Part A | 8.7
192 Part A Fundamentals
Table 8.6 In-air magnetic fields from static VED and HED sources submerged at a depth h in a deep ocean (after [8.12])
H x
H y
H z
VED 0
0
0
HED
p z xy
4 4
Ä
2
.h z/.2.h z/ 2 C 3 2 /
r 3
p z
4
.z h/
r 3 C
x 2
4
Ä
2
.h z/.2.h z/ 2 C 3 2 /
r 3
1
r 2 C .h z/r
p x y
4r 3
where r D
p
x 2 C y 2 C .z h/ 2 Note: A negative value for z is used for in-air computations
Table 8.7 Electric field subsurface-to-subsurface propagation formulas for the quasi-static range, when j 0 j j 1
jj and h and z
E x
E y
E z
VED
p z x
2 3
2
0
e
”.zCh/
p z y
2 3
2
0
e
”.zCh/
p z
2 3
2
0
2 e
”.zCh/
VMD
3m z y
2 5 e
”.zCh/
3m z x
2 5 e
”.zCh/
0
HED
p x
2 5 .x
2 2y
2 /e
”.zCh/
3p x xy
2 5 e
”.zCh/
p x x
2 3
2
0
2 e
”.zCh/
HMD
3m x xy
2 5 e
”.zCh/
m x
2 5 .y
2 2x
2 /e
”.zCh/
m x y
2
0
2 3 e
”.zCh/
where D
p
x 2 C y 2 and is the seawater conductivity
Table 8.8 Magnetic field subsurface-to-subsurface propagation formulas for the quasi-static range, when j 0 j j 1
jj and h and z
H x
H y
H z
VED
p z y
2 3
2
0
2 e
”.zCh/
p z x
2 3
2
0
2 e
”.zCh/
0
VMD
3m z x
2 5 e
”.zCh/
3m z y
2 5 e
”.zCh/
9m z
2 2 5 e
”.zCh/
HED
3p x xy
2 5 e
”.zCh/
p x
2 5 .x
2 2y
2 /e
”.zCh/
3p x y
2 2 5 e
”.zCh/
HMD
m x
2 5 .y
2 2x
2 /e
”.zCh/
3m x xy
2 5 e
”.zCh/
3m x x
2 5 e
”.zCh/
where D
p
x 2 C y 2 and is the seawater conductivity.
components in Table 8.2 yields the total in-water magnetic field for VED and HED sources submerged in
a deep ocean.
For completeness, the equations for the total in-air
electric fields from static VED and HED sources submerged in a deep ocean are listed in Table 8.5. The total
in-air magnetic fields for the same case are found in Table 8.6. Tables 8.5 and 8.6 are the total in-air equations
and are not added to any others. It should be noted that
a negative value for z is to be used when computing the
fields above the ocean surface. The deep ocean equations in Tables 8.3–8.6 were reduced from those found
in [8.12] for a shallow sea.
The equations in Tables 8.1 through 8.6 apply for
the special case where the sources are static or quasiDC. They are valid for frequencies that correspond to
seawater skin-depths that are large compared to the distance between the source and observation point, and
their depths below the surface. At the other extreme,
analytic equations have been formulated for the special case where the separation between the source and
observation point is much greater than a seawater skindepth and their depths. These equations were derived
by taking the asymptotic limits for the Sommerfeld
formulations.
The condition that the submerged source is many
seawater skin-depths from the in-water observation
point is equivalent to 1 jj. In addition, if h
and z and the quasi-static range applies, j 0 j j 1,
the subsurface-to-subsurface electric field equations in
192 Part A Fundamentals
Table 8.6 In-air magnetic fields from static VED and HED sources submerged at a depth h in a deep ocean (after [8.12])
H x
H y
H z
VED 0
0
0
HED
p z xy
4 4
Ä
2
.h z/.2.h z/ 2 C 3 2 /
r 3
p z
4
.z h/
r 3 C
x 2
4
Ä
2
.h z/.2.h z/ 2 C 3 2 /
r 3
1
r 2 C .h z/r
p x y
4r 3
where r D
p
x 2 C y 2 C .z h/ 2 Note: A negative value for z is used for in-air computations
Table 8.7 Electric field subsurface-to-subsurface propagation formulas for the quasi-static range, when j 0 j j 1
jj and h and z
E x
E y
E z
VED
p z x
2 3
2
0
e
”.zCh/
p z y
2 3
2
0
e
”.zCh/
p z
2 3
2
0
2 e
”.zCh/
VMD
3m z y
2 5 e
”.zCh/
3m z x
2 5 e
”.zCh/
0
HED
p x
2 5 .x
2 2y
2 /e
”.zCh/
3p x xy
2 5 e
”.zCh/
p x x
2 3
2
0
2 e
”.zCh/
HMD
3m x xy
2 5 e
”.zCh/
m x
2 5 .y
2 2x
2 /e
”.zCh/
m x y
2
0
2 3 e
”.zCh/
where D
p
x 2 C y 2 and is the seawater conductivity
Table 8.8 Magnetic field subsurface-to-subsurface propagation formulas for the quasi-static range, when j 0 j j 1
jj and h and z
H x
H y
H z
VED
p z y
2 3
2
0
2 e
”.zCh/
p z x
2 3
2
0
2 e
”.zCh/
0
VMD
3m z x
2 5 e
”.zCh/
3m z y
2 5 e
”.zCh/
9m z
2 2 5 e
”.zCh/
HED
3p x xy
2 5 e
”.zCh/
p x
2 5 .x
2 2y
2 /e
”.zCh/
3p x y
2 2 5 e
”.zCh/
HMD
m x
2 5 .y
2 2x
2 /e
”.zCh/
3m x xy
2 5 e
”.zCh/
3m x x
2 5 e
”.zCh/
where D
p
x 2 C y 2 and is the seawater conductivity.
components in Table 8.2 yields the total in-water magnetic field for VED and HED sources submerged in
a deep ocean.
For completeness, the equations for the total in-air
electric fields from static VED and HED sources submerged in a deep ocean are listed in Table 8.5. The total
in-air magnetic fields for the same case are found in Table 8.6. Tables 8.5 and 8.6 are the total in-air equations
and are not added to any others. It should be noted that
a negative value for z is to be used when computing the
fields above the ocean surface. The deep ocean equations in Tables 8.3–8.6 were reduced from those found
in [8.12] for a shallow sea.
The equations in Tables 8.1 through 8.6 apply for
the special case where the sources are static or quasiDC. They are valid for frequencies that correspond to
seawater skin-depths that are large compared to the distance between the source and observation point, and
their depths below the surface. At the other extreme,
analytic equations have been formulated for the special case where the separation between the source and
observation point is much greater than a seawater skindepth and their depths. These equations were derived
by taking the asymptotic limits for the Sommerfeld
formulations.
The condition that the submerged source is many
seawater skin-depths from the in-water observation
point is equivalent to 1 jj. In addition, if h
and z and the quasi-static range applies, j 0 j j 1,
the subsurface-to-subsurface electric field equations in