Finally, the general model can be rewritten using the following expression:
1 1
sin[ 1
] 12
12 15
MP p
lon lon BT
a t
c t
π −
= + +
+ ≤
LL
6
3.2.3 Latitude Influence Function: The external energy
from solar radiation is one of the primary energy sources to create the ecosystem differences. The primary control of
climate at the global level is variation in solar energy, which is related to latitude. The amount of solar radiation generally
decreases from the equator to poles, partly due to increases in the angle of incidence of the atmosphere. As a result, LST is
reducing with the latitude increasing, which has two reasons: 1 Solar radiation angle increasing; 2 Atmosphere effective
thickness changing. As illustrated in Figure 3, for the same solar radiation, the received amount per unit area in low
latitudes is larger than in high-latitude areas.
Figure 3. The relationship between solar radiation and solar radiation angle
To simplify the latitude influence function, in this work only considers the impact of solar radiation angle is considered.
Let’s assume that two beams of sunlight, squares of side length L and parallels to the equatorial plane, shine on the two
latitudes
1
ϕ
,
2
ϕ
, respectively. As in Figure 3b, in the latitude plane, the Earth radius and its projection line defines a
right triangle. Based on the geometry relationship of triangles, the vertical height of the bottom and top of the sunlight can be
calculated by:
_ _
sin sin
sin
L T
H R
H R
R
ϕ ϕ
ϕ ϕ
ϕ ϕ
= =
= + Δ
7 On the basis of this assumption, the difference of the bottom
and top of the sunlight is just the height of sunlight, as in equation 8:
_ _
T L
H H
L
ϕ ϕ
− =
8 Similarly, equation 9 can be derived:
1 1
1 2
2 2
[sin sin
] [sin
sin ]
R L
R L
ϕ ϕ
ϕ ϕ
ϕ ϕ
+ Δ − =
+ Δ −
=
9 Where
1
ϕ
Δ
、
2
ϕ
Δ
are central angles, which are close to zero, caused by the height of sunlight. From equation 9:
1 1
1 2
2 2
sin sin
sin sin
ϕ ϕ
ϕ ϕ
ϕ ϕ
+ Δ − =
+ Δ −
10 Based on product to sum equation, equation 10 can be derived:
1 1
1 1
1 2
2 2
2 2
sin cos
cos sin
sin sin
cos cos
sin sin
ϕ ϕ
ϕ ϕ
ϕ ϕ
ϕ ϕ
ϕ ϕ
Δ + Δ −
= Δ
+ Δ
−
11 Because
1
ϕ
Δ
、
2
ϕ
Δ
are close to zero,
1
cos
ϕ
Δ
、
2
cos
ϕ
Δ
will close to 1, and equation 12 can be proved:
1 1
2 2
cos sin
cos sin
ϕ ϕ
ϕ ϕ
Δ =
Δ
12 The arc of a sector can be calculated by the following:
s s
s
l r
θ
=
13 Where
s
l
is the arc of the sector,
s
r
is the radius of the sector and
s
θ
is its central angle. The ratio of the solar radiation received in different latitudes
can be calculated by:
1 2
2 1
2 1
1 2
2 2
2 2
2 2
2 2
1 1
1 1
2
sin cos
lim [ ]
[ ]
sin cos
G S
l R
G S
l R
ϕ ϕ
ϕ ϕ
ϕ ϕ
ϕ ϕ
ϕ ϕ
ϕ ϕ
ϕ ϕ
ϕ ϕ
Δ → Δ →
Δ Δ
Δ =
= =
= =
= Δ
Δ Δ
14 Where
i
G
ϕ
、
i
S
ϕ
、
i
l
ϕ
are the solar radiation, shining area and arc length of the sunlight beam of the latitude
i
ϕ
. Therefore, the latitude influence function can be simplified as
the following equation:
2
[cos ]
p
BT a
lat b
= +
15 Where a, b are regression coefficients.
3.2.4 Seasonal Influence Function: Obviously, LST
