Seismic Wave Propagation

LL F L F = k *  L/L (Hooke’s Law) k = Young’s modulus Rand quartzite strain Elastic Materials

Acoustic Waves F = Mass*Acceleration (Newton rules!) u u+ du x x + dx A F(x) F (x+dx) Net force = F(x+dx) – F(x) =  F = [  *A*dx] *acc  F/  x =  * acc * A] but F = k *  u/  x (Hookes law) and S = F/A, so K *  u/  x  x =  *  u 2 /  t 2 i.e. wave equation where V= (k  ) 1/2

Seismic Wave Equation k *  u/  x  x =  *  u 2 /  t 2 or d 2 u/dx 2 = (  /k) du 2 /dt 2 which has solutions of the form u = A e +ikt + Be -ikt

Seismic Waves Body waves Surface Waves “Ground Roll”Rayleigh Love P S

Rayleigh Waves

Seismic Wave Speeds MaterialP wave Velocity (m/s)S wave Velocity (m/s) Air332 Water Petroleum Steel Concrete Granite Basalt Sandstone Limestone Sand (Unsaturated) Sand (Saturated) Clay Glacial Till (Saturated)

Seismic Velocities

Wave Simulations Colorado School of Mines Tutorial Colorado School of Mines Tutorial Body waves in half space

Huygen’s Principle

Snell’s Law* (acoustic) *Fermat’s Least Time Principle

Snell’s Law (elastic) sin  1 = sin  2 = sin  3 Vp 1 Vp 2 Vs 3

Critical Refraction sin i c = sin (90º) V 1 V 2 sin i c = V 1 V 2

Seismic Refraction Waves passing from slow to fast medium

Snell’s Law

Reflection & Refraction T, msec Direct Reflected Refracted Distance, km

Refraction: Two Layers

Engineering Quad 1/27/00 12 gauge source 1m geophone spacing, 1 m minimum offset Travel Time Curve (Shotpoint Gather

Refraction: 3 layer case

Refraction: Dipping Layer

Refraction: Hidden Layers

Refraction: Lateral Offset

Refraction: Many Layers

Refraction: Multiple Layers

Refraction: Linear V(z)

Refraction: + Gradients V z z x x t

Refraction: - Gradients V zz t x x LVZ Low velocity zone

Diffraction

Reflection: Basic Geometry T 2 = X 2 + ( 2Z/V) 2 V 2 T 2 = X 2 + T 0 2 V 2

Reflection Hyperbola

Reflection from Dipping Layer

SHOT * Reflection display convention

Sample shot gathers offend split spread

Seismic Waves Body waves Surface Waves “Ground Roll”Rayleigh Love P S

Rayleigh Waves

Ground Roll (Rayleigh Waves) x t Dispersion Ewing, Jardetzky and Press (1957)