The
B
matrix is formed similarly to the sensor space imple-
mentation:
B
B
B
P
P
=
1
2
0
0
B
x
X
x
Y
x
Z
y
X
y
Y
y
Z
z
X
z
Y
Pi
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
( ) 3 3
×
=
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
z
Z
i
i
∂
The full ground covariance between the two points of
interest is:
A
B
=
+
−
−
1
1
(
)
Σ
Σ Σ
Σ Σ
Σ Σ Σ
G
a
T
mod ue
≡
+
(
)
G G
G
T
G
T
B A
1
12
12
2
Now the relative error covariance between
P1
and
P2
is
obtained from:
− −
= +
Σ
Σ Σ Σ Σ
rel
G G G G
T
12
1
2
12
12
.
Experiment
This section describes an implementation of Sensor-Space
ULEM
for a lidar collection. Included are a description of the
datasets and metadata, the
ULEM
generation and exploitation,
and an analysis of the
ULEM
predicted uncertainties when
compared to ground control.
Datasets
An airborne lidar dataset was collected over Purdue University
(West Lafayette, Indiana) in March 2013 by Woolpert, Incorpo-
rated (Dayton, Ohio) using a Leica ALS70-HP system, flown at
an altitude of roughly 1,800 meters AGL and generating mul-
tiple north-south lidar passes over the campus. Woolpert pro-
vided the
NGA
SGC
the lidar ground points from this collection
in
LAS
format. Woolpert also provided the aircraft trajectory
and orientation data, consisting of
GPS
/
INS
navigation solutions
for aircraft positions and orientations, and their uncertainties.
Ground truth was provided by a survey conducted on
the Purdue University campus in the spring of 2010, which
consists of many lidar-identifiable points on the ground, on
roofs, and on the tops of parking garages. The
SGC
identified
and mensurated 52 of these truth points in the lidar dataset,
for use as check points to validate the results.
It should be noted that this experiment was performed on
data received from Woolpert which had not gone through
their final data-adjustment steps. The goal was to
test the
a priori
error estimates from
ULEM
, with
a future goal of integrating these
a priori
error
estimates into the final
ULEM
adjustment process
and generating
a posteriori
error estimates for
evaluation. So, the true horizontal errors shown in
the results are significantly larger than the errors
that the customer would see in the final Woolpert
product.
ULEM Generation
The
SGC
-developed in-house software (
ULEM
Gen)
takes input
LAS
files, as well as necessary lidar
metadata, and generates
ULEM
metadata which is
added to the
LAS
files using the
LAS VLR
format.
Once the
ULEM
LAS
files are generated,
ULEM
-
based exploitation tools can compute predicted
uncertainties for the points in the file.
ULEM
Gen is
currently based on a physical sensor model consistent with
the general characteristics of many commercial lidar sensors
and was initially tested with Optech lidar systems. However,
it was determined that the physical sensor model used in
ULEM
Gen is also compatible with the Leica system. Table 1
lists the metadata used as input to
ULEM
Gen. Below each entry
is given the
ULEM
parameter(s) to which the entry contributed.
The uncertainties in Table 1 are based on the best estimates
that the
SGC
and Woolpert currently have, but they could be
updated with additional input from the sensor manufacturer.
In addition to the values in Table 1, the platform trajectory,
the uncertainty on that trajectory, and the point clouds of
interest (
LAS
format) were also passed into
ULEM
Gen.
The Historical Z Correction deserves further explana-
tion. In addition to the metadata from the Purdue collection,
Woolpert provided the
SGC
with a listing of height corrections
for 13 different projects using the same lidar system, which
were applied to their final products after comparing to ground
control. The
RMS
of these corrections was 27.5 cm. The source
of the errors causing the needed corrections was not known
by Woolpert or the
SGC
, except that the correction was only
needed in the local Z-direction. Since this correction could
not be attributed to any particular system (e.g., range,
GPS
), the
RMS
of the historical local Z correction was used as the uncer-
tainty estimate for
Δ
U
(
LCS
parameter vertical component).
Assessment of ULEM Predictions at Check Points
The output from ULEMGen consisted of modified
LAS
files,
now with added
VLR
s containing Sensor-space
ULEM
metadata.
The file sizes were roughly 120
MB
each, and the change in size
due to the addition of the
ULEM
metadata was about 0.1 percent.
These
ULEM
LAS
files were used as input in
SGC ULEM
exploita-
tion software, which generates a
ULEM
-predicted ground covari-
ance for each set of 3
D
coordinates provided. Using the mensu-
rated coordinates, predicted ground covariance matrices were
computed for each check point using the
ULEM
metadata in the
LAS
files. Additional metrics were derived, including CE90 (Cir-
cular Error 90
th
percentile) for horizontal predicted uncertainty,
and
LE
90 (Linear Error 90
th
percentile) for vertical predicted
uncertainty, per mensurated point.
CE
90 and
LE
90 were used to
be consistent with current
NGA
practices, but the error covari-
ances could easily be scaled to conform to other accuracy
specifications. Note that mensuration error was included in the
calculation of the predicted ground covariance matrices of the
mensurated check points. The average point spacing for the
dataset was 38 cm, so a mensuration sigma of 19 cm (one-half
the average point spacing, based on
SGC
experience with men-
suration from point clouds) was included in the propagation for
each of the horizontal terms. This is necessary because the lidar
samples the surface, and there will not (in all likelihood) be a
lidar return from the exact location of a check point. Therefore,
there is uncertainty in the ability of the human to measure the
T
able
1.
ULEMG
en
I
nput
M
etadata
for
the
W
oolpert
P
urdue
D
ataset
.
IMU Alignment Angles (deg)
Orientation
(
θ
1
,
θ
2
,
θ
3
)
Omega: 90.0, Phi: 0.0, Kappa: 0.0
Beam Divergence (mrad, 1/e radius)
Orientation
(
θ
1
,
θ
2
)
Omega = Phi = 0.075
Parts-per-million (GPS base station)
LCS
(
Δ
x
,
Δ
y
,
Δ
z
)
Horizontal and vertical: 3
Base Station Uncertainties (cm, 1-sigma)
LCS
(
Δ
E
,
Δ
N
,
Δ
U
)
East, North: 2 each; Up: 4
Lever Arm Uncertainties (cm, 1-sigma)
Position
(
Δ
x
,
Δ
y
,
Δ
z
)
X, Y, Z: 1 per axis
Boresight Uncertainties (mrad, 1-sigma)
Orientation
(
θ
1
,
θ
2
,
θ
3
)
Omega: 0.026, Phi: 0.026, Kappa: 0.035
Scanner Uncertainty (mrad, 1-sigma)
Orientation
(
θ
1
,
θ
2
,
θ
3
)
Omega = Phi = Kappa = 0.0524
Range Uncertainty (1-sigma) 1 cm
Range
(
Δ
r
)
Historical Z Correction (cm, 1-sigma)
LCS
(
Δ
U
)
27.5
554
July 2015
PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING
