NDT.net May 2003, Vol. 8 No.05 |
Spectral-Analysis-of-Surface waves (SASW) method is a non-destructive and non-intrusive method that by means of measuring Rayleigh wave dispersion can determine the vertical variation of material stiffness. In this paper, computer simulation of actual SASW tests, are used to show conflicting between existing filtering criteria. Also clarify that they yield inconsistent dispersion curves when more than one propagation mode participated in the wave field. To solve the uncertainty caused by unwrapping phase angles in the SASW analysis procedure, a computer program for simulation of the original Steady-State Rayleigh wave was developed. An overview of the theoretical aspects and field procedures of the surface wave method is briefly presented.
The spectral-analysis-of-surface waves (SASW) method is a seismic technique for determining in-situ elastic moduli and thickness of layered systems, such as soils and pavements. The test is performed from the ground surface and thus requires no boreholes. Measurements are made at strain levels below 0.001 percent, where elastic properties of soil are essentially independent of strain amplitude as can be seen from Fig.1. A schematic of the testing setup of the SASW method is presented in Fig. 2. The method consists of the generation of surface waves by an impulsive or random-noise load at the surface of the soil. Vibration signals are then detected by a pair of receivers spaced a distance D apart. The detected signals are transformed to the frequency domain using Fast Fourier Transform (FFT) analysis to obtain the phase difference of the transfer function (or cross-spectrum) between receivers as a function of frequency. The phase difference is then used to construct the experimental dispersion curve of the site. The inversion of dispersion curve provides information on the variation of shear wave velocity with depth.
Fig 1: Typical variation of in-situ shear modulus with shearing strain amplitude at the start of cyclic loading (After Stokoe and Hoar, 1977). |
The SASW method has the following advantages: it is nondestructive, it is performed from the surface of the system, the setup and procedure are simple, and it has the potential of being fully automated. The main disadvantage of the SASW method at this time is the process of inversion of the dispersion curve. The inversion process used until recently is an iterative procedure based on forward modeling of soil layers. In this procedure, the theoretical dispersion curve is matched to the experimental curve obtained in the field which is a time-consuming process and an experienced operator is needed. Hossian and Drnevich (1989), Addo and Robertson (1992), Yuan and Nazarian (1993) have developed automated techniques based on optimization theory. Meier and Rix (1993) and Trefor and Gucunski (1995), later introduced an automated inversion procedure using Neural Networks which are trained using a large data base of theoretical dispersion curves.
Fig 2: Layout of the testing setup of the SASW method (after Roesset et al 1990). |
The common drawback of the all above methods is that they are based on the assumption that the first Rayleigh mode is the predominant propagation mode. Gucunski and Woods (1992), Roesset et al. (1991), and Tokimatsu et al. (1992) have suggested the use of a simulated dispersion curve to overcome the difficulty associated with multi-mode propagation. In this method, the inversion procedure is performed by comparing the dispersion curve obtained from SASW field tests in the usual manner with the dispersion curve obtained from analytical or numerical simulation of the actual SASW tests used in the field. The propose of this paper is to show simulation of SASW test and to introduce new filter criterion.
Surface wave tests are simulated by applying a harmonic load on the surface of the soil layer or pavement. Simulation of SASW tests involves the solution of an axisymmetric wave propagation problem in which the source is represented by a disk load on the surface of a layered soil system (Fig. 3)
Fig 3: An axisymmetric model of a soil system with circular loading. |
The mathematical formulation of wave propagation in a layered system as used in this study is based on the stiffness matrix approach. The stiffness matrix approach is similar to the stiffness matrix or displacement method in structural analysis. The stiffness matrix of soil layer, as an extension of the transfer matrix (Thomson 1950, Haskell 1953), was derived by Kausel and Roesset (1981), and in slightly different form presented by Wolf and Obernhuber (1982) and Wolf (1985).
In this approach, the external forces that are applied at the layer boundaries are related to the their displacement through a stiffness matrix. The stiffness matrix is function of both frequency and wave number. For a soil system consisting of several layers, the stiffness matrices of individual layers are assembled, in a fashion similar to that in structural analysis, to form the global stiffness matrix [K]. The global stiffness matrix relates displacements {u} to external forces {P} in the frequency-wave number domain as follows:
(1) |
To find the solution for arbitrary loads using Eq. 1, the loads need to be transformed from the spatial coordinate domain to the frequency-wave number domain. In axisymmetric loading, a common procedure is to expand the loading function in terms of Fourier series in the circumferential direction q (n=0, 1, ....) and in terms of Bessel functions in the radial direction r. This transformation can be written as:
(2) |
Where
(3) |
P(r,q) equals loading vector in the spatial domain with components in the radial, tangential and vertical direction, and q(k,n) equals loading vector in the frequency-wave number domain. In the symmetric case:
(4) |
And for an antisymmetric case
(5) |
and
(6) |
Where J_{n}(kr) is the Bessel function of the first kind of order n. In the axisymmetric case, coordinate q and the corresponding displacement v can be eliminated. Considering a uniform vertical circular loading with intensity of p_{o} and radius R_{o,} the corresponding loading in wave number domain for which only the zeroth symmetric Fourier term (n=0) need to be considered, can be written as
(7) |
(8) |
The following two identities of the Bessel functions are used:
(9) |
which in integral form is equal to
(10) |
and
(11) |
Using Eq.11, Eq. 8 is formulated as
(12) |
For the axisymmetric case, only u and w displacement components exist. The displacements u(k) and w(k) in the frequency-wave number domain can be obtained by solving Eq.1 for the loading vector q(k). The loading vector q has only one nonzero component, the vertical component at the surface q_{o}, given by Eq.12.
The displacement amplitudes at the surface can be expressed in terms of flexibility coefficients as
(13) |
Where F_{uw} and F_{ww} are elements of [k]^{-1} that are condensed at the surface.
To obtain the displacements in the spatial domain, inverse Hankel transform is applied to displacements in the wave number domain. The relation for this operation is
(14) |
Where U_{s} is the displacement vector in the spatial domain and U_{w} the displacement vector in the wave number domain. In axisymmetric, case only u and w displacements exist and therefore, the second row and column of matrices D and C_{n} can be eliminated. Surface displacements in spatial domain (u_{s}, w_{s}) can be found in term of displacements in the wave number domain by the following relation:
(15) |
Only the vertical surface displacement is of interest in SASW tests. It can be obtained by making use of the identity of Eq.9 and substituting Eq.12 leading to
(16) |
Combining Eq.12 and 17
(17) |
Knowing the surface vertical displacement, the wave length of the simulated dispersion curve can be evaluated by two ways. The first method evaluates an average wavelength from the displacement curve at each particular frequency (see Fig. 4). This procedure should be repeated for a set of frequencies. The second method evaluates the transfer function H(f) between two prescribed points representing receiver locations at each frequency. The transfer function is the ratio of the spectrum of the far receiver to the spectrum of the near receiver. Then the phase difference between these two prescribed points is easily evaluated. This phase shift can be translated into travel time by the following relations.
Fig 4: Surface vertical displacements in the spatial domain and evaluation of the wavelength of the simulated dispersion curve (Gucunski and Woods, 1992). |
For a travel time equal to the period of the wave (T), the phase difference is 360 degrees.
Therefore for each frequency the travel time between receivers can be expressed by
(18) |
where
f = frequency,Because the travel distance is represented by the receiver spacing D, the Rayleigh wave velocity V_{ph} is simply calculated by
(19) |
and the corresponding wavelength can be determined by
(20) |
By repeating the above procedure for every frequency, the Rayleigh wave velocity corresponding to each wave length is evaluated and the dispersion curve is determined.
The dispersion curve generated according to equations [19] and [20] are filtered by applying the suggested criterion. These criteria are applied to ensure that the short wavelengths picked up by the receivers are eliminated because their amplitudes might be reduced drastically and consequently they might be dominated by ambient noise. Also, long Rayleigh wavelengths are eliminated because they might not be fully developed or be contaminated by body waves when they arrive at the near receiver. However, these criteria are insufficient to ensure the condition that measured wave field is comprised of one surface mode only as will be discussed in next section.
To perform the simulation of SASW test, a computer program was developed. The program calculates the exact flexibility coefficient in the wave number domain by inverting the stiffness matrix. Then using numerical integration, Equation [17] is evaluated. The only approximation introduced in this procedure is the use of the trapezoidal integration method. Errors due to this approximation are minimized by reducing the interval of integration. The program has the capability of considering layered system in three different ways:
The program was tested by finding simulated dispersion curve for the actual layered soil system reported by Nazarian and Desai (1993). The material properties used in the computer simulation are given in Table 1 and Fig. 5. The frequency range used in the analysis was from 0.1 to 520 Hz. The excitation frequencies from 0.1 to 50 Hz and from 51 to 520 were increased by increments of 0.1 and 1 Hz, respectively. Phase angle spectra are generated for 1, 2, 4, 8, 16, 32, 64, 128 m receivers spacings. Simulated relative phase angle spectra for these receiver spacings are given in Fig. 6. The simulated dispersion curves for the site obtained with applying the different filtering criteria are shown in Fig. 7. It can be seen that different dispersion curves were obtained for this site depending on the filtering criterion used. This may be due to the participation of more than one surface wave mode or other type of motions that can not be distinguished by available criteria. Therefore, it is necessary to find a method that enables the separation of modes from each other. Such a procedure will significantly assist the interpretation of SASW results.
Fig 5: Composite profile (from Yuan and Nazarian): (a) Stiffness profile; (b) Tip resistance; (c) Friction ratio; (d) Material profile. |
Fig 6: Simulated relative phase angle spectra (in degrees) for various spacings Source to near receiver distance equals receiver spacings. | Fig 7: Dispersion curves from available filtering criteria. |
Layer | Thickness (m) | Shear wave velocity (m/s) | Unit Weight (kN/m^{3}) | Poisson’s ratio | Damping ratio |
Sandy Silt | 3 | 110 | 20 | 0.45 | 0.05 |
Silty Clay | 5 | 135 | 20 | 0.45 | 0.05 |
Sandy Silt | 11 | 175 | 20 | 0.45 | 0.05 |
Sand | Inf | 320 | 20 | 0.45 | 0.05 |
Table 1: Material properties of simulated layered soil site. |
In order to find a reliable dispersion curve the following criterion is introduced. Since the SASW test is usually performed for several receiver spacings, therefore the dispersion data from different spacing are overlapped over wide frequency ranges and for each frequency several dispersion data may be available. These dispersion data must be combined to generate the average dispersion. In the new filtering criterion, the data points at same frequency but obtained from two different receiver spacings are rejected if they do not match each other. The dispersion curve obtained in this way is shown in Fig.8. Fig. 9 shows the experimental dispersion curve reported by Nazarian and Desai (1993). This dispersion curve is obtained by the following procedures:
By comparing Fig. 8 and Fig. 9 one can conclude that the simple proposed filtering criteria yields a dispersion curve similar to the one obtained by Nazarian and Desai (1993) using cumbersome analysis procedures.
Fig 8: Dispersion curve from proposed filter criterion. | Fig 9: Dispersion curve by Nazarian and Desai (1993). |
To check the results of the SASW simulation with proposed filter criteria a program for simulation of a Steady-State Rayleigh wave test was developed. The numerical simulation of the Steady-State Rayleigh wave test can be performed by calculating of the vertical displacement as a function of distance for each frequency. In-phase positions of surface motion are then found by the same procedure as in the case of original Steady-State Rayleigh wave test. The wavelength and phase velocity are calculated based on the average of two to three cycles of surface wave motion.
Fig 10: Simulated vertical motion at surface for case 1 at different frequencies. | Fig 11: Simulated vertical motion at surface for case 2 at different frequencies. | Fig 12: In-phase positions of surface motion for case 1 at different frequencies. | Fig 13: In-phase positions of surface motion for case 2 at different frequencies. |
The surface motions for the two four-layer models listed in Table 2 are computed and compared with the results obtained for the sites by Tokimatsu et al., 1993. The stiffness of these sites varies irregularly with depth. The surface motions for these sites are calculated by the developed simulation program. Figs. 10, 11 show the surface motion. These curves indicate that there are significant variations of the amplitude of motion with distance from source. In the frequency range below cut-off frequencies of higher Rayleigh modes, the variation results from a contamination of Rayleigh waves with direct and reflected body waves. The in-phase positions of surface motion for the two soil systems are calculated at each frequency and are shown in Figs. 12, 13. The dispersion curves derived by simulation of the Steady-State Rayleigh wave test for both sites are given in Figs. 14, 15. The dispersion curves for the above sites as reported by Tokimatsu et al. (1992) were derived by simulation of the SASW test. The theoretical formulae used in the simulation process were based on the transfer matrix method proposed by Thomson (1950). In this method phase lag of vertical and horizontal particle motions between the sensors at each frequency are calculated. Then the apparent phase velocity for vertical and horizontal motion are determined though Equations 18 to 20.
Fig 14: Dispersion curve for case 1 from diferrent procedures. | Fig 15: Dispersion curve for case 2 from diferrent procedures. | Fig 16: a)-Variation of Phase Velocity with Frequency for Case 1 b)-Variation of Wavelength with Phase Velocity for Case 1 from Tokimatsu et al., 1992. | Fig 17: a)-Variation of Phase Velocity with Frequency for Case 2 b)-Variation of Wavelength with Phase Velocity for Case 2 from Tokimatsu et al., 1992. |
Layer number |
Shear Wave Velocity (m/s) Case (1) (2) | Thickness (m) | Damping ratio | Poisson’s ratio | Unit Weight (kN/m^{3}) | |
1 | 180 | 80 | 2 | 0.005 | 0.35 | 18 |
2 | 120 | 180 | 4 | 0.005 | 0.35 | 18 |
3 | 180 | 120 | 8 | 0.005 | 0.35 | 18 |
4 | 360 | 360 | ---------- | 0.005 | 0.35 | 18 |
Table 2: Soil properties for Four-layer system |
The reported dispersion curves for the two four-layer systems are show in Figs. 16, 17. By comparing the results it is found that there is good agreement between the simulated dispersion curves from the Steady-State.
The Rayleigh wave has different modes of propagation, and thus may have different velocities at any frequency. The participation of each mode varies depending on the soil’s stratification as well as the frequency. Generally, inversion procedures require the calculation of the theoretical dispersion curve. Available procedures assume that the first mode of the Rayleigh wave is the predominant one and the dispersion curve of the first mode is then compared with the field dispersion curve. For irregular stratification, i.e., when the shear wave velocity does not increase with depth, higher Rayleigh modes provide a significant, and in many cases, a dominant influence on the overall wave propagation along the surface of a system. The inversion of the dispersion curve for such sites should not be guided solely by the theoretical first Rayleigh mode. Simulation of SASW tests is one remedy for this problem. In this method, the inversion procedure is performed by comparing the dispersion curve obtained from SASW field tests in the usual manner with the dispersion curve obtained from analytical or numerical simulation of the actual SASW tests used in the field. The existing filtering criteria unable to constract representive field or simulated dispersion curves. In order to find a reliable dispersion curve a new criterion is introduced. To evaluate the test and data analysis procedures of the SASW method, a computer program for simulation of actual tests was successfully developed. The implementation and applicability of the computer simulation were verified and demonstrated. To solve the uncertainty caused by unwrapping phase angles in the SASW analysis procedure, a computer program for simulation of the original Steady-State Rayleigh wave was developed.
© NDT.net - info@ndt.net | |Top| |