In this research, the theoretical precision of drift com-
pensation and affine compensation is elaborated, and the
advantage of drift compensation in both theory and practice is
proved. First, the physical mechanism of the drift compensa-
tion model is studied. Subsequently, a stereo pair-based block
adjustment model is developed. To derive analytical equa-
tions of theoretical precision, the
GCPs
are designed to be de-
ployed in the first strip. The block adjustment is simplified to
orientation in steps. Then, the theoretical precision about the
lateral overlap is deduced for both affine compensation and
drift compensation. Four strips of
ZY-3
triplet stereo images
were used for the experiment to validate the role of tie points
and the change of geolocation errors with the strip numbers
for both affine and drift compensation.
Error Compensation of
RFM
The
RFM
is a replacement sensor model for the rigorous sen-
sor model (
RSM
), which describes the relationship between
the normalized image coordinates (
R, C
) and the normalized
object coordinates (
P, L,
and
H
) through ratios of third-degree
polynomials (Tao and Hu, 2001)
C
a P L H
b P L H
ijk
j i i k k
k
i
i
j
j
ijk
j i i k k
k
i
i
j
j
=
− −
= = =
− −
= = =
∑∑∑
∑∑
0 0 0
3
0 0 0
3
0 0 0
3
0 0
∑
∑∑∑
∑∑
=
− −
= = =
− −
= =
R
c P L H
d P L H
ijk
j i i k k
k
i
i
j
j
ijk
j i i k k
k
i
i
j
j
=
∑
0
3
(1)
where
a
ijk
,
b
ijk
,
c
ijk
,
d
ijk
are the coefficients of the cubic polyno-
mials, and
b
000
=
b
000
= 1. The geopositioning uncertainty of
the
RSM
propagates into the
RFM
because the rational polyno-
mial coefficients (
RPCs
) are calculated by least-squares estima-
tion using dense object-image grids, which are established by
the
RSM
when terrain-independent scenarios are applied.
The errors of
RSM
consist of attitude, orbit, and
IOP
. Be-
cause of the precise orbit determination, the orbit accuracy
is at the centimeter level (Luthcke
et al.
2003), which is not
significant compared with the meter or sub-meter ground
sample distance. A routine geometric calibration can keep the
accuracy of
IOPs
within subpixels. The attitude errors can be
divided into high-frequency, such as attitude oscillation (Ja-
cobsen, 2017), and low-frequency misalignment, which plays
a major role in geopositioning uncertainty. The misalign-
ment between the navigation system and the camera system
is dynamic during flight, even if the misalignment has been
calibrated at the specific field. The attitude errors of three
linear cameras have different drift rates in orbit, which means
that the convergence angles of triplet stereo change (Pan
, et
al.,
2016). Meanwhile, the bore sight of multiple star trackers
shows time-dependent deformation (Hashmall
et al.,
2004;
Wang
et al.,
2016). Likewise, the alignment between the Earth
observation camera and star trackers might also be dynamic,
because of thermal variance (Schwind
et al.,
2012), which
depends on the exposure and satellite position. However, the
attitude error of several sequential scenes can be modeled as a
shift (Ikonos within 50 km (Grodecki and Dial, 2003) or drift)
IRS-P5 approximately 1,500 km (Gupta
et al.,
2008) model,
because the thermal condition changes slowly during the
imaging process.
The attitude errors of the
RSM
could be compensated as:
t
t
0 1
= +
t
t
0 1
= +
t
t
0 1
∆
∆
∆
φ φ φ
ω ω ω
κ κ κ
( )
= +
( )
( )
(2)
where
t
is time, and the pitch angle error
φ
causes displace-
ment in the line direction; the roll angle
ω
introduces dis-
placement in the sample direction, and the yaw angle
κ
creates sample-dependent displacement in both the line and
sample directions. Because of the narrow
FoV
angle, the maxi-
mum displacement ratio between the pitch and yaw angle is
equal to the tangent of a half-
FoV
(Pan
, et al.,
2016). Hence, the
yaw angle error is not as significant as the pitch and roll angle
errors. After eliminating the yaw angle, the drift compensa-
tion model in the image space is:
∆
∆
c e e l
r e e l
= + ⋅
= + ⋅
0 1
3 4
(3)
where
e
0
,
e
1
is the drift compensation model of the roll angle;
e
3
,
e
4
is the drift compensation model of the pitch angle (Fra-
ser and Hanley, 2005);
Δ
c
=
c
–
s
;
Δ
r
=
r
–
l
;
l,s
are the mea-
sured image coordinates in the line and sample directions,
and
r,c
are the calculated image coordinates in the row and
column directions. Comparing this with the affine compensa-
tion in the image space (Fraser and Hanley, 2005)
∆
∆
c e e l e s
r e e l e s
= + ⋅ + ⋅
= + ⋅ + ⋅
0 1
2
3 4
5
(4)
Equation 3 eliminates the sample-dependent parameters
e
2
,
e
5
. For single stereo pairs, the drift compensation model is
not superior to the affine compensation, because only one ad-
ditional
GCP
is required to compensate the yaw angle and prin-
cipal distance errors. However, the affine compensation model
could not achieve homogenous accuracy for a large block
because of random error accumulation. On the contrary, drift
compensation can overcome this disadvantage. The theoreti-
cal precision is investigated carefully in the following section.
Block Adjustment
In the block adjustment, the tie points are one of the major
observations whose number and distribution have significant
effects on theoretical precision (Förstner and Wrobel, 2016).
To illustrate the error propagation, the derived observations
and calculated image coordinates
r
,
c
, are used to analyze the
variance, instead of direct observations. The observation func-
tion of affine compensation in the image space is:
v
=
Ax – l
(5)
where
A
is the design matrix,
x
is an unknown, and
l
is a
column vector of constants. For any point
P
i
in a scene, the
observation equation is
v
v
l s
l s
e
e
e
e
e
e
si
li
i
i
i
i
=
1
0 0 0
0 0 0 1
0
1
2
3
4
5
−
−
−
s c
l r
p
p
i
i
i
i
ci
ri
,
(6)
792
December 2018
PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING
