Fractal surfaces of synthetical DEM generated by GRASS GIS module r.surf.fractal from ETOPO1 raster grid

Öz

The research problem is about to generate artificial fractal landscape surfaces from the Digital Elevation Model (DEM) using a stochastic algorithm by Geographic Resources Analysis Support System Geographic Information System (GRASS GIS) software. Fractal surfaces resemble appearance of natural topographic terrain and its structure using random surface modelling. Study area covers Kuril-Kamchatka region, Sea of Okhotsk, North Pacific Ocean. Techniques were included into GRASS GIS modules (r.relief, d.rast, r.slope.aspect, r.mapcalc) for raster calculation, processing and visualization. Module 'r.surf.fractal' was applied for generating synthetic fractal surface from ETOPO1 DEM GeoTIFF using algorithm of fractal analysis. Three tested dimensions of the fractal surfaces were automatically mapped and visualized. Algorithm of the automated fractal DEM modelling visualized variations in steepness and aspect of the artificially generated slopes in the mountains. Controllable topographic variation of the fractal surfaces was applied for three dimensions: dim=2.0001, 2.0050, 2.0100. Auxiliary modules were used for the visualization of DEMs (d.rast, r.colors, d.vect, r.contour, d.redraw, d.mon). Modules 'r.surf.gauss' and 'r.surf.random' were applied for artificial modelling as Gauss and random based mathematical surfaces, respectively. Univariate statistics for fractal surfaces were computed for comparative analysis of maps representing continuous fields by module 'r.univar': number of cells, min/max, range, mean, variance, standard deviation, variation coefficient and sum. The paper includes 9 maps and GRASS GIS codes used for visualization.

Anahtar Kelimeler

• GRASS GIS
• Fractal surface
• Digital elevation model
• Grid
• ETOPO1

ETOPO1 raster verisinden sentetik SYM fraktal yüzeylerinin GRASS GIS r.surf.fractal modülü ile elde edilmesi

Öz

Araştırma problemi, GRASS GIS yazılımı ile stokastik bir algoritma kullanılarak Sayısal Yükseklik Modeli'nden (SYM) yapay fraktal yüzeylerin üretilmesidir. Fraktal yüzeyler, doğal topografik arazinin görünümüne ve yapısına rastgele yüzey modellemesi kullanarak benzerler. Çalışma alanı Kuril-Kamçatka bölgesini, Okhotsk Denizi'ni, Kuzey Pasifik Okyanusu'nu kapsamaktadır. Raster hesaplama, işleme ve görselleştirme için kullanılan yöntemler GRASS GIS modüllerini (r.relief, d.rast, r.slope.aspect, r.mapcalc) içermektedir. Fraktal analiz algoritması kullanılarak ETOPO1 DEM GeoTIFF'den sentetik fraktal yüzey oluşturmak için 'r.surf.fractal' modülü uygulanmıştır. Fraktal yüzeylerin test edilen üç boyutu otomatik olarak haritalanmış ve görselleştirilmiştir. Otomatik fraktal DEM modellemesinin algoritması kullanılarak dağlık alanlarda yapay olarak üretilen yamaçların dikliği ve yönü bakımından oluşturulan varyasyonlar ile görselleştirmeler yapılmıştır. Fraktal yüzeylerin kontrol edilebilir topografik varyasyonu üç boyut için uygulanmıştır: dim = 2.0001, 2.0050, 2.0100. DEM'lerin görüntülenmesi için yardımcı modüller kullanılmıştır (d.rast, r.colors, d.vect, r.contour, d.redraw, d.mon). Yapay modelleme için 'r.surf.gauss' ve 'r.surf.random' modülleri Gauss ve rasgele tabanlı matematiksel yüzeyler olmak üzere sırasıyla uygulanmıştır. Fraktal yüzeyler için tek değişkenli istatistikler 'r.univar' modülüne göre sürekli alanları temsil eden haritaların karşılaştırmalı analizi için hesaplanmıştır: hücre sayısı, min / maks, aralık, ortalama, varyans, standart sapma, varyasyon katsayısı ve toplam. Makalede 9 harita ve görselleştirme için kullanılan GRASS GIS kodları bulunmaktadır.

Anahtar Kelimeler

• GRASS GIS
• Fraktal yüzey
• Sayısal yükseklik modeli
• Grid
• ETOPO1

Kaynakça

1. Amante, C., & Eakins, B. W. (2009). ETOPO1 1 Arc-Minute Global Relief Model: Procedures, Data Sources and Analysis. NOAA Technical Memorandum NESDIS NGDC-24. National Geophysical Data Center, NOAA.
2. Breiman, L., & Friedman, J. H. (1985). Estimating optimal transformations for multiple regression and correlation. Journal of the American Statistical Association, 80(391), 580–598.
3. Briggs, J. (1992). Fractals: The Patterns of Chaos: A new aesthetic of art, science, and nature. New York, Simon and Schuster. pp. 192.
4. Burrough, P. (1981). Fractal dimensions of landscapes and other environmental data. Nature, 294(5838), 240-242.
5. Cressie, N. (1993). Statistics for Spatial Data, revised edition, Wiley Series in Probability and Statistics. In Applied Probability and Statistics. Wiley & Sons New York.
6. De La Re, A., Abad, F., Camahort, E., & Juan, M. C. (2009). Tools for procedural generation of plants in virtual scenes. In International Conference on Computational Science. pp. 801-810. Springer, Berlin, Heidelberg.
7. Duchesnay E., & Löfstedt T. (2019). Statistics and Machine Learning in Python. Release 0.2. http://www.python.org, Accessed: 12 April 2020.
8. Duhamel, P. & Vetterli, M. (1990). Fast Fourier transforms: a tutorial review and a state of the art. Signal Processing, 19, 259–299.

9. Dutton, G. H. (1981). Fractal enhancement of cartographic line detail. The American Cartographer, 8(1), 23-40.
10. Edgar, G. (2007). Measure, topology, and fractal geometry. Springer Science & Business Media.
11. Evans, I. S. (1972). General geomorphometry, derivatives of altitude, and descriptive statistics. Spatial analysis in geomorphology, 17-90.
12. Evans, I. S. (1979). An integrated system of terrain analysis and slope mapping: Final Report on U.S. Army Grant DA-ERO-591-73-g0040. Statistical characterization of altitude matrices by computer. Department of Geography, University of Durham, England, pp. 192.
13. Evans, I. S., & Cox, N. J. (2005). Global variations of local asymmetry in glacier altitude: separation of north–south and east–west components. Journal of glaciology, 51(174), 469-482.
14. Everitt, B. (1998). The Cambridge Dictionary of Statistics. Cambridge, UK: Cambridge University Press.
15. Falconer, K., (2003). Fractal Geometry: Mathematical Foundations and Applications (2nd ed.). John Wiley & Sons, pp. 332.
16. Feder, J. (2013). Fractals. Springer Science & Business Media. pp. 11.
17. Friedman, J. H. (1991). Multivariate adaptive regression splines. The Annals of Statistics, 19(1), 1–67.
18. Frigo, M., & Johnson, S. G. (1997). The fastest Fourier transform in the west. Technical Report MIT-LCS-TR- 728, Massachusetts Institute of Technology Cambridge.
19. Frigo, M., & Johnson, S. G. (1998). FFTW: An adaptive software architecture for the FFT. Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing, 3, pp. 1381-1384.
20. Frigo, M. (1999). A Fast Fourier Transform Compiler. Proceedings of the 1999 ACM SIGPLAN Conference on Programming Language Design and Implementation pp. 169-180.
21. Gauger, S., Kuhn, G., Gohl, K., Feigl, T., Lemenkova, P., & Hillenbrand, C.-D. (2007). Swath-bathymetric mapping. Reports on Polar and Marine Research, 557, pp. 38- 45.
22. Gloaguen, R., Marpu, P. R., & Niemeyer, I. (2007). Automatic extraction of faults and fractal analysis from remote sensing data. Nonlinear Processes in Geophysics, 14, pp. 131-138.
23. Gonzales-Barron, U., & Butler, F. (2005). A comparison of visual assessment and digital fractal texture analysis of bread-crumb features. In Using Cereal Science and Technology for the Benefit of Consumers. pp. 395-400. Woodhead Publishing.
24. Gordon, N. (2000). Introducing fractal geometry. pp. 71.
25. Ibanez, J. J., Arnold, R. W., & Ahrens, R. J. (2009). The fractal mind of pedologists (soil taxonomists and soil surveyors). Ecological complexity, 6(3), 286-293.
26. Imre, A. R., Novotný, J., & Rocchini, D. (2012). The Korcak-exponent: a non-fractal descriptor for landscape patchiness. Ecological complexity, 12, 70-74.
27. Johnson, S. G., & Frigo, M. (2006). A modified split-radix FFT with fewer arithmetic operations. IEEE Transactions on Signal Processing, 55(1), 111-119.
28. Klaučo, M., Gregorová, B., Stankov, U., Marković, V., & Lemenkova, P. (2013). Determination of ecological significance based on geostatistical assessment: a case study from the Slovak Natura 2000 protected area. Open Geosciences, 5(1), 28-42.
29. Klaučo, M., Gregorová, B., Stankov, U., Marković, V., & Lemenkova, P. (2015). Land planning as a support for sustainable development based on tourism: A case study of Slovak Rural Region. Environmental Engineering and Management Journal, 2(16), 449-458.
30. Kuhn, G., Hass, C., Kober, M., Petitat, M., Feigl, T., Hillenbrand, C. D., Kruger, S., Forwick, M., Gauger, S., & Lemenkova, P. (2006). The response of quaternary climatic cycles in the South-East Pacific: development of the opal belt and dynamics behavior of the West Antarctic ice sheet. Bremerhaven, Germany, 94.
31. Lemenkova, P., Promper, C., & Glade, T. (2012). Economic assessment of landslide risk for the Waidhofen ad Ybbs region, Alpine Foreland, Lower Austria.
32. Lemenkova, P. (2015a). Google Earth web service as a support for GIS mapping in geospatial research at universities. In Web-Technologies in the Educational Space. Problems, Approaches, Perspectives. pp. 24.
33. Lemenkova, P. (2015b). Innovations in the Geoscience Research: Classification of the Landsat TM Image Using ILWIS GIS for Geographic Studies. Prospects for the Higher School Development, pp. 60-63.
34. Lemenkova, P. (2015c). Data Capture for Seafloor Bathymetric Mapping Using Software Caris Hips, GMT and ArcGIS. Actual Problems of the Modern Machinery, pp. 111-117. Lemenkova, P. (2018a). Factor Analysis by R Programming to Assess Variability Among Environmental Determinants of the Mariana Trench. Turkish Journal of Maritime and Marine Sciences, 4(2), 146-155. Lemenkova, P. (2018b). R Scripting Libraries for Comparative Analysis of the Correlation Methods to Identify Factors Affecting Mariana Trench Formation. Journal of Marine Technology and Environment, 2, 35-42.
35. Lemenkova, P. (2019a). Processing Oceanographic Data by Python Libraries Numpy, Scipy and Pandas. Aquatic Research, 2(2), 73–91.
36. Lemenkova, P. (2019b). Statistical Analysis of the Mariana Trench Geomorphology Using R Programming Language. Geodesy and Cartography, 45(2), 57-84. Lemenkova, P. (2019c). An empirical study of R applications for data analysis in marine geology. Marine Science and Technology Bulletin, 8(1), 1-9.
37. Lemenkova, P. (2019d). Numerical Data Modelling and Classification in Marine Geology by the SPSS Statistics. International Journal of Engineering Technologies, 5(2), 90-99.
38. Lemenkova, P. (2019e). Geospatial Analysis by Python and R: Geomorphology of the Philippine Trench, Pacific Ocean. Electronic Letters on Science and Engineering, 15(3), 81-94.
39. Lemenkova, P. (2019f). Topographic surface modelling using raster grid datasets by GMT: example of the Kuril–Kamchatka Trench, Pacific Ocean. Reports on Geodesy and Geoinformatics, 108(1), 9-22.
40. Lemenkova, P. (2019g). Testing Linear Regressions by StatsModel Library of Python for Oceanological Data Interpretation. Aquatic Sciences and Engineering, 34, 51-60. Lemenkova, P. (2019h). Geomorphological modelling and mapping of the Peru-Chile Trench by GMT. Polish Cartographical Review, 51(4), 181-194.
41. Lemenkova, P. (2019i). Regression Models by Gretl and R Statistical Packages for Data Analysis in Marine Geology. International Journal of Environmental Trends, 3(1), 39-59.
42. Lemenkova, P. (2019j). AWK and GNU Octave Programming Languages Integrated with Generic Mapping Tools for Geomorphological Analysis. GeoScience Engineering, 65(4), 1-22.
43. Lemenkova, P. (2019k). GMT Based Comparative Analysis and Geomorphological Mapping of the Kermadec and Tonga Trenches, Southwest Pacific Ocean. Geographia Technica, 14(2), 39-48.
44. Lemenkova, P. (2019l). Automatic Data Processing for Visualising Yap and Palau Trenches by Generic Mapping Tools. Cartographic Letters, 27(2), 72-89. Lemenkova, P. (2020a). Visualization of the geophysical settings in the Philippine Sea margins by means of GMT and ISC data. Central European Journal of Geography and Sustainable Development, 2(1), 5-15.
45. Lemenkova, P. (2020b). GMT-based geological mapping and assessment of the bathymetric variations of the Kuril-Kamchatka Trench, Pacific Ocean. Natural and Engineering Sciences, 5(1), 1-17.
46. Lemenkova, P. (2020c). GMT Based Comparative Geomorphological Analysis of the Vityaz and Vanuatu Trenches, Fiji Basin. Geodetski list, 74(1), 19-39.
47. Malinverno, A. (1990). A simple method to estimate the fractal dimension of a self‐affine series. Geophysical Research Letters, 17(11), 1953-1956.
48. Mandelbrot, B. (1967). How long is the coast of Britain? Statistical self-similarity and fractional dimension. Science, 156, 636-638.
49. Mandelbrot, B. (1975). Stochastic models for the Earth's relief, the shape and the fractal dimension of the coastlines, and the number-area rule for islands. Proceedings of the National Academy of Sciences, 72(10), 3825-3828.
50. Mandelbrot, B. (1982). The Fractal Geometry of Nature. San Francisco, W. H. Freeman & Co. pp. 460.
51. Mandelbrot, B. (2004). Fractal aspects of the iteration of z→ λz (1-z) for complex λ and z. In Fractals and Chaos. pp. 37-51. Springer, New York, NY.
52. Mark, D. M., & Aronson, P. B. (1984). Scale-dependent fractal dimensions of topographic surfaces: an empirical investigation, with applications in geomorphology and computer mapping. Journal of the International Association for Mathematical Geology, 16(7), 671-683.
53. Muzy, J. F., Bacry, E., & Arneodo, A. (1993). Multifractal formalism for fractal signals: The structure-function approach versus the wavelet-transform modulus-maxima method. Physical review E, 47(2), 875.
54. Panchev, S. (1971). Random Functions and Turbulence. Pergamon, Oxford.
55. Pecknold, S., Lovejoy, S., Schertzer, D., Hooge, C., & Malouin, J. F. (1993). The simulation of universal multifractals, Cellular Automata.
56. Peleg, S., Naor, J., Hartley, R., & Avnir, D. (1983). Multiple-resolution texture analysis and classification. Computer Science Department, University of Maryland, College Park. Pickover, C. A. (1995). Generating extraterrestrial terrain. IEEE Computer Graphics and Applications, 15(2), 18-21.
57. Prusinkiewicz, P., & Hammel, M. (1993). A fractal model of mountains and rivers. In Graphics Interface. 93, pp. 174-180. Canadian Information Processing Society.
58. R Core Team (2017). R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. https://www.R-project.org/
59. Santner, T. J., Williams, B. J., & Notz, W. I. (2003). The design and analysis of computer experiments. New York: Springer.
60. Saupe, D. (1988). Algorithms for random fractals. In The science of fractal images. pp. 71-136. Springer, New York, NY.
61. Scheidegger, A. (1970). Theoretical Geomorphology. Springer Verlag.
62. Schenke, H. W., & Lemenkova, P. (2008). Zur Frage der Meeresboden-Kartographie: Die Nutzung von AutoTrace Digitizer für die Vektorisierung der Bathymetrischen Daten in der Petschora-See. Hydrographische Nachrichten, 25(81), 16–21.
63. Schertzer, D., & Lovejoy, S. (1991). Non-linear Variability in Geophysics: Scaling and Fractals. Kluwer Academic Publishers.
64. Schertzer, D., & Lovejoy, S. (1993). Nonlinear Variability in Geophysics 3: Scaling and Multifractal Processes, Institut D'études Scientifiques de Cargèse. Institut d'études scientifiques de Cargèse.
65. Suetova, I., Ushakova, L., & Lemenkova, P. (2005). Geoinformation mapping of the Barents and Pechora Seas. Geography and Natural Resources, 4, 138-142.
66. van Pabst, J. V. L., & Jense, H. (1996). Dynamic terrain generation based on multifractal techniques. In High Performance Computing for Computer Graphics and Visualisation. pp. 186-203. Springer, London.
67. van Rossum, G., & Drake Jr, F. L. (1995). Python reference manual. Amsterdam: Centrum voor Wiskunde en Informatica, Amsterdam.
68. Weatherall, P., Marks, K. M., Jakobsson, M., Schmitt, T., Tani, S., Arndt, J. E., Rovere, M., Chayes, D., Ferrini, V. & Wigley, R. (2015). A new digital bathymetric model of the world's oceans. Earth and Space Science, 2(8), 331-345.
69. URL-1: An Ivy Generator, http://graphics.uni-konstanz.de/~luft/ivy_generator (Accessed: 11 May 2020)
70. URL-2: FracTree, http://archives.math.utk.edu/software/msdos/fractals/fractree (Accessed: 11 May 2020)
71. URL-3: LStudio, http://algorithmicbotany.org (Accessed: 11 May 2020)
72. URL-4: Xfrog, http://www.xfrog.com/ (Accessed: 11 May 2020)
73. URL-5: FFTW, http://www.fftw.org/ (Accessed: 12 April 2020)