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

Alemam, Mustafa K.; Yong, Bin; Mohammed, Abubakar S.

doi:10.2478/arsa-2022-0002

Figures & Tables

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

Algorithm steps of the ANN stage (binary numbers)

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

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 number of neurons

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

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
Broadcast	Sat. clocks	~5 ns RMS ~2.5 ns SD	Real time	--	Daily
Ultra-rapid (predicted half)	Orbits	~5 cm	Real time	Four times/day	15 min
Ultra-rapid (predicted half)	Sat. clocks	~3 ns RMS ~1.5 ns SD	Real time	Four times/day	15 min
Ultra-rapid (observed half)	Orbits	~3 cm	3–9 h	Four times/day	15 min
Ultra-rapid (observed half)	Sat. clocks	~150 ps RMS ~50 ps SD	3–9 h	Four times/day	15 min
Rapid	Orbits	~2.5 cm	17–41 h	One time/day	15 min
Rapid	Sat. and stn clocks	~75 ps RMS ~25 ps SD	17–41 h	One time/day	5 min
Final	Orbits	~2.5 cm	12–18 days	One time/week	15 min
Final	Sat. and stn clocks	~75 ps RMS ~20 ps SD	12–18 days	One time/week	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
		σ_x (m)	MSE (m)	NOF	σ_Y (m)	MSE (m)	NOF	σ_Z(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
		σ_x (m)		σ_y (m)		σ_z (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
		σ_x (m)	MSE (m)	NOF	σ_Y (m)	MSE (m)	NOF	σ_Z (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
		σ_x (m)		σ_y (m)		σ_z (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
		σ_x (m)	MSE (m)	NOF	σ_Y (m)	MSE (m)	NOF	σ_Z (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
		σ_x (m)		σ_y (m)		σ_z (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
		σ_x (m)	MSE (m)	NOF	σ_Y (m)	MSE (m)	NOF	σ_Z (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
		σ_x (m)		σ_y (m)		σ_x (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
Station	σ_x(m)	σ_Y(m)	σ_Z(m)	σ_P(m)	Elapsed time (s)	σ_x(m)	σ_Y(m)	σ_Z(m)	σ_P(m)	Elapsed time(s)	σ_x(m)	σ_Y(m)	σ_Z(m)	σ_P(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
	σ_m(X)(m)	σ_m(Y)(m)	σ_m(Z)(m)	σ_m(P)(m)	σ_m(X)(m)	σ_m(Y)(m)	σ_m(Z)(m)	σ_m(P)(m)	σ_m(X)(m)	σ_m(Y)(m)	σ_m(Z)(m)	σ_m(P)(m)	σ_m(X)(m)	σ_m(Y)(m)	σ_m(Z)(m)	σ_m(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
Station	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
Station	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
Station	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
			°	′	″	°	′	″	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
		σ_x (m)	MSE (m)	NOF	σ_Y (m)	MSE (m)	NOF	σ_Z (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
		σ_x (m)		σ_y (m)		σ_z (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

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

Figures & Tables

Figure 1.

Figure 1.

Figure 3.

Figure 4.

Figure 5.

Figure 6.

Figure 7.

Figure 8.

Figure 9.

Figure 10.

Characteristics of IGS satellite ephemerides and clock products (2019)

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

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

The tested parameters in the ANN algorithm

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

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

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

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

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

Data sources

The parameters involved in the ANN algorithm

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

Paradigm

My account