GEOL 335.3
Refraction Seismic Method
Intercept times and apparent velocities;
Critical and crossover distances;
Hidden layers;
Determination of the refractor velocity and depth;
The case of dipping refractor
Inversion methods:
Hagedoorn plus-minus method;
Generalized Reciprocal Method;
Travel-time continuation.
Reading:
➢
Reynolds, Chapter 5
➢
Shearer, Chapter 4
➢
Telford et al., Sections 4.7.9, 4.9
GEOL 335.3
Refraction Seismic Method
Uses travel times of refracted arrivals to derive:
1) Depths to velocity contrasts (“refractors”);
2) Shapes of refracting boundaries;
3) Seismic velocities.
GEOL 335.3
Apparent Velocity
Relation to wavefronts
Apparent velocity, Vapp, is the velocity at which the
wavefront sweeps across the geophone spread.
Because the wavefront also propagates upward,
Vapp, ≥ Vtrue:
AC=
BC
sin 

V app =
V
.
sin 
2 extreme cases:
➢
θ = 0: Vapp = ∞;
➢
θ = 90°: Vapp = Vtrue.
Apparent propagation direction
C
θ
wa v e
front
Propaga
tio
directio n
n
A
B
GEOL 335.3
Two-layer problem
One reflection and one refraction
t
At pre-critical
offsets,
record direct wave
and reflection
In post-critical domain,
record direct wave,
refraction, and reflection
d:
Reflecte
al
critic
t
s
o
p
pre-critical
t0
ec
Dir
t
t=
Head
x
t =t 0 =t 0 x p2
V2
x
=x p1
V1
xcritical
x
xcrossover
S
cted
i
Refl
e
h1
Direct
Ref
ra
V1
ic
c te d
Headwave
V2>V1
GEOL 335.3
Travel-time relations
Two horizontal layers
x
S
R
i
h1
V1
ic
S'
A
B
V2>V1
R'
For a head wave (“often called refraction”):
p=
1
V2
cos i c =1− pV 1 2
sin i c = pV 1
“linear moveout”
term
h1
t =2
 p  x−2 h1 tan i c =t 0  px
V 1 cos i c
h1
2h
t 0=2
1− pV 1 sin i c = 1 cos i c
V 1 cos i c
V1
intercept time
this also equals sin i
For a reflection (we'll use this later):
pV 1=sin i
x
tan i =
2h1
t =2


h12
V1
x
2
hyperbolic
moveout”
4 h1  x

=
V1
2
2
here, p is variable and controlled
By arbitrary angle i
GEOL 335.3
Critical and cross-over
distances
xcritical
S
h1
xcrossover
V1
ic
A
B
V2>V1
Critical distance:
x critical =2 h1 tan i c =2 h1
V 1/ V 2
 1−V
1 /V 2 
Cross-over distance:
t direct  x crossover =t headwave  x crossover 
x crossover
x crossover
=t 0
V1
V2
t0
x crossover =
1/ V 1−1/V 2
“slownesses”
2
=
2 h1 V 1
V
2
2
2
−V 1
GEOL 335.3
Multiple-layer case
(Horizontal layering)
p is the same
1
critical ray p=
V refractor
parameter;
t0 is
accumulated
across the
layers:
n
t =∑
k=1
2 hk
cos i k  px
Vk
sin i k = p V k in any layer
Critical p determined here
GEOL 335.3
Dipping Refractor Case
shooting down-dip
x
S
hd
R
x(cosa-sinα tanic)
xsinα
ic
ic
R'
A
B
xsinα/cosic
hu
V1
α (dip)
V2>V1
t=
t=
t=
2h d
1
1 x sin 
cos i c 
x cos −sin  tan i c 
V1
V2
V 1 cos i c
sinic
[
2h d
V1
x
cos i c 
cos  cos i c−sin sin i c sin 
V1
V 1 cos i c V 2
]
2h d
x
cos i c  cos sin i c sin  cos i c 
V1
V1
t=
2h d
x
cos i c  sin i c 
V1
V1
would change to '-' for up-dip recording
GEOL 335.3
Refraction Interpretation
Reversed travel times
One needs reversed recording (in opposite directions) for
resolution of dips.
The reciprocal times, TR, must be the the same for
reversed shots.
Dipping refractor is indicated by:
Different apparent velocities (=1/p, TTC slopes) in the two
directions;
determine V2 and α (refractor velocity and dip).
➢
Different intercept times.
determine hd and hu (interface depths).
➢
TR
t
pd=
pu=
sin i c 
V1
sin i c −
V1
2h u cosi c
V1
2h d cosi c
V1
S1
slope= p 1=
1
V1
x
S2
GEOL 335.3
Determination of Refractor
Velocity and Dip
Apparent velocity is Vapp = 1/p, where p is the ray
parameter (i.e., slope of the travel-time curve).
Apparent velocities are measured directly from the
observed TTCs;
Vapp = Vrefractor only in the case of a horizontal layering.
For a dipping refractor:
V1
sin i c 
V1
V u=
sini c −
➢
Down dip: V d =
(slower than V2);
➢
Up-dip:
(faster).
From the two reversed apparent velocities, ic and α
are determined:
i c =sin−1
V1
,
Vd
i c −=sin−1
V1
V u.
1
−1 V 1
−1 V 1
i c = sin
sin
,
2
Vd
Vu
1
−1 V 1
−1 V 1
= sin
−sin
.
2
Vd
Vu
From ic, the refractor velocity is:
V1
V 2=
.
sin i c
GEOL 335.3
Determination of Refractor Depth
From the intercept times, td and tu, refractor depth is
determined:
V 1 td
,
2 cos i c
V t
h u= 1 u .
2 cos i c
hd =
x
S
hd
R
ic
ic
A
B
hu
V1
α (dip)
V2>V1
GEOL 335.3
Delay time
Consider a nearly horizontal, shallow interface with
strong velocity contrast (a typical case for weathering
layer).
In this case, we can separate the times associated with the
source and receiver vicinities: tSR = tSX + tXR.
Relate the time tSX to a time along the refractor, tBX:
tSX = tSA – tBA + tBX = tS Delay+x/V2.
t S Delay =
hs
h tan i c
hs
h cosic
SA BA
−
=
− s
=
1−sin 2 i c = s
V 1 V 2 V 1 cos i c
V2
V 1 cos i c
V 1.
Note that V2=V1/sinic
Thus, source and receiver delay times are:
h cosi c
t S , R Delay = s ,r
V 1.
and t SR=t S Delay t R Delay 
x
S
hs ic
R
hr
h/cosic
A
B
htanic
X
V1
V2>V1
SR
V 2.
GEOL 335.3
Plus-Minus Method
(Weathering correction; Hagedoorn)
Assume that we have recorded two headwaves in
opposite directions, and have estimated the velocity of
overburden, V1.
How can we map the refracting boundary?
S1
S2
D(x)
t
TR
tS2 D
tS1 D
V1
x
S1
Solution:
➢
Profile S1 → S2:
➢
Profile S2 → S1:
S2
D
x
t t D ;
V2 S
 S S −x 
t S D= 1 2
t S t D.
V2
t S D=
1
1
2
2
Form PLUS travel-time:
t PLUS =t S Dt S D=
1
Hence:
2
S1 S 2
t S t S 2t D =t S S 2t D.
V2
1
1
t D = t PLUS −t S
2
2
1
S2
1
2
.
To determine ic (and depth), still need to find V2.
GEOL 335.3
Plus-Minus Method
(Continued)
To determine V2:
this is a constant!
Form MINUS travel-time:
t MINUS =t S D −t S D=
1
Hence:
2
2 x S1S2
−
t s −t s .
V2
V2
1
slope[ t MINUS  x ]=
2
2
.
V2
The slope is usually estimated by using the
Least Squares method.
Drawback of this method – averaging over the
pre-critical region.
S1
D(x)
S2
V1
TR
t
tS2 D
tS1 D
x
S1
D
S2
GEOL 335.3
Generalized Reciprocal
Method (GRM)
Introduces offsets ('XY') in travel-time readings in the
forward and reverse shots;
so that the imaging is targeted on a compact interface region.
Proceeds as the plus-minus method;
Determines the 'optimal' XY:
1) Corresponding to the most linear velocity analysis function;
2) Corresponding to the most detail of the refractor.
XY
S1
TR
S2
D
t
tS2 D
tS1 D
V1
x
S1
The velocity analysis function:
t V=
1
t −t t
2 S D S D S
1
2
1
S2
S2
D
XY

should be linear, slope = 1/V2;
The time-depth function:
t D=


1
XY
t S D t S D−t S S −
.
2
V2
1
2
1
2
this is related to the desired depth:
hD=
tD V 1 V 2
V
2
2
−V 1
2
GEOL 335.3
Head-wave “migration”
(travel-time continuation) method
“Migration” refers to transforming the spacetime picture (travel-time curves here) into a
depth image (position of refractor).
Refraction (head-wave) migration:
Using the observed travel times, draw the headwave wavefronts in depth;
Identify the surface on which:
t forward  x , z t reversed  x , z =T R
This surface is the position of the refractor.
x
S1
S2
z
Constant travel
times from S2
P
Constant travel
times from S1
GEOL 335.3
Phantoming
Refraction imaging methods work within the region
sampled by head waves, that is, beyond critical
distances from the shots;
In order to extend this coverage to the shot points,
phantoming can be used:
Head wave arrivals are extended using time-shifted
picks from other shots;
However, this can be done only when horizontal
structural variations are small.
t
Phantom arrivals
x
GEOL 335.3
Hidden-Layer Problem
Velocity contrasts may not be visible in refraction
(first-arrival) travel times. Three typical cases:
Low-velocity layers;
Relatively thin layers on top of a strong
velocity contrast;
Short travel-time branch may be missed with sparse
geophone coverage.