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Deep earth imaging
Published in Rajib Biswas, Recent Developments in Using Seismic Waves as a Probe for Subsurface Investigations, 2023
The travel time of a seismic wave between source and receiver solely depends on the medium's velocity structure through which the wave propagates. Therefore, subsurface structure in a seismic travel time inversion is represented by variations in P- or S-wave velocity (or slowness). Velocity variations may be defined by a set of interfaces whose geometry is varied to satisfy the data. These variations are represented by a set of constant velocity blocks or nodes with a specified interpolation function or a combination of velocity and interface parameters (Figure 1.6). The most appropriate choice will depend on the a priori information (e.g., known faults or other interfaces), whether or not the data indicates the presence of interfaces (e.g., reflections, mode conversions), whether data coverage is adequate to resolve the trade-off between interface position and velocity, and finally, the capabilities of the inversion routine. There are several methods for parameterization of the subsurface model. Some of them are discussed next.
Acoustic and Ultrasonic Waves in Elastic Media
Published in Sourav Banerjee, Cara A.C. Leckey, Computational Nondestructive Evaluation Handbook, 2020
Sourav Banerjee, Cara A.C. Leckey
Here, the term x−ct is the phase and remain constant to retain the constant value of the plane wave potential ϕ. It is apparent that the phase wave velocity c is the slope of the equi-phase lines as shown in Fig. 4.2. It is timely to introduce another parameter that will be valuable later is the slowness. Slowness is the inverse of the phase wave velocity. Hence, phase slowness can be defined as s=1/c=k/ω and its unit is sec/m in SI unit.
Elastic waves in fractured isotropic and anisotropic media
Published in Xia-Ting Feng, Rock Mechanics and Engineering, 2017
Laura J. Pyrak-Nolte, Siyi Shao, Bradley C. Abell
where ρ is the density of the medium. Slowness is the reciprocal of velocity. The solutions for Equation 23 correspond to the quasi P- and quasi S-waves. σ3 can be expressed in terms of σ1 and the elastic components as:
Reflection of inhomogeneous waves at the surface of a cracked porous solid with penny-shaped inclusions
Published in Waves in Random and Complex Media, 2022
The displacement of solid particles in the cracked porous solid due to the presence of an incident wave and four reflected waves is expressed as follows. where are the excitation factors for reflected waves relative to incident wave. The complex vector defines the polarization and phase shift of the motion of solid particles for incident wave (k = 0) and reflected waves (k = 1, 2, 3, 4). The displacements of the fluid particles in the host medium and inclusions are calculated from the Equations (11) and (12) using the wave-specific values of the matrices and . The incident wave is specified by the vector and reflected wave is specified with the vector . The horizontal slowness of incident and reflected waves are always unchanged (Snell's law). The vertical slowness of incident wave is . The propagation of incident wave toward the boundary is ensured by the condition . The vertical slowness of reflected waves is . The decay of reflected waves along positive z-direction is ensured by the condition .
Reflection of plane waves in a functionally graded thermoelastic medium
Published in Waves in Random and Complex Media, 2021
We consider the incidence of QL or QT wave on the surface of the functionally graded half space making an angle with the positive direction of the z-axis, as shown in Figure 1. The incident QL or QT wave will generate reflected QL, reflected QT, and reflected T-mode wave in the half space z > 0. So, the displacement components and the field quantity will take the following form where in which and are the horizontal and vertical slowness vector for the incident wave, and are the horizontal and vertical slowness vector for the reflected wave.
Corrected procedure for reflection of harmonic plane waves in a transversely isotropic piezothermoelastic medium: inhomogeneous propagation of incident and reflected waves
Published in Waves in Random and Complex Media, 2021
Following [18], the characteristic Equation (7) is solved for , with This expression of involves only one unknown, i.e. β. Thus, in (7), the coefficients as well as the roots ( become the functions of β only. That means, for any incident wave, will be a complex function of β only. For real β, the equation is solved, numerically. This β, along with , determines the propagation velocity v as Thus completed slowness vector represents the inhomogeneous incidence and provides the common horizontal slowness (p) for Snell's law. This specification of enables to define the attenuation angle (), homogeneous attenuation () and inhomogeneous attenuation () for incidence of inhomogeneous waves in TIPTh medium. In slowness vector , the component denotes the vertical slowness of incident wave.