Abstract

The Earth’s topography generally obeys fractal statistics; after either one- or two-dimensional Fourier transforms the amplitudes have a power-law dependence on wave number. The slope gives the fractal dimension, and the unit wave-number amplitude is a measure of the roughness. In this study, digitized topography for the state of Oregon (∼7 points/km) has been used to obtain maps of fractal dimension and roughness amplitude. The roughness amplitude correlates well with variations in relief and is a promising parameter for the quantitative classification of landforms. The spatial variations in fractal dimension are low and show no clear correlation with different tectonic settings. For Oregon the mean fractal dimension from a two-dimensional spectral analysis is D = 2.586, and for a one-dimensional spectral analysis the mean fractal dimension is D = 1.487, which is close to the Brown noise value D = 1.5. Synthetic two-dimensional images have also been generated for a range of D values. For D = 2.6, the synthetic image has a mean one-dimensional spectral fractal dimension D = 1.58, which is consistent with our results for Oregon. This approach can be easily applied to any digitized image that obeys fractal statistics.

© 1990 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. B. Mandelbrot, “How long is the coast of Britain? Statistical self-similarity and fractional dimension,” Science 156, 636–638 (1967).
    [CrossRef] [PubMed]
  2. B. Mandelbrot, The Fractal Geometry of Nature (Freeman, San Francisco, Calif., 1982).
  3. B. Mandelbrot, “Stochastic models for the Earth’s relief, the shape and the fractal dimension of the coastlines, and the number-area rule for islands,” Proc. Natl. Acad. Sci. USA 72, 3825–3828 (1975).
    [CrossRef]
  4. D. L. Turcotte, “Fractals and fragmentation,” J. Geophys. Res. 91, 1921–1926 (1986).
    [CrossRef]
  5. K. Aki, “A probabilistic synthesis of precursory phenomena,” in Earthquake Prediction, D. W. Simpson, P. G. Richards, eds. (American Geophysical Union, Washington, D.C., 1981), pp. 566–574.
  6. D. L. Turcotte, “Fractals in geology and geophysics,” Pure Appl. Geophys. 131, 171–196 (1989).
    [CrossRef]
  7. M. F. Goodchild, “Fractals and the accuracy of geographical measures,” Math. Geol. 12, 85–98 (1980).
    [CrossRef]
  8. B. Mandelbrot, “Self-affine fractal sets, I: the basic fractal dimensions,” in Fractals in Physics, L. Pietronero, E. Tosatti, eds. (Elsevier, Amsterdam, 1986), pp. 3–15.
  9. R. F. Voss, “Random fractals: characterization and measurement,” in Scaling Phenomena in Disordered Systems, R. Pynn, A. Skjeltrop, eds. (Plenum, New York, 1985), pp. 1–11.
  10. R. F. Voss, “Random fractal forgeries,” in Fundamental Algorithms for Computer Graphics, Vol. F17 of NATO ASI Series, R. A. Earnshaw, ed. (Springer-Verlag, Berlin, 1985), pp. 805–835.
    [CrossRef]
  11. C. Cox, H. Sandstrom, “Coupling of internal and surface waves in water of variable depth,” J. Oceanogr. Soc. Jpn. 20, 499–513 (1962).
  12. F. P. Bretherton, “Momentum transport by gravity waves,” Q. J. R. Meteorol. Soc. 95, 213–243 (1969).
    [CrossRef]
  13. B. A. Warren, “Transpacific hydrographic sections at lats. 43° S and 28° S: the SCORPIO Expedition. II. Deep water,” Deep-Sea Res. 20, 9–38 (1973).
  14. T. H. Bell, “Statistical features of sea-floor topography,” Deep-Sea Res. 22, 883–892 (1975).
  15. T. H. Bell, “Mesoscale sea floor roughness,” Deep-Sea Res. 26A, 65–76 (1979).
    [CrossRef]
  16. G. I. Barenblatt, A. V. Zhivago, Y. P. Neprochnov, A. A. Ostrovskiy, “The fractal dimension: a quantitative characteristic of ocean-bottom relief,” Oceanology 24, 695–697 (1984).
  17. C. G. Fox, D. E. Hayes, “Quantitative methods for analyzing the roughness of the seafloor,” Rev. Geophys. 23, 1–48 (1985).
    [CrossRef]
  18. D. Gilbert, V. Courtillot, “Seasat altimetry and the South Atlantic geoid, I. Spectral analysis,” J. Geophys. Res. 92, 6235–6248 (1987).
    [CrossRef]
  19. G. Balmino, K. Lambeck, W. M. Kaula, “A spherical harmonic analysis of the Earth’s topography,” J. Geophys. Res. 78, 478–481 (1973).
    [CrossRef]
  20. D. L. Turcotte, “A fractal interpretation of topography and geoid spectra on the Earth, Moon, Venus, and Mars, in Proceedings of the Lunar Planetary Science Conference 17, Part 2, J. Geophys. Res. 92 suppl., E597–E601 (1987).
  21. R. F. Voss, “Fractals in nature: from characterization to simulation,” in The Science of Fractal Images, H. Peitgen, D. Saupe, eds. (Springer-Verlag, New York, 1988), pp. 21–70.
    [CrossRef]
  22. J. Huang, D. L. Turcotte, “Fractal mapping of digitized images: application to the topography of Arizona and comparisons with synthetic images,” J. Geophys. Res. 94, 7491–7495 (1989).
    [CrossRef]

1989 (2)

D. L. Turcotte, “Fractals in geology and geophysics,” Pure Appl. Geophys. 131, 171–196 (1989).
[CrossRef]

J. Huang, D. L. Turcotte, “Fractal mapping of digitized images: application to the topography of Arizona and comparisons with synthetic images,” J. Geophys. Res. 94, 7491–7495 (1989).
[CrossRef]

1987 (2)

D. L. Turcotte, “A fractal interpretation of topography and geoid spectra on the Earth, Moon, Venus, and Mars, in Proceedings of the Lunar Planetary Science Conference 17, Part 2, J. Geophys. Res. 92 suppl., E597–E601 (1987).

D. Gilbert, V. Courtillot, “Seasat altimetry and the South Atlantic geoid, I. Spectral analysis,” J. Geophys. Res. 92, 6235–6248 (1987).
[CrossRef]

1986 (1)

D. L. Turcotte, “Fractals and fragmentation,” J. Geophys. Res. 91, 1921–1926 (1986).
[CrossRef]

1985 (1)

C. G. Fox, D. E. Hayes, “Quantitative methods for analyzing the roughness of the seafloor,” Rev. Geophys. 23, 1–48 (1985).
[CrossRef]

1984 (1)

G. I. Barenblatt, A. V. Zhivago, Y. P. Neprochnov, A. A. Ostrovskiy, “The fractal dimension: a quantitative characteristic of ocean-bottom relief,” Oceanology 24, 695–697 (1984).

1980 (1)

M. F. Goodchild, “Fractals and the accuracy of geographical measures,” Math. Geol. 12, 85–98 (1980).
[CrossRef]

1979 (1)

T. H. Bell, “Mesoscale sea floor roughness,” Deep-Sea Res. 26A, 65–76 (1979).
[CrossRef]

1975 (2)

B. Mandelbrot, “Stochastic models for the Earth’s relief, the shape and the fractal dimension of the coastlines, and the number-area rule for islands,” Proc. Natl. Acad. Sci. USA 72, 3825–3828 (1975).
[CrossRef]

T. H. Bell, “Statistical features of sea-floor topography,” Deep-Sea Res. 22, 883–892 (1975).

1973 (2)

B. A. Warren, “Transpacific hydrographic sections at lats. 43° S and 28° S: the SCORPIO Expedition. II. Deep water,” Deep-Sea Res. 20, 9–38 (1973).

G. Balmino, K. Lambeck, W. M. Kaula, “A spherical harmonic analysis of the Earth’s topography,” J. Geophys. Res. 78, 478–481 (1973).
[CrossRef]

1969 (1)

F. P. Bretherton, “Momentum transport by gravity waves,” Q. J. R. Meteorol. Soc. 95, 213–243 (1969).
[CrossRef]

1967 (1)

B. Mandelbrot, “How long is the coast of Britain? Statistical self-similarity and fractional dimension,” Science 156, 636–638 (1967).
[CrossRef] [PubMed]

1962 (1)

C. Cox, H. Sandstrom, “Coupling of internal and surface waves in water of variable depth,” J. Oceanogr. Soc. Jpn. 20, 499–513 (1962).

Aki, K.

K. Aki, “A probabilistic synthesis of precursory phenomena,” in Earthquake Prediction, D. W. Simpson, P. G. Richards, eds. (American Geophysical Union, Washington, D.C., 1981), pp. 566–574.

Balmino, G.

G. Balmino, K. Lambeck, W. M. Kaula, “A spherical harmonic analysis of the Earth’s topography,” J. Geophys. Res. 78, 478–481 (1973).
[CrossRef]

Barenblatt, G. I.

G. I. Barenblatt, A. V. Zhivago, Y. P. Neprochnov, A. A. Ostrovskiy, “The fractal dimension: a quantitative characteristic of ocean-bottom relief,” Oceanology 24, 695–697 (1984).

Bell, T. H.

T. H. Bell, “Mesoscale sea floor roughness,” Deep-Sea Res. 26A, 65–76 (1979).
[CrossRef]

T. H. Bell, “Statistical features of sea-floor topography,” Deep-Sea Res. 22, 883–892 (1975).

Bretherton, F. P.

F. P. Bretherton, “Momentum transport by gravity waves,” Q. J. R. Meteorol. Soc. 95, 213–243 (1969).
[CrossRef]

Courtillot, V.

D. Gilbert, V. Courtillot, “Seasat altimetry and the South Atlantic geoid, I. Spectral analysis,” J. Geophys. Res. 92, 6235–6248 (1987).
[CrossRef]

Cox, C.

C. Cox, H. Sandstrom, “Coupling of internal and surface waves in water of variable depth,” J. Oceanogr. Soc. Jpn. 20, 499–513 (1962).

Fox, C. G.

C. G. Fox, D. E. Hayes, “Quantitative methods for analyzing the roughness of the seafloor,” Rev. Geophys. 23, 1–48 (1985).
[CrossRef]

Gilbert, D.

D. Gilbert, V. Courtillot, “Seasat altimetry and the South Atlantic geoid, I. Spectral analysis,” J. Geophys. Res. 92, 6235–6248 (1987).
[CrossRef]

Goodchild, M. F.

M. F. Goodchild, “Fractals and the accuracy of geographical measures,” Math. Geol. 12, 85–98 (1980).
[CrossRef]

Hayes, D. E.

C. G. Fox, D. E. Hayes, “Quantitative methods for analyzing the roughness of the seafloor,” Rev. Geophys. 23, 1–48 (1985).
[CrossRef]

Huang, J.

J. Huang, D. L. Turcotte, “Fractal mapping of digitized images: application to the topography of Arizona and comparisons with synthetic images,” J. Geophys. Res. 94, 7491–7495 (1989).
[CrossRef]

Kaula, W. M.

G. Balmino, K. Lambeck, W. M. Kaula, “A spherical harmonic analysis of the Earth’s topography,” J. Geophys. Res. 78, 478–481 (1973).
[CrossRef]

Lambeck, K.

G. Balmino, K. Lambeck, W. M. Kaula, “A spherical harmonic analysis of the Earth’s topography,” J. Geophys. Res. 78, 478–481 (1973).
[CrossRef]

Mandelbrot, B.

B. Mandelbrot, “Stochastic models for the Earth’s relief, the shape and the fractal dimension of the coastlines, and the number-area rule for islands,” Proc. Natl. Acad. Sci. USA 72, 3825–3828 (1975).
[CrossRef]

B. Mandelbrot, “How long is the coast of Britain? Statistical self-similarity and fractional dimension,” Science 156, 636–638 (1967).
[CrossRef] [PubMed]

B. Mandelbrot, The Fractal Geometry of Nature (Freeman, San Francisco, Calif., 1982).

B. Mandelbrot, “Self-affine fractal sets, I: the basic fractal dimensions,” in Fractals in Physics, L. Pietronero, E. Tosatti, eds. (Elsevier, Amsterdam, 1986), pp. 3–15.

Neprochnov, Y. P.

G. I. Barenblatt, A. V. Zhivago, Y. P. Neprochnov, A. A. Ostrovskiy, “The fractal dimension: a quantitative characteristic of ocean-bottom relief,” Oceanology 24, 695–697 (1984).

Ostrovskiy, A. A.

G. I. Barenblatt, A. V. Zhivago, Y. P. Neprochnov, A. A. Ostrovskiy, “The fractal dimension: a quantitative characteristic of ocean-bottom relief,” Oceanology 24, 695–697 (1984).

Sandstrom, H.

C. Cox, H. Sandstrom, “Coupling of internal and surface waves in water of variable depth,” J. Oceanogr. Soc. Jpn. 20, 499–513 (1962).

Turcotte, D. L.

D. L. Turcotte, “Fractals in geology and geophysics,” Pure Appl. Geophys. 131, 171–196 (1989).
[CrossRef]

J. Huang, D. L. Turcotte, “Fractal mapping of digitized images: application to the topography of Arizona and comparisons with synthetic images,” J. Geophys. Res. 94, 7491–7495 (1989).
[CrossRef]

D. L. Turcotte, “A fractal interpretation of topography and geoid spectra on the Earth, Moon, Venus, and Mars, in Proceedings of the Lunar Planetary Science Conference 17, Part 2, J. Geophys. Res. 92 suppl., E597–E601 (1987).

D. L. Turcotte, “Fractals and fragmentation,” J. Geophys. Res. 91, 1921–1926 (1986).
[CrossRef]

Voss, R. F.

R. F. Voss, “Random fractals: characterization and measurement,” in Scaling Phenomena in Disordered Systems, R. Pynn, A. Skjeltrop, eds. (Plenum, New York, 1985), pp. 1–11.

R. F. Voss, “Random fractal forgeries,” in Fundamental Algorithms for Computer Graphics, Vol. F17 of NATO ASI Series, R. A. Earnshaw, ed. (Springer-Verlag, Berlin, 1985), pp. 805–835.
[CrossRef]

R. F. Voss, “Fractals in nature: from characterization to simulation,” in The Science of Fractal Images, H. Peitgen, D. Saupe, eds. (Springer-Verlag, New York, 1988), pp. 21–70.
[CrossRef]

Warren, B. A.

B. A. Warren, “Transpacific hydrographic sections at lats. 43° S and 28° S: the SCORPIO Expedition. II. Deep water,” Deep-Sea Res. 20, 9–38 (1973).

Zhivago, A. V.

G. I. Barenblatt, A. V. Zhivago, Y. P. Neprochnov, A. A. Ostrovskiy, “The fractal dimension: a quantitative characteristic of ocean-bottom relief,” Oceanology 24, 695–697 (1984).

Deep-Sea Res. (3)

B. A. Warren, “Transpacific hydrographic sections at lats. 43° S and 28° S: the SCORPIO Expedition. II. Deep water,” Deep-Sea Res. 20, 9–38 (1973).

T. H. Bell, “Statistical features of sea-floor topography,” Deep-Sea Res. 22, 883–892 (1975).

T. H. Bell, “Mesoscale sea floor roughness,” Deep-Sea Res. 26A, 65–76 (1979).
[CrossRef]

J. Geophys. Res. (4)

D. L. Turcotte, “Fractals and fragmentation,” J. Geophys. Res. 91, 1921–1926 (1986).
[CrossRef]

D. Gilbert, V. Courtillot, “Seasat altimetry and the South Atlantic geoid, I. Spectral analysis,” J. Geophys. Res. 92, 6235–6248 (1987).
[CrossRef]

G. Balmino, K. Lambeck, W. M. Kaula, “A spherical harmonic analysis of the Earth’s topography,” J. Geophys. Res. 78, 478–481 (1973).
[CrossRef]

J. Huang, D. L. Turcotte, “Fractal mapping of digitized images: application to the topography of Arizona and comparisons with synthetic images,” J. Geophys. Res. 94, 7491–7495 (1989).
[CrossRef]

J. Oceanogr. Soc. Jpn. (1)

C. Cox, H. Sandstrom, “Coupling of internal and surface waves in water of variable depth,” J. Oceanogr. Soc. Jpn. 20, 499–513 (1962).

Math. Geol. (1)

M. F. Goodchild, “Fractals and the accuracy of geographical measures,” Math. Geol. 12, 85–98 (1980).
[CrossRef]

Oceanology (1)

G. I. Barenblatt, A. V. Zhivago, Y. P. Neprochnov, A. A. Ostrovskiy, “The fractal dimension: a quantitative characteristic of ocean-bottom relief,” Oceanology 24, 695–697 (1984).

Proc. Natl. Acad. Sci. USA (1)

B. Mandelbrot, “Stochastic models for the Earth’s relief, the shape and the fractal dimension of the coastlines, and the number-area rule for islands,” Proc. Natl. Acad. Sci. USA 72, 3825–3828 (1975).
[CrossRef]

Proceedings of the Lunar Planetary Science Conference 17 (1)

D. L. Turcotte, “A fractal interpretation of topography and geoid spectra on the Earth, Moon, Venus, and Mars, in Proceedings of the Lunar Planetary Science Conference 17, Part 2, J. Geophys. Res. 92 suppl., E597–E601 (1987).

Pure Appl. Geophys. (1)

D. L. Turcotte, “Fractals in geology and geophysics,” Pure Appl. Geophys. 131, 171–196 (1989).
[CrossRef]

Q. J. R. Meteorol. Soc. (1)

F. P. Bretherton, “Momentum transport by gravity waves,” Q. J. R. Meteorol. Soc. 95, 213–243 (1969).
[CrossRef]

Rev. Geophys. (1)

C. G. Fox, D. E. Hayes, “Quantitative methods for analyzing the roughness of the seafloor,” Rev. Geophys. 23, 1–48 (1985).
[CrossRef]

Science (1)

B. Mandelbrot, “How long is the coast of Britain? Statistical self-similarity and fractional dimension,” Science 156, 636–638 (1967).
[CrossRef] [PubMed]

Other (6)

B. Mandelbrot, The Fractal Geometry of Nature (Freeman, San Francisco, Calif., 1982).

K. Aki, “A probabilistic synthesis of precursory phenomena,” in Earthquake Prediction, D. W. Simpson, P. G. Richards, eds. (American Geophysical Union, Washington, D.C., 1981), pp. 566–574.

B. Mandelbrot, “Self-affine fractal sets, I: the basic fractal dimensions,” in Fractals in Physics, L. Pietronero, E. Tosatti, eds. (Elsevier, Amsterdam, 1986), pp. 3–15.

R. F. Voss, “Random fractals: characterization and measurement,” in Scaling Phenomena in Disordered Systems, R. Pynn, A. Skjeltrop, eds. (Plenum, New York, 1985), pp. 1–11.

R. F. Voss, “Random fractal forgeries,” in Fundamental Algorithms for Computer Graphics, Vol. F17 of NATO ASI Series, R. A. Earnshaw, ed. (Springer-Verlag, Berlin, 1985), pp. 805–835.
[CrossRef]

R. F. Voss, “Fractals in nature: from characterization to simulation,” in The Science of Fractal Images, H. Peitgen, D. Saupe, eds. (Springer-Verlag, New York, 1988), pp. 21–70.
[CrossRef]

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (5)

Fig. 1
Fig. 1

Gray-scale map of the digitized topography for Oregon. The data resolution is ∼7 points/km.

Fig. 2
Fig. 2

Plots of one-dimensional spectral energy density versus wave number selected from three regions in Oregon with different tectonic and geomorphic settings: (a) Willamette lowland, (b) Wallowa Mountains, (c) Klamath Falls.

Fig. 3
Fig. 3

Plots of mean spectral energy density versus radial wave number for four typical 32 × 32 point subregions in Oregon. The linear trend on a log–log plot indicates a power-law fractal distribution.

Fig. 4
Fig. 4

Gray-scale maps for Oregon: (a) fractal dimension, (b) roughness amplitude. There is generally limited systematic variation in the fractal dimension; however, the roughness amplitude is sensitive to texture changes.

Fig. 5
Fig. 5

Synthetic fractal images on a 512 × 512 grid: (a) white noise without fractal filtering, (b) filtered with β = 1 and D = 3.0, (c) filtered with β = 1.8 and D = 2.6.

Tables (2)

Tables Icon

Table 1 Regional Averages over One-Dimensional Profiles of Oregon Topography

Tables Icon

Table 2 Summary of Mean Fractal Dimensions Estimated by One-Dimensional and Two-Dimensional Spectral Analysis for Both the Topography of Oregon and Synthetic Images

Equations (22)

Equations on this page are rendered with MathJax. Learn more.

N i = C / r i D ,
N = C / r D ,
P i = N i r i = C r i 1 D .
S ( k ) k β .
V ( L ) = 1 L 0 L [ h ( x ) h ¯ ] 2 d x ,
V ( L ) L 2 H .
σ ( L ) = [ V ( L ) ] 1 / 2 L H .
N n = L σ n L n h n = n 2 σ n σ ,
σ n σ = σ ( L / n ) σ ( L ) = 1 n H ,
N n = n 2 H = ( L / L n ) 2 H .
D = 2 H .
S ( L ) = L V ( L ) L β L 1 + 2 H ,
D = 5 β 2 .
H s t = n = 0 N 1 m = 0 N 1 h n m exp [ 2 π i N ( s n + t m ) ] ,
r = ( s 2 + t 2 ) 1 / 2 .
S 2 j = L 2 N j I N j | H s t | 2 ,
S 2 j k j 1 β .
D 2 = 3 H
S 2 ( L ) L 2 V ( L ) L 1 + β L 2 + 2 H ,
D 2 = 7 β 2
p ( y ) d y = 1 ( 2 π ) 1 / 2 e y 2 / 2 d y .
H s t * = H s t / k s t β / 2 .

Metrics