The data collected from seismic tests can be analyzed in both time and frequency domains. In the time domain, group or pulse velocities are evaluated; whereas, phase velocities and the attenuation coefficients are evaluated in the frequency domain. Wave velocity is related to the strength, stiffness, state of stress, and density of the medium. On the other hand, wave attenuation is an indicator of fractures, cementation, de-cementation, and compaction.
The propagation of low-strain mechanical waves is a perturbation phenomenon that assesses the state of materials without causing permanent effects. When the wavelength is larger than the internal scale of the material, such as the brick size in a masonry wall, the wave velocity and attenuation coefficient can be defined for an equivalent continuum. The group shear and compressional wave velocities VS and VP are (Graff 1975):
where G is shear modulus, M is constraint modulus and p is mass density of the medium. The velocities of compressional waves and shear waves are related through Poisson’s ratio of the medium. If the medium is homogeneous, all the frequency components of a pulse travel at the same speed VS or VP. However if the medium is dispersive, the propagation of each frequency component has a different velocity. The velocity of the pulse is referred to as group velocity; whereas, the velocity of each frequency component is called phase velocity.
For low excitation frequencies, two thirds of the energy introduced into a medium from a circular rigid plate converts to surface waves (Richart et al. 1970). In conventional seismic surveys, surface waves are not used because they mask the reflection and refraction events from body waves. The use of surface waves for shallow applications has been improving since the development of the spectral analysis of surface waves (SASW) method. The depth of penetration of surface waves is proportional to the pulse wavelength. Surface waves practically penetrate to a depth of two times their wavelength. However for
material characterization, their effective depth of penetration is approximately one-third their wavelength because the maximum displacements induced by surface waves take place close to the surface. The phase velocity of Rayleigh waves (R-wave) velocity Vph at any frequency f is related to distance Ax and the phase difference A0between the receivers by:
Ъ=п % 
The wavelength X as a function of frequency and phase velocity is given by
X(f) = f 
As seismic waves propagate, their amplitude decreases with distance due to geometric spreading of the wave front and the intrinsic attenuation of the medium. Geometric attenuation is the result of the increasing surface area of the wave front as it propagates outward. In an ideal material, the amplitude of the wave front decreases only due to geometric spreading because the intrinsic attenuation is zero. However, in real materials, part of the elastic energy is absorbed. Intrinsic attenuation can be expressed as
A = g" “Ir 2 – r 1
A, where a is the attenuation coefficient, the wave amplitudes at distances r, and r2 from the source are A, and A2, respectively.
The two-dimensional Fourier transform (2D-FFT) has been used to compute dispersion curves of multimode signals (Zerwer et al. 2001). The arrangement of multiple-channel measurements into a matrix permits the use of the 2D-FFT, which generates a matrix of complex numbers. The contour plot of the magnitude of the 2D-FFT renders a plot of frequency versus wavenumber k = 2n/X. Different waves are identified in the contour plot as a sequence of peaks. Peaks associated with non-dispersive waves plot as straight lines that pass through the origin; whereas, peaks corresponding to dispersive waves plot as curved lines with non-zero intercepts. Positive wavenumbers represent waves traveling in the forward direction (away from the source); whereas, negative wavenumbers represent waves traveling in the backward direction (towards the source).
Fuzzy-based methods have increasingly been used in a variety of civil and infrastructure-engineering problems from the evaluation of concrete and steel structures to water and wastewater applications (Liang et al. 2001; Najjaran et al. 2004). Fuzzy logic provides a language with semantics to translate qualitative knowledge into numerical reasoning. In many problems, the available information about the likelihoods of various items is vaguely known or assessed; hence, the information in terms of either measured data or expert knowledge is imprecise to justify the use of deterministic numbers. The strength of fuzzy logic is that it provides a rational and systematic approach to decision making through the integration of descriptive knowledge (e. g., very high, high, very low, low) and numerical data into a fuzzy model and uses approximate reasoning algorithms to propagate the uncertainties throughout the decision process. A fuzzy model contains three distinguished features: fuzzy numbers instead of, or in addition to numerical variables; relations between the variables in terms of IF-THEN rules; and an inference mechanism. A fuzzy number describes the relationship between an uncertain quantity x and a membership function |i(x) є [0,1]. A rule base determines the relationships between the inputs and outputs of a system using linguistic antecedent and consequent propositions in a set of IF-THEN rules. The inference mechanism uses approximate reasoning algorithms and the relationships to infer the outputs for given inputs.