6. Refraction Seismic Surveys

Write down/ scetch the law of refraction.

= Snells Law (p.106 sketch)

sin i_1 / sin i_2 = v_1 / v_2

What is critical incidence?

specific angle at which a seismic wave, traveling from a medium with a lower velocity to a medium with a higher velocity, is refracted along the boundary between the two media. This angle is known as the critical angle.

What is critical distance?

- the minimum distance from the seismic source at which the refracted wave first arrives at the surface = point where the refracted wave arrives before the direct wave

- important for depth and velocity analysis

What is a head wave?

- is a plane wave

- travels along the boundary at the higher velocity of the lower layer and re-emerges into the upper layer at a critical angle.

(When a seismic wave encounters a boundary where the lower layer has a higher velocity than the upper layer, it can be refracted along this boundary into -> head wave is the one that travels back upwards the surface .)

What is intercept time?

time at which a refracted seismic wave first arrives at a geophone or seismometer

including the time spent traveling along the boundary between layers

What is delay time?

- delay time δt_SR is the difference between the travel-time t_SR and a hypothetical travel-time along a plane interface at depth

δt_SR = δt_S + δt_R

or

- time it takes for seismic waves to travel through the Earth

Name 4 methods of ray-tracing

- shooting methods

- bending methods

- network solver/ shortest path

- eikonal solver

For arbitory plane wavefront: how is apparent velocity related to ray parameter and true velocity?

p = 1/ v_apparent = sin (alpha)/ v_true

p ... horizontal slowness of the wave

v_apparent ... aparrent velocity

alpha. .. angle of wave incidence

v_true ... true velocity in medium

Name essential condition to record a head wave (with source and receiver on surface of half-space)

long-offset recordings (at least five times of the (expected) interface depth)

=>

Source-receiver distance > Critical distance

Dipping layer case: how does velocity v2 relate to apparent velocities V_u and V_d?

for dipping layer case:

true v_2 is the harmonic mean of up-dip and down-dip apparent velocities!

v_2 = V2_mean * cos(alpha)

V2_mean = 2/ (1/V_u + 1/V_d)

alpha ... incidence angle

eikonal solver method

a solution of the eikonal equation ( ∣∇T(x)∣ = 1/v(x) ) on a grid to compute travel-times.

+ always a global minimum

+ solution includes ray paths to any point in the model

- difficult to compute arbitary seismic phases.

T(x) ... travel time of wave at position x

v(x) ... velocity of the wave at position x

∇T(x) ... gradient of travel time

network solver/shortest path method

Fermats principle of shortest travel time:

algorithm computes the shortest path to any given node in a tree-like network

+ always a solution

+ always a global minimum

+ any point in model reached

- difficult to compute arbitary seismic phases

- inaccurate

bending method

adjusts ray paths to minimize the travel time by bending the ray at each interface between layers.

+ fast and always a solution

- inaccurte and difficult

- not essencially global minimum

- difficult to compute arbitary seismic phases

shooting method

ray tracing from the source to the receiver by iteratively adjusting the initial angle of the ray until it matches the desired travel path.

+ highly accurate+

+ computing of arbitary seismic phases (wave types)

- no global minimum

- target hitting difficulties

critical angle

critical angle (i_c) is given by:

sin⁡(i_c) = v_1\v_2

where v_1 is the velocity of the upper medium and v_2 is the velocity of the lower medium.

a seismic wave that hits the boundary at the critical angle, generates a head wave that travels along the boundary at the higher velocity of the lower medium and then is refracted back upwords with at the same critical angle.

arbitary seismic phases

different seismic wave types

EX: P-waves, S-waves, Surface waves ...