index, we also track the
LOS
refracted direction in different
atmospheric layers. Therefore, the proposed method is more
conformal to the
LOS
propagation characteristics in the strati-
fied atmosphere. By compensating the atmospheric refraction
error in the rigorous collinear geometric model, the geometric
positioning accuracy of satellite remote sensing image with-
out ground control points can be improved.
Since DMC3/TripleSat constellation was launched on 10
July 2015, this study has been continually experimenting with
the three 1 m resolution optical satellites. The atmospheric
refraction correction algorithm has been integrated into the
DMC3/TripleSat rigorous geometric model, which is used to
produce the Level-1A basic image products.
Atmospheric Refraction Geolocation Error
The stratified Earth’s atmosphere comprises mainly gas mol-
ecules, water vapor, and aerosols. When the Sun’s ray is reflect-
ed by ground objects through the atmosphere to the imaging
detectors in-orbit, the propagation direction of reflection rays
shall be deviated due to atmospheric refraction. Conversely,
from the view of detectors of imaging payload, the atmospheric
refraction results in a deviation of
LOS
from its original propa-
gation direction and lead to an atmospheric refraction geolo-
cation error. In the following, we will model the geolocation
error with rigorous mathematic formulas. In Figure 1, suppose
atmospheric refraction does not exist, the line
dSP
represents
the
LOS
of one detector with view angle
α
, where Point
d
refers
to the detector (the distance from the imaging detector to the
linear array principal Point
o
is also expressed as
d
), Point
S
is the perspective center of optical imaging payload, and Point
P
is the intersection of
LOS
and the Earth’s ellipsoid. This line
dSP
also contains Point
P0
, which is the intersection of
LOS
and the top atmosphere. In order to describe the ray tracking
geometric algorithm of
LOS
vector, it is assumed the Earth’s at-
mosphere is a single layer homogeneous spherical atmosphere
only consisting of gas molecules and water vapor:
h
is the
atmosphere thickness and
n
is the atmospheric refraction index
(
n
>1). Point
P1
is the intersection of
LOS
deviated by atmo-
spheric refraction and the Earth ellipsoid. The line of
d1SP1
(dashed line) is the atmospheric refraction bias compensated
LOS
. Also, in Figure 1, we define
f
as the focal length,
H
as
satellite platform height, and
R
as the Earth’s mean radius. The
ground surface distance between point
P
and
P1
is the atmo-
spheric refraction geolocation error that would exist in rigorous
geometric model if atmospheric refraction is not considered.
Next, we will calculate the atmospheric refraction geoloca-
tion error with a rigorous mathematic formula using symbols
predefined in Figure 1. According to the sine law we have the
following formulas:
R
R H
sin sin(
)
α
β
=
+
−
180
(1)
⇒ =
+
arcsin
sin
β
α
R H
R
(2)
where
α
is the off-view angle, and
β
is the incident angle of
LOS
to the earth ellipsoid without considering the atmospher-
ic refraction error. In the same way, the incident angle of
LOS
to the top atmosphere can be calculated:
i
R H
R h
=
+
+
arcsin
sin
α
(3)
where
h
is the atmosphere thickness.
According to the Snell refraction law, the refraction angle
r
is as follows:
n r
i
sin sin
=
(4)
⇒ =
arcsin
sin
r
i
n
(5)
where
n
is the refraction index,
i
is the incident angle, and
r
is
the refraction angle.
We define
θ
as
∠
SOP
,
θ
0 as
∠
SOP
0 as
∠
P
0
OP
1 and
∆
θ
as
∠
P
1
OP
. Since
OP
0 =
R
+
h
,
OP
=
R
, and the refraction angle
r
is known in
∆
P
0
OP
1, and
θ
1 can be calculated:
θ
1
=
+
−
arcsin
(
)sin
R h i
R
r
(6)
Known as the off-view angle
α
, the incident angle
i
and
β
in
∆
SOP
0,
θ
0 can be calculated:
= −
θ β α
(7)
= −
θ
α
0
i
(8)
Combining the above equations, angle
∆
θ
(radian) and arc
length
are calculated.
Figure 1. LOS propagated in single layer spherical atmosphere.
428
June 2016
PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING
