ALAN D. HOWARD Department of Environmental Sciences, University of Virginia, Charlottesville, Virginia 22903 Simulation Model of Stream Capture ABSTRACT ture should have its greatest relative impor- Because the importance of the process of cap- tance in early stages of drainage basin ture, or piracy, in the formation of stream net- evolution. works is difficult to evaluate by field or map studies, an indirect approach is used in this pa- INTRODUCTION per to investigate capture, through the use of a Many features of stream networks have been simulation model involving capture within rec- closely predicted by theoretical and simulation tangular stream networks on a square matrix. models involving random processes (for exam- The simulation rules make the probability of ple, the topological theory of Shreve, 1966, capture of a stream by a lower adjacent stream 1967, 1969, and the simulation models of Leo- proportional to the advantage in gradient of the pold and Langbein, 1962; Schenck, 1963; potential path of capture between the streams Smart and others, 1967; Howard, 1971). The compared to the present gradient of the higher success of random models in predicting stream stream. Stream elevations are assumed to be topology and order-ratio statistics probably defined by the same type of pattern observed in arises because of the multiplicity of causes in- natural stream networks, that is, a linear rela- volved in the development of natural stream tionship between the logarithms of gradient networks (Krumbein and Shreve, 1970, p. 40; and drainage area. The slope of this relation- Howard, 1971). ship, Z, is variable in nature and is the main Observed deviations of the numerical prop- adjustable parameter in the simulation model. erties of stream networks from present random Simulation of capture must start from assigned theories and simulations (Smart and others, initial network patterns; random walk networks 1967; Smart, 1969, p. 1770-1771; James and and parallel drainage are among those used for Krumbein, 1969, p. 550-551; Krumbein and initial networks. Shreve, 1970, p. 78) may be due to systematic For a given value of Z, the statistical proper- influences on the development of stream net- ties of networks (for example, stream numbers, works. Capture, or piracy, could be one such length and area ratios, and shape factors) influence. formed after repeated captures are nearly the Although numerous instances of inferred same for a wide range of assigned initial net- past or impending capture are common in the works. However, when the value of Z changes geologic literature, their numbers are small during capture, the statistical properties of the compared to the total number of streams resultant networks may depend upon the type (Small, 1970, p. 236), and most of these cap- of change, so that properties may be partially tures are due to structural and stratigraphic inherited from earlier stages of basin evolution. causes (for example, the drainage diversions Both the networks simulated by capture and along the Catskill Escarpment, New York). natural networks have similar slight deviations The rarity of present captures is due to the from topological randomness. The capture rather direct courses that most streams follow simulations more closely predict many proper- from their origin to their junction with another ties of natural networks than do completely ran- stream or their termination, coupled with the dom methods of simulation, such as the random large junction angles between streams (adja- walk. In addition, several parameters in the cap- cent streams are seldom closely parallel, and ture-simulated networks exhibit a consistent large streams are separated from one another trend with respect to the parameter Z that ap- by low-order streams; see Lubowe, 1964). But pears to occur also in natural networks. These the rarity of captures in long-established net- correspondences between the capture model works may be due, in part, to frequent capture and natural networks suggest that capture may early in the development of the drainage net- be an important natural process. However, cap- work when indirect courses, determined by ir- Geological Society of America Bulletin, v. 82, p. 1355-1376, 6 figs., May 1971 1355 Downloaded from http://pubs.geoscienceworld.org/gsa/gsabulletin/article-pdf/82/5/1355/3433062/i0016-7606-82-5-1355.pdf by guest on 27 September 2021 1356 A.D. HOWARD—SIMULATION MODEL OF STREAM CAPTURE regularities on the original surface, were lated). The restrictions to four flow directions straightened by capture, or an originally paral- and the assumption of uniform drainage density lel drainage net on a sloping surface was con- have been used in previous simulation models verted into a dendritic network by abstraction (Leopold and Langbein, 1962; Schenck, 1963; (Gilbert, 1877, p. 141; Horton, 1945, p. 333- Smart and others, 1967; Howard, 1971). 349; Small, 1970, p. 242). In either case the numerical properties of these networks may have been influenced by capture. Probabilistic Modelling of Capture The importance of capture in natural stream In natural stream networks, capture occurs networks is difficult to evaluate through field or through a large number of processes, for exam- map study because of the slowness with which ple, by headward erosion and reduction of di- stream networks evolve and the subtlety of the vides, by subterranean capture or abstraction, evidence for past captures (Small, 1970, p. 236- and by breaching of a divide by a meandering 250). Therefore, an indirect approach is used stream (Crosby, 1937; Thornbury, 1969, p. here. First, a simulation model of stream cap- 147-154; Gilbert, 1877, p. 141; Lauder, 1968; ture is developed, and the effects of the parame- Small, 1970, p. 236-250). All of these pro- ters of the model upon the drainage basin cesses produce discrete capture, for part of a properties are examined. The main parameter stream network changes its point of entry into of the model is the exponent of proportionality the rest of the network (or becomes tributary to between stream gradients and the correspond- a different network). Although these processes ing drainage area. This parameter also can be differ in detail and will vary in importance with measured in natural stream networks, so that relief and rock type, in each case the path of the correlation between variations in this capture is steeper than the original stream parameter and changes in stream network prop- course. At the immediate site of potential cap- erties can be compared in natural and simulated ture of one stream (the captive) by another (the streams. captor), three situations may occur ("captor" and "captive" are adopted from Gresswell, THE CAPTURE MODEL 1967, p. 211): 1. Capture is impossible: The gradient of the The computer model simulates the evolution potential path of capture is uphill (negative). of drainage basins by successive captures of one 2. Capture is disadvantageous: The gradient of stream by another, starting from an assigned the captive is greater than that along the poten- initial drainage pattern. The probability of cap- tial path of capture. ture is determined by a function which depends 3. Capture is advantageous: The gradient of upon the gradient relationships at the potential the captive is less than that along the potential site of capture. path of capture. In natural stream networks capture would be Model Network expected if, and only if, capture were advanta- The drainage network is represented by geous. The ratio R of the gradient along the stream segments ordered on a 40 X 40 matrix. potential path of capture Gc to that of the cap- One stream segment originates from each posi- tive at that point Gd conveniently measures the tion in the interior of the matrix (a total of 382 possibility of capture, for capture is impossible positions). Matrix points on the four edges if R is negative, it is disadvantageous if R lies are drainage exits; that is, no streams originate between zero and unity, and the potential path from these points, and all streams terminate at of capture becomes increasingly advantageous one of these locations. The stream segments are as R ranges above unity. constrained to flow either east, north, south, or A probabilistic approach to capture is neces- west from each interior position. Because of sary in the simulations because interstream these restrictions, all networks have an equal slopes are not modelled explicitly. In natural total length of streams (1444 units). Figure 1 streams the breaching of a divide, on the sur- shows typical drainage patterns developed on face or subterraneously, is a prerequisite to cap- this matrix. For simplicity of calculations each ture. The model assumes that, within a natural stream segment is assumed to receive drainage stream system in a region of fairly uniform from one unit area of surrounding slope (that drainage density, divide relief, and geology, is, a uniform drainage density of unity is postu- the frequency of discrete capture is an increas- Downloaded from http://pubs.geoscienceworld.org/gsa/gsabulletin/article-pdf/82/5/1355/3433062/i0016-7606-82-5-1355.pdf by guest on 27 September 2021 -.!«J S = A Afte;- Capture Contour Interval 2.0 -t S=A™ After Capture Contour Interval 1.0 Headword Growth A-II-A S = A™ After Capture Contour Interval LO -1.2 After Capture Contour Interval 0.5 Figure 1. Initial stream matrices and changes pro- nations), and curved lines are contours. Only those net- duced by capture. Rectangular connected lines are works which are tributary to the drainage exits on the stream channels, circles are drainage exits (stream termi- right half of the lower matrix edge are illustrated. Downloaded from http://pubs.geoscienceworld.org/gsa/gsabulletin/article-pdf/82/5/1355/3433062/i0016-7606-82-5-1355.pdf by guest on 27 September 2021 1358 A.D.
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