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ERTH3120 Exploration and Mining Geophysics

Basic Concepts in Seismology

1 Seismic Wave Types

Existence of different wave types

The existence of different seismic wave types can be understood at various levels. Inhistorical earthquake seismology, it was empirically noted that following an earth-quake, distinct bursts of energy arrived at recording stations at different times. Thefirst main burst of energy was referred to as a P-wave and the second as an S-wave.For shallower earthquakes, this was often followed by a long string of energy referredto as the surface waves. Figure 1 shows these wave arrivals on a typical seismogramof distant earthquakes, recorded at the UQ Charters Towers site.

Figure 1: Typical seismogram of distant earthquakes, recorded at the UQ ChartersTowers station.The smaller quake was in Santa Cruz (M = 5.1), and the largerone was in Indonesia (M = 5.9). The arrival times of P, S, surface (L) events areindicated. For both events, the S-P time is about 3.5 minutes, indicating that theearthquake is about 2200 km away. Both events are shallow (< 50km) and henceproduce strong surface waves

ERTH3021 Basic Seismology - 2

The elastic wave equation

The existence of different seismic wave types can also be predicted from a purely math-ematical viewpoint. It is well known that the general wave equation describing thetransmission of some function f through space at a speed v is given by

∂2f

∂t2= v2

(∂2f

∂x2+

∂2f

∂y2+

∂2f

∂z2

). (1)

In seismology, the elastic wave equation describes how seismic waves travel throughthe earth. In its most general form it is slightly more complex then Equation 1, becauseit includes more than one wave type. However, it can be broken down into simplerequations which describe particular waves.

For example, stress-strain analysis allows us to identify a compressional (i.e. push/pull)disturbance (θ), which is the divergence of ground displacement. The particular equa-tion which describes how such a compressional disturbance transmits through the earthis

∂2θ

∂t2=

(K + 4µ3

ρ

)(∂2θ

∂x2+

∂2θ

∂y2+

∂2θ

∂z2

). (2)

Comparing Equations 1 and 2 it is clear that this compressional wave is travelling at avelocity

vP =

√K + 4µ

3

ρ(3)

This wave is the P wave identified by earthquake seismologists. The parameters in thevelocity term are considered in detail below.

Similar mathematical analysis also reveals the existence of a rotational or shear distur-bance (ζ). (This is the curl of the displacement.) The wave equation describing thetransmission of this shear is

∂2ζ

∂t2=

(µ

ρ

)(∂2ζ

∂x2+

∂2ζ

∂y2+

∂2ζ

∂z2

). (4)

Again, comparing Equations 1 and 4 it is clear that this shear wave is travelling at avelocity

vS =

√µ

ρ(5)

Other solutions lead to the prediction of various surface waves (Rayleigh, Love, Stonely,etc). The practical significance of all these solutions relates to the type of particle mo-tion involved, and the velocities of the waves.

Exploration Geophysics Laboratory University of Queensland


Particle motion of body waves

P waves and S waves are called body waves. They can travel within the body of theearth.

The most basic characteristic of the different waves relates to the way the particles ofrock move as the wave passes. As shown by Figure 2, when a P wave passes, theparticles move back and forward in the direction of the wave travel. For the S wave,the particle motion is in a plane perpendicular to this direction.

Figure 2: Particle motions for body waves travelling from left to right. P-wavemotion is in the direction of travel. S-wave motion is perpendicular to direction oftravel.



Particle motion of surface waves

As the name suggests surface waves must propogate close to the surface. Two commonvariations are the Rayleigh wave (ground roll) and the Love wave. As shown by Figure3, when a Rayleigh wave passes, the particles move in a retrograde elliptical fashion. TheLove wave requires a surface low-velocity layer. The particle motion is transverse, withhigh amplitude near the surface, and decreasing to zero at the base of the layer.

Figure 3: Particle motions for surface waves. Rayleigh-wave motion is like a back-flip (technically retrograde elliptical). Love-wave motion is transverse in a surfacelayer, with amplitude decreasing towards the base of the layer.



2 Body Wave Velocities

When we find the solutions of the seismic wave equation we obtain the velocities ofpropagation of the different wave types. As noted above, for the body waves we have

vP =

√K + 4µ

3

ρ(6)

vS =

√µ

ρ(7)

Here K is the bulk modulus (or incompresibility), µ is the shear modulus (or rigidity)and ρ is the density. These three physical properties can all be measured in a laboratory.If we consider realistic values of these parameters we deduce two important practicalresults.

(i) VP /VS Ratio and Poisson’s ratio

For many competent rocks the P-wave velocity is about twice the S-wave velocity. Theprecise value of the VP /VS ratio (termed γ) is a useful indicator of rock strength. Forexample in granites γ could be as low as 1.7, while in softer rocks (shales) it might be2.3. In the near surface weathered zone, very high values of γ can be observed (5-10).

Geotechnical and mining engineers prefer to use Poisson’s ratio (σ) to describe rockstrength. It can be derived from the velocity ratio via

σ =γ2 − 2

2(γ2 − 1)(8)

Substitution indicates that stronger rocks could have σ around 0.25, while weaker rockscould have σ around 0.4.

Finally, note that γ (or σ) are generally more indicative of rock strength than vP or vS

individually.



(ii) Body-wave velocities in the presence of fluids

Fluids have zero rigidity. That is µ is zero. Fluids also have reduced (but non-zero)incompresibility K. Consideration of the above equations immediately reveals that:

• P-waves will travel in a fuid (although more slowly than in rock).

• S-waves will not travel in a fluid.

The second of these results was very important in the successful prediction that theearth’s outer core was fluid (Figure 4).

Figure 4: S-waves travelling from an earthquake are not observed beyond an epi-central distance of 103 ◦. This is because they cannot travel through the liquid outercore.

In applied seismology, S-waves will not travel through the ocean, and this is relevantin marine exploration. The different behaviour of P and S waves in porous rocks is alsovery important (and sometimes poorly understood). Porous rocks are typically at least80% rock (if porosity is 20 %). The S-waves can travel through the rock matrix, essen-tially ignoring the fluid. That is, S-waves respond only to the geology, whilst P-wavesare influenced both by geology and fluid content. This difference is very important inthe exploration for porous rocks (e.g. hydrocarbon reservoirs, groundwater aquifers).



Consider the example in Figure 5, which shows a porous limestone reservoir overlainby a shale cap rock. In the left hand model, the limestone holds water, and on the rightit holds hydrocarbon, which has significantly reduced the P-wave velocity contrast atthe boundary. This renders the interface invisible to P-waves. However, the S velocityis unaffected, meaning that the reservoir can be mapped using S-waves. (The relevantconcept of reflection coefficient is explored in more detail below.)

Figure 5: Porous limestone reservoir overlain by shale cap rock. In the left handmodel, the limestone holds water. Introduction of hydrocarbon to the reservoir(right) reduces the P-wave velocity, but does not change the S-wave velocity signif-icantly.

3 Surface Wave Velocities

Rayleigh waves

In the simplest case of a homogenous earth, the Rayleigh wave travels along the sur-face at a velocity of about vR = 0.9vS . In the real-earth situation, there are normally lay-ers of different velocity near the surface. In this case, the highest-frequency Rayleighwaves travel at a velocity of 0.9vS1 (i.e. they travel close to the surface). Waves withprogressively lower frequency penetrate deeper, and hence sense layers with highervelocity. That is, lower-frequency Rayleigh waves travel with higher velocities. Thisphenomenon (velocity dependent on frequency) is called dispersion. On a seismic shotrecord, this is manifested as a band of Rayleigh-style energy, rather than a single welldefined event (see Figure 6). This more complicated mix of various Rayleigh waveevents is generally referred to as ground roll.



Figure 6: Vibroseis shot record from central Australia. The energy labelled A in-dicates the low velocity end of the dispersive ground roll band. Other ground-rollcomponents with higher velocity (lower slope) are also visible (labelled B).

Love waves

As noted above, propagation of Love waves requires a surface low-velocity layer. TheLove wave is dispersive, with velocities lying between the S-wave velocities in thesurface layer and the underlying layer. More complex love waves exist in multi-layeredsystems.

Interactions between surface and body waves

The bulk of reflection seismology is carried out using vertical-component geophones.Due to ray bending near the surface (Snell’s Law, see below), reflected (and refracted)waves are travelling almost vertically when they arrive at the geophone. This meansthat normal vertical-component reflection surveys record predominantly reflected andrefracted P-waves, and also ground roll (because it is elliptical and has a strong verticalcomponent). These normal surveys do not record reflected or refracted S-waves, or



Love waves, because their particle motion is close to horizontal.

Because the ground roll travels relatively slowly across the surface, it can often inter-fere with reflected P-waves from deep interfaces. In Figure 6, this is seen over much ofthe central part of the record. Methods have been developed to reduce this interference(geophone groups, velocity filtering).

Multi-component recording

A less common mode of recording uses 3-component geophones, which are able torecord all wave types (P, S, ground roll, Love waves.) This is a topic of considerableresearch over the past 20 years. Because of the different response of the earth to P andS waves (e.g. fluids), integrated P and S reflection using 3-component geophones canadd significantly to the geological characterisation of the subsurface.

4 Snell’s Law and the Ray Parameter

As derived in ERTH2020, a simple application of Fermat’s principle of least time leadsto Snell’s Law, which describes the direction taken by all seismic waves at a boundarybetween two different materials. Consider the simple case of a P wave incident on aboundary at an angle i1. Snells Law requires

sin i1v1

=sin i2v2

(9)

where v1 and v1 are the velocities on either side of the boundary, and i2 is the anglemade by the transmitted ray with the normal. In applied seismology we call the quan-tity ( sini

v) the ray parameter (p), and say that the ray parameter must remain constant

for a ray and its offspring (reflections, transmissions, conversions).

Mode conversion

When a P wave hits an interface at non-normal incidence, a shearing component isapplied to the interface, and this yields mode-converted S-waves (both reflected andtransmitted). Conversely an incident S-wave can impart a component of compressionon the interface, yielding mode-converted P waves. Snell’s law controls the directiontaken by all waves, via a logical extension of Equation 9, as shown in Figure 7.

Critical refraction

When the incident ray hits the interface at a special angle, then the transmitted raycan travel at an angle of 90 ◦ to the normal (Figure 8). This special angle is called thecritical angle (iC) and the phenomenom is referred to as critical refraction. From Snell’slaw, we can see that the critical angle is defined by sin iC = v1/v2. Note that criticalrefraction can only occur if v2 > v1. This is the normal situation in the earth, butvelocity inversions are a lso relatively common. The phenomenon of critical refractionis the basis of the so-called seismic refraction method of exploration.



Figure 7: An incident P-wave (at non-normal incidence) produces reflected andtransmitted P and S waves, whose directions are controlled by Snell’s law. All raysin the figure have the same ray parameter ( sin i

v ).

Figure 8: Critical refraction occurs when an incident wave hits an interface at thecritical angle.



5 Seismic reflection and seismic refraction methods

In general, whenever we initiate a seismic source at or near the surface, there is po-tential for many of the wave types discussed above to be produced, either directlyor by conversion. In applied seismology, there are two distinct exploration methodswhich use body waves. These are referred to as the seismic reflection and seismic refrac-tion methods. The distinction is based on the wave path which is being analysed. SeeFigure 9.

Seismic reflection uses reflected energy to construct geological imagery, commonlyover the depth range 50m - 50km. Seismic refraction is widely used to map the weath-ering zone (0-50m) although the method is sometimes also used at crustal scale. It isinteresting that the major use of refraction is to provide near-surface models to be usedin reflection processing.

Figure 6, discussed earlier, was a production record from a reflection survey. It containsboth reflection events (green, pink) and refraction events (yellow, blue).

Note that there are also other specialised techniques (e.g. MASW) which analyse sur-face waves to provide near-surface information.

Figure 9: In applied seismology, two important body-wave paths are analysed, andthese require distinct interpretation methods. A seismic source generates reflectedwaves (solid) and refracted waves (dashed). In geophysics the latter term generallyrefers to waves which have undergone critical refraction along a boundary.



6 Seismic reflection amplitudes

Commercially, the most important branch of applied seismology is seismic reflection,which has been heavily used in hydrocarbon exploration and more recently in hard-rock, engineering and environmental geophysics. A fundamental parameter which in-dicates whether an interface will be a good target for seismic reflection is the reflectioncoefficient.

Near-normal incidence

When a P wave strikes an interface at an angle close to normal, the reflection coefficientis defined as the ratio of reflected amplitude to incident amplitude.

c =Ar

Ai

=Z2 − Z1

Z2 + Z1

. (10)

Here Z is the acoustic impedance of a layer defined as product of density and velocity.

Z = ρv (11)

Most sedimentary interfaces (sand / shale etc) have a relatively small reflection co-efficient (< 0.1), implying that the amplitude of the reflected wave is < 10% of theincident-wave amplitude. More effective reflecting interfaces include unconformities,the water bottom, and coal. These can have reflection coefficients of magnitude 0.3-0.5.

We can a lso define a transmission coefficient, which describes the ratio of the transmitted-wave amplitude to the incident-wave amplitude.

t =At

Ai

= 1− c =2Z1

Z2 + Z1

. (12)

Reflection polarity

Consider Equation 10. If the second material has higher acoustic impedance (basicallyis ’harder’), Z2 > Z1 and c is positive. Conversely, if Z2 < Z1, then c is negative.Ultimately this controls whether the recorded pulse appears as a positive voltage ornegative voltage. Figure 10 gives a simple example for land recording on a geophone.

(Note that for marine recording on a hydrophone the voltages would be the oppositeto that on a geophone.)



Figure 10: Consider a compressional source (explosion, hammer) which generatesa compressive pulse (push). The sketch shows the two cases for reflection from aninterface having positive c (left) and negative c (right). In each case the resultantreflection polarity is indicated at the bottom. Recall that a geophone gives positivevoltage if ’pushed’ from above, and negative voltage if pushed from below.

Non-normal incidence

Equations 10 and 12 in theory hold for normal incidence. In practice they are generallyusable for angles up to about 20-30 ◦. Rigorous analysis of non-normal incidence isachieved using the Zoeppritz equations (equivalent to Knott equations). A commonlyused approximation is the Shuey approximation which is accurate up to about 40 ◦.

The variation of reflectivity with incident angle (i.e. with offset) is the basis of theconcept of AVO (Amplitude vs offset) which has been widely used in exploration.

7 Physical basis of seismic refraction

As indicated in Figure 9 the rays exploited in seismic refraction are formed by criticalrefraction at the interface. It is not intuitively obvious why a ray should then returnupward to the geophone. In fact, from the point at which the critical ray strikes theboundary, a set of new waves termed head waves arise, and these travel back to the sur-face also at the critical angle (Figure 11). The mechanism for these waves is interestingand is further explained in a separate note (headwaves.pdf ).

Figure 11: The seismic refraction method is based on the generation of a critically-refracted wave which skims along an interface. In turn, this results in head waveswhich travel back to the surface, also at the critical angle.



8 Seismic refraction amplitudes

The so-called head-wave coefficient provides an indication of the amplitude of the refrac-tion arrival. It is a complex expression involving densities, P velocities, and S velocitiesin the layers on either side of the interface.

It is interesting that head-wave coefficients are larger for low-contrast interfaces (i.e. asv1/v2 approaches unity. Intuitively this makes sense when ray orientations are consid-ered (Figure 12)

Finally, note that for a compressional source, the headwave will always arrive as acompression at the base of the geophone, causing a negative-voltage. This often helpsidentification of the refraction arrival time.

Figure 12: In high-contrast situations (left) the head wave has a very different ori-entation to the critically refracted ray, resulting in smaller amplitude transfer. Con-versely, for low-contract situations, orientations are similar, and amplitude transferis greater.


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