GPS Admissible Errors in Positioning Inventory Plots for Forest Structure Studies
F. Mauro 1 R. Valbuena 1 A. García 1 J.A. Manzanera 1
1 Research group for Sustainable Management. TECNATURA (E.T.S.I. Montes) Faculty of
Forestry, Ciudad Universitaria, S/N 28040 Madrid, Spain. +34-913366401.
f.mauro@alumnos.upm.es, r.valbuena@upm.es, antonio.garcía.abril@upm.es
josenatonio.manzanera@upm.es
Introduction
Sampling designs based on georeferenced field plots are commonly affected by positional
errors caused by the survey instrumentation’s positional errors. But these errors are not the
only factor to be taken into account when selecting among diverse commercial topographic
equipment. Other restrictions should be considered, for instance the price of the equipment,
availability of processing software, or a user-friendly interface in accordance with the
surveyor’s expertise. There are basically two constraints related to the accuracy. One is the
positioning error itself, whereas the other one is the admissible error in a specific work. Many
authors have studied differences on positional errors among diverse GPS receivers measuring
under different forest conditions (Næsset 2002; Tuceck 2002; Rodriguez 2007). This paper
focuses on assessing the admissible errors, rather than quantifying the accuracy itself. We
propose a methodology for evaluating whether the Global Positioning System (GPS)
positional errors can be accepted when studying forest structure from circular field plots.
This study was done when a systematic sampling based on circular plots was being designed
to validate a set of models relating forest structure and Light Detection And Ranging
(LiDAR) variables. The study responds to a need for quantifying whether or not the positional
errors would affect the results of this validation. The models to be validated were computed
previously and they describe forest structure by means of tree height distribution.
Due to the above-mentioned positional errors, the location of circular plots is finally shifted
from its true situation in the field. Hence, we may consider the existence of both the real
circular plot and the shifted circular plot whose coordinates were obtained from the GPS
survey. The LiDAR variables for the models that we are validating are going to be extracted
around the GPS point. If the forest structure of a real field plot is different from the forest
structure around the GPS point, the use of this plot in the validation process might be
incorrect. Nevertheless, we accepted a certain amount of error, so that a receiver was
considered invalid for the survey when expected differences were significant in more than 5%
of the observations.
Error tolerance is seriously affected by plot radius, as we may expect the tolerance to be
higher as we increase plot size. On the other hand, as we are studying structures, it is
necessary to have a minimum number of trees in the majority of the plots. But big sizes
dramatically increase the cost and the effort needed for each plot. For these reasons, we had to
find a compromise between the number of trees available per plot and the decrease of error
tolerance and the costs of field surveying.
Objectives
In this study, we aim at establishing and applying a repeatable methodology for assessing the
minimum plot size which would not be susceptible to changes due to GPS positional errors
when studying forest structure.
Material and methods
Study area
For applying this methodology, we selected the Scots pine (Pinus sylvestris L.) forest of
Valsain (approx. 40º 49’ N, 4º 1’ W) located in Segovia, Spain. The study site is situated in
mountainous relief, in the skirts of the north face of Guadarrama Range, with a maximum
elevation of 2428 m asl. These forest stands are naturally regenerated and shelterwood
managed, with rotation periods of 120 years. A long regeneration period of 80 years makes it
possible to find a great variety of sizes and ages within a stand.
Six rectangular plots of 60 x 40 m were selected in order to represent all the different forest
structure types present. Two of them were located in regeneration stands with parent trees.
Another one was established in a regular young stand. The age of this stand was around 60
years. Finally, three more plots were located in mature and extra-mature stands with ages
between 100-135 years. In one of these plots natural regeneration was already starting.
Every tree within plot higher than 1.3 m was georeferenced with internal accuracies of 1-2
cm. The height, diameter, crown height and crown diameter were recorded for each tree.
Theoretical framework
The theoretical framework of this methodology is the definition of conditional probabilities,
since the probability of obtaining a mistake in our results is conditioned by the error
occurrence in GPS surveying. For each plot radius (r, m) at one meter intervals from 5 to 16
m, we computed the probability of a GPS error (e, cm) to motivate an estimation of a forest
structure type different than the measured one 𝑃(𝑋 = 𝐷𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑡)𝑟 . Accordingly, X is a
binary variable containing a logical statement on whether or not the structure in the shifted
plot is different than the structure in the measured field plot. As observed in (1), we can
calculate the probability of obtaining a GPS error which generates a mistaken outcome 𝑃(𝑋 =
𝐷𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑡 ∩ 𝜀 = 𝑒)𝑟 , as the product between the probability of obtaining such error from the
measurement 𝑃(𝜀 = 𝑒), and the probability of obtaining a different result once we had that
given error 𝑃(𝑋 = 𝐷𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑡|𝜀 = 𝑒)𝑟 .
𝑃(𝑋 = 𝐷𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑡 ∩ 𝜀 = 𝑒)𝑟 = 𝑃(𝑋 = 𝐷𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑡 ∣ 𝜀 = 𝑒)𝑟 ∗ 𝑃(𝜀 = 𝑒)
(1)
Hence, this probability of obtaining different structures 𝑃(𝑋 = 𝐷𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑡)𝑟 , could be
expressed (2) as the sum of all possible combinations in which we both obtain an error from
the GPS and this error leads us to compute a different structure 𝑃(𝑋 = 𝐷𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑡 ∩ 𝜀 = 𝑒)𝑟 .
𝑃(𝑋 = 𝐷𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑡)𝑟 = ∑∞
𝑒=0 𝑃(𝑋 = 𝐷𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑡 ∩ 𝜀 = 𝑒)𝑟
(2)
The equation (1) allowed us to split the calculations into two simpler problems. First, we
estimated the probability of error occurrence when taking GPS measurements 𝑃(𝜀 = 𝑒). And
second, we estimated the probability of a given error to obtain a prediction of forest structure
type different than the real one 𝑃(𝑋 = 𝐷𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑡|𝜀 = 𝑒)𝑟 .
Field data to characterize the GPS errors 𝑷(𝜺 = 𝒆)
The GPS device employed in this study was a Topcon Hiper-pro receiver. Topcon Hiper Pro
is a geodetic multi-channel dual-frequency receiver observing GPS/GLONASS, pseudorange
and carrier phase L1-L2. It allows Static/Rapid Static surveying as well as Real Time
Kinematic (RTK). The nominal precision provided by the vendor is 3 mm + 0.5 ppm times
the baseline length for horizontal coordinates when working on Static & Rapid Static mode
(Topcon 2006).
We established 23 landmarks with a methodology similar to the one described by Næsset
(2002), five of which were located in the absence of canopy cover. They were considered as
control points since not any obstacle was disturbing GPS signal reception at their position.
Moreover, we assured their results matched with the nominal precision given by the vendor
by comparing them from two different and independent observations. The other 18 markers
were distributed under the canopy cover ranging all the structure types in the forest. The
coordinates of these points were obtained by terrestrial traverse surveys starting at the control
points. Once the ground-truth coordinates were known, a GPS-GLONASS static observation
was recorded at every landmark. The phase differential correction was done in postprocessing using the data from a permanent reference station located at a distance of 13 km.
We calculated GPS errors as the difference between the ground truth XY coordinates and
those measured with GPS as (3) and (4). We did not consider errors in height since our
methodology computes forest structure from points located using only XY coordinates,
regardless of the Z coordinate.
𝜀𝑥𝑖 = 𝑋𝐺𝑃𝑆 – 𝑋𝑟𝑒𝑎𝑙 (3)
𝜀𝑦𝑖 = 𝑌𝐺𝑃𝑆 − 𝑌𝑟𝑒𝑎𝑙 (4)
Probability of error occurrence P(ε=e)
We analyzed the presence of systematical errors, differences in magnitude between X and Y
errors and correlations between X and Y errors, finding no evidence of either one. For this
reason, we assumed that error distribution was similar in every direction and we only studied
the magnitude of the errors. We therefore computed error magnitudes as the Euclidean
distances between GPS and ground-truth points.
We searched for a density function which fits properly to the errors found. As a result, we
were able to describe the probability of having an error 𝑃(𝜀 = 𝑒) by means of such density
function. We selected a Gamma function because it is flexible and its domain is in all real
positive numbers, since an error magnitude cannot be negative. The Chi-square goodness-offit test showed a p-value of 0.54 so we therefore further accepted that fitted function as a
descriptor of the probability for error occurrence 𝑃(𝜀 = 𝑒) in (1) and we used it in the
subsequent simulation.
Failure rate for a given error 𝑷(𝑿 = 𝑫𝒊𝒇𝒇𝒆𝒓𝒆𝒏𝒕|𝜺 = 𝒆)𝒓
The probability of obtaining GPS coordinates from points with different structure when the
error is e was obtained by simulation, using the dataset from the six large rectangular plots. In
order to prevent edge effects, we inner-buffered every rectangular plot at a distance (b, m)
determined as the sum of the radius and a certain distance selected looking at the GPS error
detected. Then for each radius 30 random points were distributed within the corresponding
buffer zone in every large plot. Each one of them was considered the center of a “simulated
circular plot center”. For every “simulated circular plot center” all the trees within a radius
were selected as they would be selected in the field if these points were real plot centers.
Then, around every “simulated field plot center”, 20 “simulated shifted plot centers” were
randomly distributed in every direction for each magnitude of error e =0.1, 0.2,…b meters. As
the plot center shifted, trees within circular plot may have changed, therefore perhaps
changing the structure as well. We used a two-sample Kolmogorov-Smirnov test for
evaluating whether the shift generates a different sample.
For each e and r, we recorded the proportion of simulations for which the structure within the
“simulated field plot” and the structure within the “simulated shifted plot” were found
significantly different. These proportions are conditional failure rates, which we regarded as
conditional probabilities to obtain a different structure type 𝑃(𝑋 = 𝐷𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑡|𝜀 = 𝑒)𝑟 . As
mentioned above, these failure rates are conditioned to a specific error e.
Results
As we can see in Figure 1, the susceptibility of error tolerance to an increase in plot size was
confirmed as expected. Failure rate tended to increase when the magnitude of the error did.
The authors wish to emphasize that, within the magnitude of error which our GPS actually
showed in the field work, a circular plot of r = 10 m is far from reaching the threshold
tolerance we fixed. This chart shows only the effect of the structure. It does not include the
error detected so it could be used to check the errors of other devices as well. Nevertheless, in
this case, it was not necessary to integrate this factor, in view of the failure rates within the
observed error range.
Conditional Failure rate vs error e
Conditional Failure rate
0.35
0.3
Radius 10 m
0.25
Observed error range.
0.2
Radius 8 m
0.15
0.1
0.05
Radius 6 m
0
0
1
2
3
4
5
Error e (m)
6
7
8
9
Tolerance threshold
Figure 1. Conditional failure rates observed when simulating GPS positional errors from circular plots
of diverse radius.
Discussion
The methodology we present in this paper was demonstrated reliable for evaluating the effect
of positional GPS errors when studying forest structures employing circular field plots. It
requires itself a big effort in data acquisition, which would make it inappropriate for small
inventories, since an important amount of field work is necessary for computing the error
distribution and georeferencing such large plots in which every tree should be located with
centimetric precision. Those inputs, error distribution and georeferenced plots, are inputs or
outputs of other works, so it could be possible to recycle or adapt old datasets to reduce the
ad-hoc work. For example georeferenced plots are commonly used to study growth and
competition models. For instance, Mette et al (2009) used a set of 39 plots of 0.25 ha each to
evaluate a growth simulator, whereas Næsset (2002), Tuceck (2002) and Rodriguez (2007)
provided different error estimations which might be applied for this methodology as well. The
datasets employed in those studies could be complemented and used to fit error distributions
and to apply this methodology in their corresponding environments and GPS receivers.
Furthermore a cost-benefit study of inventory work performed over large areas or from
airborne LiDAR sensors would clearly benefit from a previous analysis of the effect of GPS
positional error.
Further study would be needed in order to analyze whether or not these results would show
significant differences among diverse forest structure types. If a sampling design stratified by
forest structure is desired, the workload increases and a specific error function and plots
would be needed for every structure.
A similar simulation showed that the radius needed to ensure a minimum number of trees in
the majority of the plots should be at least 18 m in this case. Such a radius should be
unsusceptible to the GPS errors in view of our results. Therefore, this radius would meet both
constraints: the minimum number of trees needed per plot and the restriction about the
sensitivities to the errors. Working with this plot radius ensures that the field plot and the
LiDAR datasets can be correctly related and that the noise introduced by the GPS errors is
tolerable for the validation process.
Conclusion
This paper explains a reliable methodology which may be repeated in other forest studies
pursuing an evaluation of the decrease in failure occurrence when augmenting plot size. This
quantification includes the effect of the GPS methodology actually used in the field work.
Therefore, it presents a practicable and realistic measurement which may be compared with
the extra costs and effort which an increase in plot size causes. Finally, in this particular study
the minimum number of trees was more restrictive than the susceptibility to the GPS error,
though it might not be the case in further applications.
Literature Cited
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mixed Silver fir–Norway spruce stands in South-West Germany. Ecological modeling.
Næsset, E. 2002. Assessing point accuracy of DGPS under forest canopy before data acquisition, in
the field and after postprocessing.
Topcon 2006) Hiper-Pro operator´s manual. P/N 7010-0681. Rev C. Topcon Positioning Systems Inc.,
California
Tuceck, J. 2002. Forest canopy influence on the precision of location with GPS receivers. Journal of
forest science.
Rdríguez, J. 2006. Comparison of GPS receiver accuracy and precision in forest environments.
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