1 Methods

1.1 Study area

Table 1.1: Properties of the study areas maisC and maisD, including the number of directional (N direct) and omnidirectional (N omni) stations and the mean distance to the closest station (distance), the area enclosed by the station set-up (core area) as well as the number of transmitters used for testing (N tags) and the height above ground they were attached (height tags).
site maisC maisD
location Vierlinden, Brandenburg, Germany Döbberin, Brandenburg, Germany
elevation 6-8 m asl 47-63 m asl
vegetation agriculture, tree lines, hedges, ditches with reed agriculture, tree lines, lakes with reed and woods
core area 20 ha 16 ha
N direct (distance) 10 (175 m) 8 (170 m)
N omni (distance) 10 (155 m) 0
height of station 6 m 8 m
N testtags 4 3
height testtags 0.5m, 1m, 1.5m, 2m 0.5m, 1m, 1.5m

1.2 Ground-truth data

Table 1.2: Properties of test tracks used to compare position estimations and errors for maisC (C1-3, Ctest) and maisD (D1-3, Ctest). N points is the number of recorded GPS fixes (summarized in 2-second intervals) and N tags the number of used transmitters.
track ID date duration (h) length (km) N points N tags
C1 2023-10-05 0.73 3.33 1259 3
C2 2023-10-09 2.99 10.59 5371 4
C3 2023-10-10 3.00 11.74 5401 4
Ctest 2023-10-13 1.05 5.07 1744 4
D1 2023-09-07 4.32 7.30 3406 3
D2 2023-09-16 1.81 3.30 3219 3
D3 2023-09-28 2.36 6.63 4250 3
Dtest 2023-09-13 1.47 3.96 2646 3

1.3 Bearing estimation

The bearings computed using the described method (x-axis) were compared to bearings computed from the positions recorded using a GPS track (y-axis), where the color indicated the error in bearing estimation. Received signals were matched to the GPS locations using the nearest time, hence GPS locations where no signal was received were not used for calibration. In the example, 10,543 signal detections with GPS location matches resulted in a median error of 4.5° and a root mean square error (RMSE) of 8.3°. While the median error indicated that the antenna might be rotated slightly by around 4.5 during the tests, the RMSE showed the variance in bearing estimates. Specifically the bearings near perpendicular to the antenna axes, i.e., 0°, 90°, 180° and 270°, had the largest errors. This was due to falsely selecting the antenna with the second strongest signal reception, which might happen due to the reception on the back lobe of the antenna.

Example of linear bearing model applied to one station in maisC. x-axis = estimated bearings, y-axis = (true) bearings based on GPS locations from ground-truth data.

Figure 1.1: Example of linear bearing model applied to one station in maisC. x-axis = estimated bearings, y-axis = (true) bearings based on GPS locations from ground-truth data.

1.4 Distance estimation

Fig. 1.2 showed a calibration curve of an exponential power distance model fitted to the underlying calibration data. The received signals and GPS locations were matched based on the nearest time, as described in the previous section. The curve was optimized using non-linear least squares. In the shown example of the directional station, a calibration track of 7,371 detected GPS fixes achieved a median distance error of 32.7 m with a root mean square error of 49.7 m. For the omnidirectional station, a calibration track of 13,014 detected GPS fixes achieved a median distance error of 24.4 m with a root mean square error of 60.2 m. The smaller distance error for omnidirectional stations is due to their uniform antenna characteristic compared to the directional characteristic. The error in distance estimation was mainly caused by topography and vegetation in the test area as well as transmitter orientation.

Exponential power distance model fit for one directional (*left*) and one omnidirectional (*right*) station in maisC.

Figure 1.2: Exponential power distance model fit for one directional (left) and one omnidirectional (right) station in maisC.

1.5 Angulation

For the example we used three directional stations \(s_0\), \(s_1\) and \(s_2\) each receiving the same signal of a transmitter. Based on the received signals strengths, \(s_0\) selected the southern as the main and eastern as second antenna. Using the received signal strengths and the configured maximum gain difference \(\Delta m_ = 20\), the bearing was computed as follows:

\[ \begin{align} \Delta \omega_{s_0} &= (90° - 90° * \frac{-67 -(-68)}{20})~/~2 = 42.8° \\ \omega_{s_0} &= 180° - 42.8° = 137.2° \end{align} \]

The bearing of station \(s_2\) was computed analog resulting in \(\omega_{s_2} = 274.5°\). For \(s_1\) no bearing could be estimated, since neither the eastern nor the western antenna detected the signal and hence could not be selected as second antenna to the main southern antenna. In terms of distance estimates in the presented illustrative example a fixed optimized curve of \(d = 4 * 0.94^{p}\) was used (note that in a real world application, each station would have an individually fixed model). This results in distance estimations \(d_{s_0} = 4 * 0.94^{-67} = 252.6~m\) and \(d_{s_2} = 4 * 0.94^{-71} = 323.6~m\). Bearing and distance estimations are then used to compute pairwise intersections, which yielded one intersection in the presented example.

Exemplified position finding using angulation based on bearings.

Figure 1.3: Exemplified position finding using angulation based on bearings.

1.6 Antenna beams

The antenna beam computations were performed using a detection range of \(r = 1,300 m\) and a gain normalization of \(p_n = 90 dB\). First, the gain normalization was used to compute per-antenna weights \(W\) and the summed station weight \(w\):

\[ \begin{align} W_{s_0} &= (-89, -68, -67, -83) + 90 = (1, 22, 23, 7) \\ w_{s_0} &= \sum W_{s_0} = 53 \end{align} \]

Second, the relative positions \(\Delta x\) and \(\Delta y\) were computed using the weight difference on the respective axis divided by the station weight and multiplied with half the detection range.

\[ \begin{align} \Delta x_{s_0} &= \frac{W_{s_0, 1} - W_{s_0, 3}}{w_{s_0}} * \frac{p}{2} = \frac{22-7}{53} * 650.0 = 184.0 m \\ \Delta y_{s_0} &= \frac{W_{s_0, 0} - W_{s_0, 2}}{w_{s_0}} * \frac{p}{2} = \frac{1-23}{53} * 650.0 = -269.8 m \end{align} \]

Third, the absolute locations were computed by adding the station coordinates, taking into account any configured rotation of the station.

For every station with at least one received signal, a station location and a weight value are obtained, indicated by the colored plus symbol. Finally, using the weight \(w\) of the stations, a combined position (weighted mean) was computed illustrated by the diamond marker.

Exemplified position finding using antenna beams based on antenna geometry.

Figure 1.4: Exemplified position finding using antenna beams based on antenna geometry.

1.7 Multilateration

For multilateration, the distance estimates per station were computed using the aforementioned exponentially fitted function \(dist = 4 * 0.94^{power}\). This resulted in the distance estimates
\(d_{S0} = 4 * 0.94^{-67} = 252.6 m\),
\(d_{S1} = 4 * 0.94^{-78} = 499.0 m\) and
\(d_{S2} = 4 * 0.94^{-71} = 323.6 m\).
The figure shows the distances as radius for the circles drawn around the center of the stations. An initial position is computed by the inverse distance weighted average of of the station positions. \(x_0 = 0.437 * (200, 400) + 0.221 * (600, 700) + 0.341 * (800, 200) = (493.4, 398.1)\) Starting at this initial position the location is optimized by minimizing the summed distance error using the Nelder-Mead method, resulting in the final position \((442.9, 243.5)\).

Exemplified position finding using multilateration based on distances.

Figure 1.5: Exemplified position finding using multilateration based on distances.

1.8 Covered area

Theoretical areas where an estimated position can be located (based on the respective calculation per method). If the signal of a transmitter outside the area is detected, the estimated position will fall within the area. For directional antenna beams (direct ab) we used a detection range of 800 m for maisC and 900 m for maisD (see results section), for directional angulation (direct an) we assumed a maximum distance of 2*450 m which matches a maximum signal strength of approx. -80 dB, and for omnidirectional multilateration (omni ml) we assumed a maximum distance of 450 m. Note that the threoretical area does not imply that all true positions within this area can be estimated by the respective method. Copyright map data: OpenStreetMap contributors

Figure 1.6: Theoretical areas where an estimated position can be located (based on the respective calculation per method). If the signal of a transmitter outside the area is detected, the estimated position will fall within the area. For directional antenna beams (direct ab) we used a detection range of 800 m for maisC and 900 m for maisD (see results section), for directional angulation (direct an) we assumed a maximum distance of 2*450 m which matches a maximum signal strength of approx. -80 dB, and for omnidirectional multilateration (omni ml) we assumed a maximum distance of 450 m. Note that the threoretical area does not imply that all true positions within this area can be estimated by the respective method. Copyright map data: OpenStreetMap contributors

1.9 Station cover

Exemplified calculation of detection probability (left) and station cover (summed up detection probabilities across all stations, right) for directional antennas in maisC. Copyright map data: OpenStreetMap contributors

Figure 1.7: Exemplified calculation of detection probability (left) and station cover (summed up detection probabilities across all stations, right) for directional antennas in maisC. Copyright map data: OpenStreetMap contributors

2 Results

2.1 Estimated positions

maisC 0.5m

Position errors (PE) of estimated positions per testtrack (columns) and method (rows) for a test transmitter at 0.5 m above ground in maisC. Ground-truth positions are displayed and colored by PE. Positions that could not be estimated are colored in gray. Note the log10 scaling of the PE color scale.

Figure 2.1: Position errors (PE) of estimated positions per testtrack (columns) and method (rows) for a test transmitter at 0.5 m above ground in maisC. Ground-truth positions are displayed and colored by PE. Positions that could not be estimated are colored in gray. Note the log10 scaling of the PE color scale.

maisC 1m

Position errors (PE) of estimated positions per testtrack (columns) and method (rows) for a test transmitter at 1 m above ground in maisC. Ground-truth positions are displayed and colored by PE. Positions that could not be estimated are colored in gray. Note the log10 scaling of the PE color scale.

Figure 2.2: Position errors (PE) of estimated positions per testtrack (columns) and method (rows) for a test transmitter at 1 m above ground in maisC. Ground-truth positions are displayed and colored by PE. Positions that could not be estimated are colored in gray. Note the log10 scaling of the PE color scale.

maisC 1.5m

Position errors (PE) of estimated positions per testtrack (columns) and method (rows) for a test transmitter at 1.5 m above ground in maisC. Ground-truth positions are displayed and colored by PE. Positions that could not be estimated are colored in gray. Note the log10 scaling of the PE color scale.

Figure 2.3: Position errors (PE) of estimated positions per testtrack (columns) and method (rows) for a test transmitter at 1.5 m above ground in maisC. Ground-truth positions are displayed and colored by PE. Positions that could not be estimated are colored in gray. Note the log10 scaling of the PE color scale.

maisC 2m

Position errors (PE) of estimated positions per testtrack (columns) and method (rows) for a test transmitter at 2 m above ground in maisC. Ground-truth positions are displayed and colored by PE. Positions that could not be estimated are colored in gray. Note the log10 scaling of the PE color scale.

Figure 2.4: Position errors (PE) of estimated positions per testtrack (columns) and method (rows) for a test transmitter at 2 m above ground in maisC. Ground-truth positions are displayed and colored by PE. Positions that could not be estimated are colored in gray. Note the log10 scaling of the PE color scale.

maisC all meth.

Overview whether positions in maisC could be estimated by all methods ('yes', orange) or not ('no', grey) per testtrack (columns) and test transmitter (rows, the label depicts the transmitter's height above ground in m). Note that transmitter 2 was not used for test track C1.

Figure 2.5: Overview whether positions in maisC could be estimated by all methods (‘yes’, orange) or not (‘no’, grey) per testtrack (columns) and test transmitter (rows, the label depicts the transmitter’s height above ground in m). Note that transmitter 2 was not used for test track C1.

maisD 0.5m

Position errors (PE) of estimated positions per testtrack (columns) and method (rows) for a test transmitter at 0.5 m above ground in maisD. Ground-truth positions are displayed and colored by PE. Positions that could not be estimated are colored in gray. Note the log10 scaling of the PE color scale.

Figure 2.6: Position errors (PE) of estimated positions per testtrack (columns) and method (rows) for a test transmitter at 0.5 m above ground in maisD. Ground-truth positions are displayed and colored by PE. Positions that could not be estimated are colored in gray. Note the log10 scaling of the PE color scale.

maisD 1m

Position errors (PE) of estimated positions per testtrack (columns) and method (rows) for a test transmitter at 1 m above ground in maisD. Ground-truth positions are displayed and colored by PE. Positions that could not be estimated are colored in gray. Note the log10 scaling of the PE color scale.

Figure 2.7: Position errors (PE) of estimated positions per testtrack (columns) and method (rows) for a test transmitter at 1 m above ground in maisD. Ground-truth positions are displayed and colored by PE. Positions that could not be estimated are colored in gray. Note the log10 scaling of the PE color scale.

maisD 1.5m

Position errors (PE) of estimated positions per testtrack (columns) and method (rows) for a test transmitter at 1.5 m above ground in maisD. Ground-truth positions are displayed and colored by PE. Positions that could not be estimated are colored in gray. Note the log10 scaling of the PE color scale.

Figure 2.8: Position errors (PE) of estimated positions per testtrack (columns) and method (rows) for a test transmitter at 1.5 m above ground in maisD. Ground-truth positions are displayed and colored by PE. Positions that could not be estimated are colored in gray. Note the log10 scaling of the PE color scale.

maisD all meth.

Overview whether positions in mais D could be estimated by all methods ('yes', orange) or not ('no', grey) per testtrack (columns) and test transmitter (rows, the label depicts the transmitter's height above ground in m).

Figure 2.9: Overview whether positions in mais D could be estimated by all methods (‘yes’, orange) or not (‘no’, grey) per testtrack (columns) and test transmitter (rows, the label depicts the transmitter’s height above ground in m).

2.2 Correlations between predictors

direct ab - maisC

Correlations between predictors (Ac = number of receiving antennas, Sc = number of receiving stations, cover = station cover (proxy for detection probability), maxSig = maximum received signal (dB), Weight = summed up normalized signals), including correlation coefficients (upper panels) and histograms (diagonal panels) for directional antenna beams in maisC. For comparison, the log10-scaled response variable PE is given as well (PE_log10).

Figure 2.10: Correlations between predictors (Ac = number of receiving antennas, Sc = number of receiving stations, cover = station cover (proxy for detection probability), maxSig = maximum received signal (dB), Weight = summed up normalized signals), including correlation coefficients (upper panels) and histograms (diagonal panels) for directional antenna beams in maisC. For comparison, the log10-scaled response variable PE is given as well (PE_log10).

direct ab - maisD

Correlations between predictors (Ac = number of receiving antennas, Sc = number of receiving stations, cover = station cover (proxy for detection probability), maxSig = maximum received signal (dB), Weight = summed up normalized signals), including correlation coefficients (upper panels) and histograms (diagonal panels) for directional antenna beams in maisD. For comparison, the log10-scaled response variable PE is given as well (PE_log10).

Figure 2.11: Correlations between predictors (Ac = number of receiving antennas, Sc = number of receiving stations, cover = station cover (proxy for detection probability), maxSig = maximum received signal (dB), Weight = summed up normalized signals), including correlation coefficients (upper panels) and histograms (diagonal panels) for directional antenna beams in maisD. For comparison, the log10-scaled response variable PE is given as well (PE_log10).

direct an - maisC

Correlations between predictors (Sc = number of receiving stations, cover = station cover (proxy for detection probability), maxSig = maximum received signal (dB)), including correlation coefficients (upper panels) and histograms (diagonal panels) for directional angulation in maisC. For comparison, the log10-scaled response variable PE is given as well (PE_log10).

Figure 2.12: Correlations between predictors (Sc = number of receiving stations, cover = station cover (proxy for detection probability), maxSig = maximum received signal (dB)), including correlation coefficients (upper panels) and histograms (diagonal panels) for directional angulation in maisC. For comparison, the log10-scaled response variable PE is given as well (PE_log10).

direct an - maisD

Correlations between predictors (Sc = number of receiving stations, cover = station cover (proxy for detection probability), maxSig = maximum received signal (dB)), including correlation coefficients (upper panels) and histograms (diagonal panels) for directional angulation in maisD. For comparison, the log10-scaled response variable PE is given as well (PE_log10).

Figure 2.13: Correlations between predictors (Sc = number of receiving stations, cover = station cover (proxy for detection probability), maxSig = maximum received signal (dB)), including correlation coefficients (upper panels) and histograms (diagonal panels) for directional angulation in maisD. For comparison, the log10-scaled response variable PE is given as well (PE_log10).

omni ab - maisC

Correlations between predictors (Sc = number of receiving stations, cover = station cover (proxy for detection probability), maxSig = maximum received signal (dB), Weight = summed up normalized signals), including correlation coefficients (upper panels) and histograms (diagonal panels) for omnidirectional antenna beams in maisC. For comparison, the log10-scaled response variable PE is given as well (PE_log10).

Figure 2.14: Correlations between predictors (Sc = number of receiving stations, cover = station cover (proxy for detection probability), maxSig = maximum received signal (dB), Weight = summed up normalized signals), including correlation coefficients (upper panels) and histograms (diagonal panels) for omnidirectional antenna beams in maisC. For comparison, the log10-scaled response variable PE is given as well (PE_log10).

omni ml - maisC

Correlations between predictors (Sc = number of receiving stations, cover = station cover (proxy for detection probability), maxSig = maximum received signal (dB)), including correlation coefficients (upper panels) and histograms (diagonal panels) for omnidirectional multilateration in maisC. For comparison, the log10-scaled response variable PE is given as well (PE_log10).

Figure 2.15: Correlations between predictors (Sc = number of receiving stations, cover = station cover (proxy for detection probability), maxSig = maximum received signal (dB)), including correlation coefficients (upper panels) and histograms (diagonal panels) for omnidirectional multilateration in maisC. For comparison, the log10-scaled response variable PE is given as well (PE_log10).

2.3 Position error prediction

pPE model

Predicted mean PE (pPE, based on 4000 repetitions) from the model, namely the median pPE (triangle) including 50 % (thick bar) and 95 % CI (thin bar), as well as the distribution (polygon, only center and bottom panels) for all present combinations of Ac-Sc per site and method. For predictions, maxSig, (weight) and cover were set to their respective raw data mean per Ac-Sc combination.

Figure 2.16: Predicted mean PE (pPE, based on 4000 repetitions) from the model, namely the median pPE (triangle) including 50 % (thick bar) and 95 % CI (thin bar), as well as the distribution (polygon, only center and bottom panels) for all present combinations of Ac-Sc per site and method. For predictions, maxSig, (weight) and cover were set to their respective raw data mean per Ac-Sc combination.

pPE Ctest, Dtest

Predicted PE (pPE, based on 4000 repetitions) for position estimates from test tracks *Ctest* and *Dtest*, namely the median pPE (triangle) including 50 % (thick bar) and 95 % CI (thin bar), as well as the distribution (polygon, only center and bottom panels). Positiones are summarized per present combinations of Ac-Sc per site and method. Note that this is based on raw data (not mean predictions per Ac-Sc combination), therefore showing more variance and larger CIs.

Figure 2.17: Predicted PE (pPE, based on 4000 repetitions) for position estimates from test tracks Ctest and Dtest, namely the median pPE (triangle) including 50 % (thick bar) and 95 % CI (thin bar), as well as the distribution (polygon, only center and bottom panels). Positiones are summarized per present combinations of Ac-Sc per site and method. Note that this is based on raw data (not mean predictions per Ac-Sc combination), therefore showing more variance and larger CIs.

diff PE - pPE Ctest, Dtest

Difference between raw PE and predicted PE (pPE, based on 4000 repetitions) for position estimates from test tracks *Ctest* and *Dtest*, namely the median difference (triangle) including 50 % (thick bar) and 95 % CI (thin bar), as well as the distribution (polygon, only center and bottom panels). Positiones are summarized per present combinations of Ac-Sc per site and method. Note that this is based on raw data (not mean predictions per Ac-Sc combination), therefore showing more variance and larger CIs.

Figure 2.18: Difference between raw PE and predicted PE (pPE, based on 4000 repetitions) for position estimates from test tracks Ctest and Dtest, namely the median difference (triangle) including 50 % (thick bar) and 95 % CI (thin bar), as well as the distribution (polygon, only center and bottom panels). Positiones are summarized per present combinations of Ac-Sc per site and method. Note that this is based on raw data (not mean predictions per Ac-Sc combination), therefore showing more variance and larger CIs.

pPE model q50

Predicted median PE (pPE50, based on 4000 repetitions) from the model, namely the median pPE (triangle) including 50 % (thick bar) and 95 % CI (thin bar), as well as the distribution (polygon, only center and bottom panels) for all present combinations of Ac-Sc per site and method. For predictions, maxSig, (weight) and cover were set to their respective raw data mean per Ac-Sc combination. This could be done for any desired quantile (e.g. 65 % quantile instead of 50 % quantile).

Figure 2.19: Predicted median PE (pPE50, based on 4000 repetitions) from the model, namely the median pPE (triangle) including 50 % (thick bar) and 95 % CI (thin bar), as well as the distribution (polygon, only center and bottom panels) for all present combinations of Ac-Sc per site and method. For predictions, maxSig, (weight) and cover were set to their respective raw data mean per Ac-Sc combination. This could be done for any desired quantile (e.g. 65 % quantile instead of 50 % quantile).

pPE Ctest, Dtest q50

Predicted median PE (pPE50, based on 4000 repetitions) for position estimates from test tracks *Ctest* and *Dtest*, namely the median pPE (triangle) including 50 % (thick bar) and 95 % CI (thin bar), as well as the distribution (polygon, only center and bottom panels). Positiones are summarized per present combinations of Ac-Sc per site and method. Note that this is based on raw data (not mean predictions per Ac-Sc combination), therefore showing more variance and larger CIs.

Figure 2.20: Predicted median PE (pPE50, based on 4000 repetitions) for position estimates from test tracks Ctest and Dtest, namely the median pPE (triangle) including 50 % (thick bar) and 95 % CI (thin bar), as well as the distribution (polygon, only center and bottom panels). Positiones are summarized per present combinations of Ac-Sc per site and method. Note that this is based on raw data (not mean predictions per Ac-Sc combination), therefore showing more variance and larger CIs.