Lee M. Liberty Associate Research Professor Boise State University

     Dates  Mar 17, 19, 21 15-20 minute oral presentation Metrics: presentation, style (professionalism), organization, accuracy, references Include: history of topic, theory, approach to addressing/solving topic, relevance to industry/society, how do we measure/acquire needed data, how do we process the observed data ◦ ◦ ◦ ◦ ◦ Topics: ◦ Earthquake amplification - Site response method comparisons (Gribler) ◦ Earthquake stress fields (Terbush) Beam forming (Demonte) Prestack depth migration (Lindsay)? Seismo-electric/electroseismic effects (Hetrick)?

The spectrum of seismology on volcanic studies from tilt meters to geophones (Miller) Subsalt seismic imaging (Moeller)

 Vertical (time) resolution

>>improves with greater bandwidth

 Horizontal resolution >>improves with greater range of incidence angles (more complete diffraction curves) >>MUST migrate data to get this resolution

 The travel time to each geophone for the direct wave in the first layer is:

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

 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

   A refraction refers to the wave energy that travels below the upper interface at the velocity of the lower layer. The refraction travels horizontally along the lower interface, then travel back toward the earth’s surface. The refracted wave can only be received after the critical distance x crit beyond the critical angle i crit .

 The ray path and the travel time for the refracted wave for a 2-layer model can be derived as

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

 Travel times for the direct, reflected, and refracted waves for a 2-layer model

V

            1

j V t i

 1

j t i

          1/2

When a refractor dips, the slope of the traveltime curve does not represent the "true" layer velocity: •shooting updip , i.e. geophones are on updip side of shot, apparent refractor velocity is higher •shooting downdip velocity is lower apparent To determine both the layer velocity and the interface dip, forward and reverse refraction profiles must be acquired.

Dipping Layer

V

1 Recall our dipping layer problem: Letting

j

= 2

h

1 we can alternatively write

h

1  Here, distance from the shotpoint to an image point

j

is 

t



x

2  4

h

1

x

cos    2

V

1  4

j

2 

x

2  4

jx

sin 

V

1

Complex solution to complex geologic problems

Reflected/Refracted Waves

A) A compressional wave upon an interface at an oblique angle , , incident is split into four phases : P and S waves reflected back into the original medium; P and S waves refracted into other medium.

For a wave traveling from material of velocity V 1 into velocity V 2 according to material, ray paths are refracted Snell’s law .

i 1 = angle of incidence i 2 = angle of refraction

 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.

ray wavefront (t 0 ) wavefront (t1) 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 Reflected wave Refracted wave, velocity v 2 sin θ 1 = BC/AC sin sin θ 1 / sin θ 2 θ 2 = AD/AC = BC/AD but: BC = v 1 D t and AD = v 2 D 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 v 2 < v 1

sin θ 1 / sin θ 2 = v 1 / v 2 θ 1 θ 1 θ 2 v 2 > v 1

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

sin θ 1 / sin θ 2 = v 1 / v 2 θ 1 θ 2 v 2 Critical angle: θ 2 = 90 > v 1

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

= acoustic impedance (density x velocity)

 What is a typical reflection coefficient?

Vp=2400 ρ=2.4

Vp=2600 ρ=2.5

R = r 2 r 2 v 2 v 2 – r 1 + r 1 v 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

Vp=2600 ρ=2.5

Vp=2400 ρ=2.4

R = r 2 r 2 v 2 v 2 – r 1 + r 1 v 1 v 1 R=0.06=6% (amplitude) R = r 2 r 2 v 2 v 2 – r 1 + r 1 v 1 v 1 R=-0.06=-6% (amplitude)

 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.

            R (θ) = a Δα / α + b Δρ / ρ + c Δβ/β where: a = 1 /( 2 cos 2θ), = (1 + tan 2θ)/2 b = 0.5 - [(2β2/α2 ) sin 2θ] c = -(4β2/α2) sin 2θ α = (α1 + α2)/2 β = (β1 + β2)/2 ρ = (ρ1 + ρ2)/2 Δα = α2 - α1 Δβ = β2 - β1 Δρ = ρ2 - ρ1 θ = (θi + θt)/2, where θt = arcsin [(α2/α1) sin θi] The Aki, Richards and Frasier approximation is written as three terms, the first involving P-wave velocity, the second involving density, and the third involving S-wave velocity.

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

NMO correction

linear hyperbolic

 Average velocity is simply twice the depth divided by the two way, vertical traveltime: Z  V ave = 2Z T 0 T 0

Because of differences in the velocity within each layer, the raypaths bend at each boundary. The velocity between each boundary and the surface does not have a constant velocity.

The shape of the reflection curve is not truly hyperbolic.

Σ

The importance of CDP-Stack 1-Enhancement of the signal 2- Attenuate the random noise 3- Control the multiples 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:

t x



v stack x

2 

v st

.

2 

v RMS t o

2 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:

v RMS

2 

v

1 2

t

1

t

1 

v

2 

t

2 2

t

2  

t

3

v

3  2

t

3 ...

 ...

t 1 t 2 t 3 Surface

v

1

v

2

v

3 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.

v Average



Z

1

t

1  

Z

2

t

2  

t

3

Z

3   ...

...

t 1 t 2 t 3 Surface Z 1 Z 2 Z 3

4- Interval Velocity:

v v

It is average velocity within a certain bed or layer , usually obtained from sonic log survey.

int int

erval erval

 

Z

2

t

2

Z

3

t

3 

Z

1 

t

1 

Z

2 

t

2 ……….For layer 2 ……….For layer 3 t 1 t 2 t 3 Layer 2 Layer 3 Surface Z 1 Z 2 Z 3 5- Instantaneous Velocity: It is average velocity at a certain point within a layer or a geological formation.

v Inst

.

 2785

m

/

Sec

.

Layer 1 Surface

v Inst

.

 2445

m

/

Sec

.

Layer 2

v Inst

.

 3925

m

/

Sec

.

Layer 3