All
Allometry,t scalingli andd scale-specificityl
ifi itoff cirques,ilandslides
l d lidand other landforms

Ian S. Evans, Durham University
CSIS Seminar
14 October
October, 2009.
2009 The University of Tokyo

CONTENTS• Connections…Japan• Discontinuities: specific (&general) geomorphometry• Allometry; cirques (allometric scaling)• 95% confidence intervals on regression
allometric coefficients (power exponents)• Alternative RMA / SD line coefficients• Cirque development; the ‘buzzsaw’• Scale specificity …second broad theme• Landslide size distributions• Conclusions

JAPANESE PAPERS (in English)
cited in my PhD on Geomorphometry & Glaciation• IImamura G 1937 Past
P t glaciersl i& presentt topographyth off thethJapanese Alps. Science Reports of Tokyo B. D. , C7, 61 pp.
Di
Dimensionlessi lH
Hypsometrict i IIntegraltl …cirquesi• Hoshiai M & Kobayashi K 1957 A theoretical discussion on
the so-called ‘snowline’… Japanese J. of Geology & Geogr.
28 (1-3),(1 3) 61
61-75.
75• K
Kobayashibhi K 1958 Quaternary
Q tglaciationl i ti off theth JapanJ
Alps.
AlJ. of Fac. Liberal Arts & Sci, Shinshu Univ. 8 (II), 13-67.
• Higuchi K & Iozawa T 1971 Atlas of perennial snow patches
in Central Japan
Japan. Water Research Laboratory
Laboratory, Nagoya
University, 81pp.

SOME INFLUENTIAL JAPANESE PAPERS(in English), to 1981…quantitative• Yatsu E. 1965 On the longitudinal profile of the graded river.
p y Union 36, 655-663.
Trans. Amer. Geophys.
• Hirano M. 1968 A mathematical model of slope development…
J. of Geosciences, Osaka City Univ. 11 (2), 13-52.
• Sunamura T. 1977 A relationship between wave-induced cliff
erosion and erosive force of waves. J. Geology 85, 613-618.
• Tokunaga E. 1978 Considerations on the composition ofdrainage networks... Geogr. Reports, Tokyo Metro. Univ. 13, 127 (& 1972
27.
1972, 6/7
6/7, 39
39-49)
49)• Ohmori H. 1978 Relief structure of the Japanese mountains
and their stages in geomorphic dedevelopment.
elopment Bull.
B ll Dept
Dept. Geogr
Geogr.
Univ. Tokyo 10, 31-85.
• Yoshikawa,
Yoshikawa T.,
T Kaizuka,
Kaizuka S
S. & Ota
Ota, YY. 1981 The landforms ofJapan. University of Tokyo Press Crustal movements.

GEOMORPHOMETRY
International Conference at Zurich,A
Augustt 2009;2009
g at –
Proceedings
http://geomorphometry.org/
Please register there!
Geomorphometry Society;next biennial conference: 2011
(Location to be announced in late 2009.)

In manylandscapes,pwe
can recognize
discontinuities:
in altitude,
gradient, aspect& curvatures

Minar & Evans2008
Geomorphology

Discontinuities relate to breaks in the continuity of form and process,and typically arise because of rock contrasts,

or events in the historical development of the landscape.
These discontinuities can be joined up – currently, rather subjectively(a satisfactory automated routine is a continuing research challenge)
to outline Elementary Forms (segments, units, facets)of the land surface.
These in turn can be associated with neighbours, with which theirdevelopment is related, to define specific landforms such as
cirques drumlins
cirques,
drumlins, dunes
dunes, landslides
landslides, volcanoes
volcanoes, valley-sides
valley-sides…
When completely delimited, landforms can be measured and theirposition, size and shape (including gradient) can be analysed.
This is specific geomorphometry, of which a large part is relating
shape and size of delimited forms – the study of allometry or isometry…

Nine Stages in a (specific) geomorphometric analysis:
1. Conceptualisation of landform types
2. Precise operational definition
3. Complete delimitation from surrounding land
4. Measurement of position, direction, size, gradient, shape & context
5. Calculation of derived indices, ratios
6. Assessment of frequency distributions; transformation - check effects
7. Mapping & spatial distributional analyses
8. Interrelation of attributes, e.g. shape v. size or position
9. Interpretation cf. genesis & chronology

Evans 1987, ‘The morphometry of specific landforms’. In V. Gardiner (Ed.)
te at o a Geo
Geomorphologyo p o ogy 1986
986 Parta t II
International
Techniques have changed, but all nine stages still apply…
P S if you don’t
P.S.
don t like / do not find discontinuities
discontinuities, use general geomorphometry
geomorphometry…
…first, allometry of cirques…

ALLOMETRY
Many landforms develop allometrically, that is they change
shape as size increases. In all but the most dynamic situations
this can be tested only by considering variation with size at agiven time, i.e. static allometry, as was proposed for cirques
originallyi i ll ffor a smallll populationl ti (15) iin C
Colorado.
l d It isi nowpossible to test this for several cirque populations, each muchbibiggerthan
th iin Ol
Olyphant’s
h t’ (1977) original
i i l study.
t dResults are presented here for different regions of British
Columbia, Britain, Romania, Scandinavia, and Spain.
g
pplots of horizontal and vertical dimensions againstg
Logarithmic
overall size are presented. They show that, as size increases,
cirque length increases faster than vertical dimensions
dimensions.
This is confirmed wherever the 95% confidence intervals on
exponents do not overlap – which is consistent across regions.

Cirques of various grades of development.

1: Podragu in the Făgăraş
Mts. , a classic cirque witha large rock basin lake.

2: Vârtopul de Vest, a welldefined cirque in the Făgăraş
Mts.; it has a good headwall
and floor, but no lake.

3: Răţiţiş,3Răţiţi a definite
d fi itcirque in theCălimani Mts. witha moderate
headwall curved
around an
outsloping floor.
Photos;Marcel Mindrescu.
(See Evans 2006b

4: Ursu in the Căpăţânii Mts
Mts.,
Geomorphology’ for
5 Balota,
5.
Balota on Muntele
M ntele Mic (Ţarc(Ţarcu),) ‘Geomorphology’
a poor cirque because of thea marginal cirque because of a
Welsh examples).
gentle headwall and irregular floor. very poor headwall.

Cirque definition mapof Iezer Mountains,
after Mindrescu 2006

Defining thevariables:i bllength of median axis,
andd width,idth ini green.

Axial aspect = 9ºPlan closure = 19+(360-219)=160º
Marcel Mindrescu
Height range = crest maximum to threshold:Wall height = highest part of headwall

F = cirque focus(middle of threshold)threshold).
1, 2, 3, 4, 5, 6 =locations where altitudeinterpolated.
(Vertical) Amplitude =fall from top of median axis
to bottom,
i.e. Alt(6) – Alt(1)Height range = highest
altitude on crest– lowest (on threshold)threshold),i.e. Alt(4) – Alt(1)10 = line of maximum(head)wall height.

Romania: mountain ranges with cirques.
Blue = many,y, yellowy= few.

Marcel Mindrescu

Maramures

Northingg, middle
e of media
an axis, kkm

5300

Northern Romania

Rodna

Calimani

5200
Bihor

Transylvanian Alps

5100
CindrelFagarasRetezat
I
Iezer

Tarcu
Godeanu

5000
5100

Bucegi(Southern
Carpathians)

Parang

5200
5300
5400
Easting, middle of median axis, km

Cirques of the Romanian Carpathians

DATA:
Marcel Mindrescu
Stefan cel Mare U.,
SuceavaSize = cube root of (length x width x amplitude).
llengthth ((m))width (m)amplitude (m)2000

Length (m)Width (m)Amplitude (m)llengthth regressioniwidth regression
amplitude regression

2000

Allometric plot

1000

length regression
width regression
amplitude regression

Allometric plot

1000

500

500

200

200
100

631 cirques,
Romania200

500

Size (m)1000

132 cirques,
Iezer & Northern

100
200

500

Size (m)Scales are logarithmic. Vertical & horizontal scales differ.

1000

Regression gradients would be the same, if shape did not change with size.
Length (m)Width (m)Amplitude (m)2000

length regression
width regression
amplitude regression

Allometric plot

2000

1000

length regression
width regression
amplitude regression

Length (m)Width (m)Amplitude (m)Allometric plot

1000

500
500

200
200

293 cirques,Western Romania

100

200

500

Size (m)206 cirques,iFagaras Mountains

100

1000

200

500

Size (m)1000

n.b. best to have many cirques, for precise estimates of regression coefficients

Exponents for logarithmic (power) regressions
off size
i variables
i bl on overallll cirqueisizei_______________________________________________________________
Variable
expon
expon.
95% conf.
conf R2, % Northern Fagaras Retezat-G
Length1.095

1.06-1.13

88| 1.12

1.06

1.08

Width

1.043

0.99-1.09

72| 1.06

1.09

1.08

Amplitude

0.861
0 861

0 82-0 91
0.82-0.91

68| 0.82
0 82

0 84
0.84

0 79
0.79

Height range 0.871

0.83-0.91

72| 0.860.85

0.81Wall height

0.78-0.92

50| 0.81

0.82

0.74

0.852

_________________________________________________________________
95% confidence intervals and R2 measures of fit for all 631 cirques in Romania
are given on the left. Exponents are given on the right for three larger regions.

Length exponents are significantly above 1.0, i.e. length increases fasterthan size as a whole. Amplitude and other vertical dimensions haveexponents significantly below 1
1.0;0; vertical growth of cirques lags behindhorizontal growth, their shape changes with size… they are allometric.

ROMANIA: 95% confidence limits on regression coefficients (allometric exponents)

L
LengththWestern
FagarasIezer & NorthROMANIA

Width

WesternWt
FagarasIezer & NorthROMANIAIf 95% confidence limits do not
overlap, coefficients are significantly
diffdifferentat theh 1% llevell(a rough test…)

Height range
Western
FagarasIezer & NorthROMANIA

Amplitude

Western
FagarasIezer & NorthROMANIA.6.81
isometry

Allometric exponents

1.2

Allometry: change in cirque shape with size.
L
Lengthth andd widthidth grow fasterf t thanth verticalti l amplitude.
lit dlength (m)length regression

length (m)length regression

width
idth ((m))

widthidth regressioni

width
idth ((m))

widthidth regressioniamplitude (m)amplitude regression

amplitude (m)amplitude regression

2000
2000
1000

1000

500

500

200
200
100

260 cirques,
Wales
200

500

Size (m)1000

158 Lake District
cirques, England

100

200

500

Size (m)1000

Length (m)Width (m)Height range (m)2000

le n g th (m )w id th (m )a m p litu de (m )length regression
width regression
height range regression

Allometric plot

20 00

1000

10 00

500

50 0

200

20 0

206 cirques,
Gallego basin,
S
Spanishi hP
Pyrenees

100
200

500

Size (m)leng th regres sion
w id th re g res s io n
a m p litu d e re gre ss io n

10 0

1 98 C ayo o sh cirq ue s , B .C .

1000
20 0

Pyrenees data: J.M. Garcia-Ruiz et al.

500

Size (m )1000

Length (m)Width (m)Amplitude (m)2000

Length (m)Width (m)Amplitude (m)length regression
width regression
amplitude regression

Allometric plot

Allometric plot
2000

1000

1000

500

500

200

200

222 cirques
cirques,
Bendor Range,
B. C. Coast Mtns.

100
200

500

length regression
width regression
amplitude regression

1000

126 cirques
cirques,
Shulaps Range,
B. C. Coast Mtns.

100
200

Size (m) Provisional data sets – I.S. Evans

500

1000

Size (m)British Columbia Coast Mountains
LengthBendorShulaps
CayooshWidth
BendorShulaps
Cayooshy

Height range
BendorShulaps
CayooshAmplitude
BendorShulaps
Cayoosh.6.8

1
isometryy

Allometric exponents

1.2

Western Europe
LengthPyrenees
N. Scandinavia
Lake District
Wales

Width

Pyrenees
N. Scandinavia
Lake District
Wales

Height range
Pyrenees
N. Scandinavia
Lake District
Wales

Amplitude

N. Scandinavia
Lake District
Wales

.6

.8

1
y
isometry

Allometric exponents

1.2

ALTERNATIVE RELATIONSHIPS• Traditionally, allometric relations are logarithmic
regressions of one size component, y, treated as
dependent, on overall size, x.
• ThisThi hash theth advantagedt
that
th t theth exponents
t off
length, width & amplitude sum to 3.0;
if one >1.0,>1 0 another must <1.0.
<1 0• Alternatively, if we give y and x equal status, we may
prefer to calculate the standard deviation line (reduced
major axis, geometric mean regression).
• This will always be steeper than the regression of y on xx.
• Thus there is no constraint on the exponent sum, which
will be greater where scatter is greater (R2 is lower).
• bootstrap provides 95% confidence limits.

WHAT DIFFERENCE DOES IT MAKE?

Romanian cirque width
3.3

Lake District cirque amplitude

Exponents & confidence limits:
Regression 1.015 0.939 - 1.091

2.70

SD line 1.228 1.181 - 1.276

2.50loog amplitude

Logarrithm of width (m)3.1

Exponents & confidence limits:
Regression 0.736 0.646 - 0.825
SDline 0
0.929
929 0
0.830
830 - 1.028
1 028

2.9

27
2.7

2.30

2.10

2.5
log of width (m)SD lineRegression

2.3
2.3

2.5
2.7
2.9
Logarithm of size (m)3.1

log amplitude
SD lineRegression
2.30

2.50

2.70
log size2.90

bigger difference where scatter greater…
big scatter here for amplitude, 8 cirques < 90 m:
coefficients not well specified, data editing needed.
N. Scandinavia cirque length
3.5

Exponents & confidence limits:
Regression 0.835 0.723 - 0.947
SD line 1.561 1.407 - 1.716

Exponents
Et & confidencefidlilimits:
itRegression 1.177 1.106 - 1.248
SD line 1.445 1.371 - 1.520
3.1

3.1

2.9

2.9

2.7

lo
og amplituddelog lengthh

3.3

N. Scandinavia cirque amplitude

2.7
2.5

2.5
2.3
2.1

2.3
1.9
2.1

log length
SD lineRegression

1.9
2.3

2.5

2.7
2.9
logsize

3.1

3.3

log amplitude
SD lineRegression
2.3

2.5

2.7
2.9
logsize

3.1

3.3

Pyrenees

N. Scandinavia

Lake District

Wales

The

log length

Data:

log widthhL

log ammplitudeWlog
g he ight rang
ge

Amp
HR
log size

log size

log size

log size

More scatter in N
N. Scandinavia data set - outlierswith low amplitudes – data problems…

sizealternative exponents:
RMA (geometric regression,
regression SD line)
exponents for Length, Width, & Amplitude:
Pyrenees

1.310 1.204 0.994 (HR)* *N ScandinaviaN.

1 445 1.255
1.445
1 255 1.561
1 561 (unreliable)

Lake District

1.231 1.210 0.929 * *Wales1.207 1.186 1.156 no significant difference

Romania1.167 1.228 1.044 * *Bendor, B.C.

1.211 1.140 1.113 *Shulaps B
Shulaps,
B.C.
C

1 138 1.206
1.138
1 206 1.009
1 009 *

Cayoosh, B.C.
1.169 1.224 1.042 * ** * significantly less (95% confidence intervals do not overlap, L or W)* significantly different at .05 level. Essentially confirms previous analysis.
Amplitude exponents are smaller than L & W exponents,
but some 95% confidence limits now overlap.

LengthWestern
FagarasIezer & NorthROMANIA

Returning to –ROMANIA

Width

Western
FagarasIezer & NorthROMANIA

95% confidence limits onregression coefficients((allometric exponents)p)

Height range
Western
FagarasIezer & NorthROMANIA

LengthPyrenees
N. Scandinavia
L k Di
Lake
District
ti t
Wales

AmplitudepWestern
FagarasIezer & NorthROMANIA

Width

.6

.8

1
isometry

1.2

Allometric exponents

Western Europe

Pyrenees
N. Scandinavia
Lake District
W l
Wales

Height range
Pyrenees
N. Scandinavia
Lake District
Wales

LengthBendorBdShulaps
CayooshAmplitude

British Columbia
Width Coast Mountains

N. Scandinavia
Lake District
Wales

.6

BendorShulaps
Cayoosh.8

1
isometry

Allometric exponents

Height range
BendorShulaps
CayooshAmplitude
BendorShulaps
Cayoosh.6.8

1
isometry

1.2

1.2

SUMMARY TABLE
Exponents for logarithmic (power) regressions of size variables on overall size
nr. of cirques Length
Width Ampl. Height range
Data source
Nn Scandinavia * 541 1.18
Nn.
1 18
.99
99
.84
84
.65
65 S.
S Hassinen
Hassinen, cf
cf. 1998
Blanca, Rockies
15 1.14
1.20.66 Olyphant 1977
Lake DD. *158 1.17
1 17
1 10
1.10
.74
74
.75
75 Evans and Cox 1995
C. Span. Pyrenees 260 1.17
1.06
.77 J.M. García-Ruiz et al. 2000yB.C.
198 1.10
1.05.85.83 Evans & McClean, 1995
Cayoosh,Shulaps, B.C.
126 1.06
1.07.87.83 Evans, unpubl.
Maritime Alps
432 1.08
1.08.84 Federici and Spagnolo, 2004
Bendor, B.C.
222 1.12
0.98.91.84 Evans, unpubl.
Romania
631 1.10
1.04.86.87 M. Mindrescu, Ph. D.
W l *
Wales
260 1.12
1 12
.98
98
.90
90
.91
91 Evans
E
2006
Ben Ohau, N.Z.
94
.99
1.00
1.01 cf. M. Brooks et al. 2006

* in ‘old massifs’: the others are in active orogenic belts.
Evans et al. & Mindrescu data are based on identical cirque definitions.
E
Exponents
t off 1.00
1 00 iindicate
di t equall change,hi.e.
i isometryit(found only in Ben Ohau, N.Z.).

Results are consistent in confirming the static allometry of
glaciall i l cirques:illarger cirquesi
are relatively
l ti l llonger anddbbroader,d morethan they are deeper. Power coefficients for length and width are
generallyll aboveb
1.0,
1 0 while
hil ththose ffor depthd th are significantly
i ifi
tl b
below.
l
I
In
most regions the length exponent exceeds the width exponent: henceth allometrythellt cannott beb explainedl i db
by llateralt l coalescenceloff cirques.
i
All length exponents are significantly above 1.0, and all
d th exponents
depth
t are significantly
i ifi
tl below,b lwhetherh th verticalti l didimensioniis expressed as height range, axial amplitude or headwall height.
RelationsRl tibbetweentllengthth andd width,idth hhowever, vary b
betweent
ranges.
It is inferred that cirque headwall retreat is faster thancirqueiddeepening.
iY
Yett many cirquesihhave deepdlakes
l k thatth t attest
tt t totconsiderable cirque deepening; this means that cirque development
i allll thinthree didimensionsi
iis considerable.
id blThis study shows the importance of considering confidence
iintervalstl whenh making
ki conclusions
l i
about
b t relative
l ti rates
t off change;hand of checking consistency between regions.

Headwall recession and floor deepening:
DATA BASED MODELS of CIRQUE DEVELOPMENT :DATA-BASED
means for the 52 cirques in each of five equal size classes in Wales.

(Evans(E
ans 2006
2006,
Geomorphology)
This data-based
data based
generalization
uses floor andcrest altitudes &max and min
max.
min.
gradients.
Plan: development ofa hypothetical
average mid-height
mid height
contour, using planclosure width
closure,width, andhalf the length.

profileplan

Where glaciation was asymmetric, headward
extension of valleys or divide displacement by ae km sshowso s coconsiderables de ab e headwall
ead a retreat
e ea (c(cirqueque
few
lengthening):
1.5 to 3 km

Terrace, West Central B.C., Hanson 1924

up to 2.5 km

Shulaps Range, B.C., Evans 1972

2 to 5 km

Bendor Range, B.C., Evans 1972

up to 1 km

Kenai & Talkeetna, S. Alaska, Tuck 1935

up to 2.5 km

Sa. Nevada, Brocklehurst & Whipple 2002

0.9 to 4.4 km Kyrgyz Ra., C. Asia, Oskin & Burbank 2005

Faster headwall recession implies support for the ‘buzzsaw hypothesis’ of
rapid glacial erosion limiting the height of many mountain ranges
ranges.
Instances of complete range truncation are, however, hard to find:coalescent and back-to-back cirques are common,
common but only occasionally
do intervening ridges seem to have been removed.
Cirques are rarely more than 2 km long or wide. It is interestingthat cirques in plateau areas, where range truncation has clearly notoccurred, are not dissimilar in size to those in more dissected mountains
with back-to-back cirques, where the buzzsaw hypothesis might beapplicable.
The hypothesis implies that only very small areas can rise highabove snowline (firnline, ELA). Yet analysis of the World Glacier Inventory
shows
h
th
thatt many glaciersl irisei >1000
1000 m aboveb
ELA
ELA, especially
i ll iin th
theHimalayas and Tien Shan but also in other orogenic belts.
At the extreme,
extreme icefields over 3000 km2 in area are found in
Patagonia, SE & C Alaska, & the Karakoram. It is likely that rapid
Quaternary uplift in these areas carried mountains quickly through the zoneof rapid glacial erosion, into that of cold ice frozen to its bed.

There is evidence for a glacial buzzsaw,
i partsint off theth westernt
USA andd theth Andes:A da parallelism between present and former ELAs
and summit altitudes (Mitchell & Montgomery 2006).
But this is by no means universal; many presentday glaciers start well above ELA :a horizontal buzzsaw would not accountf many resultsfor
lt fromfWorld
W ld GlacierGl i IInventory.
t(& application to Japan limited?)
Cirque headwall retreat may limit
y increase in relief,any
but downward erosion of cirque floors andglacial troughs is a major part of the erosionthat affects orogen development (uplift pattern).

INTERIM CONCLUSIONS : cirqueq allometryyLarge cirques differ in shape and gradient from small ones.
T ki llarger cirques
Takingias hhavingi ddevelopedld ffurther,theach dimension can be plotted against an overall size
measure to express static allometry or isometry.
Vertical dimensions increase more slowly than do horizontal
horizontal,
and length usually increases faster than width, but length –width relations vary between areas.
(exponents: length > width > height)
The allometric nature of cirque development is supported
on theth basisb i off a sett off largeliinventoriest i off cirques.
i
Acknowledgements to Marcel Mindrescu,
Nick Cox, J.M. Garcia-Ruiz and S. Hassinen.

As well as such variation in shapep with size, landforms usuallyy show alimited range of size. This is clear for glacial cirques, which vary over
only one order of magnitude.
As larger cirques
develop in length
and width morethan in depth,
th showtheyhscalingbehaviour as well
as scalespecificity.
Cl l thClearlythese
behaviours canbe combined.

Cirque scale-specificity: low standard deviations (StDev).
LENGTH

log10 scale:metres: geometric arithmetic

skewness:

Region

N

mean

StDev

median mean

mean

initiallog10Western

293

2.77

.174

591

586

635

.941

.018

Fagaras

206

2.79

.173

592

613

664

1.077

.155

Iezer & N 132

2.79

.175

610

624

679

1.595

.379

ROMANIA 631

2.78

.174

596

603

654

1.190

.137

Bendor

222

2.84

.210

705

698

785

1.772

.086

p
Shulaps

126

2.85

.204

730

710

797

2.416

.364

Cayoosh 198

2.85

.206

670

709

798

1.674

.328

Pyrenees 206

2 66
2.66

.228
228

450

453

519

1 290
1.290

.042
042

N. Scand. 541 2.86

.243

750

724

842

1.690

-.198

Lake D
D.

158

2 75
2.75

.190
190

535

562

620

1 363
1.363

.184
184

Wales

260

2.78

.182

610

603

667

2.453

.323

Frequency distributions of drumlin length (& width).
Samplep sizes: Britain ((n = 37,043),) Ireland ((n = 21,940),) and combined ((n = 58,983).
)
Clark, C.D.
ett al.
l 2009Quaternary
Science
Reviews 28
677–692“the abrupt
lowerbound andlongppositivetail in thefrequency
histograms,
histogramsthe 100 m
initialbumpscaling”

Evans 2003, in ‘Concepts & modelling in geomorphology’(see IAG web site)
also discussed scale-specificity in aeolian bedforms,
glacial & fluvial bedforms, karst, slopes,
tectonic, volcanic and submarine features… & landslides
In earlier papers (e.g. Evans & McClean 1995 ZfG SB 101),
he discussed deviations from thefractal (~scale-free) model

Pike
Pike,
1980

Again,
g , scaling…
g+ scale-specific characteristics

n.b. THRESHOLDS(transition sizes)

Breaks in the crater depth: diameter scaling
relation; the morphologic transition from
simple to complex craters.
(a) 230 craters on Mars, showing larger simple
craters on p
plains than on ‘cratered terrain’.
(b) Based on 203 mare craters and 136 uplandcraters on the Moon. Simple craters follow asimilar relation for maria and for uplands (asfor the two divisions of Mars), but complex
craters average 12% deeper in uplands.
(c) Summary of the relationships on threeplanets and the Moon. The transition size
increases as gravity decreases.

LengthWidth

Scale-specific landslides?
H i htHeightLogarithmic histograms of length;mean width, and altitude rangeof 3424 landslide masses
in south-central Japan(data: Sugai and Ohmori,cf. 1994 Trans. Japanese
Geomorph. Union).

Length:arithmetic mean 446 m
geometric mean 409 m
mode & median 400 m
mean log 2.61
SD log 0.18
(log10 of metres)

Logarithmic quantile
quantile-quantile
quantile plot of landslide mass length insouth-central Japan (data supplied by Sugai and Ohmori).
The quantiles are the data ordered by magnitude.
Each quantile is plotted against the corresponding quantile of the Gaussian
frequency distribution with the same mean, standard deviation and sample
size.
i
Li
Linearity
it showshththe excellent
ll t matcht h off thi
this llarge (3424) ddatat sett tto
the log-normal frequency distribution.

Next slide shows Dependence of landslide probability
densities p on landslide area AL, for three landslide inventories:(1) 11 111 landslides triggered by the 17 January 1994 Northridge
earthquake in California, USA (squares)(Harp and Jibson, 1995, 1996);(2) 4233 landslides triggered by a snowmelt event in the Umbria
regioni off Italy
It l iin JJanuary 1997 (circles)( i l ) (Cardinali(C di li ett al.,l 2000)2000);(3) 9594 landslides triggered by heavy rainfall from Hurricane
Mitch in Guatemala in late October and early November 1998
(filled diamonds) (Bucknam et al., 2001).

logarithmic axes (A)linear axes

Length 10 m

100 m

1 km

“there is strong evidence
that [these 3 inventories]
are substantially completeso that the rollover at small
areas is well documented
documented.
The three studies werecarried out within hours to acouple of months of thelandslide events.
Therefore wasting
Therefore,
processes did not obliteratesmaller landslides and theboundaries of even smalllandslides were distinct.”…nearlynearly complete forlandslides with lengthscales greater than 15 m
(AL ≈ 225 m2)

400 m2 is most frequent
Proposedplandslide pprobability
y distribution is the best fit to the three landslide
inventories of the three-parameter inverse-gamma distribution with parameter values
ρ= 1·40, a = 1·28 . 10−3 km2, s = −1·32 . 10−4 km2
Malamud et al., ESPL 2004

1 m³

Malamud et al. (2004) note thatrockfalls / rockslides (left) do notshow a rollover: frequency
increases down to very small
ffeatures(<( 1 m³)³)

Distributions for historical events (right)( g )deviate from model (lines) because of
inventory incompleteness…
(top three are the ‘single-event’‘single event’ inventories).
inventories)

The cumulative magnitude
magnitude-frequency
frequency
curve for the landslides from the 370 km²
Loughborough Inlet area (50.5 °N, B.C.)
in the 18 November 2001 storm.
The landslides above 10 000 m² arewellll d
described
ib d bby a power law
l
with
ith
a slope of about −1.24. Several curves
would fit the remainder of the data
data.
(Guthrie & SG Evans NHESS 2004)

LoughboroughLhbh IInletl t llandslides
d lid plottedl tt dagainst the double Pareto distribution using
maximum likelihood estimation
estimation.
The rollover point as determined by Stark
and Hovius (2001), t , is 8882 m², however,there is substantial variation around thisnumber and 10 000 m² is an equally good fit.

“The rollover“Thlleffect
ff t ini magnitude-frequency
it d fdistributions is not merely an artefact of censoring, but
represents a physical manifestation of the conditions
under which the landslides occur.
In the case of coastal BC watersheds, the rollover
seems to occur at or near 10 000 m²
m , nearly 1
1.5
5 ordersof magnitude larger than our minimum recordedl d lid size.
landslidei
We note that for total disturbed areas below about
630 m² there may remain a censoring effect.”((Guthrie & SG Evans NHRSS 2004))

CONCLUSIONS Landslides
CONCLUSIONS:
L d lid• For landslides there is debate about limits to their scaling
behaviour, with differing interpretations of the ‘rollover’
rollover in size
sizefrequency plots.
• Si
Size distributions
di t ib tishowhpower scaling
li only
l withinithi limits
li it(between ‘rollover’ size and largest feasible feature)• Landslides, like cirques, may be scale-specific –especially forsingle events or particular types
types…
• The lower size limit reflects a threshold ‘critical mass’.
• The upper size limit reflects the frame, the available slope relief.
• More large,
large complete inventories would be welcome
welcome…

Geomorphology; The Future?
• Huge numbers of observations – both process and form…
• More precise dating, over long as well as short timescales –replacementlt off speculation,l ti
iin llong-termtlandscapel dddevelopment.
ltp survey
y of gground materials, to some depth.
p• Rapid• Precise knowledge of the 1 m land surface globally;
already 90 m, soon 30 m, 5 m if resources were focused.
• No excuse for not testing morphological ideas widely
widely…
• Better equipment; process rates over broader range of timescales.
• Further convergence of models and data.

Avoid:• Too small a data set• Too much ‘black box analysis• Ignoring spatial & broader context• Thinking your study area is typical of the whole world…
• Leaving your students behind your technical andypprogress.
ganalytical

CONCLUSIONS: General• Many fluvial features scale over many orders of magnitude.
Scaling (e.g. fractal model) is more important for hydrology &fluvial landforms, but always has limits (if only grain size, & size of Earth!)• Cirques are scale-specific but also scale allometrically.
• Bedforms (dunes, drumlins…) are also scale-specific.
Is allometryy also generalg
for them?
• Scale specificity is important because it relates either toprocess thresholds or to the scale of controlling frameworks(e.g. whole valley-side, for mass movements)• I hypothesize that all landforms show some scale
scalespecificity: there are good process reasons for limits totheir scaling behaviour.
• You lucky people! In the digital / GIS / DEM / Lidar / GPS /Google Earth world
world, it is a great time to be a geomorphometrist(& a field earth scientist).