1.1 Array measurement of microtremors

Port of Spain (POS) lies on an alluvial fan deposit and forms a costal aquifer with a high water table comprising poorly sorted gravels, sand, clay, and boulders. The part of today's downtown Port of Spain closest to the sea was once an area of tidal mudflats covered by mangroves which have been reclaimed by anthropological

means. Recent studies suggested a peak ground acceleration of about 0.6 g on rock sites for a 2475 year return period in POS. Salazar et al. [1] presented a high-resolution grid of H/V spectral ratios employing 1181 mobile microtremors at Port of Spain in order to retrieve the S-wave fundamental periods of vibration of the soil and proposed a microzonation map for the city and the correspondent seismic coefficients for building design. So the main objective of this article is to validate such periods through an alternative geophysical method, namely, the microtremors array, and use them to develop a liquefaction hazard map for the city based on the same H/V spectral ratios of microtremors.

Nine microtremors array were done at nine sites in POS located on recent alluvium and reclaimed land (Figure 1). The objective of the microtremors array is to obtain the shear wave velocity (VS) profile at the site by locating seven sensors that measure vertical ambient vibrations in a circular configuration (see Figure 2 and Table 1). Then the main idea of deploying an array is to compare the motion of sensor (1) located in the center of the circle with the motion on sensors (2–4) separated by a distance equal to the radii "r" of the circle located in the vertexes of the triangle. Through the comparison of the vertical motion that comprises Rayleigh waves, it is possible to work backward retrieving the VS and thickness of the soil and bedrock layers through genetic inversions. So, with the VS profiles, it is possible to obtain the soil amplification factors to be incorporated in seismic codes for building design.

To perform the microtremors array, we use a Tokyo Sokushin 9-channel SAMTAC 802-H and the sensors VSE-15D6 with a flat response between 0.1 and 100 Hz and a 24-bit recording system of Δt = 0.01 s equivalent to 100 samples per second; the sensors measured micromotion in terms of velocity (Figure 3). We recorded a total of 25 min for each array in a silent environment.

Since we had a recorder with nine channels, we locate in location number 1 three channels corresponding to two horizontal and one vertical component to compute

the H/V spectral ratios [4], while in the remaining location numbers (2–7), we

Top: general microtremors array configuration in a plan view. The sensors 1, 2, 3, and 4 represent the array with the largest radii r, while the sensors 1, 5, 6, and 7 represent the array with the smallest radii r/2. Since we had a recorder of nine channels, we set in location number 1 three channels corresponding to two horizontal and one vertical component, while the remaining numbers (2–7) we locate just one vertical sensor. Bottom:

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

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

The comparison of the recorded vertical motion is made in the frequency domain via application of the spatial autocorrelation method (SPAC) [5, 6]. The first step in the SPAC method is to compute the cross spectra Sij through Fourier transformation of the signal, and then the autocorrelation function R1j of the sensor

ð1Þ

j = (2, 3, 4) in the vertex of the triangle with central site 1 yields as follows:

1.2 Spectral autocorrelation (SPAC) and dispersion curve

photo of an array at Mucurapo Secondary School (site 3X in Figure 1).

locate just one vertical sensor.

Figure 2.

55

where f denotes frequency domain.

#### Figure 1.

Geological map of Port of Spain. The locations of the nine microtremors array are depicted by a red solid triangle and a number with an X (e.g., 1X–9X, see Table 1); the corresponding thickness of sediments above the bedrock retrieved from the genetic inversions is depicted in cursive numbers. Water well data near the arrays are depicted with a solid brown circles (see thickness and soil layers classification in Figure 6 and Table 5); open triangles denote boreholes reaching the bedrock; see elevation model of section A-A in Figure 6. Arrows in the clockwise direction around the Queen's Park Savannah indicate the flow of constant traffic in the roundabout of 500 m radii.

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

#### Figure 2.

means. Recent studies suggested a peak ground acceleration of about 0.6 g on rock sites for a 2475 year return period in POS. Salazar et al. [1] presented a high-resolution grid of H/V spectral ratios employing 1181 mobile microtremors at Port of Spain in order to retrieve the S-wave fundamental periods of vibration of the soil and proposed a microzonation map for the city and the correspondent seismic coefficients for building design. So the main objective of this article is to validate such periods through an alternative geophysical method, namely, the microtremors array, and use them to develop a liquefaction hazard map for the city based on the

Nine microtremors array were done at nine sites in POS located on recent alluvium and reclaimed land (Figure 1). The objective of the microtremors array is to obtain the shear wave velocity (VS) profile at the site by locating seven sensors that measure vertical ambient vibrations in a circular configuration (see Figure 2 and Table 1). Then the main idea of deploying an array is to compare the motion of sensor (1) located in the center of the circle with the motion on sensors (2–4) separated by a distance equal to the radii "r" of the circle located in the vertexes of the triangle. Through the comparison of the vertical motion that comprises Rayleigh waves, it is possible to work backward retrieving the VS and thickness of the soil and bedrock layers through genetic inversions. So, with the VS profiles, it is possible to obtain the soil amplification factors to be incorporated in seismic codes for

To perform the microtremors array, we use a Tokyo Sokushin 9-channel SAMTAC 802-H and the sensors VSE-15D6 with a flat response between 0.1 and 100 Hz and a 24-bit recording system of Δt = 0.01 s equivalent to 100 samples per second; the sensors measured micromotion in terms of velocity (Figure 3). We

Since we had a recorder with nine channels, we locate in location number 1 three channels corresponding to two horizontal and one vertical component to compute

Geological map of Port of Spain. The locations of the nine microtremors array are depicted by a red solid triangle and a number with an X (e.g., 1X–9X, see Table 1); the corresponding thickness of sediments above the bedrock retrieved from the genetic inversions is depicted in cursive numbers. Water well data near the arrays are depicted with a solid brown circles (see thickness and soil layers classification in Figure 6 and Table 5); open triangles denote boreholes reaching the bedrock; see elevation model of section A-A in Figure 6. Arrows in the clockwise direction around the Queen's Park Savannah indicate the flow of constant traffic in the roundabout of 500 m radii.

recorded a total of 25 min for each array in a silent environment.

same H/V spectral ratios of microtremors.

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building design.

Figure 1.

54

Top: general microtremors array configuration in a plan view. The sensors 1, 2, 3, and 4 represent the array with the largest radii r, while the sensors 1, 5, 6, and 7 represent the array with the smallest radii r/2. Since we had a recorder of nine channels, we set in location number 1 three channels corresponding to two horizontal and one vertical component, while the remaining numbers (2–7) we locate just one vertical sensor. Bottom: photo of an array at Mucurapo Secondary School (site 3X in Figure 1).

the H/V spectral ratios [4], while in the remaining location numbers (2–7), we locate just one vertical sensor.

#### 1.2 Spectral autocorrelation (SPAC) and dispersion curve

The comparison of the recorded vertical motion is made in the frequency domain via application of the spatial autocorrelation method (SPAC) [5, 6]. The first step in the SPAC method is to compute the cross spectra Sij through Fourier transformation of the signal, and then the autocorrelation function R1j of the sensor j = (2, 3, 4) in the vertex of the triangle with central site 1 yields as follows:

$$R\_{\iota\iota}(\mathcal{J}) = \frac{S\_{\iota\iota}(\mathcal{J})}{[S\_{\iota\iota}(\mathcal{J})S\_{\iota\iota}(\mathcal{J})]^{\text{bs}}} \tag{1}$$

where f denotes frequency domain.


See general plan view and photo of an array in Figure 2. Maximum wavelength is calculated from dispersion curves on Figures 5 and 7 as λ = co/f, where co is the phase velocity and f denotes frequency in Hz.

#### Table 1.

Selected sites for the microtremors array in Port of Spain (see location in Figure 1).

Then the direction average of the autocorrelation function ρ (SPAC) for the three sensors separated by a radii r gives:

$$\rho(f;r) = \frac{1}{3} \sum\_{\substack{j=2 \\ j \neq 2}}^{4} \mathcal{R}\_{lj}(f;r) \tag{2}$$

maximum radii between 20 and 40 m depending on the available space at the site of interest, and we repeat the procedure for a small array that corresponds to r/4 and r/8. We present the results of the SPAC at the Queen's Park Savannah array (site 1X

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

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

To obtain the observed Rayleigh wave velocity co(f), a 0-order Bessel function of

2πfr coð Þf

(3)

(4)

in Figure 1), for the radii of 5, 10, 20, and 40 m (Figure 4); we calculated the average of the SPAC for 81.92 s of stationary parts of the signal. The SPAC with values of about +1.0 means that the wave motion at short frequencies is very similar regardless of the aperture of the array, and the SPAC decreases as the frequency increases; the negative value in the SPAC represents change of polarity in the wave

Example of velocity history (cm/s) for the vertical component of the microtremors array in Mucurapo Secondary School. See the number of channel in the left upper part of each record in accordance to the array configuration in Figure 2. Channel 1 corresponds to the sensor located in the center; sensors 2, 3, and 4 correspond to the radii of 40 m and sensors 5, 6, and 7 to the radii of 20 m in the vertices of the triangles.

ρð Þ¼ f;r Jo xð Þ¼ Jo

Employing the argument x of the Bessel function, the phase velocity yields:

coð Þ¼ <sup>f</sup> <sup>2</sup>πfr x

Figure 5 shows the resulting dispersion curve (phase velocity) for the Queen's Park Savannah array through the SPAC employing a 0-order Bessel function of first kind. To get a single dispersion curve, we averaged the four common parts of each phase velocity from the different array sizes and added the single reliable parts of the dispersion curves corresponding to the maximum and minimum array sizes. Arrays with a bigger aperture are able to retrieve the velocities for low frequencies (long period) of motion and subsequently retrieve a deeper soil structure; arrays with smaller aperture are able to retrieve the velocities for high frequencies (shorter

motion for longer frequencies (shorter periods).

first kind Jo(x) is used as follows:

Figure 3.

57

The vertical motion is also compared with sensor (1) and sensors (5–7) which corresponds to a radius equal to "r/2" (Figure 2). We used several aperture

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

#### Figure 3.

Example of velocity history (cm/s) for the vertical component of the microtremors array in Mucurapo Secondary School. See the number of channel in the left upper part of each record in accordance to the array configuration in Figure 2. Channel 1 corresponds to the sensor located in the center; sensors 2, 3, and 4 correspond to the radii of 40 m and sensors 5, 6, and 7 to the radii of 20 m in the vertices of the triangles.

maximum radii between 20 and 40 m depending on the available space at the site of interest, and we repeat the procedure for a small array that corresponds to r/4 and r/8.

We present the results of the SPAC at the Queen's Park Savannah array (site 1X in Figure 1), for the radii of 5, 10, 20, and 40 m (Figure 4); we calculated the average of the SPAC for 81.92 s of stationary parts of the signal. The SPAC with values of about +1.0 means that the wave motion at short frequencies is very similar regardless of the aperture of the array, and the SPAC decreases as the frequency increases; the negative value in the SPAC represents change of polarity in the wave motion for longer frequencies (shorter periods).

To obtain the observed Rayleigh wave velocity co(f), a 0-order Bessel function of first kind Jo(x) is used as follows:

$$\rho(f, r) = f o(\infty) = J\_o \left(\frac{2\pi f r}{c\_o(f)}\right) \tag{3}$$

Employing the argument x of the Bessel function, the phase velocity yields:

$$
\omega\_o(f) = \frac{2\pi f r}{\varkappa} \tag{4}
$$

Figure 5 shows the resulting dispersion curve (phase velocity) for the Queen's Park Savannah array through the SPAC employing a 0-order Bessel function of first kind. To get a single dispersion curve, we averaged the four common parts of each phase velocity from the different array sizes and added the single reliable parts of the dispersion curves corresponding to the maximum and minimum array sizes. Arrays with a bigger aperture are able to retrieve the velocities for low frequencies (long period) of motion and subsequently retrieve a deeper soil structure; arrays with smaller aperture are able to retrieve the velocities for high frequencies (shorter

ð2Þ

Then the direction average of the autocorrelation function ρ (SPAC) for the

See general plan view and photo of an array in Figure 2. Maximum wavelength is calculated from dispersion curves

The vertical motion is also compared with sensor (1) and sensors (5–7) which

corresponds to a radius equal to "r/2" (Figure 2). We used several aperture

three sensors separated by a radii r gives:

Table 1.

56

Array number Location Size of the array

1X Queen's Park Savannah 40 m

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2X Nelson Mandela Park 40 m

School

4X Federation Park 20 m

Authority)

6X Sea Lots 40 m

Home

8X Woodford Square 40 m

9X St. James Hospital 20 m

3X Mucurapo Secondary

5X Port Area (Licensing

7X St. Dominic's Children's

Large circle

(first) 10 m (second)

(first) 10 m (second)

40 m (first) 10 m (second)

(first) 5 m (second)

40 m (first) 10 m (second)

(first) 10 m (second)

20 m (first) 5 m (second)

(first) 10 m (second)

(first) 5 m (second)

on Figures 5 and 7 as λ = co/f, where co is the phase velocity and f denotes frequency in Hz.

Selected sites for the microtremors array in Port of Spain (see location in Figure 1).

(radii r)

Small circle

20 m (first) 5 m (second)

20 m (first) 5 m (second)

20 m (first) 5 m (second)

10 m (first) 2.5 m (second)

20 m (first) 5 m (second)

20 m (first) 5 m (second)

10 m (first) 2.5 m (second)

20 m (first) 5 m (second)

10 m (first) 2.5 m (second)

Maximum wavelength λ (m)

393 9.8

379 9.5

447 11.2

280 14

570 14.3

213 5.3

77 3.9

248 6.2

117 5.9

λ/max array size

Figure 4. Spatial autocorrelation coefficient (SPAC) for the Queen's Park Savannah array of microtremors (point 1X in Figure 1).

period) with a better resolution for soil structures near the surface. Note that each phase velocity in the arrays has unreliable parts for very low- and high-frequency components of motion due to the aperture radii used in each case; in other words, an array has a limited frequency band of usefulness between fmin and fmax that is dependent on its aperture. Rayleigh waves are dispersive and their velocities decrease with frequency; the reliable parts of each phase velocity must follow such trend eliminating in the average calculation the increase of velocity at low and high frequencies of motion. We noticed that the maximum wavelength at which the phase velocity can be estimated is about 10 times the radii r of the arrays at Queen's Park Savannah (Figure 5); the minimum wavelength is about 2 times the radii r of the arrays [7]. To obtain the average velocity at each frequency of motion f, we used N frequencies equally separated by the value of Δ<sup>f</sup> in terms of a logarithm scale as follows:

$$\log \Delta\_f = \frac{\log f\_{\max} - \log f\_{\min}}{N - 1} \tag{5}$$

with n bits in series of 0 and 1 defining a priori lower and upper bound limits for the shear wave velocity and thickness of the layer (e.g., 200–600 m/s and 10–100 m, respectively). Each bit represents a gene, and a series of bits concatenated represents a chromosome. So, an optimal solution is searched using the chromosome that best matches the soil model represented by the experimental phase velocity curve developed using the microtremors array after applying the SPAC method. In this work we employed the method of Yamanaka and Ishida [8]. The reproduction of the initial population to a new population relies on the fitness function of each individual applying the three genetic operations modulated by the selection, crossover, and mutation; crossover acts to generate a good, new model with the combination of good parts of chromosomes of two parents; in the mutation operation, a gene is reversed (e.g., from 1 to 0 or vice versa). The mutation procedure is necessary to escape trapping at local minimum

Dispersion curve of Rayleigh wave (phase velocity) for Queen's Park Savannah array (point 1X in Figure 1). We select the average velocity (open circles) of the four radii r to be used in the genetic inversion. The thin solid lines depict the wavelength that corresponds to 10–30 times and 2 times the radii of the arrays. The red thick solid line depicts the theoretical phase velocity curve that corresponds to best individual (soil profile in Figure 6) after searching the optimum solution via genetic algorithms; fmax and fmin are the maximum and

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

The selection process begins declaring a misfit function ϕ<sup>k</sup> for a k individual is

coð Þ� f ccð Þf σcð Þf <sup>2</sup>

(6)

<sup>ϕ</sup><sup>k</sup> <sup>¼</sup> <sup>1</sup> <sup>N</sup> <sup>∑</sup> N i¼1

solutions.

59

Figure 5.

minimum reliable frequencies for the aperture arrays.

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

defined as follows:

where fmax and fmin are the maximum and minimum reliable frequencies for the aperture arrays, respectively (Figure 5). In our case, we generally set the value of N = 20 and select the corresponding velocity which belongs to the nearest frequency in such interval.

#### 1.3 Inversion of phase velocities through genetic algorithms (GAs)

Genetic algorithms (GAs) are mathematical simulations based on biological evolution of natural selection rules. The soil parameters are digitized to gene type Estimation of Shear Wave Velocity Profiles Employing Genetic Algorithms and the Diffuse Field… DOI: http://dx.doi.org/10.5772/intechopen.85129

#### Figure 5.

period) with a better resolution for soil structures near the surface. Note that each phase velocity in the arrays has unreliable parts for very low- and high-frequency components of motion due to the aperture radii used in each case; in other words, an array has a limited frequency band of usefulness between fmin and fmax that is dependent on its aperture. Rayleigh waves are dispersive and their velocities decrease with frequency; the reliable parts of each phase velocity must follow such trend eliminating in the average calculation the increase of velocity at low and high frequencies of motion. We noticed that the maximum wavelength at which the phase velocity can be estimated is about 10 times the radii r of the arrays at Queen's Park Savannah (Figure 5); the minimum wavelength is about 2 times the radii r of the arrays [7]. To obtain the average velocity at each frequency of motion f, we used N frequencies equally separated by the value of Δ<sup>f</sup> in terms of a logarithm scale as follows:

Natural Hazards - Risk, Exposure, Response, and Resilience

Spatial autocorrelation coefficient (SPAC) for the Queen's Park Savannah array of microtremors (point 1X in

logΔ<sup>f</sup> <sup>¼</sup> log f max � log f min

Genetic algorithms (GAs) are mathematical simulations based on biological evolution of natural selection rules. The soil parameters are digitized to gene type

1.3 Inversion of phase velocities through genetic algorithms (GAs)

in such interval.

58

Figure 4.

Figure 1).

where fmax and fmin are the maximum and minimum reliable frequencies for the aperture arrays, respectively (Figure 5). In our case, we generally set the value of N = 20 and select the corresponding velocity which belongs to the nearest frequency

<sup>N</sup> � <sup>1</sup> (5)

Dispersion curve of Rayleigh wave (phase velocity) for Queen's Park Savannah array (point 1X in Figure 1). We select the average velocity (open circles) of the four radii r to be used in the genetic inversion. The thin solid lines depict the wavelength that corresponds to 10–30 times and 2 times the radii of the arrays. The red thick solid line depicts the theoretical phase velocity curve that corresponds to best individual (soil profile in Figure 6) after searching the optimum solution via genetic algorithms; fmax and fmin are the maximum and minimum reliable frequencies for the aperture arrays.

with n bits in series of 0 and 1 defining a priori lower and upper bound limits for the shear wave velocity and thickness of the layer (e.g., 200–600 m/s and 10–100 m, respectively). Each bit represents a gene, and a series of bits concatenated represents a chromosome. So, an optimal solution is searched using the chromosome that best matches the soil model represented by the experimental phase velocity curve developed using the microtremors array after applying the SPAC method. In this work we employed the method of Yamanaka and Ishida [8]. The reproduction of the initial population to a new population relies on the fitness function of each individual applying the three genetic operations modulated by the selection, crossover, and mutation; crossover acts to generate a good, new model with the combination of good parts of chromosomes of two parents; in the mutation operation, a gene is reversed (e.g., from 1 to 0 or vice versa). The mutation procedure is necessary to escape trapping at local minimum solutions.

The selection process begins declaring a misfit function ϕ<sup>k</sup> for a k individual is defined as follows:

$$\phi\_k = \frac{1}{N} \sum\_{i=1}^{N} \left[ \frac{c\_o(f) - c\_\epsilon(f)}{\sigma\_\epsilon(f)} \right]^2 \tag{6}$$

where N is the number of observed data that correspond to the number of the discrete frequencies used in the analysis (see Eq. (5)), co(f) is the observed Rayleigh wave velocity retrieved from the SPAC, cc(f) is the calculated Rayleigh wave velocity, and σ<sup>c</sup> is the standard deviation of the calculated velocity based on the average of all n individuals that constitute a population. Note that cc(f) is obtained theoretically employing the Haskell [9] model for plane waves using the VS and the thickness of layers produced by the genetic reproduction.

Then a fitness function fit is based on the misfit function as follows:

$$
\hat{f}\mathbf{t}\_k = \frac{\mathbf{1}}{\phi\_k} \tag{7}
$$

that the stiffness of soil increases with the depth, with an overlap in the VS ranges between two consecutive soil layers to take into account the possibility of velocity reversal when increasing depth. The best model is considered the average for all models that fits into the 10% of average misfit, so we were able to calculate the standard deviation σ for the VS and thickness H of each layer.

Left: boreholes at Queen's Park Savannah. TD, terminal depth; MSL, mean sea level. Note that the total depth of the boreholes must be accounted above and below the MSL (e.g., depth ≈ 210 + 50 = 260 feet or 80 m). Cross section A-A is located in Figure 1 depicting boreholes reaching the bedrock in the Queen's Park Savannah (after [2]). Right: Shear wave velocity profile obtained via genetic inversion of phase velocity at Queen's Park Savanna (point 1X in Figure 1). The best model (thick black line) is considered the average for good models that fits into the 10% of average misfit (thin gray lines) in the final round of successive genetic inversions (Table 2). T

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

b.We perform a second round of GAs selecting a narrow lower and upper bound limits for VS for the bedrock from step (a) as the mean standard deviation σ (e.g., 1705 116); instead we select a broad lower and upper bound during the genetic reproduction of shear wave velocity and thickness of soil layers as shown in Table 2 (Vs = 100–600, 200–700, and 300–800 m/s for the first, second, and third layer, respectively, and thickness H = 5–50 m for all layers).

c. We perform a third round of genetic inversion fixing the VS and thickness of the soil layers within the mean standard deviation σ calculated in (b) and select broad lower and upper bound limits for the VS of bedrock half space (Table 2), namely, 1000–2200 m/s. Then a new value of VS and its standard

d.We perform again step (b) with a new narrow lower and upper bound limits for VS for the bedrock σ of the mean of step (c) and selecting again, a broad

deviation for the bedrock are estimated in this step.

Figure 6.

61

denotes the fundamental period of the soil.

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

lower and upper bounds of VS and thickness of soil layers.

In the inversion, the soil model that fits the observed data must have a high value of fitness and survives to a greater extent to the next generation, while the models with a low value of fitness (bad ones) are replaced by newly generated models.

It is noted that some authors (e.g., [10]) suggest that the dispersion curve is not carrying information (or very limited) of the velocity and the position of the bedrock; in such cases both parameters are badly constrained if we set a broad lower and upper bound of VS for the half space in the bedrock during one round of a GAs' process. According to the seismic refraction data for the region [11], the bedrock yields a VS of 2000 m/s. However, such VS value was obtained for the Cariaco sedimentary basin at north eastern Venezuela which is located 225 km away from POS. In order to validate the VS of 2000 m/s proposed by Schmitz et al. [11], we extended the original GAs employing successive rounds of inversions.

#### 1.4 Successive rounds of genetic inversion

We applied successive rounds of GAs for the array at Queen's Park Savannah and St. Dominic's Children's Home (see site 1X and 7X, Figure 1) due to the following reasons:


Then, the procedure for the successive genetic inversions for the Queen's Park Savannah site is as follows:

a.We perform the first round of GAs with broad lower and upper bound limits for both, the soil deposits and the bedrock, namely, Vs = 100–600, 200–700, and 300–800 m/s for the first, second, and third layer, respectively, and thickness H=5–50 m for all soil layers, and a Vs = 1000–2200 m/s for the bedrock (Table 2). The P-wave velocity was calculated from the S-wave velocity using empirical relation determined by Kitsunezaki et al. [12]. Generally, we assume

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

#### Figure 6.

where N is the number of observed data that correspond to the number of the discrete frequencies used in the analysis (see Eq. (5)), co(f) is the observed Rayleigh wave velocity retrieved from the SPAC, cc(f) is the calculated Rayleigh wave velocity, and σ<sup>c</sup> is the standard deviation of the calculated velocity based on the average of all n individuals that constitute a population. Note that cc(f) is obtained theoretically employing the Haskell [9] model for plane waves using the VS and the

thickness of layers produced by the genetic reproduction.

Natural Hazards - Risk, Exposure, Response, and Resilience

Then a fitness function fit is based on the misfit function as follows:

fitk <sup>¼</sup> <sup>1</sup> ϕk

In the inversion, the soil model that fits the observed data must have a high value of fitness and survives to a greater extent to the next generation, while the models with a low value of fitness (bad ones) are replaced by newly generated models.

It is noted that some authors (e.g., [10]) suggest that the dispersion curve is not

We applied successive rounds of GAs for the array at Queen's Park Savannah

a. In the Queen's Park Savanna, there is water well information (see Figure 6) to compare with the genetic inversion results. The array site is also located inside of a busy roundabout of 500 m radii, so presumably the constant source of the energy of microtremors is guaranteed in this case due to constant traffic

b.St. Dominic's Children's Home has the shortest period among the array sites, and it is located 500 m from the roundabout (Figure 1). So it would be easy to get a reliable shear wave velocity on bedrock according to the aperture array

Then, the procedure for the successive genetic inversions for the Queen's Park

a.We perform the first round of GAs with broad lower and upper bound limits for both, the soil deposits and the bedrock, namely, Vs = 100–600, 200–700, and 300–800 m/s for the first, second, and third layer, respectively, and thickness H=5–50 m for all soil layers, and a Vs = 1000–2200 m/s for the bedrock (Table 2). The P-wave velocity was calculated from the S-wave velocity using empirical relation determined by Kitsunezaki et al. [12]. Generally, we assume

and St. Dominic's Children's Home (see site 1X and 7X, Figure 1) due to the

carrying information (or very limited) of the velocity and the position of the bedrock; in such cases both parameters are badly constrained if we set a broad lower and upper bound of VS for the half space in the bedrock during one round of a GAs' process. According to the seismic refraction data for the region [11], the bedrock yields a VS of 2000 m/s. However, such VS value was obtained for the Cariaco sedimentary basin at north eastern Venezuela which is located 225 km away from POS. In order to validate the VS of 2000 m/s proposed by Schmitz et al. [11], we

extended the original GAs employing successive rounds of inversions.

activity in clockwise direction (see arrows in Figure 1).

size and a very shallow structure (Table 1).

c. We want to compare the results of (a) and (b).

1.4 Successive rounds of genetic inversion

following reasons:

Savannah site is as follows:

60

(7)

Left: boreholes at Queen's Park Savannah. TD, terminal depth; MSL, mean sea level. Note that the total depth of the boreholes must be accounted above and below the MSL (e.g., depth ≈ 210 + 50 = 260 feet or 80 m). Cross section A-A is located in Figure 1 depicting boreholes reaching the bedrock in the Queen's Park Savannah (after [2]). Right: Shear wave velocity profile obtained via genetic inversion of phase velocity at Queen's Park Savanna (point 1X in Figure 1). The best model (thick black line) is considered the average for good models that fits into the 10% of average misfit (thin gray lines) in the final round of successive genetic inversions (Table 2). T denotes the fundamental period of the soil.

that the stiffness of soil increases with the depth, with an overlap in the VS ranges between two consecutive soil layers to take into account the possibility of velocity reversal when increasing depth. The best model is considered the average for all models that fits into the 10% of average misfit, so we were able to calculate the standard deviation σ for the VS and thickness H of each layer.



employing broad lower and upper bound limits for the sediments and vice versa

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

When applying the methodology above to the Queen's Savannah Park array, for each iteration, the total number of unknown parameters yields four velocities and three thicknesses, searching an optimal combination for them in the inversion that matches the experimental phase velocity presented in Figure 5. These parameters were digitized as 8-bit binary strings, setting the population size at n = 30 individuals, with a crossover probability of 0.7 and an initial mutation probability of 0.01, terminating the iterations at the 100th generation. Since the algorithm used initial random numbers finding the global minimum solution, we performed for each round of GAs 5 iterations (or inversions) that indeed had different initial random numbers with a total of 15,000 soil models in each round. The final model was selected as an acceptable solution if its average misfit was less

The GAs' inversion yields a value of VS = 2032 104 m/s for the bedrock for the sixth round according to (e) above, which is basically the same value of 2000 m/s proposed by Schmitz et al. [11]. The results also yielded a first layer of VS = 403 m/s with a thickness of 32 m, a second layer of VS = 513 m/s with a thickness of 30 m, and a third layer of VS = 645 m/s with a thickness of 12 m that are classified as sand and clay. The soil profile is presented in Figure 6. Then the total thickness yields 75 m above a half space constituted by a shear wave velocity of nearly 2000 m/s. We validated our results with the depth of bedrock of about 80 m (260 feet in Figure 6) in this area reported by the Water and Sewerage Authority (WASA) [2]. Note that the thickness of 32 7 m of the first layer is similar to the sandy-clay first layer of 36 m with the water well profile presented in Figure 6; however, some differences are found to the second and third layer. We attribute such differences due to the fact that such water well information is 200 m apart from the array site. The

It is noted a good match between the experimental and calculated (theoretical) phase velocity via application of the Haskell [9] model for plane waves employing the final model presented in Table 2. This confirms the effectiveness of the genetic

The authors tested secondly the successive rounds of GAs performing an array at St. Dominic's Children's Home (see site 7X, Figure 1) with the shortest period of 0.22 s among the arrays (see Figure 10h). Then it would be suitable to find the VS for the bedrock for a shallower and a simple soil structure. The results are presented in Table 3. It is noted that we found also a value near 2000 m/s for the bedrock when applying the GAs at this site. As it was expected, for the St. Dominic's Children's Home case, the GAs converge faster than for the Queen's Park Savannah case due to

If we fix the VS for the bedrock as VS 2032 104 m/s taken from round number six in Table 2 from GAs in Queen's Park Savanah and perform one round of GAs for St. Dominic's Children's Home, we found practically the same optimal model for the

Further seven microtremors array were made in Port of Spain and distributed in the City (Figure 1 and Table 1); for such cases we fix in the GAs'scheme the VS 2008 124 m/s in the half space according to round 4 in Table 3. The proportion of the maximum wave length and the array size lay between 4 and 14 for all measurements (see Table 1). So we assured that the search limits for the thickness of the soil deposits in the GAs' process yield less than the penetration of the

Rayleigh waves for each array. A commonly adopted criterion is that the maximum investigation depth is half of the maximum wavelength [15]. Appropriate search limits were decided after several trial runs. The results of the GAs' inversion are

through several rounds of inversions in a subsequent manner.

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

bedrock in this case is found at the boulders' level.

soil profile from the previous process above (Table 4).

than 10% [13, 14].

scheme (Figure 5).

63

a simple and shallower soil structure.

σ denotes the standard deviation and α denotes infinite thickness on half space. The arrows indicate the bedrock or sediments information that is used in the subsequent round of GAs.

#### Table 2.

Example of successive rounds of genetic inversion, search limits, and optimal final model for Queen's Park Savannah site 1X.

e. The schemes (b–d) is repeated till when we find the mean VS for the soil deposits and bedrock inside of the range Vs mean σ of a previous round when selecting a narrow lower and upper bound in the bedrock (e.g., rounds 4 and 6 in Table 2).

Then the successive rounds of inversion are based on the effect of fixing the bedrock properties while searching the optimum solutions in one round of GAs

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

employing broad lower and upper bound limits for the sediments and vice versa through several rounds of inversions in a subsequent manner.

When applying the methodology above to the Queen's Savannah Park array, for each iteration, the total number of unknown parameters yields four velocities and three thicknesses, searching an optimal combination for them in the inversion that matches the experimental phase velocity presented in Figure 5. These parameters were digitized as 8-bit binary strings, setting the population size at n = 30 individuals, with a crossover probability of 0.7 and an initial mutation probability of 0.01, terminating the iterations at the 100th generation. Since the algorithm used initial random numbers finding the global minimum solution, we performed for each round of GAs 5 iterations (or inversions) that indeed had different initial random numbers with a total of 15,000 soil models in each round. The final model was selected as an acceptable solution if its average misfit was less than 10% [13, 14].

The GAs' inversion yields a value of VS = 2032 104 m/s for the bedrock for the sixth round according to (e) above, which is basically the same value of 2000 m/s proposed by Schmitz et al. [11]. The results also yielded a first layer of VS = 403 m/s with a thickness of 32 m, a second layer of VS = 513 m/s with a thickness of 30 m, and a third layer of VS = 645 m/s with a thickness of 12 m that are classified as sand and clay. The soil profile is presented in Figure 6. Then the total thickness yields 75 m above a half space constituted by a shear wave velocity of nearly 2000 m/s. We validated our results with the depth of bedrock of about 80 m (260 feet in Figure 6) in this area reported by the Water and Sewerage Authority (WASA) [2]. Note that the thickness of 32 7 m of the first layer is similar to the sandy-clay first layer of 36 m with the water well profile presented in Figure 6; however, some differences are found to the second and third layer. We attribute such differences due to the fact that such water well information is 200 m apart from the array site. The bedrock in this case is found at the boulders' level.

It is noted a good match between the experimental and calculated (theoretical) phase velocity via application of the Haskell [9] model for plane waves employing the final model presented in Table 2. This confirms the effectiveness of the genetic scheme (Figure 5).

The authors tested secondly the successive rounds of GAs performing an array at St. Dominic's Children's Home (see site 7X, Figure 1) with the shortest period of 0.22 s among the arrays (see Figure 10h). Then it would be suitable to find the VS for the bedrock for a shallower and a simple soil structure. The results are presented in Table 3.

It is noted that we found also a value near 2000 m/s for the bedrock when applying the GAs at this site. As it was expected, for the St. Dominic's Children's Home case, the GAs converge faster than for the Queen's Park Savannah case due to a simple and shallower soil structure.

If we fix the VS for the bedrock as VS 2032 104 m/s taken from round number six in Table 2 from GAs in Queen's Park Savanah and perform one round of GAs for St. Dominic's Children's Home, we found practically the same optimal model for the soil profile from the previous process above (Table 4).

Further seven microtremors array were made in Port of Spain and distributed in the City (Figure 1 and Table 1); for such cases we fix in the GAs'scheme the VS 2008 124 m/s in the half space according to round 4 in Table 3. The proportion of the maximum wave length and the array size lay between 4 and 14 for all measurements (see Table 1). So we assured that the search limits for the thickness of the soil deposits in the GAs' process yield less than the penetration of the Rayleigh waves for each array. A commonly adopted criterion is that the maximum investigation depth is half of the maximum wavelength [15]. Appropriate search limits were decided after several trial runs. The results of the GAs' inversion are

e. The schemes (b–d) is repeated till when we find the mean VS for the soil deposits and bedrock inside of the range Vs mean σ of a previous round when selecting a narrow lower and upper bound in the bedrock (e.g., rounds 4 and 6 in Table 2).

σ denotes the standard deviation and α denotes infinite thickness on half space. The arrows indicate the bedrock or

Example of successive rounds of genetic inversion, search limits, and optimal final model for Queen's Park

sediments information that is used in the subsequent round of GAs.

Natural Hazards - Risk, Exposure, Response, and Resilience

Table 2.

62

Savannah site 1X.

Then the successive rounds of inversion are based on the effect of fixing the bedrock properties while searching the optimum solutions in one round of GAs


presented in Figures 7 and 8. It is observed a good match between the experimental and calculated (theoretical) phase velocity for all array sites. The soil profiles containing the VS and thickness resulting from the microtremors array analysis are plotted in Figure 8. The shear wave velocity in the POS sediments yields from 51 to 750 m/s and the bedrock is located at 28 to 225 m depth with shallow structures in the peripheries near the hills and deeper structures toward the south of the city at the Port Area (Figure 1). It is worth mentioning that at the Port Area, very

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

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

Mean shear wave velocity (VS) profiles after application of the genetic inversion at eight sites of microtremors array in Port of Spain. The best model (thick black line) is considered the average for good models that fits into the 10% of average misfit (thin gray lines). (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). The sites are ordered from top to bottom from the largest to the shortest fundamental period of soil T (s). See locations of microtremors array in Figure 1.

Figure 8.

65

σ denotes the standard deviation and α denotes infinite thickness on half space. The arrows indicate the bedrock or sediments information that is used in the subsequent round of GAs.

#### Table 3.

Example of successive genetic inversion, search limits, and optimal final model for St. Dominic's Children's Home site 7X.


σ denotes the standard deviation and α denotes infinite thickness on half space. We fix the bedrock velocity according to the results of the Succesive Inversion for Queen's Park Savannah (Table 2).

#### Table 4.

Example of genetic inversion, search limits, and optimal final model for St. Dominic's Children's Home site 7X.

#### Figure 7.

Theoretical (line) and experimental (open circles) phase profiles after application of the genetic inversion at eight sites of microtremors array in Port of Spain. (a) Port Area (5X), (b) Mucurapo Secondary School (3X), (c) Sea Lots (8X), (d) Nelson Mandela Park (2X), (e) Woodford Park (8X), (f) Federation Park (4X), (g) St. James hospital (9X), and (h) St. Dominic's Children's home (7X). See locations of microtremors array in Figure 1.

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

presented in Figures 7 and 8. It is observed a good match between the experimental and calculated (theoretical) phase velocity for all array sites. The soil profiles containing the VS and thickness resulting from the microtremors array analysis are plotted in Figure 8. The shear wave velocity in the POS sediments yields from 51 to 750 m/s and the bedrock is located at 28 to 225 m depth with shallow structures in the peripheries near the hills and deeper structures toward the south of the city at the Port Area (Figure 1). It is worth mentioning that at the Port Area, very

#### Figure 8.

σ denotes the standard deviation and α denotes infinite thickness on half space. The arrows indicate the bedrock or

Example of successive genetic inversion, search limits, and optimal final model for St. Dominic's Children's

σ denotes the standard deviation and α denotes infinite thickness on half space. We fix the bedrock velocity according to

Example of genetic inversion, search limits, and optimal final model for St. Dominic's Children's Home site 7X.

Theoretical (line) and experimental (open circles) phase profiles after application of the genetic inversion at eight sites of microtremors array in Port of Spain. (a) Port Area (5X), (b) Mucurapo Secondary School (3X), (c) Sea Lots (8X), (d) Nelson Mandela Park (2X), (e) Woodford Park (8X), (f) Federation Park (4X), (g) St. James hospital (9X), and (h) St. Dominic's Children's home (7X). See locations of microtremors array in

sediments information that is used in the subsequent round of GAs.

Natural Hazards - Risk, Exposure, Response, and Resilience

the results of the Succesive Inversion for Queen's Park Savannah (Table 2).

Table 3.

Table 4.

Figure 7.

Figure 1.

64

Home site 7X.

Mean shear wave velocity (VS) profiles after application of the genetic inversion at eight sites of microtremors array in Port of Spain. The best model (thick black line) is considered the average for good models that fits into the 10% of average misfit (thin gray lines). (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). The sites are ordered from top to bottom from the largest to the shortest fundamental period of soil T (s). See locations of microtremors array in Figure 1.

consolidated sediments with VS of about 700 m/s constitute the thicker layers with more than 100 m above the bedrock.

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

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

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

(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.

Figure 9.

67
