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Quantifying source areas through land surface curvature and shape
Journal of Hydrology, 57 (1982) 359--373
359
Elsevier Scientific Publishing Company, Amsterdam -- Printed in The Netherlands
[11 Q U A N T I F Y I N G SOURCE AREAS T H R O U G H LAND S U R F A C E C U R V A T U R E AND SHAPE
RICHARD G. HEERDEGEN and MAX A. BERAN
Department of Geography, Massey University, Palmerston North (New Zealand) Institute of Hydrology, Wallingford, Oxon OXIO 8BB (Great Britain) (Received April 7, 1981 ; accepted for publication June 10, 1981)
ABSTRACT Heerdegen, R.G. and Beran, M.A., 1982. Quantifying source areas through land surface curvature and shape. J. Hydrol., 57: 359--373. The theory of source areas is concerned with those hydrologically active localities said to be responsible for quick-response flood runoff. This paper describes a technique of map analysis identifying such localities as areas with convergent flow paths and retarding overland slope. _Measures of curvature and slope are constructed and used in a pilot-seale regression analysis with flood response quantities as dependent variables. No large improvement in predictive ability over conventional catchment characteristics are found although the relative importance of the variables, the derived curvature maps and the techniques employed will be of interest to modellers.
1. INTRODUCTION
The description of landform, especially its description in numerical terms, is an area of apparent c o m m o n interest for the geomorphologist and applied hydrologist. However, in reality the two are pulling in separate directions. The geomorphologist regards landform as the o u t p u t from an evolving system of erosion and deposition and an object of interest in its own right. On the other hand, the hydrologist's interest centres on the key that landform offers in the prediction and regionalisation of hydrological variables such as river flood or low-flow statistics. This paper resides firmly in the hydrological tradition because numerical indices of landform are developed and used to estimate flood response characteristics. However, the particular qualities of the catchment landform which are considered in this paper owe their origin to the hydrologist's idea of source areas, those restricted hydrologically-active localities which are said to be responsible for much o f the quick-response r u n o f f during flood events, as well as to the geomorphologists, concern with linking landform and landforming processes. Section 2 reviews source area theory and sets o u t the guiding principles underlying the development of the catchment characteristics.
0022-1694/82/0000--0000/$02.75 © 1982 Elsevier Scientific Publishing Company
360
The actual derivation of these characteristics which index the contour (plan) and slope or profile (sectional) curvature of each point on the basin, as well as the familiar magnitude and direction of the surface slope, is described in Section 3. Results in terms of mapped characteristics and correlations and regressions with flood statistics and unit hydrograph model parameters are described in Section 4.
2. PREVIOUS STUDIES IN SOURCE A R E A IDENTIFICATION
2.1. Landform characteristics from maps In the 35 years since Horton (1945) showed that certain characteristics of landform and stream network could be quantified, earth scientists have produced a veritable dictionary of catchment characteristics. These are well reviewed in a recent article b y Gardiner and Park (1978}. Now that the euphoria of being able to digitise and analyse b y means of computers has passed, along with the realisation that meaningful causal relationships are not so easily produced, scientists have either given up such analysis or have continued to proceed with optimistic caution. Much of the morphometric analysis of landform has been conducted at a micro-level, using arduously obtained field measurements. Analysis of the measurements has resulted in descriptive statistical treatments of the profile form including the gradient and curvature of the flow path, a slope profile terminology (Savigear, 1956; Young 1964), and the formulation of processforming relationships covering the whole range of landforms in varying climatic regions. In contrast to the extensive literature produced from the detailed approaches to landform, little attempt has been made to quantify the shape of the land surface without resorting to field measurement, i.e. from national mapping. The first attempts were initiated b y those seeking a means of landscape quantification for terrain analysis (Wood and Snell, 1960; Carr and van Lopik, 1962; Brink et al., 1966; Speight, 1968). Troeh's (1965) paper described h o w landform shape could be quantified using contour maps while Greysukh (1967) outlined the use of digital computers in landform description, followed b y papers in a similar vein a decade later (Druet, 1973; van Asch and Steijn, 1973; Penteado and Hulke, 1974). A recent paper b y Evans (1980) treads a similar path to that of Section 3.3 and demonstrates the fundamental role o f the four shape parameters to many applications. One of the major problems of working with small-scale maps and photographs has been the difficulty of maintaining consistency in the light of changing cartographic definition of contours, stream networks and stream lengths. The inconsistency from map to map has led to a large b o d y of
361 literature on such a fact, b u t apart from various methods of adjusting for such differences, little success has been achieved in overcoming the problem.
2. 2. Theories o f runoff formation The theory of the disposition of precipitation from slopes, especially in periods when the infiltration capacity of the soil is exceeded and "surface" runoff occurs, has been the subject of another b o d y of literature. Chorley (1978) reviews some of the many models which have been developed by soil physicists, groundwater hydrologists, and hydraulicians. Geomorphologists have been much more intent on explaining the landform runoff--erosion cycle, although Carson and Kirkby (1972) have redressed the balance in their b o o k b y approaching the development of slope forms through quantitative process studies. There are two major schools of thought about how storm runoff is generated. The two theories differ at both the elementary and the catchment scale. In the earlier theory, due to H o r t o n (1945), runoff occurs as overland flow when the infiltration capacity Las been exceeded. In the source area theory it is recognised that, at least in small catchments in the temperate zone, runoff is generated in a near-surface layer of soil and vegetation over a less permeable sub-surface layer (Musgrave and Holtan, 1964; Kirkby and Chorley, 1967). However, both of these processes act at the elementary scale and entail processes to which landform is largely irrelevant. On the catchment scale, runoff in Horton's theory is deemed to take place uniformly over the entire area. The source area theory recognises the observational evidence for hydrologically more active localities within a catchment (Dunne and Black, 1970; Engman and Rogowski, 1974). These include not only channels and their surrounding riparian zone b u t also contributing areas which are often variable in extent and expand as the storm continues. It is this partial area concept which offers the scope for introducing map analysis because of its dependence on the hypothesis that the shape of the land is an important causal factor in the temporal and spatial disposition of storm runoff. Many analyses have shown that catchments with increasing slope (however measured) will produce hydrographs with shorter times to peak and higher proportion volume of runoff, all other things being equal. Such measures of slope have usually involved a channel-slope measurement of some form although mean catchment slope is often used. But to date not much attempt has been made to use information from the whole of the catchment space. In doing so, the location of areas of different gradient, say flat or very steep areas, may take on much greater significance than measures of mean slope.
2.3. Landform and flood response From a hydrological point of view, landscape shape seems to have much relevance to the source of contributing areas of the catchment. Researchers
362
in the U.S.A. in the 1960's (Amorocho and Orlob, 1961; Betson, 1964) found that the percentage of watershed areas contributing to runoff varied from as little as 5% to an extreme of 85%, with low percentages being the norm and only rising to a major part during very extreme events. Many subsequent papers have dealt with this partial-area concept, b u t its applicability has been limited b y problems of measurement. Source areas could most properly be thought o f as areas of surface runoff concentration of hollow shape with perhaps some proximity to the existing permanent channel network, although Amerman (1965) found that runoff seemed to originate in seemingly random fashion from ridge-tops, valley slopes and valley bottoms. Kirkby and Chorley (1967) enumerate the types of area which are most frequently saturated (i.e. source areas) as being those adjacent to flowing streams, lines of greatest slope convergence, local concavities, and areas of thin or less permeable soil cover, with the most favoured zones for overland flow being in valley b o t t o m s and in particular in stream-head hollows. Carson and Kirkby (1972) suggest, however, that not only is there not a positive correlation between concave slope profiles and contour concavity, that is hollows, b u t that slope profiles traced down the true slope (lines of maximum steepness) on a map tend to end in major valleys and not in the smaller lateral channels. Consequently, it could be argued that contributing areas are almost impossible to define spatially. However, in a detailed field experiment, Anderson and Burt (1978) found that convergent flow occurred in relatively steep hollows simply because of the contour pattern, and that not only were the hollow zones sources of throughflow b u t also they remained saturated during periods of low streamflow. Soil surveyors recognise the likelihood of gleyed soils at these points which tend to be more impermeable. Parsons (1979) states that strong contour concavity (short radius of curvature) is positively correlated with strong profile concavity, but much less so when the curvatures are less. He also suggests that if slope curvature is the more dominant curvature factor affecting runoff processes, then contour curvature, by its relationship with slope curvature, would produce effects at a very localised level perhaps causing concentrations of water locally within the whole slope. The general findings of this survey would tend to support the rather random location of hollows in most catchments.
3. D E R I V A T I O N O F C A T C H M E N T A N D F L O W C H A R A C T E R I S T I C S
3.1. Background to the study
This study is part of a continuing effort to extend and improve the results of a nationwide analysis of floods contained in the Flood Studies Report (NERC, 1975). This report presented regression equations to predict unit hydrograph time to peak and standard percentage runoff from various catch-
363 ment characteristics representing size, slope, drainage, soil type, land use and climate. As with all such studies some individual catchment predictions were considerably in error and the main object of this exercise was to test whether these outliers could be reduced by a better numerical description of the landform. These new landform catchment characteristics are based on the supposition, in turn deduced from the source area concept, that a significant proportion of flood r u n o f f derives from convergent, retarding-flow localities. The following subsections describe the method of calculating the necessary plan and slope curvatures, and the slope vector (aspect) and gradient of the surface. 3.2. Map digitisation and surface fitting The shear impracticality of large-scale field measurements produces a reliance on existing forms of data, and the contour map is the source of most potential information. Derivation of slope characteristics such as plan and slope curvature, gradient and slope vector can be determined at individual locations from a c o n t o u r map, but catchment~wide derivation of such values requires a more generalised approach. Using an interpolative computermapping routine, SACM, a regularly spaced grid of elevation spot heights were produced from digitised contour information. This results in a reduction in information, with the loss of individual contour crenulations, and efforts to minimise this by increasing the density of the spot-height grid merely produces a non-rational topography. The grid replicates the general form of the surface as if subjected to a low-pass filter whose properties are a function of contour spacing and topographic detail. Once a grid of spot heights had been created each spot could then be located three-dimensionally: x and y as plan coordinates, and z as elevation. By focussing on one grid point it was hoped that its elevation and those of the neighbouring points would yield directly values of plan and slope curvature, gradient and slope vector. Each grid point in turn would then be inspected to produce maps of the quantities for the whole catchment. 3.3. Calculation o f slope parameters This proved to be unsuccessful so a two-step procedure was adopted. The first step was to fit a function of known mathematical form -- in this case a quadric surface -- to the 3 × 3 matrix of grid points surrounding a given point. The second step was to compute the landform parameters from the properties of the surface. Difficulties were experienced in fitting the nine points to a general quadric (step 1) which permitted a folded surface. A restricted five-parameter surface was adopted finally which has the equation: z = ax 2 + b y 2 + e x y + d x + e y which was solved in the following way.
(1)
364 The nine elevations identified, Z7
Z8
Z9
Z4
Z5
Z6
Z1
Z2
Z3
from
which
the
surface
w a s t o be d e s c r i b e d w e r e
w i t h Zs h a v i n g z e r o h e i g h t , a n d all o t h e r e i g h t p o i n t s b e i n g o f relative h e i g h t t o it. All p o i n t s w e r e o n a r e g u l a r s q u a r e grid, w h o s e side in t h e c a t c h m e n t s u s e d v a r i e d f r o m 2 3 . 5 t o 1 0 0 m. T h i s v a r i a t i o n w a s n e c e s s a r y t o allow f o r c o n s t r a i n t s i m p o s e d o n t h e size o f t h e grid m a t r i x b y t h e c o m p u t e r programs. F r o m eq. 1 t h e f o l l o w i n g s y s t e m o f e q u a t i o n s w a s d e r i v e d : U1 =
Z 1 "4-Z 3 -4-Z 4 q - Z 6 q - Z 7 + Z 9
v2 = z 1 + z 2
+z 3 +z 7 +z s +z 9
va = z 1 - - z 3 - z
7 +z 9
(2)
V4
:
--Zl
-~ Z 3 - - Z 4 -~ Z 6 - - Z 7 "4-Z 9
V5
=
--Zl
- - Z 2 - - Z 3 --~Z 7 --~- ZS -~- Z9
f r o m w h i c h t h e u n k n o w n s in eq. 1 w e r e s o l v e d as f o l l o w s : a =
(0.3vl -- 0.2r2)/G 2
t
b =
(0.3v2 -- 0.2vl)/G2
t(3)
c = v3/4G2;
d = Va/6G;
and
e = Vs/6G
w h e r e G is t h e grid size, i.e. d i s t a n c e b e t w e e n t h e s p o t h e i g h t s in m e t r e s . I n s t e p 2 t h e f o u r p a r a m e t e r s - - s l o p e v e c t o r (1 ° - - 3 6 0 ° ), g r a d i e n t , p l a n o r contour curvature, and slope or flow path curvature -- were then calculated using the following formulae: -1 ( - - e / - - d ) - - 9 0 ( - - d / i d [ )
[ s l o p e v e c t o r (° )]
=
180--tan
[ g r a d i e n t ( m m -1 ]
=
(d 2 + e 2 )0.s
[ p l a n c u r v a t u r e ( m -1 )]
= e (cd -- 2ae)/(d 2 + e 2 )l.s
(4) (5) (6)
Fig. 1. Croasdale Beck at Croasdale Flume (71003), grid reference SD706546. A. Contour map (heights in metres). B. Areas of concave slope curvature with a radius of curvature less than 100 m. C. Areas of tight concave contour curvature ( ~ 100-m radius) are shown in black. The unshaded and shaded squares correspond to all squares in F. D. Concave plan curvature less than 250-m radius. E. Concave plan curvature less than 500-m radius. F. Concave plan curvature less than 1000-m radius.
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[slope curvature (m -1 )] = 2 (ad 2 + be 2 + cde)/(d: + e 2 )(1.0 + d 2 + e 2 ) 1. s
(7) Having established a catchment~wide matrix of slope parameters, mapping and statistical analysis were then carried out.
3.4. Data analysis With the aid of a plotting package, there is an almost limitless range of options available for the portrayal of the data. But the problem was not one of portrayal but of analysis -- that is, how is the spatial distribution of a parameter in one catchment compared with that in another? There are some very sophisticated analytical techniques available but as with all analysis, the o u t p u t has to have some physical reality. Will it distinguish, for example, between areas of strong concavity adjacent to stream channels in one catchm e n t and those in the high interfluves in another? Areas within each catchment that were considered to have one or more of the features that characterise source areas were identified by means of a "Boolean chop", that is, listing or mapping only those grid points where the value of the feature was either above or below certain threshold values, or in some cases, between two values. For example, the maps o f Croasdale Beck show areas where the radius of concave slope curvature {reciprocal of curvature) is less than 100 m (Fig. 1B), and the succeeding maps (Fig. 1C--F) show incremental areas of decreasing concave plan curvature from 100 to 1000 m radius. Similarly, maps of combined parameters, such as slope curvature and plan curvature (Fig. 2C) or gradient and plan curvature (Fig. 2D--F), can be drawn to show spatially related areas. Maps combining slope gradient and aspect, drawn with arrows, probably do not produce any more information than can be interpreted from a good contour map, but the information is easily reproduced a n d analysed (Fig. 2B). Thus source areas, if one accepts a definition of both plan and slope concave curvature, enhanced by steepening gradient, can be identified and mapped. But meaningful spatial analysis seems to be much less attainable. Fig. 2. River Bourne at Hadlow (40006), grid reference TQ632497. A. Contour map (heights in metres). B. Generalised slope direction and gradient (arrow shaft length is proportional to slope angle). Some of the perimeter points slope outwards because of lack of boundary definition, and the main channel often shows reverse slope because of its gentle gradient relative to the surrounding area. C. Areas of c o n c a v e s l o p e and plan curvature with radii less than 500 m, which are generally adjacent to the streams. D. Areas of steep gradient ( ~ 250 m km -1 ) and tight concave plan curvature ( ~ 250-m radius) which are located primarily in headwater areas. E. Areas with same plan curvature ( ~ 250-m radius) as D but including less steep areas (gradient ~ 100 m km -1 ). F. Areas with same gradient as E ( ~ 100 m km -1 ) but more open concave plan curvature ( ~ 500-m radius).
367
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368 A visual estimate of the intensity and spatial distribution of the curvature feature in each catchment was made to predict the tendency to over- or underestimate flood response and subsequently compared with known catchment variables such as average time-to-peak and flood magnitude. The complete lack of success in establishing any relationships in this "double blind" but subjective test only served to underline the futility of trying more sophisticated spatial and analytical techniques. Less subjective tests were tried and these are described in the next section.
( ~ ~
2 8 0 3 3 1 1
~
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• 330123304 @/~
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I
•40006
100km 1
Fig. 3. England and Wales: location of catchments. 4. RESULTS The objects of the study had been partly fulfilled in that the practicality of developing characteristics indexing landform shape had been demonstrated. The other important objective of testing whether the new characteristics were capable of improving predictions of flood characteristics remained to be tested and is discussed in this section. Three flood characteristics were considered:
369 TP
= (time to peak of the 1-hr. unit hydrograph)
SPR
= (standard percentage r u n o f f being the ratio of r u n o f f to gross rainfall in storms, reduced to a standard depth and antecedent catchment wetness)
MAF = (mean annual flood) All three variables are extensively discussed in the F l o o d S t u d i e s R e p o r t (NERC, 1975) in which t h e y appear as pivotal variables in recommended procedures for flood estimation at an ungauged site. The catchments selected for study (Fig. 3) were among the smaller ones used in t h a t study, for reasons o f economy in map digitisation. Also catchments were selected from among those which were worst predicted by the existing regression equations i.e. MAF furthest removed from MAF, etc., so that the predictive process could be improved, not merely repeated. The following equations were used to derive parameters in the catchments studied. TP
= 46.6,S1085 -°'4 ,L0.14 ,RSMD-0.4 ,URB-2
(8)
SPR
= 95.5,SOIL + 0.12,URB
(9)
MAF
= C , AREA0.94 ,STMFRQ0.27 , S O I L 1.23 ,RSMD 1.03, (1 + LAKE) -°'ss , $ 1 0 8 5 °'16
(10)
where $1085
= channel slope between 10% and 85% of the main channel length
(m km -1 )
URB
= proportion of catchment under urban development
RSMD
= 5-yr. return period 1-day effective rainfall
(mm)
L
= main stream length
(km)
SOIL
= index of winter rainfall acceptance potential
AREA
= topographic catchment area
STMFRQ = stream frequency LAKE
(km 2 ) (junctions/km 2)
= proportion of catchment draining through a lake
C
= regionally varying multiplying factor ranging from 0.0153 in eastern Britain to 0.0315 in southwestern Britain Note: $1085 is the only one that can be regarded as directly indexing some aspect of landform. The use of " o u t l i e r " catchments has resulted in small differences in the data sets available for the different tests.
370 TABLE I Landform indices Variable number
Variable acronym
Description
XI
GMED
X2
GLQ
X3
GVQ
X4
GFT2
Xs
GST20
median gradient, i.e. 50% of the catchment is steeper and 50% flatter than X 1 lower quartile gradient, i.e. 25% of the catchment is flatter than X 2 upper quartile gradient, i.e. 25% of the catchment is steeper than X 3 proportion of catchment flatter than 2° proportion of catchment steeper than 20 °
Gradients:
Slope curvatures: X6
SCMED
X7
SCLQ
Xs
SCLT500
median slope curvature, i.e. 50% of the catchment has a slope radius of curvature less than X 6 lower quartile slope curvature, i.e. 25% of the catchment has a slope radius of curvature less than X 7 proportion of catchment with slope radius of curvature less than 500 m
Plan curvatures: X9
PCMED
X10
PCLQ
Xll
PCLT500
median plan curvature, i.e. 50% of the catchment has a plan radius of curvature less than X 9 lower quartile plan curvature, i.e. 25% o f the catchment has a plan radius of curvature less than X10 proportion of catchment with less than 500 m (tight valley contours)
T h e i n i t i a l a p p r o a c h as m e n t i o n e d in S e c t i o n 3 . 4 c o n s i s t e d o f i n f o r m a l l y ranking the catchments according to an eye estimate of the number and location of source areas. This subjective approach failed to reveal any obvious conclusions; clearly the cause of the poor prediction was not due solely to the landform elements that were visible from the maps. The next stage was to develop numerical indices from the maps and perform correlations and regression analyses to try to extract linkages between f l o o d a n d c a t c h m e n t c h a r a c t e r i s t i c s . T h e s e n u m e r i c a l i n d i c e s a r e l i s t e d in T a b l e I, d i v i d e d i n t o g r a d i e n t a n d t h e t w o t y p e s o f c u r v a t u r e g r o u p s . A l s o i n c l u d e d in t h e d a t a s e t w e r e t h e s t a n d a r d c a t c h m e n t c h a r a c t e r i s t i c s l i s t e d above, and observed and estimated flood characteristics.
371 The within-group intercorrelations are considered first. Six slope characteristics including $1085 were available. This channel slope characteristic, which was employed in NERC (1975) r e c o m m e n d e d equations was quite well correlated with all five overland slope indices; 0.73, 0.70, 0.75, --0.72, 0.83 with X 1 to X s . It appears that $ 1 0 8 5 is slightly better correlated with indices of "steepness" than with indices of "flatness" X2 and X4. In fact X4, measuring catchment flatness stands out as separate from the other four overland slope indices with lower correlations o f - - 0 . 5 5 , --0.52, --0.55 and --0.68, with X1, X2, Xa and X s , respectively, elsewhere values of 0.9 and above being the rule. The two curvature groups (X6--X 8 and X9--Xll ) display similar behaviour. The median measure in both cases is relatively poorly correlated with the indices focussing on "tight" curvatures which are themselves intercorrelated. Median plan curvature in particular appears almost totally uncorrelated with the other two plan curvature based characteristics. There is also very little correlation between the two groups, for e x a m p l e - - 0 . 1 2 between X 6 and X9, 0.39 between X 7 and X10, and 0.37 between Xs and Xxl. It appears that there is no tendency for plan curvature to be dictated b y slope curvature thus supporting Parson's (1979) conclusion. Arguing from first principles, linkages would be expected between gradient along the flow path and slope curvature as "tight" slope curvature could not occur without steep slopes at the upper end. In fact only X s , which focusses totally on this aspect displays a high correlation, 0.77 with X s . There is virtually no correlation between any of the slope gradient measures and any of the plan curvature indices, the highest value among all 3 × 6 possible values barely exceeding 0.3. Flow variables were constructed based upon the observed and estimated values of TP, SPR and MAF. Poor correlations between these and the landform indices confirmed the visual impression of the initial test that there was no immediate explanation of outliers in the presence or absence of potential source areas. For example, a multiple correlation on the residual from the MAF equation succeeded in explaining only 15% of the variance with still lower values for other residuals. One might expect to see the major enhancement to predictive ability in the two process-oriented hydrological variables TP and SPR. The first is well correlated with $ 1 0 8 5 in the total Flood Studies Report data set, with a correlation of --0.81. In this data set the correlation with $1085 is much poorer, --0.46, b u t nevertheless rather better than most of the derived overland slope measures. The exception is X4 which focusses on the flat areas and is spectacularly better correlated than the others, with a correlation of 0.73. Multiple regressions involving overland slope, X , , plan curvature, X9, and urban fraction would yield useful predictions although whether in practice these would represent an improvement over the present recommendation involving $1085 would require further testing. Nearly 95% of the variance in $1085 can be explained b y using one of each of the groups of landform indices in a multiple regression.
372 The major predictor of SPR, the volumetric response, is the soil index, SOIL. There is some correlation between SOIL and the gradient variables which is to be expected given t h a t the soil classes were in part determined by slope gradient. This is reflected in the gradient correlations with SPR of ~ 0 . 5 . The correlations with curvature variables are lower -- prediction of SPR from curvature variables alone would succeed in explaining less than 20% of the variance which does not lend much support to the contributing area theory. However, the landform variables add a useful extra-explained sum of squares of the SOIL variable alone, raising the multiple correlation coefficient from 0.67 to 0.73.
5. SUMMARY Contour maps contain more inherent information about landform shape than any other existing source, and digitising of the contours produces a data file which can easily be reproduced. But definition of landform shape from such information appears to be impracticable unless a uniform matrix of spot heights is substituted for contours. Mathematical algorithms can then be utilised on the spot heights to produce shape parameters such as slope and plan curvature, slope gradient and slope aspect. Computer drawn maps using all or parts of this information can be produced, and individual results are very worthwhile. However, substantial difficulties prevent any spatial variations and comparisons from being ascertained. Indices of shape and gradient are used to predict flood response, and show that in part t h e y do reflect the sorts of responses that would be expected. However, their predictive ability is thwarted to some extent by the generalisation of slope form which has to be undertaken to produce the parameters. Further studies could well derive methods of obtaining data which more accurately reflect the nature of the landform, and in so doing utilise the vast a m o u n t of information currently stored on contour maps.
ACKNOWLEDGEMENTS The authors thank the Director of the Institute of Hydrology, Wallingford, for permission to publish this paper, and, in the case of the first author, for the opportunity to be a visiting scientist there. Acknowledgement is also given to the Council of Massey University, Palmerston North, New Zealand, for leave to pursue this project. Thanks are also due to many members of the Institute's staff for their assistance in the execution of this project. The study was carried out as part of a commission for the U.K. Ministry of Agriculture, Fisheries and Food.
373 REFERENCES Amerman, C.R., 1965. The use of unit-source watershed data for runoff prediction. Water Resour. Res., 1 (4): 499--507. Amorocho, J. and Orlob, G.T., 1961. An evaluation of inflow--runoff relations in hydrologic studies. University of California, Berkeley, Calif., Water Resour. Cent., Contrib. 4 1 , 49pp. Anderson, M.G. and Burt, T.P., 1978. Towards more detailed field monitoring of variable source areas. Water Resour. Res., 14(6): 1123--1131. Betson, R.P., 1964. What is watershed runoff?. J. Geophys. Res., 69(8): 1541--1552. Brink, A.B., Mabbutt, J.A., Webster, R. and Beckett, P.H.T., 1966. Report of the working group on land classification and data storage. Mil. Eng. Exp. Estab., Christchurch, England, Rep. 940. Carr, D.D. and van Lopik, J.R., 1962. Terrain quantification, Phase 1 : Surface geometry measurements. U.S. Air Force Cambridge Res. Lab., Bedford, Mass., Rep. A F C R L 63-208. Carson, M.A. and Kirkby, M.J., 1972. Hillslope Form and Process. Cambridge University Press, London, 475 pp. Chorley, R.J., 1978. The hillslope hydrological cycle. In: M.J. Kirkby (Editor), Hillslope Hydrology. Wiley, New York, N.Y., pp. 1--42. Druet, Y., 1973. L'utilisation de l'ordinateur pour l'analyse morphom~trique du bassin versant en rue de la planification du r~seau hydrom~trique national du base. Rev. G~omorphol. Dyn., 22 : 173--178. Dunne, T. and Black, R.D., 1970. Partial area contributions to storm runoff in a small New England watershed. Water Resour. Res., 6(5): 1296--1311. Engman, E.T. and Rogowski, A.S., 1974. A partial area model for storm flow routing. Water Resour. Res., 10(3): 464--472. Evans, I.S., 1980. An integrated system of terrain analysis and slope mapping. Z. Geomorphol., Neue Folge, 36: 274--295. Gardiner, V. and Park C.C., 1978. Drainage basin morphometry: review and assessment. Progr. Phys. Geogr., 2: 1--35. Greysukh, V.L., 1967. The possibility of studying land forms by means of digital computers. Soy. Geogr., 8: 137--149. Horton, R.E., 1945. Erosional development of streams and their drainage basins: hydrophysical approach to quantitative morphology. Bull. Geol. Soc. Am., 56: 275--370. Kirkby, M.J. and Chorley, R.J., 1967. Throughflow, overland flow and erosion. Bull. Int. Assoc. Sci. Hydrol., 12(3): 5--21. Musgrave, G.W. and Holtan, H.M., 1964. Infiltration. In: V.T. Chow (Editor), Handbook of Applied Hydrology. McGraw-Hill, New York, N.Y., pp. 12.1--12.30. NERC (Natural Environment Research Council), 1975. Flood Studies Report, Vol 1. Nat. Environ. Res. Counc., London. Parsons, A.J., 1979. Plan form and profile form of hillslopes. Earth Surf. Process., 4: 395--402. Penteado, M.M. and Hulke, S.D., 1974. A computer method for quantitative drainage basin analysis. Not. Geomorfol., 14: 16--75. Savigear, R.A.G., 1956. Technique and terminology in the investigation of slope forms. Slopes Comm. Rep., 1 : 66--75. Speight, J.G., 1968. Parametric description of land form. In: G.A. Stewart (Editor), Land Evaluation. Macmillan of Australia, Sydney, N.S.W., pp. 239--250. Troeh, F.R., 1965. Landform equations fitted to contour maps. Am. J. Sci., 263: 616-627. van Asch, T.W.J. and Steijn, H., 1973. Computerised morphometric analysis of drainage basins from maps. Geogr. Inst., Rijksuniv. Utrecht, Utrecht, Publ. Set. B51. Wood, W.F. and Snell, J.B., 1960. A quantitative system for classifying landforms. Quartermaster Research and Engineering Command, U.S. Army, Tech. Rep. EP-124. Young, A., 1964. Slope profile analysis. Slopes Comm. Rep., 3: 45--66.
359
Elsevier Scientific Publishing Company, Amsterdam -- Printed in The Netherlands
[11 Q U A N T I F Y I N G SOURCE AREAS T H R O U G H LAND S U R F A C E C U R V A T U R E AND SHAPE
RICHARD G. HEERDEGEN and MAX A. BERAN
Department of Geography, Massey University, Palmerston North (New Zealand) Institute of Hydrology, Wallingford, Oxon OXIO 8BB (Great Britain) (Received April 7, 1981 ; accepted for publication June 10, 1981)
ABSTRACT Heerdegen, R.G. and Beran, M.A., 1982. Quantifying source areas through land surface curvature and shape. J. Hydrol., 57: 359--373. The theory of source areas is concerned with those hydrologically active localities said to be responsible for quick-response flood runoff. This paper describes a technique of map analysis identifying such localities as areas with convergent flow paths and retarding overland slope. _Measures of curvature and slope are constructed and used in a pilot-seale regression analysis with flood response quantities as dependent variables. No large improvement in predictive ability over conventional catchment characteristics are found although the relative importance of the variables, the derived curvature maps and the techniques employed will be of interest to modellers.
1. INTRODUCTION
The description of landform, especially its description in numerical terms, is an area of apparent c o m m o n interest for the geomorphologist and applied hydrologist. However, in reality the two are pulling in separate directions. The geomorphologist regards landform as the o u t p u t from an evolving system of erosion and deposition and an object of interest in its own right. On the other hand, the hydrologist's interest centres on the key that landform offers in the prediction and regionalisation of hydrological variables such as river flood or low-flow statistics. This paper resides firmly in the hydrological tradition because numerical indices of landform are developed and used to estimate flood response characteristics. However, the particular qualities of the catchment landform which are considered in this paper owe their origin to the hydrologist's idea of source areas, those restricted hydrologically-active localities which are said to be responsible for much o f the quick-response r u n o f f during flood events, as well as to the geomorphologists, concern with linking landform and landforming processes. Section 2 reviews source area theory and sets o u t the guiding principles underlying the development of the catchment characteristics.
0022-1694/82/0000--0000/$02.75 © 1982 Elsevier Scientific Publishing Company
360
The actual derivation of these characteristics which index the contour (plan) and slope or profile (sectional) curvature of each point on the basin, as well as the familiar magnitude and direction of the surface slope, is described in Section 3. Results in terms of mapped characteristics and correlations and regressions with flood statistics and unit hydrograph model parameters are described in Section 4.
2. PREVIOUS STUDIES IN SOURCE A R E A IDENTIFICATION
2.1. Landform characteristics from maps In the 35 years since Horton (1945) showed that certain characteristics of landform and stream network could be quantified, earth scientists have produced a veritable dictionary of catchment characteristics. These are well reviewed in a recent article b y Gardiner and Park (1978}. Now that the euphoria of being able to digitise and analyse b y means of computers has passed, along with the realisation that meaningful causal relationships are not so easily produced, scientists have either given up such analysis or have continued to proceed with optimistic caution. Much of the morphometric analysis of landform has been conducted at a micro-level, using arduously obtained field measurements. Analysis of the measurements has resulted in descriptive statistical treatments of the profile form including the gradient and curvature of the flow path, a slope profile terminology (Savigear, 1956; Young 1964), and the formulation of processforming relationships covering the whole range of landforms in varying climatic regions. In contrast to the extensive literature produced from the detailed approaches to landform, little attempt has been made to quantify the shape of the land surface without resorting to field measurement, i.e. from national mapping. The first attempts were initiated b y those seeking a means of landscape quantification for terrain analysis (Wood and Snell, 1960; Carr and van Lopik, 1962; Brink et al., 1966; Speight, 1968). Troeh's (1965) paper described h o w landform shape could be quantified using contour maps while Greysukh (1967) outlined the use of digital computers in landform description, followed b y papers in a similar vein a decade later (Druet, 1973; van Asch and Steijn, 1973; Penteado and Hulke, 1974). A recent paper b y Evans (1980) treads a similar path to that of Section 3.3 and demonstrates the fundamental role o f the four shape parameters to many applications. One of the major problems of working with small-scale maps and photographs has been the difficulty of maintaining consistency in the light of changing cartographic definition of contours, stream networks and stream lengths. The inconsistency from map to map has led to a large b o d y of
361 literature on such a fact, b u t apart from various methods of adjusting for such differences, little success has been achieved in overcoming the problem.
2. 2. Theories o f runoff formation The theory of the disposition of precipitation from slopes, especially in periods when the infiltration capacity of the soil is exceeded and "surface" runoff occurs, has been the subject of another b o d y of literature. Chorley (1978) reviews some of the many models which have been developed by soil physicists, groundwater hydrologists, and hydraulicians. Geomorphologists have been much more intent on explaining the landform runoff--erosion cycle, although Carson and Kirkby (1972) have redressed the balance in their b o o k b y approaching the development of slope forms through quantitative process studies. There are two major schools of thought about how storm runoff is generated. The two theories differ at both the elementary and the catchment scale. In the earlier theory, due to H o r t o n (1945), runoff occurs as overland flow when the infiltration capacity Las been exceeded. In the source area theory it is recognised that, at least in small catchments in the temperate zone, runoff is generated in a near-surface layer of soil and vegetation over a less permeable sub-surface layer (Musgrave and Holtan, 1964; Kirkby and Chorley, 1967). However, both of these processes act at the elementary scale and entail processes to which landform is largely irrelevant. On the catchment scale, runoff in Horton's theory is deemed to take place uniformly over the entire area. The source area theory recognises the observational evidence for hydrologically more active localities within a catchment (Dunne and Black, 1970; Engman and Rogowski, 1974). These include not only channels and their surrounding riparian zone b u t also contributing areas which are often variable in extent and expand as the storm continues. It is this partial area concept which offers the scope for introducing map analysis because of its dependence on the hypothesis that the shape of the land is an important causal factor in the temporal and spatial disposition of storm runoff. Many analyses have shown that catchments with increasing slope (however measured) will produce hydrographs with shorter times to peak and higher proportion volume of runoff, all other things being equal. Such measures of slope have usually involved a channel-slope measurement of some form although mean catchment slope is often used. But to date not much attempt has been made to use information from the whole of the catchment space. In doing so, the location of areas of different gradient, say flat or very steep areas, may take on much greater significance than measures of mean slope.
2.3. Landform and flood response From a hydrological point of view, landscape shape seems to have much relevance to the source of contributing areas of the catchment. Researchers
362
in the U.S.A. in the 1960's (Amorocho and Orlob, 1961; Betson, 1964) found that the percentage of watershed areas contributing to runoff varied from as little as 5% to an extreme of 85%, with low percentages being the norm and only rising to a major part during very extreme events. Many subsequent papers have dealt with this partial-area concept, b u t its applicability has been limited b y problems of measurement. Source areas could most properly be thought o f as areas of surface runoff concentration of hollow shape with perhaps some proximity to the existing permanent channel network, although Amerman (1965) found that runoff seemed to originate in seemingly random fashion from ridge-tops, valley slopes and valley bottoms. Kirkby and Chorley (1967) enumerate the types of area which are most frequently saturated (i.e. source areas) as being those adjacent to flowing streams, lines of greatest slope convergence, local concavities, and areas of thin or less permeable soil cover, with the most favoured zones for overland flow being in valley b o t t o m s and in particular in stream-head hollows. Carson and Kirkby (1972) suggest, however, that not only is there not a positive correlation between concave slope profiles and contour concavity, that is hollows, b u t that slope profiles traced down the true slope (lines of maximum steepness) on a map tend to end in major valleys and not in the smaller lateral channels. Consequently, it could be argued that contributing areas are almost impossible to define spatially. However, in a detailed field experiment, Anderson and Burt (1978) found that convergent flow occurred in relatively steep hollows simply because of the contour pattern, and that not only were the hollow zones sources of throughflow b u t also they remained saturated during periods of low streamflow. Soil surveyors recognise the likelihood of gleyed soils at these points which tend to be more impermeable. Parsons (1979) states that strong contour concavity (short radius of curvature) is positively correlated with strong profile concavity, but much less so when the curvatures are less. He also suggests that if slope curvature is the more dominant curvature factor affecting runoff processes, then contour curvature, by its relationship with slope curvature, would produce effects at a very localised level perhaps causing concentrations of water locally within the whole slope. The general findings of this survey would tend to support the rather random location of hollows in most catchments.
3. D E R I V A T I O N O F C A T C H M E N T A N D F L O W C H A R A C T E R I S T I C S
3.1. Background to the study
This study is part of a continuing effort to extend and improve the results of a nationwide analysis of floods contained in the Flood Studies Report (NERC, 1975). This report presented regression equations to predict unit hydrograph time to peak and standard percentage runoff from various catch-
363 ment characteristics representing size, slope, drainage, soil type, land use and climate. As with all such studies some individual catchment predictions were considerably in error and the main object of this exercise was to test whether these outliers could be reduced by a better numerical description of the landform. These new landform catchment characteristics are based on the supposition, in turn deduced from the source area concept, that a significant proportion of flood r u n o f f derives from convergent, retarding-flow localities. The following subsections describe the method of calculating the necessary plan and slope curvatures, and the slope vector (aspect) and gradient of the surface. 3.2. Map digitisation and surface fitting The shear impracticality of large-scale field measurements produces a reliance on existing forms of data, and the contour map is the source of most potential information. Derivation of slope characteristics such as plan and slope curvature, gradient and slope vector can be determined at individual locations from a c o n t o u r map, but catchment~wide derivation of such values requires a more generalised approach. Using an interpolative computermapping routine, SACM, a regularly spaced grid of elevation spot heights were produced from digitised contour information. This results in a reduction in information, with the loss of individual contour crenulations, and efforts to minimise this by increasing the density of the spot-height grid merely produces a non-rational topography. The grid replicates the general form of the surface as if subjected to a low-pass filter whose properties are a function of contour spacing and topographic detail. Once a grid of spot heights had been created each spot could then be located three-dimensionally: x and y as plan coordinates, and z as elevation. By focussing on one grid point it was hoped that its elevation and those of the neighbouring points would yield directly values of plan and slope curvature, gradient and slope vector. Each grid point in turn would then be inspected to produce maps of the quantities for the whole catchment. 3.3. Calculation o f slope parameters This proved to be unsuccessful so a two-step procedure was adopted. The first step was to fit a function of known mathematical form -- in this case a quadric surface -- to the 3 × 3 matrix of grid points surrounding a given point. The second step was to compute the landform parameters from the properties of the surface. Difficulties were experienced in fitting the nine points to a general quadric (step 1) which permitted a folded surface. A restricted five-parameter surface was adopted finally which has the equation: z = ax 2 + b y 2 + e x y + d x + e y which was solved in the following way.
(1)
364 The nine elevations identified, Z7
Z8
Z9
Z4
Z5
Z6
Z1
Z2
Z3
from
which
the
surface
w a s t o be d e s c r i b e d w e r e
w i t h Zs h a v i n g z e r o h e i g h t , a n d all o t h e r e i g h t p o i n t s b e i n g o f relative h e i g h t t o it. All p o i n t s w e r e o n a r e g u l a r s q u a r e grid, w h o s e side in t h e c a t c h m e n t s u s e d v a r i e d f r o m 2 3 . 5 t o 1 0 0 m. T h i s v a r i a t i o n w a s n e c e s s a r y t o allow f o r c o n s t r a i n t s i m p o s e d o n t h e size o f t h e grid m a t r i x b y t h e c o m p u t e r programs. F r o m eq. 1 t h e f o l l o w i n g s y s t e m o f e q u a t i o n s w a s d e r i v e d : U1 =
Z 1 "4-Z 3 -4-Z 4 q - Z 6 q - Z 7 + Z 9
v2 = z 1 + z 2
+z 3 +z 7 +z s +z 9
va = z 1 - - z 3 - z
7 +z 9
(2)
V4
:
--Zl
-~ Z 3 - - Z 4 -~ Z 6 - - Z 7 "4-Z 9
V5
=
--Zl
- - Z 2 - - Z 3 --~Z 7 --~- ZS -~- Z9
f r o m w h i c h t h e u n k n o w n s in eq. 1 w e r e s o l v e d as f o l l o w s : a =
(0.3vl -- 0.2r2)/G 2
t
b =
(0.3v2 -- 0.2vl)/G2
t(3)
c = v3/4G2;
d = Va/6G;
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e = Vs/6G
w h e r e G is t h e grid size, i.e. d i s t a n c e b e t w e e n t h e s p o t h e i g h t s in m e t r e s . I n s t e p 2 t h e f o u r p a r a m e t e r s - - s l o p e v e c t o r (1 ° - - 3 6 0 ° ), g r a d i e n t , p l a n o r contour curvature, and slope or flow path curvature -- were then calculated using the following formulae: -1 ( - - e / - - d ) - - 9 0 ( - - d / i d [ )
[ s l o p e v e c t o r (° )]
=
180--tan
[ g r a d i e n t ( m m -1 ]
=
(d 2 + e 2 )0.s
[ p l a n c u r v a t u r e ( m -1 )]
= e (cd -- 2ae)/(d 2 + e 2 )l.s
(4) (5) (6)
Fig. 1. Croasdale Beck at Croasdale Flume (71003), grid reference SD706546. A. Contour map (heights in metres). B. Areas of concave slope curvature with a radius of curvature less than 100 m. C. Areas of tight concave contour curvature ( ~ 100-m radius) are shown in black. The unshaded and shaded squares correspond to all squares in F. D. Concave plan curvature less than 250-m radius. E. Concave plan curvature less than 500-m radius. F. Concave plan curvature less than 1000-m radius.
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[slope curvature (m -1 )] = 2 (ad 2 + be 2 + cde)/(d: + e 2 )(1.0 + d 2 + e 2 ) 1. s
(7) Having established a catchment~wide matrix of slope parameters, mapping and statistical analysis were then carried out.
3.4. Data analysis With the aid of a plotting package, there is an almost limitless range of options available for the portrayal of the data. But the problem was not one of portrayal but of analysis -- that is, how is the spatial distribution of a parameter in one catchment compared with that in another? There are some very sophisticated analytical techniques available but as with all analysis, the o u t p u t has to have some physical reality. Will it distinguish, for example, between areas of strong concavity adjacent to stream channels in one catchm e n t and those in the high interfluves in another? Areas within each catchment that were considered to have one or more of the features that characterise source areas were identified by means of a "Boolean chop", that is, listing or mapping only those grid points where the value of the feature was either above or below certain threshold values, or in some cases, between two values. For example, the maps o f Croasdale Beck show areas where the radius of concave slope curvature {reciprocal of curvature) is less than 100 m (Fig. 1B), and the succeeding maps (Fig. 1C--F) show incremental areas of decreasing concave plan curvature from 100 to 1000 m radius. Similarly, maps of combined parameters, such as slope curvature and plan curvature (Fig. 2C) or gradient and plan curvature (Fig. 2D--F), can be drawn to show spatially related areas. Maps combining slope gradient and aspect, drawn with arrows, probably do not produce any more information than can be interpreted from a good contour map, but the information is easily reproduced a n d analysed (Fig. 2B). Thus source areas, if one accepts a definition of both plan and slope concave curvature, enhanced by steepening gradient, can be identified and mapped. But meaningful spatial analysis seems to be much less attainable. Fig. 2. River Bourne at Hadlow (40006), grid reference TQ632497. A. Contour map (heights in metres). B. Generalised slope direction and gradient (arrow shaft length is proportional to slope angle). Some of the perimeter points slope outwards because of lack of boundary definition, and the main channel often shows reverse slope because of its gentle gradient relative to the surrounding area. C. Areas of c o n c a v e s l o p e and plan curvature with radii less than 500 m, which are generally adjacent to the streams. D. Areas of steep gradient ( ~ 250 m km -1 ) and tight concave plan curvature ( ~ 250-m radius) which are located primarily in headwater areas. E. Areas with same plan curvature ( ~ 250-m radius) as D but including less steep areas (gradient ~ 100 m km -1 ). F. Areas with same gradient as E ( ~ 100 m km -1 ) but more open concave plan curvature ( ~ 500-m radius).
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368 A visual estimate of the intensity and spatial distribution of the curvature feature in each catchment was made to predict the tendency to over- or underestimate flood response and subsequently compared with known catchment variables such as average time-to-peak and flood magnitude. The complete lack of success in establishing any relationships in this "double blind" but subjective test only served to underline the futility of trying more sophisticated spatial and analytical techniques. Less subjective tests were tried and these are described in the next section.
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Fig. 3. England and Wales: location of catchments. 4. RESULTS The objects of the study had been partly fulfilled in that the practicality of developing characteristics indexing landform shape had been demonstrated. The other important objective of testing whether the new characteristics were capable of improving predictions of flood characteristics remained to be tested and is discussed in this section. Three flood characteristics were considered:
369 TP
= (time to peak of the 1-hr. unit hydrograph)
SPR
= (standard percentage r u n o f f being the ratio of r u n o f f to gross rainfall in storms, reduced to a standard depth and antecedent catchment wetness)
MAF = (mean annual flood) All three variables are extensively discussed in the F l o o d S t u d i e s R e p o r t (NERC, 1975) in which t h e y appear as pivotal variables in recommended procedures for flood estimation at an ungauged site. The catchments selected for study (Fig. 3) were among the smaller ones used in t h a t study, for reasons o f economy in map digitisation. Also catchments were selected from among those which were worst predicted by the existing regression equations i.e. MAF furthest removed from MAF, etc., so that the predictive process could be improved, not merely repeated. The following equations were used to derive parameters in the catchments studied. TP
= 46.6,S1085 -°'4 ,L0.14 ,RSMD-0.4 ,URB-2
(8)
SPR
= 95.5,SOIL + 0.12,URB
(9)
MAF
= C , AREA0.94 ,STMFRQ0.27 , S O I L 1.23 ,RSMD 1.03, (1 + LAKE) -°'ss , $ 1 0 8 5 °'16
(10)
where $1085
= channel slope between 10% and 85% of the main channel length
(m km -1 )
URB
= proportion of catchment under urban development
RSMD
= 5-yr. return period 1-day effective rainfall
(mm)
L
= main stream length
(km)
SOIL
= index of winter rainfall acceptance potential
AREA
= topographic catchment area
STMFRQ = stream frequency LAKE
(km 2 ) (junctions/km 2)
= proportion of catchment draining through a lake
C
= regionally varying multiplying factor ranging from 0.0153 in eastern Britain to 0.0315 in southwestern Britain Note: $1085 is the only one that can be regarded as directly indexing some aspect of landform. The use of " o u t l i e r " catchments has resulted in small differences in the data sets available for the different tests.
370 TABLE I Landform indices Variable number
Variable acronym
Description
XI
GMED
X2
GLQ
X3
GVQ
X4
GFT2
Xs
GST20
median gradient, i.e. 50% of the catchment is steeper and 50% flatter than X 1 lower quartile gradient, i.e. 25% of the catchment is flatter than X 2 upper quartile gradient, i.e. 25% of the catchment is steeper than X 3 proportion of catchment flatter than 2° proportion of catchment steeper than 20 °
Gradients:
Slope curvatures: X6
SCMED
X7
SCLQ
Xs
SCLT500
median slope curvature, i.e. 50% of the catchment has a slope radius of curvature less than X 6 lower quartile slope curvature, i.e. 25% of the catchment has a slope radius of curvature less than X 7 proportion of catchment with slope radius of curvature less than 500 m
Plan curvatures: X9
PCMED
X10
PCLQ
Xll
PCLT500
median plan curvature, i.e. 50% of the catchment has a plan radius of curvature less than X 9 lower quartile plan curvature, i.e. 25% o f the catchment has a plan radius of curvature less than X10 proportion of catchment with less than 500 m (tight valley contours)
T h e i n i t i a l a p p r o a c h as m e n t i o n e d in S e c t i o n 3 . 4 c o n s i s t e d o f i n f o r m a l l y ranking the catchments according to an eye estimate of the number and location of source areas. This subjective approach failed to reveal any obvious conclusions; clearly the cause of the poor prediction was not due solely to the landform elements that were visible from the maps. The next stage was to develop numerical indices from the maps and perform correlations and regression analyses to try to extract linkages between f l o o d a n d c a t c h m e n t c h a r a c t e r i s t i c s . T h e s e n u m e r i c a l i n d i c e s a r e l i s t e d in T a b l e I, d i v i d e d i n t o g r a d i e n t a n d t h e t w o t y p e s o f c u r v a t u r e g r o u p s . A l s o i n c l u d e d in t h e d a t a s e t w e r e t h e s t a n d a r d c a t c h m e n t c h a r a c t e r i s t i c s l i s t e d above, and observed and estimated flood characteristics.
371 The within-group intercorrelations are considered first. Six slope characteristics including $1085 were available. This channel slope characteristic, which was employed in NERC (1975) r e c o m m e n d e d equations was quite well correlated with all five overland slope indices; 0.73, 0.70, 0.75, --0.72, 0.83 with X 1 to X s . It appears that $ 1 0 8 5 is slightly better correlated with indices of "steepness" than with indices of "flatness" X2 and X4. In fact X4, measuring catchment flatness stands out as separate from the other four overland slope indices with lower correlations o f - - 0 . 5 5 , --0.52, --0.55 and --0.68, with X1, X2, Xa and X s , respectively, elsewhere values of 0.9 and above being the rule. The two curvature groups (X6--X 8 and X9--Xll ) display similar behaviour. The median measure in both cases is relatively poorly correlated with the indices focussing on "tight" curvatures which are themselves intercorrelated. Median plan curvature in particular appears almost totally uncorrelated with the other two plan curvature based characteristics. There is also very little correlation between the two groups, for e x a m p l e - - 0 . 1 2 between X 6 and X9, 0.39 between X 7 and X10, and 0.37 between Xs and Xxl. It appears that there is no tendency for plan curvature to be dictated b y slope curvature thus supporting Parson's (1979) conclusion. Arguing from first principles, linkages would be expected between gradient along the flow path and slope curvature as "tight" slope curvature could not occur without steep slopes at the upper end. In fact only X s , which focusses totally on this aspect displays a high correlation, 0.77 with X s . There is virtually no correlation between any of the slope gradient measures and any of the plan curvature indices, the highest value among all 3 × 6 possible values barely exceeding 0.3. Flow variables were constructed based upon the observed and estimated values of TP, SPR and MAF. Poor correlations between these and the landform indices confirmed the visual impression of the initial test that there was no immediate explanation of outliers in the presence or absence of potential source areas. For example, a multiple correlation on the residual from the MAF equation succeeded in explaining only 15% of the variance with still lower values for other residuals. One might expect to see the major enhancement to predictive ability in the two process-oriented hydrological variables TP and SPR. The first is well correlated with $ 1 0 8 5 in the total Flood Studies Report data set, with a correlation of --0.81. In this data set the correlation with $1085 is much poorer, --0.46, b u t nevertheless rather better than most of the derived overland slope measures. The exception is X4 which focusses on the flat areas and is spectacularly better correlated than the others, with a correlation of 0.73. Multiple regressions involving overland slope, X , , plan curvature, X9, and urban fraction would yield useful predictions although whether in practice these would represent an improvement over the present recommendation involving $1085 would require further testing. Nearly 95% of the variance in $1085 can be explained b y using one of each of the groups of landform indices in a multiple regression.
372 The major predictor of SPR, the volumetric response, is the soil index, SOIL. There is some correlation between SOIL and the gradient variables which is to be expected given t h a t the soil classes were in part determined by slope gradient. This is reflected in the gradient correlations with SPR of ~ 0 . 5 . The correlations with curvature variables are lower -- prediction of SPR from curvature variables alone would succeed in explaining less than 20% of the variance which does not lend much support to the contributing area theory. However, the landform variables add a useful extra-explained sum of squares of the SOIL variable alone, raising the multiple correlation coefficient from 0.67 to 0.73.
5. SUMMARY Contour maps contain more inherent information about landform shape than any other existing source, and digitising of the contours produces a data file which can easily be reproduced. But definition of landform shape from such information appears to be impracticable unless a uniform matrix of spot heights is substituted for contours. Mathematical algorithms can then be utilised on the spot heights to produce shape parameters such as slope and plan curvature, slope gradient and slope aspect. Computer drawn maps using all or parts of this information can be produced, and individual results are very worthwhile. However, substantial difficulties prevent any spatial variations and comparisons from being ascertained. Indices of shape and gradient are used to predict flood response, and show that in part t h e y do reflect the sorts of responses that would be expected. However, their predictive ability is thwarted to some extent by the generalisation of slope form which has to be undertaken to produce the parameters. Further studies could well derive methods of obtaining data which more accurately reflect the nature of the landform, and in so doing utilise the vast a m o u n t of information currently stored on contour maps.
ACKNOWLEDGEMENTS The authors thank the Director of the Institute of Hydrology, Wallingford, for permission to publish this paper, and, in the case of the first author, for the opportunity to be a visiting scientist there. Acknowledgement is also given to the Council of Massey University, Palmerston North, New Zealand, for leave to pursue this project. Thanks are also due to many members of the Institute's staff for their assistance in the execution of this project. The study was carried out as part of a commission for the U.K. Ministry of Agriculture, Fisheries and Food.
373 REFERENCES Amerman, C.R., 1965. The use of unit-source watershed data for runoff prediction. Water Resour. Res., 1 (4): 499--507. Amorocho, J. and Orlob, G.T., 1961. An evaluation of inflow--runoff relations in hydrologic studies. University of California, Berkeley, Calif., Water Resour. Cent., Contrib. 4 1 , 49pp. Anderson, M.G. and Burt, T.P., 1978. Towards more detailed field monitoring of variable source areas. Water Resour. Res., 14(6): 1123--1131. Betson, R.P., 1964. What is watershed runoff?. J. Geophys. Res., 69(8): 1541--1552. Brink, A.B., Mabbutt, J.A., Webster, R. and Beckett, P.H.T., 1966. Report of the working group on land classification and data storage. Mil. Eng. Exp. Estab., Christchurch, England, Rep. 940. Carr, D.D. and van Lopik, J.R., 1962. Terrain quantification, Phase 1 : Surface geometry measurements. U.S. Air Force Cambridge Res. Lab., Bedford, Mass., Rep. A F C R L 63-208. Carson, M.A. and Kirkby, M.J., 1972. Hillslope Form and Process. Cambridge University Press, London, 475 pp. Chorley, R.J., 1978. The hillslope hydrological cycle. In: M.J. Kirkby (Editor), Hillslope Hydrology. Wiley, New York, N.Y., pp. 1--42. Druet, Y., 1973. L'utilisation de l'ordinateur pour l'analyse morphom~trique du bassin versant en rue de la planification du r~seau hydrom~trique national du base. Rev. G~omorphol. Dyn., 22 : 173--178. Dunne, T. and Black, R.D., 1970. Partial area contributions to storm runoff in a small New England watershed. Water Resour. Res., 6(5): 1296--1311. Engman, E.T. and Rogowski, A.S., 1974. A partial area model for storm flow routing. Water Resour. Res., 10(3): 464--472. Evans, I.S., 1980. An integrated system of terrain analysis and slope mapping. Z. Geomorphol., Neue Folge, 36: 274--295. Gardiner, V. and Park C.C., 1978. Drainage basin morphometry: review and assessment. Progr. Phys. Geogr., 2: 1--35. Greysukh, V.L., 1967. The possibility of studying land forms by means of digital computers. Soy. Geogr., 8: 137--149. Horton, R.E., 1945. Erosional development of streams and their drainage basins: hydrophysical approach to quantitative morphology. Bull. Geol. Soc. Am., 56: 275--370. Kirkby, M.J. and Chorley, R.J., 1967. Throughflow, overland flow and erosion. Bull. Int. Assoc. Sci. Hydrol., 12(3): 5--21. Musgrave, G.W. and Holtan, H.M., 1964. Infiltration. In: V.T. Chow (Editor), Handbook of Applied Hydrology. McGraw-Hill, New York, N.Y., pp. 12.1--12.30. NERC (Natural Environment Research Council), 1975. Flood Studies Report, Vol 1. Nat. Environ. Res. Counc., London. Parsons, A.J., 1979. Plan form and profile form of hillslopes. Earth Surf. Process., 4: 395--402. Penteado, M.M. and Hulke, S.D., 1974. A computer method for quantitative drainage basin analysis. Not. Geomorfol., 14: 16--75. Savigear, R.A.G., 1956. Technique and terminology in the investigation of slope forms. Slopes Comm. Rep., 1 : 66--75. Speight, J.G., 1968. Parametric description of land form. In: G.A. Stewart (Editor), Land Evaluation. Macmillan of Australia, Sydney, N.S.W., pp. 239--250. Troeh, F.R., 1965. Landform equations fitted to contour maps. Am. J. Sci., 263: 616-627. van Asch, T.W.J. and Steijn, H., 1973. Computerised morphometric analysis of drainage basins from maps. Geogr. Inst., Rijksuniv. Utrecht, Utrecht, Publ. Set. B51. Wood, W.F. and Snell, J.B., 1960. A quantitative system for classifying landforms. Quartermaster Research and Engineering Command, U.S. Army, Tech. Rep. EP-124. Young, A., 1964. Slope profile analysis. Slopes Comm. Rep., 3: 45--66.