Geoscience Reference
In-Depth Information
3.6.3.1
Matching Distance
Here we use Eq.
3.12
for deriving the equation defining the matching distance
R
.
Substituting Eq.
3.164
into Eq.
3.12
yields
ˇ
ˇ
ˇ
ˇ
R
D
@r
ŒG
o
.r;R/
C
G
a
.r;R/
ˇ
ˇ
ˇ
ˇ
rDR
e
ˇjU.R/j
1
C
e
ˇjU.R/j
F.R/
@ln n
J
@r
@
(3.170)
From Eqs.
3.12
and
3.169
it is not difficult to find that
ˇ
ˇ
ˇ
ˇ
R
D
ˇ
j
U.R/
j
0
C
@ln n
J
@r
˛
o
.a/
2DR
2
1
1
C
e
ˇjU.R/j
F.R/
(3.171)
In deriving Eq.
3.171
we used the inversion of Eq.
3.9
:
˛.a/e
ˇU.R/
˛.a;R/
D
R
e
ˇU
.
r
0
/
dr
0
r
0
2
˛.a/
4D
1
Now we must find the derivatives of G
o
and G
a
. We will do this first for a
nonsingular attractive potential. From the definition of G
o
we find
ˇ
ˇ
ˇ
ˇ
rDR
D
e
ˇjU.R/j
ˇ
j
U.R/
j
0
@G
o
.r;R/
@r
Next (see the definition of E
o
),
ˇ
ˇ
ˇ
ˇ
rDR
D
ˇ
j
U.R/
j
0
Z
@G
a
.r;R/
@r
1
p
dx
e
x
.x
C
x
0
/
p
x
C
x
0
q
1
r
2
R
2
ˇjU.R/j
Z
dx
e
x
.x
C
ˇ
j
U.r
/
j
/
r
2
R
3
.x
C
x
0
/
2
p
C
q
1
p
x
C
x
0
r
2
R
2
ˇjU.R/j
r
1
Z
dx
e
x
.x
C
x
0
/
p
x
C
x
0
2
p
r
2
R
2
F.R/
D
ˇ
j
U.R/
j
For nonsingular potentials this expression simplifies
ˇ
ˇ
ˇ
ˇ
rDR
D
ˇ
j
U.R/
j
0
C
R
3
.1
C
2.ˇ
j
U.a/
j
ˇ
j
E
0
j
/
e
ˇjE
0
j
q
1
a
2
@G
a
.r;R/
@r
a
2
R
2
