Velocity Models

Chapter 1 Introduction

5.1 Velocity Models

in Figure 5.2. The forces are applied at the model’s center on the surface nodes as described in the Section 5.2. For a detailed description of the code, refer to Aagaard (1999).

Depth(m) Density(Kg/m³) V_p(m/s) V_s(m/s)

0−3 1748 306 220

3−7 1907 673 310

7−31 1907 673 322

31−97 1957 1421 502

97−113 2065 1794 666

113−413 2065 1894 680

half −space 2500 5505 2258

Table 5.1 Material properties for soil under Millikan Library from literature search.

steps required. Instead, the velocity model was modified to the linearly interpolated material properties shown in Table 5.2.

Depth(m) Density(Kg/m³) V_p(m/s) V_s(m/s)

0 1907 673 322

112 2065 1894 680

321 2065 1894 680

577 2500 5505 2258

1500 2500 5505 2258

Table 5.2 Material properties used for the first test of the Finite Element Code (FEC). There is a linear interpolation between the given data points, and this model will be referred to as the

“Smooth” model.

Furthermore, velocity models were also obtained using the Neighbourhood Al- gorithm of Sambridge (1999a,b) and forward modelling with the Green’s function method of Hisada (1994, 1995). Gueguen et al. (2000) have successfully modelled the pull-out tests (an experiment where the top of a structure is displaced from its resting position and released instantaneously) of a model structure at the Volvi test site located near Thessaloniki, Greece, for distances closer than 50 meters from the source using the Green’s function method of Hisada (1994, 1995). In this work, the Green’s function code was able to match the shape of the generated displacements

very well, however, the predicted amplitudes varied by up to 50% from the measured ones. These amplitude differences may be due to inaccurate force estimates (from the model structure used) or soil velocity differences between the real Earth and their velocity model. The latter is more probable, as the force estimates are easier to con- strain than the small 3-D velocity structure variations that can affect the observed amplitudes.

Due to the possibility that the estimated forces can be inaccurate, to emphasize the fitting of the shape of the measured displacements over their absolute amplitudes, an error function for normalized displacements is used in this study. At this stage, it should be clarified that since the observation points and the FEC nodes are usually not located at the same distance from the source, a polynomial is used to fit the synthetics and to interpolate both the phase and amplitude at the locations of the observations. For the displacement plots shown at the end of the chapter, the node amplitude values are also provided by simple dots, to qualitatively show the reader the adequacy of the polynomials used. From now on in this chapter, the interpolated data points will be referred to as either the predicted or synthetic data points or waveforms, where as the results from the experiments will be referred to as the observed data points or waveforms. The displacement errors between the predicted and the observed waveforms are calculated for the six non-nodal components of the EW and the NS radial lines shown in Figure 4.1 for both shaking directions. The mean displacement error,E_A, is computed by averaging the six Root Mean Squared (RMS) errors for each normalized component line. The waveform and synthetic with the largest amplitudes for each line are normalized to 1, and the remainder of the data points are normalized accordingly. The ratio of the maximum data amplitude to the maximum synthetic amplitude is called the normalizing factor. Subsequently, the synthetics for each line are re-normalized to obtain the lowest RMS error. These normalizations generate a normalizing factor for each component in each line, and a mean normalizing factor (N_F) is computed from the average of the individual factors.

The phase velocities of the radiated signal are independent of force magnitude and only depend on velocity structure. Therefore, the synthetics’ phase difference with

those of the measured waveforms at each station, is a real measure of the differences in velocity between the soil and the model. To eliminate some errors due to the finite size of the building and the point forces used in the model, the phase error between the data and synthetics is calculated relative to the first instrument in any one of the instrument lines. Subsequently, the mean phase error, EP, is computed from the average of the six relative RMS errors for the non-nodal components of the radial lines. The mean model error, ET, is then computed for a velocity model by:

E_T =E_P + 2πE_A

where the normalized total amplitude error is multiplied by a factor of 2π to make the phase and amplitude errors similar in amplitude. Then, the Neighbourhood Algo- rithm, in conjunction with theHisada (1994, 1995) code to forward model synthetics, were used to find the best-fitting velocity models utilizing only the previously men- tioned normalized total displacement error and the total phase error. Various runs were performed for each set-of parameters by implementing different model error schemes (eg. phase error only, amplitude error only, and the above mentioned combi- nation), where the initial velocity model parameters were set according to plausible and realistic properties. Since only relatively few waveforms are available, it was found that only a limited number of variables can be resolved. If more than 6 vari- ables are specified, the model does not converge on any solution. As a result, the various layer densities were fixed to be those of Table 5.1, and the density for layer 2 was chosen to be either 1957 or 2065 Kg/m³ for the different runs. Moreover, the half-space properties were chosen from the same table, with the exception of the shear wave speed, which was modified to be that of the equivalent Poisson solid (ν = 0.25, λ = µ, V_P = √

3V_S). This was done, as the half-space velocity was first determined by approximating it from other sources, but it was later found that Duke and Leeds (1962) also provides an estimate for it which is close to the Poisson solid value, but never mentions the source of the data. This estimate, together with the results from the Neighbourhood Algorithm for the top layers led to the modification of the half-

space to the value of a Poisson solid. The following are variables to be determined by the Neighbourhood Algorithm: Layer 1 thickness, V_P, V_S; Layer 2 thickness, V_P, VS. For the second layer, the obtained ratio for VP and VS is very close to that of a Poisson solid, and for simplicity it is assumed that the second layer is a Poisson solid. Tables 5.3 and 5.4 summarize the two solutions obtained for the optimal 2 layer velocity models.

Depth(m) Density(Kg/m³) V_p(m/s) V_s(m/s)

45 1907 710 376

375 1957 1410 814

1500 2500 5505 3178

Table 5.3 Material properties obtained from the use of the Neighbourhood algorithm for the first of the two converging solutions. There is no linear interpolation between the given data points, as this model is a layered model. This model will be referred to as model Layer2A.

Depth(m) Density(Kg/m³) V_p(m/s) V_s(m/s)

45 1907 830 440

375 1957 1590 843

1500 2500 5505 3178

Table 5.4 Material properties obtained from the use of the Neighbourhood algorithm for the second of the two converging solutions. There is no linear interpolation between the given data points, as this model is a layered model. This model will be referred to as model Layer2B.