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Seismo 1: Body waves Barbara Romanowicz Ins%tut de Physique du Globe de Paris, Univ. of California, Berkeley Les Houches, 6 Octobre 2014 Seismology • Seismology is the study of the generation and propagation of elastic waves through the Earth • Wave generation: source seismology – Earthquakes – Tremor/micro-seismicity – Explosions/mine collapses – Ocean-solid earth interactions (Earth’s hum..) • Wave propagation: structural seismology – Oil/gas/mineral exploration – Crust, Mantle, Core structure u(t) = S(t) ! P(t) ! I(t) Typically : Z (verFcal) , N, E (2 horizontals oriented orthogonally) VonReubeur Paschwitz 1889 “Reading the report on this earthquake in NATURE (June 13, p. 162), I was struck by its coincidence in Fme with a very singular perturbaFon registered by two delicate horizontal pendulums at the Observatories of Potsdam and Wilhelmshaven.” VonRebeur Paschwitz, E., Nature, 40, July 25, 1889 Global Seismology • Seismology is the most powerful technique to sample and constrain the present day physical structure of the Earth’s interior. – Seismic waves are generated by natural earthquakes and bounce around the earth’s interior – For large enough earthquakes (M~5) they can be observed as far as the antipode of the earthquake • Seismology was essential in defining the 1D structure of the Earth Global seismology • Powerful imaging techniques (seismic tomography) have been developed to infer physical properties inside the earth’s mantle and core from observations of records of natural earthquakes (“seismograms”)– similarly to techniques used in medicine (CAT-scan, IRM..ultrasound..) Surface wave P S SS Body waves Energy travels through the Interior of the earth Rays bend and reflect due to variations of physical properties of the Earth’s interior "-> Snell’s law Two types of body waves: P waves and S waves P waves faster than S waves S waves do not propagate through liquids Surface wave P S SS Surface waves Energy travels along the surface of the earth and decays with depth Dispersive: speed varies with frequency Arrive after the main body waves Contain most of the long period energy radiated by an earthquake Large amplitudes dominate seismograms Amplitude Distance (x) Period = Frequency -‐1 T = 1/f Wavelength = Wavenumber-‐1 λ= 1/k Speed (c): c = λf = f/k 10 From Stein and Wysession, 2003 What are elastic waves? • In an elastic continuum, perturbing the position of one particle will result in a restoring force that will accelerate that particle back and past the equilibrium position – In an elastic medium, stress and strain are linearly related (for small strains, and assuming perfect elasticity) – In 1D, this is the familiar Hooke’s Law: F = -‐ k x The constant k is the spring stiffness • Perturbation in position of one particle will accelerate its neighbors Constitutive relationship • In 1D: – F=-kx • In 3D, x becomes the strain tensor, and F becomes the stress tensor: k=3 l=3 ! ij = cijkl " kl # $$ cijkl " kl k=1 l=1 10/7/14 13 Tractions and the stress tensor • The stress tensor describes forces acting across infinitesimal planes within a continuum (i.e. tractions) • Its units are: N/m2 a.k.a. Pa • Can vary with position, describes both compression / estension and shearing • Stress tensor is symmetric, in 3D has 6 independent components Pressure: direcFon depends on surface 10/7/14 Gravity: direcFon always down 15 Internal deformation • How do we describe (small) internal deformations? 10/7/14 CIDER 2014 -‐ Seismology #1 -‐ Ved Lekic 16 Strain and Rotation tensors • No net rotation only if ∂ui/∂xj – ∂uj/∂xi = 0 • Decompose deformation into Strain and Rotation matrices: Strain tensor is symmetric + Rotation tensor is anti-symmetric • Strain is a dimensionless quantity (ΔL/L ) • Strain due to passage of seismic waves are typically <10-5 10/7/14 1 !ij = (ui, j + u j,i ) 2 17 Elasticity: Linear stress-strain relationship ! ij = cijkl" kl Hooke s law Strains <10-4 • cijkl is the elastic tensor – Reduced from 81 to 21 independent elements (symmetries) • Isotropic material: properties are the same in all directions: – there are only 2 independent elements, λ and µ (Lamé Parameters) ! ij = "#ij$ kk + 2µ$ij Isotropic medium • In an isotropic medium, wave speed depends on NEITHER the direction of propagation NOR polarization direction • Relationship between stress and strain is simple: ! ij = "#ij$ kk + 2µ$ij µ= 10/7/14 ! xy 2" xy • Shear modulus: measure of the resistance of material to shear. • Bulk modulus: measure of incompressibility of a material. Given by hydrostatic stress divided by fractional volume change. • Poisson’s ratio: lateral contraction of a cylinder divided by longitudinal extension. 19 Isotropic medium Described by distribution of (ρ, λ, µ) within the medium or: (ρ, κ, µ) = “elastic parameters” or: (ρ ,α ,β) P velocity: S velocity: λ + 2µ α= ρ µ β= ρ Bulk sound velocity: ! <" κ Vφ = ρ 4 2 Vφ = α − β 3 2 2 Momentum Seismic Wave Equation 2 " ui ! 2 = " j # ij "t • Homogeneous momentum equation governs seismic waves outside of source regions • Summation convention: repeated index ! ij = "# ij$k uk + µ($i u j + $ j ui ) + manipulaFon and a vector idenFty: • Constitutive relation for isotropic medium, where we used 2εjk = (∂uk/∂xj + ∂uk/∂xj) • Seismic Wave Equation for an isotropic medium 10/7/14 21 Seismic Wave Equation Gradient: Vector describing the direction and magnitude of change in a quantity 10/7/14 Divergence: Scalar describing volume change Curl: Vector describing infinitesimal rotation (think paddle wheel) 22 Homogeneous medium • In a homogeneous medium, or in the high-frequency approximation, the gradients of λ and μ can be neglected: • Taking the divergence of the equation (remember ): Compressional wave (Pwave) propagating at speed α • Taking the curl of the equation (remember and ): " Shear wave (S-wave) involving no volume change propagating at speed β 10/7/14 23 Polarisation of P and S waves S waves are shear waves No volume change, shear and rotaFon P waves and S waves have different velociFes P waves are compressional waves cause volume change – compression And rarefacFon No rotaFon Similar to sound waves in air From Stein and Wysession, 2003 Terminology θ Great c ircle Terminology Terminology Shallow earthquake From Stein and Wysession, 2003 Plane waves • SoluFons to the P and S equaFons are waves: u(x,t)= f(x ± vt), v = velocity • Plane waves: s is the slowness vector • ParFcle displacement at a frequency ω: (Figure from Stein and Wysession, 2003) k is the wavenumber vector 10/7/14 30 Wavefronts and raypaths Ray theory • Simple and fast: – Used extensively in earthquake location, focal mechanisms, inversion for structure in crust and mantle • Shortcomings – High frequency approximation: fails at long periods – Does not predict non geometrical effects i.e. diffracted waves, head waves – Limitations in predicting effects of heterogeneity on waveforms Horizontal & vertical slownesses • Slowness: u = 1 / c • Horizontal slowness: p = ΔT/ΔX • Vertical slowness: 10/7/14 • We can measure how quickly a wave sweeps across a small, dense array of seismometers figures from Shearer, 2009 33 Snell’s Law – the ray parameter For the wavefield to remain continuous along interface between regions with different velocities, waves will change direction in addition to their speed. 10/7/14 34 Snell’s Law – the ray parameter For wavefield to remain continuous along interface between regions with different velocities, waves will change direction in addition to their speed. • Snell’s Law: sin !1 sin ! 2 = =p v1 v2 • Ray parameter (horizontal slowness) remains the same along the ray path figure from garnero.asu.edu 10/7/14 35 Layered / shell medium By specifying a ray parameter and a starting location, we fix the ray-path through a layered or spherically-symmetric Earth The deepest depth the ray can reach is one where: sin 90 1 = =p v1 v1 10/7/14 figure from garnero.asu.edu 36 In general, some of the energy is transmitted, some reflected, and, in the P-SV case, some converted SH case P-SV case Angles of reflection/transmission depend only on velocities Amplitudes depend on impedance (ρ v) Travel-time curves figures from Shearer, 2009 • T(X) = τ(p) + pX Ray parameter (horizontal slowness) is the slope of the traveltime curve • Different rays observed at different distances • Shallowing of slope dT/dX with distance slowness 10/7/14 decreases with depth • 38 Ray paths in spherical geometry, when velocity increases with depth Seismic wavefield P waves S waves Seismic waves take a variety of paths and sample all parts of the Earth They provide most complete and direct information on presentday state of the deep interior 10/7/14 40 Wavefronts and raypaths 10/7/14 41 Triplications figures from Shearer, 2009 • Depth ranges in which velocity increases rapidly with depth can produce triplications: multiple rays (with different horizontal slownesses) arrive at the same time • Transition zone discontinuities produce triplications • At epicentral distances > 30° rays bottom in the lower mantle and no triplications are 10/7/14 (typically) observed 42 Steep gradients, such as upper-mantle discontinuities, create triplications Shadow zones Rays cannot bottom in a depth range where velocity decreases with depth* Low-velocity zones produce shadow zones Prominent shadow zone in PKP waves due to low velocities in the outer core 10/7/14 figures from Shearer, 2009 44 Low velocity layers create shadow zones Naming body waves P or S means K means I or J P or S transmission thru mantle P transmission thru outer core means P or S transmission thru inner core i = reflection off inner core c = reflection off CMB Basics: paths Basics: paths Example: ParFcle moFon plots for SKS+SKKS and Sdiff From Stein and Wysession, 2003 Basics: paths Basics: paths Basics: paths Basics: paths Basics: paths Basics: paths PredicFons from IASP91 model Shearer, 1996 • Densité moyenne de la terre ρ = 5.515 kg/m3 • Densité des roches en surface ~ 3 kg/m3 – => densité augmente avec la profondeur – => La terre ne peut être enFèrement composée de roches – Emil Wiechert (1898) suggère que la terre est comme un météorite géant avec un noyau de fer-‐nickel qui, plus lourd que les roches s’est enfoncé au centre de la terre par un processus similaire à la séparaFon du fer de sa gangue dans une enclume. D’où l’idée d’une terre” à deux couche”s avec un noyau de fer/nickel entouré d’une couche de roche. The1910 earth La terre circa as seen ~1910 Proof of the existence of a core: Oldham (1906) Richard Oldham 1858-1936 First director of the Seismological Service of India ? ? S P Radius of the core estimated by Oldham: ~2600km Oldham, 1906 Liquid state of the core • Suggested for a long time from the study of earth’s solid tides, which are significantly larger than what would be expected if the earth was rigid throughout • Confirmed by the absence of observations of S waves beyond the distance of penetration into the core • Jeffreys (1926) combined tide data and S wave data to construct the first proof of lack of rigidity in the core Sir Harold Jeffreys 1891-1989 Mean rigidity of the earth (from solid tides) is lower than that of the mantle calculated from S wave travel times) “The rigidity of the Earth’s central core”, Geophys. J. (1926) P wave shadow zone Shadow zones due to the presence of the liquid core Discovery of the inner core: 1936 Observes “P-like” phases in the shadow of the core Calls the new seismic phase P’ Writes a paper whose title is the shortest ever: “ P’ ” Inge Lehmann 1888-1993 ->Existence of a region at the center of the core with different elastic properties than the external part of the core.