Geoscience Reference
In-Depth Information
Fig. 3. The position of the three sensors and the intersection of three spheres related to the
sensors
Now, the obtained value
x
is put in the one of the equations (27). We select the first one:
R
2
−
x
0
2
R
1
y
2
z
2
R
1
−
+
+
=
⇒
4
x
0
R
2
−
x
0
2
R
1
y
2
z
2
R
1
−
+
=
−
.
(29)
4
x
0
For convenience, the right side of the second equality of (29) is showed by
R
cir
.
Thus, the
intersection of the two spheres is a circle with the following equation:
y
2
z
2
R
cir
,
+
=
(30)
R
2
−
R
1
4
x
0
=
Which is located in the plane
x
.
Then the intersection of this circle and the third sphere should be obtained. The third sphere
has the center C and radius
R
3
. So its equation is:
2
x
2
z
2
R
3
.
+ (
y
−
y
0
)
+
=
(31)
The left side of the equation (31) is extended, and the circle equation is used in it:
x
2
2
z
2
y
0
+
x
2
y
2
z
2
+ (
y
−
y
0
)
+
=
−
2
yy
0
+
+
+
R
2
−
2
R
1
y
0
+
R
cir
=
−
2
yy
0
+
+
⇒
4
x
0
R
2
−
R
1
4
x
0
2
y
0
+
R
cir
+
y
=
.
(32)
2
y
0