A Note on Stream OrderingAuthor(s): John LewinSource: Area, Vol. 2, No. 2 (1970), pp. 32-35Published by: The Royal Geographical Society (with the Institute of British Geographers)Stable URL: http://www.jstor.org/stable/20000444 .

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32 Conservation, Stability and Management

Schumm, S. A. and R. W. Lichty, 1965. Time, space and causality in geomorphology. American J. of Sci., 263, 110-199.

Sorensen, T., 1961. The development of coast profiles on a receding coast protected by Groynes. Proc. 7th Conf. Coastal Engineering, Ed. J. W. Johnson, 836-846.

Strahler, A. N., 1956. Quantitative slope analysis. Bull. Geol. Soc. America, 67, 571-96. Tansley, A. G., 1935. The use and misuse of vegetational concepts and terms. Ecology, 16, 284-307. Tansley, A. G., 1949. The British Islands and their vegetation, 2nd Ed. Cambridge Univ. Press,

Cambridge. Watt, A. S., 1947. Pattern and Process in the plant community. J. Ecol. 35, 1-22. Young, A., 1961. Characteristic and limiting slope angles. Ziets. fur Geomorphologie, 5, 126

131.

A note on stream ordering John Lewin, University College of Wales, Aberystwyth

The number of stream ordering procedures now available is such that it is more than ever necessary to decide on the best to use. Apart from the intrinsic elegance

of the various quantitative relationships revealed, a practical criterion against which they may be judged is the degree to which order is proportional to stream status as manifested in size of catchment, number and lengths of contributing streams, stream discharge, or channel dimensions (Haggett and Chorley 1969, Strahler 1957). With the exception of stream or alternatively link numbers (the only purely topological property), none of these is likely to bear a consistent relationship with order under varying environmental conditions and scales. To avoid arbitrary or ambiguous definition, it seems preferable to have an ordering system for stream status based entirely on topological criteria.

In this respect, what is probably the best-known procedure, Strahler's modification of Horton's ordering system, is not entirely satisfactory. Consider the two networks in Figure IA. A and B represent end-members of a set of topologies with the same number of outer links or first-order tributaries (16) and therefore, in addition, the same number of inner links (n-I= 15). Intuitively they appear to be of the same status. The pinnate pattern has a single trunk stream and only reaches a Strahler-order of 2, while the dendritic pattern is one having no streams that Smart (1967) called 'excess', so that the highest possible Strahler-order is produced (here 4) and no low-order tributaries are 'wasted' by joining those of a higher order. Neither A- nor C-type streams are common in nature and in randomly generated networks their probability is also low.

However, streams may deviate topologically from the most probable pattern and they may not-closely follow a Hortonian law of stream numbers. An order ing system ought to relate to stream status unaffected by the way in which streams or links are combined together.

Shreve's proposal to use simply the number of first-order links as an index of

magnitude is most useful in this respect (Shreve 1966), as was a comparable proposal by Scheidegger (1965). Such numbers are very easily counted, although

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Stream Ordering 33

the well-known difficulties in defining the sources of real networks on the basis of varying map, photograph or survey evidence remain. They do, in fact, correlate with other catchment characteristics better than does Strahler-order, presumably because of the sort of topological variation already indicated. Thus product moment correlation coefficients for twelve catchments in Wales which are largely unmodified by human activity are shown in Table 1. Furthermore,

Table 1

Shreve Strahler magnitude order

Area 0.88 0.61

Mean flow 1964/5 0.94 0.59

Max. flow 1964/5 0.87 0.40

Using a t-test with 10 degrees of freedom, only coefficients of above 0.6 are significant at the 5 per cent level.

Shreve magnitude is precisely proportional to the number of links in a way that Strahler order is not. It will be interesting to see, in future work, whether correlation coefficients for disparate morphological and hydrological data will be consistently maximized using magnitude rather than some other status measure. Topological status may prove not to be consistently equivalent to flow

status, for example. Whether it is better to reduce the large numbers in Shreve magnitude by some form of logarithmic transformation, or to put up with them so as to record the differences between large-order basins at as low a level as those between small, is a matter of preference. There is certainly reason to give close attention to high-order basins, because of their intrinsic morphometric properties (Milton 1966, Smart 1967), and because for practical purposes it may be the larger streams that really concern the hydrologist.

Some further development of the Shreve system may also be suggested. Shreve-magnitude, based only on outer segments, neglects the fact that inner links gather water as well. 'Interbasin areas' (Schumm 1956), and also the lengths of inner links (Shreve 1969, James and Krumbein 1969), may be only about half those of first-order streams, but this is not negligible. The reason for the size differentiation is also a little puzzling: one possible explanation may be that if networks characteristically develop headwards literally by bifurcation, then outer links may be longer on average because many of them are in a critical pre-bifurcation state. Headward-growth simulation models and studies are curiously rare in the literature (the writer only knows of his own published and hybrid attempts), and it is possibly too early as yet to decide that inner and outer links represent separate populations. It seems reasonable at this stage-and certainly justifiable from a purely topological viewpoint-to include inner links as of equal status to outer.

In terms of magnitude this makes little practical difference, since the numbers of each are invariably related, but what it does reintroduce is the possibility of

assessing topological structure as well as magnitude. Suppose Shreve's magnitude system is varied as follows. Streams above which are two links become second order, above which there are four links are third order, and so on. The system is thus logarithmic one to the base 2. If the number of links or segments per order

is then plotted on log2 graph paper, A- and B-type networks appear contrasted

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34 Stream Ordering

as in the inset to Figure IB. A-type show a regularly increasing number of

links per order after first order; B-type a decrease. The total number of inner

links is, of course, dependent on the number of outer ones, but it is their distri

bution by order that provides an interesting key to topological structure. Graphs for a number of simulated or natural networks are shown in Figure B.

Leopold and Langbein's random walk model needs some subjective interpret ation because it contains numerous instances where three streams join at a point,

Alv A - type

B-type

B

2586

8 X E 128 A A

4 X 0

L2 64 E

\B 1 2 3 4 5

o segment order 32 -

C

a ~~~~~~~ -A~~~~~rroyo de los Frijoles c 16- (Leopold and Miller 1956.) E

* ~~~Nant Bran, Wales Random walk model 8- (SN 93) (Leopold and Langbein'1962.)

E ----Hightower Creek (Horton 1945.)

4

2- B-type; n =128 first order streams.

0 0 2 4 6 8 10

segment order

Fig. 1

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Stream Ordering 35

an uncommon natural occurrence. What is interesting from these graphs is that many natural networks have A-type tendencies at high-order levels if not at low-orders. These levels may be of some significance in hydrological studies relating topology to storm size and flood magnitude, as well as further defining the actual structure of real networks.

Previously network structure has been approachable through bifurcation ratios, which were additionally useful as a guide to 'real' magnitude, for example in systems of low order but high bifurcation ratio. Shreve-magnitude is, unaided, a simple yet satisfactory index of stream status. In adding an indication of structure to a measure of magnitude, it is hoped that this paper will suggest further re-exploration of such relationship as were revealed in Hortonian analysis but using an alternative, and in the writer's view improved, ordering system as the basis.

References Haggett, P. and Chorley, R. J., 1969. Network analysis in geography. Horton, R. E., 1945. Erosional development of streams and their drainage basins: hydrophy

sical approach to quantitative morphology. Bull. Geol. Soc. Amer., 56, 275-370. James, W. R. and Krumbein, W. C., 1969. Frequency distribution of stream link lengths.

Journ. Geol., 77, 544-65. Leopold, L. B. and Langbein, W. B., 1962. The concept of entropy in landscape evolution.

U.S.G.S. Professional Paper, 500-A. Leopold, L. B. and Miller, J. P., 1956. Ephemeral streams: hydraulic factors and their relation

to the drainage net. U.S.G.S. Professional Paper, 282-A. Lewin, J., 1969. The Yorkshire Wolds: a study in geomorphology. Univ. of Hull Dept. of

Geography Occ. papers, 11. Milton, L. E., 1966. The geomorphic irrelevance of some drainage net laws. Aust. Geog. Stud.,

4, 89-95. Scheidegger, A. E., 1965. The algebra of stream order numbers. U.S.G.S. Professional Paper

525-B, 187-9. Schumm, S. A., 1956. The evolution of drainage systems and slopes in badlands at Perth

Amboy, New Jersey. Bull. Geol. Soc. Amer., 67, 597-646. Shreve, R. L., 1966. Statistical law of stream numbers. Journ. Geol., 74, 17-37. Shreve, R. L., 1967. Infinite topologically random channel networks, Journ. Geol., 75, 178-86. Shreve, R. L., 1969. Stream lengths and basin areas in topologically random channel networks.

Journ. Geol., 77, 397-414. Smart, J. S., 1967. A comment on Horton's law of stream numbers. Water Resources Research,

3, 773-6. Smart, J. S., 1968. Statistical properties of stream lengths. Water Resources Research, 4, 1001

14. Smart, J. S. et. al., 1967. Digital simulation of channel networks. LA.S.H. Berne Assembly

Symposium on River Morphology, 87-98. Strahler, A. N., 1957, Quantitative analysis of watershed geomorphology. Trans. Amer. Geophys.

Un., 38, 913-20.

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