Measures of Central Tendency: Mean, Median and Mode
Fundamentals of Seismic Refraction Theory
1. Fundamentals of Seismic
Refraction
Theory, Acquisition, and Interpretation
Craig Lippus
Manager, Seismic Products
Geometrics, Inc.
December 3, 2007
2. Geometrics, Inc.
• Owned by Oyo Corporation,
Japan
• In business since 1969
• Seismographs, magnetometers,
EM systems
• Land, airborne, and marine
• 80 employees
5. Fundamentals of Seismic
Waves
A. What is a seismic wave?
A. Transfer of energy by way of
particle motion.
Different types of seismic waves are
characterized by their particle motion.
6. Three different types of
seismic waves
• Compressional (“p”) wave
• Shear (“s”) wave
• Surface (Love and
Raleigh) wave
Only p and s waves (collectively referred to
as “body waves”) are of interest
in seismic refraction.
7. Compressional (“p”) Wave
Identical to sound wave – particle
motion is parallel to propagation
direction.
Animation courtesy Larry Braile, Purdue University
8. Shear (“s”) Wave
Particle motion is perpendicular
to propagation direction.
Animation courtesy Larry Braile, Purdue University
9. Velocity of Seismic Waves
Depends on density elastic moduli
4µ
K+
Vp = 3 µ
Vs =
ρ ρ
where K = bulk modulus, µ = shear
modulus, and ρ = density.
10. Velocity of Seismic Waves
Bulk 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.
14. Q. How well does water
carry shear waves?
A. It doesn’t.
15. 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”.
16. p-wave velocity vs. s-
wave velocity
p-wave velocity must always
be greater than s-wave
velocity. Why?
4µ
K+
3
Vp 2 ρ K 4
= = +
Vs 2
µ µ 3
ρ
K and µ are always positive numbers, so Vp
is always greater than Vs.
17. 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?
18. 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?
A. Elastic moduli tend to increase with density also,
and at a faster rate.
19. 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.
20. Q. How do we use seismic
waves to understand the
subsurface?
21. Q. How do we use seismic
waves to understand the
subsurface?
A. Must first understand wave
behavior in layered media.
22. Q. What happens when a
seismic wave encounters a
velocity discontinuity?
23. 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!!
24. Q. What happens when a
seismic wave encounters a
velocity discontinuity?
25. Five important
concepts
• Seismic Wavefront
• Ray
• Huygen’s Principle
• Snell’s Law
• Reciprocity
29. 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.
32. Q. What is a ray?
A. Also referred to as a “wavefront
normal” a ray is an arrow
perpendicular to the wave front,
indicating the direction of travel at
that point on the wavefront. There
are an infinite number of rays on a
wave front.
34. 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.
36. 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.
43. Snell’s Law
Critical Refraction
At Critical Angle of incidence ic, angle of
refraction r = 90o
sin(ic ) V 1
=
sin 90 V 2
V1
sin(ic ) = (2)
V2
V1 (3)
ic = sin −1
V2
44. Snell’s Law
Critical Refraction
At Critical Angle of incidence ic, angle of
refraction r = 90o
45. Snell’s Law
Critical Refraction
At Critical Angle of incidence ic, angle of
refraction r = 90o
46. Snell’s Law
Critical Refraction
Seismic refraction makes use of
critically refracted, first-arrival
energy only. The rest of the wave
form is ignored.
47. 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.
49. T 1 = x /V 1
ac cd df
T2 = + +
V1 V 2 V1
h
ac = df =
cos(ic )
bc = de = h tan(ic )
cd = x − bc − de = x − 2h tan(ic )
2h x − 2h tan(ic )
T2 = +
V 1 cos(ic ) V2
2h 2h tan(ic ) x
T2 = − +
V 1 cos(ic ) V2 V2
1 sin(ic ) x
T 2 = 2 h − +
V 1 cos(ic ) V 2 cos(ic ) V 2
50. V2 V 1 sin(ic ) x
T 2 = 2 h
V 1V 2 cos(ic ) V 1V 2 cos(ic ) + V 2
−
V 2 − V 1 sin(ic ) x
T 2 = 2h
V 1V 2 cos(ic ) + V 2
V2
− sin(ic )
x
T 2 = 2hV 1 V 1 +
V 1V 2 cos(ic ) V 2
V1
sin ic = (Snell’s Law)
V2
1
− sin(ic )
T 2 = 2hV 1
sin(ic ) + x
V 1V 2 cos(ic ) V 2
51. 1 − sin 2 (ic ) x
T 2 = 2hV 1 +
V 1V 2 sin(ic ) cos(ic ) V 2
cos 2 (ic ) x
T 2 = 2hV 1
V 1V 2 sin(ic ) cos(ic ) + V 2
cos(ic ) x
T 2 = 2h
V 2 sin(ic ) + V 2
From Snell’s Law,
V 1 = V 2 sin(ic )
2h cos(ic ) x
T2 = + (4)
V1 V2
72. Using Seismic Refraction
to Map the Subsurface
Xc V 2 − V 1 T iV 1
Depth = = (6)
2 V 2 +V1 V1−1
2 cos(sin )
For layer parallel to
V2
surface
{
Depth
73. Summary of Important
Equations
For refractor
sin i V 1 parallel to surface
= (1) Snell’s Law
sin r V 2
2h cos(ic ) x
T2 = + (4)
V1 V2
V1
sin(ic ) = (2)
V2 Xc V 2 − V 1
h= (5)
2 V 2 +V1
V1
ic = sin −1 (3)
V2 Ti V 1
h=
−1 V 1 (6)
2 cos sin
V2
80. Important Rule of Thumb
The Length of the Geophone
Spread Should be 4-5 times
the depth of interest.
81. Dipping Layer
Defined 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.
82. Dipping Layer
If layer is dipping (relative to ground
surface), opposing travel time curves
will be asymmetrical.
Updip shot – apparent velocity > true velocity
Downdip shot – apparent velocity < true velocity
84. Dipping Layer
V 1md = sin(ic + α )
V 1mu = sin(ic − α )
ic + α = sin −1 V 1md
ic − α = sin −1 V 1mu
1
ic = (sin −1 V 1md + sin −1 V 1mu )
2
1
α = (sin −1 V 1md − sin −1 V 1mu )
2
85. Dipping Layer
From Snell’s Law,
V1
V2 =
sin(ic )
V 1Tiu
2 cos(ic )
Du =
cos α
V 1Tid
2 cos(ic )
Dd =
cos α
86. Dipping Layer
The 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.
2V 2UV 2 D
V2 = cos α
V 2U + V 2 D
88. What if V2 < V1?
sin i V 1
Snell’s Law =
sin r V 2
89. What if V2 < V1?
sin i V 1
Snell’s Law =
sin r V 2
90. What if V2 < V1?
If V1>V2, then as i increases, r
increases, but not as fast.
91. 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.
92. 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.
93.
94.
95.
96.
97.
98.
99.
100. 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
102. Delay Time Method
x
V1
V2
hA AB hA tan(ic ) hB tan(ic ) hB
TAB ≅ + − − +
V 1 cos(ic ) V 2 V2 V2 V 1 cos(ic )
103. Delay Time Method
x
V1
V2
hA AB hA tan(ic ) hB tan(ic ) hB
TAB ≅ + − − +
V 1 cos(ic ) V 2 V2 V2 V 1 cos(ic )
hA AP hA tan(ic ) hP tan(ic ) hP
TAP ≅ + − − +
V 1 cos(ic ) V 2 V2 V2 V 1 cos(ic )
104. Delay Time Method
x
V1
V2
hA AB hA tan(ic ) hB tan(ic ) hB
TAB ≅ + − − +
V 1 cos(ic ) V 2 V2 V2 V 1 cos(ic )
hA AP hA tan(ic ) hP tan(ic ) hP
TAP ≅ + − − +
V 1 cos(ic ) V 2 V2 V2 V 1 cos(ic )
hB BP hB tan(ic ) hP tan(ic ) hP
TBP ≅ + − − +
V 1 cos(ic ) V 2 V2 V2 V 1 cos(ic )
106. t 0 = TAP + TBP − TAB
hA AP hA tan(ic ) hP tan(ic ) hP h B BP hB tan(ic ) hP tan(ic ) hP
t0 = + − − + + + − − +
V 1 cos(ic ) V 2 V2 V2 V 1 cos(ic ) V 1 cos(ic ) V 2 V2 V2 V 1 cos(ic )
hA AB hA tan(ic ) hB tan(ic ) hB
− + − − +
V 1 cos(ic ) V 2 V2 V2 V 1 cos(ic )
AP + BP − AB 2 hp 2hP tan(ic )
t0 = + −
V2 V 1 cos(ic ) V2
107. But from figure above, AB = AP + BP. Substituting, we get
AP + BP − AP − BP 2 hp 2hP tan(ic )
t0 = + −
V2 V 1 cos(ic ) V2
or
2hp 2hP tan(ic )
t0 = −
V 1 cos(ic ) V2
108.
1 sin(ic )
t 0 = 2h
p
−
V 1 cos(ic ) V 2 cos(ic )
V2 V 1 sin(ic )
t 0 = 2hp
−
V 1V 2 cos(ic ) V 1V 2 cos(ic )
V2
V1 sin(ic )
t 0 = 2hpV 1 −
V 1V 2 cos(ic ) V 1V 2 cos(ic )
Substituting from Snell’s Law, sin ic = V 1
V2
1
sin ic sin(ic )
t 0 = 2hpV 1 −
V 1V 2 cos(ic ) V 1V 2 cos(ic )
109.
1
sin ic sin(ic )
t 0 = 2hpV 1 −
V 1V 2 cos(ic ) V 1V 2 cos(ic )
Multiplying top and bottom by sin(ic)
1 sin 2 (ic )
t 0 = 2 h pV
1
−
V 1V 2 sin(ic ) cos(ic ) V 1V 2 sin(ic ) cos(ic )
cos 2 (ic )
t 0 = 2hpV 1
V 1V 2 sin(ic ) cos(ic )
cos(ic )
t 0 = 2h p
V 2 sin(ic )
cos(ic )
t 0 = 2hp
V 2 sin(ic )
111. to 2hp cos(ic ) hp cos(ic )
Delay time at point P = DTP = = = (9)
2 2V 1 V1
112. Reduced Traveltimes
x
Definition:
T’AP = “Reduced Traveltime” at point P for a source at A
T’AP=TAP’
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.
113. Reduced Traveltimes
x
From the above figure, T’AP is also equal to TAP minus the
Delay Time. From equation 9, we then get
to
T ' AP = TAP − DTP = TAP −
2
114. Reduced Traveltimes
x
Earlier, we defined to as
t0 = T AP + T BP − T AB (7)
Substituting, we get
to TAP + TBP − TAB (10)
T ' AP = TAP − = TAP −
2 2
115. Reduced Traveltimes
Finally, rearranging yields
T AB (T AP − T BP )
T ' AP = + (11)
2 2
The above equation allows a graphical determination of the T’
curve. TAB is called the reciprocal time.
116. Reduced Traveltimes
T AB (T AP − T BP )
T ' AP = +
2 2
The first term is represented by the dotted line below:
117. Reduced Traveltimes
T AB (T AP − T BP )
T ' AP = +
2 2
The numerator of the second term is just the difference in the
traveltimes from points A to P and B to P.
118. Reduced Traveltimes
T AB (T AP − T BP )
T ' AP = +
2 2
Important: The second term only applies to refracted arrivals. It
does not apply outside the zone of “overlap”, shown in yellow
below.
119. Reduced Traveltimes
T AB (T AP − T BP )
T ' AP = +
2 2
The 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.
120. We can now calculate the delay time at point P. From Equation 10,
we see that
to
T ' AP = TAP − (10)
2
According to equation 8
to hp cos(ic )
= (8)
2 V1
So
t0 hp cos(ic )
T ' AP = TAP − = TAP − (12)
2 V1
Now, referring back to equation 4
2h cos(ic ) x
T2 = + (4)
V1 V2
121. It’s fair to say that
2hp cos(ic ) x
TAP ≅ + (13)
V1 V2
Combining equations 12 and 13, we get
hp cos(ic ) 2hp cos(ic ) x hp cos(ic )
T ' AP = TAP − = + −
V1 V1 V2 V1
Or
hp cos(ic ) x
T ' AP = + (14)
V1 V2
122. Referring back to equation 9, we see that
hp cos(ic )
DTp = (9)
V1
Substituting into equation 14, we get
hp cos(ic ) x x
T ' AP = + = DTp +
V1 V2 V2
Or
x
DTp = T ' AP − (15)
V2
Solving equation 9 for hp, we get
D TPV 1
hP = (16)
c o s (ic)
123. We know that the incident angle i is critical when r is 90o.
From Snell’s Law,
sin i V 1
=
sin r V 2
sin ic V 1
=
sin 90 V 2
V1
sin ic =
V2
V1
−1
ic = sin
V 2
124. Substituting back into equation 16,
DTpV 1
hp = (16)
cos(ic )
we get
DTpV 1
hp =
−1 V 1
cos sin (17)
V 2
125. In summary, to determine the
depth to the refractor h at any
given point p:
129. T (T AP − T BP )
4. Per equation 11, T ' A P =
AB
+ ,
2 2
divide the reciprocal time TAB by 2.
130. T (T AP − T BP )
5. Per equation 11, T ' A P = 2 +
AB
,
2
add ½ the difference time at each
point P to TAB/2 to get the reduced
traveltime at P, T’AP.
131. 6. Fit a line to the reduced
traveltimes, compute V2 from slope.
132. 7. Using equation 15,
x
DTp = T ' AP − (15)
V2
Calculate the Delay Time DT at
P1, P2, P3….PN
133. 8. Using equation 17,
DTpV 1
hp = (16)
−1 V 1
cos sin
V 2
Calculate the Depth h at P1, P2,
P3….PN
