatively, another option
may be to manually pick several dispersion curves covering the whole area with high
number of velocity estimates, then average them to estimate the mean and standard
deviation.
Variability of measurements due to different source-receiver configurations is a com-
bination of aleatory and epistemic uncertainty however epistemic uncertainty would be
prevailing in case of lateral heterogeneity of soil deposits. Since standard surface wave
testing intrinsically assumes 1D modeling, the uncertainty associated to different source-
receiver configurations should be assumed aleatory since sites with pronounced 2D effects
should be disregarded.

3.4.1 Active measurements

Although uncertainty estimation is necessary, there is not yet any protocol shared inter-
nationally on the best way to measure it from active surface wave analysis.

Independent processing of several recordings acquired with the same layout (shot and receivers) can be used to obtain different estimates of the experimental dispersion curve. These values can then be used to estimate mean and standard deviation of the phase velocity at each frequency. In this case the observed variability is only associated to repeatability of the test and observed discrepancies are mainly due to the effect of ambient vibrations. It is often much lower than the variability taking into account the natural variability of the ground and the uncertainties linked to the choice of the processing technique.

The variability of the results on the experimental dispersion curve can be estimated by considering different source locations for the same array, as well as using subsets of receivers instead of all receivers. With this approach, the uncertainties due to different factors are lumped together (e.g. lateral variations; effect of higher modes; near-field effects; signal-to-noise ratio). This way of uncertainty estimation is probably the one that is most similar to the uncertainty estimation for passive measurement (see below).

**Feedback from InterPacific project** Although no international consensus have yet been reached on the best way to estimate uncertainties for active surface wave analysis, the feedback from several international blind analysis projects (among them the InterPacific one, see Garofalo et al. 2016a) shows that, within the reliable wavelength band, inter-analyst uncertainties, estimated in terms of Coefficient of Variarion (COV), on the average phase velocity range from 5 to 10% depending on the sites (Fig. 27). The uncertainty on dispersion estimates provided by each analysist range between 5 and 20% depending on the site and the analyst’s (Fig. 28).

3.4.2 Passive surveys

For passive experiments, dispersion curves are most often obtained by averaging the
dispersion estimates from the various time blocks. Such uncertainties are related to time
variation of ambient vibration wavefield properties, in relation with the array capability to
separate simultaneous propagating waves across the array, with the lateral variation of the
subsurface, and with near-field effects.

This can be achieved by analyzing, frequency by frequency, the histogram of the
velocity estimates and removing the outliers to get a Gaussian distribution. Mean and
standard deviation can then be estimated at each frequency. Alternatively, another option
may be to manually pick several dispersion curves covering the whole area with high
number of velocity estimates, then average them to estimate the mean and standard
deviation.
<figure><img src="image_30.png" /><figcaption>**Fig. 27** Variability of estimated average dispersion curves from 12 different operators on the same active and passive dataset for the Grenoble site (top) and corresponding Coefficient of Variation (bottom) with evidence of the significant frequency band (gray line) (Garofalo et al. 2016a)</figcaption></figure>

<figure><img src="image_31.png" /><figcaption>**Fig. 28** Uncertainty (coefficient of variation) on estimated dispersion curves for 12 different operators on the same active and passive dataset for the Mirandola, Grenoble and Cadarache sites. Each colour refers to an analyst</figcaption></figure>

## 3.5 Combination of active and passive surveys

Combination of different datasets (active and passive with different array apertures) can provide an experimental dispersion curve over a broad frequency band. Branches of the dispersion curve from different dataset should overlap each other in the common frequency bands (see Fig. 29). A poor overlap may be associated to several causes (e.g. different
<figure><img src="image_30.png" /><figcaption>**Fig. 27** Variability of estimated average dispersion curves from 12 different operators on the same active and passive dataset for the Grenoble site (top) and corresponding Coefficient of Variation (bottom) with evidence of the significant frequency band (gray line) (Garofalo et al. 2016a)</figcaption></figure>

<figure><img src="image_31.png" /><figcaption>**Fig. 28** Uncertainty (coefficient of variation) on estimated dispersion curves for 12 different operators on the same active and passive dataset for the Mirandola, Grenoble and Cadarache sites. Each colour refers to an analyst</figcaption></figure>

## 3.5 Combination of active and passive surveys

Combination of different datasets (active and passive with different array apertures) can provide an experimental dispersion curve over a broad frequency band. Branches of the dispersion curve from different dataset should overlap each other in the common frequency bands (see Fig. 29). A poor overlap may be associated to several causes (e.g. different
<figure><img src="image_32.png" /><figcaption>Fig. 29 Example of overlap between branches estimated from active and passive data (Mirandola site, InterPACIFIC project)</figcaption></figure>

modes are retrieved, presence of lateral variation, lack of spectral resolution, pitfalls in the processing). In any case, a poor overlap implies a poor reliability of the analysis.

As mentioned in 3.4, estimated uncertainties do not always take the same variability into account. When combining dispersion curves with uncertainties of different origins, attention should be paid to homogenize these uncertainties in order not to give artificially a preponderant weight to the curve(s) with smaller uncertainties.

# 4 Inversion

## 4.1 Introduction

The inversion process is aimed at searching the best subsurface model whose forward response fits well the experimental data. It is based on a proper definition of the misfit function that is minimized in the inversion. Typically, for surface wave analysis, the misfit function is a norm of the distance between the experimental dispersion curve(s) or SPatial Autocorrelation Curves) and the theoretical dispersion curves associated to a given subsurface velocity model.

However, like most other surface geophysical problems, the surface wave inverse problem is non-linear, mathematically ill-posed and it is affected by solution non-uniqueness whereby several models may provide an equally good fit to the experimental data. For this reason, it is strongly recommended that the solution is provided in conjunction with a quantification of the uncertainty arising from solution non-uniqueness. The degree of ill-posedness is also strongly dependent upon the specific algorithm that is used to invert the dispersion curve and on the constraints that may be enforced on the calculated model (e.g. smoothness, layer thicknesses, etc.). This may be represented as an estimation of uncertainty bounds on the best solution (under the forced hypothesis of a linear problem with unique solution), or preferably as a set of equivalent solutions (i.e. solutions providing similar misfit).

The inversion process requires the description of the ground model via a series of physical and mechanical parameters linked to the experimental data by physical
<figure><img src="image_32.png" /><figcaption>Fig. 29 Example of overlap between branches estimated from active and passive data (Mirandola site, InterPACIFIC project)</figcaption></figure>

modes are retrieved, presence of lateral variation, lack of spectral resolution, pitfalls in the processing). In any case, a poor overlap implies a poor reliability of the analysis.

As mentioned in 3.4, estimated uncertainties do not always take the same variability into account. When combining dispersion curves with uncertainties of different origins, attention should be paid to homogenize these uncertainties in order not to give artificially a preponderant weight to the curve(s) with smaller uncertainties.

# 4 Inversion

## 4.1 Introduction

The inversion process is aimed at searching the best subsurface model whose forward response fits well the experimental data. It is based on a proper definition of the misfit function that is minimized in the inversion. Typically, for surface wave analysis, the misfit function is a norm of the distance between the experimental dispersion curve(s) or SPatial Autocorrelation Curves) and the theoretical dispersion curves associated to a given subsurface velocity model.

However, like most other surface geophysical problems, the surface wave inverse problem is non-linear, mathematically ill-posed and it is affected by solution non-uniqueness whereby several models may provide an equally good fit to the experimental data. For this reason, it is strongly recommended that the solution is provided in conjunction with a quantification of the uncertainty arising from solution non-uniqueness. The degree of ill-posedness is also strongly dependent upon the specific algorithm that is used to invert the dispersion curve and on the constraints that may be enforced on the calculated model (e.g. smoothness, layer thicknesses, etc.). This may be represented as an estimation of uncertainty bounds on the best solution (under the forced hypothesis of a linear problem with unique solution), or preferably as a set of equivalent solutions (i.e. solutions providing similar misfit).

The inversion process requires the description of the ground model via a series of physical and mechanical parameters linked to the experimental data by physical
relationships. Section 4.2 describes the specificity of the parameterization for surface wave inversion.

Several inversion approaches have been proposed to search for the set(s) of parameters best representing the real ground characteristics. These methods can be grouped into two broad categories: local search methods and global search methods. Their applications to surface wave inversion are respectively described in Sects. 4.3 and 4.4.

## 4.2 Parameterization

### 4.2.1 Parameters

Surface wave analysis is typically implemented by modeling the subsurface as a vertical stack of homogeneous and isotropic linear elastic layers. Each layer is characterized by 4 parameters: thickness (h, except for the half-space), mass density (ρ) and two elastic parameters (often implicitly expressed in terms of velocity of propagation of shear and compression waves, $V_S$ and $V_P$, linked together via the Poisson’s ratio, $v$). If attenuation is modeled, quality factors (Qp and Qs) are also included in the parameterization. Quality factors are also denoted as damping ratios (D) in geotechnical engineering; the relation between these two quantities being $Q = 1/2D$.

Layer thicknesses and shear wave velocities have the most influence on the dispersion curve, while P-wave velocity and mass density have a much smaller influence. The inverted parameters are thus often reduced to thickness and shear-wave velocity of each layer. The mass density and the other elastic parameter, which is either the Poisson’s ratio or the P-wave velocity, are then fixed on the basis of a-priori information or using standard values from expert judgment. Alternatively, $V_P$ can be left as free parameter, and the Poisson’s ratio has to be limited to reasonable bounds, depending on the expected soil conditions. This last strategy has the great advantage to prevent the occurrence of physically unrealistic $V_P-V_S$ pairs.

A large difference is however observed on the dispersion curves between unsaturated (low $V_P$, low $V_S$) and saturated (high $V_P$, low $V_S$) soils. The wrong choice of a-priori model parameters between values for saturated and unsaturated soils may cause very significant errors in the inversion process and the obtained shear wave velocity model may be unrealistic. A reasonable estimate of water table condition at the site is then essential for the analysis of surface wave data (see Sect. 4.2.4).

Expected ranges of mass density and Poisson ratio values for different geomaterials are reported in Table 2. More accurate estimates can be obtained if lithological information is available.

### 4.2.2 Number of layers

The number of layers must be fixed carefully in order to avoid over-parameterization (i.e. too many layers) that leads to an unreliable final model because of the insufficient available information to constrain the inversion (i.e. over-determined inversion). On the other side,

<table><caption>**Table 2** Expected values of a-priori parameters for soils and rocks</caption><thead><tr><td>Geo-material</td><td>Poisson ratio</td><td>Mass density (kg/m³)</td></tr></thead><tbody><tr><td>Soil (unsaturated)</td><td>0.15–0.35</td><td>1200–1800</td></tr><tr><td>Soil (saturated)</td><td>0.47–0.49</td><td>1500–2100</td></tr><tr><td>Rock</td><td>0.20–0.25</td><td>2100–2800</td></tr></tbody></table>
relationships. Section 4.2 describes the specificity of the parameterization for surface wave inversion.

Several inversion approaches have been proposed to search for the set(s) of parameters best representing the real ground characteristics. These methods can be grouped into two broad categories: local search methods and global search methods. Their applications to surface wave inversion are respectively described in Sects. 4.3 and 4.4.

## 4.2 Parameterization

### 4.2.1 Parameters

Surface wave analysis is typically implemented by modeling the subsurface as a vertical stack of homogeneous and isotropic linear elastic layers. Each layer is characterized by 4 parameters: thickness (h, except for the half-space), mass density (ρ) and two elastic parameters (often implicitly expressed in terms of velocity of propagation of shear and compression waves, $V_S$ and $V_P$, linked together via the Poisson’s ratio, $v$). If attenuation is modeled, quality factors (Qp and Qs) are also included in the parameterization. Quality factors are also denoted as damping ratios (D) in geotechnical engineering; the relation between these two quantities being $Q = 1/2D$.

Layer thicknesses and shear wave velocities have the most influence on the dispersion curve, while P-wave velocity and mass density have a much smaller influence. The inverted parameters are thus often reduced to thickness and shear-wave velocity of each layer. The mass density and the other elastic parameter, which is either the Poisson’s ratio or the P-wave velocity, are then fixed on the basis of a-priori information or using standard values from expert judgment. Alternatively, $V_P$ can be left as free parameter, and the Poisson’s ratio has to be limited to reasonable bounds, depending on the expected soil conditions. This last strategy has the great advantage to prevent the occurrence of physically unrealistic $V_P-V_S$ pairs.

A large difference is however observed on the dispersion curves between unsaturated (low $V_P$, low $V_S$) and saturated (high $V_P$, low $V_S$) soils. The wrong choice of a-priori model parameters between values for saturated and unsaturated soils may cause very significant errors in the inversion process and the obtained shear wave velocity model may be unrealistic. A reasonable estimate of water table condition at the site is then essential for the analysis of surface wave data (see Sect. 4.2.4).

Expected ranges of mass density and Poisson ratio values for different geomaterials are reported in Table 2. More accurate estimates can be obtained if lithological information is available.

### 4.2.2 Number of layers

The number of layers must be fixed carefully in order to avoid over-parameterization (i.e. too many layers) that leads to an unreliable final model because of the insufficient available information to constrain the inversion (i.e. over-determined inversion). On the other side,

<table><caption>**Table 2** Expected values of a-priori parameters for soils and rocks</caption><thead><tr><td>Geo-material</td><td>Poisson ratio</td><td>Mass density (kg/m³)</td></tr></thead><tbody><tr><td>Soil (unsaturated)</td><td>0.15–0.35</td><td>1200–1800</td></tr><tr><td>Soil (saturated)</td><td>0.47–0.49</td><td>1500–2100</td></tr><tr><td>Rock</td><td>0.20–0.25</td><td>2100–2800</td></tr></tbody></table>
the number of layers has to be sufficient to reproduce adequately the variation with depth, especially in the shallow portion of the model. Typically, a preliminary assessment with the adoption of different trial parameterizations is necessary to achieve the right balance, since no specific rules can be given and the choice is reserved to the expertise of the analyst. Variability of the results of the inversion due to different layer-model associated to expert choice is a typical example of epistemic uncertainty. However, a recent paper by Cox and Teague (2016) provides guidance on one method of systematically selecting trail parameterizations when performing inversions without the benefit of a-priori information to inform the choice of subsurface layering.

In any case it is recommended to repeat the inversion process by using different parameterizations to assess the influence on the final result. If a similar misfit is obtained, the solution with the minimum number of layers has to be preferred to improve the sensitivity to model parameters and the consequent reliability of the results. Remaining residuals between the observed dispersion and the dispersion predicted for the final model will indicate at which depth additional layer interfaces might be required (Fig. 30).

An over-parameterization of the model can be accepted as a different strategy if additional constraints are used to regularize the solution (smoothing, reference model, a-priori information). However, the constraints need to be carefully established on the basis of subsurface conditions. For example, when no large velocity contrasts are expected (e.g. in case of smooth gradient profiles of the stiff soils and rock sites), it may be advisable to increase the number of layers, but also to impose additional constraints on their depths and/or velocities (e.g. fixed depth and increasing thickness with depth following geometric progression or exponential laws; linear or power law increase of velocity with depth,...). Another example concerns the presence of low velocity layers, which may be forced in the solution when evidences of such structure are present (shape of the dispersion curve or a priori information). Nonetheless, the number of velocity reversals should be limited (ideally no more than one or two) as otherwise this will affect the non-uniqueness of the inversion problem.

### 4.2.3 Minimum and maximum depths

The minimum and the maximum retrieved wavelengths should be taken into account during the definition of the parameterization. For the inversion of the fundamental mode of Rayleigh wave, the thickness of the first layer should be greater than half minimum

<figure><img src="image_33.png" /><figcaption>**Fig. 30** Choosing the number of layers; numerical example with a 3-layer model dispersion curve inverted with two layers (a) and three layers (b). In a, misfit between observed (black) and modelled (colors) curves particularly at high frequencies indicate that adding a layer at shallower depths will improve the results</figcaption></figure>
the number of layers has to be sufficient to reproduce adequately the variation with depth, especially in the shallow portion of the model. Typically, a preliminary assessment with the adoption of different trial parameterizations is necessary to achieve the right balance, since no specific rules can be given and the choice is reserved to the expertise of the analyst. Variability of the results of the inversion due to different layer-model associated to expert choice is a typical example of epistemic uncertainty. However, a recent paper by Cox and Teague (2016) provides guidance on one method of systematically selecting trail parameterizations when performing inversions without the benefit of a-priori information to inform the choice of subsurface layering.

In any case it is recommended to repeat the inversion process by using different parameterizations to assess the influence on the final result. If a similar misfit is obtained, the solution with the minimum number of layers has to be preferred to improve the sensitivity to model parameters and the consequent reliability of the results. Remaining residuals between the observed dispersion and the dispersion predicted for the final model will indicate at which depth additional layer interfaces might be required (Fig. 30).

An over-parameterization of the model can be accepted as a different strategy if additional constraints are used to regularize the solution (smoothing, reference model, a-priori information). However, the constraints need to be carefully established on the basis of subsurface conditions. For example, when no large velocity contrasts are expected (e.g. in case of smooth gradient profiles of the stiff soils and rock sites), it may be advisable to increase the number of layers, but also to impose additional constraints on their depths and/or velocities (e.g. fixed depth and increasing thickness with depth following geometric progression or exponential laws; linear or power law increase of velocity with depth,...). Another example concerns the presence of low velocity layers, which may be forced in the solution when evidences of such structure are present (shape of the dispersion curve or a priori information). Nonetheless, the number of velocity reversals should be limited (ideally no more than one or two) as otherwise this will affect the non-uniqueness of the inversion problem.

### 4.2.3 Minimum and maximum depths

The minimum and the maximum retrieved wavelengths should be taken into account during the definition of the parameterization. For the inversion of the fundamental mode of Rayleigh wave, the thickness of the first layer should be greater than half minimum

<figure><img src="image_33.png" /><figcaption>**Fig. 30** Choosing the number of layers; numerical example with a 3-layer model dispersion curve inverted with two layers (a) and three layers (b). In a, misfit between observed (black) and modelled (colors) curves particularly at high frequencies indicate that adding a layer at shallower depths will improve the results</figcaption></figure>
available experimental wavelength while the maximum reached depth about half of the maximum available experimental wavelength or even less in stratigraphic conditions involving large impedance contrast at depth. In the latter case, consistency of the inverted velocity structure and the site resonance frequency may help in constraining the maximum resolvable depth.

### 4.2.4 Use of a-priori and external information

Any a-priori information available on the site is of paramount importance for the surface wave analysis. Useful information can be retrieved from any previous study performed in the area, geological maps, other independent surveys as borehole measurements or other geophysical and geotechnical tests.

**Stratigraphic log** If the stratigraphy, or any other borehole measurements, of the site is available, this should be used to define the optimum model parameterization. The interfaces of the initial model can be assumed as detected in the stratigraphy. Expected values of $V_S$ for each lithotype identified in the stratigraphic log could be selected on the basis of literature data (see Table 3 for example). However, it should be noted that that geological interfaces not always correspond to detectable velocity variations, and that P-wave and S-wave velocity are not necessarily subject to the same stratigraphy (e.g. in the case of water saturated sediments).

In uncemented coarse-grained soils, the velocity of propagation of the shear wave is strongly affected by effective confinement and as such it is dependent on depth. Several relationships between $V_S$ and depth can be found in the literature that can be used to define narrower ranges compared to those reported in Table 3 (e.g. Zimmer et al. 2007; Mitchell and Soga 2005).

**P-wave velocity** AP-wave seismic refraction survey performed on the same array of the active-source surface wave test (see Sect. 2.1) can be very useful as it may provide a clear indication of water table depth in soil deposits. Indeed the latter is associated to a sharp transition between unsaturated (few hundred meters per second) and saturated (higher than 1450 m/s) values of P-wave velocity of propagation. The sharp transition is easily identified as a major refractor. Like surface wave techniques, seismic refraction also suffers from non-uniqueness of the solution, and another option is to invert jointly for the dispersion characteristics and for the P-wave first arrivals (see Appendix 6).

**Resonance frequency and/or H/V curve** When available and when the experimental dispersion curve gives information down to the H/V frequency peak, this frequency may be jointly inverted with the dispersion curves (see Appendix 7). As well, part of or the whole

<table><caption>**Table 3** Expected values of shear wave velocity for soils and rocks</caption><thead><tr><td>Geomaterial</td><td>V<sub>S</sub> (m/s)</td></tr></thead><tbody><tr><td>Soft clay</td><td>80–200</td></tr><tr><td>Stiff clay</td><td>200–600</td></tr><tr><td>Loose sand</td><td>80–250</td></tr><tr><td>Dense sand</td><td>200–500</td></tr><tr><td>Gravel</td><td>300–900</td></tr><tr><td>Weathered rock</td><td>600–1000</td></tr><tr><td>Competent rock</td><td>1200–2500</td></tr></tbody></table>
available experimental wavelength while the maximum reached depth about half of the maximum available experimental wavelength or even less in stratigraphic conditions involving large impedance contrast at depth. In the latter case, consistency of the inverted velocity structure and the site resonance frequency may help in constraining the maximum resolvable depth.

### 4.2.4 Use of a-priori and external information

Any a-priori information available on the site is of paramount importance for the surface wave analysis. Useful information can be retrieved from any previous study performed in the area, geological maps, other independent surveys as borehole measurements or other geophysical and geotechnical tests.

**Stratigraphic log** If the stratigraphy, or any other borehole measurements, of the site is available, this should be used to define the optimum model parameterization. The interfaces of the initial model can be assumed as detected in the stratigraphy. Expected values of $V_S$ for each lithotype identified in the stratigraphic log could be selected on the basis of literature data (see Table 3 for example). However, it should be noted that that geological interfaces not always correspond to detectable velocity variations, and that P-wave and S-wave velocity are not necessarily subject to the same stratigraphy (e.g. in the case of water saturated sediments).

In uncemented coarse-grained soils, the velocity of propagation of the shear wave is strongly affected by effective confinement and as such it is dependent on depth. Several relationships between $V_S$ and depth can be found in the literature that can be used to define narrower ranges compared to those reported in Table 3 (e.g. Zimmer et al. 2007; Mitchell and Soga 2005).

**P-wave velocity** AP-wave seismic refraction survey performed on the same array of the active-source surface wave test (see Sect. 2.1) can be very useful as it may provide a clear indication of water table depth in soil deposits. Indeed the latter is associated to a sharp transition between unsaturated (few hundred meters per second) and saturated (higher than 1450 m/s) values of P-wave velocity of propagation. The sharp transition is easily identified as a major refractor. Like surface wave techniques, seismic refraction also suffers from non-uniqueness of the solution, and another option is to invert jointly for the dispersion characteristics and for the P-wave first arrivals (see Appendix 6).

**Resonance frequency and/or H/V curve** When available and when the experimental dispersion curve gives information down to the H/V frequency peak, this frequency may be jointly inverted with the dispersion curves (see Appendix 7). As well, part of or the whole

<table><caption>**Table 3** Expected values of shear wave velocity for soils and rocks</caption><thead><tr><td>Geomaterial</td><td>V<sub>S</sub> (m/s)</td></tr></thead><tbody><tr><td>Soft clay</td><td>80–200</td></tr><tr><td>Stiff clay</td><td>200–600</td></tr><tr><td>Loose sand</td><td>80–250</td></tr><tr><td>Dense sand</td><td>200–500</td></tr><tr><td>Gravel</td><td>300–900</td></tr><tr><td>Weathered rock</td><td>600–1000</td></tr><tr><td>Competent rock</td><td>1200–2500</td></tr></tbody></table>
H/V curve can be inverted, which requires the use of a dedicated solver. If not possible, the transfer function corresponding to the resulting $V_S$ profile should be computed and its consistency with the measured H/V frequency peak checked.

## 4.3 Local search methods: linearized inversions

### 4.3.1 Principle

In local search methods, the solution is searched with an iterative process starting from an initial model (in green in Fig. 31a). The model is progressively adjusted at successive iterations in order to minimize the value of the misfit function (Fig. 31b). Figure 31c shows the comparison among the experimental data and the forward response of the initial model and the final model.

Local search methods usually compute the forward response for the model at each iteration. Moreover, at each iteration the Jacobian matrix is usually also computed. The Jacobian matrix is the sensitivity matrix of the measure (the dispersion or SPAC curve) to the model parameters. It is used to determine how to modify the model to decrease the misfit, and also gives an indication on the resolution of each parameter: the lower the sensitivity, the worse that parameter is resolved.

### 4.3.2 Initial model

The choice of the initial model is crucial in local search methods. The result of the inversion may be strongly dependent on the adopted initial model, as the solution may be trapped in a local minimum of the misfit function. It is strongly recommended that the

<figure><img src="image_34.png" /><figcaption>**Fig. 31** Example of local search method result. **a** Final model (in red) and initial model (in green). **b** The misfit as function of iterations. **c** Comparison among the experimental data (in blue), the forward response of the initial model (in green) and the forward response of the final model (in red) (Mirandola site, InterPACIFIC project)</figcaption></figure>
H/V curve can be inverted, which requires the use of a dedicated solver. If not possible, the transfer function corresponding to the resulting $V_S$ profile should be computed and its consistency with the measured H/V frequency peak checked.

## 4.3 Local search methods: linearized inversions

### 4.3.1 Principle

In local search methods, the solution is searched with an iterative process starting from an initial model (in green in Fig. 31a). The model is progressively adjusted at successive iterations in order to minimize the value of the misfit function (Fig. 31b). Figure 31c shows the comparison among the experimental data and the forward response of the initial model and the final model.

Local search methods usually compute the forward response for the model at each iteration. Moreover, at each iteration the Jacobian matrix is usually also computed. The Jacobian matrix is the sensitivity matrix of the measure (the dispersion or SPAC curve) to the model parameters. It is used to determine how to modify the model to decrease the misfit, and also gives an indication on the resolution of each parameter: the lower the sensitivity, the worse that parameter is resolved.

### 4.3.2 Initial model

The choice of the initial model is crucial in local search methods. The result of the inversion may be strongly dependent on the adopted initial model, as the solution may be trapped in a local minimum of the misfit function. It is strongly recommended that the

<figure><img src="image_34.png" /><figcaption>**Fig. 31** Example of local search method result. **a** Final model (in red) and initial model (in green). **b** The misfit as function of iterations. **c** Comparison among the experimental data (in blue), the forward response of the initial model (in green) and the forward response of the final model (in red) (Mirandola site, InterPACIFIC project)</figcaption></figure>
inversion procedure be repeated starting with different initial models and any a priori
information is included (see Sect. 4.2 for parameterization choice).

If no a priori information is available, a simple approach to build a first tentative profile
is to use the measured dispersion curve itself, based on the relation between wavelength (λ)
and penetration depth: V<sub>S</sub> values are defined from phase velocity values, possibly
increased by 5% or 10% (difference between Rayleigh phase velocity and V<sub>S</sub>); in a
homogeneous medium depths values are defined from a fraction of the wavelength. The
exact fraction linking wavelength and depth depends on the velocity structure and cannot
be defined a priori. However, this exact fraction is not required to define a rough initial
model, and used values range between 0.4 and 0.8. This approach was used in the past to
directly estimate the shear wave velocity profile from surface wave data avoiding a formal
solution of the inverse problem. Actually, it better represents the average V<sub>S</sub> value from the
surface to the relevant depth. It can be shown that it provides reasonable results especially
when the fundamental Rayleigh mode is dominant in the propagation.

Figure 32 shows an example where the initial V<sub>S</sub> profile is defined as a 6-layer-over-
half-space model, with velocities and interface depths estimated to best fit the phase-
velocity versus wavelength/2.5 curve. This initial model was used for the local search in
the example of Fig. 31.

4.3.3 Convergence criteria

The convergence criteria are the inversion parameters that stop the inversion iterative
process. The final model at the last iteration represents the inversion result.

Greatest decreases of the misfit usually occur in the first few iterations, after which the
difference between observed and computed dispersion characteristics is only slightly
reduced. Carrying on the inversion on a large number of iterations may either be useless, or
may even introduce artifacts if the problem is over-parameterised. The iteration is therefore
usually stopped after a limited number of iterations.

<figure><img src="image_35.png" /><figcaption>**Fig. 32** Example of definition of the initial model based on λ/2.5 approach (Mirandola site, InterPACIFIC project)</figcaption></figure>
inversion procedure be repeated starting with different initial models and any a priori information is included (see Sect. 4.2 for parameterization choice).

If no a priori information is available, a simple approach to build a first tentative profile is to use the measured dispersion curve itself, based on the relation between wavelength (λ) and penetration depth: V<sub>S</sub> values are defined from phase velocity values, possibly increased by 5% or 10% (difference between Rayleigh phase velocity and V<sub>S</sub>); in a homogeneous medium depths values are defined from a fraction of the wavelength. The exact fraction linking wavelength and depth depends on the velocity structure and cannot be defined a priori. However, this exact fraction is not required to define a rough initial model, and used values range between 0.4 and 0.8. This approach was used in the past to directly estimate the shear wave velocity profile from surface wave data avoiding a formal solution of the inverse problem. Actually, it better represents the average V<sub>S</sub> value from the surface to the relevant depth. It can be shown that it provides reasonable results especially when the fundamental Rayleigh mode is dominant in the propagation.

Figure 32 shows an example where the initial V<sub>S</sub> profile is defined as a 6-layer-over-
half-space model, with velocities and interface depths estimated to best fit the phase-
velocity versus wavelength/2.5 curve. This initial model was used for the local search in
the example of Fig. 31.

4.3.3 Convergence criteria

The convergence criteria are the inversion parameters that stop the inversion iterative
process. The final model at the last iteration represents the inversion result.

Greatest decreases of the misfit usually occur in the first few iterations, after which the
difference between observed and computed dispersion characteristics is only slightly
reduced. Carrying on the inversion on a large number of iterations may either be useless, or
may even introduce artifacts if the problem is over-parameterised. The iteration is therefore
usually stopped after a limited number of iterations.

<figure><img src="image_35.png" /><figcaption>**Fig. 32** Example of definition of the initial model based on λ/2.5 approach (Mirandola site, InterPACIFIC project)</figcaption></figure>
The other criteria usually defined for stopping linearized inversions is a misfit threshold below which the improvement of the forward dispersion characteristics is considered to be smaller than the uncertainties affecting the data. The inversion then stops when the misfit is lower than the defined threshold.

### 4.3.4 Assessment of results

The results obtained with the solution of the inverse problem should be carefully inspected to assess their reliability. In particular, the following points deserve attention.

**Residuals** Remaining residuals between the observed dispersion and the dispersion predicted for the final model will indicate the reliability of the model. A poor overlap in a given wavelength range is an indication of lack of reliability at a certain depth range.

**Depth range** The thickness of the first layer and the maximum investigation depth should be consistent with the available wavelengths in the dispersion curve (see Sect. 4.2.3), which themselves should be consistent with the resolution of the receiver array(s) (see Sects. 2.1.2 and 3.1.2).

**Consistency with a priori and/or external information** If a priori or external information is available (type of materials or depths from borehole logs, P-wave velocities or water table position, resonance frequency or H/V curves), it is important to check the consistency of the results with this information in order to make sure the final inversion results are generally agreeable with the a-priori information.

### 4.3.5 Uncertainties

Uncertainties on the results of a linearized inversion may be estimated from the analysis of the Jacobian (sensitivity) matrix: the higher the sensitivity, the better that parameter is resolved. The Jacobian can also be used to propagate uncertainties on the dispersion curve to uncertainties on the shear wave velocity profile (Lai et al. 2005).

Alternatively, uncertainties may be evaluated from the different results obtained with different initial models, or different constraints on the less influent parameters (mass density, P-wave velocity, Poisson’s ratio). Eventually different results due to different inversion algorithms may be evaluated.

## 4.4 Global search algorithms

### 4.4.1 Principle

In global search methods, such as Monte Carlo methods, a probability density distribution (e.g. uniform or normal) is defined for each model parameter (green boundaries in Fig. 33a). The model parameter space is then sampled following a specific search algorithm. For each sample profile the forward response is computed and the misfit with respect to experimental data is obtained. Inference is then made on the population of generated profiles to identify those that represent acceptable solutions. In contrast with the single model produced by linearized inversion, the result of global search methods is a set of many acceptable profiles rather than the single profile associated to the minimum misfit value. This allows the assessment of the uncertainty related to non-uniqueness of the solution.
The other criteria usually defined for stopping linearized inversions is a misfit threshold below which the improvement of the forward dispersion characteristics is considered to be smaller than the uncertainties affecting the data. The inversion then stops when the misfit is lower than the defined threshold.

### 4.3.4 Assessment of results

The results obtained with the solution of the inverse problem should be carefully inspected to assess their reliability. In particular, the following points deserve attention.

**Residuals** Remaining residuals between the observed dispersion and the dispersion predicted for the final model will indicate the reliability of the model. A poor overlap in a given wavelength range is an indication of lack of reliability at a certain depth range.

**Depth range** The thickness of the first layer and the maximum investigation depth should be consistent with the available wavelengths in the dispersion curve (see Sect. 4.2.3), which themselves should be consistent with the resolution of the receiver array(s) (see Sects. 2.1.2 and 3.1.2).

**Consistency with a priori and/or external information** If a priori or external information is available (type of materials or depths from borehole logs, P-wave velocities or water table position, resonance frequency or H/V curves), it is important to check the consistency of the results with this information in order to make sure the final inversion results are generally agreeable with the a-priori information.

### 4.3.5 Uncertainties

Uncertainties on the results of a linearized inversion may be estimated from the analysis of the Jacobian (sensitivity) matrix: the higher the sensitivity, the better that parameter is resolved. The Jacobian can also be used to propagate uncertainties on the dispersion curve to uncertainties on the shear wave velocity profile (Lai et al. 2005).

Alternatively, uncertainties may be evaluated from the different results obtained with different initial models, or different constraints on the less influent parameters (mass density, P-wave velocity, Poisson’s ratio). Eventually different results due to different inversion algorithms may be evaluated.

## 4.4 Global search algorithms

### 4.4.1 Principle

In global search methods, such as Monte Carlo methods, a probability density distribution (e.g. uniform or normal) is defined for each model parameter (green boundaries in Fig. 33a). The model parameter space is then sampled following a specific search algorithm. For each sample profile the forward response is computed and the misfit with respect to experimental data is obtained. Inference is then made on the population of generated profiles to identify those that represent acceptable solutions. In contrast with the single model produced by linearized inversion, the result of global search methods is a set of many acceptable profiles rather than the single profile associated to the minimum misfit value. This allows the assessment of the uncertainty related to non-uniqueness of the solution.
<figure><img src="image_36.png" /><figcaption>**Fig. 33** Example of global search method result. **a** In *red* the profile with minimum misfit value. The *green dashed lines* represent the boundaries of the search area. The profiles, from *blue* to *yellow* according to their misfit values, are all the accepted profiles; **b** in *red* experimental data and the line from *blue* to *yellow* correspond to the forward response of the accepted profiles; **c** comparison between experimental data (in *red*) and the forward response of the profile with minimum misfit value (in *blue*) (Mirandola site, InterPACIFIC project)</figcaption></figure>

In the example of Fig. 33, the profiles (Fig. 33a) whose forward response provides a misfit that passes a statistical test that accounts for experimental uncertainties and model parameterization (Fig. 33b) are selected. In Fig. 33c, the comparison between the experimental data (in red) and the forward response of the profile with minimum misfit value is reported.

The probability of getting a reliable solution with global search methods is related to the size of the population of tentative models. Pure Monte Carlo approaches require a very large number of samples, which are randomly generated. The size of the population should be large enough to properly sample the model parameter space, but small enough to limit computational cost.

Many optimization approaches have been proposed to limit the number of trial profiles (e.g. Latin hypercube sampling, genetic algorithms, simulated annealing, neighbourhood algorithm...). Basically, in all modern global optimization algorithms, after a relatively broad initial explorative random search, the parameter space is iteratively refined and progressively restricted around one or several regions potentially including the absolute minimum of the misfit function.

### 4.4.2 Definition of the parameter space

As already mentioned, any a-priori information should be taken into account in the design of the parameter space. The width of the parameter interval and the number of layers defines the size of the model space and should be selected according to the level of a priori knowledge available. Very narrow ranges of model parameters strongly constrain the solution, which may fall into a local minimum of the misfit function therefore not taking advantage of the global search approach, so very narrow ranges should be used only when very reliable a priori information is available. On the other side, very large ranges increase
<figure><img src="image_36.png" /><figcaption>**Fig. 33** Example of global search method result. **a** In *red* the profile with minimum misfit value. The *green dashed lines* represent the boundaries of the search area. The profiles, from *blue* to *yellow* according to their misfit values, are all the accepted profiles; **b** in *red* experimental data and the line from *blue* to *yellow* correspond to the forward response of the accepted profiles; **c** comparison between experimental data (in *red*) and the forward response of the profile with minimum misfit value (in *blue*) (Mirandola site, InterPACIFIC project)</figcaption></figure>

In the example of Fig. 33, the profiles (Fig. 33a) whose forward response provides a misfit that passes a statistical test that accounts for experimental uncertainties and model parameterization (Fig. 33b) are selected. In Fig. 33c, the comparison between the experimental data (in red) and the forward response of the profile with minimum misfit value is reported.

The probability of getting a reliable solution with global search methods is related to the size of the population of tentative models. Pure Monte Carlo approaches require a very large number of samples, which are randomly generated. The size of the population should be large enough to properly sample the model parameter space, but small enough to limit computational cost.

Many optimization approaches have been proposed to limit the number of trial profiles (e.g. Latin hypercube sampling, genetic algorithms, simulated annealing, neighbourhood algorithm...). Basically, in all modern global optimization algorithms, after a relatively broad initial explorative random search, the parameter space is iteratively refined and progressively restricted around one or several regions potentially including the absolute minimum of the misfit function.

### 4.4.2 Definition of the parameter space

As already mentioned, any a-priori information should be taken into account in the design of the parameter space. The width of the parameter interval and the number of layers defines the size of the model space and should be selected according to the level of a priori knowledge available. Very narrow ranges of model parameters strongly constrain the solution, which may fall into a local minimum of the misfit function therefore not taking advantage of the global search approach, so very narrow ranges should be used only when very reliable a priori information is available. On the other side, very large ranges increase
the computational time and decrease the chance of getting close to the absolute minimum of the misfit function. A possible strategy is starting with large limits and repeats the inversion by narrowing them on the basis of the first run results.

### 4.4.3 Definition of the results

Contrarily to linearized inversion the result of global search methods is a set of acceptable profiles selected according to their misfit. The size of this set can be defined according to the method that is used to make inference on the population of trial models. Optimization methods may converge toward a very limited number or even individual final models, other methods may select the acceptable models on the basis of statistical tests or to simple criteria based on misfit threshold of accepted model number.

The solution space (that represents the misfit as a function of the model parameters) may provide a useful insight about the goodness of the model space mapping.

### 4.4.4 Assessment of results

Assessment of results are based on the same criteria as for the linearized inversion (see Sect. 4.3.4).

### 4.4.5 Uncertainties

The boundaries of the area filled by all the accepted profiles could be considered as uncertainty boundaries of the profile with minimum misfit (the red profile in Fig. 33a). Statistics on the selected profiles would require a clear assessment of epistemic and aleatory uncertainty related to the various ingredients of the inversion process (data uncertainty, ground model parameterization, etc.). Quantification of variability of the final ground models are thus still an open issue and certainly deserves further investigation in future research.

When further forward calculations are to be computed in the frame of a study (e.g. assessment of seismic response for seismic risk assessment), we suggest using a representative set of acceptable profiles.

## 4.5 Advantages and disadvantages of each type of inversion

In general global search methods have to be preferred over local search methods, as the latter may provide results strongly dependent on the first tentative profile adopted for the inversion. Moreover, global search methods allow easier investigation of the uncertainties associated to solution non-uniqueness. On the other hand, they are usually more time and computational resource consuming.

Both approaches can be combined, by using a global search sampling the model space and refining the solution close to the minima of the misfit function using a local search algorithm (e.g. Poggi et al. 2012).
the computational time and decrease the chance of getting close to the absolute minimum of the misfit function. A possible strategy is starting with large limits and repeats the inversion by narrowing them on the basis of the first run results.

### 4.4.3 Definition of the results

Contrarily to linearized inversion the result of global search methods is a set of acceptable profiles selected according to their misfit. The size of this set can be defined according to the method that is used to make inference on the population of trial models. Optimization methods may converge toward a very limited number or even individual final models, other methods may select the acceptable models on the basis of statistical tests or to simple criteria based on misfit threshold of accepted model number.

The solution space (that represents the misfit as a function of the model parameters) may provide a useful insight about the goodness of the model space mapping.

### 4.4.4 Assessment of results

Assessment of results are based on the same criteria as for the linearized inversion (see Sect. 4.3.4).

### 4.4.5 Uncertainties

The boundaries of the area filled by all the accepted profiles could be considered as uncertainty boundaries of the profile with minimum misfit (the red profile in Fig. 33a). Statistics on the selected profiles would require a clear assessment of epistemic and aleatory uncertainty related to the various ingredients of the inversion process (data uncertainty, ground model parameterization, etc.). Quantification of variability of the final ground models are thus still an open issue and certainly deserves further investigation in future research.

When further forward calculations are to be computed in the frame of a study (e.g. assessment of seismic response for seismic risk assessment), we suggest using a representative set of acceptable profiles.

## 4.5 Advantages and disadvantages of each type of inversion

In general global search methods have to be preferred over local search methods, as the latter may provide results strongly dependent on the first tentative profile adopted for the inversion. Moreover, global search methods allow easier investigation of the uncertainties associated to solution non-uniqueness. On the other hand, they are usually more time and computational resource consuming.

Both approaches can be combined, by using a global search sampling the model space and refining the solution close to the minima of the misfit function using a local search algorithm (e.g. Poggi et al. 2012).
# 5 Applications and use of surface waves survey for earthquake engineering studies

This section provides some examples of application and use of surface wave surveys for various objectives, from the simplest one (e.g. $V_{S,30}$ determination) to more complex one (1D soil response computation). For these examples, possible survey layouts are proposed. They have to be considered as possible layouts, not as a strict procedure: other approaches may be suitable as well.

For each example, we will comment what could be considered as reliable outcomes from surface wave methods, but also what should be consider with care. These comments are partly based on the learnings from the InterPacific project (Garofalo et al. 2016a, b).

This section aims to provide information for geophysicists that will implement surveys, but also for clients and analysts that will use their results.

## 5.1 Soil class and $V_{S,30}$ determination

$V_{S,30}$ i.e. the time-average shear-wave velocity in the topmost 30 m, is a parameter adopted by several building codes, i.e. EC8 and IBC, for the seismic soil classification. $V_{S,30}$ is also widely used for the implementation of Ground Motion Prediction Equations (GMPEs) and therefore, this parameter is typically needed to characterize the sites of accelerometric networks.

For a given $V_S$ profile, the $V_{S,30}$ is estimated as:

$$V_{S,30} = \frac{30 \text{ m}}{\sum_{i=1}^{N} \frac{H_i}{V_{S,i}}} \quad (2)$$

where N is the number of layers of the $V_S$-profile down to the depth of 30 m, while $H_i$ and $V_{S,i}$ are the thickness and the shear-wave velocity of each layer respectively.

Several studies have shown that calculated $V_{S,30}$ from the shear wave velocity profiles of surface wave analysis is robust and in very good agreement with estimates obtained with other seismic methods (e.g. borehole methods such as cross-hole method, down-hole method, P-S suspension logging).

Since $V_{S,30}$ is a parameter related to the average properties of the subsurface, some authors have suggested some direct relationship between the $V_{S,30}$ and the phase velocity of the dispersion curve at a given wavelength. Using these formulas, the formal inversion of surface wave data is not necessary. These approaches can be used for a rough and fast estimate of $V_{S,30}$ or to double-check the values calculated from the shear wave velocity profile. According to these relationships, the $V_{S,30}$ can be estimated as the phase velocity at a given wavelength times a correcting factor (e.g. Brown et al. 2000; Martin and Diehl 2004; Albarello and Gargani 2010; Comina et al. 2011). A simple approximation of these expressions is given by:

$$V_{S,30} = V_{R,[40-45]} \quad (3)$$

in which $V_{R,[40-45]}$ is the phase velocity of the Rayleigh wave fundamental mode for wavelength ranging from 40 to 45 m. Generally, $V_{R,40}$ is more appropriate at sites with shallow groundwater and $V_{R,45}$ is more appropriate at sites with deep groundwater.

Therefore, $V_{S,30}$ can be calculated from the shear wave velocity profile obtained from the whole procedure of surface wave method processing (including the inversion), but also estimated directly from the dispersion curves which is a reliable and stable quantity,
# 5 Applications and use of surface waves survey for earthquake engineering studies

This section provides some examples of application and use of surface wave surveys for various objectives, from the simplest one (e.g. $V_{S,30}$ determination) to more complex one (1D soil response computation). For these examples, possible survey layouts are proposed. They have to be considered as possible layouts, not as a strict procedure: other approaches may be suitable as well.

For each example, we will comment what could be considered as reliable outcomes from surface wave methods, but also what should be consider with care. These comments are partly based on the learnings from the InterPacific project (Garofalo et al. 2016a, b).

This section aims to provide information for geophysicists that will implement surveys, but also for clients and analysts that will use their results.

## 5.1 Soil class and $V_{S,30}$ determination

$V_{S,30}$ i.e. the time-average shear-wave velocity in the topmost 30 m, is a parameter adopted by several building codes, i.e. EC8 and IBC, for the seismic soil classification. $V_{S,30}$ is also widely used for the implementation of Ground Motion Prediction Equations (GMPEs) and therefore, this parameter is typically needed to characterize the sites of accelerometric networks.

For a given $V_S$ profile, the $V_{S,30}$ is estimated as:

$$V_{S,30} = \frac{30 \text{ m}}{\sum_{i=1}^{N} \frac{H_i}{V_{S,i}}} \quad (2)$$

where N is the number of layers of the $V_S$-profile down to the depth of 30 m, while $H_i$ and $V_{S,i}$ are the thickness and the shear-wave velocity of each layer respectively.

Several studies have shown that calculated $V_{S,30}$ from the shear wave velocity profiles of surface wave analysis is robust and in very good agreement with estimates obtained with other seismic methods (e.g. borehole methods such as cross-hole method, down-hole method, P-S suspension logging).

Since $V_{S,30}$ is a parameter related to the average properties of the subsurface, some authors have suggested some direct relationship between the $V_{S,30}$ and the phase velocity of the dispersion curve at a given wavelength. Using these formulas, the formal inversion of surface wave data is not necessary. These approaches can be used for a rough and fast estimate of $V_{S,30}$ or to double-check the values calculated from the shear wave velocity profile. According to these relationships, the $V_{S,30}$ can be estimated as the phase velocity at a given wavelength times a correcting factor (e.g. Brown et al. 2000; Martin and Diehl 2004; Albarello and Gargani 2010; Comina et al. 2011). A simple approximation of these expressions is given by:

$$V_{S,30} = V_{R,[40-45]} \quad (3)$$

in which $V_{R,[40-45]}$ is the phase velocity of the Rayleigh wave fundamental mode for wavelength ranging from 40 to 45 m. Generally, $V_{R,40}$ is more appropriate at sites with shallow groundwater and $V_{R,45}$ is more appropriate at sites with deep groundwater.

Therefore, $V_{S,30}$ can be calculated from the shear wave velocity profile obtained from the whole procedure of surface wave method processing (including the inversion), but also estimated directly from the dispersion curves which is a reliable and stable quantity,
avoiding the need to perform the inversion, which is the most complex and time-consuming step in surface wave analysis.

One common pitfall of this very simple approach is due to the possible misinterpretation of the phase velocity of a higher mode as the fundamental mode. It is therefore necessary to collect sufficient information to ensure that the fundamental mode is correctly identified over the frequency band of interest.

A rather complete survey layout to achieve this requirement could be the acquisition of 3 consecutive arrays (e.g. circle arrays using 8 or 10 sensors) with a diameter (aperture) of 20, 50 and 125 m, plus a MASW line, involving 24 geophones with 2 m geophone spacing. This layout will ensure the estimation of a reliable dispersion curve, with a satisfactory overlap between the dispersion curves from passive arrays and MASW linear array.

## 5.2 Other commonly used proxies

Other parameters are also used, especially within the framework of GMPEs formulations, in order to achieve a better representation of site features. A non-exhaustive list of such parameters follows:

* $f_0$: fundamental resonance frequency of the site,

* $H_{800}$: depth at which the shear wave velocity profile exceeds the value of 800 m/s (i.e. the position of a conventional seismic bedrock).

* $H_{\text{bedrock}}$: depth of the geological bedrock, that is to say the depth corresponding to the highest contrast within the shear wave velocity profile. This contrast usually corresponds to the one that causes the $f_0$ resonance and also corresponds to a lithology change.

* $V_{\text{moy}}$: mean shear wave velocity between surface and bedrock (as defined in $H_{\text{bedrock}}$ definition).

* $V_{\text{bedrock}}$: shear wave velocity of the bedrock.

* $V_{\text{qwl}}$: a frequency-dependent estimation of the average shear wave velocity at a depth corresponding to ¼ of the wavelength.

The $f_0$ value can rather easily be obtained from single station ambient vibration measurements performed with a 3-component sensor (see Sesame Guidelines 2004). Often, these data can be extracted by sensors in the passive array used for surface wave analyses.

The other parameters can be calculated from the shear wave velocity profile. Taking into account solution non-uniqueness it is recommended that the uncertainty bounds on these parameters be evaluated by repeating the calculation for a set of equivalent solution (see Sect. 4.4.5).

The results of the InterPacific benchmark (Garofalo et al. 2016a), show that, while the average velocities within the first tens of meters are rather well determined by surface wave methods, the determination of $H_{\text{bedrock}}$ and $V_{\text{bedrock}}$ have higher uncertainties. In order to quantify this uncertainty, it is recommended to perform multiple inversions using various hypotheses and parameterizations. Performing a joint inversion involving both dispersion curves and value of fundamental frequency $f_0$ or a-priori information may also reduce the uncertainty. Regarding the $V_{\text{bedrock}}$ parameter, if the “bedrock” geological formation is outcropping near the studied site, it may be useful to perform a survey on the outcropping area to obtain a reference value for the bedrock velocity.

In order to determine the velocity profiles needed for the determination of above mentioned parameters (except for $f_0$ which only needs a single-station measurement), one can realize a MASW line (for example involving 24 geophones with 2 m geophone
avoiding the need to perform the inversion, which is the most complex and time-consuming step in surface wave analysis.

One common pitfall of this very simple approach is due to the possible misinterpretation of the phase velocity of a higher mode as the fundamental mode. It is therefore necessary to collect sufficient information to ensure that the fundamental mode is correctly identified over the frequency band of interest.

A rather complete survey layout to achieve this requirement could be the acquisition of 3 consecutive arrays (e.g. circle arrays using 8 or 10 sensors) with a diameter (aperture) of 20, 50 and 125 m, plus a MASW line, involving 24 geophones with 2 m geophone spacing. This layout will ensure the estimation of a reliable dispersion curve, with a satisfactory overlap between the dispersion curves from passive arrays and MASW linear array.

## 5.2 Other commonly used proxies

Other parameters are also used, especially within the framework of GMPEs formulations, in order to achieve a better representation of site features. A non-exhaustive list of such parameters follows:

* $f_0$: fundamental resonance frequency of the site,

* $H_{800}$: depth at which the shear wave velocity profile exceeds the value of 800 m/s (i.e. the position of a conventional seismic bedrock).

* $H_{\text{bedrock}}$: depth of the geological bedrock, that is to say the depth corresponding to the highest contrast within the shear wave velocity profile. This contrast usually corresponds to the one that causes the $f_0$ resonance and also corresponds to a lithology change.

* $V_{\text{moy}}$: mean shear wave velocity between surface and bedrock (as defined in $H_{\text{bedrock}}$ definition).

* $V_{\text{bedrock}}$: shear wave velocity of the bedrock.

* $V_{\text{qwl}}$: a frequency-dependent estimation of the average shear wave velocity at a depth corresponding to ¼ of the wavelength.

The $f_0$ value can rather easily be obtained from single station ambient vibration measurements performed with a 3-component sensor (see Sesame Guidelines 2004). Often, these data can be extracted by sensors in the passive array used for surface wave analyses.

The other parameters can be calculated from the shear wave velocity profile. Taking into account solution non-uniqueness it is recommended that the uncertainty bounds on these parameters be evaluated by repeating the calculation for a set of equivalent solution (see Sect. 4.4.5).

The results of the InterPacific benchmark (Garofalo et al. 2016a), show that, while the average velocities within the first tens of meters are rather well determined by surface wave methods, the determination of $H_{\text{bedrock}}$ and $V_{\text{bedrock}}$ have higher uncertainties. In order to quantify this uncertainty, it is recommended to perform multiple inversions using various hypotheses and parameterizations. Performing a joint inversion involving both dispersion curves and value of fundamental frequency $f_0$ or a-priori information may also reduce the uncertainty. Regarding the $V_{\text{bedrock}}$ parameter, if the “bedrock” geological formation is outcropping near the studied site, it may be useful to perform a survey on the outcropping area to obtain a reference value for the bedrock velocity.

In order to determine the velocity profiles needed for the determination of above mentioned parameters (except for $f_0$ which only needs a single-station measurement), one can realize a MASW line (for example involving 24 geophones with 2 m geophone
spacing) for high frequency and shallow layers, then deploy several passive arrays, with successive apertures starting from relatively small (15–20 m) up to a value equal to the double of the desired investigation depth. A very large passive array could help in the determination of the bedrock velocity.

## 5.3 Ground response analysis

One of the current applications of surface wave methods is the determination of shear wave velocity profiles in order to compute 1D site response. We recommend that a set of acceptable velocity model is estimated.

When the geophysicist provides a set of possible solutions to the engineer responsible for site response simulation, a common error is to group all the profiles on a single plot and to consider three inferred profiles for the computations: the mean of all profiles; the mean plus and minus one standard deviation (all computed “depth by depth”). This will lead to an overestimation of the uncertainty. Indeed both outer profiles will not correspond to the measured surface wave dispersion curve, which is a robust parameter and represents a signature of the site.

When representative profiles are built for ground response analysis, they should always be tested by checking the corresponding theoretical dispersion curve and fundamental frequency against the available experimental ones. Similarly, when randomization approaches (e.g. Toro 1995) are adopted, it is necessary to select only “randomized” profiles that fit the measured values of dispersion curves and $f_0$ (Teague and Cox 2016).

The suggested approach to account for solution non-uniqueness is to repeat the ground response analyses for the individual profiles selected during the inversion process. This approach allows a direct evaluation of the final uncertainty associated to the seismic site response of the site. This uncertainty may be significantly smaller than the one associated to the shear wave velocity profile (e.g. Fig. 34). Indeed the profiles that match the experimental dispersion curve are likely equivalent also in terms of ground response to vertically propagating shear waves, which are considered in 1D analyses.

<figure><img src="image_37.png" /><figcaption>**Fig. 34** Incidence on uncertainty of velocity profiles inferred from surface wave experiment of 1D soil response computation. Left set of velocity profiles from inversion. One can see that for a given depth, the variability can be high. Right corresponding 1D soil response. The differences between acceleration response spectra at the surface are limited (Foti et al. 2009)</figcaption></figure>
spacing) for high frequency and shallow layers, then deploy several passive arrays, with successive apertures starting from relatively small (15–20 m) up to a value equal to the double of the desired investigation depth. A very large passive array could help in the determination of the bedrock velocity.

## 5.3 Ground response analysis

One of the current applications of surface wave methods is the determination of shear wave velocity profiles in order to compute 1D site response. We recommend that a set of acceptable velocity model is estimated.

When the geophysicist provides a set of possible solutions to the engineer responsible for site response simulation, a common error is to group all the profiles on a single plot and to consider three inferred profiles for the computations: the mean of all profiles; the mean plus and minus one standard deviation (all computed “depth by depth”). This will lead to an overestimation of the uncertainty. Indeed both outer profiles will not correspond to the measured surface wave dispersion curve, which is a robust parameter and represents a signature of the site.

When representative profiles are built for ground response analysis, they should always be tested by checking the corresponding theoretical dispersion curve and fundamental frequency against the available experimental ones. Similarly, when randomization approaches (e.g. Toro 1995) are adopted, it is necessary to select only “randomized” profiles that fit the measured values of dispersion curves and $f_0$ (Teague and Cox 2016).

The suggested approach to account for solution non-uniqueness is to repeat the ground response analyses for the individual profiles selected during the inversion process. This approach allows a direct evaluation of the final uncertainty associated to the seismic site response of the site. This uncertainty may be significantly smaller than the one associated to the shear wave velocity profile (e.g. Fig. 34). Indeed the profiles that match the experimental dispersion curve are likely equivalent also in terms of ground response to vertically propagating shear waves, which are considered in 1D analyses.

<figure><img src="image_37.png" /><figcaption>**Fig. 34** Incidence on uncertainty of velocity profiles inferred from surface wave experiment of 1D soil response computation. Left set of velocity profiles from inversion. One can see that for a given depth, the variability can be high. Right corresponding 1D soil response. The differences between acceleration response spectra at the surface are limited (Foti et al. 2009)</figcaption></figure>
A key parameter for ground response analyses is material damping, which is typically introduced on the basis of laboratory tests. However they may be not sufficiently representative of the response of the soil deposit. Although not yet fully included in the state of the practice, surface wave attenuation can be exploited to obtain damping parameters (Foti et al. 2014). Appendix 10 reports on possible options for this application.

**Acknowledgements** The Authors are grateful to Carlo G. Lai for reviewing the manuscript and for the precious suggestions on several issues that have been incorporated in the text. The Interpacific Project has been financed by the Research & Development Program SIGMA funded by EdF, Areva, CEA and ENEL and by CASHIMA project, funded by CEA, ILL and ITER. Partial funding for participants from the Politecnico di Torino was provided by the ReLUIS 3 project, sponsored by the Italian Civil Protection Agency. Partial funding for participants from the University of Texas was provided by U.S. National Science Foundation (NSF) Grant CMMI-1261775. However, any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the funding agencies.

**Open Access** This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

## References

Albarello D, Gargani G (2010) Providing NEHRP soil classification from the direct interpretation of effective Rayleigh-wave dispersion curves. Bull Seismol Soc Am 100(6):3284–3294

Bard P-Y, Cadet H, Endrun B, Hobiger M, Renalier F, Theodulidis N, Ohrnberger M, Fäh D, Sabetta F, Teves-Costa P, Duval A-M, Cornou C, Guillier B, Wathelet M, Savvaidis A, Köhler A, Burjanek J, Poggi V, Gassner-Stamm G, Havenith HB, Hailemikael S, Almeida J, Rodrigues I, Veludo I, Lacave C, Thomassin S, Kristekova M (2010) From non-invasive site characterization to site amplification: recent advances in the use of ambient vibration measurements. Geotech Geol Earthq Eng 17:105–123

Bolt BA (1987) Earthquakes. W. H Freeman, New York

Brown L, Diehl J, Nigbor R (2000) A simplified method to measure average shear-wave velocity in the top 30 m ($V_{S,30}$). In: Proceedings of the 6th international conference on seismic zonation, pp 1–6

Comina C, Foti S, Boiero D, Socco LV (2011) Reliability of $V_{S,30}$ evaluation from surface waves tests. J Geotech Geoenviron Eng 137(6):579–588. doi:10.1061/(ASCE)GT.1943-5606.0000452

Cox BR, Teague DP (2016) Layering ratios: a systematic approach to the inversion of surface wave data in the absence of A-priori information. Geophys J Int 207:422–438. doi:10.1093/gji/ggw282

Foti S, Comina C, Boiero D, Socco LV (2009) Non-uniqueness in surface wave inversion and consequences on seismic site response analyses. Soil Dyn Earthq Eng 29(6):982–993

Foti S, Parolai S, Albarello D, Picozzi M (2011) Application of surface wave methods for seismic site characterization. Surv Geophys 32(6):777–825

Foti S, Lai CG, Rix G, Strobbia C (2014) Surface wave methods for near-surface site characterization. CRC Press, London

Garofalo F, Foti S, Hollender F, Bard PY, Cornou C, Cox BR, Ohrnberger M, Sicilia D, Asten M, Di Giulio G, Forbriger T, Guillier B, Hayashi K, Martin A, Matsushima S, Mercerat D, Poggi V, Yamanaka H (2016a) InterPACIFIC project: comparison of invasive and non-invasive methods for seismic site characterization. Part I: Intra-comparison of surface wave methods. Soil Dyn Earthq Eng 82:222–240

Garofalo F, Foti S, Hollender F, Bard PY, Cornou C, Cox BR, Dechamp A, Ohrnberger M, Perron V, Sicilia D, Teague D, Vergniault C (2016b) InterPACIFIC project: comparison of invasive and non-invasive methods for seismic site characterization. Part II: Inter-comparison between surface-wave and borehole methods. Soil Dyn Earthq Eng 82:241–254

Lai CG, Foti S, Rix GJ (2005) Propagation of data uncertainty in surface wave inversion. J Environ Eng Geophys 10(2):219–228

Martin AJ, Diehl JG (2004) Practical experience using a simplified procedure to measure average shear-wave velocity to a depth of 30 meters ($V_{S,30}$). In: Proceedings of the 13th world conference on earthquake engineering, Vancouver, BC., Canada, August 1–6, 2004, paper no. 952

Mitchell JK, Soga K (2005) Fundamentals of soil behavior. Wiley, London
A key parameter for ground response analyses is material damping, which is typically introduced on the basis of laboratory tests. However they may be not sufficiently representative of the response of the soil deposit. Although not yet fully included in the state of the practice, surface wave attenuation can be exploited to obtain damping parameters (Foti et al. 2014). Appendix 10 reports on possible options for this application.

**Acknowledgements** The Authors are grateful to Carlo G. Lai for reviewing the manuscript and for the precious suggestions on several issues that have been incorporated in the text. The Interpacific Project has been financed by the Research & Development Program SIGMA funded by EdF, Areva, CEA and ENEL and by CASHIMA project, funded by CEA, ILL and ITER. Partial funding for participants from the Politecnico di Torino was provided by the ReLUIS 3 project, sponsored by the Italian Civil Protection Agency. Partial funding for participants from the University of Texas was provided by U.S. National Science Foundation (NSF) Grant CMMI-1261775. However, any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the funding agencies.

**Open Access** This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

## References

Albarello D, Gargani G (2010) Providing NEHRP soil classification from the direct interpretation of effective Rayleigh-wave dispersion curves. Bull Seismol Soc Am 100(6):3284–3294

Bard P-Y, Cadet H, Endrun B, Hobiger M, Renalier F, Theodulidis N, Ohrnberger M, Fäh D, Sabetta F, Teves-Costa P, Duval A-M, Cornou C, Guillier B, Wathelet M, Savvaidis A, Köhler A, Burjanek J, Poggi V, Gassner-Stamm G, Havenith HB, Hailemikael S, Almeida J, Rodrigues I, Veludo I, Lacave C, Thomassin S, Kristekova M (2010) From non-invasive site characterization to site amplification: recent advances in the use of ambient vibration measurements. Geotech Geol Earthq Eng 17:105–123

Bolt BA (1987) Earthquakes. W. H Freeman, New York

Brown L, Diehl J, Nigbor R (2000) A simplified method to measure average shear-wave velocity in the top 30 m ($V_{S,30}$). In: Proceedings of the 6th international conference on seismic zonation, pp 1–6

Comina C, Foti S, Boiero D, Socco LV (2011) Reliability of $V_{S,30}$ evaluation from surface waves tests. J Geotech Geoenviron Eng 137(6):579–588. doi:10.1061/(ASCE)GT.1943-5606.0000452

Cox BR, Teague DP (2016) Layering ratios: a systematic approach to the inversion of surface wave data in the absence of A-priori information. Geophys J Int 207:422–438. doi:10.1093/gji/ggw282

Foti S, Comina C, Boiero D, Socco LV (2009) Non-uniqueness in surface wave inversion and consequences on seismic site response analyses. Soil Dyn Earthq Eng 29(6):982–993

Foti S, Parolai S, Albarello D, Picozzi M (2011) Application of surface wave methods for seismic site characterization. Surv Geophys 32(6):777–825

Foti S, Lai CG, Rix G, Strobbia C (2014) Surface wave methods for near-surface site characterization. CRC Press, London

Garofalo F, Foti S, Hollender F, Bard PY, Cornou C, Cox BR, Ohrnberger M, Sicilia D, Asten M, Di Giulio G, Forbriger T, Guillier B, Hayashi K, Martin A, Matsushima S, Mercerat D, Poggi V, Yamanaka H (2016a) InterPACIFIC project: comparison of invasive and non-invasive methods for seismic site characterization. Part I: Intra-comparison of surface wave methods. Soil Dyn Earthq Eng 82:222–240

Garofalo F, Foti S, Hollender F, Bard PY, Cornou C, Cox BR, Dechamp A, Ohrnberger M, Perron V, Sicilia D, Teague D, Vergniault C (2016b) InterPACIFIC project: comparison of invasive and non-invasive methods for seismic site characterization. Part II: Inter-comparison between surface-wave and borehole methods. Soil Dyn Earthq Eng 82:241–254

Lai CG, Foti S, Rix GJ (2005) Propagation of data uncertainty in surface wave inversion. J Environ Eng Geophys 10(2):219–228

Martin AJ, Diehl JG (2004) Practical experience using a simplified procedure to measure average shear-wave velocity to a depth of 30 meters ($V_{S,30}$). In: Proceedings of the 13th world conference on earthquake engineering, Vancouver, BC., Canada, August 1–6, 2004, paper no. 952

Mitchell JK, Soga K (2005) Fundamentals of soil behavior. Wiley, London
Okada H (2003) The microtremor survey method. Geophysical monograph series. Society of Exploration Geophysics, Tulsa

Park CB, Miller RD, Xia J (1999) Multichannel analysis of surface waves. Geophysics 64:800–808

Peterson J (1993) Observations and modeling of seismic background noise. U.S. Geol. Surv. Open-File Rept. 93-322-95

Poggi V, Fäh D, Burjanek J, Giardini D (2012) The use of Rayleigh wave ellipticity for site-specific hazard assessment and microzonation. An application to the city of Luzern (Switzerland). Geophys J Int 188(3):1154–1172

Schramm KA, Abbott RE, Asten MW, Bilek S, Pancha AP, Patton HJ (2012) Broadband Rayleigh-wave dispersion curve and shear wave velocity structure for Yucca Flat, Nevada. Bull Seismol Soc Am 102(4):1361–1372. doi:10.1785/0120110296

SESAME team (2004) Guidelines for the implementation of the H/V spectral ratio technique on ambient vibrations: measurements, processing and interpretation (Deliverable No. D23.12), WP12. SESAME European research project

Socco L, Foti S, Boiero D (2010) Surface wave analysis for building near surface velocity models: established approaches and new perspectives. Geophysics 75(5):A83–A102

Teague DP, Cox BR (2016) Site response implications associated with using non-unique vs profiles from surface wave inversion in comparison with other commonly used methods of accounting for vs uncertainty. Soil Dyn Earthq Eng. doi:10.1016/j.soildyn.2016.07.028

Toro GR (1995) Probabilistic models of site velocity profiles for generic and site-specific ground-motion amplification studies. Technical report no 779574, Upton

Wielandt E (1993) Propagation and structural interpretation of non-plane waves. Geophys J Int 113(1):33–45. doi:10.1111/j.1365-246X.1993.tb02527.x

Yong A, Martin A, Stokoe K, Diehl J (2013) ARRA-funded $V_{S,30}$ measurements using multi-technique approach at strong-motion stations in California and Central-Eastern United States. Reston, Virginia, U.S. Geol. Surv. Open-File Rept. 2013-1102, 59

Zimmer MA, Prasad M, Mavko G, Nur A (2007) Seismic velocities of unconsolidated sands: part 1—pressure trends from 0.1 to 20 MPa. Geophysics 72(1):E1–E13. doi:10.1190/1.2399459
Okada H (2003) The microtremor survey method. Geophysical monograph series. Society of Exploration Geophysics, Tulsa

Park CB, Miller RD, Xia J (1999) Multichannel analysis of surface waves. Geophysics 64:800–808

Peterson J (1993) Observations and modeling of seismic background noise. U.S. Geol. Surv. Open-File Rept. 93-322-95

Poggi V, Fäh D, Burjanek J, Giardini D (2012) The use of Rayleigh wave ellipticity for site-specific hazard assessment and microzonation. An application to the city of Luzern (Switzerland). Geophys J Int 188(3):1154–1172

Schramm KA, Abbott RE, Asten MW, Bilek S, Pancha AP, Patton HJ (2012) Broadband Rayleigh-wave dispersion curve and shear wave velocity structure for Yucca Flat, Nevada. Bull Seismol Soc Am 102(4):1361–1372. doi:10.1785/0120110296

SESAME team (2004) Guidelines for the implementation of the H/V spectral ratio technique on ambient vibrations: measurements, processing and interpretation (Deliverable No. D23.12), WP12. SESAME European research project

Socco L, Foti S, Boiero D (2010) Surface wave analysis for building near surface velocity models: established approaches and new perspectives. Geophysics 75(5):A83–A102

Teague DP, Cox BR (2016) Site response implications associated with using non-unique vs profiles from surface wave inversion in comparison with other commonly used methods of accounting for vs uncertainty. Soil Dyn Earthq Eng. doi:10.1016/j.soildyn.2016.07.028

Toro GR (1995) Probabilistic models of site velocity profiles for generic and site-specific ground-motion amplification studies. Technical report no 779574, Upton

Wielandt E (1993) Propagation and structural interpretation of non-plane waves. Geophys J Int 113(1):33–45. doi:10.1111/j.1365-246X.1993.tb02527.x

Yong A, Martin A, Stokoe K, Diehl J (2013) ARRA-funded $V_{S,30}$ measurements using multi-technique approach at strong-motion stations in California and Central-Eastern United States. Reston, Virginia, U.S. Geol. Surv. Open-File Rept. 2013-1102, 59

Zimmer MA, Prasad M, Mavko G, Nur A (2007) Seismic velocities of unconsolidated sands: part 1—pressure trends from 0.1 to 20 MPa. Geophysics 72(1):E1–E13. doi:10.1190/1.2399459