Modeling geomorphic processes

PHYSICA Physica D 77 (1994) 229-237

ELSEVIER

Modeling geomorphic processes Donald L. Turcotte* Department o f Geological Sciences, CorneU University, Snee Hall, Ithaca, N Y 14853-1504, USA

Abstract

The surface of the earth is largely molded by erosional and depositional processes, these processes yield complex structures but within the complexity there is order. The concept of fractals was first introduced in terms of the self-similarity of landforms and spectral studies of both topography and bathymetry often yield self-affine fractals. Drainage networks play an essential role in the evolution of landforms and drainage networks are a classic example of fractal trees. A number of models have recently been proposed for the evolution of drainage networks; a very simple approach that is quite realistic and reproduces observed statistics is based on diffusion limited aggregation. The erosion necessary to produce landforms is attributed to floods with the great floods doing most of the erosion. The statistics of floods and droughts is quite controversial with the applicability of Hurst exponents uncertain; an essential question is whether extreme floods obey fractal (power law) statistics or logarithmic statistics. A tremendous amount of data on paleoweather and climate resides in the sedimentary record; but there are always uncertainties with regard to the completeness of the sedimentary record. Modeling should be possible using the basic concepts of time series but the applicable time series are more likely to be log normal than Gaussian. The statistics of extreme events under such circumstances remain uncertain.

I. Introduction

It is now recognized that many natural processes obey fractal statistics. There are several definitions of fractals but the simple numberlength scaling can form the basis for almost all natural applications, this can be written N = Cr -° ,

(1)

where D is the fractal dimension and N is the n u m b e r objects with a linear dimension r for a discrete distribution and the number with a

* E-mail: [email protected](e-mail)

linear dimension greater than r for a continuous distribution. The concept of fractals was introduced in terms of the length of the west coast of Great Britain by Mandelbrot [1], his result is given in Fig. la. The length P of the coast line is given as a function of the length r of the measuring rod. Similar results are obtained for the length of topographic contours on topographic maps, three examples are also given in Fig. 1 for diverse geological settings. This application of fractals illustrates several features of most applications to natural phenomena. First, the applicable range for fractal statistics has both upper and lower bounds. Second, the fractal dimension

0167-2789/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved S S D I 0167-2789(94)00086-6

D.L. Turcotte / Physica D 77 (1994) 229-237

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a self-affine fractal results with constraints on the value of /3 and D = ½ ( 5 - / 3 ) . Spectral expansions of the global topography of the earth [3] and Venus have been carried out, results are given in Fig. 2. In both cases a good correlation with/3 = 2 (D = 1.5) is found, i.e. Brown noise. This is somewhat surprising since erosion and deposition are dominant in the evolution of many landforms on the earth, these are essentially absent on Venus so that tectonic processes are dominant. This suggests that the tectonic processes that build topography and the erosional and depositional processes that destroy topography both give Brown noise statistics. Once again the fractal dimension of topography does not appear to be diagnostic.

2. Drainage networks

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F i g . 1. (a) L e n g t h P o f t h e w e s t coast of G r e a t B r i t a i n as a f u n c t i o n o f t h e l e n g t h r o f t h e m e a s u r i n g r o d [1], ( b ) - ( d ) l e n g t h s P o f s p e c i f i e d t o p o g r a p h i c c o n t o u r s in s e v e r a l m o u n t a i n b e l t s as a f u n c t i o n o f t h e l e n g t h r o f t h e m e a s u r i n g r o d .

(b) 3,000f1. contour of the Cobblestone Mountain quadrangle, Transverse Ranges, CA; (c) 10,000ft. contour of the Byers Peak quadrangle, Rocky Mountains, Colorado; (d) 1,000 ft. contour of the Silver Bay quadrangle, Adirondack Mountains, NY. Correlations are with the fractal relation (1). is not diagnostic; the fractal dimensions of topographic contours are not sensitive to the geological setting. The fractal dimension of relatively smooth old mountains (i.e. the Adirondacks) is not significantly different from that of jagged young mountains (i.e. the Transverse Ranges). Coastlines and topographic contours are examples of self-similar fractals. Topography (and bathymetry) is also an example of self-affine fractals. If a spectral analysis results in a powerlaw relation between the spectral power density S and the wave number k [2],

Essential features of most landforms on the earth are drainage networks. Drainage networks are classic examples of fractal trees. It is standard practice to use the Strahler [4] ordering system. When two like-order streams meet they form a stream with one higher order than the original. Thus two first-order streams come in to form a second-order stream, two second-order streams combine to form a third-order stream, and so forth. The bifurcation ratio R b is defined by

N. R b - Nn+, ,

(3)

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is the mean length of streams of order R b and R r are found to be nearly for a range of stream orders in a basin [5]. From (1) the fractal dimendrainage network is

D.L. Turcotte I Physica D 77 (1994)229-237

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Fig. 3. (a) The drainage network in the Volfe and Bell Canyons, San Gabriel Mountains, near Glendora, California obtained from field mapping [6]. (b) Illustration of a D L A model for a drainage network [19].

232

D.L. Turcotte / Physica D 77 (1994) 229-237

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An example of a drainage network is given in Fig. 3a; this is the drainage network in the Volfe and Bell Canyons, San Gabriel Mountains near Glendora, California [6]. The number-length statistics for this network are given in Fig. 4a; a good correlation with (5) is obtained taking D = 1.81. Leopold et al. [7] have obtained similar results for the entire United States, a good correlation with (5) is also found with D = 1.83. Drainage networks are in general fractal with little variation in the fractal dimension. Once again the fractal dimension of the drainage network is not diagnostic of its geological setting. A variety of models have been proposed to simulate drainage networks. Leopold and Lang10 3 0

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bein [8] and Schenck [9] hypothesized that streams follow random walks, but the results bore little resemblance to actual drainage networks. Other probabilistic models for the generation of channel networks were introduced by Shreve [10,11], Scheidegger ]12], Smart and Surkan [13], Seginer [14], and Howard [15]. Smart [16] reviewed much of this work. In the past five years there has been a rebirth of interest in the problem. Kondoh and Matsushita [17], Meakin et al. [18], and Masek and Turcotte [19] have introduced diffusion-limited aggregation (DLA) models. Stark [20] developed a model based on self-avoiding percolation clusters. Chase [21] and Inaoka and Takayasu [22] introduced models based on self-organized criticality. Willgoose et al. [23] and Kramer and Marder [24] developed advection-diffusion models based on equations that yield a network structure. In their D L A model Masek and Turcotte [19] considered a square grid of points and introduced seed cells on one boundary of the grid. Cells are allowed to accrete to the evolving network if one (and only one) of the four nearest neighbor cells is part of the network. A cell is introduced randomly on the grid and follows a random walk on the grid until it either (i) accretes to the evolving grid, (ii) is lost from the grid, or (iii) occupies a forbidden site. A simulation carried out on a 256 x 256 grid is illustrated in Fig. 3b, 20,000 random walkers have been introduced. The simulated and actual drainage networks are reasonably similar. The numberlength statistics for the simulated network are given in Fig. 4b; a correlation with (5) is obtained taking D = 1.85 in good agreement with real drainage networks. The random walkers can be interpreted as floods that flow over a relatively flat surface until they find a gully. When the flood enters the gully it further erodes the gully and extends the network, headwards. This type of headward gully evolution has often been proposed for actual drainage networks [25].

D.L. Turcotte / Physica D 77 (1994) 229-237

3. Flood statistics

An important question in geomorphology concerns which floods dominate erosion. Is erosion dominated by the 10 year, the 100 year, or the very largest floods? The answer to this question depends upon whether extreme flood probabilities have an exponential or power-law dependence on time. Flood-frequency statistics also have a variety of other implications. Land-use regulations and flood control projects are based on extrapolations for future floods. A wide variety of statistical techniques have been applied to the evaluation of the flood hazard, but the data base for establishing the validity of the statistics is quite limed. A speculative hypothesis is that floods also obey fractal statistics [26]. The peak discharge during a flood lk is a measure of its intensity. If floods are fractal then f ' = C T eI ,

(6)

where fr is the peak discharge in a time interval T (under some circumstances H can be referred to as the Hausdorff measure). This scale invariant distribution can also be expressed in terms of the ratio F of the peak discharge over a 10-year interval to the peak discharge over a one-year interval. With self-similarity the parameter F is then also the ratio of the 100-year peak discharge to the 10-year peak discharge. The parameters H and F can be related by F = I0 n .

(7)

Care must be taken in defining individual floods, it is not appropriate to take yearly maxima in f" with a minimum one month interval preferred. As two specific examples we consider station 1-1805 on the Middle Branch of the Westfield River in Goss Heights, MA for the period 19111960 [27] and station 11-0980 in the Arroyo Seco near Pasadena, CA for the period 1914-1965 [28]. The reason that these stations were chosen is that they were used as benchmarks by Benson [29] who applied a variety of geostatistical distributions to flood-frequency forecasting. Floods

233

are considered independent only if the peak flows are separated by more than one month. For a 50 year record, the 50 largest values of lkm are ordered, the largest lkm is assigned a period T - - 5 0 years, the second largest flood a period T = 5 0 / 2 = 2 5 years, the third largest flood a period T = 50/3 = 16.7 years, and so forth. The results for the two stations are given in Fig. 5. For station 1-1805 the fractal fit (F) gives H = 0.51 and F = 3.3. For station 11-0980 the fractal fit gives H = 0.87 and F = 7.4. Also included in these figures are the six statistical correlations given by Benson [29]: the two parameter gamma (Ga), Gumbel (Gu), log Gumbel (LGu), log normal (LN), Hazen (H), and IogPearson type III (LP). For large floods the fractal predictions (F) correlate best with the log Gumbel (LGu) while the other statistical techniques predict longer recurrence time for very serious floods. The fractal and log Gumbel are essentially power-law correlations whereas the others are essentially logarithmic. The log Pearson type III (LP) is the federally approved distribution for evaluating the flood hazard in the United States. The values of the Hausdorff measure H and flood intensity factor F for the ten benchmark stations considered by Benson [29] are given in Table 1. These results show that there are clear regional trends in the values of F. The values in the southwest are systematically high; this can be attributed to the arid conditions and the rare tropical storm that causes severe flooding. The values in the Pacific northwest are low; this can be attributed to the maritime climate. Since F is equivalent to a fractal dimension, D = 2 - log F, this may be a case in which the fractal dimension of floods are diagnostic of climate.

4. Sedimentary statistics

The statistics of floods are directly related to the statistics of many sedimentary sequences. An important question is whether sedimentation is

234

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Table 1 Values of the Hausdorff measure H and flood intensity factor F for the ten benchmark stations considered by Benson [29]. Station 1-1805

2-2185 5-3310 6-3440 6-8005 7-2165 8-1500 10-3275 11-0980 12-1570

Site

H

F

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3.31 3.47 2.95 3.31 3.47 4.27 5.24 4.13 7.4 2.04

D.L. Turcotte I Physica D 77 (1994) 229-237

dominated by the very largest floods. With power-law flood statistics the very largest floods are likely to be dominant. With logarithmic flood statistics intermediate sized floods will dominate. Sedimentary rocks are generally composed of a sequence of layers, each layer representing a distinct sedimentation event. Each layer is often composed of an upward gradation from coarse grained sediments to fine grained sediments, individual layers are generally separated by well defined bedding planes. Several recent studies of the thickness statistics of turbidite deposits have been carried out Rothman et al. [30] carried out direct measurements on an outcrop of the Kingston Peak Formation near the southern end of Death Valley, CA. These results are given in Fig. 6a; an excellent correlation with (1) is obtained taking D = 1.39. Hiscott et al. [31] have studied a volcaniclastic turbidity deposit in the lzu-Bonin forearc basin off shore of Japan. Layer thicknesses were obtained from formation-microscanner images in the middle to upper Oligocentre part of the section. Results for two DSDP holes located 75 km apart are given in Fig. 6b; a good correlation with (1) is obtained taking D = 1.12. Turbidite deposits are associated with slumps off the continental margin. These avalanche-like events can be directly associated with the sandslide analogy to self-organized criticality [30]. Thus the fractal distribution of turbidite layers can be directly associated with a selforganized critical mechanism. Fractal correlations of sedimentary sequences are not restricted to turbidite deposits. Stolum [32] has obtained fractal statistics for the Middle Jurassic Tilje Formation in the Northsea Haltenbanken basin; these sediments were deposited in a marine shelf environment. H e found values of D = 0.71, 0.80. Malamud and Turcotte [33] obtained fractal statistics for sandy-bed thicknesses in the shallow marine environment of the Late Devonian Ithaca Formation, New York, with D = 1.41. In several cases fractal dimensions for thick-

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ness statistics are greater than one. The constructions illustrated in Fig. 7 show that D can in fact be greater than one. The standard Cantor set is illustrated in Fig. 7a; one layer, N, = 1, with thickness r 1 = 1/3, two layers, N z = 2, with thickness r 2 = 1/9, four layers, N 3 = 4, with thickness r 3 = 1 / 2 7 . From (1) D = ( l n 2 ) / ( l n 3 ) = 0 . 6 3 0 9 . In Fig. 7b a stretched cantor set is illustrated. At each step the remaining segments are stretched

D.L. Turcotte / Physica D 77 (1994) 2 2 9 - 2 3 7

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by a factor of two before being further subdivided. This gives N~ = 1 with r I = 1/3, N z = 2 with r 2 = 2 / 9 , N 3 = 4 with r 3 = 4 / 2 7 . From (1) D = (ln 2)/[ln(3/2)] = 1.710. The length L of the set is unbounded, L--~o~ as r---> 0. For real data sets with both upper and lower bounds on r, this construction illustrates that values of D greater than one are acceptable. The fractal behavior of stratigraphic sequences has also been demonstrated using spectral techniques. Hewett [34] gave results for a densityporosity log in a well through a late Mioceneearly Pleistocene sandstone formation deposited in a deep submarine fan. He showed that the spectra of the well log correlated with (2) taking /3---0.71. Similar results have been reported by Todoeschuck and Jensen [35] and by Todoeschuck et al. [36]. Hewett [34] also developed a fractal based interpolation technique for determining the porosity distribution in reservoirs. The vertical spectra was taken from analyses of well logs, the horizontal spectra from the typical Brown noise spectra of topography. The phases in the threedimensional spectral expansion were constrained from well-log data; accurate and realistic sedimentary structures were found. These results provide further evidence that sedimentary structures obey fractal statistics. An interesting possibility is that the statistics

of sedimentary layers can be the statistics of floods. If this provide an improved basis assessments. It may also be paleoclimates in this way.

directly related to can be done it will for flood hazard possible to study

5. Conclusions

Observations that fractal statistics are applicable to a variety of geomorphic processes have many implications. Empirically such observations can contribute for example to estimates of the flood hazard and to interpolate sedimentary structures using well-log data. But more important, the observations provide important clues to the underlying physical processes. Progress is being made in understanding the development of drainage networks and their associated topography. However, a successful model for the erosional evolution of topography remains an elusive goal. Progress is also being made in the understanding of the coupled processes of sediment erosion, transport and deposition. If the frequency-magnitude statistics of floods are fractal this has important implications for hazard mitigation. The frequency-thickness statistics of sedimentary layers may also be fractal under a variety of circumstances. A testable hypothesis is

D.L. Turcotte / Physica D 77 (1994) 229-237

t h a t flood statistics can be d e r i v e d from sedim e n t a r y layer statistics.

References [1] B. Mandelbrot, Science 156 (1967) 636. [2] D.L. Turcotte, Fractais and Chaos in Geology and Geophysics (Cambridge Univ. Press, Cambridge, 1992). [3] R.H. Rapp, Geophys. J. Int. 99 (1989) 449. [4] A.N. Strahler, Trans. Am. Geophys. Un. 38 (1957) 913. [5] R.E. Horton, Geol. Soc. Am. Bull. 56 (1945) 275. [6] J.C. Maxwell, Project NR 389-042, Technical Report 19, Columbia University, Dept. of Geology, 1960. [7] L.B. Leopold, M.G. Wolman and J.P. Miller, Fluvial Processes in Geomorphology (Freeman, San Francisco, 1964). [8] L.B. Leopold and W.B. Langbein, US Geol. Survey Prof Paper 500A, 1962. [9] H. Schenck, J. Geophys. Res. 68 (1963) 5739. [10] R.L. Shreve, J. Geol. 74 (1966) 17. [11] R.L. Shreve, J. Geol. 75 (1967) 178. [12] A.E. Scheidegger, Int. Assoc. Sci. Hydrol. Bull. 12 (1967) 15. [13] J.S. Smart and A.J. Surkan, Water Resour. Res. 3 (1967) 963. [14] I. Seginer, Water Resour. Res. 5 (1969) 591. [15] A.D. Howard, Geol. Soc. Am. Bull. 82 (1971) 1355, [16] J.S. Smart, Adv. Hydrosci. 8 (1972) 305. [17] H. Kondoh and M. Matsushita, J. Phys. Soc. Japan 55 (1986) 3289. [18] P. Meakin, J. Feder and T. Jossang, Physica A 176 (1991) 409. [19] J.G. Masek and D.L. Turcotte, Earth Planet. Sci. Lett. 119 (1993) 379.

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[20] C. Stark, Nature 352 (1991) 423. [21] C.G. Chase, Geomorphology 5 (1992) 39. [22] H. Inaoka and H. Takayasu, Phys. Rev. E 47 (1993) 899. [23] G. Willgoose, R.L. Bras and I. Rodriguez-Iturbe, Water Resour. Res. 27 (1991) 1671. [24] S. Kramer and M. Marder, Phys. Rev. Lett, 68 (1992) 205. [25] S.A. Schumm, M.P. Mosley and W.E. Weaver, Experimental Fluvial Geomorphology (Wiley, New York, 1987). [26] D.L. Turcotte and L. Green, Stochastic Hydrol. Hydraul. 7 (1993) 33. [27] A.R. Green, US Geol. Survey, Water Supply Paper 1671 (1964). [28] I.E. Young, and R.W. Cruff, US Geol. Survey, Water Supply Paper 1685 (1967). [29] M.A. Benson, Water Resour. Res. 4 (1968) 891, [30] D.H. Rothman, J. Grotzinger and P. Flemings, J. Sed. Petrol. A 64 (1994) 59. [31] R.N. Hiscott, A Colella, P. Pezard, M.A. Lovell and A. Malinverno, Proc. Ocean Dril. Program Sci. Results 126 (1992) 75. [32] H.H. Stolum, in: Reservoir Characterization II, L.W. Lake, H.B. Carroll and T.C. Wesson, eds., Vol. 579 (Academic Press, San Diego, 1991). [33] B.D. Malamud and D.L. Turcotte, Geol. Soc. Am. Abstracts Programs 24, 7, A357, 1992. [34] T.A. Hewett, Society of Petroleum Engineers, Paper 15386, 1986. [35] J.P. Todoeschuck and O.G. Jensen, Geophys. 53 (1988) 1410. [36] J.P. Todoeschuck, O.G. Jensen and S. Labonte, Geophys. 55 (1990) 480.