Seismic imaging is the topic of this presentation Based on Seismic Inversion by by Schuster, G.T.

1. From observed data on the
surface to the subsurface image
Kamal Aghazade
Reference: Seismic Inversion by Schuster, G.T

2. As a seismic explorer what are we looking for?
Look at this real Earth model from Angola
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3. As a seismic explorer what are we looking for?
Look at this real Earth model from Angola
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4. As a seismic explorer what are we looking for?
Look at this real Earth model from Angola
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5. As a seismic explorer what are we looking for?
Look at this real Earth model from AngolaNow suppose this is our earth model
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6. As a seismic explorer what are we looking for?
Look at this real Earth model from AngolaNow suppose this is our earth model
We have No idea about the
subsurface.
But we want to explore it!
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7. As a seismic explorer what are we looking for?
Look at this real Earth model from AngolaNow suppose this is our earth model
We have No idea about the
subsurface.
But we want to explore it!
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8. As a seismic explorer what are we looking for?
Look at this real Earth model from AngolaNow suppose this is our earth model
We have No idea about the
subsurface.
But we want to explore it!
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9. As a seismic explorer what are we looking for?
Look at this real Earth model from AngolaNow suppose this is our earth model
We have No idea about the
subsurface.
But we want to explore it!
How?
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10. As a seismic explorer what are we looking for?
Look at this real Earth model from AngolaNow suppose this is our earth model
We have No idea about the
subsurface.
But we want to explore it!
How?
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11. As a seismic explorer what are we looking for?
Look at this real Earth model from AngolaNow suppose this is our earth model
We have No idea about the
subsurface.
But we want to explore it!
How?
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12. As a seismic explorer what are we looking for?
Look at this real Earth model from AngolaNow suppose this is our earth model
We have No idea about the
subsurface.
But we want to explore it!
How?
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13. As a seismic explorer what are we looking for?
Look at this real Earth model from AngolaNow suppose this is our earth model
We have No idea about the
subsurface.
But we want to explore it!
How?
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14. As a seismic explorer what are we looking for?
Look at this real Earth model from AngolaNow suppose this is our earth model
We have No idea about the
subsurface.
But we want to explore it!
How?
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15. We have data set.
What is the next step?
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16. We have data set.
What is the next step?
After some
processing
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17. We have data set.
What is the next step?
After some
processing
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18. We have data set.
What is the next step?
After some
processing
Stacked section
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19. We have data set.
What is the next step?
After some
processing
Stacked section
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20. We have data set.
What is the next step?
After some
processing
Stacked section
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21. We have data set.
What is the next step?
After some
processing
Stacked section
What is the solution?
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22. 41

23. What is migration?
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24. What is migration?
Lets start from the first,
when we propagate
waves into subsurface.
• We want to image
subsurface.
• We want to
estimate
reflectivity model.
• How it can be
done?
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25. What is migration?
Lets start from the first,
when we propagate
waves into subsurface.
• We want to image
subsurface.
• We want to
estimate
reflectivity model.
• How it can be
done?
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26. What is migration?
Lets start from the first,
when we propagate
waves into subsurface.
• We want to image
subsurface.
• We want to
estimate
reflectivity model.
• How it can be
done?
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27. What is migration?
Lets start from the first,
when we propagate
waves into subsurface.
• We want to image
subsurface.
• We want to
estimate
reflectivity model.
• How it can be
done?
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28. What is migration?
Lets start from the first,
when we propagate
waves into subsurface.
• We want to image
subsurface.
• We want to
estimate
reflectivity model.
• How it can be
done?
Forward and Adjoint
modeling using Green’s
function, which direction?
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29. What is migration?
Lets start from the first,
when we propagate
waves into subsurface.
• We want to image
subsurface.
• We want to
estimate
reflectivity model.
• How it can be
done?
Forward and Adjoint
modeling using Green’s
function, which direction?
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30. What is migration?
Lets start from the first,
when we propagate
waves into subsurface.
• We want to image
subsurface.
• We want to
estimate
reflectivity model.
• How it can be
done?
Forward and Adjoint
modeling using Green’s
function, which direction?
Given s-x coordinates, the
source wavelet and velocity
model our goal with
forward modeling is to use
Green’s theorem to find
pressure field.
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32. 51

33. 52

34. 53

35. 54

36. How can we use Green’s function?
What is its rule in forward
modeling?
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37. How can we use Green’s function?
What is its rule in forward
modeling?
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38. 57

39. 58

40. Now we have this, sounds cumbersome 
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41. Now we have this, sounds cumbersome 
If only outgoing waves are considered and the surface S0 is infinity we
have:
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42. Now we have this, sounds cumbersome 
If only outgoing waves are considered and the surface S0 is infinity we
have:
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43. Now we have this, sounds cumbersome 
If only outgoing waves are considered and the surface S0 is infinity we
have:
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44. Look at this relation:
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45. Look at this relation:
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46. Look at this relation:
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47. Look at this relation:
So What?
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48. Look at this relation:
So What?
applying the appropriate Green’s function to the Helmholtz equation and
integrating over the volume is the inverse operator to the Helmholtz
equation.
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49. Look at this relation:
So What?
applying the appropriate Green’s function to the Helmholtz equation and
integrating over the volume is the inverse operator to the Helmholtz
equation.
Welcome to the Green’s function world 
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50. I’m really sorry, this is not the end 
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51. I’m really sorry, this is not the end 
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52. I’m really sorry, this is not the end 
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53. I’m really sorry, this is not the end 
Has cumbersome computation!!
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54.  Lippmann-Schwinger solution
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55.  Lippmann-Schwinger solution
This equation has cumbersome computation too!!
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56. 75

57. We have two methods for finding an
approximate solution:
 Neumann series solution
 Born approximation
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58. We have two methods for finding an
approximate solution:
 Neumann series solution
 Born approximation
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59. We have two methods for finding an
approximate solution:
 Neumann series solution
 Born approximation
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60. We have two methods for finding an
approximate solution:
 Neumann series solution
 Born approximation
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62. 81

63. What is Born approximation?
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64. The first Born approximation, or simply the Born
approximation, is obtained by approximating the scattered
field by the first-order term of the Neumann series
in Neumann solution:
(0)
P P
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65. The first Born approximation, or simply the Born
approximation, is obtained by approximating the scattered
field by the first-order term of the Neumann series
in Neumann solution:
(0)
P P
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66. The first Born approximation, or simply the Born
approximation, is obtained by approximating the scattered
field by the first-order term of the Neumann series
in Neumann solution:
(0)
P P
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67. The first Born approximation, or simply the Born
approximation, is obtained by approximating the scattered
field by the first-order term of the Neumann series
in Neumann solution:
(0)
P P
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68. The first Born approximation, or simply the Born
approximation, is obtained by approximating the scattered
field by the first-order term of the Neumann series
in Neumann solution:
(0)
P P
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69. The first Born approximation, or simply the Born
approximation, is obtained by approximating the scattered
field by the first-order term of the Neumann series
in Neumann solution:
(0)
P P
Lets go on with linear algebra 
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70. We now have this equation:
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71. We now have this equation:
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72. We now have this equation:
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73. We now have this equation:
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74. We now have this equation:
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75. We now have this equation:
Least Squares solution
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76. We now have this equation:
Least Squares solution
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77. We now have this equation:
Least Squares solution
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78. We now have this equation:
Least Squares solution
This is Least Squares Migration
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79. General Imaging Algorithms
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80. General Imaging Algorithms
The goals of seismic imaging and migration are to estimate
slowness and reflectivity distribution, respectively.
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81. General Imaging Algorithms
The goals of seismic imaging and migration are to estimate
slowness and reflectivity distribution, respectively.
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82. General Imaging Algorithms
The goals of seismic imaging and migration are to estimate
slowness and reflectivity distribution, respectively.
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86. Slowness model in terms of Green’s function
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87. Slowness model in terms of Green’s function
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88. Please stay with me and look at this formulae
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89. Please stay with me and look at this formulae
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90. Please stay with me and look at this formulae
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91. Please stay with me and look at this formulae
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92. Please stay with me and look at this formulae
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94. 113

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96. Next presentation is LSM
Have a best migrated day.
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