Simulation Model of Stream Capture

Home , Stream capture

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

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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-

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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.

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ing function of the ratio R' of the average be assigned initially, and assumptions must be gradient between the adjacent streams Gc' to made about the evolution of these elevations as the downstream gradient of the captive G

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correspond to natural networks in which the posed of the 38 parallel streams formed by set- captures occur infrequently enough that almost ting all interior segments to flow southward complete regrading (due to addition or with- (abbreviated PAR in Fig. 3). drawal of drainage area) occurs between cap- Process of Capture tures. Regrading generally increases the elevations Because the capture model is unavoidably of beheaded channels relative to the drainage complex, the model rationale follows an exposi- exits (the gradient increases due to loss of tion of the simulation rules. Briefly stated, cap- drainage area). Such an increase of relative ele- ture occurs according to the following scheme: vation might occur in natural networks without a point in the interior of the matrix (a potential aggradation if the average amount of erosion captive) is selected at random, and its capture within the network between captures were by one of the surrounding streams occurs (or greater than the relative changes in elevation fails to occur) with a probability proportional to occasioned by capture. the function /"defined above. Additional matrix Thus the assumptions regarding elevation points are examined, and occasional captures changes limit the model to analogy with infre- occur, until the process is terminated. Specifi- quent captures within eroding natural basins cally, the process proceeds as follows: characterized by constant parameters K and Z. 1. All interior matrix locations (potential cap- Few natural areas are likely to have had a corre- tives) are examined in a random order, without sponding history of infrequent captures during replacement, for the possibility of capture. continuous erosion in homogeneous rock. After this examination, if capture is to continue, However, if the process of capture in natural a new random listing of the matrix points is streams is equifinal, so that the pattern of drain- constructed and examined in sequence. Thus age networks continuously adjusts toward a capture occurs in generations. unique equilibrium with the gradient relation- 2. Having selected a potential captive, the ships, then the capture model can be used to probability of capture of the stream by the four investigate equilibrium drainage patterns in surrounding matrix locations to the east, west, natural networks, if the other assumptions of north, and south (potential captors) is eval- the model are reasonable. The question of uated by the function P. This probability is, of equifinality is discussed below (seeEffects of Ini- course, zero if the stream now flows directly to tial Network Configuration). the potential captor. Junction of more than two streams at a single matrix location was made Initial Networks illegal in order to be able to unequivocally esti- Because the capture model involves modifi- mate the order of branching within the net- cations within pre-existing networks, the initial work; therefore, the probability of capture by a configuration of the network on the matrix potential captor is equal to zero if the captor must be assigned. A wide range of initial drain- already receives two incoming streams. age patterns was assumed in order to evaluate 3. If none of the four probabilities of capture the effects of initial conditions upon subsequent is greater than zero, the next matrix location in capture. the random listing is examined. If only one path Most capture simulations started from sample of capture with nonzero probability exists, it is networks generated by one of three random examined for the occurrence of capture (next simulation models: the random walk model step); if two potential paths of capture exist, (Smart and others, 1967) and the A-II-A and then one of them is selected at random. A-II (P = 0.5) head ward growth models 4. A random number from a uniform distri- (Howard, 1971). The statistical characteristics bution between zero and unity is generated and of all three models are summarized in Howard compared with the probability of capture; if the (1971). Some properties of these networks are random number is greater than the probability indicated in Figure 3, with the processes of gen- of capture, then no capture occurs, and a new eration abbreviated by RW, A-II-A, and A-II, matrix point is selected. respectively. In the capture simulations investi- 5. If the random number is less than the prob- gating the effects of the parameter Z, one of ability of capture, the direction of the captive each of these random networks was used as an stream segment is changed to drain into the initial network, and, in addition, one simulation captor location; the drainage area and elevation at each value of Z started from a network com- of the affected parts of the network are adjusted

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Figure 3. Average dimensionless properties of capture-generated and natural networks plotted versus the area-gradient exponent, Z. Average properties for double episodes of capture (.see text) are plotted at the final value of Z. Also shown are the average and 95-percent confidence limits for the population mean for several types of random network models. Each point for * 8 natural and capture-generated networks is the average of about 120 second-order networks or 25 third-order networks, but 95- percent confidence limits are omitted for clarity. Horizontal dashed lines show the expected value for an infinite topologically random channel network. The probability parameter, W, has a value of 0.5 for all capture simulations pictured here. -.2 -4 -.6 -8 -10 Z

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to compliance with equation 2; another matrix as random sampling with replacement. In ran- location is selected, and the process is con- dom sampling with replacement, some matrix tinued until capture is terminated when only a points would be sampled several times before fixed number of interior matrix locations are others would be sampled once; therefore, a ran- sites of potential capture, or when an arbitrary dom component would be introduced into the number of captures is performed. model which would not depend upon the gradi- In all simulations in which the exponent Z ent relationships within the basin. In the (equation 2) was negative, the number of cap- adopted sampling procedure, the temporal ran- tures within each generation decreased with dom component within the model resides al- successive generations, on the average. If the most solely in the probability function process of capture were continued indefinitely (although the stochastic sampling within gener- in such cases, the number of possible captures ations introduces desired randomness in areal would reach zero, but the last few captures, sampling which minimizes directional biases occurring in situations with low probability, and interactions). would consume much computer time. Because MEASURED PARAMETERS the percentage change of network properties that would be produced by these few captures All streams, natural and simulated, were is small, the process was terminated when cap- analyzed according to the Horton-Strahler ture remained possible at only 80 of the 1444 method of stream ordering (Strahler, 1952, p. interior matrix locations (about 5.5 percent). 1120), and information on stream numbers, Termination before completion of all possible stream lengths, and drainage areas were tabu- captures is likely to correspond to the scattered lated by order. The portion of these data that occurrences of impending capture in ancient are presented here is determined by two con- natural networks. Fifteen to twenty generations siderations. were usually required for termination. Firstly, the simulated networks can be com- Re-evaluation of the drainage area and eleva- pared with natural networks only by enumera- tion of each location which would be affected tive or dimensionless properties. Hence only by a capture (using equation 2) requires consid- such parameters as stream numbers, length, and erable computer time due to the difficulty of area ratios are discussed here. identifying affected portions of the captor and Secondly, comparisons between natural and captive basins. Therefore, the drainage areas simulated networks of these dimensionless and and elevations of the entire matrix are re- enumerative parameters for basins of a given evaluated only after one-fifth of all possible cap- order are reasonable only if the simulated net- tures within the matrix actually occur. works are sufficiently large that the size of ba- Because drainage area and elevations are not sins of that order is not limited by the evaluated after every capture, a correlation is dimensions of the matrix. For example, basins introduced between the probability of succes- of order 4 or greater may be unrepresentatively sive captures. In most cases this correlation is small, or missing, on stream networks gene- small because the captures occur in headwater rated on a 40 X 40 matrix. However, this size areas or involve only channel straightening, re- of matrix appears to be sufficiently large so that sulting in only small additions or deletions of the average statistical properties of second- and drainage area to affected basins; therefore, third-order simulated basins are nearly equal to changes of gradient are small, except in the those of the same order generated on a very immediate vicinity of the capture. Correspond- large (nearly infinite) matrix by the same proc- ingly, captures might occur frequently enough ess: in rapidly evolving natural drainage networks 1. Simulated networks developed by purely so that complete regrading does not occur be- random processes on a 40 X 40 matrix (for tween captures. example, by random walk or headward The method of sampling of interior matrix growth) are close to being topologically ran- points assumed in this model (that is, random dom (defined in Shreve, 1966, p. 27; see tests sampling without replacement during succes- for randomness in Howard, 1971). In addition, sive generations) more closely approximates the average values of topological properties of the natural process, in which every stream loca- all second- and third-order basins formed by tion is continuously subject to possible capture, these random processes on a 40 X 40 matrix are than would a simpler sampling procedure, such very close to the expected values in infinitely

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large topologically random networks (defined in Shreve, 1967, p. 178; tests in Howard, 1971). Similarly, the total numbers of first-, second-, and third-order streams formed by ran- N,2 dom processes on a 40 X 40 matrix decrease by a factor of about 4 between order as expected in infinite topologically random networks (Shreve, 1967, p. 182; Howard, 1971). 15 20 „ 25 30 35 X4 2. The average topological properties of second- and third-order basins generated by the capture process on a 40 X 40 matrix, in general, deviate from expected values for infinite topologically random networks in proportion to the chi-square goodness-of-fit statistics from tests for topological randomness on the same matrices (Fig. 4). 3. The average topological properties of second- and third-order basins formed by the cap- 0 5 10 15 20 , 25 30 35 40 ture model deviate from those expected for infinite topologically random networks by having an excess of contributing lower order streams. If the second- or third-order basins were limited in size by the dimensions of the matrix, an average deficiency of contributing

streams would be likely (and actually occurs for 40 - statistics of fourth- and fifth-order streams). 20 - 4. An increase in matrix size to 50 X 50 in randomly generated networks (increasing the number of stream segments by about 1.6 times) has little effect upon the topological and dimen- Figure 4. Relationships between chi-square statistics sionless ratio properties of second- and third- showing departure from topological randomness (hori- order networks. zontal axis) and departures from expected values for properties of infinite topologically random networks Therefore, all second- and third-order basins (vertical axis) for networks generated on a 40 x 40 ma- generated by capture on the 40 X 40 matrix trix. Each circle shows values for all appropriate net- were sampled, and the average values of several works generated at a single value of Z in enumerative and dimensionless properties capture-generated networks, while plus-signs indicate values for randomly generated networks (RC, RW, A-II, were calculated. The following notations were or A-II-A models). Dashed lines show the expected val- used: ues for infinite topologically random networks, while dotted lines show critical values in chi-square tests. The Njk Average number of jth-order streams in subscripts for chi-square tests for topological random- all kth-order networks developed on the 40 X ness indicate the number of sources in tested networks 40 matrix. (4-6). The Xn statistic results from tests for the expected ratio of 4 between total numbers of stream net- Njk/N(j+ok The average, for all kth-order works of successive orders in an infinite topologically networks on the matrix, of the ratio of the num- random network. This test was applied to the total number of jth-order streams to the number of (j + ber of first-, second-, and third-order networks generated on 40 x 40 matrices. Scatter in the diagram is produced l)-order streams in each network (the ratio is primarily by two factors: (1) the topological tests and calculated within each network, and then aver- values plotted on the two axes are derived from different aged between networks). populations within the same matrices (for example, networks with 6 sources and third-order basins); (2) the L

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taken, and the average ratio is calculated be- was investigated by subjecting four types of ini- tween networks.) tial networks to capture at several values of Z. The probability parameter was fixed at a value A j i)k/Ajk The average ratio of average ( + of 0.5. contributing drainage areas, as with length, Values of Z greater than zero (gradient in- above. creasing downstream) produce unstable net- The average of the ratio of the aver- works in which the number of captures per age length of exterior links (first-order streams) generation reaches a steady state, because cap- to the average length of interior links (portions ture increases the relative elevation of the cap- of streams between junctions) in all kth-order tor at the point of capture, making recapture by networks. Statistics are presented for third- adjacent streams with smaller drainage areas order networks only. advantageous. When Z equals zero, capture eventually S.D. Stream directness, that is, the ratio of (1) forms straight, unbranched streams leading the length along a stream from the farthest from symmetrical divides, because only the point of the network divide to the mouth of the length of flow, and not the size of the stream, network to (2) the straight-line distance. This determines the elevation differences within the parameter, measured only on third-order net- network. Indirect courses, therefore, are works, is similar to sinuosity, except that it con- straightened. siders larger scale wanderings, that is, the For Z less than zero (as in natural stream "straight-line distance" does not follow the networks), gradient decreases downstream. general valley trend as is done when measuring Such stream networks exhibit an "economy of sinuosity. scale": addition of new drainage area by cap- %T.J. Percentage of trans junctions, that is, the ture results in a general lowering within the percentage frequency of occurrence, when fol- network after re-evaluation of elevations. The lowing the path of greatest magnitude up- portion of the stream network beheaded by stream, of the entrance of successive tributaries capture is concomitantly raised in relative ele- from opposite sides of the main channel (James vation, and, therefore, is susceptible to further and Krumbein, 1969, p. 547). Measurements capture, reinforcing the tendency for the con- were continued upstream to the beginning of centration of drainage into a few large basins. second-order streams, whereas James and Generally, capture with negative Z forms Krumbein (1969, p. 549) terminated their branched networks. measurements upon reaching magnitude 10. However, the geometry of the generated networks varies with the value of Z (Fig. 1), Kk The average, for all kth-order networks, although in all cases capture generates a stream of the total number of links (exterior and in- system with few strong interstream gradients. terior) in the network times the drainage area For slightly negative Z, stream networks are of the network, divided by the square of the elongate and nearly parallel, and the drainage total length of channel within the network (after patterns are fairly regular and symmetric. Net- Shreve, 1967, p. 185). Measured only on third- works are more pear shaped, or irregular, with order networks. more negative Z. Greatest modification of the original drainage pattern occurs for the less F.R. The form ratio is defined as the ratio of negative values of Z; this is reflected both in the the total network area to the square of the dis- degree of alteration (Fig. 1) and in the total tance from the network mouth to the farthest number of captures (Fig. 5). point of the divide (after Horton, 1932, p. 351), and gives a relative measure of the com- The average dimensionless network proper- pactness of the drainage basin. This ratio was ties generally show strong dependence upon measured and averaged for all third-order net- the value of Z prevailing during capture (Fig. works. 3). Many of the relationships are not mono- tonic; several show a maximum or a minimum near a Z of-1/2. The reasons for most of the RESULTS observed functional relationships are uncertain; deduction of these relationships from the basic Effects of Model Parameters premises of the model probably would be hope- The effect of the area-gradient exponent Z lessly complex in view of the three-dimensional

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structure of the drainage networks and the in- capture-generated networks differing only in clusion of stochastic processes. In general, initial source network often show greater varia- properties involving first- and second-order tion the more negative the value of Z, reflecting streams (for example, Ni2, L23/Li3) show greater inheritance of features from the origi- stronger variation with Z than do relationships nal network (Fig. 3). Finally, the number of between second- and third-order streams (for captures during the simulation decrease as Z example, A33/A23)- becomes more negative (Fig. 5). The effect of variation in the probability As might be expected, the initial networks parameter ( W in equation 1) upon network which are developed by random processes (ran- properties was investigated at three values of dom walk and headward growth models) and Z(-.15, -.3, -.6). The effects of increasing W simple geometrical models like the parallel from 0.5 to 30 upon network properties is gen- drainage, in general, are modified extensively erally slight or nonsystematic (Fig. 6). Because by capture, because the gradient relationships fewer advantageous captures are bypassed at within the network play no part in their forma- higher values of W, future simulation models tion. The headward growth networks, how- might employ high values of the probability ever, are less modified by capture than are the parameter, or, in the limiting case, automati- random walk models, for they are already cally allow any advantageous capture to occur highly symmetrical and regular (Figs. 1 and 5). (subject to the other restrictions of the model, The degree of inheritance was further inves- such as the two-junction rule). tigated by subjecting a single network to two successive episodes of capture at different val- Effects of Initial Network Configuration ues of Z. The four networks subjected to cap- Of considerable importance in the interpreta- ture at Z of -.6 were further modified at Z of tion of stream patterns is the degree to which -.15. Similarly the four networks which were the process of capture destroys the initial struc- formed by capture at Z of -. 15 were subse- ture of the network, that is, the lack or presence quently subjected to capture at Z of -.6. The of inherited features in capture-generated net- statistical properties of the resulting networks works. The degree of inheritance depends both after the double episodes of capture are shown upon the geometry of the initial network and in Figure 3 at the final value of Z. the value of the exponent Z The simulations with a sudden change in the As noted previously, courses of main streams value of Z within the network are not proposed and divides remain only slightly changed for to be realistic representations of erosional his- highly negative Z, but are greatly altered for tory; changes in Z within natural networks less negative values. Statistical properties of would be gradual. However, in order to simu- late capture with a gradual change in Z, the * Initial Networks: simulation rules must specify the rate at which 1400 • RW capture occurs relative to the rate at which Z ' . • PAR " * A-ll changes. The step change investigated here is EDO T A-II-A • one end of this spectrum; the sole purpose of these simulations is to investigate further the 1000 • importance of inherited network features in the capture model. T • 800 Stream networks first subject to capture at No of Captures i negative values of Z near zero are only slightly

600 • modified if Z becomes more negative, due to T low probabilities of capture within the network ' (Fig. 3). However, if the value of Zis changed from more to less negative, the statistical properties of the networks are altered to values close to those resulting from a single episode of capture of the initial network at the final value of Z, due to a large number of additional captures. Z This failure of symmetry and lack of equifinality Figure 5. Number of captures occurring during simulations on 40 x 40 matrices. The various symbols (independence from initial conditions) is one indicate the type of initial networks. of the most striking results of the capture simu-

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Figure 6. Effects of increase of the probability parameter, W, upon average network properties. Connected points start from the same initial network: solid lines connect networks formed by capture from RW network with the area-gradient exponent, Z at —.15, dotted lines connect captures from A-II network at a Z of —.3, and dashed lines connect captures from 4.2 RW network at a Z of —.6.

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counted for in the simulation model, such as of observed over predicted number of net- structure, lithologic differences, microclimatic works in second-order classes (Table 1). How- variations, or geomorphic processes other than ever, natural and simulated networks capture; (4) if the natural networks show a simi- correspond less closely in the relative abun- lar inheritance of properties from earlier stages dances of third-order networks with 5 and 6 in drainage basin evolution as occurs in the first-order tributaries. By comparison, the ran- simulation model, then differences in the past domly generated networks (A-II, A-II-A, and evolution of relief within the basin would pro- RW) approach much closer to topological ran- duce variation in the statistical properties plot- domness in the same tests (Howard, 1971, Ta- ted in Figure 3 among networks with the same ble 4). present value of Z. The natural networks sampled in the present The degree of explanation of natural stream study appear to deviate more strongly from networks provided by the capture simulations is topological randomness than do samples re- best evaluated by comparison with theoretical ported by other authors, possibly due to unin- and simulation models involving random pro- tentional biases in sampling or measuring or to cesses, for example, the random topology the- the selection rules discussed above. For exam- ory of Shreve (1966, 1967, 1969) and the ple, the large number of basins sampled by random walk and headward growth simulation Smart (1969) closely approach topological ran- models. domness. Because Smart's basins were not al- The capture-generated networks more ways selected within areas of uniform drainage closely predict the dimensionless and enumera- pattern, inhomogeneities of bedrock and struc- tive properties of second- and third-order natu- ture may have considerably affected stream ral networks within their common range of Z patterns, thus introducing random (unac- than do the randomly generated networks. This counted-for) variation in network topology is especially clear in ^he case of the following which would mask any effects of systematic network properties: KS, S.D., L2j/Li3, Ni3, process, such as capture. and Le3/L;3 (Fig. 3). However, the randomly On the other hand, Krumbein and Shreve simulated networks are more successful than (1970, p. 78) narrowly accepted the hypothesis the capture simulations in predicting the %TJ. of topological randomness in a goodness-of-fit statistic. test by ambilateral classes for 153 basins of Because many of the dimensionless proper- magnitude 5 in an area of fairly uniform li- ties of the capture-simulated networks exhibit a thology in eastern Kentucky (goodness-of-fit maximum or minimum at Z of about -1/2, statistic of 5.9 with a 95 percent critical value these properties show little correlation with re- of 6.0). A sample of size comparable to that spect to Z within the range of Z of natural net- used in the similar test in the present paper works. The corresponding properties of natural (370 basins, Table 1) might have resulted in networks likewise exhibit little systematic trend rejection. with Z. However, for those dimensionless Additionally, the average magnitude of 90 properties of the capture simulations which do third-order basins in the Krumbein and Shreve show a clear trend in this range of Z(%TJ., study (1970, Table 2.2 gives Ni3 as 13.7) is F.R., KS, and S.D.), the natural networks ap- considerably greater than the expected value of pear to follow a similar trend (neither the the- 11 for an infinite topologically random net- ory of topological randomness nor the random work, and is similar to average values observed models of generation would predict systematic in the present study (grand mean for 342 third- relationships between network properties and order basins is 13.4). the parameter Z). One intriguing feature of the capture- The natural and capture-generated networks generated networks is the close approach to can be tested against the hypothesis of topologi- randomness in the narrow range of Znear -1/2 cal randomness by tests of goodness-of-fit to the (Table 1). In this same range many basin prop- predicted frequency of topological or ambilat- erties reach a maximum or minimum (Fig. 3), eral classes (Smart, 1969) for networks with 4, and the enumerative properties of second- and 5, and 6 first-order tributaries (Table 1). Both third-order basins most closely approach those the natural and capture-generated networks in of the random models and of the expected val- the same range of Zfail these tests for topologi- ues for infinite topologically random networks. cal randomness, and both show a general excess The deviations of the properties of capture-

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TABLE 1. CHI-SQUARE TESTS FOR TOPOLOGICAL RANDOMNESS OF NATURAL AND CAPTURE-GENERATED NETWORKS WITH 4, 5, OR 6 SOURCES*

4 Sources by Topological Classes X N /

Capture-Generated Networks No. Observed 43 33 16 21 5 118t Z = - .15 No. Expected 23.6 23.6 23.6 23.6 23.6 118 Chi-Square 15.95t 3.74 2.45 .29 14.66 37.08 (9.49) § No. Observed 35 39 20 35 15 144 7. — - .3 No. Expected 28.8 28.8 28.8 28.8 28.8 144 Chi-Square 1.33 3.61 2.69 1.33 6.61 15.58 (9.49) No. Observed 46 35 40 33 22 176 Z = - .45 No. Expected 35.2 35.2 35.2 35.2 35.2 176 Chi-Square 3.31 .00 ^65 .14 4.95 9.06 (9.49) No. Observed 36 37 31 36 18 158 Z = - .6 No. Expected 31.6 31.6 31.6 31.6 31.6 158 Chi-Square .61 .92 .33 .61 5.85 8.01 (9.49) No. Observed 46 31 43 43 14 177 Z = - .9 No. Expected 35.4 35.4 35.4 35.4 35.4 177 Chi-Square 3.17 .55 1.63 1.63 12.94 19.92 (9.49) No. Observed 37 28 45 33 11 154 Z = -1.2 No. Expected 30.8 30.8 30.8 30.8 30.8 154 Chi-Square 1.25 .25 6.55 .16 12.73 20.93 (9.49) Z = - .3 No. Observed 117 111 91 104 55 478 to -.6, No. Expected 95.6 95.6 95.6 95.6 95.6 478 lumped. II Chi-Square 4.79 2.48 .22 .74 17.24 25.47 (9.49) Natural Networks No. Observed 119 96 118 122 78 533 Z = - .44 No. Expected 106.6 106.6 106.6 106.6 106.6 533 Chi-Square 1.44 1.05 1.22 2.22 7.67 12.16 (9.49)

generated networks from topological random- be captured by the adjacent stream in the ness and from randomly generated networks is chosen direction, unless a loop would be due to the restrictions placed upon the occur- formed in the resulting network. A new matrix rence of capture, that is, the provision that cap- location is selected, and the process is con- ture occurs only under advantageous tinued until the visual appearance of the origi- conditions. If this condition is relaxed, so that nal network is greatly altered (about 10,000 captures occur randomly within the network captures on a 40 X 40 matrix). The resulting irrespective of the gradient relationships at the networks resemble those generated by the ran- site of capture, the statistical properties of the dom walk process. generated networks (random capture networks, abbreviated RC in Fig. 3) are similar to CONCLUSIONS those produced by other methods of random Many of the dimensionless properties of simulation. natural streams are better simulated by the cap- Random capture networks are formed by ture model introduced in this paper than by modifying an arbitrary existing network by random models, suggesting that capture pro- selecting a matrix location in the interior of the cesses may be important in stream network matrix at random. One of the two possible di- development, especially in the early stages. rections of capture is selected at random, and However, the evidence is insufficient to be con- the stream at the selected location is allowed to sidered proof for several reasons:

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TABLE 1. (continued)

5 Sources by Ambilateral Classes

Capture-Generated Networks No. Observed 74 14 2 90 Z = - .15 No. Expected 51.43 25.71 12.85 90 Chi-Square 9.91 5.34 9.16 24.41 (5.99) No. Observed 76 17 9 102 Z = - .3 No. Expected 58.28 29.14 14.57 102 Chi-Square 5.39 5.06 2.13 12.57 (5.99) No. Observed 75 22 11 108 2 = - .45 No. Expected 61.71 30.86 15.42 108 Chi-Square 2.86 2.54 1.27 6.67 (5.99) No. Observed 74 29 10 113 Z-- .6 No. Expected 64.57 32.28 16.14 113 Chi-Square 1.38 .33 2.33 4.05 (5.99) No. Observed 87 13 10 110 Z = - .9 No. Expected 62.85 31.43 15.71 110 Chi-Square 9.28 10.80 2.07 22.15 (5.99) No. Observed 89 13 8 110 Z = -1.2 No. Expected 62.85 31.43 15.71 110 Chi-Square 10.88 10.80 3.78 25.43 (5.99) Z = - .3 No. Observed 225 68 30 323 to -.6, No. Expected 184.56 92.28 46.12 323 lumped. II Chi-Square 8.86 6.39 5.64 20.89 (5.99) Natural Networks No. Observed 241 82 47 370 Z = - .44 No. Expected 211.4 105.7 52.9 370 Chi-Square 4.14 5.32 .64 10.10 (5.99)

1. The degree of improvement in explana- ACKNOWLEDGMENTS tion of natural network properties provided by This study was supported by a National the capture model over random models is gen- Science Foundation Institutional Grant at the erally slight enough that the improvement University of Virginia. Early studies were sup- might have arisen from random factors in selec- ported by the Geography Department at the tion of natural networks. Johns Hopkins University. The author wishes 2. Systematic processes other than capture to acknowledge the helpful comments by R. L. in natural networks might produce the ob- Shreve and J. S. Smart. The tireless efforts of served deviations of the stream properties from James Cole in analysis of natural stream net- random models. One such process might be works are appreciated greatly. modification of junction angles (Schumm, 1956, p. 617-620). 3. Correlations of capture-generated net- REFERENCES CITED work properties with the exponent Zmay result from the geometrical structure of the model Crosby, I. B. Methods of stream piracy: J. Geol., Vol. 45, p. 465-486, 1937. rather than from capture related assumptions, Gilbert, G. K. Report on the geology of the Henry for example, from the equidistant spacing of Mountains, Utah: U.S. Geol. Survey of the streams or from the direction-of-flow restric- Rocky Mountains Region (Powell), 160 p., tions. 1877.

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TABLE 1. (continued)

6 Sources bv Ambikteral Classes

Capture-Generated Networks No. Observed 52 8 12 1 2 1 76 Z = - .15 No. Expected 28 95 14.47 14.47 7.23 7.23 3.62 76 Chi-Square 18 36 2.89 .42 5.37 3.79 1.89 32.73 (11.07) No. Observed 43 12 12 5 7 2 81 Z = - .3 No. Expected 30 86 15.42 15.42 7.71 7.71 3.86 81 Chi-Square 4 78 .76 .76 .95 .07 .89 8.21 (11.07) No. Observed 34 17 10 2 10 4 77 Z = - .45 No. Expected 29 33 14.66 14.66 7.33 7.33 3.66 77 Chi-Square 74 ^37 1.48 3.88 .97 .03 7.48 (11.07) No Observed 44 13 16 0 8 1 82 Z = - .6 No. Expected 31.23 15.61 15.61 7.81 7.81 3.90 82 Chi-Square 5 22 .44 .01 7.81 .00 2.16 15.64 (11.07) No. Observed 50 14 10 1 8 1 84 Z = - .9 No. Expected 32.00 15.99 15.99 8.00 8.00 4.00 84 Chi-Square 10 13 .25 2.25 6,12 .00 2.25 21.00 (11.07) No. Observed 57 15 10 2 7 0 91 Z = —1.2 No. Expected 34 66 17.33 17.33 8.66 8.66 4.33 91 Chi-Square 14 40 .31 3.10 5.12 .32 4.33 27.58 (11.07) Z = - .3 No. Observed 121 42 38 7 25 7 240 to —.6, No. Expected 91 42 45.70 45.70 22.85 22.85 11.42 240 lumped. II Chi-Square 9 57 .30 1.29 10.99 .20 1.71 24.08 (11.07) Natural Networks No. Observed 148 35 38 31 21 9 282 Z = - .44 No. Expected 107 4 53.7 53.7 26.8 26.8 13.6 282 Chi-Square 15 34 6.51 4.59 .64 1.27 1.46 29.80 (11.07)

* For explanation of tests by topological and ambilateral classes see Smart (1969). t Underlined chi-square values due to excess of observed over expected, t Marginal totals in this column. § Critical value at 95% level of significance. II All networks with given number of sources from the 12 40 X 40 matrices generated with a Z of -.3, -.45, and -.6.

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