# Seismic Waves

Introduction
Seismology is the study of the passage of elastic waves (see below) through the earth. Earthquake seismology is the best tool to study the interior of the earth.

When an earthquake or explosion occurs, part of the energy released is as elastic waves that are transmitted through the earth.

The waves are then detected and recorded by seismograms, which measure, amplify and record the motion of the ground.

The information is then used to determine earthquake locations, the subsurface structures and etc.

This pendulum-mounted seismograph records horizontal motion. The mass is coupled to the Earth by means of a pendulum and a pivot is attached to a rod to constrain the mass to move in the horizontal direction only.

The spring-mounted seismograph records the vertical ground motion. A spring is attached to the mass which is connected to a rod. The rod is attached to a pivot to constrain the mass to move in an up and down direction only.

Basic Physics

There is some basic terminology and physics that describe the various aspects of wave form and motion.

The wavelength (λ) is the distance between two adjacent points on the wave that have similar displacements, one wavelength is the distance between successive crest.

Amplitude (A) of the wave is the maximum displacement of the particle motions, or the height of the ripple crest.

Period (T) is the time it takes for two successive waves to pass a reference point or the motion to complete one cycle.

The cycle of seismic waves or repetitions in a given unit of time is called frequency (f). Frequency and period are related by this relationship:

f = 1 / T   [unit: hertz (Hz) or 1/s]

The speed in which the wavefront (or ripple crest) travel can be detected if the time the wavefront takes to reach a known distance is recorded:

V = distance / time [unit: m/s]

Or if wavelength and frequency are known:

V= f λ

Elastic Moduli

Elasticity is the behavior of a material that when subjected to a stress (force/area), deforms and changes shape (strain), but returns to original shape when the stress is removed.

The manner and speed of seismic waves travel through material is controlled by their elastic properties.

The linear relationship between applied stress, σ, and resulting strain ε is:

σ = Eε

E is the constant of proportionality called an elastic modulus.

We are concerned with two types of deformation – uniform compression or expansion, and shear deformation:

The original volume (V0) change to final volume (VF) when compared to the pressure change is called bulk modulus (K). The bulk modulus is a measure of the incompressibility of the material:

K = V0(P-P0)/(V0-VF)

When deforming a solid state by simple shear, a shear strain (γ) is induced by applying a shear stress, σ. The ratio of these quantities is the rigidity modulus (G):

G = σ/ γ

Units of elastic modulus are the same as pressure – i.e. MPa or GPa.

Seismic Waves

There are two different types wave produced by an earthquake: body waves and surface waves.

Body Waves

·       Body waves are seismic waves that travel through the body of the earth.

·       Body waves are reflected and transmitted at interfaces where seismic velocity and/or density change, and they obey Snell's law.

The two different types of body waves are:

·       P-Waves (P stands for primary or pressure or push-pull). These waves are also called longitudinal waves or compressional waves due to particle compression during their transport. These waves involve compression and rarefaction of the material as the wave passes through is but not rotation. P-wave is transmitted by particle movement back and forth along the direction of propagation of the wave. The most correct description of P-waves is it is a dilational or irrotational waves.

·       P-waves has the greatest speed and appears first on seismograms.

·       S-Waves (S stands for secondary or shear or shake). Also known as transverse waves, because particle motions are transverse to the direction of movement of the wavefront, or perpendicular to the ray. These waves involve shearing and rotation of the material as the wave passes through it, but not volume change.

·       S-waves have speeds less than P-waves, and appear on seismograms after P-waves.

Surface Waves

·       Surface waves are seismic waves that are guided along the surface of the Earth and the layers near the surface.

·       These waves do not penetrate the deep interior of the earth, and are normally generated by shallow earthquakes (nuclear explosions do not generate these surface waves).

·       Surface waves are larger in amplitude and longer in duration than body waves.

·       These waves arrive at seismograph after the arrival of P- and S-waves because of their slower velocities. The two different surface waves are:

·       Rayleigh waves or descriptively called "ground roll" in exploration seismology. The particle motion of this wave is confined to a vertical plane containing the direction of propagation and retrogrades elliptically. The particle displacements are greatest at the surface and decrease exponentially downward. Rayleigh waves show dispersion, and its velocity is not constant but varies with wavelength. This wave is similar to how ocean waves propagate.

·       VR < VS

·       Period is typically ~ 20 s, with wavelength or ~ 100km

·       Love waves (named for A.E.H. Love, who discovered them) travel by a transverse motion of particles that is parallel to the ground surface. This wave is somewhat similar to S-waves.

·       Love waves cannot exist in a uniform solid, and can only occur when there is a general increase of S- wave velocity with depth.

·       Their existence is another proof of the Earth’s vertical inhomogeneity.

·       The particle motion is transverse and horizontal.

·       Generally, Love wave velocities are greater than Rayleigh waves, so Love waves arrive before Rayleigh waves on seismograph.

Seismic Wave Velocities

·       The velocities of P- and S-waves are given below in terms of the density (ρ) and elastic coefficients of a material:

Vp = √((K+4/3G)/ρ)

Vs =√(G/ρ)

·       If we note that the bulk modulus (K) and the rigidity modulus (G) are always positive, then evidently the velocity of P-waves must always be greater than S-waves.

·       Shear waves (S-waves) cannot propagate through liquid. This is evident when we substitute G = 0 for liquids, then the velocity of S-waves goes to zero.

·       This is how it was determined that the outer core consists of liquid.

·       Some times you will come across the bulk sound velocity:

VΦ =√(K/ρ)

= √(Vp2 -4/3Vs2)

·       Also, Vp and Vs are related via Poisson’s ratio (r).

·       When a rod is stretched it becomes longer but narrower, the ratio to the lateral to longitudinal strain is Poisson’s ratio.

·       The ratio of Vp to Vs is given by:

Vp/Vs = [2(1-r)/(1-2r)]1/2

·       For most rocks, r ~ 0.25, so Vp ~ 1.7 Vs.

There are a few more general rules to the velocity ranges of common materials:

o  Unsaturated sediments have lower values than saturated sediments.

o  Unconsolidated sediments have lower values than consolidated sediments.

o  Velocities are very similar in saturated, unconsolidated sediments.

o  Weathered rocks have lower values than similar rocks that are unweathered.

o  Fractured rocks have lower values than similar rocks that are unfractured.

Below is a list of velocity estimation of common waves:

For rocks can plot V v. density:

More generally, Birch observed a general relationship between density and seismic wave velocity which helps us establish the composition of the Earth:

Now see more detailed notes on seismic waves, and you might like also like to look at the associated practical (optional).

# o  One can study subsurface velocity and layer interface structure by analyzing the first arrival times of P-waves (longitudinal or compressional waves) at the surface of the earth. This technique is termed seismic refraction.

o  Applications of subsurface imaging include:

1.   locating buried archeological sites,

2.  assessing subsurface geological hazards,

3.  defining aquifer geometry

4.  exploring for fossil fuel and other natural resources.

Seismic P-Wave Behavior

·       When a ray encounters an inhomogeneity in its travels, for example a lithological contact with another rock, the incident ray transforms into several new rays. A reflected wave enters and exits at the same angle measured to the normal of the boundary - angle of incidence equals angle of reflection.

o  From Snell's Law, a ray path is dependent on the wave velocities through different layers.

o  For refraction seismology, the critical angle is the most important angle value to understand. If angle (r) equals 90 degrees, then the refracted wave propagates along the boundary interface.

o  If r = 90, then sin(r) = 1, and the critical angle (ic) is given by:

ic = sin-1(V1/V2)

o  As the critically refracted wave propagates along the boundary, according to Huygen's Theory of Wavelets, the primary critically refracted wave acts as a source for new secondary wave fronts and ray paths.

o  These secondary ray paths exit back to the surface at the critical angle.

Simple Refraction Model

o  Two Horizontal Layers - In the ideal world (of engineering), refraction seismology is most easily understood through a horizontal two layer model.

Seismic waves are generated from a source (e.g. a sledge hammer, explosion, air gun….).

o  Since P-waves travel at the fastest speeds, the first seismic signal received by a geophone represents the P-wave arrival.

o  Five P-waves are of interest in refraction seismology:

o  direct

o  diving

o  reflected

o  refracted

o  The direct wave propagates along the atmosphere-upper layer (called layer1) boundary.

o  A transmitted wave through lower layer (layer 2) is termed a diving wave.

o  A reflected wave enters with the same angle of incidence as exit angle.

o  If the incident wave hits at the critical angle, the critically refracted head wave travels along the layer 1-layer 2 interface.

o  Refracted waves propagate from the interface as the head wave progresses, with exit angles equal to the critical angle.

o  With arrival time data collected, arrival times for P-waves are noted or computed from the seismographs.

o  Arrival times can be represented on a travel-time graph or T-X plot, that is P-wave arrival times (usually in milliseconds) verses distance (geophone location).

o  This plot shows that at small distances (x) from the source, the direct wave arrives first.

o  At distances up to the critical distance only the direct ray, and weakly (sub-critically) reflected rays arrive at the geophone. The reflected rays are always later than the direct ray.

o  At the critical distance, direct waves and the first refracted ray arrives. Its amplitude is stronger than the reflected ray, but is still later than the direct ray.

o  At some distance (the cross over distance), the refracted ray arrives first, since it has traveled at V2 for long enough in the interface so as to catch up the direct ray.

o  From the travel-time curve we can calculate:

o  velocities of P-wave propagation through layers 1 and 2 (V1 and V2)

o  thickness of layer 1 (H1).

o  To obtain these values, combination of equations and interpretation from the T-X plot is required.

o  The travel time of the direct wave is given by:

t DIRECT = x/V1

o  So V1 can be obtained from the slope of the direct arrivals, which passes through the origin.

o  The travel time for a reflected ray is given by:

tREFLECTION = (x2 + 4H12)1/2/V1

o  This is the equation for a hyperbola, where H1 is the layer thickness.

o  The travel time for the refracted wave is given by:

tREFACTED = x/V2 + 2H1(V22 –V12)1/2/(V1V2)

o  See detailed notes and Fowler for full derivations.

o  The equation for t REFRACTED is that of a straight line ( y = mx + c). The slope gives 1/V2 and the intercept on the t axis (i.e. when x=0) enables H1 to be determined from:

H1 = t(x=0)(V1V2)/2(V22 –V12)1/2

Two Layer Dipping Model

o  When discussing dipping layers, one wants to quantify the amount of dip. For a simple case of two dipping layers, seismic refraction can be utilized to calculated dip of the layers.

o  For a given survey profile, sources must be located at the beginning of the profile (forward shot) and at the end of the profile (reverse shot).

o  P-wave arrival times for both forward and reverse shots can be plotted on a T-X plot.

o  From the Principle of Reciprocity, time required for a ray to travel along the forward and reverse shot should be the same, since the ray pathways are the same.

o  From the T-X plot, V1 and V2 velocities for forward and reverse shots can be calculated, as well as the time-intercepts for forward and reverse refracted waves.

Kearey & Brooks (1984) show how this geometry can be analyzed to get h, θ, etc.

Horizontal Multi-Layer Model

o  Why only stop with interpretation of two horizontal layers?

o  Calculation of layer velocities and thicknesses for multi-layers requires patience with many equations chock full of algebra and trigonometry.

o  Please refer to Kearey & Brooks(1984), Fowler (1990) for these equations. Interpretation of T-X plots remains the same.

o  Each layer yields an interpolated refracted wave slowness, and time intercept used to calculate layer thickness.

o  This approach leads to understanding why seismic rays are reflected back to the surface on Earth as V increases generally with depth:

Problems and Limitations

o  The preceding models assume planar boundary interfaces. Conformable sequences of sedimentary rock may form planar boundaries. However, erosion and uplift easily produce irregular boundary contacts. More sophisticated algorithms can process refraction surveys where irregular interfaces might be expected.

o  Profile length and source energy limit the depth penetration of the refraction method. Typically, a profile can only detect features at a depth of one-fifth survey length.

Thus, refraction imaging of the Moho would require profile lengths of over one hundred kilometers; an difficult experiment.

o  Larger sources could be utilized for greater depth detection, but certain sources (e.g. explosives) may cause problems in urban areas.

o  Refraction depends on layers to increase in velocity with depth. In the hidden slow layer senario, a buried layer is overlain by a faster layer. No critical refraction will occur along the boundary interface.

Thus, refraction will not easily detect the slow layer. All is not lost since reflection seismology could detect the slower layer.

o  Seismograms require careful analysis to pick first arrival times for layers. If a thin layer produces first arrivals which cannot easily be identified on a seismogram, the layer may never be identified. Thus, another layer may be misinterpreted as incorporating the hidden layer. As a result, layer thicknesses may increase.

# Reflection Seismology

Reflection seismology began to take prominence in the 1920s to begin to locate salt domes, an indication where oil would be found.

The reflection method soon replaced the refraction after it was proved with numerous successes, the most visible in the petroleum industry.

The key is to develop an equation which represents the time it takes for a particular ray to travel through this single layer. First, the seismic velocity through the layer of material that the wave is propagating needs to be lower than the layer directly below, which we will assume is infinitely thick.

Therefore, just by simple time-velocity relation and geometry:

This can be re-written (dropping the subscripts) as:

V2t2 = x2 +4h2

so

V2t2/4h2 – x2/4h2 = 1

which has a hyperbolic form:

Now, What Does That Arrival Time Mean Anyway?

Well, the first thing to note is what you can do with the hyperbola.

A hyperbola has an asymptote along which the hyperbola approaches. The equation of this line is

Therefore, the asymptote for the travel time curve has a slope of the reciprocal of the velocity.

Another approach to analysing the data is to get velocity and thickness from a plot of x2 v t2. Now recall:

By squaring both sides, the equation resembles closely the equation of a straight line.

The slope of the line is the reciprocial of the square of the velocity. The intercepts gives h via:

Exploration Seismology

In the exploration industry there are many ways of processing reflection data so as to provide more information about the near sub-surface. This is beyond this course, but you may read more non-examinable material, and also in the following text taken from the Signalworks Pty. Ltd web site.

## (from Signalworks Pty. Ltd)

### Introduction

Reflection seismology is a technique for imaging the geological structure beneath the earth's surface using sound energy. The technique is used primarily for oil exploration. An acoustic energy source at the surface transmits an acoustic signal into the earth, which reflects some of the energy back toward the surface at each geological interface. An array of geophones or hydrophones detects the faint signals reflected back to the surface, which are recorded for later processing. The raw data is very noisy and uninterpretable, requiring extensive processing to produce an image of the earth's interior.

### Seismic Data Acquisition.

Figure 1 illustrates the process of marine seismic data acquisition. The survey ship trails an acoustic source (usually compressed air 'guns') and a string of hydrophones, called a streamer. The streamer is usually about 4000m in length and contains groups of hydrophones spaced typically every 15m. When the air guns are fired, releasing a pulse of compressed air, a pressure pulse radiates in an approximately spherical wavefront through the water and into the earth. The semi-circles in figure 1 indicate the position the wavefront at regular intervals in time (say every 100mS). When the wavefront reaches a reflecting geological boundary, some of the wavefront energy is reflected back towards the surface (light grey semi-circles). This echoed acoustic energy is sensed by the hydrophones and recorded on the ship for later processing.

To simplify seismic acquisition models, the energy received at a hydrophone can be considered to have travelled along a linear raypath from the source, into the earth, then reflecting from the boundary back to the hydrophone. Raypaths from the source to four hydrophones are shown in figure 1. The raypaths are perpendicular to the wavefronts.

### Principles of Acoustic Imaging.

Acoustic imaging in its simplest form consists of measuring the time taken by a pulse to travel from a source to a reflector and back to a receiver. Repeating these measurements over a range of positions allows an image of the reflecting surface to be formed. Figure 2 shows the configuration of a simple imaging system. In practice, noise and imaging distortions require more elaborate data acquisition configurations and data processing techniques to achieve accurate imaging.

#### Figure 2. Simple Acquisition Configuration.

Ideally, the simple acquisition configuration could be used to produce the acoustic image shown in figure 3. Each geological interface reflects some of the acoustic signal so that each trace shows a pulse corresponding to each reflector, with an increasing reflector depth resulting in an increasing time delay on the corresponding pulse.

a)
..........b)

### Imaging Problems and Solutions.

The simple imaging technique shown in figure 2 was used in the early days of seismic imaging, but produced poor results. The main problems were:

a) Noise -- the reflection energy is usually small after travelling a large distance and bouncing off a weak reflector. Spurious noise in the earth, air and recording electronics can swamp the reflection signal.

b) Multiples -- the raypaths not only travelled from source to receiver with one bounce off a reflector, but also followed paths making several intermediate bounces between reflectors and producing a travel time out of proportion to the reflector depth. Events on the image associated with raypaths making multiple bounces are called 'multiples' and should be removed from the image.

c) Source Pulse Shape -- the source pulse may not be sharp enough to produce a high resolution image and may vary in shape from shot to shot. (The activation of the source to produce a pulse is termed a 'shot'.)

d) Positioning of Dipping Reflectors -- the acoustic image is produced by displaying the trace at each record location vertically on the image. If a reflector is dipping, the raypath reflection point does not lie vertically below the record location, but is offset to one side. Further processing is required to correctly position the acoustic image.

a)
..........b)

#### Figure 4. a) Noisy Image and .. b) Detail of First Trace.

Figure 4 shows the effect of noise on the image. The reflected acoustic pulses are recorded from the hydrophones with a peak amplitude of 1mV. The noisy image shown in the figure has had random noise added with a normal amplitude distribution, mean value of 0mV and standard deviation of 0.5mV. The noise has nearly completely masked the reflection energy. The reflections cannot be discerned on the extracted trace shown in figure 4 (b).

Adding together repeated records taken at the same location can be used to improve the signal to noise ratio. Figure 5 shows a series of 32 repeated records. The reflected energy at 156mS and 416mS can be vaguely made out on this display, but would be difficult from a single trace. This figure also shows the result of 'stacking' these records. Stacking involves summing each trace and normalising the resultant summed trace. The reflection energy is reinforced and the random noise tends to cancel in the stacked trace (figure 5 (b)), resulting in an increased signal to noise ratio (S/N).

a)
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a)
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#### Figure 6 a) Raypath of 'Multiple' Energy and .. b) Recorded Trace with Multiple at 312mS.

Figure 6 (a) shows the raypath of acoustic energy making two bounces off reflector 1 between the source and receiver. The recorded pulse of this energy is termed a 'multiple' and can be seen at 312mS on the recorded trace of figure 6 (b). To obtain an acoustic image resembling the reflecting layers, multiples must be removed as they are mis-positioned on the image. The pulses of energy that travel directly from source to receiver with a single bounce off the reflectors are termed 'primaries' and produce proportional images of the geology.

a)
..........b)

#### Figure 7. a) Common Depth Point Acquisition Configuration and .. b) CDP Gather.

Figure 7 (a) shows the data aquisition configuration that allows multiple energy to be identified and removed during processing. This is called the Common Depth Point (CDP) method because the data is repeatedly recorded over increasing source to receiver offsets, but with the raypaths reflecting off the same depth location on each geological surface. The CDP gather shown in figure 7 (b) shows the recorded traces for all source / receiver pairs. As the source to receiver offset increases, the length of the raypath bouncing off a reflector increases and the pulse is recorded at a larger time delay. The curved line of pulses on the gather corresponding to a particular reflector is called an 'event', and its shape is determined by the reflector's depth and the acoustic velocity along the raypaths.

It is the shape of the event that allows multiple events to be identified and removed by 2D filtering. The ideal shape of these events is hyperbolic and is called a Normal Move Out (NMO) curve. When the geological layers are flat and have constant acoustic velocity, the events have an accurate NMO shape. As the geology becomes more complex with sloping layers and rapid velocity variations, the events deviate from the ideal shape.

a)
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#### Figure 8. a) NMO Corrected CDP Gather and .. b) Trace Produced by Stacking the Gather.

The process used to filter out the multiples is called 'stacking'. This is a two stage process involving distorting the gather so that the primary events become flat (termed 'NMO correction'), then summing each trace to produce a single stacked trace. The stacked trace is also usually rescaled by a factor of 1/N, where N is the number of traces added in the stack.

The shallow primary reflector has been flattened in the gather, but the NMO correction has stretched out the pulse in the long offset traces. This is called 'NMO stretch' and will reduce the sharpness of the corresponding stacked pulse. This is seen in the 156mS event in figure 8 (b) when compared to the ideal event shape in figure 6 (b). To reduce the problem, regions of excessive NMO stretch are zeroed ('muted') before stacking.

The multiple event at about 312mS is not flattened by the primary NMO correction and has reduced amplitude on the stack trace. Figure 8 (b) shows that the multiple amplitude has been reduced by about 50% while the primary amplitudes have been preserved. This performance can be improved by increasing the range of offsets recorded in the gather and increasing the sharpness (or resolution) of the pulses.

a)
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#### Figure 9. a) CDP Gather (Sharp Acoustic Pulse) and .. b) Stacked Trace.

Figure 9 shows the NMO corrected CDP gather and stacked trace produced using a sharper acoustic pulse. The sharp pulse has a dominant period of 25mS compared to 51mS used previously. The multiple on the stacked trace is reduced to around a quarter of the amplitude of the primary events.

a)
..........b)

#### Figure 10. a) Raw Seismic Wavelet and .. b) Wavelet after Shaping.

Seismic sources usually produce non-ideal wavelet (or pulse) shapes, often having several oscillations over a broad wavelet and inconsistent shapes from shot to shot. A raw wavelet such as shown in figure 10 (a) can be filtered to remove oscillations and sharpen the pulse to produce a shaped wavelet shown in figure 10 (b). An ideal sharp wavelet improves the resolution and interpretability of the acoustic image.

#### Figure 11. The Reflection Point for a Dipping Reflector is Offset from the Middle of the Source / Receiver Pair.

Figure 11 shows the raypath from a near offset source / receiver pair down to a dipping reflector. The reflection point does not lie beneath the centre of the source / receiver where it is plotted on a stacked trace section. The process of repositioning dipping reflectors is called 'migration', and the output of this process is a 'migrated section'. Migration also corrects 'diffractions', which are hyperbola shaped events appearing on stack sections and emanating from sharp discontinuities in the geology. Migration can be performed on a stack section by summing amplitudes along a hyperbolic curve and placing the scaled sum at the apex of the hyperbola. This can also be viewed as collapsing diffractions to a point over the entire stack section. The shape of the summing hyperbolas varies over the section and is a function of the depth and shallower acoustic velocities. The velocity distribution determined from earlier stacking velocity analyses can be used to control the migration process.

a)
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#### Figure 12. a) Stacked and .. b) Migrated Seismic Sections.

Figure 12 (a) shows a stacked section with a steeply dipping reflector mis-positioned. The migrated section (figure 12 (b)) shows the dipping reflector re-positioned in the up-dip direction and with a steeper slope.