2.1 H/V interpretations: the Diffuse Field Approach and the ellipticity of Rayleigh waves

Despite some authors have performed a joint inversion of the phase velocity and the H/V observed spectral ratios [16], we preferred to validate our Vs profile retrieved from the GAs generating synthetics' H/V ratios via application of the Diffuse Field Approach (DFA) and compare them with the observed ones. If we incorporate the observed H/V curve in a joint inversion, we would force a priori the soil profiles to fit with such curve, issue that the new interpretation of DFA would validate completely in a separate manner.

The soil profiles' results by the GAs' inversion of the previous section are validated via two alternative analyses: (i) the theoretical H/V ratios inferred from the Diffuse Field Approach (DFA) and the observed H/V ratios; (ii) the theoretical H/V ellipticity of Rayleigh waves.

Recently a new interpretation has been proposed and formulated by Sánchez-Sesma et al. [17, 18] and Perton et al. [19] based on a Diffuse Field Approach that the H/ V ratios on microtremors can be interpreted as the square root of the ratio of the sum of horizontal displacements for horizontal unit harmonic loads Im[G11] and Im[G22] and the imaginary part of vertical displacement for a vertically applied unit harmonic load, Im[G33], when both the source and the receiver are the same, as follows:

$$\frac{H(o)}{V(o)} = \sqrt{\frac{Im[\mathbf{G}\_{11}(\mathbf{x}, \mathbf{x}; o)] + Im[\mathbf{G}\_{22}(\mathbf{x}, \mathbf{x}; o)]}{Im[\mathbf{G}\_{33}(\mathbf{x}, \mathbf{x}; o)]}}\tag{8}$$

where ω denotes the circular frequency, x denotes the position vectors for source and receiver which are the same, and the indices (11, 22, and 33) denote the displacement and the direction of the unit applied load, respectively (e.g., 1, northsouth; 2, east-west; 3, up-down). Such calculations of the imaginary part of Green's function G in Eq. (8) are performed by the conventional discrete wavenumber summation method developed by Bouchon [20]. Then, the input data to compute H/V synthetics based on this method are the compressional and shear wave velocity, the density, the thickness, and the quality factor of each soil layer that can be retrieved in our case from GAs from the previous section. The details of the method can be found in Sánchez-Sesma et al. [18]. Equation (8) implies energy equipartition of the 3D wave field in space for a distribution of random sources. This interpretation has been revised by Kawase et al. [21] showing that the DFA approach explains well the observed H/V ratios of microtremors in Japan. Such new interpretation depends on the contribution of all waves considered in the Green's function, namely Rayleigh, Love, and body waves.

Konno and Ohmachi [3] and Bonnefoy-Claudet et al. [22] have demonstrated that the H/V curves exhibit in most cases a single peak due to the ellipticity of the fundamental mode of Rayleigh waves through 1D noise simulation; the vanishing of the vertical component occurs nearly to the fundamental resonance period of S waves where a sharp S-wave impedance contrast exists larger than 3.0 between the surface layers and the underlying stiffer formations and when the sources are near and surficial.

Estimation of Shear Wave Velocity Profiles Employing Genetic Algorithms and the Diffuse Field… DOI: http://dx.doi.org/10.5772/intechopen.85129

We calculated the observed horizontal-to-vertical spectral ratio (H/V) employing the resultant vector of the orthogonal north-south and east-west components of motion and averaging the results for all the stationary parts selected for each record (details of the digital processing of single mobile microtremors are explained [1]). To compute the synthetics' H/V ratios employing the DFA in Eq. (8), we adopted for the surface sediments above the bedrock a low-quality factor of 5.0 for all frequencies to incorporate the effects of total water saturation (since water table in POS can be found just at the surface) yielding high attenuation on wave propagation [23, 24] and a quality factor of 50 for the bedrock [25].

We present the imaginary parts of Green's functions Im[G11] and Im[G33] in Figure 9a and the H/V synthetics (see Eq. (8)) based on the DFA in Figure 9b at Queen's Park Savannah. A good agreement is found among the amplification calculations cited before for both, the fundamental period of vibration and the shape of the overall observed H/V ratios. Despite the fundamental period of 0.57 s can be explained by the ellipticity pattern depicted in Figure 9c, it is noted that the DFA

#### Figure 9.

consolidated sediments with VS of about 700 m/s constitute the thicker layers with

2.1 H/V interpretations: the Diffuse Field Approach and the ellipticity of

the H/V observed spectral ratios [16], we preferred to validate our Vs profile retrieved from the GAs generating synthetics' H/V ratios via application of the Diffuse Field Approach (DFA) and compare them with the observed ones. If we incorporate the observed H/V curve in a joint inversion, we would force a priori the soil profiles to fit with such curve, issue that the new interpretation of DFA would

Despite some authors have performed a joint inversion of the phase velocity and

The soil profiles' results by the GAs' inversion of the previous section are validated via two alternative analyses: (i) the theoretical H/V ratios inferred from the Diffuse Field Approach (DFA) and the observed H/V ratios; (ii) the theoretical H/V

Recently a new interpretation has been proposed and formulated by Sánchez-Sesma et al. [17, 18] and Perton et al. [19] based on a Diffuse Field Approach that the H/ V ratios on microtremors can be interpreted as the square root of the ratio of the sum of horizontal displacements for horizontal unit harmonic loads Im[G11] and Im[G22] and the imaginary part of vertical displacement for a vertically applied unit harmonic load,

where ω denotes the circular frequency, x denotes the position vectors for source and receiver which are the same, and the indices (11, 22, and 33) denote the displacement and the direction of the unit applied load, respectively (e.g., 1, northsouth; 2, east-west; 3, up-down). Such calculations of the imaginary part of Green's function G in Eq. (8) are performed by the conventional discrete wavenumber summation method developed by Bouchon [20]. Then, the input data to compute H/V synthetics based on this method are the compressional and shear wave velocity, the density, the thickness, and the quality factor of each soil layer that can be retrieved in our case from GAs from the previous section. The details of the method

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Im G½ �þ <sup>11</sup>ð Þ x; x;ω Im G½ � <sup>22</sup>ð Þ x; x;ω Im G½ � <sup>33</sup>ð Þ x; x;ω

(8)

Im[G33], when both the source and the receiver are the same, as follows:

can be found in Sánchez-Sesma et al. [18]. Equation (8) implies energy

function, namely Rayleigh, Love, and body waves.

and surficial.

66

equipartition of the 3D wave field in space for a distribution of random sources. This interpretation has been revised by Kawase et al. [21] showing that the DFA approach explains well the observed H/V ratios of microtremors in Japan. Such new interpretation depends on the contribution of all waves considered in the Green's

Konno and Ohmachi [3] and Bonnefoy-Claudet et al. [22] have demonstrated that the H/V curves exhibit in most cases a single peak due to the ellipticity of the fundamental mode of Rayleigh waves through 1D noise simulation; the vanishing of the vertical component occurs nearly to the fundamental resonance period of S waves where a sharp S-wave impedance contrast exists larger than 3.0 between the surface layers and the underlying stiffer formations and when the sources are near

more than 100 m above the bedrock.

Natural Hazards - Risk, Exposure, Response, and Resilience

validate completely in a separate manner.

Hð Þ ω <sup>V</sup>ð Þ <sup>ω</sup> <sup>¼</sup> s

2. Results and discussion

ellipticity of Rayleigh waves.

Rayleigh waves

(a) Imaginary part of Green's function (Im G11 and Im G33 in Eq. (8)) via application of the Diffuse Field Approach (DFA) for Queen's Park Savanna (Point 1X in Figure 1); (b) H/V observed spectral ratio (mean) and H/V synthetics spectral ratios via application of DFA; (c) ellipticity of Rayleigh waves for the first mode of vibration—note that absolute values of ellipticity are drawn; and (d) absolute Fourier velocity spectrum for horizontal (N-S and E-W) and vertical components of motion. Diagram of ellipticity pattern taken from Konno and Ohmachi [3]. The fundamental period of soil T is indicated by the arrow in the H/V spectral ratios.

yields a more robust interpretation since the amplification factor cannot be measured employing the ellipticity approach.

Also, it is interesting that the change in the ellipticity pattern depicted in Figure 9c clearly reflects the change of particle motion from prograde to retrograde at the fundamental period of vibration observed for the theoretical and experimental calculations. The trough in the vertical component confirms the analysis causing the peak observed in the H/V ratios (Figure 9d).

Take the theoretical fundamental period T in seconds of a homogenous soil profile over a rigid base equal to:

$$T(\mathfrak{s}) = \frac{4H}{\overline{\mathfrak{Vs}}} \tag{9}$$

where H is the thickness of the sediments above the bedrock and Vs is the average shear wave velocity. Introducing the values of H and Vs in Eq. (9) as 75 m and 489 m/s resulting from the GAs' inversion (Section 1.4), the period T yields 0.60s coinciding fairly well with the one obtained by the observed H/V spectral ratio technique and the one predicted by the diffuse wave field theory and the ellipticity pattern of Rayleigh waves.

The analysis for the remaining eight microtremors array sites is presented in Figure 10. The fundamental periods are well explained for all sites due to ellipticity pattern in the wave motion of microtremors; the DFA confirmed the effectiveness of the application of the H/V spectral technique and the GAs for the city of POS. The deeper profiles are found in the coastal areas (5X) at the Port with a total of about 225 m of sediments and a fundamental period of 1.4 s, this in accordance with water well information at the Port presented in Table 5 that no bedrock is identified at 100 m depth. Sites in the foot of hills yield the shallower profiles of 25–30 m with fundamental periods less than 0.3 s. Intermediate periods between 0.4 and 1.0 s are found in downtown areas yielding depths between 60 and 110 m. For all array sites, Vs varies from 50 to 2000 m/s, including the bedrock.

An interesting feature of the H/V ratios can be seen for the three sites located in the coastal areas, namely, the Licensing Authority (Port Area), Mucurapo Secondary School, and Sea Lots (Figure 10a–c, sites 3X, 5X, and 6X, respectively, Figure 1). Short period components between 0.1 and 0.3 s yield a very low amplification or a de-amplification at the three sites. We attribute such phenomena due to the presence of a thin rigid layer in the surface with Vs of about 600 m/s; such feature was introduced in the search limits for the top layers in the GAs at these


sites. We have evidence of existing stiff layers near the surface as it is corroborated by the well logs reported by WASA near the array sites (see Hard Sand deposit in Table 5). We attribute the high Vs on the top due to compaction works, deck

Left: H/V spectral ratio (observed-mean and synthetic via application of Diffuse Field Approach (DFA)). Center: ellipticity of Rayleigh waves for the first mode of vibration; note that absolute (ABS) values of ellipticity are drawn. Right: absolute Fourier velocity spectrum for horizontal (N-S and E-W) and vertical components of motion. The sites are ordered from top to bottom from the largest to the shortest fundamental period of soil T indicated by the arrows in the H/V spectral ratios. (a) Port Area (5X), (b) Mucurapo Secondary School (3X), (c) Sea Lots (8X), (d) Nelson Mandela Park (2X), (e) Woodford Square (8X), (f) Federation Park (4X), (g) St. James hospital (9X), and (h) St. Benedict's Children's home (7X). See locations of microtremors array in

Estimation of Shear Wave Velocity Profiles Employing Genetic Algorithms and the Diffuse Field…

DOI: http://dx.doi.org/10.5772/intechopen.85129

Figure 10.

Figure 1.

69

Table 5. Water well for the Port Area (License Office) site 5X (Figure 1). Estimation of Shear Wave Velocity Profiles Employing Genetic Algorithms and the Diffuse Field… DOI: http://dx.doi.org/10.5772/intechopen.85129

#### Figure 10.

yields a more robust interpretation since the amplification factor cannot be mea-

Also, it is interesting that the change in the ellipticity pattern depicted in Figure 9c clearly reflects the change of particle motion from prograde to retrograde at the fundamental period of vibration observed for the theoretical and experimental calculations. The trough in the vertical component confirms the analysis causing

Take the theoretical fundamental period T in seconds of a homogenous soil

T sðÞ¼ <sup>4</sup><sup>H</sup>

where H is the thickness of the sediments above the bedrock and Vs is the average shear wave velocity. Introducing the values of H and Vs in Eq. (9) as 75 m and 489 m/s resulting from the GAs' inversion (Section 1.4), the period T yields 0.60s coinciding fairly well with the one obtained by the observed H/V spectral ratio technique and the one predicted by the diffuse wave field theory and the

The analysis for the remaining eight microtremors array sites is presented in Figure 10. The fundamental periods are well explained for all sites due to ellipticity pattern in the wave motion of microtremors; the DFA confirmed the effectiveness of the application of the H/V spectral technique and the GAs for the city of POS. The deeper profiles are found in the coastal areas (5X) at the Port with a total of about 225 m of sediments and a fundamental period of 1.4 s, this in accordance with water well information at the Port presented in Table 5 that no bedrock is identified at 100 m depth. Sites in the foot of hills yield the shallower profiles of 25–30 m with fundamental periods less than 0.3 s. Intermediate periods between 0.4 and 1.0 s are found in downtown areas yielding depths between 60 and 110 m. For all array sites,

An interesting feature of the H/V ratios can be seen for the three sites located

in the coastal areas, namely, the Licensing Authority (Port Area), Mucurapo Secondary School, and Sea Lots (Figure 10a–c, sites 3X, 5X, and 6X, respectively, Figure 1). Short period components between 0.1 and 0.3 s yield a very low amplification or a de-amplification at the three sites. We attribute such phenomena due to the presence of a thin rigid layer in the surface with Vs of about 600 m/s; such feature was introduced in the search limits for the top layers in the GAs at these

Thickness (feet/m) Description –7/0–2 Clay fill –20/2–6 Sand + gravel –25/6–8 Hard sand –110/8–34 Sand + boulders –115/34–35 Brown clay –200/35–61 Clay and boulders

200–251/61–77 Gravel with streaks of clay

251–338/77–103 Sand + boulders

Water well for the Port Area (License Office) site 5X (Figure 1).

Vs (9)

No bedrock is identified

sured employing the ellipticity approach.

profile over a rigid base equal to:

ellipticity pattern of Rayleigh waves.

Table 5.

68

Vs varies from 50 to 2000 m/s, including the bedrock.

the peak observed in the H/V ratios (Figure 9d).

Natural Hazards - Risk, Exposure, Response, and Resilience

Left: H/V spectral ratio (observed-mean and synthetic via application of Diffuse Field Approach (DFA)). Center: ellipticity of Rayleigh waves for the first mode of vibration; note that absolute (ABS) values of ellipticity are drawn. Right: absolute Fourier velocity spectrum for horizontal (N-S and E-W) and vertical components of motion. The sites are ordered from top to bottom from the largest to the shortest fundamental period of soil T indicated by the arrows in the H/V spectral ratios. (a) Port Area (5X), (b) Mucurapo Secondary School (3X), (c) Sea Lots (8X), (d) Nelson Mandela Park (2X), (e) Woodford Square (8X), (f) Federation Park (4X), (g) St. James hospital (9X), and (h) St. Benedict's Children's home (7X). See locations of microtremors array in Figure 1.

sites. We have evidence of existing stiff layers near the surface as it is corroborated by the well logs reported by WASA near the array sites (see Hard Sand deposit in Table 5). We attribute the high Vs on the top due to compaction works, deck

constructions at the Port/Coastal Area, and/or a high degree of consolidation due to the constant presence of heavy weight (containers) that are located at these sites for shipping purposes. Note that the DFA predicted very well the H/V ratios in such circumstances as well, for both, the fundamental period and the overall shape of the transfer function. It is noted that this consolidated layer at the top of the Port Area behaves as a low pass filter and does not have an influence in the fundamental period of motion of the whole soil system; such feature was corroborated performing the DFA without the stiff top layer at Mucurapo Secondary School (see Figure 10b). Sea Lots site at the South East of POS (see Figure 1 at site 6X) is characterized by the lowest VS of 50 m/s for all array sites that correspond to a swamp area overlaid by stiff deposits. Such low values of VS have been observed in sedimentary stratigraphy of natural intertidal flats [26].

factor of 5.0 for all frequencies to incorporate the effects of total water saturation. We plotted two kinds of theoretical SH transfer functions, namely, case (1) up + down amplification with refraction and reflection in bedrock and case (2) only up wave amplification. In both cases the 1-D SH wave amplification replicates the fundamental period of the observed H/V ratios; however, in most of the cases, the overall shape of the H/V ratios differs mainly at long period components for case (1) and for short period components for case (2). It is noticed that a level of amplification yield between three and five yields at the predominant peak. This level of

Estimation of Shear Wave Velocity Profiles Employing Genetic Algorithms and the Diffuse Field…

An important parameter in the modification of seismic waves propagating toward the surface is the composition of the near-surface soil layers. In different building codes around the world, the average shear wave velocity of the upper 30 m (VS30) has been adopted to characterize the response of seismic waves to the

In first instance, we compared the VS30 obtained from our microtremors array observation and the ones estimated by the empirical formulas of Matsuoka et al.

We calculated the VS30 from our microtremor results using the following

VS<sup>30</sup> <sup>¼</sup> <sup>30</sup> ∑<sup>N</sup> i¼1 hi Vi

where hi and Vi denote the thickness (in meters) and the shear wave velocity of

We classified the sites (1X–5X, 8X) as a Gravelly Terrace, Sea Lots (site 6X) as a Reclaimed Land, and St. Dominic's Children's Home to the East (7X) and St. James

empirical formulas to estimate VS30 (m/s) for the Gravelly Terrace (Eq. (11)), the

logVS30 ¼ 2:493 þ 0:072logEv þ 0:027logSp � 0:164logDm � 0:122ð Þ σ (11)

where Ev is the elevation (m), Sp refers to the Tangent of Slope\*1000, Dm yields the distance (km) from mountain or hill, and σ denotes the standard deviation. We took Dm as the shortest distance to the Northern Range or the Laventille Metalimestone foothills (Figure 1). The results are presented in Figure 12. In general the estimated VS30 from the empirical formulas of Matsuoka et al. [27] estimates well the velocities obtained by the GA's from our array measurements in

microtremors array profiles with the ones estimated by Allen and Wald [28] using the topographic slope as a proxy of site conditions employing the USGS Web Server (earthquake.usgs.gov/hazards/apps/vs30/). We retrieved the correspondent predicted VS30 at the location of each microtremors array. The most noticeable difference is observed for the mountain foot slope in St. James (site 9X). However, we did not find a good correlation when comparing with soil types proposed by

the range of �σ (standard deviation). We also compared the VS30 of our

logVS30 ¼ 2:373 � 0:124logDm � 0:123ð Þ σ (12)

logVS30 ¼ 2:602 � 0:092ð Þ σ (13)

Hospital to the West (9X) as Mountain Foot Slope sites (see Figure 1). The

Reclaimed Land (Eq. (12)), and the Mountain Foot Slope (Eq. (13)) yield:

(10)

[27] employing 2000 sites in Japan based on geomorphological units.

the ith layer; N is the total number of soil layers respectively.

amplification is referred to the bedrock motion.

3. VS30 and fundamental period

DOI: http://dx.doi.org/10.5772/intechopen.85129

influence of near-surface strata.

formula:

71

#### 2.2 H/V ratios and 1-D theoretical transfer function for SH-waves

Figure 11 depicts the comparison between the 1-D SH wave amplification employing the Vs profiles obtained by the GAs and the H/V observed spectral ratios. We also adopted for the surface sediments above the bedrock a low-quality

#### Figure 11.

Comparison of H/V spectral ratios and 1-D SH wave transfer function; case (1) only up wave amplification and case (2) up + down amplification with refraction and reflection in bedrock. (a) Queen's Park Savannah array (1X), (b) Port Area (5X), (c) Mucurapo Secondary School (3X), (d) Sea Lots (8X), (e) Nelson Mandela Park (2X), (f) Woodford Square (8X), (g) Federation Park (4X), (h) St. James hospital (9X), and (i) St. Benedict's Children's home (7X). The fundamental period of soil T is indicated by the arrows.

Estimation of Shear Wave Velocity Profiles Employing Genetic Algorithms and the Diffuse Field… DOI: http://dx.doi.org/10.5772/intechopen.85129

factor of 5.0 for all frequencies to incorporate the effects of total water saturation. We plotted two kinds of theoretical SH transfer functions, namely, case (1) up + down amplification with refraction and reflection in bedrock and case (2) only up wave amplification. In both cases the 1-D SH wave amplification replicates the fundamental period of the observed H/V ratios; however, in most of the cases, the overall shape of the H/V ratios differs mainly at long period components for case (1) and for short period components for case (2). It is noticed that a level of amplification yield between three and five yields at the predominant peak. This level of amplification is referred to the bedrock motion.
