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Determining the area corrections affecting the map areas in GIS applications

Faruk Yildirim
Faruk Yildirim
 y 
Fatih Kadi
Fatih Kadi
   | 11 dic 2021
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Article Category: Original Article
Publicado en línea: 11 dic 2021
Páginas: 9 - 17
Recibido: 08 jun 2021
Aceptado: 02 nov 2021
DOI: https://doi.org/10.2478/rgg-2021-0003
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© 2021 Faruk Yildirim et al., published by Sciendo
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Figure 1

Auxiliary sphere
Auxiliary sphere

Figure 2

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

Figure 3

Comparison of the area distortion for equal area projections
Comparison of the area distortion for equal area projections

Figure 4

Comparison of UTM and Karney method
Comparison of UTM and Karney method

Figure 5

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

Figure 6

The effect of ellipsoidal height on area (graphical display)
The effect of ellipsoidal height on area (graphical display)

Figure 7

The effect of slope factor on the parcel area
The effect of slope factor on the parcel area

Examination of methods

Methods Areas Fmethod [m2] Difference (Fmethod − Fe) [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
mp = mx = my [m] 0.2 0.4 1 2 5 10 20 50
∑S2[ha] Edge [m] Area Root mean square error of the MF 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°) fconformal 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 fequal area - 0.0001%–0.0020%
Scale of the map fscale 1:1000–1:250,000 - 0.0077%–1.9292%
Area (Karney method) in ellipsoid F ≈ fequal area
Ellipsoid height Fh 100–3000 m + 0.0031%–0.0942%
Slope of terrain Fs 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°] (fconformal) (F – f) (4) formula F (Karney) AEAC (fequal_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
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