1. Acquire the conjugate points (
x
,
y
) and (
x
′
,
y
′
) by matching
the images of satellites B and A, respectively, and calcu-
late the ground coordinates (
X
,
Y
,
Z
) corresponding to
(
x
′
,
y
′
) in satellite A by using the geometric model defined
in the first part of Equation (6) and the
SRTM
-
DEM
. Then,
select (
x
,
y
,
X
,
Y
,
Z
) as the control points for satellite B.
2. Calculate the original interior orientation parameters
a
i
,
b
j
of satellite B with the parameters measured at the labora-
tory before launch. Then, take the original
a
i
,
b
j
(
i
,
j
≤
5) as
known, and utilize the control points in step 1 to solve for
R
u
for satellite B based on the second part of Equation 6 by
using the least squares method.
3. Take
R
u
obtained in step 2 as known, and utilize the
control points in Step 1 to solve
a
i
,
b
j
(
i
,
j
≤
5) based on
the second part of Equation 6 by using the least squares
method.
4. Update the geometric model of satellite B by using the
solved parameters in steps 2 and 3 and calculate the
ground coordinates (
X
l
,
Y
l
,
Z
l
) and (
X
r
,
Y
r
,
Z
r
) corre-
sponding to the conjugate points (
x
l
,
y
l
) and (
x
r
,
y
r
) in the
adjacent
CCD
arrays of satellite B with the updated geo-
metric model and
SRTM
-
DEM
. Finally, select (
x
l
,
y
l
,
X
′
,
Y
′
,
Z
′
) and (
x
r
,
y
r
,
X
′
,
Y
′
,
Z
′
) as the new control points for the
geometric calibration of satellite B, where (
X
′
,
Y
′
,
Z
′
) =
X X Y Y Z Z
r
r
r
1
1
1
2
2
2
+
+
+
,
,
.
5. Use the control points obtained in steps 1 and 4 and
recalculate
R
u
and
a
i
,
b
j
(
i
,
j
≤
5) to update the calibration
parameters of satellite B.
6. Repeat steps 2-5 until the differences between the calibra-
tion parameters solved twice are less than the threshold.
From the above principles and procedure, the accuracy of
geometric cross-calibration can be approximately determined
as follows, and it is mainly limited by the matching accuracy
of the conjugate points, accuracy of the reference image (e.g.,
an image from the abovementioned satellite A), and differ-
ence in the imaging angles:
δ
δ
δ
δ
cal
match prior
angle
=
+
+
2
2
2
(10)
where
δ
cal
denotes the accuracy of cross-calibration,
δ
match
is
the matching accuracy,
δ
prior
is the accuracy of the reference
image, and
δ
angle
can be calculated using Equation 1.
Results and Analysis
Study Areas and Data Sources
Two datasets (Datasets A and B) were adopted for validating
the proposed method: Dataset A included the multitemporal
images of a single satellite, i.e., Yaogan-4, whereas Dataset B
contained the multi-satellite images of ZY3-01 and ZY02C.
Table 1and list Table 2 information related to the satellites
and experimental images, respectively.
In Dataset A, the geometric model of 08-07 2010-Tianjin
was built by using the calibration parameters obtained in
2010; the direct positioning accuracy was better than 20 m,
and the accuracy of the interior orientation parameters was
better than 0.6 pixels. However, because the camera param-
eters of Yaogan-4 were adjusted in January 2011, the above-
mentioned calibration parameters obtained in 2010 were not
applicable to the other three images, i.e., 05-15-2011-Henan,
11-14-2011 Tianjin, and 08-09 2012-08-Tianjin. The orienta-
tion parameters after January 2011 needed to be recalibrated.
On the basis of the side angles of -08-17 2010-Tianjin and
11-14-2011-Tianjin, as given in Table 2, the deviation in the
intersection caused by the elevation error can be calculated
according to Equation 11:
X
=
Δ
h
(tan(–14.88) – tan(–13.98)) = –0.017
Δ
h
.
(11)
The deviation in the intersection of the conjugate points
could be more precisely estimated as follows with the geo-
metric models of 08-17-2010-Tianjin and, -11-14-2011-Tianjin,
as shown in Figure 2 a:
1. Given pixel
p
0
in 98-17-08-17-Tianjin, if its height in the
90-m resolution
SRTM
-
DEM
(90 m-
SRTM
) is assumed to be
h
, then its corresponding location (
X
,
Y
,
Z
) in the
WGS84
coordinate system can be calculated using the geometric
model of 08-17-2010-Tianjin.
2. Image coordinates (
x
,
y
)
h
of pixel
p
1
in 11-14-2011-Tianjin
corresponding to (
X
,
Y
,
Z
) can be calculated using the
geometric model of 11-14-2011-Tianjin.
3. The height of
p
0
can be changed from
h
to
h
+
Δ
h
; then,
steps 1 and 2 are repeated to obtain the image coordinates
(
x
,
y
)
h +
Δ
h
of pixel
p
1
in 11-14-2011-Tianjin. Next, the de-
viation in the intersection of the conjugate points caused
by height error
Δ
h
is
Δ
(
x,y
) = (
x
,
y
)
h +
Δ
h
– (
x
,
y
)
h
.
As shown in Figure 1b, when the deviation in the intersec-
tion caused by the height errors ranged from 0 to 100 m,
90 m-
SRTM
showed an accuracy that is better than 16 m (
Δ
h
= 16 m in Equation 11; Consortium for Spatial Information,
2012) and may cause a deviation in the intersection of 0.15
pixels, which is very close to the value of 0.14 pixels cal-
culated using Equation 11. Therefore, the height error of 90
m-
SRTM
can be neglected in geometric cross-calibration, and
08-17 2010-Tianjin and 11-14 2011-Tianjin can be adopted
for geometric cross-calibration. Further, 05-15-2011-Henan
and 08-09-2012-Tianjin were used to validate the geometric
cross-calibration method; the 1:2 000 scale digital ortho-
photo model (
DOM
) and
DEM
of the Henan and Tianjin regions
(shown in Figure 3) were used as control data for validation.
Table 1. Information related to the Yaogan-4, ZY3-01, and
ZY02C satellites.
Average
altitude
(km)
Focal
length
(m)
Ground–
sample
distance (m)
CCD
array
information
Swath
width
(km)
Yaogan-4
650
3.3
2
12288 × 10 µm 25
ZY3-01
505
1.7
2.1
24576 × 7 µm 52
ZY02C
780
3.3
2.36
12288 ×10 µm 29
*The sensors onboard Yaogan-4, ZY3-01, and ZY02C all contain
three CCD arrays.
Table 2. Information related to the experimental images.
Satellite ID
GSD
(m)
Imaging time
Side
angle (°)
Dataset A
Yaogan-4 2010-08-17-Tianjin 2
2010-08-17
−14.88
Yaogan-4 2011-05-15-Henan 2
2011-05-15
16.69
Yaogan-4 2011-11-14-Tianjin 2
2011-11-14
−13.98
Yaogan-4 2012-08-09-Tianjin 2
2012-08-09
5.51
Dataset B
ZY3-01 ZY3_Tianjin
2.1 2013-03-09
0.00
ZY02C 02C_Tianjin
2.36 2013-02-20
0.00
488
August 2018
PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING
