Determining the area corrections affecting the map areas in GIS applications

Yildirim, Faruk; Kadi, Fatih

doi:10.2478/rgg-2021-0003

Figures & Tables

Satellite imagery of forest parcels (left) and geographical boundaries of the Kütahya Regional Directorate of forestry and forestry operation directorates in ITRF96 datum (right)

Comparison of the area distortion for equal area projections

The effect of digitisation technique on the area in the scale map

The effect of ellipsoidal height on area (graphical display)

The effect of slope factor on the parcel area

Examination of methods

Methods	Areas F_method [m²]	Difference (F_method − F_e) [m²]	Δλ = 0°	Δ φ = 0°	λ = (0°; 90°)	φ = (0°; 90°)	t
Danielsen	63,758,202,714,811.400	0.000	√	√	√	√	0.000
Freire	-	-	√	√	-	-	7.906
Gillissen	-	-	√	√	-	-	0.125
Karney	63,758,202,714,811.400	0.000	√	√	√	√	0.031
Kimerling	63,900,265,931,354.400	142,063,216,543.047	√	√	-	-	0.000
Sjoberg	63,758,202,714,811.400	0.000	√	√	√	√	0.281
Tseng	-	-	-	√	-	-	0.000

The effect of scale-related accuracy of measurement on the area (for regular polygon parcels)

Scale of the map			1000	2000	5000	10,000	25,000	50,000	100,000	250,000
m_p = m_x = m_y [m]			0.2	0.4	1	2	5	10	20	50

∑S²[ha]	Edge [m]	Area	Root mean square error of the M_F area depending on the scale [m²]
0.08	10	100 m²	2	4	10	20	50	100	200	500
8	100	1 ha	20	40	100	200	500	1000	2000	5000
32	200	4 ha	40	80	200	400	1000	2000	4000	10,000
200	500	25 ha	100	200	500	1000	2500	5000	10,000	25,000
450	750	55 ha	150	300	750	1500	3750	7500	15,000	37,500
800	1000	100 ha	200	400	1000	2000	5000	10,000	20,000	50,000
3200	2000	400 ha	400	800	2000	4000	10,000	20,000	40,000	100,000
20,000	5000	2500 ha	1000	2000	5000	10,000	25,000	50,000	100,000	250,000
80,000	10,000	100 km²	2000	4000	10,000	20,000	50,000	100,000	200,000	500,000
180,000	15,000	225 km²	3000	6000	15,000	30,000	75,000	150,000	300,000	750,000
320,000	20,000	400 km²	4000	8000	20,000	40,000	100,000	200,000	400,000	1,000,000
720,000	30,000	900 km²	6000	12,000	30,000	60,000	150,000	300,000	600,000	1,500,000

The effect of scale-related accuracy of measurement on the area (forest parcels) in AEAC projection

Scale of the map		1000	2000	5000	10,000	25,000	50,000	100,000	250,000
mp = mx = my [m]		0.2	0.4	1	2	5	10	20	50

ΣS²[m²]	Parcels	Root mean square error of the area depending on the scale [m²]
33,022,713.72	P1	812.683	1625.367	4063.417	8126.834	20,317.084	40,634.169	81,268.338	203,170.845
43,907,616.54	P2	937.098	1874.196	4685.489	9370.978	23,427.446	46,854.891	93,709.782	234,274.456
25,256,098.93	P3	710.719	1421.439	3553.597	7107.193	17,767.983	35,535.967	71,071.934	177,679.835
49,422,346.14	P4	994.207	1988.413	4971.033	9942.067	24,855.167	49,710.334	99,420.668	248,551.670
30,571,936.51	P5	781.945	1563.891	3909.727	7819.455	19,548.637	39,097.274	78,194.548	195,486.369
37,908,657.94	P6	870.731	1741.463	4353.657	8707.314	21,768.285	43,536.570	87,073.139	217,682.848
42,306,252.37	P7	919.851	1839.701	4599.253	9198.506	22,996.264	45,992.528	91,985.056	229,962.639
50,596,221.46	P8	1005.945	2011.889	5029.723	10,059.445	25,148.614	50,297.227	100,594.455	251,486.136
15,412,135.51	P9	555.196	1110.392	2775.981	5551.961	13,879.903	27,759.805	55,519.610	138,799.025
34,040,110.57	P10	825.107	1650.215	4125.537	8251.074	20,627.685	41,255.370	82,510.739	206,276.848
36,106,507.87	P11	849.782	1699.565	4248.912	8497.824	21,244.560	42,489.121	84,978.242	212,445.604
10,698,760	P12	462.575	925.149	2312.873	4625.745	11,564.363	23,128.727	46,257.453	115,643.634
16,397,451.31	P13	572.668	1145.337	2863.342	5726.683	14,316.708	28,633.417	57,266.834	143,167.085
25,863,964.9	P14	719.221	1438.443	3596.107	7192.213	17,980.533	35,961.066	71,922.131	179,805.328
47,247,724.79	P15	972.088	1944.175	4860.438	9720.877	24,302.192	48,604.385	97,208.770	243,021.925

Rate of total area corrections

Area corrections	Formulas	Test range	−/+	Rate of correction (%)
Area in the map (UTM, 3°)	f_conformal	100 m2−900 km2 for regular polygon parcels100 m2−900 km2 for forest parcels {{100\,{{\rm{m}}^2} - 900\,{\rm{k}}{{\rm{m}}^2}\,{\rm{for}}\,{\rm{regular}}\,{\rm{polygon}}\,{\rm{parcels}}} \over {100\,{{\rm{m}}^2} - 900\,{\rm{k}}{{\rm{m}}^2}\,{\rm{for}}\,{\rm{forest}}\,{\rm{parcels}}}}
Area in AEAC projection	f_{equal area}		-	0.0001%–0.0020%
Scale of the map	f_scale	1:1000–1:250,000	-	0.0077%–1.9292%
Area (Karney method) in ellipsoid	F ≈ f_{equal area}
Ellipsoid height	F_h	100–3000 m	+	0.0031%–0.0942%
Slope of terrain	F_s	5%–70%	+	0.1249%–22.0656%

The effect of ellipsoidal height on area (for forest parcels)

Forest parcels	Area [m²]	Average ellipsoidal height of the parcels [m]

		100	250	500	1000	1500	2000	2500	3000
		Area corrections [m²]
P1	28,570,391	897.030	2242.574	4485.148	8970.295	13,455.443	17,940.591	22,425.738	26,910.886
P2	24,240,429	761.081	1902.702	3805.405	7610.810	11,416.214	15,221.619	19,027.024	22,832.429
P3	19,454,332	610.811	1527.028	3054.055	6108.110	9162.166	12,216.221	15,270.276	18,324.331
P4	18,201,894	571.488	1428.720	2857.440	5714.880	8572.320	11,429.761	14,287.201	17,144.641
P5	16,730,816	525.300	1313.251	2626.502	5253.003	7879.505	10,506.007	13,132.508	15,759.010
P6	15,951,082	500.819	1252.047	2504.094	5008.189	7512.283	10,016.378	12,520.472	15,024.567
P7	14,808,969	464.960	1162.399	2324.799	4649.598	6974.397	9299.196	11,623.995	13,948.794
P8	12,920,318	405.661	1014.154	2028.307	4056.615	6084.922	8113.229	10,141.537	12,169.844
P9	11,066,953	347.471	868.678	1737.355	3474.710	5212.066	6949.421	8686.776	10,424.131
P10	10,550,252	331.248	828.120	1656.240	3312.481	4968.721	6624.962	8281.202	9937.443
P11	9,498,533.6	298.227	745.568	1491.136	2982.271	4473.407	5964.542	7455.678	8946.813
P12	7,246,670.6	227.525	568.812	1137.625	2275.250	3412.875	4550.500	5688.124	6825.749
P13	6,575,802.9	206.462	516.154	1032.308	2064.616	3096.924	4129.233	5161.541	6193.849
P14	5,774,547.5	181.304	453.261	906.522	1813.045	2719.567	3626.090	4532.612	5439.134
P15	3,673,390.7	115.334	288.335	576.670	1153.341	1730.011	2306.682	2883.352	3460.023

Projections and parameters used in the study

Map projections	L₀ Central longitude	B₀ Central latitude	B₁ Standard parallel	B₂ Standard parallel
UTM (3° of longitude in width)	30°E	0° (Equator)	-	-
AEAC	30°E	30.05°N	34.45°N	34.85°N
EAC	30°E	30.05°N	-	-
BEAC	30°E	0° (Equator)	-	-
BEA	30°E	0° (Equator)	-	-
LAEA	30°E	30.05°N	-	-
SEA	30°E	0° (Equator)	-	-

The effect of ellipsoidal height on area (for regular polygon parcels)

Area values	Average ellipsoidal height of the parcel [m]

	100	250	500	1000	1500	2000	2500	3000

	Area corrections [m²]
100 m²	0.003	0.008	0.016	0.031	0.047	0.063	0.078	0.094
1 ha	0.314	0.785	1.570	3.140	4.710	6.279	7.849	9.419
4 ha	1.256	3.140	6.279	12.559	18.838	25.118	31.397	37.677
52 ha	7.849	19.623	39.246	78.493	117.739	156.986	196.232	235.479
55 ha	17.268	43.171	86.342	172.684	259.027	345.369	431.711	518.053
100 ha	31.397	78.493	156.986	313.972	470.958	627.943	784.929	941.915
400 ha	125.589	313.972	627.943	1255.887	1883.830	2511.774	3139.717	3767.661
2500 ha	784.929	1962.323	3924.647	7849.294	11,773.940	15,698.587	19,623.234	23,547.881
100 km²	3139.717	7849.294	15,698.587	31,397.174	47,095.761	62,794.349	78,492.936	94,191.523
225 km²	7064.364	17,660.911	35,321.821	70,643.642	105,965.463	141,287.284	176,609.105	211,930.926
400 km²	12,558.870	31,397.174	62,794.349	125,588.697	188,383.046	251,177.394	313,971.743	376,766.091
900 km²	28,257.457	70,643.642	141,287.284	282,574.568	423,861.852	565,149.137	706,436.421	847,723.705

Comparison of UTM, area reduction (F – f), Karney method and AEAC projection

Parcel number	Area calculation [m²]				Area distortion [m²]

	Conform map projection		Ellipsoid geographical coordinates	Equal area projection	Area reduction formula	Equal area projection	Conform map projection

	UTM [3°] (f_conformal)	(F – f) (4) formula	F (Karney)	AEAC (f_{equal_area})	Karney– [f–(F – f)]	Karney–AEAC	Karney–UTM
P1	28,570,421.774	118.994	28,570,400.287	28,570,390.847	140.481	9.440	21.487
P2	24,240,491.726	100.960	24,240,477.498	24,240,428.709	86.732	48.789	14.228
P3	19,454,360.215	81.026	19,454,337.971	19,454,331.754	58.782	6.217	22.244
P4	18,201,997.722	75.810	18,201,879.790	18,201,893.606	42.122	13.816	117.932
P5	16,730,817.085	69.683	16,730,812.149	16,730,815.813	64.747	3.664	4.936
P6	15,951,105.646	66.435	15,951,099.137	15,951,081.911	59.926	17.226	6.509
P7	14,809,228.918	61.679	14,808,973.667	14,808,969.185	193.572	4.482	255.251
P8	12,920,294.789	53.812	12,920,293.910	12,920,317.541	52.933	23.631	0.879
P9	11,067,018.225	46.093	11,066,945.786	11,066,952.772	26.346	6.986	72.439
P10	10,550,351.283	43.941	10,550,250.549	10,550,251.662	56.793	1.113	100.734
P11	9,498,509.712	39.561	9,498,505.310	9,498,533.562	35.159	28.252	4.402
P12	7,246,694.410	30.182	7,246,670.634	7,246,670.599	6.406	0.035	23.776
P13	6,575,866.610	27.388	6,575,812.176	6,575,802.912	27.046	9.264	54.434
P14	5,774,651.499	24.051	5,774,533.261	5,774,547.548	94.187	14.287	118.238
P15	3,673,397.840	15.299	3,673,391.983	3,673,390.666	9.442	1.317	5.857

Area distortions of equal area projections

Parcel number	Area calculation [m²]

	Conformal map projection	Ellipsoid geographical coordinates	Area distortion (Karney-equal area map projections)

	UTM (3°)	(Karney method)	AEAC	EAC	BEAC	BEA	LAEA	SEA
P1	28,570,421.774	28,570,400.287	9.440	90.265	90.247	1.854	5.603	6.880
P2	24,240,491.726	24,240,477.498	48.789	495.779	496.021	37.717	42.016	177.158
P3	19,454,360.215	19,454,337.971	6.217	64.581	64.516	7.223	6.395	32.869
P4	18,201,997.722	18,201,879.790	13.816	135.762	135.712	25.911	18.612	114.817
P5	16,730,817.085	16,730,812.149	3.664	32.262	32.260	4.938	4.220	20.196
P6	15,951,105.646	15,951,099.137	17.226	156.574	156.824	15.139	15.271	66.613
P7	14,809,228.918	14,808,973.667	4.482	44.089	44.112	5.986	5.158	27.267
P8	12,920,294.789	12,920,293.910	23.631	249.194	248.966	37.360	29.865	174.311
P9	11,067,018.225	11,066,945.786	6.986	58.247	58.261	4.470	5.488	16.470
P10	10,550,351.283	10,550,250.549	1.113	18.177	18.236	7.909	4.080	39.129
P11	9,498,509.712	9,498,505.310	28.252	294.518	294.698	28.161	27.681	133.011
P12	7,246,694.410	7,246,670.634	0.035	5.766	5.679	0.175	0.103	3.432
P13	6,575,866.610	6,575,812.176	9.264	93.417	93.364	6.931	7.786	32.686
P14	5,774,651.499	5,774,533.261	14.287	136.143	136.038	16.496	14.751	72.715
P15	3,673,397.840	3,673,391.983	1.317	13.831	13.764	0.586	0.922	3.230
Σ			188.519	1888.605	1888.698	200.856	187.951	920.784

Determining the area corrections affecting the map areas in GIS applications

Figures & Tables

Figure 1

Figure 2

Figure 3

Figure 4

Figure 5

Figure 6

Figure 7

Examination of methods

The effect of scale-related accuracy of measurement on the area (for regular polygon parcels)

The effect of scale-related accuracy of measurement on the area (forest parcels) in AEAC projection

Rate of total area corrections

The effect of ellipsoidal height on area (for forest parcels)

Projections and parameters used in the study

The effect of ellipsoidal height on area (for regular polygon parcels)

Comparison of UTM, area reduction (F – f), Karney method and AEAC projection

Area distortions of equal area projections

Paradigm

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