Lee M. Liberty Research Professor Boise State University

 Oct – 2 or 3 per day  15-minute oral presentation  Include: history of topic, theory, approach to addressing/solving topic, relevance to industry/society Topic? Andrew: seismic monitoring of volcanoes Tate: glacial monitoring using seismology Aiada: Seismoelectric/electroseismic (Hinz dissertation) Travis: Fat ray path/WET inversions Marlon: Nuclear seismology Will: Episodic Tremor and Slip Dmitri: Deep sea subsalt imaging (Brazil)

calculate the depth to your refracted boundary cross over distance, the critical distance and the intercept time Use the Redpath, 1973 reference

 Snell's law defines the critical angle of incidence, a, by:

 If we now let X = 0, then T becomes the intercept time, T i, and we can rewrite the last expression as:  or T i =

Minimum distance where refractions are generated

 At the crossover distance x cross the travel times to the point are the same for the direct wave and the refracted wave, so we have

velocity and amplitude

Shot record showing relation between refraction/reflecti on Direct wave Refracted wave reflections Surface waves Reflections are asymptotic to refracted/direct arrivals

 The travel time for the reflected wave for a 2-layer model can be derived as follows. Start with the ray path and the knowledge of Snell’s law we have SAR=v 1 *t reflect SA= A v 1 *t reflect 2 A

 Average velocity is simply twice the depth divided by the two- way, vertical travel time:  V ave = Z 2Z T0T0 T0T0 But, we usually want to estimate depth (Z)

Assuming straight ray paths: V av = average velocity above the interface Z= depth from the surface to the interface t= one-way travel from surface to interface But, we do not know V i or the ray paths

Different type of seismic velocities 1- Stacking velocity: It is a velocity that gives best stack of the seismic signals, usually drive from the following equation: 2- Root mean square velocity: If subsurface made of horizontal layers with interval velocities v 1,v 2,v 3,… and one way time t 1, t 2, t 3,…, VRMS gives by the following formula:

t1t1 t2t2 t3t3 Surface Note: 1- RMS velocity is usually higher than average velocity by approximately %5. 2- RMS velocity map gives a first indication of velocity variations. 3- Average Velocity: It is usually obtained from well velocity surveys. t1t1 t2t2 t3t3 Surface Z1Z1 Z2Z2 Z3Z3

4- Interval Velocity: It is average velocity within a certain bed or layer, usually obtained from sonic log survey. t1t1 t2t2 t3t3 Surface Z1Z1 Z2Z2 Z3Z3 Layer 2 Layer 3 ……….For layer 2 ……….For layer 3 5- Instantaneous Velocity: It is average velocity at a certain point within a layer or a geological formation. Surface Layer 2 Layer 3 Layer 1

But, because of Snell’s Law: V RMS = root mean square velocity V i = seismic velocity of the i th layer t i = two-way travel time within the i th layer

For n=2 V 1 = Stacking velocity for reflection from upper interface [V RMS (1)] V 2 = Stacking velocity for reflection from lower interface [V RMS (2)] Dix, 1955 (p.130 of Sheriff and Geldart, 1995) V int =ΔZ/ Δt V int = average velocity between two interfaces ΔZ= thickness of layer between the two interfaces Δt= one-way travel time between the two interfaces

Introduction to Geophysics-KFUPM

 Stacking velocity is the velocity obtained by taking the best-fit hyperbola through a reflector (not necessarily through T 0 ), assuming a constant-velocity model.  T 2 =T The stacking velocity is determined by computer velocity analyses, and is used to correct the CDP data for normal moveout (NMO). x2x2 v stack 2

 For flat layers that are “well-behaved” (only gradual velocity changes):  v stack ≈ v rms ≈ v ave  >>generally within 3%, nearly always within 5%

t 2,v 2 t 3,v 3 t 4,v 4 t 5,v 5 t 1,v 1 v stack

Shot record showing relation between refraction/reflecti on Direct wave Refracted wave reflections Surface waves Reflections are asymptotic to refracted/direct arrivals Amplitudes reflect density and velocity changes What about amplitudes?

Waves spread in all directions to form a sphere with a surface area of 4  R 2 E = 1/ (4  R 1 2 ) The amplitude equals the square root of the energy density, so the ratio is: A 1 /A 0 = sqrt(4  R 0 2 )/sqrt(4  R 1 2 ) A 1 /A 0 = R 0 /R 1 If we assume R 0 is a unit radius, the amplitude at radius R is A 1 =A 0 /R R A0A0 A1A1

 Q = dissipation (attenuation) constant or seismic quality factor

 Amplitude also is lost due to friction as the wave moves through the material.  A 1 = A 0 e -  f t/Q  f = frequency, t = traveltime  Q = dissipation (attenuation) constant or seismic quality factor (the fraction of energy lost per cycle) Q is dispersive because the rate of attenuation increases with frequencydispersive  Soils, sediment: Q ~ 20 to 100  Sedimentary rock: Q ~  Crystalline rock: Q ~

Higher freqs are attenuated with depth. Time (seconds)

 Automatic gain control (equalize running average) Use only if you do not care about reflection amplitude  Inverse correction A = At 2

Correcting the signal for spherical spreading and attenuation will cause the noise to also increase.

A) A compressional wave, incident upon an interface at an oblique angle, is split into four phases: P and S waves reflected back into the original medium; P and S waves refracted into other medium. Reflected/Refracted Waves For a wave traveling from material of velocity V 1 into velocity V 2 material, ray paths are refracted according to Snell’s law. i 1 = angle of incidence i 2 = angle of refraction angular dependence

 A wavefront is a surface that joins all the points at which motion is just beginning.  A ray, or ray path, is a line perpendicular to the wavefront.  The ray shows the direction of wave propagation at that portion of the wavefront.

wavefront (t 0 ) wavefront (t1) ray Every point on a wavefront is the source of a new wave that travels outward in spherical shells. The location of a wavefront at t 1 can be determined by extrapolating the wave at time t 0 forward.

Incident wave, velocity v 1 Refracted wave, velocity v 2 v2 < v1v2 < v1 sin θ 1 = BC/AC sin θ 2 = AD/AC sin θ 1 / sin θ 2 = BC/AD but: BC = v 1  t and AD = v 2  t Snell’s Law: sin θ 1 / sin θ 2 = v 1 /v 2 or sin θ 1 / v 1 = sin θ 2 /v 2 Reflected wave

sin θ 1 / sin θ 2 = v 1 /v 2 v2 > v1v2 > v1 θ1θ1 θ1θ1 θ2θ2

v2 > v1v2 > v1 θ1θ1 θ2θ2 Critical angle: θ 2 = 90

 Seismic waves are mechanical perturbations that travel in the Earth at a speed governed by the acoustic impedance of the medium in which they are travelling. The acoustic (or seismic) impedance, Z, is defined by the equation: Seismic wavesacoustic impedance where V is the seismic wave velocity and ρ is the rock density.wave velocitydensity  When a seismic wave travelling through the Earth encounters an interface between two materials with different acoustic impedances, some of the wave energy will reflect off the interface and some will refract through the interface.reflectrefract

 For normal (vertical) incidence, there is no mode conversion, and the amplitude of the reflected wave is given by: R =  2 v 2 –  1 v 1  2 v 2 +  1 v 1  v = acoustic impedance (density x velocity)

 What is a typical reflection coefficient? Vp=2400 ρ=2.4 Vp=2600 ρ=2.5 R =  2 v 2 –  1 v 1  2 v 2 +  1 v 1 R=0.06=6% (amplitude) T=??

 What is a typical reflection coefficient? Vp=2400 ρ=2.4 Vp=2600 ρ=2.5 R =  2 v 2 –  1 v 1  2 v 2 +  1 v 1 R=0.06=6% (amplitude) Vp=2600 ρ=2.5 Vp=2400 ρ=2.4 R =  2 v 2 –  1 v 1  2 v 2 +  1 v 1 R=-0.06=-6% (amplitude) ρ=density

sin θ 1p / sin θ 2s = v 1p /v 2s v2 > v1v2 > v1 θ 1p θ 1s θ 2s P S P P S

 When a wave reaches a boundary with another solid of a different density, it can either reflect off of the boundary at some angle, or pass through it.  Snell’s Law tells us that the reflected or transmitted angle will change  The Zoeppritz Equations describe how likely a wave traveling through the earth is to be reflected at a boundary between two different layers of earth, or to be bent when passing from one layer of earth to another.  The expressions for the reflection and transmission coefficients are found by applying appropriate boundary conditions to the wave equation. The resulting formulas are known as the Zoeppritz equations.

The values for reflection coefficients are determined by the angle of incidence, and by the density (  ) and wave velocities (v) for each layer. The plot shows how the coefficients change with angle of incidence, from  =0, where the wave is traveling perpendicular to the boundary, to  =90, where the wave is parallel (grazing incidence).

 Ray parameter There are 4 equations with 4 unknowns (and six independent elastic parameters) Although they can be solved, they do not give an intuitive understanding for how the reflection amplitudes vary with the rock properties (e.g., density and velocity). The notation used for each coefficient "R P " or "T S “ indicates whether it is a reflection or transmission coefficient, the second letter indicates whether the incident wave is P or S. The sizes of the four coefficients R P, R S, T P, and T S are related to how the energy of a P-wave is distributed when it reaches an interface. The coefficients R S and T S can be appended with (v) or (h). This is because an S-wave can oscillate either in a plane containing a vertical line (v) or one containing a horizontal line (h). Only the former can generate or be derived from P-waves.

 R(θ) = a Δα / α + b Δρ / ρ + c Δβ/β (valid for <30 o angles)  where: a = 1/(2cos2θ), = (1 + tan2θ)/2  b = [(2β2/α2) sin2θ]  c = -(4β2/α2) sin2θ  α = (α1 + α2)/2 = p-wave velocity  β = (β1 + β2)/2 = s-wave velocity  ρ = (ρ1 + ρ2)/2 = density  Δα = α2 - α1  Δβ = β2 - β1  Δρ = ρ2 - ρ1  θ = (θi + θt)/2, where θt = arcsin[(α2/α1) sinθi]  Aki, Richards and Frasier approximation: written as three terms, the first involving P-wave velocity, the second involving density, and the third involving S-wave velocity.

 Look at Shuey (1985) “A simplification of the Zoeppritz equations”