Geology 5660/6660
Applied Geophysics
5 Feb 2016
Last time: Refraction Plus-Minus; Reflection:
• Plus-Minus Method for arbitrarily changing h:
2
V2 =
m where m– is slope of a plot of t1i–t2i vs xi for SP1, 2 and
geophones i
• Seismic Reflection Travel-Times have eqns of a
hyperbola. For single layer over halfspace,
x 2 4h12 has intercept 2h1/V1 and
2
t = 2+ 2
slope of the asymptote is 1/V1!
V1
V1
• Defined Normal Move-Out (NMO) as the reflection
travel-times at distance x minus t at x = 0:
tNMO =
x 2 + 4h12 2h1
V1
V1
For Mon 08 Feb: Burger 167-199 (§4.2-4.3)
© A.R. Lowry 2016
Distance
water
Two-Way Travel Time
shale
gas sand
shale
Shifting the seismic reflection amplitude to where it would be
(in two-way travel-time) for zero-offset produces an image…
Note that this approach is VERY different from the model of
velocity structure we generate from the refraction method!
Reflection from a second layer interface over half-space:
x/2
V1
h1
V2
h2
Can derive using Snell’s law but easier to consider that:
• For x = 0,
 intercept of hyperbola
• For x  , asymptotic to the layer 2 refraction
2
2
x 2h1 V2 -V1
t= +
V2
V1V2
Equation for the hyperbola then is
After some algebra we have
æ 2h V 2 -V 2 ö
1 ÷
V çt - 1 2
ç
÷
V1V2
è
ø
2
2
2
ì é æ
ùüï
2
2 ö
ï
V
-V
1 ÷
- x12 = í2 êh1 ç1- 2
+ h2 úý
ç
úûï
V1V2 ÷ø
ïî êë è
þ
2
And you might see where this could start to get complicated
for 3, 4, … layers
This is part of why industry seismic reflection processing
historically did not go after full seismic velocity analysis
but instead took shortcuts to imaging of structures…
A Dipping Reflector:
V1
h1


Geometrically, this is equivalent to rotating the axis
of the reflector by the dip angle . This rotates the
hyperbola on the travel-time curve by  = tan-1(–2h1sin)
and has equation
V12 t 2 - x 2 = 4h1(h1 - x sina )
65
60
55
Time (ms)
70
Example: V1 = 1500, h = 45 m,  = 8°
-40.00
-20.00
0.00
20.00
40.00
Distance (m)
Helpful Hint: Reflect does not model dipping layers, but
the Table 4-6 Excel spreadsheet does!
Getting velocity structure: The x2 – t2 method
If we have travel-times from a reflection, can plot t2 vs x2
to get parameters of thickness & velocity from slope
and intercept of the resulting line fit!
Recall that the goal of geophysics is to invert for
parameters (in this case, velocity and thickness)
Given travel-times from a reflection, can plot t2 vs x2
to get parameters of thickness & velocity from slope and
intercept of the resulting line fit.
For the single layer case:
2
x 2 + 4h12
1
h
t=
Þ t 2 = 2 x 2 + 4 12
V1
V1
V1
So for slope m, intercept t02 of a line fit:
t2
m=
t02
4h12
= 2
V1
x2
1
V1 =
m
t0V1
h1 =
2
1
V12
Note x2 – t2 method is also useful for dipping layers!

down-dip
limb
t
t2
x = 2h1 sina
x
x2
2
4h
sin
a
4h
t 2 = 2 - 1 2 x + 21
V1
V1
V1
x2
mean
slope
1
2
up-dip V1
limb
Note this is the approach taken to
inverting data in Reflect!