Seismic Waves - Faculty Personal Homepage- KFUPM

Theoretical principles of seismic wave propagation
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A solid body can be deformed by the application of an external force. If
the solid is perfectly elastic, it will return to its original shape once
that force is removed. In the context of exploration seismology, the earth
can generally be considered as perfectly elastic because the stresses
generated by seismic exploration activities are too small to permanently
deform subsurface rocks. The elastic limit is the maximum stress that can
be applied to a solid without permanently deforming it.
When an impulsive or transitory stress is applied to a finite area on the
surface of an elastic solid, a strain is generated in the immediately
adjacent subvolume. The strained subvolume then transfers stress to
adjacent interior areas within the solid, which generates strains in
surrounding subvolumes. In this fashion an impulsive stress propagates
through a solid as an elastic wave. Elastic waves that propagate in the
earth are known as seismic waves.
The exploration seismologist is primarily interested in seismic body waves
that propagate through the earth's interior; however, seismic energy can
also propagate as surface waves. Seismic surface waves are generally
regarded as a form of noise in seismic exploration because they only
contain information about the very near surface. For land reflection
surveys, however, surface waves may have amplitudes that are significantly
greater than the amplitudes of the sought after body waves. Also, depending
on the depth of interest, these surface waves may arrive at the seismic
sensors at nearly the same time as the desired signals. In these
situations, suppression of surface waves during acquisition and/or their
removal during processing become important considerations of the seismic
experiment.
Seismic body waves can be subdivided into two distinct wave types on the
basis of the direction of the seismic wave propagation relative to the
direction that particles in the medium are displaced during propagation.
For compressional waves, particles in the medium move parallel to the
propagation direction. For shear waves, medium particles move perpendicular
to the propagation direction. Within a medium, compressional waves
propagate faster than shear waves. For this reason, compressional waves are
usually referred to as primary waves, or P-waves, and shear waves are also
referred to as secondary waves, or S-waves.
The seismic velocity of a medium is a function of its elasticity and can be
expressed in terms of its elastic constants. For a homogeneous, isotropic
medium, the seismic P-wave velocity Vp is given by
[pic] where [pic]is the shear modulus, k is the bulk modulus and [pic]is the
density of -the medium. Using the same notation, the S-wave velocity Vs is
given by
[pic] As the names imply, the shear modulus is proportional to the shear strength
of the medium, while the bulk modulus describes the incompressibility of
the medium. The elastic constants have real values and are never negative,
so a comparison of the expressions for Vp and Vs supports the observation
that P-waves always travel faster than S-waves. Also, the expression for Vs
implies that shear wave propagation can only be supported by solid bodies
with shear strength. Fluids such as air and water do not support the
propagation of S-waves. In any case, most seismic reflection surveys are
exclusively concerned with P-wave acquisition, processing and
interpretation.
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Figure 1: Huygen's Principle is a kinematic description for calculating the
position of a wavefront at time (t + Dt) from the position of the wavefront
at time t. Huygen's Principle does not address the amplitude of the
wavefront.
Seismic wave behavior can sometimes be understood intuitively in terms of
ray path theory. First, consider a wave front at a particular time t inside
a medium of velocity V. (A wave front is a constant phase surface inside a
moving wave.) According to Huygen's Principle, the location of the wave
front at a later time (t + Dt) can be calculated by assuming that the wave
front will advance by a distance V*Dt from each point along the current
wave front. By using every point on the wave front at time t as the origin
of an arc with radius V*Dt, the position of the wavefront at time (t + Dt)
can be constructed from the envelope of the arcs. In other words, each
point along the wavefront at time t acts as an independent wave source, and
the wavefront at time (t + Dt) can be constructed by summing all the
individual source contributions (Figure 1). A line segment from a point on
the time t wavefront to the position where its arc touches the time (t +
Dt) wavefront defines a raypath of the seismic wave. Note that the raypath
is perpendicular to both wavefronts. A raypath that extends from a seismic
source to a detector is referred to as a travel path.
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Figure 2: Refraction and reflection of a ray path at planer impedance
boundary.
Two important concepts in seismic propagation are reflection and
refraction. Figure 2 shows a two-layer model with a higher velocity layer
over a lower velocity layer. In the figure, a down-going ray in the upper
layer is partitioned into an up-going reflected ray and a down-going
refracted ray at the layer boundary. For this simple homogeneous model, the
raypath geometry can be specified by three principles: (1) raypaths in a
constant velocity medium are straight. (2) At an impedance boundary, the
reflected raypath is reflected at an angle equal to the angle of incidence.
(3) At an impedance boundary, a change in velocity will cause the
transmitted ray path to bend or refract. The refraction angle can be
calculated using Snell's law, which is given by
[pic] where V1 and V2 are velocities in the upper and lower layers, [pic]is the
angle of the incident raypath with respect to the vertical, and [pic]is the
angle of transmission of the refracted raypath with respect to the
vertical. Note that Snell's law is also valid for the reflection angle if
V2 on the right hand side of the equation is replaced by V1. Figure 2 is
incomplete for the general elastic case because it neglects the phenomenon
of seismic mode conversions. For the general case, each layer should have a
P-wave velocity and an S-wave velocity. Then if the incident ray in the
diagram represents a down-going P-wave, it will be partitioned into four
waves at the layer boundary: a reflected P-wave, a refracted P-wave, a
reflected converted S-wave and a refracted converted S-wave. The angles of
reflection and refraction for the converted raypaths can also be calculated
using Snell's law.
For the purposes of exploration geology, the local geology of a sedimentary
basin can frequently be represented by a simple "layer-cake" model
consisting of a stack of homogeneous layers with planar upper and lower
surfaces. As was the case in Figure 2, all changes in density and acoustic
velocity in a layer-cake model are confined to the layer interfaces. When a
seismic wave encounters an interface, it is partitioned into a reflected
wave that bounces off the interface and a refracted wave that crosses the
interface, but may change its propagation direction.
Seismic velocities vary greatly with the type of rock or medium. P-wave
velocities of sedimentary rocks range from 1500 m/s for water, to 4500 m/s
for salt, and between 800 and 5000 m/s for sandstones. These materials,
together with shales and carbonates, are the main components of the world's
sedimentary basins. Oil and gas deposits are almost exclusively located in
porous sedimentary rocks (such as sandstones and carbonates) and may be
held in place by impermeable sedimentary rocks (such as salt and shale).
Using seismic methods, it is usually possible to derive estimates of
subsurface seismic velocities. Unfortunately, even perfect knowledge of the
seismic velocity does not provide a unique identification of rock type. The
seismic velocity of a sedimentary rock may vary depending on its fluid
content, precise mineral composition, degree of compaction, and strength of
cementation, among other factors; as a result, the velocity ranges of
different rock types overlap. | |[pic]
Seismic Deformation
When an earthquake fault ruptures, it causes two types of deformation:
static; and dynamic. Static deformation is the permanent displacement of
the ground due to the event. The earthquake cycle progresses from a fault
that is not under stress, to a stressed fault as the plate tectonic motions
driving the fault slowly proceed, to rupture during an earthquake and a
newly-relaxed but deformed state.
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Typically, someone will build a straight reference line such as a road,
railroad, pole line, or fence line across the fault while it is in the pre-
rupture stressed state. After the earthquake, the formerly stright line is
distorted into a shape having increasing displacement near the fault, a
process known as elastic rebound.
Seismic Waves
The second type of deformation, dynamic motions, are essentially sound
waves radiated from the earthquake as it ruptures. While most of the plate-
tectonic energy driving fault ruptures is taken up by static deformation,
up to 10% may dissipate immediately in the form of seismic waves.
The mechanical properties of the rocks that seismic waves travel through
quickly organize the waves into two types. Compressional waves, also known
as primary or P waves, travel fastest, at speeds between 1.5 and 8
kilometers per second in the Earth's crust. Shear waves, also known as
secondary or S waves, travel more slowly, usually at 60% to 70% of the
speed of P waves.
P waves shake the ground in the direction they are propagating, while S
waves shake perpendicularly or transverse to the direction of propagation. Although wave speeds vary by a factor of ten or mor