Testing the PDEM to Control the Precision of an IFSAR DEM
JOSE F. ZELASCO1, KEVIN ENNIS†, BARBARA SULPIS1, JOSE LUIS FERNÁNDEZ
AUSINAGA2
1Facultad de Ingeniería - Universidad de Buenos Aires
Paseo Colón 850 – Ciudad de Buenos Aires
ARGENTINA
2Facultad de Ciencias Exactas – Universidad del Centro de la Pcia. de Bs. As.
Arroyo Seco, B7000 Tandil, Provincia de Buenos Aires
ARGENTINA
Abstract: The Perpendicular Distance Estimation Method (PDEM), a method to estimate the precision of a Digital
Elevation Model (DEM), in “Effectiveness of Geometric Quality Control Using a Distance Evaluation Method”
was tested assuming independent normal random for the vertical and horizontal error, considering isotropic this
horizontal error. Here the PDEM is tested in the case of a DEM obtained by IFSAR technology, also, assuming
independent normal random errors, but this time in three independent directions. Because vertical direction is
correlated with one horizontal direction, -the range axis direction-, a rotation has to be done and so, three
independent directions are obtained.
Keywords: GIS geometric accuracy assessment; IFSAR DEM geometric accuracy, Perpendicular Distance
Evaluation Method (PDEM); IFSAR DEM quality evaluation.
Received: July 9, 2022. Revised: October 16, 2023. Accepted: November 18, 2023. Published: December 24, 2023.
1. Introduction and Description of
Challenges.
A Digital Elevation Model (DEM) is a Digital
Surface Model (DSM) of a terrain surface. It can be
defined as a set of points, laid out on a regular square
grid or on a triangular grid, where the altitudes are
known in the vertices. The elevation anywhere else is
obtained by interpolation. In the literature, often, the
evaluation of the precision of a given DEM is
obtained by “comparison” with a reference DEM
considered “much more precise or exact” estimating
the standard deviation of the discrepancies between
them. To know the precision of different methods to
obtain a DEM allows one to choose the method that
has the best relation precision / price.
To carry out this comparison, two methods stand out
in the literature: the measurement of vertical
distances between the models and the comparison
with benchmark points.
Zelasco2019 [17] describes these methods and
establishes their limitations and drawbacks.
Briefly: the measurements of these vertical distances
are affected by the slope of the reference DEM; and
the corresponding points of these benchmarks may be
subject to particular conditions or the benchmarks
which have identifiable corresponding points are not
a representative sample of the surface, Hirano et al.
2003 [8].
Also, in Zelasco2019 [17] the Perpendicular Distance
Evaluation Method is formally presented and tested
assuming independent normal random for the vertical
and horizontal error and considering isotropic the
horizontal error.
The purpose of this article is to test the PDEM
evaluating the geometric quality of a DEM obtained
employing IFSAR technology. For the rest of the
document, the given DEM will be referred to as the
evaluated DEM (e-DEM). The reference DEM (r-
DEM) is a real DEM of 100,000 points. The e-DEM
is obtained by simulated errors from the r-DEM.
Therefore, e-DEM errors are known. The PDEM
evaluates the errors of the e-DEM by estimating the
standard deviation in relation to the surface of the r-
DEM. To test the PDEM it suffices to compare the
estimated error values obtained by the PDEM with
the known error values. As in Zelasco2019 [17], here,
is assumed that the measurement errors are
independent random variables with components in
three orthogonal directions, and because the vertical
direction is correlated with range axis a rotation has
to be done given the beam axis, the azimuth axis and
the axis normal to the previous two (y’; the rotated
range axis).
The rest of this paper is organized as follows: Section
2 provides the background of this study. Section 3
presents a brief description of the proposed method.
Section 4 explains the error correlation of the e-DEM
obtained by the IFSAR technology and the
reformulation. Section 5 describes the experiments.
Section 6 evaluates the results. Section. Finally,
section 7 gives some concluding remarks regarding
this study.
Engineering World
DOI:10.37394/232025.2023.5.21
Jose F. Zelasco, Kevin Ennis,
Barbara Sulpis, Jose Luis Fernández Ausinaga
E-ISSN: 2692-5079
187
Volume 5, 2023
2. Background and Previous Problem
Treatments.
The evaluation of a DEM error is an important
topic, in the literature several attempts have been
made for the evaluation of a DSM (e-DSM) error
relating to a more precise reference DSM (r-DMS).
Zelasco2019 [17] mention many previous works as
Guptill and Morrison 1995 [5]; Harvey 1997 [6];
Laurini and Milleret-Raffort 1993 [10]; Ubeda and
Servigne 1996 [15], and other several solutions
concerning the DEM´s quality proposed on Dunn et
al. 1990 [2], Lester and Chrisman 1991 [9], etc. Also,
analogous problems were studied for horizontal
errors in maps in Abbas 1994 [1]; Grussenmeyer et
al. 1994 [4]; Hottier 1996 [8]. They successfully use
Hausdorff distance to evaluate the errors in maps, but
this does not work in two dimensions in the cases
when the components have different errors. Anyway,
no solution to the simultaneous evaluation of vertical
and horizontal errors is proposed.
The DEM Quality Assessment chapter in Maune
2001 [13], states that horizontal accuracy, although
recognized as a part of DEM quality, is generally
considered difficult to evaluate in the absence of an
image coincident with the e-DEM (check points or
benchmarks), or of clearly discernable surface
features.
Zelasco 2019 [17] explains the lack, or at least
scarcity, of work devoted to the horizontal accuracy
of a DEM.
Zelasco et al. 2013 [16] is mostly a users’ tutorial
of the method.
In Zelasco 2019 [17] there is a formal presentation
of the mechanics of the PDEM. The PDEM allows us
to get statistical information about the horizontal and
vertical standard deviations, only the perpendicular
component of a point from the e-DEM to the r-DEM
surface is required.
3. A brief description of the PDEM
Resuming Zelasco 2019 [17], the PDEM, unlike
the vertical distance methods, produces vertical
standard deviation results without a systematic error
in the vertical direction and allows to obtain the
horizontal variance under the condition of sufficient
surface roughness.
The error is the vector function which denotes the
discrepancy between both surfaces, and is defined for
each point
i
M
and its homologous point
i
P
in the r-
DEM. The points
i
M
define the e-DEM surface, but
their homologous points
i
P
are not vertices of the r-
DEM surface triangles. If they were, their
identification would be easy, and our problem would
be trivial. However, we want to deal with the more
common case in which the homologous points are not
readily identifiable. The error vector is assumed to be
the result of three stochastically independent
components,
,,
x y z
e e e
one in each of the basic axes
of the x, y, z coordinate system. Notice that this error
vector is not constrained to be vertical, nor
necessarily orthogonal to any of the surfaces.
For each
i
P
= [xi, yi, zi]T, the error vector
,,
i i i i i i
x y M x y P
, i 1,2,…,n cannot
usually be determined because of the difficulties in
establishing the homologous point
i
P
. However,
even if the homologous point
i
P
cannot be identified,
the triangle Ti r-DEM containing it, can usually be
identified.
Fundamental property: An important property is
that the projection of the error vector
,,
i i i i i i
x y M x y P
on a unitary vector Ni
orthogonal to the surface Ti r-DEM remains
invariant if the point
i
P
is replaced by any other point
Q
Ti.
For the projection of the error vector on Ni the
scalar product is
(1)
where
i
M
is the normal projection of
i
M
relative to the surface of the triangle Ti, the point of
the triangle determined by the line normal to that
triangle and which passes through
i
M
.
For any point
Q
belonging to the surface of the
triangle Ti, we define a vector, which it will be called
Q
, the projection of the difference
i
MQ
on N
coincides with the projection, on N, of
,,
i i i i i i
x y M x y P
(2)
Both projections are equal to
ii
MM
. The
fundamental property resulting from relation (1) is
the reason for the choice of the name PDEM.
This relation implies that the length of the
projection of the error vector may be computed
Engineering World
DOI:10.37394/232025.2023.5.21
Jose F. Zelasco, Kevin Ennis,
Barbara Sulpis, Jose Luis Fernández Ausinaga
E-ISSN: 2692-5079
188
Volume 5, 2023
knowing only the triangle Ti r-DEM that contains
i
P
, even without knowing the exact position of
i
P
.
Consequently, the variance of
ii
MM
is
′
′
(3)
where
cos , cos , cos
are known, because
they are the direction cosines of the normal unit
vector, obtained from the data of the surface triangle
Ti r-DEM, with
2 2 2
cos cos cos 1
(4).
An estimator for this
2
Ni
is therefore the
ii
MM
; if n observations were available, the
estimator would be:
2
1
n
ii
j
MM
n
(5)
For different triangles we have different normal
vectors, and different values of
2
Ni
. Moreover, we
have one observation for each, and one estimate for
each. This gives us one relation for each. In these
expressions, the coefficients:
2 2 2
cos , cos , cos
i i i
(4), are known, and
2 2 2
,
x y z
are unknown, and they are what we
need to estimate. Our n expressions (4) give us an
observation matrix, or design matrix
2 2 2
1 1 1
2 2 2
2 2 2
2 2 2
2 2 2
cos cos cos
cos cos cos
cos cos cos
cos cos cos
i i i
n n n
M
(6)
The n expressions (4) allow us to establish n
estimates for
2
Ni
, which form a vector
1
2
2
2
2
2
i
n
N
N
N
N
L
(7)
We may now estimate
222
,,
x y z
as if they
were the parameters of an ordinary linear regression
of the output variable
2
Ni
as a linear function of the
three variables
2 2 2
cos , cos , cos
i i i
, given by
(4). We have at our disposal n points, and the
estimates may be obtained by the usual least squares
regression method. We seek the values of
222
,,
x y z
, which we may write as a vector
2
22
2
x
y
z
(8)
and we seek to minimize the sum of squares of
differences
. (9)
A complete discussion about the method is shown
in Zelasco 2019 [19].
4. Error Correlation of the DEM
Obtained by IFSAR: PDEM Reformu-
lation.
The IFSAR geometry, see El-Taweel 2007 [3],
Redadaa and Benslama, 2005 [14], Massonnet et al
1996 [11], and Massonnet and Feigl, 1995 [12]
carries a correlation between axes (range axis) and
the axis (Fig. 1).
It gets a diagonal covariance-matrix from a rotation
around the axis. This rotation joins the axis with
the direction of the radar.
This hypothesis is justified by the formulas presented
in El-Taweel 2007 [3] and Redadaa and Benslama,
2005 [14] as well as Massonnet et al, 1996 [11].
After the rotation, in the new referential, the axis
parallel to the trajectory of the satellite does not
Engineering World
DOI:10.37394/232025.2023.5.21
Jose F. Zelasco, Kevin Ennis,
Barbara Sulpis, Jose Luis Fernández Ausinaga
E-ISSN: 2692-5079
189
Volume 5, 2023
change its position, the and
z
directions
correspond to the normal plane of the trajectory and
the random variables in the three directions are not
correlated.
Figure 1
B
is the base (distance between antennas
1A
and
2A
),
H
is the height of the antennae
1A
and
R
is
the distance from
1A
to the target
C
.
The value of z (height of the target) is given by:
, where
H
and
R
are known and
is incognito.
So: where
R
,
B
,
r
(
r
is functional to the phase-difference and the
wave length) are known.
In the triangle of vertices , and formed by the
two antennas and the target,
rR
is the distance
between A2 and , the angle in is α, and
then,
.
Therefore:
,
and
Finally:
(10)
sin Ry
(11)
.
5. Description of the Experiment
The r-DEM is composed of a set of points that
permits the construction of a net of
K
triangles. Each
triangle
KkTk,,1
belongs to a plane
defined by a unitary vector
Kk
k,,1
.
The following needs to be determined:
- the coordinates of the center of each triangle
expressed in the rotated reference system,
a value of standard deviation , and for
each direction in which the random variables are
independent. Based on these values, random
values that follow the normal law (noise) are
determined. These new random values will be
added to the coordinates of the center of mass of
the triangles in the direction of the 3 axes
expressed in this new reference. The new
coordinate points constitute the (simulated) e-
DEM.
the coordinates of the normal unitary vector
to each triangle in the new referential
the distance that separates each point from the e-
DEM (simulated to evaluate), to the plane that
contains the corresponding triangle
the normal deviation of the points of the e-DEM
in the directions of each coordinate of the rotated
system: , and .
Table 1 shows the results with triangles selected by
considering the angle they have with the Z axis.
Results are presented, so that for each simulation they
show:
The value of
used to produce noise on each
axis,
The PDEM estimation,
The standard deviation of the estimation (over 20
times),
The relative error (standard deviation/noise) of the
estimations.
Fig. 2 shows the histograms of the PDEM,
considering the angle, only with the Z axis.
Figure 2: Previous Method PDEM Histogram.
Some results in Table 1 are not satisfactory for
x
.
To improve these results, we did other simulations
(A, B, C, D, E, F) selecting triangles with respect to
the 3 axes (not only to the Z axis)
For each simulation, we selected a different number
of triangles with respect to each axis, indicating the
triangle slope interval (FI_ALFA, FI_BETA o
FI_GAMA) in relation to each axis (X, Y, and Z)
respectively.
x
y
z
K
x
y
z
Engineering World
DOI:10.37394/232025.2023.5.21
Jose F. Zelasco, Kevin Ennis,
Barbara Sulpis, Jose Luis Fernández Ausinaga
E-ISSN: 2692-5079
190
Volume 5, 2023
Table 1
Simulation A produced the best results. For this
simulation, 500 triangles were selected for each axis.
The angle that forms the x axis with each of its 500
triangles does not exceed 60 degrees. The same
criterion is taken into account for the y axis. It can be
observed that in the direction of the x axis, the results
are very satisfactory, except for the first two tests.
Therefore, in the following section we only evaluate
this simulation.
6. Evaluation of the Results
The average values in the three axes are given by:
N
nXX n
;
N
nYY n
;
N
nZZ n
The standard deviation of
(variance-covariance
matrix after rotation of a θ angle to nullify the
correlation) is defined as:
21
2N
nnn
where
n
is the real value of the sample – very
similar to the value used to produce the sampling of
the errors (noise) – and
n
is the value obtained by
the PDEM.
We did several tests using different values to produce
the noise (errors that produce the simulated e-DEM
of study). This enabled us to study the behavior of the
PDEM for different relations of noise in the axes.
With the same values of noise, we performed each
test over 20 times with different random values.
As it was said, six simulations were performed. The
one with the best results (Simulation A), is obtained
by selecting triangles related to the 3 axes in equal
proportion (500 triangles in each case, it means a total
of 1500) so that for the X and Y’ axes, the triangle
slope does not exceed 60 degrees and in the Z’ axis it
does not exceed 45 degrees.
For simulation A:
Table 2 shows the results with triangles selected by
considering the angle they have with the three axes;
And again:
The values of
used to produce noise in each
axis,
The PDEM estimation,
The standard deviation of the estimations (over 20
times),
The relative error of the estimation.
Table 2: simulation A
Figure 3 shows the histograms obtained for the
simulations A.
The histograms show how, after the rotation needed
to nullify the correlation, the triangle slopes are
modified. It is noteworthy that since the rotation is
performed around the X axis, the first histogram
remains unchanged.
Figure 3: Reformulated Method PDEM Histogram
(Simulation A).
The histograms show how, after the rotation needed
to nullify the correlation, the triangle slopes are
modified. It is noteworthy that since the rotation is
Engineering World
DOI:10.37394/232025.2023.5.21
Jose F. Zelasco, Kevin Ennis,
Barbara Sulpis, Jose Luis Fernández Ausinaga
E-ISSN: 2692-5079
191
Volume 5, 2023
performed around the X axis, the first histogram
remains unchanged.
7. Conclusions
In the case of DEM’s obtained by IFSAR technology,
the PDEM provides error estimation for the three not
correlated directions: the beam axis, the azimuth axis
and the axis normal to the previous two (y’; the
rotated range axis). As expected, the results are
excellent in the direction of the beam axis. Since a
rotation must be done to find the uncorrelated
directions the unevenness of the surface artificially
increases in the range axis direction therefore the
error estimations are very good in the direction of the
y' axis. The estimation error in the azimuth direction
is compromised because it depends on enough
irregular terrain conditions.
The PDEM provides a useful tool for evaluating the
error of Digital Surface Models in the general case,
as it was shown in previous works, and also when the
DEM’s are obtained by IFSAR technology.
Acknowledgement:
We thank Mr. Hugo RYCKEBOER for his comments
and suggestions.
We extend our appreciation to Mr. Bernard Meyer for
the real DEM used for testing the method.
References:
[1] Abbas, Iyad., 1994. Vector database and map
error: Problems with spot control; an
alternative method based on Hausdorff's
distance: The linear control (Base de données
vectorielles et erreur cartographique:
problèmes posés par le contrôle ponctuel; une
méthode alternative fondée sur la distance de
Hausdorff: le contrôle linéaire). Thesis (PhD).
Université Paris 7, France.
[2] Dunn R., Harrison A. R. and White J. C., 1990.
Positional accuracy and measurement error in
digital databases of land use: An empirical
study. International Journal of Geographic
Information Systems, Vol 4 No 4, pp. 385-398.
El–Taweel, 2007. Enhanced Model for
Generating 3-D Images from RADAR
Interferometric Satellite Images. WSEAS
Transactions on Environment and
Development. Issue 3, Volume 3,. ISSN: 1790-
5079.
[3] Grussenmeyer P., Hottier P. and Abbas I., 1994.
The topographic control of a map or a database
constituted by photogrammetric means (Le
contrôle topographique d'une carte ou d'une base
de données constituées par voie
photogrammétrique). Journal XYZ Association
Française de Topographie, No 59, pp. 39-45.
[4] Guptill, S. C. and J. L. Morrison., 1995.
Elements of spatial data quality. Oxford:
Elsevier.
[5] Harvey, F.,. Quality Needs More Than
Standards. In: M.F. Goodchild, and R.
Jeansoulin, eds. Data Quality in Geographic
Information: From Error to Uncertainty. Paris:
Hermes, 1998, 37–42.
[6] Hirano, A., Welch R. and Lang H., 2003.
Mapping from ASTER stereo image data : DEM
validation and accuracy assessment. ISPRS
Journal of Photogrammetry and Remote
Sensing, No 57, pp. 356-370.
[7] Hottier, P., 1996. Geometric quality of the
planimetry. Punctual control and linear control.
Dossier: The notion of precision in the GIS.
(Qualité géométrique de la planimétrie.
Contrôle ponctuel et contrôle linéaire. Dossier:
La notion de précision dans le GIS). Journal
Géomètre, Ordre des Géomètres-Experts
Français, No 6, pp. 34-42.
[8] Lester, M., and Chrisman, N., 1991. Not all
slivers are skinny: a comparison of two methods
for detecting positional error in categorical
maps. Proceedings of GIS/ L1S'91 Atlanta
conference, October 28-November 1, No 1, pp.
648–657.
[9] Laurini, R., and Milleret-Raffort F. Les bases de
données en géomatique. Paris: HERMES. 1993.
[10] Massonnet D., Vadon H. and Rossi M., 1996.
Reduction of the need for phase unwrapping in
radar interferometry. IEEE Transaction on
Geosciences and Remote Sensing No 34, pp.
489-497.
[11] Massonnet D. and Feigl K. L., 1995. Radar
interferometry and its applications to changes in
the Earth surface. Reviews of Geophysics No 36,
pp.441-500.
[12] Maune, D. F., 2001. Digital Elevation Model
Technologies and Applications: The DEM Users
Manual. ASPRS
[13] Redadaa S. and Benslam M., 2005. Topographic
SAR Interferometry: A Geolocation Algorithm.
WSEAS Transactions on Systems. Issue 12, Vol
4.
[14] Ubeda, T. and Servigne, S., Geometric and
Topological Consistency of Spatial Data.
Proceedings of the 1st International Conference
on Geocomputation, Leeds, United Kingdom,
Sep. 17-19, pp. 830-842.
Engineering World
DOI:10.37394/232025.2023.5.21
Jose F. Zelasco, Kevin Ennis,
Barbara Sulpis, Jose Luis Fernández Ausinaga
E-ISSN: 2692-5079
192
Volume 5, 2023
[15] Zelasco J. F., Julien P., Porta G., Ennis K.,
Donayo J., 2013. PDEM Estimator for Digital
Elevation Model: Utilities. International
Journal of Applied Mathematics and
Informatics, Vol 1 7, 24-31
[16] Zelasco J. F., Donayo J., 2019. Effectiveness of
Geometric Quality Control Using a Distance
Evaluation Method. International Journal of
Image and Data Fusion, Vol 10 No 4, pp. 263-
279.
Engineering World
DOI:10.37394/232025.2023.5.21
Jose F. Zelasco, Kevin Ennis,
Barbara Sulpis, Jose Luis Fernández Ausinaga
E-ISSN: 2692-5079
193
Volume 5, 2023
Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
The authors equally contributed in the present
research, at all stages from the formulation of the
problem to the final findings and solution.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
Conflict of Interest
The authors have no conflicts of interest to declare
that are relevant to the content of this article.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.en
_US
