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A preliminary report on an empirical analysis of drainage network adjustment to precipitation input
Journal of Hydrology 8 (1969) 227-238 ; © North-Holland Publishing Co., Amsterdam N o t to be reproduced by photoprint or microfilm without written permission from the publisher
A PRELIMINARY
REPORT
ON AN
EMPIRICAL ANALYSIS OF DRAINAGE NETWORK ADJUSTMENT TO PRECIPITATION
INPUT
ROY E. WILLIAMS and PHILLIP M. FOWLER
University of ldaho, Moscow, Idaho, U.S.A. Abstract: In an analysis of the effect of precipitation on drainage network adjustment and on cumulative stream discharge, seven 4th-order drainage basins in each of two areas were investigated. Both areas are covered by glacial till and have similar precipitation intensities and thunderstorm frequencies, but one area receives approximately 30 percent greater average annual precipitation than the other. Values of certain geomorphic parameters were determined for each basin, and these values were compared statistically. As might be expected from considerations based on Horton's law, the number of segments per order per unit area of a 4th-order basin in the wetter area is systematically related to that in the drier area. The analysis indicates that a basin of a given order in the drier area contains more segments of every order than does a basin of the same order in the wetter area. But on a unit area basis more segments per order per unit area of 4th-order basins occur in the wetter area. Runoff data indicate that cumulative stream discharge per segment per order per unit area should be appreciably greater in the wetter area than in the drier area. On the basis of data collected from the two areas it appears that some variable other than precipitation, materials, evapotranspiration, and precipitation intensity is operative in the adjustment process. This variable may be initial relief.
Introduction Q u a n t i t a t i v e m o r p h o m e t r i c analysis o f drainage basin a d j u s t m e n t to a c o n d i t i o n o f d y n a m i c e q u i l i b r i u m has p r o d u c e d m a n y parameters, each o f which is useful in describing some aspect o f the a d j u s t m e n t process or the characteristics o f the land surface resulting f r o m the a d j u s t m e n t process. O n e o f the first was the linear relationship between the log o f n u m b e r o f streams and o r d e r n u m b e r observed by H o r t o n (1945). He defined the slope o f this straight line to be the b i f u r c a t i o n ratio o f the basin drainage network. A high bifurcation ratio indicates greater n u m b e r s o f lower o r d e r streams relative to higher o r d e r streams. K i n g (1953, p. 748) h y p o t h e s i z e d that there should be m o r e streams in a h u m i d area than in a dry area w i t h o u t specifying w h et h er this increase in f r e q u e n c y w o u l d occur only in first o r d er streams, be a d d e d u n i f o r m l y 227
228
ROY E. WILLIAMS AND P H I L L I P M. F O W L E R
throughout all orders, or some combination thereof. Most relevant concepts have been reviewed by Strahler (1964). Strahler also discusses the use of basin properties in the examination of interbasin characteristics such as geometric similarity. From notions developed in other disciplines concerning energy distribution, it seems reasonable that a channel network would adjust uniformly throughout a basin in homogeneous materials if precipitation input were altered (Leopold and Langbein, 1962). Specifically, with increases in precipitation, that is, energy, one might expect the number of stream segments per order per unit area, (in a drainage basin in homogeneous material) to increase geometrically with decreasing order. The purpose of this paper is to examine the nature of the adjustment of drainage networks to variable energy input and the resulting effects upon stream discharge. The technique employed is somewhat similar to that utilized by Chorley (1957), however the parameters employed differ considerably.
Analytical procedure In a field study it is impossible to examine the basins in an area, change the precipitation input, then re-examine the properties of the basins. As an alternative to this procedure, the authors have selected two areas with different precipitation inputs wherein values of certain other variables are also known. It is assumed that the difference in precipitation has persisted long enough for the basins in each area to have adjusted to their respective precipitation inputs. That is, it is assumed that both areas are in a state of dynamic equilibrium under current climatic conditions. In order to control the effect of materials with respect to hydraulic conductivity, infiltration characteristics, and surface resistance, areas covered by similar materials were chosen for investigation. The selected areas are central Iowa covered by Kansan an d Wisconsinian Age glacial till and southeastern Indiana covered by Illinoisian Age glacial till (Glacial map of the United States east of the Rocky Mountains, Geol. Soc. Amer., 1959). Seven drainage basins, culminating in 4th-order streams were examined in each area. These basins are listed in Table 1. The drainage basins in central Iowa receive 30-32 inches of average annual precipitation while those basins in southeastern Indiana receive 40-42 inches of average annual precipitation (Climate of the States, U.S.W.B. series 60-12 and 60-13). It is assumed that over the extended period of time necessary for establishment of integrated drainage, local spatial variations in precipitation are negligible. Because thunderstorm frequency in Indiana is similar to that in Iowa, (U.S. Weather Bureau Tech. Paper 19), it is believed that
Indiana South Hogan Indian Creek North Hogan S o u t h F o r k Creek Arnold Creek Bear Creek Log Lick Creek
Iowa Beaver Creek Clanton Creek South River North River Squaw Creek Big Creek North Branch
decreasing area
Basins listed by
2
23 15 18 15 11 10 12
1
2 37 24 24 16 22 19 10
98 73 78 61 46 41 52
1 152 127 113 86 105 64 52
4 5 2 2 3 3 3
3
3 6 4 3 3 5 5 2
Numbers of segments by order
1 1 1 1 1 1 1
4
4 1 1 1 1 1 1 1
TABLE 1
65.97 65.56 60.60 36.35 28.38 25.70 23.32
365.84 165.14 134.32 127.37 119.89 102.16 75.57
Area (Mi.Z)
1
1.486 1.113 1.287 1.678 1.621 1.595 2.230
1
0.416 0.769 0.841 0.675 0.876 0.626 0.688
0.349 0.229 0.297 0.413 0.388 0.389 0.515
2
0.101 0.145 0.179 0.126 0.184 0.186 0.132
2
0.061 0.076 0.033 0.055 0.106 0.117 0.129
3
0.106 0.024 0.022 0.024 0.042 0.049 0.026
3
4
0.015 0.015 0.017 0.028 0.035 0.040 0.043
4
0.003 0.006 0.008 0.008 0.008 0.010 0.013
Segments/Order/ Area of 4th Order basin
577 560 605 465 484 401 511
352 434 411 500 411 350 380
Total relief (ft)
8.75 8.54 9.98 12.79 15.60 17.05 22.03
.962 2.63 3.06 3.93 3.43 3.43 5.03
Area
Relief
Values of parameters for basins in Indiana and Iowa
1
.002,575 .001,987 .002,127 .003,608 .003,349 .003,977 .004,364
1
.001,181 .001,771 .002,046 .001,351 .002,132 .001,788 .001,810
2
.000,604 .000,408 .000,490 .000,888 .000,801 .000,970 .001,007
2
.000,286 .000,334 .000,435 .000,252 .000,447 .000,531 .000,347
3
.000,105 .000,135 .000,054 .000,118 .000,219 .000,291 .000,252
3
.000,045 .000,055 .000,053 .000,048 .000,102 .000,140 .000,068
Area
Seg/Order/ Relief
4
.000,025 .000,026 .000,028 .000,060 .000,072 .000,097 .000,084
4
.000,008 .000,013 .000,019 .000,016 .000,019 .000,028 .000,034
Z
-t
©
Z
Z >
© br~ ~7
> Z > Z > .<
230
R O Y E. W I L L I A M S A N D P H I L L I P M. F O W L E R
intensity of precipitation has not produced a significant difference in drainage networks between the two areas. Values of several parameters were obtained, which it was hoped would prove useful in comparing drainage networks in the basins of the two areas. The total existing relief, area, and numbers of segments by order of each of the fourteen 4th-order basins were calculated. The average total existing relief on the Indiana till plain is 109 feet greater than that in Iowa. The definition of lst-order streams was arbitrarily chosen to be those streams which appear as intermittent on U.S. Geological Survey topographic maps. Prior to 1960 intermittent streams were identified as those streams which carry water for at least six months of the year (Thurston, 1968, personal communication). All maps used in this study were prepared prior to 1960. The decision to use intermittent streams as first order streams was based on a comparison of topographic maps and aerial photographs. Intermittent streams appear to be the most consistent, easily identified starting points for stream counts. The use of gullies or rills as first order streams introduces errors in that their ease of identification on aerial photographs varies considerably with vegetation, slope and farming practice. Furthermore, gullies and rills receive little or no base flow and would not reflect the influence of that portion of precipitation which reaches streams via ground water flow systems. Several combinations of the above geomorphic parameters for the basins in the two areas were compared. Those deemed most significant in terms of drainage network adjustment are presented in Table 1. All comparisons of parameters for the two areas were made on the following basis: the basins in Indiana (hereafter called the wet area) were arranged in order of decreasing area as were the basins in Iowa (hereafter called the dry area). This arrangement permitted comparison of the smallest basin in the wet area with the smallest basin in the dry area and similarly for all other basins. For each basin the value of any given parameter was obtained for all orders. Correlation and regression analyses were run on the paired observations of the given parameter. The data for frequency of streams by order appear in Table 1. The data indicate that number of segments per order is generally higher in the dry area than in the wet area. At first glance this might seem abnormal because some investigators have hypothesized that greater precipitation should yield concomitantly higher frequency of segments in a drainage network (King, 1953). Column 2 of Table 1, however, reveals that drainage basin areas are much larger in the dry region, thus generating greater absolute numbers of streams per basin. Thus, number of segments by order on a unit area basis, as indicated in column 3 of Table 1, portrays the condition postulated by King. That is, areas receiving greater precipitation have greater numbers of segments per order per unit area. Schumm (1965) on the other hand has
AN ANALYSIS OF DRAINAGE N E T W O R K ADJUSTMENT
231
suggested that both sediment yield and drainage density may be at a maximum in semi-arid areas. In other words, frequency of segments should be greater in semi-arid areas than in humid areas. This observation may well be explained by the predominance of surface runoff over ground water runoff (base flow) in channels in semi-arid regions. Channel networks in semi-arid regions would reflect this input of high energy water through high drainage density and possibly by other parameters. Both areas treated in this study are humid and the effect of base flow and surface runoff should not be significantly different for the two areas. The empirical relationship between variations in precipitation and size of equal order basins is of considerable interest, and will receive detailed consideration in a subsequent study. Adjustment of segments per order per unit area
The first parameter examined was number of segments per order per square mile of the 4th-order basins, and will hereafter be referred to as frequency density. It should be noted that this parameter is similar but not identical to "stream frequency" as defined by Horton (1945). Horton's value for stream frequency is the sum of segments of all orders in a given basin divided by the area of that basin. In this analysis the number of segments per order per square mile of the basin of highest order (4th-order) is more significant because the drainage network of the 4th-order basins is examined in terms of segments of each order. Values of the parameter, frequency density, for the basins in the two areas were compared through the previously described correlation and regression analysis. Because the area values in this parameter differ from basin to basin, the requirement of independence of observations is met. Frequency density within the correlation and regression framework can be utilized to determine the degree of systematic adjustment of the drainage networks in the wet area to higher precipitation with adjustment of drainage area taken into account. The coefficient of determination (r a) for the paired values of this parameter for the two areas is .85. The value is statistically significant at the .05 level of confidence. Thus with respect to this parameter there exists a high degree of systematic adjustment of the drainage network in the wet area relative to the dry area throughout all orders. Stated in other terms, this analysis indicates that if precipitation in a basin is increased, any stream segments added should be added systematically throughout all orders as opposed to an overall increase in, say, l st-order streams per unit area. The regression coefficient for the pairs of values is 2.06 and the a value is .056. Thus on the average, if any given value of the parameter for the dry
232
ROY E. WILLIAMS AND PH1LLIP M. FOWLER
area is multiplied by 2.06 and .056 added, the corresponding frequency density value for the wet area is obtained. Therefore, for the wet area, frequency density is approximately 2.06 times that for the dry area, plus the intercept value of .056. Frequency density data are presented graphically in Fig. 1.
•
22-
2.0-
i,8-
1.6-
®
® ®
1.4-
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0.6o
:® ® 0.4- ~ o O ® 0.2DRY O 0
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l
l
0.2
0.4
,
AREA
l
I
J
0.6
0.8
1.0
Segments per order per unit area for each order in each basin in the wet area versus segments per order per unit area for each order in the corresponding basin in the dry area.
233
AN ANALYSIS OF DRAINAGE NETWORK ADJUSTMENT
Stream discharge Since the precipitation in the wet area is approximately 30 percent greater than in the dry area, the percent by which precipitation is greater in the wet area is less than the percent by which number of stream segments per order per unit area is greater in the wet area. Specifically, precipitation in the wet area is approximately 1.3 times greater, whereas frequency density is slightly over 2.06 times greater, than in the dry area. This relation might lead one to the conclusion that for any given order, streams in the wet area should carry less water than do the corresponding streams in the dry area. However, according to data collected by the U.S. Geological Survey (DeWiest, 1965, p. 62) average annual runoff for the wet area is approximately 3 times that for the dry area. These data combined with the above described relation between number of segments per order per unit area for the wet and dry areas can also be utilized to compare theoretical stream discharge in the two areas. For X segments per order per unit area for a given order in the dry area, there are 2.06 X + a segments per order per unit area for the same order in the wet area. Therefore: segments per order per unit area in the wet area
2.06 X + a
segments per order per unit area in the dry area
X
If b units of runoff are supplied from a unit area to the segments of a given order in the dry area during an arbitrary period of time then b units of runoff are distributed a m o n g X segments per order per unit area. That is, b/X equals runoff or stream discharge per segment per order per unit area in the dry area during the arbitrary period of time. During this same arbitrary period of time, in the wet area, 3b units of runoff from a unit area are distributed a m o n g 2.06 X + a segments of the same order per unit area. That is, 3b/(2.06 X + a) equals runoff or stream discharge per segment per order per unit area in the wet area during the arbitrary period of time. Combining these two equations by division and considering a given order one obtains: cumulative stream discharge per segment per order per unit area in the wet area cumulative stream discharge per segment per order per unit area in the dry area
3b/(2.06X + a) b/X
1.43X
X + a/2.06"
According to this equation, cumulative stream discharge per segment per order per unit area in the wet area should be approximately 1.43 times the discharge of a corresponding segment in the dry area. Therefore, in spite of
234
ROY E. WILLIAMS AND PHILLIP M. F O W L E R
the fact that there are slightly more than 2.06 times as many segments per order per unit area in the wet area, over a long period of time considerably more discharge per segment per order per unit area should occur in the wet area than in the dry area. Given sufficient stream gaging stations in the basins in the two areas one could check out this prediction by comparing stream ftow records. Unfortunately, an insufficient number of gaging stations are available. This relation between stream discharges provides a partial answer to the question of why the percent difference in frequency density should exceed the percent difference in average annual precipitation for the two areas. If similar materials in the two areas do in fact minimize the effect of variable surface resistance and average hydraulic conductivity as causative factors, then one might expect the volume of water reaching the streams to control the frequency density. In other words, energy in the form of precipitation delivered to a drainage system may be available only in the form of runoff. However, examination of this hypothesis reveals that percent difference in average annual runoff is not equal to percent difference in frequency density for the two areas. There is a threefold difference in average annual runoff but only a twofold difference in frequency density. Combining these observations, one can summarize by saying that for the two areas, percent difference in frequency density exceeds the percent difference in average annual precipitation, but percent difference in frequency density is less than the percent difference in average annual runoff. Therefore, some variable other than evapotranspiration, materials, and precipitation intensity must be operative. This variable may be the difference in initial relief or slope between the two areas. If total initial relief were available for each order basin, it could be treated as a continuous variable in a correlation and regression framework. Analysis of this variable would probably yield more insight into the difference in frequency density between the two areas. At the time the data were gathered, however, difference in initial relief on two till plains was not envisioned as a factor governing the adjustment of stream segment frequency, thus no attempt was made to evaluate this variable. Furthermore, evaluation of this variable would indeed be difficult; only an approximation would be possible. In an effort to analyze the relationship of existing relief to drainage network adjustment a second parameter, incorporating relief of the 4th-order basins, was derived for the two areas. The decision to analyze such a parameter resulted from an interest in whether existing relief may be systematically related to differences in average annual precipitation and evapotranspiration because of the effect of cumulative discharge on stream gra-
AN ANALYSIS OF DRAlNAGE
NETWORK
ADJUSTMENT
235
dient. The parameter selected was the number o f segments per order per unit relief per unit area o f 4th-order basins, which will be referred to as relief density for the sake o f brevity. Values o f relief density are presented graphically in Fig. 2. These paired observations were also c o m p a r e d within a correlation and regression framework. The coefficient o f determination (r E) is .84. As in the case o f the first parameter, the coefficient is statistically
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Fig. 2. Segments per order per unit area per unit relief for each order in each basin in the wet area versus segments per order per unit area per unit relief for each order in the corresponding basin in the dry area.
236
ROY E. WILLIAMS AND PH1LLIP M. FOWLER
significant with 84 percent explained variation. Therefore, the drainage networks in the wet and dry areas differ systematically also with respect to this parameter. It appears, therefore, that as a drainage basin develops, there is a systematic relationship between existing relief and numbers of segments per order per unit area. Schumm (1956) presents graphically a linear relationship between drainage density (ratio of total length of all channels within a drainage basin to the area of that drainage basin) and relief ratio (total existing relief over basin length). Thus it appears that during the development of a basin, relief adjusts systematically along with a number of other basin characteristics.
Summary The data obtained in this study indicate that in the two areas investigated, segments for all orders are added systematically when average annual precipitation is increased. The nature of the addition of segments indicates that basins of a given order are smaller in a wet area than in a dry area. It should be noted that both areas investigated are humid and that the above relations may not hold for semi-arid areas where much of the stream flow is provided by surface runoff. Upon examination of the initial relationship regarding drainage network adjustment to higher precipitation input, it was concluded that streams of any given order in the wet area should carry less discharge than corresponding segments in the dry area. In order to evaluate the effect of evapotranspiration on energy available to the drainage system, runoff was examined as the adjustment mechanism in place of precipitation. However, when the observed frequency density relationship is combined with runoff data the situation is reversed; that is, the wet area stream segments are expected to carry greater discharge at any given order. This reversal is due to a greater percent difference in average annual runoff than in percent difference in frequency density between the two areas. If frequency density in the two areas differed by the same percentage that average annual runoff differs for the two areas, one could conclude that frequency density is a function of precipitation coupled with evapotranspiration. If this intuitive notion were correct, discharge through a segment per order per unit area in the wetter area would be approximately equal to that through the corresponding segment in the drier area. Since the discharges are not the same in the two areas examined, some variable other than materials, precipitation, and evapotranspiration is apparently involved in the adjustment of drainage networks to increased energy levels. The above reasoning leads to a consideration of the role played by the probable difference in initial relief between the two areas. Because the 109
AN ANALYSISOF DRAINAGE NETWORK ADJUSTMENT
237
foot average basin relief differential between the two areas may result from difference in initial relief between the two areas, precipitation may enter the wet system with greater potential energy than that of the dry. It is possible that this difference leads to higher discharge, rather than equal discharge at any given order, in the wet area even though the basins are smaller in that area. By use of the parameter, frequency density, an attempt has been made to compare stream discharge per segment per order per unit area for a given order in the two areas examined. In order for the predicted relations to be authenticated many gaging stations would have to be installed in the two areas. At present this appears to be impractical, however the authors feel that it may be possible to examine the relations through simulation procedures. This research has served as a necessary pilot project for developing a more incisive experimental design to be employed in an expanded project. It is evident that not only does a drainage network adjust to structural elements such as initial relief, but that this adjustment process is affected by external factors, each of which alters the influence of energy input into the system. The follow up project will expand the sample size to a consideration of approximately fifty different drainage areas. Hopefully, employment of multiple correlation and regression analysis will permit simultaneous assessment of the effects of all variables on adjustment of drainage basin elements and on stream flow. Such an analysis should pinpoint each element of the network which adjusts to variable energy input as well as those factors influencing the energy available to do work.
Acknowledgements The writers wish to express their thanks to Professor Stanley A. Schumm, Colorado State University for the review of this paper, which was revised considerably after consideration of his comments. We are also grateful to the University of Idaho and the Idaho Water Resources Research Institute for support of the study.
References CHORLEY,R. J., Climate and morphometry, J. Geol., 65 (1957) 628-638 Climate of the States, Climate of Indiana, Climatography of the United States No. 60-12, U.S. Dept. of Commerce, Weather Bureau (October, 1959) Climate of the States, Climate of Iowa, Climatography of the United States No. 60-13, U.S. Dept. of Commerce, Weather Bureau (October, 1959) DEWIEST,R. J. M., Geohydrology (John Wiley and Sons, New York, 1965) Glacial Map of the United States East of the Rocky Mountains, Geol. Soc. America, 1st Ed. (1959)
238
ROY E. WILLIAMS AND PHILLIP M. FOWLER
HORTON, R. E., Erosional development of streams and their drainage basins. Bull. Geol. Soc. Amer. 56, No. 3 (1945) 275-370
KING, L. C., Canons of landscape development. Bull. Geol. Soc. Amer. 64, No. 7 (1953) 721-752
LEOPOLD, L. B. and W. B. LANGBEIN, The concept of entropy in landscape evolution, U.S. Geological Survey Professional Paper 500-A (1962) Mean Number of Thunderstorm Days in the United States, Technical Paper No. 19, U.S. Dept. of Commerce, Weather Bureau (December, 1952) SCHUMM,S. A., Quaternary paleohydrology. In: The quaternary of the United States, ed. by H. E. Wright, Jr. and D. G. Frey (Princeton Univ. Press, 1965) 783-794 SCHUMM,S. A., Evolution of drainage systems and slopes in badlands at Perth Amboy, New Jersey. Bull. Geol. Soc. Amer. 67 (1956) 597-646 STRAHLER,A. N., Quantitative geomorphology of drainage basins and channel networks. In: Handbook of applied hydrology, ed. by Ven Te Chow (McGraw Hill, 1964) 4-39 to 4-76 THURSTON, R. F., Personal communication, Pacific Region Engineer, Topographic Division, U.S. Geol. Survey, Menlo Park, California 0968)
A PRELIMINARY
REPORT
ON AN
EMPIRICAL ANALYSIS OF DRAINAGE NETWORK ADJUSTMENT TO PRECIPITATION
INPUT
ROY E. WILLIAMS and PHILLIP M. FOWLER
University of ldaho, Moscow, Idaho, U.S.A. Abstract: In an analysis of the effect of precipitation on drainage network adjustment and on cumulative stream discharge, seven 4th-order drainage basins in each of two areas were investigated. Both areas are covered by glacial till and have similar precipitation intensities and thunderstorm frequencies, but one area receives approximately 30 percent greater average annual precipitation than the other. Values of certain geomorphic parameters were determined for each basin, and these values were compared statistically. As might be expected from considerations based on Horton's law, the number of segments per order per unit area of a 4th-order basin in the wetter area is systematically related to that in the drier area. The analysis indicates that a basin of a given order in the drier area contains more segments of every order than does a basin of the same order in the wetter area. But on a unit area basis more segments per order per unit area of 4th-order basins occur in the wetter area. Runoff data indicate that cumulative stream discharge per segment per order per unit area should be appreciably greater in the wetter area than in the drier area. On the basis of data collected from the two areas it appears that some variable other than precipitation, materials, evapotranspiration, and precipitation intensity is operative in the adjustment process. This variable may be initial relief.
Introduction Q u a n t i t a t i v e m o r p h o m e t r i c analysis o f drainage basin a d j u s t m e n t to a c o n d i t i o n o f d y n a m i c e q u i l i b r i u m has p r o d u c e d m a n y parameters, each o f which is useful in describing some aspect o f the a d j u s t m e n t process or the characteristics o f the land surface resulting f r o m the a d j u s t m e n t process. O n e o f the first was the linear relationship between the log o f n u m b e r o f streams and o r d e r n u m b e r observed by H o r t o n (1945). He defined the slope o f this straight line to be the b i f u r c a t i o n ratio o f the basin drainage network. A high bifurcation ratio indicates greater n u m b e r s o f lower o r d e r streams relative to higher o r d e r streams. K i n g (1953, p. 748) h y p o t h e s i z e d that there should be m o r e streams in a h u m i d area than in a dry area w i t h o u t specifying w h et h er this increase in f r e q u e n c y w o u l d occur only in first o r d er streams, be a d d e d u n i f o r m l y 227
228
ROY E. WILLIAMS AND P H I L L I P M. F O W L E R
throughout all orders, or some combination thereof. Most relevant concepts have been reviewed by Strahler (1964). Strahler also discusses the use of basin properties in the examination of interbasin characteristics such as geometric similarity. From notions developed in other disciplines concerning energy distribution, it seems reasonable that a channel network would adjust uniformly throughout a basin in homogeneous materials if precipitation input were altered (Leopold and Langbein, 1962). Specifically, with increases in precipitation, that is, energy, one might expect the number of stream segments per order per unit area, (in a drainage basin in homogeneous material) to increase geometrically with decreasing order. The purpose of this paper is to examine the nature of the adjustment of drainage networks to variable energy input and the resulting effects upon stream discharge. The technique employed is somewhat similar to that utilized by Chorley (1957), however the parameters employed differ considerably.
Analytical procedure In a field study it is impossible to examine the basins in an area, change the precipitation input, then re-examine the properties of the basins. As an alternative to this procedure, the authors have selected two areas with different precipitation inputs wherein values of certain other variables are also known. It is assumed that the difference in precipitation has persisted long enough for the basins in each area to have adjusted to their respective precipitation inputs. That is, it is assumed that both areas are in a state of dynamic equilibrium under current climatic conditions. In order to control the effect of materials with respect to hydraulic conductivity, infiltration characteristics, and surface resistance, areas covered by similar materials were chosen for investigation. The selected areas are central Iowa covered by Kansan an d Wisconsinian Age glacial till and southeastern Indiana covered by Illinoisian Age glacial till (Glacial map of the United States east of the Rocky Mountains, Geol. Soc. Amer., 1959). Seven drainage basins, culminating in 4th-order streams were examined in each area. These basins are listed in Table 1. The drainage basins in central Iowa receive 30-32 inches of average annual precipitation while those basins in southeastern Indiana receive 40-42 inches of average annual precipitation (Climate of the States, U.S.W.B. series 60-12 and 60-13). It is assumed that over the extended period of time necessary for establishment of integrated drainage, local spatial variations in precipitation are negligible. Because thunderstorm frequency in Indiana is similar to that in Iowa, (U.S. Weather Bureau Tech. Paper 19), it is believed that
Indiana South Hogan Indian Creek North Hogan S o u t h F o r k Creek Arnold Creek Bear Creek Log Lick Creek
Iowa Beaver Creek Clanton Creek South River North River Squaw Creek Big Creek North Branch
decreasing area
Basins listed by
2
23 15 18 15 11 10 12
1
2 37 24 24 16 22 19 10
98 73 78 61 46 41 52
1 152 127 113 86 105 64 52
4 5 2 2 3 3 3
3
3 6 4 3 3 5 5 2
Numbers of segments by order
1 1 1 1 1 1 1
4
4 1 1 1 1 1 1 1
TABLE 1
65.97 65.56 60.60 36.35 28.38 25.70 23.32
365.84 165.14 134.32 127.37 119.89 102.16 75.57
Area (Mi.Z)
1
1.486 1.113 1.287 1.678 1.621 1.595 2.230
1
0.416 0.769 0.841 0.675 0.876 0.626 0.688
0.349 0.229 0.297 0.413 0.388 0.389 0.515
2
0.101 0.145 0.179 0.126 0.184 0.186 0.132
2
0.061 0.076 0.033 0.055 0.106 0.117 0.129
3
0.106 0.024 0.022 0.024 0.042 0.049 0.026
3
4
0.015 0.015 0.017 0.028 0.035 0.040 0.043
4
0.003 0.006 0.008 0.008 0.008 0.010 0.013
Segments/Order/ Area of 4th Order basin
577 560 605 465 484 401 511
352 434 411 500 411 350 380
Total relief (ft)
8.75 8.54 9.98 12.79 15.60 17.05 22.03
.962 2.63 3.06 3.93 3.43 3.43 5.03
Area
Relief
Values of parameters for basins in Indiana and Iowa
1
.002,575 .001,987 .002,127 .003,608 .003,349 .003,977 .004,364
1
.001,181 .001,771 .002,046 .001,351 .002,132 .001,788 .001,810
2
.000,604 .000,408 .000,490 .000,888 .000,801 .000,970 .001,007
2
.000,286 .000,334 .000,435 .000,252 .000,447 .000,531 .000,347
3
.000,105 .000,135 .000,054 .000,118 .000,219 .000,291 .000,252
3
.000,045 .000,055 .000,053 .000,048 .000,102 .000,140 .000,068
Area
Seg/Order/ Relief
4
.000,025 .000,026 .000,028 .000,060 .000,072 .000,097 .000,084
4
.000,008 .000,013 .000,019 .000,016 .000,019 .000,028 .000,034
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230
R O Y E. W I L L I A M S A N D P H I L L I P M. F O W L E R
intensity of precipitation has not produced a significant difference in drainage networks between the two areas. Values of several parameters were obtained, which it was hoped would prove useful in comparing drainage networks in the basins of the two areas. The total existing relief, area, and numbers of segments by order of each of the fourteen 4th-order basins were calculated. The average total existing relief on the Indiana till plain is 109 feet greater than that in Iowa. The definition of lst-order streams was arbitrarily chosen to be those streams which appear as intermittent on U.S. Geological Survey topographic maps. Prior to 1960 intermittent streams were identified as those streams which carry water for at least six months of the year (Thurston, 1968, personal communication). All maps used in this study were prepared prior to 1960. The decision to use intermittent streams as first order streams was based on a comparison of topographic maps and aerial photographs. Intermittent streams appear to be the most consistent, easily identified starting points for stream counts. The use of gullies or rills as first order streams introduces errors in that their ease of identification on aerial photographs varies considerably with vegetation, slope and farming practice. Furthermore, gullies and rills receive little or no base flow and would not reflect the influence of that portion of precipitation which reaches streams via ground water flow systems. Several combinations of the above geomorphic parameters for the basins in the two areas were compared. Those deemed most significant in terms of drainage network adjustment are presented in Table 1. All comparisons of parameters for the two areas were made on the following basis: the basins in Indiana (hereafter called the wet area) were arranged in order of decreasing area as were the basins in Iowa (hereafter called the dry area). This arrangement permitted comparison of the smallest basin in the wet area with the smallest basin in the dry area and similarly for all other basins. For each basin the value of any given parameter was obtained for all orders. Correlation and regression analyses were run on the paired observations of the given parameter. The data for frequency of streams by order appear in Table 1. The data indicate that number of segments per order is generally higher in the dry area than in the wet area. At first glance this might seem abnormal because some investigators have hypothesized that greater precipitation should yield concomitantly higher frequency of segments in a drainage network (King, 1953). Column 2 of Table 1, however, reveals that drainage basin areas are much larger in the dry region, thus generating greater absolute numbers of streams per basin. Thus, number of segments by order on a unit area basis, as indicated in column 3 of Table 1, portrays the condition postulated by King. That is, areas receiving greater precipitation have greater numbers of segments per order per unit area. Schumm (1965) on the other hand has
AN ANALYSIS OF DRAINAGE N E T W O R K ADJUSTMENT
231
suggested that both sediment yield and drainage density may be at a maximum in semi-arid areas. In other words, frequency of segments should be greater in semi-arid areas than in humid areas. This observation may well be explained by the predominance of surface runoff over ground water runoff (base flow) in channels in semi-arid regions. Channel networks in semi-arid regions would reflect this input of high energy water through high drainage density and possibly by other parameters. Both areas treated in this study are humid and the effect of base flow and surface runoff should not be significantly different for the two areas. The empirical relationship between variations in precipitation and size of equal order basins is of considerable interest, and will receive detailed consideration in a subsequent study. Adjustment of segments per order per unit area
The first parameter examined was number of segments per order per square mile of the 4th-order basins, and will hereafter be referred to as frequency density. It should be noted that this parameter is similar but not identical to "stream frequency" as defined by Horton (1945). Horton's value for stream frequency is the sum of segments of all orders in a given basin divided by the area of that basin. In this analysis the number of segments per order per square mile of the basin of highest order (4th-order) is more significant because the drainage network of the 4th-order basins is examined in terms of segments of each order. Values of the parameter, frequency density, for the basins in the two areas were compared through the previously described correlation and regression analysis. Because the area values in this parameter differ from basin to basin, the requirement of independence of observations is met. Frequency density within the correlation and regression framework can be utilized to determine the degree of systematic adjustment of the drainage networks in the wet area to higher precipitation with adjustment of drainage area taken into account. The coefficient of determination (r a) for the paired values of this parameter for the two areas is .85. The value is statistically significant at the .05 level of confidence. Thus with respect to this parameter there exists a high degree of systematic adjustment of the drainage network in the wet area relative to the dry area throughout all orders. Stated in other terms, this analysis indicates that if precipitation in a basin is increased, any stream segments added should be added systematically throughout all orders as opposed to an overall increase in, say, l st-order streams per unit area. The regression coefficient for the pairs of values is 2.06 and the a value is .056. Thus on the average, if any given value of the parameter for the dry
232
ROY E. WILLIAMS AND PH1LLIP M. FOWLER
area is multiplied by 2.06 and .056 added, the corresponding frequency density value for the wet area is obtained. Therefore, for the wet area, frequency density is approximately 2.06 times that for the dry area, plus the intercept value of .056. Frequency density data are presented graphically in Fig. 1.
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233
AN ANALYSIS OF DRAINAGE NETWORK ADJUSTMENT
Stream discharge Since the precipitation in the wet area is approximately 30 percent greater than in the dry area, the percent by which precipitation is greater in the wet area is less than the percent by which number of stream segments per order per unit area is greater in the wet area. Specifically, precipitation in the wet area is approximately 1.3 times greater, whereas frequency density is slightly over 2.06 times greater, than in the dry area. This relation might lead one to the conclusion that for any given order, streams in the wet area should carry less water than do the corresponding streams in the dry area. However, according to data collected by the U.S. Geological Survey (DeWiest, 1965, p. 62) average annual runoff for the wet area is approximately 3 times that for the dry area. These data combined with the above described relation between number of segments per order per unit area for the wet and dry areas can also be utilized to compare theoretical stream discharge in the two areas. For X segments per order per unit area for a given order in the dry area, there are 2.06 X + a segments per order per unit area for the same order in the wet area. Therefore: segments per order per unit area in the wet area
2.06 X + a
segments per order per unit area in the dry area
X
If b units of runoff are supplied from a unit area to the segments of a given order in the dry area during an arbitrary period of time then b units of runoff are distributed a m o n g X segments per order per unit area. That is, b/X equals runoff or stream discharge per segment per order per unit area in the dry area during the arbitrary period of time. During this same arbitrary period of time, in the wet area, 3b units of runoff from a unit area are distributed a m o n g 2.06 X + a segments of the same order per unit area. That is, 3b/(2.06 X + a) equals runoff or stream discharge per segment per order per unit area in the wet area during the arbitrary period of time. Combining these two equations by division and considering a given order one obtains: cumulative stream discharge per segment per order per unit area in the wet area cumulative stream discharge per segment per order per unit area in the dry area
3b/(2.06X + a) b/X
1.43X
X + a/2.06"
According to this equation, cumulative stream discharge per segment per order per unit area in the wet area should be approximately 1.43 times the discharge of a corresponding segment in the dry area. Therefore, in spite of
234
ROY E. WILLIAMS AND PHILLIP M. F O W L E R
the fact that there are slightly more than 2.06 times as many segments per order per unit area in the wet area, over a long period of time considerably more discharge per segment per order per unit area should occur in the wet area than in the dry area. Given sufficient stream gaging stations in the basins in the two areas one could check out this prediction by comparing stream ftow records. Unfortunately, an insufficient number of gaging stations are available. This relation between stream discharges provides a partial answer to the question of why the percent difference in frequency density should exceed the percent difference in average annual precipitation for the two areas. If similar materials in the two areas do in fact minimize the effect of variable surface resistance and average hydraulic conductivity as causative factors, then one might expect the volume of water reaching the streams to control the frequency density. In other words, energy in the form of precipitation delivered to a drainage system may be available only in the form of runoff. However, examination of this hypothesis reveals that percent difference in average annual runoff is not equal to percent difference in frequency density for the two areas. There is a threefold difference in average annual runoff but only a twofold difference in frequency density. Combining these observations, one can summarize by saying that for the two areas, percent difference in frequency density exceeds the percent difference in average annual precipitation, but percent difference in frequency density is less than the percent difference in average annual runoff. Therefore, some variable other than evapotranspiration, materials, and precipitation intensity must be operative. This variable may be the difference in initial relief or slope between the two areas. If total initial relief were available for each order basin, it could be treated as a continuous variable in a correlation and regression framework. Analysis of this variable would probably yield more insight into the difference in frequency density between the two areas. At the time the data were gathered, however, difference in initial relief on two till plains was not envisioned as a factor governing the adjustment of stream segment frequency, thus no attempt was made to evaluate this variable. Furthermore, evaluation of this variable would indeed be difficult; only an approximation would be possible. In an effort to analyze the relationship of existing relief to drainage network adjustment a second parameter, incorporating relief of the 4th-order basins, was derived for the two areas. The decision to analyze such a parameter resulted from an interest in whether existing relief may be systematically related to differences in average annual precipitation and evapotranspiration because of the effect of cumulative discharge on stream gra-
AN ANALYSIS OF DRAlNAGE
NETWORK
ADJUSTMENT
235
dient. The parameter selected was the number o f segments per order per unit relief per unit area o f 4th-order basins, which will be referred to as relief density for the sake o f brevity. Values o f relief density are presented graphically in Fig. 2. These paired observations were also c o m p a r e d within a correlation and regression framework. The coefficient o f determination (r E) is .84. As in the case o f the first parameter, the coefficient is statistically
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236
ROY E. WILLIAMS AND PH1LLIP M. FOWLER
significant with 84 percent explained variation. Therefore, the drainage networks in the wet and dry areas differ systematically also with respect to this parameter. It appears, therefore, that as a drainage basin develops, there is a systematic relationship between existing relief and numbers of segments per order per unit area. Schumm (1956) presents graphically a linear relationship between drainage density (ratio of total length of all channels within a drainage basin to the area of that drainage basin) and relief ratio (total existing relief over basin length). Thus it appears that during the development of a basin, relief adjusts systematically along with a number of other basin characteristics.
Summary The data obtained in this study indicate that in the two areas investigated, segments for all orders are added systematically when average annual precipitation is increased. The nature of the addition of segments indicates that basins of a given order are smaller in a wet area than in a dry area. It should be noted that both areas investigated are humid and that the above relations may not hold for semi-arid areas where much of the stream flow is provided by surface runoff. Upon examination of the initial relationship regarding drainage network adjustment to higher precipitation input, it was concluded that streams of any given order in the wet area should carry less discharge than corresponding segments in the dry area. In order to evaluate the effect of evapotranspiration on energy available to the drainage system, runoff was examined as the adjustment mechanism in place of precipitation. However, when the observed frequency density relationship is combined with runoff data the situation is reversed; that is, the wet area stream segments are expected to carry greater discharge at any given order. This reversal is due to a greater percent difference in average annual runoff than in percent difference in frequency density between the two areas. If frequency density in the two areas differed by the same percentage that average annual runoff differs for the two areas, one could conclude that frequency density is a function of precipitation coupled with evapotranspiration. If this intuitive notion were correct, discharge through a segment per order per unit area in the wetter area would be approximately equal to that through the corresponding segment in the drier area. Since the discharges are not the same in the two areas examined, some variable other than materials, precipitation, and evapotranspiration is apparently involved in the adjustment of drainage networks to increased energy levels. The above reasoning leads to a consideration of the role played by the probable difference in initial relief between the two areas. Because the 109
AN ANALYSISOF DRAINAGE NETWORK ADJUSTMENT
237
foot average basin relief differential between the two areas may result from difference in initial relief between the two areas, precipitation may enter the wet system with greater potential energy than that of the dry. It is possible that this difference leads to higher discharge, rather than equal discharge at any given order, in the wet area even though the basins are smaller in that area. By use of the parameter, frequency density, an attempt has been made to compare stream discharge per segment per order per unit area for a given order in the two areas examined. In order for the predicted relations to be authenticated many gaging stations would have to be installed in the two areas. At present this appears to be impractical, however the authors feel that it may be possible to examine the relations through simulation procedures. This research has served as a necessary pilot project for developing a more incisive experimental design to be employed in an expanded project. It is evident that not only does a drainage network adjust to structural elements such as initial relief, but that this adjustment process is affected by external factors, each of which alters the influence of energy input into the system. The follow up project will expand the sample size to a consideration of approximately fifty different drainage areas. Hopefully, employment of multiple correlation and regression analysis will permit simultaneous assessment of the effects of all variables on adjustment of drainage basin elements and on stream flow. Such an analysis should pinpoint each element of the network which adjusts to variable energy input as well as those factors influencing the energy available to do work.
Acknowledgements The writers wish to express their thanks to Professor Stanley A. Schumm, Colorado State University for the review of this paper, which was revised considerably after consideration of his comments. We are also grateful to the University of Idaho and the Idaho Water Resources Research Institute for support of the study.
References CHORLEY,R. J., Climate and morphometry, J. Geol., 65 (1957) 628-638 Climate of the States, Climate of Indiana, Climatography of the United States No. 60-12, U.S. Dept. of Commerce, Weather Bureau (October, 1959) Climate of the States, Climate of Iowa, Climatography of the United States No. 60-13, U.S. Dept. of Commerce, Weather Bureau (October, 1959) DEWIEST,R. J. M., Geohydrology (John Wiley and Sons, New York, 1965) Glacial Map of the United States East of the Rocky Mountains, Geol. Soc. America, 1st Ed. (1959)
238
ROY E. WILLIAMS AND PHILLIP M. FOWLER
HORTON, R. E., Erosional development of streams and their drainage basins. Bull. Geol. Soc. Amer. 56, No. 3 (1945) 275-370
KING, L. C., Canons of landscape development. Bull. Geol. Soc. Amer. 64, No. 7 (1953) 721-752
LEOPOLD, L. B. and W. B. LANGBEIN, The concept of entropy in landscape evolution, U.S. Geological Survey Professional Paper 500-A (1962) Mean Number of Thunderstorm Days in the United States, Technical Paper No. 19, U.S. Dept. of Commerce, Weather Bureau (December, 1952) SCHUMM,S. A., Quaternary paleohydrology. In: The quaternary of the United States, ed. by H. E. Wright, Jr. and D. G. Frey (Princeton Univ. Press, 1965) 783-794 SCHUMM,S. A., Evolution of drainage systems and slopes in badlands at Perth Amboy, New Jersey. Bull. Geol. Soc. Amer. 67 (1956) 597-646 STRAHLER,A. N., Quantitative geomorphology of drainage basins and channel networks. In: Handbook of applied hydrology, ed. by Ven Te Chow (McGraw Hill, 1964) 4-39 to 4-76 THURSTON, R. F., Personal communication, Pacific Region Engineer, Topographic Division, U.S. Geol. Survey, Menlo Park, California 0968)