Waves – Topic 4 Chapters 26 Reflection & Refraction Reflection & Refraction Reflection & Refraction.
Refraction Short Course2 (1)
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Transcript of Refraction Short Course2 (1)
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Fundamentals of Seismic Refraction
Theory, Acquisition, and Interpretation
Craig LippusManager, Seismic Products
Geometrics, Inc.
December 3, 2007
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Geometrics, Inc.• Owned by Oyo Corporation, Japan• In business since 1969• Seismographs, magnetometers, EM systems• Land, airborne, and marine• 80 employees
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Located in San Jose, California
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Fundamentals of Seismic Waves
Q. What is a seismic wave?
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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?
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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.
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Compressional (“p”) WaveIdentical to sound wave – particlemotion is parallel to propagationdirection.
Animation courtesy Larry Braile, Purdue University
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Shear (“s”) WaveParticle motion is perpendicularto propagation direction.
Animation courtesy Larry Braile, Purdue University
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Velocity of Seismic WavesDepends on density elastic moduli
3
4
KVp
Vs
where K = bulk modulus, = shear modulus, and = density.
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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.
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Q. What is the rigidity of water?
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A. Water has no rigidity. Its shear strength is zero.
Q. What is the rigidity of water?
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Q. How well does water carry shear waves?
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A. It doesn’t.
Q. How well does water carry shear waves?
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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”.
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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.
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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?
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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?
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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.
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Q. How do we use seismic waves to understand the subsurface?
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Q. How do we use seismic waves to understand the subsurface?
A. Must first understand wavebehavior in layered media.
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Q. What happens when a seismic wave encounters a velocity discontinuity?
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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!!
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Q. What happens when a seismic wave encounters a velocity discontinuity?
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Five important concepts
• Seismic Wavefront• Ray• Huygen’s Principle• Snell’s Law• Reciprocity
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Q. What is a seismic wavefront?
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Q. What is a seismic wavefront?A. Surface of constant phase, like ripples on a pond, but in three dimensions.
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Q. What is a seismic wavefront?
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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.
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Seismic wavefront
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Q. What is a ray?
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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.
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Ray
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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.
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Q. What causes refraction?
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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.
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Understanding and Quantifying How Waves
Refract is Essential
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Snell’s Law
2
1
sinsin
VV
ri
(1)
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Snell’s LawIf V2>V1, then as i increases, r increases faster
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Snell’s Lawr approaches 90o as i increases
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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)
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Snell’s LawCritical Refraction
At Critical Angle of incidence ic, angle of refraction r = 90o
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Snell’s LawCritical Refraction
At Critical Angle of incidence ic, angle of refraction r = 90o
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Snell’s LawCritical Refraction
Seismic refraction makes use of critically refracted, first-arrival energy only. The rest of the wave form is ignored.
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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.
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Using Seismic Refraction to Map the Subsurface
Critical Refraction Plays a Key Role
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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
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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
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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)
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Using Seismic Refraction to Map the Subsurface
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Using Seismic Refraction to Map the Subsurface
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Using Seismic Refraction to Map the Subsurface
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Using Seismic Refraction to Map the Subsurface
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Using Seismic Refraction to Map the Subsurface
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Using Seismic Refraction to Map the Subsurface
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Using Seismic Refraction to Map the Subsurface
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Using Seismic Refraction to Map the Subsurface
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Using Seismic Refraction to Map the Subsurface
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Using Seismic Refraction to Map the Subsurface
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Using Seismic Refraction to Map the Subsurface
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Using Seismic Refraction to Map the Subsurface
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Using Seismic Refraction to Map the Subsurface
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Using Seismic Refraction to Map the Subsurface
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Using Seismic Refraction to Map the Subsurface
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Using Seismic Refraction to Map the Subsurface
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Using Seismic Refraction to Map the Subsurface
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Using Seismic Refraction to Map the Subsurface
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Using Seismic Refraction to Map the Subsurface
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Using Seismic Refraction to Map the Subsurface
Depth{
12
12
2 VVVVXcDepth
(5)
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Using Seismic Refraction to Map the Subsurface
Depth{
For layer parallel to surface
12
12
2 VVVVXcDepth
)cos(sin22
11
1
VV
VTi
(6)
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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
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)cos(sin22
11
121
VV
VTh i
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1
32
2
21
3123
2
)/1cos(sin2
)/1cos(sin
)/1cos(sin
hVV
VVV
VVTT
h
ii
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2143
1
32
421
2
211
411
24
3)/cos(sin2
)/cos(sin2)/cos(sin)/cos(sin
hhVV
VV
VhVVVVTT
hii
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Crossover Distance vs. Depth
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Depth/Xc vs. Velocity Contrast
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Important Rule of ThumbThe Length of the Geophone Spread Should be 4-5 times the depth of interest.
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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.
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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
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Dipping Layer
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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
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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
![Page 86: Refraction Short Course2 (1)](https://reader036.fdocuments.net/reader036/viewer/2022062905/577ccf4f1a28ab9e788f6bec/html5/thumbnails/86.jpg)
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
![Page 87: Refraction Short Course2 (1)](https://reader036.fdocuments.net/reader036/viewer/2022062905/577ccf4f1a28ab9e788f6bec/html5/thumbnails/87.jpg)
What if V2 < V1?
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2
1
sinsin
VV
ri
What if V2 < V1?Snell’s Law
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2
1
sinsin
VV
ri
What if V2 < V1?Snell’s Law
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If V1>V2, then as i increases, r increases, but not as fast.
What if V2 < V1?
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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.
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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.
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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
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Delay Time Methodx
V1
V2
![Page 102: Refraction Short Course2 (1)](https://reader036.fdocuments.net/reader036/viewer/2022062905/577ccf4f1a28ab9e788f6bec/html5/thumbnails/102.jpg)
Delay Time Methodx
V1
V2
)cos()tan()tan(
)cos( 12221 c
BcBcA
c
AAB
iVh
Vih
Vih
VAB
iVhT
![Page 103: Refraction Short Course2 (1)](https://reader036.fdocuments.net/reader036/viewer/2022062905/577ccf4f1a28ab9e788f6bec/html5/thumbnails/103.jpg)
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
![Page 104: Refraction Short Course2 (1)](https://reader036.fdocuments.net/reader036/viewer/2022062905/577ccf4f1a28ab9e788f6bec/html5/thumbnails/104.jpg)
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
![Page 105: Refraction Short Course2 (1)](https://reader036.fdocuments.net/reader036/viewer/2022062905/577ccf4f1a28ab9e788f6bec/html5/thumbnails/105.jpg)
Delay Time Methodx
t T T TA P B P A B0
Definition:
V1
V2
(7)
![Page 106: Refraction Short Course2 (1)](https://reader036.fdocuments.net/reader036/viewer/2022062905/577ccf4f1a28ab9e788f6bec/html5/thumbnails/106.jpg)
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
![Page 107: Refraction Short Course2 (1)](https://reader036.fdocuments.net/reader036/viewer/2022062905/577ccf4f1a28ab9e788f6bec/html5/thumbnails/107.jpg)
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
![Page 108: Refraction Short Course2 (1)](https://reader036.fdocuments.net/reader036/viewer/2022062905/577ccf4f1a28ab9e788f6bec/html5/thumbnails/108.jpg)
)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
![Page 109: Refraction Short Course2 (1)](https://reader036.fdocuments.net/reader036/viewer/2022062905/577ccf4f1a28ab9e788f6bec/html5/thumbnails/109.jpg)
)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
![Page 110: Refraction Short Course2 (1)](https://reader036.fdocuments.net/reader036/viewer/2022062905/577ccf4f1a28ab9e788f6bec/html5/thumbnails/110.jpg)
2
1sinVVic
Substituting from Snell’s Law,
10
)cos(2V
iht cp (8)
We get
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11
)cos(2
)cos(22
Ppoint at Delay timeV
ihV
ihtD cpcpoTP (9)
![Page 112: Refraction Short Course2 (1)](https://reader036.fdocuments.net/reader036/viewer/2022062905/577ccf4f1a28ab9e788f6bec/html5/thumbnails/112.jpg)
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.
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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
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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)
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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)
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Reduced Traveltimes
TT T T
A PA B A P B P
'
2 2The first term is represented by the dotted line below:
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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.
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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.
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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.
![Page 120: Refraction Short Course2 (1)](https://reader036.fdocuments.net/reader036/viewer/2022062905/577ccf4f1a28ab9e788f6bec/html5/thumbnails/120.jpg)
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)
![Page 121: Refraction Short Course2 (1)](https://reader036.fdocuments.net/reader036/viewer/2022062905/577ccf4f1a28ab9e788f6bec/html5/thumbnails/121.jpg)
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)
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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)
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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
![Page 124: Refraction Short Course2 (1)](https://reader036.fdocuments.net/reader036/viewer/2022062905/577ccf4f1a28ab9e788f6bec/html5/thumbnails/124.jpg)
Substituting back into equation 16,
)cos(1
c
Tp
iVDh p
2
11
1
sincosVV
VDh pTp
(16)
(17)
we get
![Page 125: Refraction Short Course2 (1)](https://reader036.fdocuments.net/reader036/viewer/2022062905/577ccf4f1a28ab9e788f6bec/html5/thumbnails/125.jpg)
In summary, to determine the depth to the refractor h at any given point p:
![Page 126: Refraction Short Course2 (1)](https://reader036.fdocuments.net/reader036/viewer/2022062905/577ccf4f1a28ab9e788f6bec/html5/thumbnails/126.jpg)
1.Measure V1 directly from the traveltime plot.
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2.Measure the difference in traveltime to point P from opposing shots (in zone of overlap only).
![Page 128: Refraction Short Course2 (1)](https://reader036.fdocuments.net/reader036/viewer/2022062905/577ccf4f1a28ab9e788f6bec/html5/thumbnails/128.jpg)
3.Measure the reciprocal time TAB.
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4. Per equation 11, T
T T TA P
A B A P B P'
2 2
divide the reciprocal time TAB by 2.,
![Page 130: Refraction Short Course2 (1)](https://reader036.fdocuments.net/reader036/viewer/2022062905/577ccf4f1a28ab9e788f6bec/html5/thumbnails/130.jpg)
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.
,
![Page 131: Refraction Short Course2 (1)](https://reader036.fdocuments.net/reader036/viewer/2022062905/577ccf4f1a28ab9e788f6bec/html5/thumbnails/131.jpg)
6. Fit a line to the reduced traveltimes, compute V2 from slope.
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2'
VxTD APTp
7. Using equation 15,
Calculate the Delay Time DT at P1, P2, P3….PN
(15)
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8. Using equation 17,
Calculate the Depth h at P1, P2,
P3….PN
2
11
1
sincosVV
VDh pTp (16)
![Page 134: Refraction Short Course2 (1)](https://reader036.fdocuments.net/reader036/viewer/2022062905/577ccf4f1a28ab9e788f6bec/html5/thumbnails/134.jpg)
That’s all there is to it!
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More Data is Better Than Less
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More Data is Better Than Less
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More Data is Better Than Less
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More Data is Better Than Less
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More Data is Better Than Less
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More Data is Better Than Less
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More Data is Better Than Less
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More Data is Better Than Less
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More Data is Better Than Less
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More Data is Better Than Less
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