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Geophysics

VIEWS: 17 PAGES: 8

									Subject 6: Geometry of Reflected and Refracted wave path
Lecturer: Dr. Bakhtiar Q. Aziz
Objective: Mathematically, the equation of travel time will drive for the students to
learn step by step how a path of a single ray is propagated from surface to the
interface and how the reflected and refracted travel time of this path is calculated.
The mathematical expression is calculated for both horizontal and inclined cases. In
addition, Normal move out and Dip move out will clarify for the students.

Scientific contents
1- Geometry of reflected ray path for both horizontal and dip layers.

2- Geometry of refracted ray path for both horizontal and dip layers.

3- Normal move out and dip move out and their effect on the seismic records.


References

   1.   An introduction to applied and environmental geophysics, 1997, Reynolds, J. M.
   2.   Applied and environmental geophysics, 1999, Sharma,V.,P.
   3.   Introduction to geophysical prospecting, 1988, Durbin, M. B.
   4.   www.Geophysics.net
    Geometry of reflected wave path:
         1- Horizontal Reflector
The travel time from (S) to (G) calculated by
using image point, the image of (S) is (I) at
                                                                           Time
                                                                           Time
depth (h) below the reflector.
                                                                                                 Reflected
                                                                                                 Reflected
Travel time of direct wave=TD= Distance / V                                                      Wave
                                                                                                 Wave

Suppose the travel time of reflected wave                                          t0
                                                                                   t0
from (S) to (G) is (trefl).                                                                                  Direct Wave
                                                                                                             Direct Wave

t   refl   = Distance / Velocity
Distance = SC + CG = IG due to SC = IC
From Triangle SGI IG2= X2 + (2h)2                                                                              Distance
                                                                                                               Distance
                                                       G
                                                       G                   S
                                                                           S                 XX               G Surface
                                                                                                              G Surface
IG = t * V                                                                              Direct Wave
                                                                                        Direct Wave
(tv)2= X2 + (2h)          Divide by 4h2                                            φ
                                                                                   φ
                                                                            h
                                                                            h
                                                                                           φ φ
                                                                                           φ φ        Velocity= V
t2v2                 x2
                              =1
4h2                 4h2        Hyperbola Equation                                            C
                                                                                             C
                                                                                                                Reflector
                                                                                                                Reflector
Note:
                                                                            h
                                                                            h
1- When x is very large trefl = tD                                              φ
                                                                                φ

2- Depth of the reflector is obtained by
                                                                               I
                                                                               I
plotting x=0
t2v2
            =1,
                     h = ½ * V t0     So The above equation can written as follow: t2 = x2/V2 + t02
     2
   2- Dip Reflector


The travel time from (S) to (G) calculated by
using image point, the image of (S) is (I) at                 Time
depth (h) below the reflector.
                                                                                   Reflected
Travel time of direct wave=TD= Distance / V                                        Wave
Suppose the travel time of reflected wave from
(S) to (G) is (trefl).
                                                                                             Directed
t refl = Distance / Velocity                                                                 Wave
Distance = SC + CG = IG due to SC = IC
From Triangle SGI, depend on Cosine Law                           t0=2h/V
                                                     tm   Xm
IG2 = x2 + 4h2 – 4hx Cos (  \ 2   )                                                         Distance
V2t2 = x2 + 4h2 + 4hx Sin θ                      G            S               X                G Surface
                                                                                  Directed
                                                                      90+        Wave
Divide the equation by (2h Cos θ):                                φ
                                                              h           φ φ
       vt   2 2
                  ( x  2hSin )     2
                                1
    (2hCos ) 2
                    2hCos 
                             2
                                                          h
                                                                              C
                                                                                     
                                                                                                Reflector

                                                              φ
  This is equation of hyperbola

     tm 
            2hCos
              V
                       xm  2hSin                       I
Notes:
1- The apex of the curve is always shifted toward up dip and with axis of
symmetry equal to x = -2h Sinθ.
2- The dip angle (θ) is calculated by:


                    td
A-    Sin  v(         )                     ∆td is dip move out
                    2x



                       Xm
B-       tan   v(      )
                       tm


C- Depth of the reflector calculated by:
h = ½ V to
What are ?
1- Normal move out (NMO)
2- Dip move out (DMO)


1- Normal move out (NMO)
It is the difference between time of two geophones one placed on shot point and other at
distance (x) from the shot point. It is difference between tx and to.


tx is two way slant time        and to is two way vertical time

                           to                         tx




  tx




    tn  t x  t0  ( x / v)  t0  t0         2          2
  Effect of NMO:
  1- Distort seismic record
  2- Delay arrival time
  Note: The effect of NMO is decreases with depth, because the slant
  path of the wave is approach approximately with the vertical path.




                                                                       ∆tn1
Layer 1
                                                                        ∆tn2
Layer 2

                                                                       ∆tn3
Layer 3


                                                                       ∆tn4= 0
Layer 4
NMO and horizontal distance:
NMO increases rapidly when the distance between geophone and shot point is
increased as illustrated in the following figure.

                     G0      G1         G2      G3       G4            G5

                                                                        Surface




                                                                 Reflector



                     G0     G1          G2      G3      G4            G5
                 1                                                            Distance, m
                 2
                                 t 1                                        t 1  3  2  1m sec
                 3                       t 2
                                                 t 3                        t 2  4  2  2m sec
                                                          t 4
                 4                                                      t 5 t 3  5  2  3m sec
                 5                                                           t 4  6  2  4m sec
                 6                                                           t 5  7  2  5m sec
                 7
                 8
                 9
                10
           Time, msec.
 2- Dip move out (DMO)
 It is the difference between time of two geophones one placed at up dip direction
 and other at down dip direction.


              Dip Reflector Case                                    Horizontal Reflector Case

G1                                         G2               G1                S                     G2
                        S        X                                  X                      X
          X
                                                Surface
                                                                                      Direct Wave
                                                                                  φ
                    h           φ φ
                                                                             h
                                                                                         φ φ


                                                Reflector
Up dip direction                Down dip direction

  ∆td = t Down dip – t Up dip

  ∆td = tG2 – tG1                                                ∆t = tG2 – tG1 = 0
  ∆td = 2x Sin θ / V

								
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