WATER
RESOURCES
RESEARCH,
VOL. 34, NO. 5, PAGES 1335-1343, MAY 1998
A physical explanation of the cumulative area distribution curve
Hemantha Perera and Garry Willgoose
Department of Civil, Surveyingand EnvironmentalEngineering,The Universityof Newcastle,
Callaghan,New South Wales, Australia
Abstract. A physicalexplanationfor the behaviorof the cumulativearea distribution
(CAD) basedon the Tokunagachannelnetworkmodel is given.The CAD is dividedinto
three regions.The first region,for smallareas,is dependenton hillslopeflow
accumulationpatternsand representsthe catchmentaverageof the hillslopeflow
accumulationin the diffusiveerosion-dominatedareas,upstreamreaches,of the
catchment.The secondregion representsthat portion of the catchmentdominatedby
fluvial erosion.This region is well describedby a log-loglinear power law, which results
from the scalingpropertiesof the channelnetwork.The scaleexponent,•b,is highly
sensitiveto a parameter of the Tokunagastreamnumberingscheme.The exponent•b
convergesto -0.5 for higher order Tokunaganetworksfor parametersconsistentwith
topologicalrandom networks.Small networkshave lower valuesof •b,which asymptotic
convergesto •b = -0.5 as the catchmentscaleincrease.The third region reflectsthe
lowestreachesof the channelnetwork,the scaleof the catchment,and is a boundary
effect.An explicitanalyticalsolutionto the scalingpropertiesin the secondregion is
derivedon the basisof the Tokunaganetwork model.
1.
Introduction
The cumulativedistributionof drainagearea is the percentage of the catchmentthat has an area per unit width draining
througha point which is greaterthan or equal to a givenarea.
It is a geomorphologicmeasurethat characterizespart of the
catchmenthydrologyand is a measurecommonlyusedby geo-
morphologists
andhydrologists
to characterize
the scaleinvariant structureof channelnetworks[Tarbotonet al., 1989;Rodriguez-Iturbeet al., 1992; Inaoka and Takayasu, 1993; La
Barbera and Roth, 1994]. This cumulativearea distribution
(CAD) characterizesthe scalingpropertiesof drainagearea.
The CAD and its scalingpropertiesare key componentsin the
developmentof physicallybasedmodelsthat link geomorphology and hydrology [Perera, 1997; Ibbitt et al., 1997]. Perera
[1997]showedthat slopeof the CAD in both the hillslopeand
channeldomainsis important for describingthe geomorphic
impact on subsurfacesaturationexcessrunoff generation.
Usinga digitalelevationmodel(DEM) of a catchment,the
drainagepath through,and the drainagearea per unit width of
each point in the catchmentcan be computed.For a grided
DEM the cumulativearea(A) is determinedasthe numberof
pixelswithinthe catchment,whichhavedrainagearea,a (number of pixels),drainingthroughgreaterthan or equalto some
specifieddrainagearea (a *),
this distribution as a means of characterizingthe flow aggregation structureof channelnetworks.Other researchers[Moglen and Bras, 1994; Sun et al., 1994a, b, c; Inaoka and Takayasu, 1993] have used the CAD for the calibration of their
geomorphologymodels.
Figure la showsthe CAD for the Middle Creek catchment,
whichis describedelsewhere[Willgoose,
1994],from the Pokolbin region of Australia.The CAD has a characteristictype of
behavior that is common for catchmentsin many different
regions.These distributionscan be dividedinto three regions
[Moglenand Bras, 1994] as shownin Figure 1. Region 1 representssmall drainageareasthat we postulateto be the hillslopeflow aggregation.Region 2 represents'that part of the
catchmentdominatedby channelizedflow.This regiontendsto
follow a straightline in log-logscale,obeyingthe power law
distribution as,
A (a >- a *) cr(a*)*
(2)
where •bis the scalingexponent.Region 3 consistsof the few
nodes on the main trunk
of the channel
network
near to the
catchmentoutlet. The largejumpsand stepsin region3 result
from the large tributaries that join the main trunk and contribute large percentagesof area. In this region the CAD
rapidly decreaseswith increasingdrainagearea.
This paper deals with the characterizationof the channel
(region2) and hillslope(region 1) regionsof the CAD. The
.q(a>-a*) = N,
Tokunagachannelnetworkmodel [Tokunaga,1978] is usedto
i=1
simulatea range of channelnetworksto examinethe channel
behavior.The effect of TokunagaparametersK andE • on the
N i = 0 if a i < a*,
N, = 1 if a i -> a*
scalingpropertiesof region 2 of the CAD are examined.We
where a t is the total area of the catchment and a i is the will showthat the parameterK is stronglyrelatedto the scaling
drainagearea of ith pixel. A log-logplot of A againsta* is exponent,•b,while E• showsa lesspronouncedeffect on •b.
calledthe CAD (Figure 1). Rodriguez-Iturbe
et al. [1992]used The hillslopebehavior of region 1 is studiedusing different
hypotheticalhillslopeflow patternsand different sizesof the
Copyright1998by the American GeophysicalUnion.
hillslopearea.The hillslopebehaviorof the CAD is a resultof
the average hillslope flow accumulationpattern of a catchPaper number98WR00259.
0043-1397/98/98WR-00259509.00
ment. The behaviorof region 3 of the CAD is shownto be a
1335
1336
PERERA
AND
WILLGOOSE:
(a)
104
i
• • I iiii I
i
i i i i iii I
:
-- 103
.,•
• 102
<
i
i i i •11•1
'
i
i •i
Observed
PHYSICAL
a networkthat doesnot sufferfrom thoseproblems[Tarboton,
1996] and is scaleinvariant.Tokunagaused the Strahler orderingsystem[Strahler,1952] to define ordersof the channels
but defined a different stream law. The Tokunaga model is
built as follows.
_•
!• Analytical__
The averagenumber,/,E,,, of streamsof order n entering
from sidesinto a stream of higher order k. ParametersE i
(i = 1,..., rn - 1) and K are givenby
> 10]
=
EXPLANATION
mEm_l: m_lEm_2'- icEi•_
1 -- nEn_l '- 2E1 = E•
mEm_2 -- m_lEm_3 .....
1
3El -- E2
(3)
region-1
region-2 .
i
region-3
-
mEm_j: m_lEm_l_j
.....
-
•+jE•= Ej
0.1
1
10]
102
103
Area (pixels)
nagamodelE• andK are assumed
constants,
so that Es =
Ki- •E Tokunaga
[1978]showed
thattheparameters
K = 2,
1'
-0.5
-1.5
•
-2
•
-2.5
Ej =K
Ej_•
where m is the highestorder of the catchment.In the Toku-
(b)
0
•
E2 E3
E1 E2
104
E• = 1 correspondto the mean propertiesof topologically
distinctrandom networks[Shreve,1966]. A drainagenetwork
with K = 2, E• = 1 is shownin Figure 2 [Tokunaga,1978].
3.
Channel
Behavior
The CAD generallyobeysa log-log power law in region 2
(Figure la). This power law behaviorhas been observedin
natural basins[Rodriguez-Iturbe
et al., 1992;La Barberaand
-3.5
Roth, 1994]and simulateddrainagebasins[Sunet al., 1994a,b,
-4
, , , , ,,111
i i i i iiill
i , , , ,,,,
c; Inaoka and Takayasu,1993] and appearsto be a universal
behavior of catchments.To study this universality,drainage
1
10
100
1000
Area (nodes)
networkssimulatedwith the Tokunaga model are examined,
Figure 1. (a) Observedand analyticalcumulativearea distri- togetherwith their dependenceon the Tokunagaparameters
bution of Middle Creak natural catchmentin Pokolbin region K and E•.
Two drainagenetworkswith parametersK = 2 andE • = 1
of Australia.Data from 20 m DEM. (b) Slopeof the observed
CAD between each data point (raw data) and the moving are simulated assumingtwo different zeroth-order hillslope
flowpatterns,hill 1 and hill 2 (shownin Figures3a and 3b and
averageof five data points.
describedin detail in the nextsection).K = 2 andE• = 1 are
typicalof observedvalues[Tokunaga,1978]and, as previously
boundaryeffect.Finally,an analyticalsolutionfor the CAD in noted, are consistentwith the topologicallyrandom network
region2 is presentedbasedon the Tokunagachannelnetwork model. Figure 4 displaysthe CAD of two 10th-orderdrainage
model.Usingthe analyticalsolution,we demonstratethat scal- basinswith the different hillslope flow patterns. In Figure 4,
ing exponent,4>convergesto -0.5 for higher-ordernetworks region2 of both CADs can be approximatedby a straightline
for parametersK = 2 and E• - 1, which are consistentwith showingpower law behavior with an exponentof approxi--
Raw data
•
Moving averageof 5
data points
--
topologically
randomchannelnetworks.Thisvaluefor 4>of the
topologicallyrandomchannelnetworkwasalsoobservedbyDe
Vrieset al. [1994]. The exponentfor smallnetworksgenerally
deviatesfrom thisvaluebut asymptotically
convergesto it with
increasingarea.
2.
The Tokunaga Channel Network Model
Channelnetworksfor this studyare constructedon the basis
of Tokunaga channel numbering system[Tokunaga,1978],
which is different from the Horton [1945] and Strahler[1952]
lawsof drainagecomposition.The Horton and Strahlerdrainage compositionlaws suffer from conceptualproblemswith
respectto scaleand similaritysincetheir bifurcationratiosvary
significantlywith scaleand are applicableonly for scaleinvariant, andphysically
unrealistic,structurallyHortonian•tworks
[Scheidegger,
1968;Tarboton,1996;Kirchner,1993].The Toku- Figure 2. Hypothetical drainage network with K = 2 and
nagacyclicitymodel is an alternativemethodof characterizing E• - 1 [Tokunaga,1978]. The numbersrepresentthe order.
PERERA
AND
WILLGOOSE:
PHYSICAL
EXPLANATION
1337
, ,tillIll , II,[ll[[ , ,]lllHq I , ,t,,,tI ' '''''"l
[ [1'1["1 [ lit[,,l[
mately rb = -0.5. That rbis independentof the zeroth-order
hillslopeflow pattern suggests
that the hillslopedrainagepat• 108
,
-Hill-1
,
-........ Hl11-2
tern doesnot influencethe scalingpropertiesof the CAD in
region 2, only the interceptin the vertical axisin Figure 4.
The effect of changesin the valuesof K and E • are exam- '-'1
ined next. Figure 5 showsthe CAD of the seventh-ordersim<
region-3
ulatedbasinsusingthe hill 2 zeroth-orderhillslope.Resultsfor
g 104
ß
.
•--_.
parametervaluesofK = 4, E• = 1 andK -- 2, E• = 2 are
comparedwith the resultsfor nominal parameter values of
K = 2 andE • -- 1. Figure 5 showsthat whenK increasesfrom
• 102
2 to 4, with constantE •, the slopeof region2, 4•increasesfrom
approximately-0.5 to -0.25, suggesting
an inverserelationship betweenK and rb.There is no observableeffect on the
1 ........
I ........
I ........
I ........
I ........
I ........
I ........
slopewith the changesin E•.
1
102
104
106
108
This dependenceon K of the CAD is becauseK characterArea (pixels)
izes the scalingproperties of the network. A more highly
branchednetwork will have a higher value of K so that the Figure 4. Cumulative area distributionsof 10th-order simuCAD declinesmore slowly.
lated catchmentsusingTokunagaK = 2 and E• -- 1 for the
two differenthillslopeflow patternsfrom Figure 3.
O
4.
_.,•_
-..,,•
Hillslope Behavior
In this section we examine
how the zeroth-order
basin hill-
slope flow path patternsinfluencethe form of the CAD, in
particular,region 1. Two hypotheticalhillslopedrainagepatterns are shownin Figures3a and 3b, named hill 1 and hill 2.
The main differencebetweenthesepatternsis in the form of
the lateral inflow into the first order channel.Tokunagachannel networksup to order 10 have been simulatedusingthese
different zeroth-ordernetworks.The CAD of orders1 through
order 10 are shownin Figures6 and 7.
The CAD of the first-order networksin Figures6 and 7 are
different
II
I I I I I I I I I I
I
I I 1I I I I I
IIIIIIIIII
I I I I I I I I I I
I,I I I I I I I I I I I
111 I I I I I I I I I I
• IIIIIIIIII
I'1 I I I I I I I I II
because of the differences
in the zeroth-order
hill-
slopeandflow path patterns.In region1 of Figures6 and 7, the
slopeof the CAD initially increaseswith increasingarea, with
the slope then decreasingwith further increasingarea. This
behaviorhasbeenobservedin naturalcatchments(e.g.,Figure
lb). In hillsloperegionsa high percentageof the area of the
catchmenthas a very small drainagearea, leading to fast decline in cumulativeareawith increasingdrainagearea and thus
increasingthe slopeof CAD initially on the left-hand side of
region 1. As the drainage area increases,tributariesmerge,
resultingin a faster increasein the drainagearea leading the
upward concaveregion of the curve on the right-handside of
region1 (Figurelb).
Despite the differencesin the flow path patternsusedin the
(a)
107
••
106
'•
105
•
o
<
1 04
•D
-
> 103 r
=1 K=4
'
- - -E =2, K=2
•= 10•
....... E1=1, K=2
1
(b)
•E
'•
-'E 102 •
10•
1
102
103 104 105
Area (pixels)
106
10?
Figure 3. Hypotheticalfirst-order basinswith zeroth-order Figure 5. Cumulative area distribution of seventh-order
hillslopeflowpatterns.Arrowsindicatethe flowdirections.(a) catchmentfor hill 2 hillslope flow patterns from Figure 3b
usingdifferent valuesof TokunagaK and E •.
Hill 1 and (b) hill 2.
1338
PERERA AND WILLGOOSE: PHYSICAL EXPLANATION
randomlyselectingvarioussizesof first-orderareaswith the
hill 2 drainagepattern(with 145, 181, 225, 277, 337, and 405
nodes,coveringthe rangeof first-orderareaswithinthe Pokolbin catchmentin Figure 1). Figure8 showsthat the effectof
•, 108
•106
variableareas(with the sameflow path patterns)doesnot
significantly
affect the shapeof the CAD in region 1. In a
naturaldrainagebasin,wherethe area of the hillslopes
vary,
the shapeof the region 1 of the CAD thusrepresentsthe
averagehillslopeflow accumulation
pattern.
Figuresl a, 6, and 7 showthat the hillslopebehaviorof the
• 104
• 102
CAD (region 1) are different from the channelnetworkbehaviorin region2, reflectingthe differentflow accumulation
patternson channelandhillslope.If the convergence
propertiesof the hillslopeandchannelwerethe samethenregion1
and 2 would not be distinguishable.
Thus the differencein
1
1
102
104
106
108
Area (pixels)
regions
1 and2 appears
to be a usefulindicator
of changing
andpotentiallyof erosionprocess,
asexhibFigure 6. Cumulativearea distributionof order-1through flowaggregation,
order-10networksfor hill 1 drainagepatternfrom Figure3a ited by geomorphology.
usingTokunagaK - 2 andE1 = 1. Numbersrepresentthe
orders. Broken lines show the boundaryof the different
5. Erosion Process-BasedInterpretation of the
regions.
Cumulative
Area Distribution
Willgoose
etal. [1991b]arguethatthearea-slope
relationship
twosetof networks
in Figures6 and7, thechannelaggregation of a catchment
obeysa powerlawwith a changein the expopatternsfor higher-order
networksare the same,asK andE• nent as the catchment switches from diffusive erosion domiare same.Region1 hasno effecton the shapewith increasing nance(i.e., sedimentflux • rate constantx f(slope)) for
order.We interpretthisto meanthat region1 represents
the smallerareasto fluvial erosiondominance(i.e., sediment
hillslopecomponentof the catchmentwhere scaleinvariant flux • rate constantx f(area, slope))for largeareas.The
channelnetworkaggregation
hasno influence.
Points(pixels) area-sloperelationshipof the Middle Creek catchmentis
on hillslopes
havesmallerdrainageareasrelativeto points shownin Figure9 [Willgoose,
1994].The changein the expo(pixels)on channelnetwork.The hillslopeflowpatternsdom- nent for largerdrainageareasoccursat an area of approxiinatefor smalldrainageareas,whilewith increasing
areathe mately10pixels
(200m2/m).Drainage
areas
lessthan10pixels
influenceof the hillslopedecreases
and the channelaccumu- aredominated
by diffusion,andareasmorethan10 pixelsare
lationpatterndominates.
Thusthe CAD for smalldrainage dominated by fluvial erosion. This switch from diffusion to
areasdependson hillslopeflowpath patterns.
fluvialdominanceat an area of 10 pixelscoincides
with the
In the discussion above the areas of all of the first-order
boundary
betweenregions1 and2 in the CAD of Figurel a.
basinshavebeenassumed
to be equal.In a naturaldrainage Thiscoincidence
of areabetweenthe area-slope
relationship
basin the first-orderareaswill vary randomly.To test this and CAD can be observed in most of the natural catchments used
effect,drainagenetworksup to order 6 were simulatedby byMoglenandBras[1994],suggesting
thatregion1 of theCAD
is diffusiondominatedandthatregion2 is fluviallydominated.
108
_•glon!
106
107 -
region-2
6
•10 s __:
_
-
,
•
i
105
i I tilt
I
I
i
i i i i iiI
i
••order•
102
•
•
i_••.••__orders3
•
10:
1:',,, 1 ' '.,,•,,•
,6
....
,7,
8.....
•9,,,•
I I I II
•
<•103 __••••.••order4
03
i
!•••x..• order
6
,• 104
,
I
•
order 1
• 101
1
_
10• 102 103 104 105 106 107 108
Area (pixels)
1
101
102
103
Area (pixels)
Figure 7. Cumulativearea distributionof order-1through
order-10networks
of hill 2 drainage
patternfromFigure3b Figure8. Region1 of the cumulative
areadistributions
of
using
Tokunaga
K - 2 andE• = t, Numbers
represent
the order-tthrough
order-6
networks
simulated
randomly
select-
orders.Brokenlines showthe boundaryof the different ingvarious
sizes145,225,277,237,and405nodesof first-order
regions.
areasof hill 2 drainage
pattern.
PERERA
AND
WILLGOOSE:
PHYSICAL
We thusassertthat the boundarybetweenregions1 and 2 of
the CAD
will move to the left for catchments
EXPLANATION
1339
106
with less diffu-
sion transport.This hypothesisis examinedby simulatinga
'7' 0s
drainagenetwork assuminga unit area (one pixel) for firstorder areas(exterior link areas) and interior subbasinareas
3 4
(interior link areas).This meansthat the catchmentaccumulation pattern is dominatedby channelflow. CAD of order 10
< 103
with K - 2 andE 1 - 1 are shownin Figure 10, andthe log-log
linearity extendsup to the leftmostregion.This alsoindicates
',•
• 102
that if the hillslopedrainagepattern is the same as channel
flowpattern,both hillslopeandchannelregions(regions1 and
2) of the CAD will exhibita similarbehaviorwithout break in
• 10•
slope.
This factor can be observedin the CAD (Figure 11) of a
syntheticcatchmentsimulatedby the SIBERIA basin evolu1
101 102 103 104 105 106
tion model [Willgooseet al., 1991a].This catchmentis domiArea (pixels)
nated by fluvial erosionsincethe diffusioncomponentof SIBERIA hasbeen turned off. Figure 11 showsthat the log-log Figure 10. Simulatedand analyticalcumulativearea distrilinearity is continuousalmostup to the area of two pixels,at bution of an order-10 Tokunaganetworkwith unit interior and
exterior link areasfor TokunagaK = 2 and E1 = 1.
which numerical diffusion in the SIBERIA
solver became important [Willgooseet al., 1989]. Thus region 1 is small for
catchments
with little diffusive
erosion and vice versa.
In naturaldrainagebasins,hillslopeareas(small areas)are
dominatedby diffusiveerosionand as area increasesmoving
downstreamerosionis increasinglydominatedby fluvial processes.Channelflow compositionpatternsare determinedby
fluvial erosionso that region 2 is dominatedby fluvial erosion
and region 1 is dominatedby diffusion.
MoglenandBras [1994]notedsimilarbehaviorwith a variant
of the SIBERIA model. They showedthat an increasein diffusivitymovedthe transitionfrom region 1 to region2 to the
right which is consistentwith our hypothesis.
6. Analytical Solution for the Channel Network
Domain
assumedconstantand interior link areas(ai) proportionalto
ae as ai = ca•, where c is a constant.
To simplifythe derivationbelow,we initiallyassumea• = 1,
and c = 1, so that exterior and interior link areasare unity. A
more general analyticalsolutionis given later in this section.
From (3) the Tokunagacyclicsystemgivesthe number of
n th-order channels directly draining laterally into an rnthorder channel,,,Nn, as
rnNn
= E1K(m-1)-n
(4)
Therefore the total number of channelsdirectly draining laterally into the rnth-order channelis
rn-1
It is possibleto derive an explicitanalyticalsolutionto the
scalingpropertiesof the cumulativearea distributionin the
channel network domain, when drainage networks are describedby the Tokunagamodel. This resultyieldsthe log-log
linearityobserved.The first-orderexteriorlink areas(ae) are
Z mNn
= E1q-E1Kq-... q-E1Km-3
q-E1Km-2
n=l
=El
104
•
(5)
K-1
[
[ [ I [,][j
]
I
] [][][J
t
I [ [][[IJ
]
[ [ [
Observed
103
Analytical
,,,,•
• 102
0.1
<
;> 101
',•
:
1
_
_
_
.
_
1
10
100
1000
Area/unitwidth (m)
10000
100000
0.1
i
1
10•
•02
Area (pixels)
i
i
i i iii
104
Figure 9. The area-slopedata for the Middle Creek catchment in the Pokolbinregionof Australia.The dotsare the raw Figure 11. Observedand analyticalcumulativearea distribudata from 20-m DTM, and the circlesare averageof 20 points tion of a syntheticcatchmentsimulatedby SIBERIA model
with the same area to more clearly show the mean trend usingonly fluvial erosionsinceno diffusiveerosionmodeled
[Willgoose,1994].
[1451lgoose,
1994].
1340
PERERA
AND
WILLGOOSE:
PHYSICAL
2 + E 1q-K + [(2 + E1 q-K) 2- 8K] 1/2
107
•
EXPLANATION
Q=
106
2
(13)
KnowingK andE 1, the drainagearea for anygivenorder basin
and the cumulative
_•105
area for that basin area can then be calcu-
lated.
This analyticalsolutionfor the valuesK - 2 andE1 -- 1 is
comparedwith the previouslysimulated10th-orderTokunaga
networkwith unit exteriorand interiorlink areas(Figure 10).
g 103
The fit of the analyticalsolutionto the simulationsis good,
with the calculationsfor the analyticalsolutiongiven in Ap• 10•
pendixB.
The effect of changesin the valuesof parametersK andE
V,•101
on the analyticalsolutionhas alsobeen examined.Figure 12
comparesthe analyticalCAD of 10th-orderbasinfor K -- 4,
1
E1 = 1;K=
2, E 1 = 1;K=
1.5, E 1 = 1;K=
2, E1 =
2; and K - 2, E1 = 0.5. From Figure 12, whenK increases
1
10• 102 103 104 105 !06 107
for constantE 1, the slopeqbof the CAD decreases,
and when
Area (pixels)
K decreases,qbincreases.There is no apparentchangein
Figure 12. Analytical cumulative area distributionsof or- whenE 1 changeswith constantK. This resultsare consistent
der-10 Tokunaganetworkswith unit exteriorand interior link with the previouslydiscussedsimulationresults.
areasfor different TokunagaK and E 1,
Figure 12 shows that the slope of the CAD gradually
changes,asymptoticallyreaching a constantwith increasing
area. This behaviorwas examinedby computingthe analytical
CAD by increasingthe order. The slopesof the distributions
between
areaof orderi andorderi + 1 (i = 1, 2, 3, ..- ) for
sothat the numberof linksin an m th-orderchannelL m canbe
catchments
of orders 10, 20, and 30 are computedand shown
written as
in Figure 13. As area increasesthe scalingcoefficient,qbconK m-1 - 1
vergesto -0.5 for K = 2 and E 1 = 1. De Vrieset al. [1994]
Lm= 1 + E1 K- 1
(6) deriveda relationshipof qbwith Horton's constantsand topologicaldimension.They computedqb-- -0.5 for an infinite
With all links havingunit area, the total area of an m th-order topologicallyrandomchannelnetwork,which is in agreement
basin(am) equalsthe total numberof linksin that basin.This with our analyticalresultsfor TokunagaparametersK - 2 and
is equalto the cumulativearea,mA 1 (numberof pixels),which E1 - 1. The order-10 basinin Figure 13 did not attain this
has the drainagearea (a) greaterthan or equal to the first- constantvaluebecauseof the slowconvergence.
Middle Creek
order area a 1 (wherea 1 = ae = 1 node),
behavedsimilarly(Figure lb), not attaininga constant4>bethat slowconvergence
mAl(a -->al = 1) = am= 2m•l- i
(7) causeof smallsize.Figure 13 suggests
• 104
.5•
El=
1
• ....x....K=2,
E•=O
5
and the finite size of natural catchmentscontrol variation of
wheremldb
i (i = 1 ... m) is the numberof ith-orderchannels The slope of the CAD divergesfrom the constantvalue for
in an order rn basin.
higherdrainageareasnear to the outlet (Figure 13) becauseof
In general, the cumulativearea (mAn) of an mth-order the boundaryeffectof the networks.Hancock[1997]observed
drainagebasin,whichhasa drainageareagreaterthan or equal
to the area of nth-order subbasin(an) within an mth-order
basin,can then be written as (seeAppendixA)
-0.5
mAn(a• an)= 2m•l-
1 + ml&n
--i•1 m
/..L
i 1+E1K-1
and the area of the n th-order
(8)
ß- -0.55
"8"•
_
• -0,6
subbasin is
an = 2nt.•1-- 1
(9)
,
where, from Tokunaga[1978],
ca -0.65
mP'm= 1
mP'n
= P(m-n-1)(2
+
-0.7
0
(Q(m-n-1)_p(m-n-1))Q(2q_E1_ p)
+
(Q_p)
2
P=
(11)
2
1/2
5
10
15
20
25
30
Order (i)
Figuee 13, Slope of the cumulative area distributionsfor
(12) varying
order
ofTokunaga
networks
withunitexterior
and
interior link areasusingTokunagaK -- 2 and E1 = 1.
PERERA
AND
WILLGOOSE:
PHYSICAL
EXPLANATION
1341
a similar scale-dependentvariation of tk for his experimental Table 1. AsymptoticValues of the Exponent tk for
laboratorycatchmentsas well as for the syntheticcatchments Different K and E 1 ComputedUsing Equation (20)
simulatedby the SIBERIA model.
K
E1
q[
We can derive an expressionfor the asymptoticexponent,
2
1
0.5
tkoo,
for the CAD with Tokunagaparameters.Peckham[1995]
4
1
0.19
deriveda relationshipfor tkoo
with Horton ratios,
2
2
0.58
log Rc
logR•
(14)
We evaluated this general analytical solution on the two
whereR c = limm•oo{Lm+l/Lm} is the link ratio. For equal
link lengths(14) is similar to that of De Vdeset al. [1994]. catchmentusedbefore, the Middle Creek catchmentand syntheticcatchmentsimulatedby SIBERIA model.The analytical
Combining(6) and (14) yields
and observedCADs are comparedin Figuresla and 11. These
figuresshow that the analyticalsolutionsare in good agree(15) ment with observed CADs.
K-i+E•(K
m-1)}
Rc= lira K- 1+ Ei(K
m-f-1)
m-.->oo
1/K
m-•1/K
m
+E•(1
- 1/K
m)
}
Rc= lim 1/-•-:r2:
•/•-••/•;-(•/•-- 1/Kin
)
m--->oo
Rc = K
(16)
K> 1
(17)
lim {Q}
(18)
From Tokunaga[1978]
RB =--=
rnl&n
(rn-n)-->oo
Discussion
The cumulativearea distribution(CAD) canbe dividedinto
three regions(Figure la). Region 1 representsthe hillslope
flow aggregationpattern of diffusiveerosiontransportdominated, upstream,drainageareas.Different hillslopeflow aggregationpatterns lead to different shapesfor region 1. In
general,the relationshipbetweenlog area and log cumulative
area observed
in the field is nonlinear.
In natural
basins the
hillslopeflow pathswill vary from subcatchmentto subcatchment, sothat region1 is the averageflow accumulationpattern
Substitutingfor RB and Rc in (14) yields
log (Q/K)
tkoo
= logQ
7.
(19)
of all first-order
basins within
the catchment.
Region 2 is that part of the cumulativearea distribution
dominatedby fluvial erosion and the channel network. The
Exponent tkoo
is computedfor different values of K and E1
cumulativearea distributionin region2 is well describedby a
usingin Table 1. These resultsshowthat tk is very sensitiveto
log-loglinear powerlaw and resultsfrom the scalingproperties
the parameterK but less so to the parameterEl, which is
of the channelnetwork.The scaleexponent,•k,for the Middle
compatiblewith our previousresults.
Creek basinis about -0.56. Rodriguez-Iturbe
et al. [1992]studIt canbe observedfrom Figure 12 that all the CADs display
ied the cumulativearea distributionof five drainagebasinsin
equalslopesbetweenareasof onepixeland fivepixels(area of
North America. They found the exponent•k to be approxiorder-2 network) on x axisirrespectiveof the valuesof K or
matelyequalto -0.43 for all their basins.Inaokaand Takayasu
El. The reason for this behavior is that K effects only the
[1993] found • = -0.42 for the river patternssimulatedby
number of branches in networks of order 2 and above.
their erosionmodelfor landforms.La BarbaraandRoth [1994]
The assumptionof unit area of interior and exterior links in
derivedan equationfor cumulativearea distributionbasedon
the abovederivationis not necessary.If it is assumedthat the
Horton's laws, related to the fractal propertiesof river netarea of exteriorfirstorder links,a e (wherea e = a 1), and area
works,and found an exponentof • = -0.43. Sune! al. [1994a,
of all interior links a i are related by a i = ca•, then the total
b, c] showedthat their minimum energydissipationdrainage
area of an nth-orderdrainagebasingivenin (9) canbe rewritnetwork model had an exponentof •k - -0.51.
ten as
This studyconcludes
that thisscaleexponent(•) is sensitive
an-- al[n/•l + C(nl&l- 1)]
(20) to the scalingpropertiesof the channelnetwork.In the Tokunaga network numbering model the scale behavior is conand the cumulativearea in (8) for n - 1 can be rewrittenas trolled by the parameterK. We find that the value of exponent
,•4•(a -> aO = a•[ml• q-C(ml•- 1)]
(21) is highlysensitiveto thisparameter.The exponent•kconverges
to a constantfor higher-ordercatchmentswith the topologiand forn -> 2,
callyrandomnetwork(K = 2 andE 1 = 1), giving• = -0.5.
IncreasingK from 2 to 4 decreasesthe exponent•k from approximately-0.5 to -0.25, inverselyproportional to K. The
parameterE1 does not appear to significantlyaffect scaling
mAn(a
•>an)--cal{ml&
1-1+rnld•n
-•
i=2
m/•i1+El K- 1
sothatforknown
average
values
ofal, c,•, and
El, CAD,
of
a drainagebasincan be analyticallyderived.The sole difference between(8) and (22) is the appearanceof a factor reflectingthe ratio of interior to exterior link areas.
behavior of the network and thus •.
Region 3 representsthose large drainage areas along the
main trunk of the channelnetwork.The boundaryof regions2
and 3 of the CAD move to the right when the order of the
catchmentis increased(Figures6 and7), indicatinga boundary
effect. Figure 13 showsthat the slope of the CAD decreases
rapidlywhen the area is closeto the total basinarea, indicating
that the behaviorof region 3 is a boundaryeffect. Ibbitt et al.
[1997]founda similartrend in the slopeof the CAD acrossthe
1342
PERERA
AND
WILLGOOSE:
range of areas shownin Figure 13 for the low Strahler order
network.
This paper has shownthat the scalepropertiesfor drainage
area can be simplyparameterizedwith a scaleinvariant network ordering model. We believe on the basis of observed
evidence[Rodriguez-Iturbe
et al., 1992;Inaoka and Takayasu,
1993;Sunet al., 1994a,b, c] andthe physicalexplanationbased
on Tokunaga network that this log-log linear behavior is a
universalpropertyof observedlandformsand resultsfrom the
natural self-organizingproperty of channelnetworksduring
their development.The slopeof the line appearsto be about
-0.5 for topologicallyrandom channelnetworks.
PHYSICAL
EXPLANATION
Table 2. Calculationof the Analytical Solution for an
Order-10 TokunagaNetwork With Unit Exterior and
Interior
1
2
3
4
5
6
7
8
9
10
Link Areas
174763
43691
10923
2731
683
171
43
11
3
1
349525
131071
54611
24571
11591
5611
2731
1291
515
1
1
3
11
43
171
683
2731
10923
43691
174763
1
5
21
85
341
1365
5461
21845
87381
349525
-0.609
-0.610
-0.571
-0.541
-0.532
-0.519
-0.540
-0.662
-4.504
.-.
Appendix A
A detailed derivationof the analyticalsolutionfor the CAD
based on the Tokunaga stream numbering schemeis given Appendix B
Sample calculationsof the analyticalsolution for an orbelow.From (7),
der-10 basinis givenbelow. TokunagaparametersK = 2 and
mAl(a > al = 1) = am= 2m/_l,
1 -- 1
(A1) E• = 1 are used for calculations.Exterior and interior link
The numberof linksin a second-order
channelis givenby (6) areasare assumedas a unit (one pixel).From (12) and (13),
as
2 + 1 + 2- [(2 + 1 + 2)2- 8 x 2] 1/2
P=
K-1
L2= I + ElK_ 1
(A2)
2
=1
2 + 1 + 2 + [(2 + 1 + 2)2- 8 x 2] 1/2
Q=
2
=4
Then the number of links or pixelswhich have the drainage
area (a) less than second-ordersubbasinarea (a2) within Substituting
m = 10, P = 1, Q = 4, andE • = 1 in (11) m/-rn
m th-order basinscan be given as
can be calculated for n - 1 to 10 as in Table 2.
m/-L1
q-m/-L2
1 q-E K
- m/.L2
(A3)
Now, loAn(a -> an) for n = 1 to 10 canbe calculatedasin
Table 2 by substitutingE• = 1, K = 2, and m/,rn(n = 1 to
10) in (8).
Accordingto the Tokunagamodel,
The last term, m/,r2,is the numberof links (or pixels)at the
outletsof the second-ordernetworkswhich have the drainage
n/-rl(for n = 1 to 9) = 10/-rl0-n+l
(B1)
area equal to a 2. Therefore the cumulativearea which has
drainagearea (a) greaterthan or equalto a 2 is
and the calculatedvaluesare givenin Table 2. Now, as all n/,r•
for n - 1 to 10 are known, a n valuescan be calculatedfrom
(9), as shownin Table 2.
mA
2(a-->a2)= 2mp,
1-- 1 -- mill,
1-- mill'2
1q-E1K 1 + mill'2
(A4)
For a third-orderbasin(a 3),
Acknowledgment. The researchwork presentedin this paper was
supportedby the AustralianResearchCouncil(ARC).
mA3(a->a3)
=2m/.rl-1--m/.rl--m/.r2
1 +ELK_ 1
References
De Vries, H., T. Becker, and B. Eckhardt, Power law distribution of
--m•3 l+E1 K- 1 +m•3
(AS)
In general, for an n th-order subbasin,
Geol. Soc.Am., 56, 275-370, 1945.
mAn(a
-->an)= 2mp,
1- I--rap,l--rap,
2 1 +ELK_
.....
mP'n
1 +El K- 1
= 2mla,
1- 1 +mld,
n-
'___
The area of an n th-order
Ibbitt, R. P., G. R. Willgoose,and M. J. Duncan, The Ashley River
channel network study:Channel network simulationmodels compared with data from the Ashley River, New Zealand, Misc. Rep.
+ mP'n
319, Natl. Inst. for Water and Atmos. Res., Christchurch, New
Zealand, 1997.
mld,
i 1 + E1 K- 1 '
subbasin is the total number
dischargein ideal networks,WaterResour.Res.,30, 3541-3543, 1994.
Hancock,G., Experimentaltestingof the SIBERIA landscapeevolution model, Ph.D. thesis,Dep. of Civ., Surv. and Environ. Eng.,
Univ. of Newcastle,Callaghan,Australia, 1997.
Horton, R. E., Erosionaldevelopmentof streamsand their drainage
basins:Hydrologicalapproachto quantitativegeomorphology,
Bull.
of links
in the subbasin;
Inaoka, H., and H. Takayasu,Water erosionas a fractal growthprocess,Phys.Rev.E Stat.Phys.PlasmasFluidsRelat.Interdiscip.Top.,
47(2), 899-910, 1993.
Kirchner, J. W., Statisticalinevitabilityof Horton's laws and the apparent randomnessof stream channel network, Geology,21, 591594, 1993.
La Barbera,P2;andG2R0th, invarianceandSCaling
propertiesin the
distributionof contributingarea and energyin drainagebasin,Hy-
an = 2n/-rl- 1
(A7)
drol.Processes,
8, 125-135,1994.
PERERA
AND
WILLGOOSE:
PHYSICAL
EXPLANATION
1343
Moglen, G. E., and R. L. Bras, Simulationof observedtopography
river basinsand channelnetworksusingdigital terrain data, Tech.
using a physically-based
basin evolution model, Tech. Rep. 340,
Rep. 326, Ralph M. ParsonsLab., Mass. Inst. of Technol., CamRalph M. ParsonsLab., Dep. of Civ. Eng., Mass. Inst. of Technol.,
bridge, 1989.
Cambridge,1994.
Tarboton,D. G., R. L. Bras,and I. Rodriguez-Iturbe,A physicalbasin
for drainagedensity,Geomorphology,
5, 59-79, 1992.
Peckham,S. D., Self-similarityin the three-dimensional
geomorphology and dynamicsof large river basins,Ph.D. thesis,Fac. of the Tokunaga,E., Considerationon the compositionof drainagenetworks
Grad. School, Univ. of Colo., Boulder, 1995.
and their evolution,Geogr.Rep. 13, Tokyo Metrop. Univ., 1978.
Perera,H. J., The hydro-geomorphic
modellingof sub-surface
satura- Willgoose,G. R., A physicalexplanationfor an observedarea-slopetion excessrunoff generation,Ph.D. thesis,Dep. of Civ., Surv. and
elevation relationshipfor catchmentswith decliningrelief, Water
Resour.Res., 30, 151-159, 1994.
Environ. Eng., Univ. of Newcastle,Callaghan,Australia, 1997.
Rodriguez-Iturbe,I., E. J. Ijjasz-Vasquez,R. L. Bras,and D. G. Tar- Willgoose,G. R., R. L. Bras, and I. Rodriguez-Iturbe,A physically
boton, Power law distributionsof discharge,mass, and energy in
basedchannelnetwork and catchmentevolutionmodel, Tech.Rep.
river basins, Water Resour.Res., 28, 1089-1093, 1992.
322, Ralph M. ParsonsLab., Mass. Inst. of Technol., Cambridge,
1989.
Scheidegger,A. E., Horton's law of stream numbers,WaterResour.
Res., 4, 655-658, 1968.
Willgoose,G. R., R. L. Bras, and I. Rodriguez-Iturbe,A physically
Shreve, R. L., Statistical law of stream numbers, J. Geol., 74, 17-37,
based coupled network growth and hillslope evolution model, 1,
1966.
Theory, WaterResour.Res.,27, 1671-1684, 1991a.
Strahler,A. N., Hypsometric(area-altitude)analysisof erosionalto- Willgoose, G. R., R. L. Bras, and I. Rodriguez-Iturbe, A physical
pography,Geol. $oc.Am. Bull., 63, 1117-1142,1952.
explanationof an observedlink-area sloperelationship,WaterResour. Res., 27, 1697-1702, 1991b.
Sun,T., P. Meakin, and T. Jossang,
The topographyof optimal drainage basin, WaterResour.Res., 30, 2599-2610, 1994a.
Sun, T., P. Meakin, and T. Jossang,A minimum energydissipation
H. Perera and G. Willgoose,Department of Civil, Surveyingand
model for drainage basinsthat explicitly differentiatesbetween EnvironmentalEngineering,The Universityof Newcastle,University
channelnetworksand hillslopes,Phys.A, 210, 24-47, 1994b.
Drive, Callaghan, NSW 2308, Australia. (e-mail: cegrw@cc.
Sun,T., P. Meakin, and T. Jossang,Minimum energydissipationmodel newcastle.edu.au)
for river basin geometry,Phys.Rev. E Stat. Phys.PlasmasFluids
Relat.Interdiscip.Top., 49, 4865-4872, 1994c.
Tarboton, D. G., Fractal river networks,Hortons law and Tokunaga
cyclicity,J. Hydrol.,187, 105-117, 1996.
(ReceivedSeptember19, 1997;revisedJanuary14, 1998;
Tarboton,D. G., R. L. Bras,and I. Rodriguez-Iturbe,The analysisof acceptedJanuary23, 1998.)