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Integration of Artificial Neural Network and the Optimal GNSS Satellites’ Configuration for Improving GNSS Positioning Techniques (A Case Study in Egypt)

Mustafa K. Alemam
Mustafa K. Alemam
, 
Bin Yong
Bin Yong
 y 
Abubakar S. Mohammed
Abubakar S. Mohammed
   | 22 abr 2022
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Publicado en línea: 22 abr 2022
Páginas: 18 - 46
Recibido: 07 jul 2021
Aceptado: 02 feb 2022
DOI: https://doi.org/10.2478/arsa-2022-0002
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© 2022 Mustafa K. Alemam et al., published by Sciendo
This work is licensed under the Creative Commons Attribution 4.0 International License.

Figure 1.

ANN architecture
ANN architecture

Figure 1.

Study area and IGS stations
Study area and IGS stations

Figure 3.

Methodology flowchart
Methodology flowchart

Figure 4.

Algorithm steps of the classification stage Xi, Yi, Zi: coordinates values at each NOS, PDOP and ST n: Number of the processed observations
Algorithm steps of the classification stage Xi, Yi, Zi: coordinates values at each NOS, PDOP and ST n: Number of the processed observations

Figure 5.

Algorithm steps of the ANN stage (binary numbers)
Algorithm steps of the ANN stage (binary numbers)

Figure 6.

A scheme displays the algorithm steps for producing reinitializing operation outputs (e.g., group 1)
A scheme displays the algorithm steps for producing reinitializing operation outputs (e.g., group 1)

Figure 7.

The prediction error represented by the 3D position error versus the number of hidden layers
The prediction error represented by the 3D position error versus the number of hidden layers

Figure 8.

The prediction error represented by the 3D position error versus number of neurons
The prediction error represented by the 3D position error versus number of neurons

Figure 9.

RMSE of X, Y, and Z directions and 3D position error according to the different groups of the number of initializations
RMSE of X, Y, and Z directions and 3D position error according to the different groups of the number of initializations

Figure 10.

The ANN designed from the results; its type is cascade forward net
The ANN designed from the results; its type is cascade forward net

Characteristics of IGS satellite ephemerides and clock products (2019)

Type Accuracy Latency Updates Sample interval
GPS satellite ephemerides/satellite and station clocks
Broadcast Orbits ~100 cm Real time -- Daily
Sat. clocks

~5 ns RMS

~2.5 ns SD
Ultra-rapid (predicted half) Orbits ~5 cm Real time Four times/day 15 min
Sat. clocks

~3 ns RMS

~1.5 ns SD
Ultra-rapid (observed half) Orbits ~3 cm 3–9 h Four times/day 15 min
Sat. clocks

~150 ps RMS

~50 ps SD
Rapid Orbits ~2.5 cm 17–41 h One time/day 15 min
Sat. and stn clocks

~75 ps RMS

~25 ps SD

 5 min
Final Orbits ~2.5 cm 12–18 days One time/week 15 min
Sat. and stn clocks

~75 ps RMS

~20 ps SD

Sat.: 30 s

Stn.: 5 min
GLONASS satellite ephemerides
Final Orbits ~3 cm 12–18 days Every Thursday 15 min
where: ns = nanosecond, ps = picosecond, UTC =Universal Coordinated Time, RMS = Root Mean Square errors, SD = standard deviation

The effect of different transfer functions’ constellation on ANN performance in the case of binary and decimal numbers (part 1/3)

Transfer functions (hidden–output) layer Binary numbers
X Y Z
σx (m) MSE (m) NOF σY (m) MSE (m) NOF σZ(m) MSE (m) NOF
a) Baltim station Tansig–Purelin 0.112 0.008 1 0.404 0.020 6 0.254 0.025 4
Tansig–Logsig NaN NaN 100 NaN NaN 100 NaN NaN 100
Tansig–Tansig 0.103 0.005 2 0.319 0.007 7 0.050 0.003 5
Logsig–Purelin 0.051 0.002 0 0.189 0.005 2 0.057 0.002 3
Logsig–Logsig NaN NaN 100 NaN NaN 100 NaN NaN 100
Logsig–Tansig 0.171 0.038 3 0.495 0.085 13 0.282 0.040 19
Purelin–Purelin 1.048 0.094 7 0.955 0.087 2 0.489 0.054 6
Purelin–Logsig NaN NaN 100 NaN NaN 100 NaN NaN 100
Purelin–Tansig 0.537 0.041 7 0.805 0.049 6 0.669 0.085 7
Transfer functions (hidden–output) layer Decimal numbers
X, Y and Z
σx (m) σy (m) σz (m) MSE (m) NOF
Purelin–Purelin 0.113 0.274 0.158 2 × 10-6 0
The other constellations of transfer functions NaN NaN NaN NaN 100
b) Suez station Transfer functions (hidden–output) layer Binary numbers
X Y Z
σx (m) MSE (m) NOF σY (m) MSE (m) NOF σZ (m) MSE (m) NOF
Tansig–Purelin 0.165 0.004 3 0.227 0.013 3 0.186 0.007 2
Tansig–Logsig NaN NaN 100 NaN NaN 100 NaN NaN 100
Tansig–Tansig 0.183 0.005 3 0.185 0.009 5 0.196 0.017 7
Logsig–Purelin 0.095 0.002 1 0.133 0.005 0 0.117 0.001 1
Logsig–Logsig NaN NaN 100 NaN NaN 100 NaN NaN 100
Logsig–Tansig 0.245 0.021 7 0.401 0.005 6 0.293 0.004 6
Purelin–Purelin 0.498 0.081 6 0.554 0.071 3 0.617 0.032 3
Purelin–Logsig NaN NaN 100 NaN NaN 100 NaN NaN 100
Purelin–Tansig 0.632 0.035 6 0.488 0.045 4 0.650 0.035 10
Transfer functions (hidden–output) layer Decimal numbers
X, Y and Z
σx (m) σy (m) σz (m) MSE (m) NOF
Purelin–Purelin 0.171 0.206 0.256 3 × 10-8 1
The other constellations of transfer functions NaN NaN NaN NaN 100

The effect of different transfer functions’ constellation on ANN performance in the case of binary and decimal numbers (part 2/3)

Transfer functions (hidden–output) layer Binary numbers
X Y Z
σx (m) MSE (m) NOF σY (m) MSE (m) NOF σZ (m) MSE (m) NOF
c) Helwan station Tansig–Purelin 0.042 0.007 3 0.156 0.008 3 0.123 0.009 3
Tansig–Logsig NaN NaN 100 NaN NaN 100 NaN NaN 100
Tansig–Tansig 0.065 0.005 5 0.192 0.012 5 0.099 0.004 5
Logsig–Purelin 0.048 0.004 2 0.129 0.002 3 0.086 0.010 2
Logsig–Logsig NaN NaN 100 NaN NaN 100 NaN NaN 100
Logsig–Tansig 0.275 0.013 2 0.352 0.007 6 0.356 0.032 6
Purelin–Purelin 0.520 0.027 4 0.581 0.035 4 0.492 0.009 7
Purelin–Logsig NaN NaN 100 NaN NaN 100 NaN NaN 100
Purelin–Tansig 0.452 0.008 5 0.611 0.041 5 0.470 0.036 5
Transfer functions (hidden–output) layer Decimal numbers
X, Y and Z
σx (m) σy (m) σz (m) MSE (m) NOF
Purelin–Purelin 0.157 0.305 0.276 5 × 10-7 2
The other constellations of transfer functions NaN NaN NaN NaN 100
d) Cairo station Transfer functions (hidden–output) layer Binary numbers
X Y Z
σx (m) MSE (m) NOF σY (m) MSE (m) NOF σZ (m) MSE (m) NOF
Tansig–Purelin 0.143 0.007 4 0.031 0.010 3 0.133 0.008 2
Tansig–Logsig NaN NaN 100 NaN NaN 100 NaN NaN 100
Tansig–Tansig 0.127 0.001 3 0.108 0.008 4 0.186 0.015 3
Logsig–Purelin 0.073 0.004 2 0.014 0.006 2 0.052 0.003 2
Logsig–Logsig NaN NaN 100 NaN NaN 100 NaN NaN 100
Logsig–Tansig 0.272 0.009 5 0.358 0.031 5 0.296 0.002 5
Purelin–Purelin 0.382 0.012 4 0.399 0.023 3 0.488 0.047 7
Purelin–Logsig NaN NaN 100 NaN NaN 100 NaN NaN 100
Purelin–Tansig 0.484 0.023 6 0.511 0.045 5 0.388 0.071 4
Transfer functions (hidden–output) layer Decimal numbers
X, Y and Z
σx (m) σy (m) σx (m) MSE (m) NOF
Purelin–Purelin 0.273 0.242 0.338 0.006 0
The other constellations of transfer functions NaN NaN NaN NaN 100

The tested parameters in the ANN algorithm

Test no. Transfer function ANN type NHL NON NOI
1 Transfer function X Fit net 1 10 100
2 ANN type The optimal at test 1 X 1 10 100
3 NHL The optimal at test 1 The optimal at test 2 X 10 100
4 NON The optimal at test 1 The optimal at test 2 The optimal at test 3 X 100
5 NOI The optimal at test 1 The optimal at test 2 The optimal at test 3 The optimal at test 4 X

The SD of X, Y, and Z coordinates, 3D position error, and elapsed time for three different types of ANN

Station Pattern net Fit net Cascade forward net
σx(m) σY(m) σZ(m) σP(m) Elapsed time (s) σx(m) σY(m) σZ(m) σP(m) Elapsed time(s) σx(m) σY(m) σZ(m) σP(m) Elapsed time(s)
Baltim 0.090 0.263 0.071 0.287 44.4 0.056 0.219 0.030 0.228 38.8 0.043 0.188 0.021 0.194 27.9
Suez 0.078 0.118 0.157 0.211 47.3 0.072 0.101 0.106 0.163 36.2 0.077 0.099 0.103 0.162 24.4
Helwan 0.127 0.193 0.089 0.248 45.8 0.089 0.152 0.076 0.192 37.4 0.084 0.146 0.079 0.186 27.2
Cairo 0.161 0.068 0.118 0.211 46.9 0.115 0.044 0.066 0.140 35.1 0.084 0.029 0.048 0.101 28.9
Assiut 0.227 0.224 0.448 0.550 49.7 0.179 0.198 0.431 0.507 39.5 0.147 0.179 0.425 0.484 26.8

The averages of SDs in the directions of the coordinate axes, 3D position error, and elapsed time for the four groups of initialization numbers

Station Group 1 Group 2 Group 3 Group 4
Nt values Elapsed time (h) Nt values Elapsed time (h) Nt values Elapsed time (h) Nt values Elapsed time (h)
1–10 0.02 10–100 0.15 100–1000 1.5 1000–10,000 15.5
σm(X)(m) σm(Y)(m) σm(Z)(m) σm(P)(m) σm(X)(m) σm(Y)(m) σm(Z)(m) σm(P)(m) σm(X)(m) σm(Y)(m) σm(Z)(m) σm(P)(m) σm(X)(m) σm(Y)(m) σm(Z)(m) σm(P)(m)
Baltim 0.080 0.195 0.040 0.215 0.068 0.174 0.036 0.191 0.042 0.137 0.019 0.145 0.017 0.080 0.009 0.083
Suez 0.094 0.128 0.105 0.190 0.087 0.119 0.101 0.179 0.068 0.085 0.082 0.137 0.045 0.035 0.044 0.074
Helwan 0.067 0.177 0.086 0.208 0.058 0.142 0.067 0.168 0.052 0.107 0.049 0.129 0.034 0.065 0.030 0.080
Cairo 0.083 0.044 0.035 0.100 0.066 0.036 0.036 0.084 0.047 0.030 0.036 0.066 0.026 0.023 0.024 0.043
Assiut 0.184 0.216 0.572 0.639 0.164 0.196 0.453 0.520 0.134 0.172 0.350 0.413 0.103 0.125 0.249 0.297

The coordinates’ differences between the known points and the output data, and the position errors for the three main stages and IGS final orbits in the case of GNSS-dual frequency

Station Post-processing (broadcast ephemerides) Post-processing (final orbits) Classification algorithm ANN algorithm
dX(m) dY(m) dZ(m) Position error(m) dX(m) dY(m) dZ(m) Position error(m) dX(m) dY(m) dZ(m) Position error(m) dX(m) dY(m) dZ(m) Position error(m)
Baltim 0.050 0.020 0.003 0.054 0.001 0.002 0.004 0.005 0.034 0.011 0.009 0.037 0.011 0.007 0.008 0.015
Suez 0.006 0.003 0.006 0.009 0.003 0.007 0.003 0.008 0.003 0.002 0.004 0.005 0.002 0.003 0.004 0.005
Helwan 0.047 0.040 0.019 0.065 0.015 0.015 0.011 0.024 0.024 0.021 0.024 0.040 0.017 0.016 0.021 0.031
Cairo 0.022 0.014 0.013 0.029 0.015 0.010 0.019 0.026 0.017 0.012 0.015 0.026 0.011 0.007 0.012 0.018
Assiut 0.157 0.051 0.322 0.362 0.044 0.083 0.110 0.145 0.103 0.062 0.220 0.251 0.051 0.021 0.153 0.163

Precision improvement due to applying classification and ANN algorithms in the two cases of observations

Percentage of improvement (%)
Station Single-frequency observations Dual-frequency observations
Classification algorithm ANN algorithm Classification algorithm ANN algorithm IGS final orbits
Baltim 28 79 31 72 91
Suez 37 70 44 44 11
Helwan 39 66 38 52 63
Cairo 16 50 10 38 10
Assiut 32 62 31 55 60

The differences in coordinates between the known points and the output data, and the position errors for the three main stages in the case of GNSS-single frequency

Station Post-processing (broadcast ephemerides) Classification algorithm ANN algorithm
dX(m) dY(m) dZ(m) Position error(m) dX(m) dY(m) dZ(m) Position error(m) dX(m) dY(m) dZ(m) Position error(m)
Baltim 0.118 0.309 0.046 0.334 0.090 0.220 0.047 0.242 0.031 0.06 0.021 0.071
Suez 0.145 0.262 0.134 0.328 0.104 0.141 0.112 0.208 0.052 0.054 0.063 0.098
Helwan 0.123 0.277 0.204 0.365 0.075 0.190 0.092 0.224 0.078 0.081 0.051 0.123
Cairo 0.093 0.045 0.054 0.117 0.081 0.046 0.031 0.098 0.042 0.027 0.030 0.058
Assiut 0.286 0.330 0.992 1.084 0.190 0.241 0.675 0.741 0.101 0.132 0.381 0.416

Data sources

Station Station type GNSS instrument Reference coordinates
Longitude Latitude Ellipsoidal height, m Source
° ′ ″ ° ′ ″
DRAG IGS Leica GRX1200 35 23 31.46180 31 35 35.5288 31.834 SOPAC
RAMO IGS Leica RS500 34 45 47.31050 30 35 51.38602 886.829 SOPAC
Baltim Test Trimble R8 31 04 49.15000 31 35 45.42070 31.163 CSRS
Suez Test Trimble R8 32 36 22.45620 30 07 09.53080 53.827 CSRS
Helwan Test Trimble R8 31 20 37.30370 29 51 33.72150 135.055 CSRS
Cairo Test Leica GR10 31 14 16.45330 30 02 43.33490 68.268 CSRS
Assiut Test Ashtech Z-Xtreme 31 10 19.90010 27 11 12.12040 91.420 CSRS

The parameters involved in the ANN algorithm

Epoch Goal Max_fail Min_fail Mu Learning rate
1000 0 6 1e-7 0.001 0.01

The effect of different transfer functions’ constellation on ANN performance in the case of binary and decimal numbers (part 3/3)

Transfer functions (hidden–output) layer Binary numbers
X Y Z
σx (m) MSE (m) NOF σY (m) MSE (m) NOF σZ (m) MSE (m) NOF
e) Assiut station Tansig–Purelin 0.219 0.008 3 0.327 0.010 1 0.530 0.016 6
Tansig–Logsig NaN NaN 100 NaN NaN 100 NaN NaN 100
Tansig–Tansig 0.177 0.004 4 0.294 0.013 2 0.549 0.017 5
Logsig–Purelin 0.149 0.006 2 0.225 0.007 0 0.481 0.015 5
Logsig–Logsig NaN NaN 100 NaN NaN 100 NaN NaN 100
Logsig–Tansig 0.314 0.009 5 0.470 0.023 4 0.640 0.022 7
Purelin–Purelin 0.433 0.014 4 0.402 0.036 2 0.691 0.035 14
Purelin–Logsig NaN NaN 100 NaN NaN 100 NaN NaN 100
Purelin–Tansig 0.503 0.071 4 0.630 0.610 0 0.747 0.051 15
Transfer functions (hidden–output) layer Decimal numbers
X, Y and Z
σx (m) σy (m) σz (m) MSE (m) NOF
Purelin–Purelin 0.259 0.301 0.606 0.003 7
The other constellations of transfer functions NaN NaN NaN NaN 100
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