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Theory of seismic waves
Theory of elasticity
Stress
Stress is force per unit area.
If the force is perpendicular to the area, we have a normal stress. When the force is tangential to the element of area, we have a shearing stress.
(xy denotes a stress parallel to the x-axis acting upon a surface perpendicular to the y-axis.
Because of equilibrium: (ij = (ji.
Strain
Strain is the fractional change in a dimension or shape of a body due to the application of stress.
Consider a body with dimensions of dx, dy, and dz along the x-, y-, and z-axes respectively. If this body is subjected to stress, then generally dx will change by an amount of u(x,y,z), dy by an amount of v(x,y,z), and dz by an amount of w(x,y,z). We can define the following:
Normal strains: (xx = (u/(x,
(yy = (v/(y,
(zz = (w/(z.
Shearing strains: (xy = (yx = (v/(x + (u/(y,
(yz = (zy = (w/(y + (v/(z,
(zx = (xz = (u/(z + (w/(x.
The dilatation (() is the change in volume (DV) per unit volume (V):
( ( DV/V = (xx + (yy + (zz = (u/(x + (v/(y + (w/(z.
Hooke s Law
It states that, at sufficiently small strains, the strain is directly proportional to the stress producing it. Mathematically:
( = C (,
where ( is the stress vector and ( is the strain vector. C is the elastic-constants tensor.
The elastic-constants tensor (C) is a fourth-order tensor consisting of 81 elastic constants. However, because of the symmetry relations in stress, strain, and strain energy, there can only be a maximum of 21 non-zero independent elastic constants in a medium. This number reduces, as more symmetry relations exist in the medium.
The least number of non-zero independent elastic constants exists in an isotropic medium, which has only 2 elastic constants. These are called Lame s constants l and m. m is also called the rigidity or shear modulus (HYPERLINK "HookesLaw.doc"Figure).
Elastic constants
Lame s constants l and m are defined through the following relations in an isotropic medium:
sii = lD + 2meii (i,j = x,y,z)
sij = 2meij (i,j = x,y,z, i ( j).
Young s modulus (E) is defined as: E = sxx/exx, (for uniaxial stress: syy = szz = sxy = sxz = syz = 0.)
Poisson s ratio (n) is defined as: n = -eyy/exx = -ezz/exx.
The bulk modulus (k) is defined as: k = s/D, (for hydrostatic stress: syy = szz = sxx = s.)
For a perfect fluid: m = 0, n = 0.5.
Typically: 0 < n < 0.5. It is small for hard rocks and large for soft rocks.
The wave equation
The scalar wave equation of a displacement (u) that depends only on x and t is:
(1/V2)(2u/(t2 = (2u/(x2,
where V is the wave velocity.
The general solution of this wave equation is a plane wave given by:
u = f(x Vt) + g(x + Vt),
where f and g are arbitrary functions representing two waves traveling along the x-axis in opposite directions with velocity V (HYPERLINK "Figure-Solutions.bmp"Figure).
The quantity (x ( Vt) is called the phase.
The quantity 1/V is called the slowness.
The surface on which the phase is the same is called the wavefront. The most commonly used wavefronts in geophysics are the plane and spherical.
In most cases, it is possible to approximate spherical waves by plane waves without introducing considerable errors.
Because plane waves are easy to visualize and mathematically simple, we generally assume that conditions are such that the plane-wave assumption is valid.
General aspects of seismic waves
Seismic waves are harmonic waves that contain several frequency components.
The wave velocity (V), frequency (f), and wavelength (l) are related as follows:
V = l f.
Typical wave characteristics in seismic exploration are:
Most of the reflected energy is contained within a frequency range 2 120 Hz.
The dominant frequency range is 15 - 50 Hz.
The dominant wavelength range is 30 400 m.
Huygens principle
It states that every point on a wavefront can be regarded as a secondary source that emits waves in the forward direction. The tangent to the wavefronts formed by the secondary sources defines the new wavefront. (HYPERLINK "Figure-Huygens.doc"Figure).
It is useful in drawing successive positions of wavefronts.
Fermats principle
It states that a wave will take that path which will make the traveltime stationary (i. e., maximum or minimum). Mathematically:
dT / dX = 0,
where T is the total traveltime along the wave path and X is the distance from the source to the point where the wave changes its direction (e.g., point of reflection or refraction).
In most situations in the earth, the stationary path is the minimum-time path (HYPERLINK "Figure-Fermat.doc"Figure).
Body waves
They distort the volume element of an elastic medium.
There are two types of body waves in an elastic, homogeneous, and isotropic medium: the primary (P) wave and the secondary (S) wave.
P-wave
The P-wave has a velocity:
a = [(l + 2m)/r]1/2.
Particle motion is parallel to the propagation direction for P-waves.
S-wave
The S-wave has a velocity:
b = (m/r)1/2.
Particle motion is perpendicular to the propagation direction for S-waves. Hence, there are two S-waves: SH which is parallel to the ground surface and SV, which is perpendicular to the ground surface.
Since the elastic constants (l, m) are always positive:
a > b,
and 0 ( b/a ( 1/21/2.
Typical P-wave velocities (a):
In air:
a ( 0.33 km/s.
In water:
a ( 1.5 km/s.
In water-saturated sedimentary rocks:
1.5 ( a ( 6.5 km/s.
In the weathered layer (soil):
0.4 ( a ( 0.8 km/s.
Surface waves
They exist due to the presence of a surface that separates two elastic media.
They are called surface waves because they are tied to the surface that separates the two media.
Their amplitudes decay exponentially as they get away from the surface.
Rayleigh waves
They propagate along a free surface of a solid. The ground surface is considered as a free surface in seismic exploration.
Rayleigh waves are called ground roll in seismic exploration.
The following relation is true about the velocity of Rayleigh wave (VR):
VR < b < a.
Most of the Rayleigh wave s energy is confined to 1-2 wavelengths of depth.
Tube waves
Tube waves constitute of any wave that travel in the borehole fluid or along the borehole wall in the direction of borehole axis.
They can supply information about the formation surrounding the borehole.
The most common types of tube waves are P-waves propagating in the borehole fluid Stoneley and pseudo-Rayleigh waves propagating along the borehole wall.
They can be generated by almost anything that disturbs the borehole fluid.
Anisotropy
Seismic anisotropy is the variation of seismic velocity with the direction in which it is measured or with wave polarization.
Anisotropy types are associated with the symmetry systems.
Transverse isotropy (IT) is the most common and important type of anisotropy encountered in seismic studies.
TI involves elastic properties that are the same in any direction perpendicular to an axis but are different parallel to this axis.
Two important types of TI are observed:
Vertical Transverse Isotropy (VTI) with a vertical symmetry axis. The main cause of VTI is the thin layering of shales in the subsurface
Horizontal Transverse Isotropy (HTI) with a horizontal symmetry axis. The main cause of HTI is the presence of vertical aligned fractures.
There are 5 independent elastic constants for TI.
In TI media, velocity is lowest when measured parallel to the symmetry axis and highest when measured perpendicular to the symmetry axis.
In TI media, S-wave splits into a fast (S1) wave perpendicular to the symmetry axis and a slow (S2) wave parallel to the symmetry axis.
Anisotropy (g) = (Vmax Vmin)/Vmax.
Medium effects on waves
Geometrical spreading
As the wavefront gets farther from the source, it spreads over a larger surface area causing the energy intensity (amplitude) to decrease.
Generally, the intensity (I) is related to distance (r) as follows:
I ( r-m
m = 0 for plane waves, 1 for cylindrical waves, and 2 for spherical waves.
In correcting for geometrical spreading, spherical wavefronts are assumed (m = 2).
Absorption
The transformation of elastic energy to heat as the seismic wave passes through the medium causes the amplitude to decrease.
Absorption follows the following relation:
A(x) = A0 e-(x,
where A0 and A(x) are amplitudes of a plane wavefront at two points a distance x apart,
(: Absorption coefficient.
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The distances and frequencies involved in seismic exploration are such that geometrical spreading is far more effective than absorption. Hence, we usually correct for geometrical spreading and neglect absorption (HYPERLINK "../../Slides/Chapter1.ppt"Figure).
Dispersion
It is the dependence of seismic velocity on the frequency.
Dispersion is negligible for body waves but very important for surface waves (HYPERLINK "Figure-Dispersion.doc"Figure).
Interface-related effects
When a wave encounters an abrupt change in the elastic properties (interface), some of the energy is reflected back to the incident medium and the rest is refracted (transmitted) into the other medium.
Snells law governs reflection and refraction:
sinq1/V1 = sinq2/V2 = p,
where q1: angle of incidence,
q2: angle of refraction,
V1: velocity of the incident medium,
V2: velocity of the refraction medium,
p: ray parameter.
Snell s law applies even when the waves differ (P- or S-waves) on the sides of the equal sign.
The critical angle (qc) takes place when q2 = 90(:
qc = sin-1(V1/ V2).
When q1 = qc, head waves are generated which travel along the interface.
Diffraction takes place when the wave encounters an abrupt lateral change in lithology (e.g., fault, wedge, etc.).
Snells law does not apply for diffractions and more complicated methods are used to study them (e.g., Huygens principle).
(HYPERLINK "../../Slides/Chapter1.ppt"Figure).
Amplitude partitioning at an interface
A P-wave incident on an interface between two solids will generally generate a reflected P-wave, a reflected SV-wave, a refracted P-wave, and a refracted SV-wave.
A SV-wave incident on an interface between two solids will generate a reflected P-wave, a reflected SV-wave, a refracted P-wave, and a refracted SV-wave.
A SH-wave incident on an interface between two solids will generate only a reflected SH-wave and a refracted SH-wave.
At the interface, the following boundary conditions must be satisfied:
Normal stresses must be continuous.
Tangential stresses must be continuous.
Normal displacements must be continuous.
Tangential displacements must be continuous.
The amplitudes of reflected and refracted waves are found by applying the boundary conditions at the interface and solving the resultant Zoeppritz-Knott equations.
At nonnormal incidence ((1 ( 0(), the reflection and refraction coefficients are very algebraically complicated functions of the P- and S-wave velocities and densities in the two media as well as the angles of reflection and refraction of the P- and S-waves.
At normal incidence ((1 = 0(), the reflection (R) and refraction (T) coefficients reduce to the following simple forms:
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T = 1 |R| = 2Z1/(Z1 + Z2),
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