Geoscience Reference
In-Depth Information
If the thermal transport process is stationary then Eq. (
2.24
) transforms to the
Laplace equation:
2
2
T
2
2
T
div
gradT ¼
@
=@u
þ @
=@k
ð
2
:
25
Þ
ˆ
The solution of Eq. (
2.25
) is the harmonic function of the spatial coordinates
and
ʻ
. In accordance with the Rayleigh-Jeans law (DeWitt and Nutter 1988) the brightness
temperature as a result of remote microwave measurements is T
a
(
ˆ
,
ʻ
)=T
a
(
ˆ
,
ʻ
,
μ
,
ʸ
),
where
= const). It is proposed
that for the any small area V
M
at an arbitrary point M
∈
G the brightness temperature is
a linear function of the temperature of the media:
μ
is the wavelength,
ʸ
is the observation angle and (
μ
,
ʸ
T
a
ðu; kÞ
¼A
M
þ
B
M
T
ðu; kÞ;
ðu; kÞ2
V
M
;
ð
2
:
26
Þ
where A
M
and B
M
are constants.
Formula (
2.26
) follows from the theoretical and experimental estimations of T
a
(Krapivin and Potapov 2001; Burkov and Krapivin 2009). For media which are
homogeneous in depth, the formula T
a
=
) is the media
radiation coefficient where
ʵ
is the dielectrical permeability. According to the
experimental estimations the radiance of fresh water measured by means of radi-
ometers with wavelengths of 5
κ
T is valid. Here
ʺ
=
ʺ
(
μ
,
ʸ
,
ʵ
8 cm is a linear function of T. The slope of this
-
dependence is 0.35
0.50 K/
°
C. An increase in the water salinity S from 0 to 16
‰
is
-
accompanied by a decrease in the sensitivity of the irradiation
field to the temperature
variations. This effect is observed for wavelengths from 10 to 50 cm. The sensitivity
of the irradiation
field to variations in T is a minimum when the following conditions
are realized:
μ
S
≅
700; 0
≤
T
≤
30
°
C; 0
≤
S
≤
180
‰
;0
≤ ʸ ≤
25
°
.
From Eq. (
2.26
) it follows that T
a
at every point M
∈
G satis
es the following
condition:
Z
2
p
1
2
T
a
ðu; kÞ
¼
T
a
ðu þ
r
cos a
; k þ
r
sin a
Þ
da
;
p
0
where the integral is over a circle of radius r centered at (
).
This condition is valid for any r (0 < r < r
M
) where r
M
is the radius of the area
V
M
. Therefore T
a
is a harmonic function within G. A typical task here is the search
of the harmonic function T
a
(
ˆ
,
ʻ
) within G when T
a
ðu; kÞ
¼T
a
ð
u
Þ
on the boundary
ˆ
,
ʻ
ʓ
of G (u
∈ ʓ
, u =
ˆ
+ i
ʻ
). Such a function is the real part of some analytic function
W(z) given by:
Z
1
2
W
ð
z
Þ
¼
lðfÞ=ðf
z
Þ
d
lðfÞ=ðf
ð
2
:
27
Þ
p
C
where
μ
(
ʶ
) is the real density,
ʶ ∈ ʓ
and z =
ˆ
+ i
ʻ
is an arbitrary internal point of
G. We have Re[W(u)] and Im [d
ʶ∕
(
ʶ
-
u)] =
−
cos(r,n)d
˃
/r where r is the distance
between
ʶ
and u, d
˃
is an element of
ʓ
and n is the external normal to
ʓ
.Asz
u
→
∈ ʓ
,
μ
(u) is approximated by the solution of the integral equation:
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