2.1 Introduction to waves

Transcription

1 Seismic waves

2 2.1 Introduction to waves A wave: is a periodic disturbance transmits energy through a material no permanent deformation Seismic waves: transmit elastic strain energy (stretching, tearing, bending, distortion across some surface ) generated by a seismic source (explosion, vibroseis, earthquake )

3 View 1: Snapshot of displacement v = velocity (m/s) à speed at which the wave travels A = amplitude (m) à maximum displacement from rest position λ = wavelength (m) à distance between two points with same phase (e.g., peaks, troughs)

4 View 2: Plot particle displacement through time T = period (s) à time for one complete cycle (oscillation) f = frequency (s -1, Hz) à number of cycles per second (=1/T) Angular frequency: ω = 2πf (rad s -1 )

5 View 1 View 2 20 m 20m The two views are complementary: v=d/t and v=fλ Remark: Seismic waves are sensitive to structure of similar wavelengths or greater. Question: Given a simple structure (v) and survey wave frequency, do you know the rough resolution of a survey?

6 Seismic wave propagation In seismic exploration, we determine how seismic waves have travelled from the seismic source to the detector. à changes in velocity, propagation direction, and wave amplitude indicate changes in subsurface geology. wavefront = locus of points with the same phase (expands spherically) ray = vector showing direction of travel of one point (perpendicular to wavefront)

7 Huygens principle: each point on wavefront acts as a point source spherical waves radiate outward from each point source envelope of waves is the overall wavefront Fermat s Principle: rays propagate along the path which yields the smallest travel time (principle of least time) Constant velocity

8 Hitchhike on a highway during a summer day Question: Why is your view so fuzzy on a hot summer day?

9 Types of seismic waves Body waves: travel through the interior of the Earth P-waves compressional waves (longitudinal, primary) particle motion is in the direction of propagation (e.g., sound waves) fastest seismic waves (V P ) ( see also Fig 2.7 in textbook)

10 S-waves shear waves (transverse, secondary) particle motion is perpendicular to the direction of propagation two polarizations: SH-waves and SV-waves velocity (V S ) is less than V P ( see also Fig 2.8 in textbook)

11 Surface waves: travel along interfaces, such as the ground Rayleigh waves ( ground roll ) coupling between P and S waves at an interface elliptical retrograde particle motion in vertical plane amplitude decreases exponentially velocity (V R ) is lower than V S ( see also Fig 2.10a in textbook)

12 Rayleigh waves are dispersive (V R depends on frequency) à lower frequency Rayleigh waves have a higher velocity because they extend to greater depths Seismic recordings of the collapse of the World Trade Centre (Kim et al., EOS, 2001; ~mwest/papers/ WTC_LDEO_KIM.pdf)

13 . Dispersion and frequency-dependent depth sensitivity

14 Love waves generated when a near-surface layer has a lower V S than underlying layer trapped shear waves with velocity intermediate between V S of two layers horizontal particle motion perpendicular to direction of travel dispersive ( see also Fig 2.10b in textbook)

15 seismic source releases energy as a wave as it passes through a material, it exerts stress causes small deformation (strain) strain is elastic not permanent Seismic velocities of rocks Stress = force per unit area (pressure) N/m 2 or Pa Strain = fractional change in shape dimensionless Hooke s Law: strain is proportional to the stress that produced it (linear elasticity) For a spring: F = k Δx where k is the spring constant (material property) à k governs how much stress is needed to produce a given strain

16 For seismic waves: stress-strain relationship given by the elastic parameters (moduli) of the rocks measure of strength/rigidity of the rocks force needed to shorten by ΔL: long. stress (F/A) long. strain (ΔL/L) volume change under 3D pressure: volume stress (P) ψ= K= µ= 4 3 Can show: ψ = K + µ vol. change (ΔV/V) shear strain due to shearing: shear stress (τ) shear strain (tan θ) stronger material is harder to deform à larger ψ, K, and µ

17 Longitudinal modulus (ψ): P-wave: P 3 4 K V " # $ $ $ $ % & ρ µ + = " # $ % & ρ ψ = µ + ψ = 3 4 K F a longitudinal wave (another: sound wave)

18 P-waves " K + 4 V P = 3 µ % $ ' $ ' $ ρ ' # & 1 2 S-waves " V S = µ % $ ' # ρ & 1 2 V P is always larger than V S stronger material à larger ψ, K, and µ à larger V P and V S V P and V S decrease as density increases V P and V S do not depend on frequency (non-dispersive) in fluids, µ=0 (and K>0) à only P-waves travel through fluids (V S =0) Question: Why does velocity increase with depth?

19 Poisson s Ratio (σ) σ = lateral strain / longitudinal strain can show: V V P S = & $ % 2( 1 σ) ( 1 2σ) in consolidated crustal rocks, σ ~0.25 à V P /V S ~1.7 Poisson s ratio is strongly affected by the presence of fluids, possibly age. # " 1 2

20 Global Observations mafic rocks: > 0.25 felsic rocks: < 0.25 Zandt and Ammon, 1995, Nature Mafic is used for silicate minerals, magmas, and rocks which are relatively high in the heavier elements, magnesium, iron, sodium, calcium. Felsic rocks is used for silicate minerals, magmas, and rocks which have a lower percentage of the heavier elements, and are correspondingly enriched in the lighter elements, such as silicon and oxygen, aluminum, and potassium )20

21 Factors that control V P of rocks (m/s) Air 340 Water Petroleum Sand (unsaturated) Sandstones Tertiary Carboniferous Limestones Cretaceous Jurassic Carboniferous Salt Granite Basalt Ultramafic rocks density and elastic moduli depend primarily on composition V P increases as rocks become more mafic (granite-gabbroultramafic) V P = & $ % ψ ρ # " 1 2 & $ K = $ $ % ρ # µ " 1 2

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23 Factors that control V P of rocks (m/s) Air 340 Water Petroleum Sand (unsaturated) Sandstones Tertiary Carboniferous Limestones Cretaceous Jurassic Carboniferous Salt Granite Basalt Ultramafic rocks sedimentary rocks tend to have lower V P à depends mostly on the porosity of the rocks V P increases with age à increase in rigidity with cementation

24 Effect of porosity on velocity rock is composed of rock matrix and pore space (often filled with fluid/air that has low velocity) porosity (Φ) is the fractional volume occupied by pore space velocity decreases with increasing Φ (Sherriff and Geldart, 1995)

25 Effect of porosity on velocity the overall rock properties are the average of the matrix (m) and pore fluid (f) properties, weighted by Φ EXAMPLE: for density: ρ = ρ f Φ + (1- Φ) ρ m for velocity: weight by the amount of time that seismic wave spends in the rock matrix and pore fluid (v = d/t à t = d/v) time-average equation: P f ( 1 ) 1 Φ = + Φ V V V m )25

26 Effect of depth on seismic velocity seismic velocities generally increase with depth: increased compaction reduces pore space elastic moduli increase with pressure Reconstructing basin history: ( westvanisland/index_e.php) (Sheriff and Geldart, 1995)

27 Example questions 1. A rock has 25% porosity and is filled with water (V P =1500 m/s). If V P is 3200 m/s for the rock matrix, what is the P- wave velocity of the rock? time-average equation: P f ( 1 ) 1 Φ = + Φ V V V m

28 Example questions 2. Gas-filled sandstone: Sandstone matrix 4300 m/s Pore fluid 300 m/s If the overall V P is 2200 m/s, what is the porosity? time-average equation: P f ( 1 ) 1 Φ = + Φ V V V m

29 Factors that affect seismic wave amplitude Seismic exploration: Travel time & amplitude of seismic waves Subsurface velocity structure Subsurface geology Three factors that cause seismic wave amplitudes to decrease between source and receiver: 1. Geometrical spreading 2. Multipathing 3. Scattering 4. Intrinsic attenuation (AND as waves pass into different materials, partitioning of energy at the interface affects amplitude...sections 4.1 and 4.2)

30 Imagine a light source (seismic source): Intrinsic attenuation: loss of energy to heat due to anelasticity (e.g., friction near grain boundaries) Geometrical spreading due to growing wave front surface area slow Scattering when wavelength ~ particle dimension fast Multipathing due to high & low velocity bodies (ala Fermat)

31 1. Geometrical spreading (spherical divergence) Seismic energy (E 0 ) spreads out from a point source (e.g., explosion) as a spherical wavefront the wavefront always contains a constant amount of energy (E 0 ) the energy at a point on the sphere surface is: E(r) = E 0 / 4πr 2 (where r is the distance from the source) amplitude of a seismic wave is related to energy: E(r) α A 2 since E(r) α 1/r 2 à A 2 α 1/r 2 à A α 1/r à in order to conserve energy, seismic wave amplitude must decrease by 1/r

32 Body wave propagation One person s noise is another person s signal. This is certainly true for what surface waves mean to an exploration geophysicist and to a global seismologist Amplitude decay in surface waves (as a function of r) is less than that of body wave --- (by square root of r), the main reason that we always find larger surface waves than body waves, especially at long distances. Surface Wave Propagation 32

33 2. Attenuation (absorption) seismic wave propagation is not completely elastic small fraction of seismic energy is converted to frictional heat loss of energy = decrease in amplitude characterize this with the quality factor (Q): Q Q is a measure of the fractional loss of energy per cycle (oscillation) of the seismic wave high Q mean little energy loss = 2πE Δ E

34 can show that the loss of energy will cause an exponential decrease in seismic wave amplitude over time (t): (e is the exponential constant 2.718) note that this is a function of frequency higher frequency = more cycles per second = more energy lost can also write in terms of distance traveled by wave (x): A(t) A(x) = = A 0 e A 0 e πft à amplitude decays more rapidly for: low Q or high f (short λ) Q πx Qλ (Material) (Seismic wave)

35 Example of attenuation:

36 Common approach to calculate Q Interpretation 1: Suppose A 0 represents wave amplitude, then A = A 0 e bt = A 0 e ω 0 t /(2Q ) $ ln(a) = ln(a 0 ) ω0 ' & ) t % 2Q( ln(a) intercept slope t Star Track The Q )36

37 The Earth is more attenuating than the Moon. Problem: where are P, S, Surf. waves? )37

38 seismic sources release energy over a wide range of frequencies (Kearey et al., 2002) as wave travels, the high frequency components will be more strongly attenuated over time, wave is dominated by low frequency (long wavelength) components à seismic wave will become smoother and more spread out over time (similar to thunder) for crustal rocks, Q is typically Q P > Q S à energy loss caused mostly by shear deformation

39 3. Scattering most materials contain small heterogeneities grains, mineral boundaries, pore edges, cracks, etc. some seismic energy is scattered when it encounters these features therefore amplitude will decrease

40 Wavelength effect demonstrated for P wave coda. People use source spectrum to analyze the coda and obtain information about Q and scatters about a given path