EOSC 353

Traction Vector t

• Describes internal forces inside a solid on ANY plane orientation

• Traction tells you how the material on one side of a plane pushes/pulls on the other side

• Cauchy’s formula: t(n) = Tau * n

  • also known as stress

    • normal stress: t_N = t (dot) n

    • shear stress: t_S = t - t_N * n

Stress Tensor Tau_ij

• Maps a plane’s normal vector n to the traction vector t

• Defines tractions on the coordinate planes (xy, yz, xz)

• Acts on n to produce t

• Encodes all internal forces at a point in a solid

• Stress tensor is symmetric Tau_ij = Tau_ji

  • Static equilibrium

  • 6 independent components

Principle Axes of Stress

• det(Tau - lambda * I) = 0

• The three lambdas are the max, mid, and min normal stresses

• Finds directions where traction is purely normal (no shear)

• Simplifies the stress state and reveals max/min compression

• The natural axes of the stress field

• Max shear stress: Tau_max^S = (lambda₁ - lambda₃)/2

  • max shear stress on planes 45 deg between principal axes

Deviatoric Stress Tensor

• Tau_D = Tau - Tau_M

  • Causes shape change and faulting

• Tau_M = tr(Tau)/3

  • Also known as lithostatic or hydrostatic pressure

J Tensor (symmetric and antisymmetric)

• J_ij = du_i/dx_j = e_ij + omega_ij

• Splits deformation into shape change and rigid rotation

• e = strain tensor (symmetric)

• omega = rotation tensor (antisymmetric)

Strain Tensor e

• e_ij = ½ (du_i/dx_j + du_j/dx_i)

• Measures shape change (deformation)

• e_11,22,33 = normal strain

• e_ij, where i ≠ j is shear strain

Rotation Tensor omega

• omega_ij = ½ (du_i/dx_j + du_j/dx_i)

• Represents rigid rotation of material

• omega_11,22,33 = 0

• Since omega is antisymmetric, omega_ij = -omega_ji

• nabla x u = omega

  • Rigid rotation occurs if: nabla x u ≠ 0

Dilatation

• Volume change

• If tr(e) > 0, expansion

• If tr(e) < 0, compression

Linear Stress-Strain Law (Hooke’s Law)

• Tau_ij = C_ijkl * e_kl

• Relates stress to strain through the elastic tensor C

• Describes how elastic materials deform under load

• Most general constitutive law for elastic solids

  • Due to symmetry of stress, strain, and thermodynamic → 21 independent components

  • Meant for most general solids: anisotropic

C for Isotropic Solids

• If material behaves the same in all directions (isotropic) → 2 components

• C_ijkl = lambda delta_ij * delta_kl + mu(delta_ik delta_jl + delta_il delta_jk)

  • Reduces to two Lame parameters: lambda and mu (shear modulus)

Hooke’s Law for isotropic solids

• Sub isotropic C into general law

• Tau_ij = lambda e_kk delta_ij + 2mu e_ij

• First term: volumetric stress (compression/expansion)

• Second term: shear stress

Lame Param mu (shear modulus)

• mu = Tau_xy/(2e_xy)

• Measures resistance to shear deformation

• Shear rigidity controls S-wave speed

• Large mu → stiff material

Lame Parameter lambda

• lambda = K - 2/3 mu

• Relates lambda to bulk modulus (K) and shear modulus

• Controls how pressure produces volume change

Young’s Modulus E

• E = mu (3lambda + 2mu)/(lambda + mu)

• Higher E → harder to stretch

Bulk Modulus K

• K = lambda + 2/3 mu

• Large K → nearly incompressible material

Poisson’s Ratio v

• v = lambda/(2(lambda + mu))

• v = 0.5 → fluid (no shear)

P-wave velocity alpha

• alpha = sqrt((lambda + 2mu)/rho)

• Gives P-wave velocity

• Faster than S-waves because: lambda + 2mu > mu

S-wave velocity beta

• beta = sqrt(mu/rho)

• No mu → no S-waves (fluids)

1-D Wave Equation

• d²u/dt² = c² d²u/dx²

• Simplest wave equation in 1D with speed c

Momentum Equation

• rho d²u/dt² = dTau_ij/dx_j + f_i

• Newton’s law for a continuous elastic medium

• Ignore body forces f_i for seismology

The Homogeneous Seismic Wave Equation

• Sub isotropic stress-strain law into the momentum equation

• rho du²/dt² = (lambda + 2mu) nabla(nabla dot u) - mu nabla x (nabla x u)

• The standard wave equation for a uniform elastic solid

• Take divergence of wave equation → P-wave velocity alpha

  • nabla dot u ≠ 0 → volume change

• Take curl (nabla x u) of wave equation → S-wave velocity beta

  • nabla dot (nabla x u) = 0

  • nabla x u = 0 → irrotational

Spherical Wave

• phi(r,t) = f(t-r/alpha)/r

• P-wave potential

• Decays as 1/r due to geometrical spreading

Snell’s Law

• p = u sin(theta) = sin(theta)/v

  • p is the ray parameter (horizontal slowness)

  • u = 1/v (slowness)

• p = u₁sin(theta₁) = u₂sin(theta₂)

  • Rays bend toward slower velocities

  • Relates incidence and transmission angles across a horizontal interface

Turning point Condition

• Theta = 90 deg → p = u = 1/v

• The depth where the ray becomes horizontal

  • Ray cannot go deeper

  • Ray curves downward, then upward, forming ray paths

Vertical Slowness eta

• eta = u cos(theta) = sqrt(u²-p²)

• vertical component of slowness vector

• eta = 0 at turning point

Horizontal Slowness p

• p = u sin(theta)

• at turning point, p = u sin(90) = u

Ray Geometry Relations

• dx/ds = sin(theta)

• dz/ds = cos(theta)

• dx/dz = p/sqrt(u²-p²)

Horizontal Distance Increment

• dx/dz = p/sqrt(u²-p²)

  • Horizontal step per vertical step

  • Integrate to find the full surface-to-surface distance

• x(p) = p \integral^z_tp_0 dz/sqrt(u²-p²)

  • Distance from surface to turning point

Total Surface-to-surface Distance

X(p) = 2p \int^{z_tp}_{0} dz/sqrt(u²-p²)

• Gives total epicentral distance for a ray with parameter p

• Doubles the one way distance

Total Travel Time

T(p) = 2 \int^{z_tp}_{0} u dz/sqrt(u²-p²)

• Gives total travel time for a ray with parameter p

Also given as total travel time as vertical + horizontal:

T(p) = Tau(p) + pX(p)

Layered Model Summation

X(p) = 2p summation_i (delta z_i/sqrt(u²_i - p²))

• Computes X(p) for a stack of homogeneous layers

T(p) = 2 summation_i (u_i delta z_i/sqrt(u²_i - p²))

• Computes travel time for layered models

Delay Time Function

Tau(p) = 2 summation_i (delta z_i sqrt(u²_i - p²))

• The depth-integrate vertical slowness

• Separates vertical travel from horizontal travel

• Unravels triplications

Slope of Tau(p) curve

dTau/dp = -X(p)

• Differentiate T(p) = Tau(p) + pX(p) and use dT/dp = 0

• Tau(p) always decreases with p

If dX/dp < 0: (concave up)

• Prograde: range increases as p decreases

If dX/dp > 0: (concave down

• Retrograde: range decreases as p decreases

  • Rapid velocity increase causes triplications in T(X)

  • 2 caustics form where dX/dp = 0

Final graphing knowledge

• Slope dT/dX = p

• each slope corresponds to velocity at the rays turning depth

• slopes → layer velocities

• intercepts → layer thicknesses

• Produces layer cake model

• For discrete layered models:

  • Tau(p_j) = 2 summation_i (h_i sqrt(u²_i - p_j²))

    • Can write as:

      • Tau(p₁) = 2 h₁ sqrt(u₁² - p₁²)

      • Tau(p₂) = 2 h₁ sqrt(u₁² - p₂²) + 2 h₂ sqrt(u₂² - p₂²)

Earthquake Location methods

• Grid Search: Global search, robust to bad initial searches, can handle nonlinearity

  • computationally expensive and limited by grid spacing, impractical

• Iterative Linear Inversion: fast, works well when initial guess is close, and easy to incorporate corrections

  • if starting point is bad, solution can be wrong, and requires derivatives

Uncertainty Estimation

Chi² = summation_{i=1 ⁽}t_i - t_i^p)²/std dev²_i

• Used to assess goodness of fit for earthquake location

• 95% confidence is acceptable

Total Energy Density

~E = ~E_K + ~E_W

• Total energy density for a seismic wave

• Splits wave energy into kinetic and strain (potential) energy

Kinetic Energy Density

E_K = ½ rho u²

• Kinetic energy per unit volume

• faster particle motion → more energy

Strain (potential) energy density

E_W = ½ Tau_ij e_ij

• Elastic potential energy stored in deformation

• More distortion → more stored energy

Energy in a Harmonic Wave

E_K = E_W = ¼ rho A² omega²

• Shows kinetic and strain energy are equal on average

• Harmonic waves exchange energy between motion and strain

• So Total Energy:

  • E = E_K + E_W = ½ rho A² omega²

Energy Flux

E^flux = cE = ½ c rho A² omega²

• c = alpha or beta depending on P or S-wave

• Gives energy transported per unit are per unit time

Geometrical Spreading

• Total energy flux through a ray tube must remain constant

E^flux(t₁) = E^flux(t₂)

• thus:

  • A₂/A₁ = sqrt(dS₁/dS₂)

• Relates amplitude to change in wavefront area

• Energy in a ray tube is conserved

• If wavefront expands → Amp decreases, vice versa

For spherical wavefronts:

• A is proportional to 1/r

  • classic 1/r decay of body waves

Impedance Scaling

A₂/A₁ = sqrt(rho₁c₁/rho₂c₂)

• Gives amp change when entering a new medium

• rho c is the impedance

• Lower impedance → larger amps (site amplification)

Energy Density at Surface

E(X) = 1/(4 pi u²₀X) p/cos²(theta₀) dp/dX * E_S

• Gives amp of a ray arrival in a 1-D model

Downgoing vs. upgoing waves

• Downgoing: u = f(t - px - eta z)

• Upgoing: u = f(t - px + eta z)

SH Reflection/Transmission Coefficient

S_downS_up = rho₁beta₁cos(theta₁) - rho₂beta₂cos(theta₂)/ r b c + r b c

• Gives amplitude ratio of reflected SH wave

• Interface causes partial reflections

• Equal to A(up)₁

S_downS_down = 2rho₁beta₁cos(theta₁)/ r b c + r b c

• Gives amplitude of transmitted SH wave

• Energy splits between layers

• Equal to A(down)₂

The sum of the squares of A_norm for all scattered waves will be 1:

• S_downS_up² + S_downS_down² = 1

Energy Normalized Coefficients

A_norm = A_raw sqrt(rho₂ c₂ cos(theta₂)/ rho₁ c₁ cos(theta₁))

• Normalizes amp so energy flux sums to 1

• Energy must be conserved

Green’s Function Definition

u_i(x,t) = G_ij(x, t ; x₀, t₀) f_j(x₀, t₀)

• Gives displacement at x from a unit force at x₀

• Allows any source to be built from superposition of point forces

• Green’s function = Earth’s “impulse response”

Force Couple (vector dipole)

• 2 equal and opposite forces separated by distance d

• Represents the simplest internal source that conserves momentum

• The building block of the moment tensor

• Equivalent to a small shear displacement

Double Couple

• 2 orthogonal force couples arranged to conserve both linear and angular momentum

• Represents shear slip on a fault

• Earthquakes radiate like double couples

Moment Tensor M_ij

• Encodes all possible force couples at a point

• General mathematical representation of any seismic source

• Describes orientation + strength of equivalent body forces

• Each entry = shear or normal couple in a coordinate direction

• Symmetrical, so M_ij = M_ji

Displacement from Moment Tensor

u_i(x,t) = dG_ij(x,t ; x₀,t₀)/dx_0k M_jk(x₀,t₀)

• Relates moment tensor to observed displacement

• Moment tensor acts as a spatial derivative of point forces

Fault Geometry

• Strike (phi): azimuth of the fault trace

• Dip (delta): angle from horizontal

• Rake (lambda): direction of slip within the fault plane

  • Defines orientation + slip direction of a fault

• Slip magnitude D

• Reverse (compression), normal (extension), strike-slip (horizontal motion)

• Right lateral vs. left lateral defined by relative motion of opposite blocks

Scalar Seismic Moment

M₀ = mu D A

• Measures earthquake size physically

M₀ = ½ sqrt(M_ij²)

• Computs scalar seismic moment from the full tensor

Principal Axes for moment tensor

• Eigenvectors of the moment tensor

• Directions of max compression (P), tension (T), and null (B)

• Orientation of stress release during rupture

Isotropic moment tensor (explosions)

M⁰ = 1/3 tr(M_ij)

• Extracts volume change part of the source

• Represents uniform expansion or contraction

Deviatoric moment tensor

M’ = M - M⁰

• Removes isotropic part, leaving shear-related components

• Pure shape change without volume change

CLVD Component

M^CLVD = [[-lambda/2, 0, 0],[0,lambda,0],[0,0,-lambda/2]]

• Represents compensated linear vector dipole

Complete decomposition of isotropic M

M = M⁰ + M^DC + M^CLVD