Seismic Principles: - Chapter 1
Seismic Principles: - Chapter 1
Seismic Principles: - Chapter 1
SEISMIC
PRINCIPLES
-Chapter 1
SEISMIC PRINCIPLES
In the above figure, the stress acting upon the faces can be resolved into
components. xy denotes stress parallel to the x- axis and perpendicular to y –axis.
The stress with the same subscripts is known as normal stress, and with different
subscripts are as shearing stress.
Strain:
Strain can be defines as the change occurred in shape and dimension due to
stress. There are certain fundamental type of strain, e.g. normal strain and shearing
strain.
Dilatation:
The change in volume per unit volume is called dilatation and it is represented
by .
Hook’s Law:
Elastic constant:
K = -P/
Substituting these values into the Hook’s law, the following relations can be
obtained.
Above relations are strictly within static equilibrium state, where wave
equation comes in to remove that restrictions.
Seismic Wave
The simplest form of time variation can be expressed by sine or cosine, such as:
Period, T = /V
Frequency, = V/
= + 2
If B/d = ,
2 =2 / 2 = x . = . = ½-
+ 2 + 2 1-
Surface wave include Rayleigh waves or Love waves known as ground roll,
which travel along the free surface of the solid materials.
Seismic Noise
In seismic records, there are two types of information, signal and noise.
S/N = n / n = n
Attenuation of noise is done by adding several random noise, which cancel
with each other as they are out of phase. Common depth point method is widely used
and is very effective in cancelling several kinds of noise.
Spherical Divergence
Solution is ψ = 1 f (r –Vt)
r
Elasticity
= k – 2/3 = E . (5)
(1 + )(1 - 2)
k = + 2/3 = E . (6)
(1 + )(1 - 2)
n = = E . (7)
2(1 - 2)
= . = E -1 (9)
2( + ) 2
Body wave is the wave which is transmitted through the body of material and
consisted of two waves:
Vp = k – 4/3 n = + 2
= density
Vs = n/ = E 1 .
2(1 + )
Particle motion is elliptical and retrograde, that is the motion at the top of the ellipse is toward the
source. The magnitude of the motion decreases with depth.
Vp = k/n + 4/3 = + 2 = (1 - )
Vs (1/2 - )
VP : VL : VR = 1 : 0.5773 : 0.5308
The relative portions of the energy transmitted and reflected are determined by
the contrast in the acoustic impedances of the rocks on each side of the interface. It is
difficult to precisely relate acoustic impedance to tangible rock properties but, the
harder the rock the higher is the acoustic impedance.
The acoustic impedance of a rock is the product of its density and the velocity
of longitudinal or compressional seismic wave through it, V, designated Z.
Consider a P-ray of amplitude A 0, normally incident on an interface between two
media of differing velocities and densities.
A transmitted of ray of amplitude A2 travels on through the interface in the
same direction as the incident ray, and a reflected ray of amplitude A 1 returns to the
source along the path of the incident ray.
The transmission coefficient is the ratio of the amplitude to the incident amplitude:
T = A2/A1
Snells Law
Snells Law, originally applied to light and optics, applies equally well to
seismic waves and the earth. For a reflected ray, Snells Law states that the angle
between the reflected ray and the normal to reflecting surface is equal to angle
between the reflected the incident ray and the normal to reflecting surface. In
seismology, of course, the reflecting surface is the boundary between two layers
having different acoustic impedances.
(A) Part of an obliquely incident ray is reflected at the angle of incidence. And part is
transmitted at an angle that depends on the ratio of the velocities in the two layers.
(b) A head wave is generated in the upper layer by a wave propagating through the lower
layer along the boundary.
When the velocity is higher in the underlying layer there is a particular angle
of incidence, known as the “critical angle”, c , for which that angle of refraction is
900. This gives rise to critically-refracted ray that travels along the interface at the
higher velocity V2 with equation as follows:
This wave, known as a “head wave”, passed up obliquely through the upper layer
toward the surface, as shown in figure 1-7 (b).
E X G
hO
FIG. 1-8
R
h0 = 1 (V20t2 – X2)
2
The portion of the incident energy that is not reflected is transmitted ray
travels through the second layer. The transmitted ray travels though the second layer
with changed direction of propagation, and is referred to as a refracted ray.
Snells law of refraction states that the ratio of sine of the angle to the velocity is a
constant. For a refracted P-ray;
were the subscripts refer to layer 1 and layer 2, respectively. See figure 1-7 (a).
Refraction
i,r (or 1, 2) - the angles between the normal to boundary and the rays
V0 ,V1 - velocities of different medium
FIG. 1-9
= EA + AB + BG
V0 V1 V0
= 2 ho . + X - 2 h tan 0.1
V0 cos0.1 V1 V1
= X + 2 ho - 2 ho sin0.1 sin0.1
V1 V0 cos0.1 cos0.1 V0
TE = X + 2 ho ( 1 - sin20.1 )
V1 V0 cos0.
I= 2 ho cos0.1
V0
ho = I V0 = I V0 V1 .
2cos0.1 2 V1–V0
2 2
ho = 1 I V0 V1 .
2 V21 – V20
Huygens Principle
This principles states that every point on the primary wave front surface is a
source of secondary wavelets. The position of the wave front at a later instant then is
found by constructing a surface tangent to all secondary wavelets. This concept is a
very powerful tool for understanding all types of wave propagation, from
electromagnetic waves to seismic waves.
Huygens principle, illustrated in figure 1-13, regards each point on the
advancing subsurface wave as a source that generates a new wavefront, which radiate
in all directions. It explains one of the most important mechanism by which a
propagating seismic pulse loses energy with depth.
When seismic waves strike any irregularity along a surface such as a corner or
a point where there is a sudden charge of curvature, the irregular feature acts as a
point source for radiating waves in all directions in accordance.
Figure 1-14 illustrates a buried corner at A, from which waves, exited by
radiation downward from a source at the surface, spread out in all directions paths
which are rectilinear as long as the velocity is constant. A diffracted wave reaches the
surface first at a point directly above the edge because the path is the shortest at this
point. The amplitude of a diffracted wave falls off rapidly with distance from the
nearest point to the source; diffracted events are frequently observed on seismic
records but not always recognized.
FIG. 1-14