Geology Reference
In-Depth Information
dx
d—
D
sin ™
D
p
s
I
d
z
d—
D
cos™
p
1
sin
2
™
D
p
1
p
2
=s
2
s
p
s
2
1
D
D
p
2
(9.51)
Therefore,
dx
d
z
D
dx
d—
d
—
d
z
D
p
p
s
2
(9.52)
p
2
Fig. 9.11
Travel-time fitting curve for a monotonically
increasing velocity function. Dots are observed travel
times
T
i
at locations
X
i
. The tangent to the regression
curve (
dashed line
) represents the ray parameter of the
seismic ray arriving at distance
X
from the source
This expression can be easily integrated to
determine the
x
component of points along the
seismic ray with parameter
p
, assuming a source
located at the origin of the reference frame. To
determine the range
X
associated with
p
,wemust
take into account that in laterally homogeneous
models the distance at which the upgoing seg-
ment of the seismic ray reaches the Earth's sur-
face is twice the horizontal distance between the
seismic source and the turning point. Therefore,
continuous curve
T
D
T
(
X
)ofthearrivaltime
as a function of the distance
X
from the seismic
source (that is, the
range
). By (
9.44
)wehavethat
the slope of this curve will be given by:
dT
dX
D
p
D
s.
z
max
/
(9.49)
Z
z
max
d
z
p
s
2
.
z
/
p
2
X.p/
D
2p
(9.53)
Clearly, the derivative
dT
/
dX
is a decreasing
function of the range, because the ray parameter
(hence the slowness at the turning point)
decreases with increasing range (see Fig.
9.8
).
Therefore, a plot of the
travel time curve
T
D
T
(
X
) has the typical shape illustrated in
Fig.
9.11
. In this plot, each point of the regression
curve represents a different seismic ray, whose
parameter is given by the slope of the curve.
When the seismic velocity is a known function
of
z
, it is possible to determine analytically both
the travel-time
T
and the range
X
for any value of
the parameter
p
, hence for any seismic ray from
a known source. We know that the horizontal
component of the slowness vector
s
D
s
(
z
)
coincides with
p
(Eq.
9.44
):
s
x
D
p
. Regarding
the vertical component,
s
z
, it is given by:
s
z
.
z
/
D
s.
z
/ cos ™.
z
/
D
p
s
2
.
z
/
p
2
0
A similar procedure allows to determine the
arrival time
T
. Because
dT
D
sd
— (Eq.
9.16
), we
have:
s
2
p
s
2
dT
d
z
D
dT
d—
d
—
d
z
D
(9.54)
p
2
Integrating from the Earth's surface to
z
max
and multiplying by 2, we easily obtain the arrival
time ad distance
X
:
Z
z
max
s
2
.
z
/d
z
p
s
2
.
z
/
p
2
T.p/
D
2
(9.55)
0
When the transmitting domain can be con-
sidered as a stack of homogeneous layers, the
integral solutions (
9.53
)and(
9.55
) are substituted
by sums.
We h ave :
(9.50)
At the turning point, we have that
p
D
s
and
s
z
D
0. It is easy to find expressions for
X
D
X
(
p
)
and
T
D
T
(
p
). Let us consider an infinitesimal ray
path segment
d
—. By Snell's law (
9.40
)wehave
that:
X.p/
D
2p
X
i
h
i
q
s
i
p
2
I
s
i
>p (9.56)
