The full anchor point covariance matrix,
Σ
AP
, has the form
shown in Equation 22, where
n
is the number of anchor
points,
Σ
AP
ii
is the covariance at anchor point
i
based on adjust-
able parameters (
Δ
x
i
,
Δ
y
i
,
Δ
z
i
), and
Σ
AP
ij
is the cross-covariance
between anchor points
i
and
j
.
Σ
Σ Σ Σ
Σ
Σ Σ
Σ
Σ
Σ
AP
AP AP AP
AP
AP AP
AP
AP
AP
=
11
12
13
1
22
23
2
33
3
n
n
n
sym
nn
.
Σ
AP
, where
i
i
Δ Δ
y
x
i
y
y
Σ
σ
σ
σ
σ
σ
σ
Δ Δ
Δ
Δ
Δ Δ
Δ
AP
ii
i
i
i
i
x
x
i
z
z
z
sym
=
2
2
2
.
, and
AP
Σ
σ
σ
σ
σ
σ
σ
σ
Δ Δ Δ Δ Δ Δ
Δ Δ
ij
i
j
i
j
i
j
i
j
i
j
i
j
i
x x
x y
x z
y x
y y
y z
z
=
σ
σ
x
z y
z z
j
i
j
i
j
Δ Δ Δ Δ
Δ Δ Δ Δ Δ Δ
i
, (22)
Ground-Space
ULEM
allows for a combination of the anchor
point and 3
DC
transformation models. In this mode, both the
3
DC
transformation and corrections due to anchor point trans-
lations are applied simultaneously. The full covariance matrix
for the adjustable parameters of the ground-space combined
model (
Σ
a
) in
CU
l
is that shown in Equation 23.
Σ
Σ Σ
Σ
a
l
l
l
l
sym
=
3
3
DC DC AP
AP
.
(23)
The off-diagonal block in Equation 23, (
Σ
3DC
l
AP
l
), contains
the cross-covariance between the 3
DC
transformation param-
eters and the anchor points for
CU
l
. It is expected that this
sub-matrix will be all zeros prior to any adjustment and pos-
sibly fully populated (non-zero) after adjustment.
Since both anchor points and the 3
DC
transformation can
be parameterized such that either there are no anchor points
or there are no 3
DC
transformation parameters, the combined
model can be considered the general case for Ground-Space
ULEM
adjustment and error propagation. That is, it applies
even if some components of both or all components of one
of the two individual models have no effect. For example, if
only anchor points are used, then
Σ
3DC
l
will have zero rows
and columns and
Σ
AP
l
will have 3
n
rows and columns. The
general case Ground-Space
ULEM
adjustment model is shown
in Equation 24.
i
x
y
z
x
y
z
A
i
adj
i
=
+
'
∆
x
∆
x
∆
x
AP
AP
AP
n
1
2
~
(24)
In Equation 24, ([
xy z
]
i
´
)
T
are the model-space coordinates
of point
i
after applying the 3DC transformation using Equa-
tion 20, each
Δ
x
AP
j
= [
Δ
x
AP
j
Δ
y
AP
j
Δ
z
AP
j
]
T
is a column vector
of the coordinate corrections (translations) associated with
anchor point
j
, and
A
~
i
(3×3
n
) contains weights associated with
point
i
and is parameterized as follows:
A
~
i
= [
w
i
1
I
w
i
2
I …
w
in
I],
(25)
where
w
ij
is the weight associated with anchor point
j
for
point
i
, and I is a 3×3 identity matrix. The weights may be
based on any convex combination interpolation scheme (e.g.,
normalized inverse distance weighting) where {
for arbitrary
point i,
Σ
n
j
=1
w
ij
= 1}. The distances used in computing
A
~
i
should be based on [
x y z
]
T
i
coordinates, not the transformed
([
xy z
]
i
´
)
T
. Determination of effective interpolation schemes
for scattered anchor points is an ongoing area of research.
Covariance Storage and Modeling
As with Sensor-Space
ULEM
, Ground-Space
ULEM
supports
the storage of full covariance for adjustable parameters using
direct and indirect methods. Again, the direct method simply
stores the upper-diagonal entries of the full covariance matrix,
while the indirect method stores only the block-diagonal co-
variance values of the full covariance matrix, using an
SPDCF
to model the correlations and cross-correlations. The same
formation illustrated in Equation 12 for Sensor-Space
ULEM
can also be applied to Ground-Space
ULEM
, replacing the
n
CU
s with
n
anchor points.
Exploitation
As with Sensor-Space
ULEM
, a version of modelToGround()
based on Ground-Space
ULEM
will be implemented within
the
ULEM
CSM
sensor model. The modelToGround() function
for Ground-Space
ULEM
is the application of Equation 24,
followed by a coordinate conversion to Earth-Centered-Earth-
Fixed (
ECEF
) Cartesian coordinates.
As an absolute error propagation example was provided for
Sensor-space
ULEM
, here a relative error propagation example
will be given for Ground-space
ULEM
. The example configura-
tion consists of a dataset using 3
DC
translation components,
as well as anchor points. Also, suppose no adjustment has
been performed and there are no correlations between the
anchor point and 3
DC
adjustable parameters. Two points are
measured, labeled
P1
(with coordinates
x
1
, y
1
, z
1
) and
P2
(with
coordinates
x
2
, y
2
, z
2
). The adjustable parameter covariance
matrix would appear as:
x x
σ σ
Σ
Σ Σ
Σ
a
DC DC AP
AP
=
×
×
×
=
3
3
3 3 3 3
3 3
l
l
l
l
n
sym
n n
(
) (
)
.
(
)
σ
σ σ
σ
Σ Σ
Σ
Δ Δ Δ Δ Δ
Δ
Δ Δ
Δ
y
x z
y
y z
z
2
2
2
0 0 0 0
0 0 0 0
0 0 0 0
11
12
AP AP
…
AP
AP
AP
AP
1
22
2
n
n
nn
sym
.
Σ
Σ
Σ
…
The
A
matrix would appear as:
A
A
A
P
P
=
1
2
A
A A
n
Pi
Pi
=
×
×
3
3 3 3 3
DC AP
(
) (
)
A
x
x
x
y
x
z
y
x
y
y
y
z
z
x
z
y
z
z
3DC
=
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
Δ Δ Δ
Δ Δ Δ
Δ Δ Δ
A
AP
Pi
=
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
x
x
x
y
x
z
x
x
x
y
x
z
x
i
AP
i
AP
i
AP
i
AP
i
AP
i
AP
Δ
Δ
Δ
Δ
Δ
Δ
1
1
1
2
2
2
i
AP
i
AP
i
AP
i
AP
i
AP
i
AP
i
x
x
y
x
z
y
x
y
y
y
z
y
n
n
n
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
Δ
Δ
Δ
Δ
Δ
Δ
Δ
1
1
1
x
y
y
y
z
y
x
y
y
y
z
z
x
AP
i
AP
i
AP
i
AP
i
AP
i
AP
i
A
n
n
n
2
2
2
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
Δ
Δ
Δ
Δ
Δ
Δ
P
i
AP
i
AP
i
AP
i
AP
i
AP
i
AP
z
y
z
z
z
x
z
y
z
z
z
x
n
1
1
1
2
2
2
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
Δ
Δ
Δ
Δ
Δ
Δ
∂
∂
∂
∂
z
y
z
z
i
AP
i
AP
n
n
Δ
Δ
PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING
July 2015
553
