Geology Reference
In-Depth Information
By Snell's law we have that
s
0
sin™
0
D
p
, thereby
dp
(™
0
)
D
s
0
cos™
0
d
™
0
. F -
mo , t su : an™
0
D
p
/(
s
0
p
2
)
1/2
and
d
™
0
/
dX
D
(
s
0
p
2
)
1/2
dp
/
dX
. Therefore, we
obtain the following expression for the average
energy density at distance
X
:
LJ
LJ
LJ
LJ
LJ
LJ
LJ
LJ
pE
4 X
s
0
p
2
dp
dX
h
E.X/
iD
pE
4 X
s
0
p
2
j
dX=dp
j
D
(9.68)
Fig. 9.14
Lower focal hemisphere about a seismic source
S
and band through which the energy associated with
rays having take-off angle between ™
0
and ™
0
C
d
™
0
is
The role of the term
j
dX
/
dp
j
in expression
(
9.68
) is quite intuitive. When this quantity is
small, a large number of seismic rays with differ-
ent parameters strike the Earth's surface at com-
parable distance
X
. In this instance, the energy
density and the amplitude of seismic waves attain
elevated values. By contrast, when
j
dX
/
dp
j
is
high, rays with comparable parameters distribute
on a large area, thereby the density of energy and
the amplitude are small. Finally, for
j
dX
/
dp
jD
0,
hence at the caustics, the energy density (
9.68
)
is infinite. Clearly, this result is valid only in
the infinite frequency limit of ray theory; real
amplitudes of the seismic waves and the energy
density are elevated but finite at the caustics.
So far, we have represented travel time curves
in terms of (
T
,
X
) pairs. In this context,
T
D
T
(
X
)is
not
generally a single-value function
because of triplications. An alternative represen-
tation, which does not suffer the problems asso-
ciated with triplication, is based on pairs (£,
p
),
where the quantity £ is called
delay time
and
can be calculated easily taking the intercept of a
tangent to the travel time curve:
irradiated
expect that focusing and defocusing of seismic
rays associated with geometrical spreading de-
termine the distribution of seismic energy along
the wavefronts. Let us assume that a source at
some depth below the Earth's surface irradiates
isotropically a seismic energy
E
, and consider the
rays with take-off angle between ™
0
e ™
0
C
d
™
0
(Fig.
9.14
).
These rays leave a unit sphere about the
source through a horizontal band having area
2 sin™
0
d
™
0
(Fig.
9.14
). Because the total area of
the unit sphere is 4 and the radiation of seismic
energy
is
isotropic,
the
energy
transmitted
through the band is:
1
2
E sin ™
0
d™
0
dE .™
0
/
D
(9.65)
The corresponding rays strike the Earth's
surface through a ring belt having area
2
X
(™
0
)
dX
, while the wavefront portion will have
area 2
X
(™
0
)cos™
0
dX
, because the incidence
angle at
X
coincides with the take-off angle.
Therefore,
the
energy
distributed
along
this
£.p/
D
T.X/
pX.p/
(9.69)
wavefront portion will be given by:
In fact, any point belonging to a travel time
curve can be uniquely identified by an intercept
along the vertical axis and a straight line with
appropriate slope
p
as illustrated in Fig.
9.15
.
Although less intuitive than the previous one, this
representation allows an elegant solution to the
inversion of seismic data. In the case of a later-
ally homogeneous material, we can substitute the
dE.X/
D
2
h
E.X/
i
X.™
0
/ cos ™
0
dX (9.66)
where <
E
(
X
)> is the average energy density at
X
. By the law of conservation of energy, we must
have:
dE
(
0
)
D
dE
(
X
). Therefore,
4 X .™
0
/
tan ™
0
LJ
LJ
LJ
LJ
LJ
LJ
LJ
LJ
E
d™
0
dX
h
E.x/
iD
(9.67)
