Geology Reference
In-Depth Information
line
r
D
r
(—) in the 3-D space by the following
procedure. We select an initial arbitrary direction
n
0
D
n
. Then, we link the source point
r
0
to
the point
r
1
D
r
0
C
n
0
—
0
D
r
0
C
n
0
c
(
r
0
)
T
.
At the next step, we consider the versor
n
1
Dr
T
/
jr
T
j
and link
r
1
to a new point
r
2
D
r
1
C
n
1
—
1
D
r
1
C
n
1
c
(
r
1
)
T
. t y
successive step, we set
n
k
Dr
T
/
jr
T
j
and
link the current point
r
k
to a new point
r
k
C 1
D
r
k
C
n
k
—
k
D
r
k
C
n
k
c
(
r
k
)
T
.For
T
sufficiently small, this algorithm generates a
(generally) curved line that is called a
seismic
ray
(Fig.
9.1
). Of course, we can build infinitely
many rays starting from
r
0
simply changing the
initial arbitrary direction
n
.
At any step, the parameter —
D
—
0
C
—
1
C
:::represents the total distance from the
source, so that the position vector
r
can be
considered as a function of the parameter —.
We can also build a
slowness vector
s
such that
s
Dr
T=
jr
T
j
and with magnitude
s
D
1/
c
.
Using this notation, we have that the eikonal
equation assumes the form:
an equation that allows to determine
r
D
r
(—),
hence the seismic ray geometry, directly from the
slowness field
s
D
s
(
r
). By (
9.15
)wehavethat:
s.r/
d
d—
dT
d—
d
d—
D
d
d—
d
d—
r
T.r/
Dr
D
(9.17)
Finally, using (
9.16
) we obtain a differential
equation for the ray that does not depend from
the travel time
T
:
s.r/
d
d—
d
d—
Dr
s.r/
(9.18)
This equation, which allows to determine the
seismic ray geometry as a function of the slow-
ness field, is called the
seismic ray equation
.It
can be solved easily by finite differences to obtain
the function
r
D
r
(—) given an initial direction
n
,
granted that the variables
s
and
r
s
are known at
any point. In the case of a homogeneous region,
(
9.18
) reduces to
d
2
r
/
d
—
2
D
0
, which has the gen-
eral solution:
r
(—)
D
a
—
C
b
,
a
and
b
being con-
stant vectors. This is clearly a straight line in the
direction
a
and passing through the point
r
0
D
b
.
r
T.r/
D
s .r/
(9.14)
Assuming that the position along a seismic ray
is parametrized by an equation
r
D
r
(—), where —
is distance from the source, then the infinitesimal
variation of
r
along the ray will be given by:
9.2
Geometrical Spreading
Now let us turn our attention to the amplitude
transport Eq. (
9.9
). We are going to prove that
it determines how the amplitude
A
is transported
along a seismic ray. Substituting in the first term
at the left-hand side the gradient of
T
by the
slowness vector
s
(Eq.
9.14
)wehavethat:
dr
D
r
T.
r
/
s
d—
(9.15)
In fact,
c
r
T
(
r
)
D
(1/
s
)
r
T
(
r
)isalwaysaversor
normal to the wavefront. To determine the varia-
tion of arrival time along the ray, corresponding
to an infinitesimal variation of
r
,wewillusethe
directional derivative of
T
along the tangent to the
ray at
r
:
2
T.r/
D
0 (9.19)
2
r
A.r/
s .r/
C
A.r/
r
We note that
r
A
(
r
)
s
(
r
) is at any point
r
proportional to the directional derivative of
A
in the direction of
r
T
, thereby (
9.19
) can be
considered as an ordinary differential equation
along the curved line representing the seismic
ray. If rays are described by parametric equations
r
D
r
(—), then the variation of amplitude along a
seismic ray can be expressed as a function of the
parameter —.
dT
d—
Dr
T.r/
d
d—
Dr
T.r/
r
T.
r
/
s
D
s
(9.16)
This equation confirms our previous interpre-
tation of
T
as a travel time from the seismic
source to the wavefront. Now we want to find
