Refraction Short Course2 (1)

$Page 1: Refraction Short Course2 (1)$
Fundamentals of Seismic Refraction

Theory, Acquisition, and Interpretation

Craig LippusManager, Seismic Products

Geometrics, Inc.

December 3, 2007

Geometrics, Inc.• Owned by Oyo Corporation, Japan• In business since 1969• Seismographs, magnetometers, EM systems• Land, airborne, and marine• 80 employees

Located in San Jose, California

Fundamentals of Seismic Waves

Q. What is a seismic wave?

Fundamentals of Seismic Waves

A. Transfer of energy by way ofparticle motion.

Different types of seismic waves are characterized by their particle motion.

Q. What is a seismic wave?

Three different types of seismic waves

• Compressional (“p”) wave• Shear (“s”) wave• Surface (Love and Raleigh)

wave

Only p and s waves (collectively referred toas “body waves”) are of interest in seismic refraction.

Compressional (“p”) WaveIdentical to sound wave – particlemotion is parallel to propagationdirection.

Animation courtesy Larry Braile, Purdue University

Shear (“s”) WaveParticle motion is perpendicularto propagation direction.

Animation courtesy Larry Braile, Purdue University

Velocity of Seismic WavesDepends on density elastic moduli

3

4

KVp

Vs

where K = bulk modulus, = shear modulus, and = density.

Velocity of Seismic WavesBulk modulus = resistance to compression = incompressibility

Shear modulus = resistance to shear = rigidity The less compressible a material is, the greater its p-wave velocity, i.e., sound travels about four times faster in water than in air. The more resistant a material is to shear, the greater its shear wave velocity.

Q. What is the rigidity of water?

A. Water has no rigidity. Its shear strength is zero.

Q. What is the rigidity of water?

Q. How well does water carry shear waves?

A. It doesn’t.

Q. How well does water carry shear waves?

Fluids do not carry shear waves. This knowledge, combined with earthquake observations, is what lead to the discovery that the earth’s outer core is a liquid rather than a solid – “shear wave shadow”.

p-wave velocity vs. s-wave velocity p-wave velocity must always be greater than s-wave velocity. Why?

34

34

2

2

K

K

VsVp

K and are always positive numbers, so Vp is always greater than Vs.

Velocity – density paradox Q. We know that in practice, velocity tends to be directly proportional to density. Yet density is in the denominator. How is that possible?

Velocity – density paradox

A. Elastic moduli tend to increase with density also, and at a faster rate.

Q. We know that in practice, velocity tends to be directly proportional to density. Yet density is in the denominator. How is that possible?

Velocity – density paradox Note: Elastic moduli are important parameters for understanding rock properties and how they will behave under various conditions. They help engineers assess suitability for founding dams, bridges, and other critical structures such as hospitals and schools. Measuring p- and s-wave velocities can help determine these properties indirectly and non-destructively.

Q. How do we use seismic waves to understand the subsurface?

Q. How do we use seismic waves to understand the subsurface?

A. Must first understand wavebehavior in layered media.

Q. What happens when a seismic wave encounters a velocity discontinuity?


A. Some of the energy is reflected, some is refracted.

We are only interested in refracted energy!!


Five important concepts

• Seismic Wavefront• Ray• Huygen’s Principle• Snell’s Law• Reciprocity

Q. What is a seismic wavefront?

Q. What is a seismic wavefront?A. Surface of constant phase, like ripples on a pond, but in three dimensions.

Q. What is a seismic wavefront?

The speed at which a wavefront travels is the seismic velocity of the material, and depends on the material’s elastic properties. In a homogenious medium, a wavefront is spherical, and its shape is distorted by changes in the seismic velocity.

Seismic wavefront

Q. What is a ray?

Q. What is a ray?

A. Also referred to as a “wavefrontnormal” a ray is an arrowperpendicular to the wave front,indicating the direction of travel atthat point on the wavefront. Thereare an infinite number of rays on awave front.

Ray

Huygens' Principle Every point on a wave front can be thought of as a new point source for waves generated in the direction the wave is traveling or being propagated.

Q. What causes refraction?

Q. What causes refraction?A. Different portions of the wave front reach the velocity boundary earlier than other portions, speeding up or slowing down on contact, causing distortion of wave front.



Understanding and Quantifying How Waves

Refract is Essential

Snell’s Law

2

1

sinsin

VV

ri

(1)

Snell’s LawIf V2>V1, then as i increases, r increases faster

Snell’s Lawr approaches 90o as i increases

Snell’s LawCritical Refraction

At Critical Angle of incidence ic, angle of refraction r = 90o

2

1

90sin)sin(

VVic

2

1)sin(VVic

2

11sinVVic

(2)

(3)






Seismic refraction makes use of critically refracted, first-arrival energy only. The rest of the wave form is ignored.

Principal of Reciprocity

The travel time of seismic energy between two points is independent of the direction traveled, i.e., interchanging the source and the geophone will not affect the seismic travel time between the two.

Using Seismic Refraction to Map the Subsurface

Critical Refraction Plays a Key Role

11 /VxT

1212

Vdf

Vcd

VacT

)cos( cihdfac

)tan( cihdebc

)tan(2 cihxdebcxcd

2)(12

)tan(2cos2

Vihx

iVhT c

c

22)(12

)tan(2cos2

Vx

Vih

iVhT c

c

22)(12

)cos()sin(

cos12

Vx

iVi

iVhT

c

c

c

221

1

)(21

22

)cos()sin(

cos2

Vx

iVViV

iVVVhT

c

c

c

221

122

)cos()sin(2

Vx

iVViVVhTc

c

2

1sinVVic (Snell’s Law)

221

1

2

12)cos(

)sin(2

Vx

iVV

iVV

hVTc

c

22112

)cos(

)sin()sin(

1

2Vx

iVV

iihVT

c

cc

212

)cos(2Vx

VihT c

221

2

12)cos()sin(

)(sin12Vx

iiVVihVT

cc

c

221

2

12)cos()sin(

)(cos2Vx

iiVVihVT

cc

c

222

)sin()cos(2

Vx

iVihT

c

c

)sin(21 ciVV

From Snell’s Law,

(4)





















Depth{

12

12

2 VVVVXcDepth

(5)


Depth{

For layer parallel to surface

12

12

2 VVVVXcDepth

)cos(sin22

11

1

VV

VTi

(6)

212

)cos(2Vx

VihT c

12

12

2 VVVVXch

2

11

1

sincos2VV

VTh i

Summary of Important Equations

For refractor parallel to surface

2

1

sinsin

VV

ri

2

11sinVVic

(2)

(3)

(1)

(5)

(4)

(6)

Snell’s Law

2

1)sin(VVic


)cos(sin22

11

121

VV

VTh i

1

32

2

21

3123

2

)/1cos(sin2

)/1cos(sin

)/1cos(sin

hVV

VVV

VVTT

h

ii

2143

1

32

421

2

211

411

24

3)/cos(sin2

)/cos(sin2)/cos(sin)/cos(sin

hhVV

VV

VhVVVVTT

hii

Crossover Distance vs. Depth

Depth/Xc vs. Velocity Contrast

Important Rule of ThumbThe Length of the Geophone Spread Should be 4-5 times the depth of interest.

Dipping LayerDefined as Velocity Boundary that is not Parallel to Ground Surface

You should always do a minimum of one shot at either end the spread. A single shot at one end does not tell you anything about dip, and if the layer(s) is dipping, your depth and velocity calculated from a single shot will be wrong.

Dipping LayerIf layer is dipping (relative to ground surface), opposing travel time curves will be asymmetrical.

Updip shot – apparent velocity > true velocityDowndip shot – apparent velocity < true velocity

Dipping Layer

Dipping Layer

)sin(sin21

11

11

udc mVmVi

)sin(1 cd imV

)sin(1 cimuV

dc mVi 11sin

uc mVi 11sin

)sin(sin21

11

11

ud mVmV

Dipping LayerFrom Snell’s Law,

)sin(1

2ci

VV

cos)cos(2

1

c

iu

ui

TV

D

cos)cos(2

1

c

id

di

TV

D

Dipping LayerThe true velocity V2 can also be calculated by multiplying the harmonic mean of the up-dip and down-dip velocities by the cosine of the dip.

cos222

222

DU

DU

VVVVV

What if V2 < V1?

2

1

sinsin

VV

ri

What if V2 < V1?Snell’s Law

2

1

sinsin

VV

ri

What if V2 < V1?Snell’s Law

If V1>V2, then as i increases, r increases, but not as fast.

What if V2 < V1?

If V2<V1, the energy refracts toward the normal. None of the refracted energy makes it back to the surface.

This is called a velocity inversion.

Seismic Refraction requires that velocities increase with depth.A slower layer beneath a faster layer will not be detected by seismic refraction.The presence of a velocity inversion can lead to errors in depth calculations.








Delay Time Method• Allows Calculation of Depth Beneath Each Geophone

• Requires refracted arrival at each geophone from opposite directions

• Requires offset shots

• Data redundancy is important

Delay Time Methodx

V1

V2

Delay Time Methodx

V1

V2

)cos()tan()tan(

)cos( 12221 c

BcBcA

c

AAB

iVh

Vih

Vih

VAB

iVhT

Delay Time Methodx

)cos()tan()tan(

)cos( 12221 c

PcPcA

c

AAP

iVh

Vih

Vih

VAP

iVhT

)cos()tan()tan(

)cos( 12221 c

BcBcA

c

AAB

iVh

Vih

Vih

VAB

iVhT

V1

V2

Delay Time Methodx

)cos()tan()tan(

)cos( 12221 c

PcPcB

c

BBP

iVh

Vih

Vih

VBP

iVhT

)cos()tan()tan(

)cos( 12221 c

PcPcA

c

AAP

iVh

Vih

Vih

VAP

iVhT

)cos()tan()tan(

)cos( 12221 c

BcBcA

c

AAB

iVh

Vih

Vih

VAB

iVhT

V1

V2

Delay Time Methodx

t T T TA P B P A B0

Definition:

V1

V2

(7)

ABBPAP TTTt 0

)cos(

)tan()tan()cos( 12221

0c

PcPcA

c

A

iVh

Vih

Vih

VAP

iVht

)cos(

)tan()tan()cos( 12221 c

PcPcB

c

B

iVh

Vih

Vih

VBP

iVh

)cos(

)tan()tan()cos( 12221 c

BcBcA

c

A

iVh

Vih

Vih

VAB

iVh

2120

)tan(2)cos(

2V

ihiV

hV

ABBPAPt cP

c

p

But from figure above, BPAPAB . Substituting, we get

2120

)tan(2)cos(

2V

ihiV

hV

BPAPBPAPt cP

c

p

or

210

)tan(2)cos(

2V

ihiV

ht cP

c

p

)cos(

)sin()cos(

1221

0c

c

cp

iVi

iVht

)cos(

)sin()cos(

221

1

21

20

c

c

cp

iVViV

iVVVht

)cos(

)sin()cos(

22121

1

2

10c

c

cp

iVVi

iVVVV

Vht

2

1sinVVicSubstituting from Snell’s Law,

)cos(

)sin()cos(

sin1

22121

10c

c

c

cp

iVVi

iVViVht

)cos(

)sin()cos(

sin1

22121

10c

c

c

cp

iVVi

iVViVht

Multiplying top and bottom by sin(ic)

)cos()sin(

)(sin)cos()sin(

1221

2

2110

cc

c

ccp

iiVVi

iiVVVht

)cos()sin(

)(cos221

2

10cc

cp

iiVViVht

)sin(

)cos(22

0c

cp

iViht

)sin(

)cos(22

0c

cp

iViht

2

1sinVVic

Substituting from Snell’s Law,

10

)cos(2V

iht cp (8)

We get

11

)cos(2

)cos(22

Ppoint at Delay timeV

ihV

ihtD cpcpoTP (9)

Reduced Traveltimes

Definition:T’AP = “Reduced Traveltime” at point P for a source at A

T’AP=TAP’

x

Reduced traveltimes are useful for determining V2. A plot of T’ vs. x will be roughly linear, mostly unaffected by changes in layer thickness, and the slope will be 1/V2.

Reduced Traveltimesx

From the above figure, T’AP is also equal to TAP minus the Delay Time. From equation 9, we then get

2' o

APTAPAPtTDTT P

Reduced Traveltimesx

Earlier, we defined to as

t T T TA P B P A B0 Substituting, we get

22' ABBPAP

APo

APAPTTTTtTT

(7)

(10)

Reduced Traveltimes

TT T T

A PA B A P B P

'

2 2

Finally, rearranging yields

The above equation allows a graphical determination of the T’ curve. TAB is called the reciprocal time.

(11)

Reduced Traveltimes

TT T T

A PA B A P B P

'

2 2The first term is represented by the dotted line below:

Reduced Traveltimes

TT T T

A PA B A P B P

'

2 2The numerator of the second term is just the difference in the traveltimes from points A to P and B to P.

Reduced Traveltimes

TT T T

A PA B A P B P

'

2 2Important: The second term only applies to refracted arrivals. It does not apply outside the zone of “overlap”, shown in yellow below.

Reduced Traveltimes

TT T T

A PA B A P B P

'

2 2The T’ (reduced traveltime) curve can now be determined graphically by adding (TAP-TBP)/2 (second term from equation 9) to the TAB/2 line (first term from equation 9). The slope of the T’ curve is 1/V2.

We can now calculate the delay time at point P. From Equation 10, we see that

1

)cos(2 V

iht cpo

According to equation 8

2' o

APAPtTT

1

0 )cos(2

'V

ihTtTT cpAPAPAP

So

Now, referring back to equation 4

212

)cos(2Vx

VihT c

(12)

(4)

(8)

(10)

It’s fair to say that

21

)cos(2Vx

VihT cp

AP

Combining equations 12 and 13, we get

1211

)cos()cos(2)cos('V

ihVx

Vih

VihTT cpcpcp

APAP

Or

21

)cos('Vx

VihT cp

AP

(13)

(14)

1

)cos(V

ihD cpTp

Referring back to equation 9, we see that

Substituting into equation 14, we get

221

)cos('VxD

Vx

VihT pTcp

AP

Or

2'

VxTD APTp

hD V

iP

T

c

P

1

co s( )

Solving equation 9 for hp, we get

(15)

(16)

(9)

We know that the incident angle i is critical when r is 90o. From Snell’s Law,

2

1

sinsin

VV

ri

2

1

90sinsin

VVic

2

1sinVVic

2

11sinVVic

Substituting back into equation 16,

)cos(1

c

Tp

iVDh p

2

11

1

sincosVV

VDh pTp

(16)

(17)

we get

In summary, to determine the depth to the refractor h at any given point p:

1.Measure V1 directly from the traveltime plot.

2.Measure the difference in traveltime to point P from opposing shots (in zone of overlap only).

3.Measure the reciprocal time TAB.

4. Per equation 11, T

T T TA P

A B A P B P'

2 2

divide the reciprocal time TAB by 2.,

5. Per equation 11, T

T T TA P

A B A P B P'

2 2add ½ the difference time at each

point P to TAB/2 to get the reduced traveltime at P, T’AP.

,

6. Fit a line to the reduced traveltimes, compute V2 from slope.

2'

VxTD APTp

7. Using equation 15,

Calculate the Delay Time DT at P1, P2, P3….PN

(15)

8. Using equation 17,

Calculate the Depth h at P1, P2,

P3….PN

2

11

1

sincosVV

VDh pTp (16)

That’s all there is to it!












More Data is Better Than Less











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Refraction Short Course2 (1)