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# Fundamentals of Seismic Refraction

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### Fundamentals of Seismic Refraction

1. 1. Fundamentals of Seismic RefractionTheory, Acquisition, and Interpretation Craig Lippus Manager, Seismic Products Geometrics, Inc. December 3, 2007
2. 2. Geometrics, Inc.• Owned by Oyo Corporation,Japan• In business since 1969• Seismographs, magnetometers, EM systems• Land, airborne, and marine• 80 employees
3. 3. Located in San Jose, California
4. 4. Fundamentals of Seismic Waves A. What is a seismic wave?
5. 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. 6. Three different types of seismic waves • Compressional (“p”) wave • Shear (“s”) wave • Surface (Love and Raleigh) waveOnly p and s waves (collectively referred toas “body waves”) are of interestin seismic refraction.
7. 7. Compressional (“p”) WaveIdentical to sound wave – particlemotion is parallel to propagationdirection. Animation courtesy Larry Braile, Purdue University
8. 8. Shear (“s”) WaveParticle motion is perpendicularto propagation direction. Animation courtesy Larry Braile, Purdue University
9. 9. Velocity of Seismic WavesDepends on density elastic moduli 4µ K+ Vp = 3 µ Vs = ρ ρ where K = bulk modulus, µ = shear modulus, and ρ = density.
10. 10. Velocity of Seismic WavesBulk modulus = resistance tocompression = incompressibilityShear modulus = resistance toshear = 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.
11. 11. Q. What is the rigidity ofwater?
12. 12. Q. What is the rigidity ofwater?A. Water has no rigidity. Its shearstrength is zero.
13. 13. Q. How well does watercarry shear waves?
14. 14. Q. How well does watercarry shear waves?A. It doesn’t.
15. 15. Fluids do not carry shear waves. Thisknowledge, combined with earthquakeobservations, is what lead to thediscovery that the earth’s outer core isa liquid rather than a solid – “shearwave shadow”.
16. 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 ρK and µ are always positive numbers, so Vpis always greater than Vs.
17. 17. Velocity – density paradoxQ. We know that in practice, velocitytends to be directly proportional todensity. Yet density is in thedenominator. How is that possible?
18. 18. Velocity – density paradoxQ. We know that in practice, velocitytends to be directly proportional todensity. Yet density is in thedenominator. How is that possible?A. Elastic moduli tend to increase with density also,and at a faster rate.
19. 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. 20. Q. How do we use seismicwaves to understand thesubsurface?
21. 21. Q. How do we use seismicwaves to understand thesubsurface?A. Must first understand wavebehavior in layered media.
22. 22. Q. What happens when aseismic wave encounters avelocity discontinuity?
23. 23. Q. What happens when aseismic wave encounters avelocity discontinuity?A. Some of the energy isreflected, some is refracted.We are only interested in refracted energy!!
24. 24. Q. What happens when aseismic wave encounters avelocity discontinuity?
25. 25. Five important concepts• Seismic Wavefront• Ray• Huygen’s Principle• Snell’s Law• Reciprocity
26. 26. Q. What is a seismicwavefront?
27. 27. Q. What is a seismicwavefront?A. Surface of constant phase, likeripples on a pond, but in threedimensions.
28. 28. Q. What is a seismicwavefront?
29. 29. The speed at which a wavefronttravels is the seismic velocity ofthe material, and depends on thematerial’s elastic properties. In ahomogenious medium, awavefront is spherical, and itsshape is distorted by changes inthe seismic velocity.
30. 30. Seismic wavefront
31. 31. Q. What is a ray?
32. 32. 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.
33. 33. Ray
34. 34. Huygens PrincipleEvery point on a wave front can bethought of as a new point source forwaves generated in the direction thewave is traveling or being propagated.
35. 35. Q. What causesrefraction?
36. 36. Q. What causesrefraction?A. Different portions of thewave front reach thevelocity boundary earlierthan other portions,speeding up or slowingdown on contact, causingdistortion of wave front.
37. 37. Understanding andQuantifying How Waves Refract is Essential
38. 38. Snell’s Law sin i V 1 = (1) sin r V 2
39. 39. Snell’s LawIf V2>V1, then as i increases, rincreases faster
40. 40. Snell’s Lawr approaches 90o as i increases
41. 41. Snell’s Law Critical RefractionAt Critical Angle of incidence ic, angle ofrefraction r = 90o sin(ic ) V 1 = sin 90 V 2 V1 sin(ic ) = (2) V2 V1 (3) ic = sin −1 V2
42. 42. Snell’s Law Critical RefractionAt Critical Angle of incidence ic, angle ofrefraction r = 90o
43. 43. Snell’s Law Critical RefractionAt Critical Angle of incidence ic, angle ofrefraction r = 90o
44. 44. Snell’s Law Critical RefractionSeismic refraction makes use ofcritically refracted, first-arrivalenergy only. The rest of the waveform is ignored.
45. 45. Principal of ReciprocityThe travel time of seismic energybetween two points is independent ofthe direction traveled, i.e.,interchanging the source and thegeophone will not affect the seismictravel time between the two.
46. 46. Using Seismic Refractionto Map the Subsurface Critical Refraction Plays a Key Role
47. 47. T 1 = x /V 1 ac cd dfT2 = + + V1 V 2 V1 hac = 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 ) xT2 = − + V 1 cos(ic ) V2 V2  1 sin(ic )  x T 2 = 2 h − +  V 1 cos(ic ) V 2 cos(ic )  V 2 
48. 48.  V2 V 1 sin(ic )  xT 2 = 2 h  V 1V 2 cos(ic ) V 1V 2 cos(ic )  + V 2 −     V 2 − V 1 sin(ic )  xT 2 = 2h  V 1V 2 cos(ic )  + V 2     V2   − sin(ic )  xT 2 = 2hV 1 V 1 +  V 1V 2 cos(ic )  V 2     V1sin ic = (Snell’s Law) V2  1   − sin(ic ) T 2 = 2hV 1 sin(ic ) + x  V 1V 2 cos(ic )  V 2    
49. 49.  1 − sin 2 (ic )  x T 2 = 2hV 1 +  V 1V 2 sin(ic ) cos(ic )  V 2   cos 2 (ic )  xT 2 = 2hV 1  V 1V 2 sin(ic ) cos(ic )  + V 2     cos(ic )  xT 2 = 2h  V 2 sin(ic )  + V 2    From Snell’s Law,V 1 = V 2 sin(ic ) 2h cos(ic ) xT2 = + (4) V1 V2
50. 50. Using Seismic Refractionto Map the Subsurface
51. 51. Using Seismic Refractionto Map the Subsurface
52. 52. Using Seismic Refractionto Map the Subsurface
53. 53. Using Seismic Refractionto Map the Subsurface
54. 54. Using Seismic Refractionto Map the Subsurface
55. 55. Using Seismic Refractionto Map the Subsurface
56. 56. Using Seismic Refractionto Map the Subsurface
57. 57. Using Seismic Refractionto Map the Subsurface
58. 58. Using Seismic Refractionto Map the Subsurface
59. 59. Using Seismic Refractionto Map the Subsurface
60. 60. Using Seismic Refractionto Map the Subsurface
61. 61. Using Seismic Refractionto Map the Subsurface
62. 62. Using Seismic Refractionto Map the Subsurface
63. 63. Using Seismic Refractionto Map the Subsurface
64. 64. Using Seismic Refractionto Map the Subsurface
65. 65. Using Seismic Refractionto Map the Subsurface
66. 66. Using Seismic Refractionto Map the Subsurface
67. 67. Using Seismic Refractionto Map the Subsurface
68. 68. Using Seismic Refractionto Map the Subsurface
69. 69. Using Seismic Refractionto Map the Subsurface Xc V 2 − V 1 Depth = (5) 2 V 2 +V1 {Depth
70. 70. Using Seismic Refractionto 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
71. 71. Summary of Important Equations For refractorsin i V 1 parallel to surface = (1) Snell’s Lawsin r V 2 2h cos(ic ) x T2 = + (4) V1 V2 V1sin(ic ) = (2) V2 Xc V 2 − V 1 h= (5) 2 V 2 +V1 V1ic = sin −1 (3) V2 Ti V 1 h=  −1 V 1  (6) 2 cos sin   V2
72. 72. Ti 2V 1h1 = V1 2 cos(sin −1 ) V2
73. 73.  cos(sin − 1 V 1 / V 3)  Ti 3 − Ti 2 V 2   cos(sin − 1 V 1 / V 2)   + h1h2 = 2 cos(sin − 1 V 2 / V 3)
74. 74.  cos(sin −1 V 1 / V 4) 2h 2 cos(sin −1 2 / V 4)  Ti 4 − Ti 2 cos(sin −1 V 1 / V 2) − V2 V 3h3 =   + h1 + h 2 2 cos(sin −1 V 3 / V 4)
75. 75. Crossover Distance vs.Depth
76. 76. Depth/Xc vs. Velocity Contrast
77. 77. Important Rule of Thumb The Length of the Geophone Spread Should be 4-5 times the depth of interest.
78. 78. Dipping LayerDefined as Velocity Boundarythat 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.
79. 79. Dipping LayerIf layer is dipping (relative to groundsurface), opposing travel time curveswill be asymmetrical.Updip shot – apparent velocity > true velocityDowndip shot – apparent velocity < true velocity
80. 80. Dipping Layer
81. 81. Dipping Layer V 1md = sin(ic + α ) V 1mu = sin(ic − α ) ic + α = sin −1 V 1md ic − α = sin −1 V 1mu 1ic = (sin −1 V 1md + sin −1 V 1mu ) 2 1α = (sin −1 V 1md − sin −1 V 1mu ) 2
82. 82. Dipping LayerFrom Snell’s Law, V1 V2 = sin(ic ) V 1Tiu 2 cos(ic ) Du = cos α V 1Tid 2 cos(ic ) Dd = cos α
83. 83. Dipping LayerThe true velocity V2 can also be calculatedby multiplying the harmonic mean of the up-dip and down-dip velocities by the cosine ofthe dip.  2V 2UV 2 D  V2 =  cos α  V 2U + V 2 D 
84. 84. What if V2 < V1?
85. 85. What if V2 < V1? sin i V 1Snell’s Law = sin r V 2
86. 86. What if V2 < V1? sin i V 1Snell’s Law = sin r V 2
87. 87. What if V2 < V1?If V1>V2, then as i increases, rincreases, but not as fast.
88. 88. If V2<V1, the energyrefracts toward thenormal.None of the refracted energymakes it back to the surface.This is called a velocity inversion.
89. 89. Seismic Refractionrequires that velocitiesincrease with depth.A slower layer beneath afaster layer will not bedetected by seismic refraction.The presence of a velocity inversion canlead to errors in depth calculations.
90. 90. Delay Time Method• Allows Calculation of DepthBeneath Each Geophone• Requires refracted arrival at eachgeophone from opposite directions• Requires offset shots• Data redundancy is important
91. 91. Delay Time MethodxV1V2
92. 92. Delay Time Method x V1 V2 hA AB hA tan(ic ) hB tan(ic ) hBTAB ≅ + − − + V 1 cos(ic ) V 2 V2 V2 V 1 cos(ic )
93. 93. Delay Time Method x V1 V2 hA AB hA tan(ic ) hB tan(ic ) hBTAB ≅ + − − + V 1 cos(ic ) V 2 V2 V2 V 1 cos(ic ) hA AP hA tan(ic ) hP tan(ic ) hPTAP ≅ + − − + V 1 cos(ic ) V 2 V2 V2 V 1 cos(ic )
94. 94. Delay Time Method x V1 V2 hA AB hA tan(ic ) hB tan(ic ) hBTAB ≅ + − − + V 1 cos(ic ) V 2 V2 V2 V 1 cos(ic ) hA AP hA tan(ic ) hP tan(ic ) hPTAP ≅ + − − + V 1 cos(ic ) V 2 V2 V2 V 1 cos(ic ) hB BP hB tan(ic ) hP tan(ic ) hPTBP ≅ + − − + V 1 cos(ic ) V 2 V2 V2 V 1 cos(ic )
95. 95. Delay Time MethodxV1V2 Definition: t0 = T AP + T BP − T AB (7)
96. 96. 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
97. 97. 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
98. 98.   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 )        
99. 99.   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 )     
100. 100. Substituting from Snell’s Law,  V1   sin ic =   V2We get 2hp cos(ic ) t0 = (8) V1
101. 101. to 2hp cos(ic ) hp cos(ic )Delay time at point P = DTP = = = (9) 2 2V 1 V1
102. 102. 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. Aplot of T’ vs. x will be roughly linear, mostly unaffectedby changes in layer thickness, and the slope will be1/V2.
103. 103. Reduced Traveltimes xFrom the above figure, T’AP is also equal to TAP minus theDelay Time. From equation 9, we then get to T AP = TAP − DTP = TAP − 2
104. 104. Reduced Traveltimes xEarlier, 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
105. 105. Reduced Traveltimes Finally, rearranging yields T AB (T AP − T BP ) T AP = + (11) 2 2The above equation allows a graphical determination of the T’curve. TAB is called the reciprocal time.
106. 106. Reduced Traveltimes T AB (T AP − T BP ) T AP = + 2 2The first term is represented by the dotted line below:
107. 107. Reduced Traveltimes T AB (T AP − T BP ) T AP = + 2 2The numerator of the second term is just the difference in thetraveltimes from points A to P and B to P.
108. 108. Reduced Traveltimes T AB (T AP − T BP ) T AP = + 2 2Important: The second term only applies to refracted arrivals. Itdoes not apply outside the zone of “overlap”, shown in yellowbelow.
109. 109. Reduced Traveltimes T AB (T AP − T BP ) T AP = + 2 2The T’ (reduced traveltime) curve can now be determined graphicallyby 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.
110. 110. We can now calculate the delay time at point P. From Equation 10,we see that to T AP = TAP − (10) 2According to equation 8 to hp cos(ic ) = (8) 2 V1So t0 hp cos(ic ) T AP = TAP − = TAP − (12) 2 V1Now, referring back to equation 4 2h cos(ic ) x T2 = + (4) V1 V2
111. 111. It’s fair to say that 2hp cos(ic ) x TAP ≅ + (13) V1 V2Combining equations 12 and 13, we get hp cos(ic ) 2hp cos(ic ) x hp cos(ic ) T AP = TAP − = + − V1 V1 V2 V1Or hp cos(ic ) x T AP = + (14) V1 V2
112. 112. Referring back to equation 9, we see that hp cos(ic ) DTp = (9) V1Substituting into equation 14, we get hp cos(ic ) x x T AP = + = DTp + V1 V2 V2Or x DTp = T AP − (15) V2Solving equation 9 for hp, we get D TPV 1 hP = (16) c o s (ic)
113. 113. 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 
114. 114. Substituting back into equation 16, DTpV 1 hp = (16) cos(ic )we get DTpV 1 hp =  −1  V 1  cos sin   (17)   V 2 
115. 115. In summary, to determine thedepth to the refractor h at anygiven point p:
116. 116. 1.Measure V1 directly from the traveltime plot.
117. 117. 2.Measure the difference in traveltime to point P from opposing shots (in zone of overlap only).
118. 118. 3.Measure the reciprocal time TAB.
119. 119. T (T AP − T BP )4. Per equation 11, T A P = AB + , 2 2divide the reciprocal time TAB by 2.
120. 120. T (T AP − T BP )5. Per equation 11, T A P = 2 + AB , 2add ½ the difference time at eachpoint P to TAB/2 to get the reducedtraveltime at P, T’AP.
121. 121. 6. Fit a line to the reduced traveltimes, compute V2 from slope.
122. 122. 7. Using equation 15, x DTp = T AP − (15) V2 Calculate the Delay Time DT at P1, P2, P3….PN
123. 123. 8. Using equation 17, DTpV 1 hp = (16)  −1  V 1  cos sin     V 2  Calculate the Depth h at P1, P2, P3….PN
124. 124. That’s all there is to it!
125. 125. More Data is Better Than Less
126. 126. More Data is Better Than Less
127. 127. More Data is Better Than Less
128. 128. More Data is Better Than Less
129. 129. More Data is Better Than Less
130. 130. More Data is Better Than Less
131. 131. More Data is Better Than Less
132. 132. More Data is Better Than Less
133. 133. More Data is Better Than Less
134. 134. More Data is Better Than Less