ian s. evans, durham university · columbia, britain, romania, scandinavia, and spain. loggp...

All t li d l ifi itAllometry, scaling and scale-specificity

f i l d lidof cirques, landslides

and other landforms

Ian S. Evans, Durham University

CSIS Seminar

14 October 2009 The University of Tokyo14 October, 2009. The University of Tokyo

CONTENTSCONTENTS• Connections…Japan• Discontinuities: specific (&general) geomorphometry• Allometry; cirques (allometric scaling)• Allometry; cirques (allometric scaling)• 95% confidence intervals on regression

allometric coefficients (power exponents)• Alternative RMA / SD line coefficientsAlternative RMA / SD line coefficients• Cirque development; the ‘buzzsaw’• Scale specificity …second broad theme• Landslide size distributions• Conclusions

JAPANESE PAPERS (in English)JAPANESE PAPERS (in English)cited in my PhD on Geomorphometry & GlaciationI G 1937 P t l i & t t h f th• Imamura G 1937 Past glaciers & present topography of the Japanese Alps. Science Reports of Tokyo B. D. , C7, 61 pp.

Di i l H t i I t l iDimensionless Hypsometric Integral …cirques

• Hoshiai M & Kobayashi K 1957 A theoretical discussion on the so-called ‘snowline’… Japanese J. of Geology & Geogr.28 (1 3) 61 7528 (1-3), 61-75.

K b hi K 1958 Q t l i ti f th J Al• Kobayashi K 1958 Quaternary glaciation of the Japan Alps.J. of Fac. Liberal Arts & Sci, Shinshu Univ. 8 (II), 13-67.

• Higuchi K & Iozawa T 1971 Atlas of perennial snow patches in Central Japan Water Research Laboratory Nagoyain Central Japan. Water Research Laboratory, Nagoya University, 81pp.

SOME INFLUENTIAL JAPANESE PAPERS (in English), to 1981…quantitative

• Yatsu E. 1965 On the longitudinal profile of the graded river.Trans. Amer. Geophys. Union 36, 655-663.p y

• 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 of drainage networks... Geogr. Reports, Tokyo Metro. Univ. 13, 1-27 (& 1972 6/7 39 49)27. (& 1972, 6/7, 39-49)

• Ohmori H. 1978 Relief structure of the Japanese mountains and their stages in geomorphic de elopment B ll Dept Geogrand their stages in geomorphic development. Bull. Dept. Geogr. Univ. Tokyo 10, 31-85.

• Yoshikawa T Kaizuka S & Ota Y 1981 The landforms of• Yoshikawa, T., Kaizuka, S. & Ota, Y. 1981 The landforms of Japan. University of Tokyo Press Crustal movements.

GEOMORPHOMETRY

International Conference at Zurich,

A t 2009August 2009;

Proceedings at –g

http://geomorphometry.org/

Please register there!

Geomorphometry Society;Geomorphometry Society;

next biennial conference: 2011

(Location to be announced in late 2009.)

In many landscapes, we pcan recognize discontinuities:

in altitude, gradient, aspect & curvatures& curvatures

Minar & Evans

2008

GeomorphologyGeomorphology

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)(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 their

development is related, to define specific landforms such as

cirques drumlins dunes landslides volcanoes valley-sidescirques, drumlins, dunes, landslides, volcanoes, valley-sides…

When completely delimited, landforms can be measured and their

position, 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 position8. 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.) International Geomorphology 1986 Part IIte at o a Geo o p o ogy 986 a t

Techniques have changed, but all nine stages still apply…

P S if you don’t like / do not find discontinuities use general geomorphometryP.S. if you don t like / do not find discontinuities, use general geomorphometry…

…first, allometry of cirques…

ALLOMETRYMany 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 a given time, i.e. static allometry, as was proposed for cirques

i i ll f ll l ti (15) i C l d It ioriginally for a small population (15) in Colorado. It is now possible to test this for several cirque populations, each much

bi th i Ol h t’ (1977) i i l t dbigger than in Olyphant’s (1977) original study. Results are presented here for different regions of British Columbia, Britain, Romania, Scandinavia, and Spain.

Logarithmic plots of horizontal and vertical dimensions against g p goverall size are presented. They show that, as size increases,

cirque length increases faster than vertical dimensionscirque length increases faster than vertical 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 2: Vârtopul de Vest, a well-defined cirque in the Făgăraş 3 Răţiţi d fi itMts. , a classic cirque with

a large rock basin lake.

defined cirque in the Făgăraş Mts.; it has a good headwall and floor, but no lake.

3: Răţiţiş, a definitecirque in the Călimani Mts. with a moderate headwall curved around an outsloping floor.

Photos;

4: Ursu in the Căpăţânii Mts 5 Balota on M ntele Mic (Ţarc )

Marcel Mindrescu.(See Evans 2006b

‘Geomorphology’ for4: Ursu in the Căpăţânii Mts., a poor cirque because of the gentle headwall and irregular floor.

5. Balota, on Muntele Mic (Ţarcu),a marginal cirque because of a very poor headwall.

Geomorphology’ for Welsh examples).

Cirque definition map of Iezer Mountains, after Mindrescu 2006after Mindrescu 2006

Defining the i blvariables:

length of median axis, d idth iand width, in green.

F = cirque focus (middle of threshold)(middle of threshold).

1, 2, 3, 4, 5, 6 =locations where altitude

interpolated. (Vertical) Amplitude =

fall from top of median axisfall from top of median axis to bottom,

i.e. Alt(6) – Alt(1) Height range = highest

altitude on crest – lowest (on threshold)lowest (on threshold),

i.e. Alt(4) – Alt(1)Axial aspect = 9ºPlan closure = 19+(360-219)=160ºMarcel Mindrescu

10 = line of maximum (head)wall height.

Marcel MindrescuHeight range = crest maximum to threshold: Wall height = highest part of headwall

Romania: mountain ranges with cirques.Blue = many, yellow = few. Marcel Mindrescuy, y

Maramures5300

kmNorthern Romania

Rodna

Calimanian a

xis,

k

Calimani

Bihor

5200

e of

med

ia

5100g, m

iddl

e

Transylvanian Alps

Fagaras

Retezat BucegiI

Cindrel

Nor

thin

g

(Southern Carpathians)

ParangGodeanu

Tarcu Iezer

5000

5100 5200 5300 5400 DATA:5100 5200 5300 5400Easting, middle of median axis, km

DATA:

Marcel MindrescuCirques of the Romanian Carpathians Stefan cel Mare U.,

Suceava

Cirques of the Romanian Carpathians

l th ( ) l th i Length (m) length regression

Size = cube root of (length x width x amplitude).

length (m)width (m)amplitude (m)

length regressionwidth regressionamplitude regression

2000

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


Allometric plot2000 Allometric plot

2000

1000 1000

500500

200

200

631 cirques, Romania

100 132 cirques, Iezer & Northern

100200 500 1000

Size (m)200 500 1000

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

Regression gradients would be the same, if shape did not change with size.



2000



Allometric plot2000 Allometric plot

2000

10001000

500500

200

206 i

200200

206 cirques, Fagaras Mountains100

293 cirques, Western Romania

100

200 500 1000Size (m)

200 500 1000Size (m)

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

Exponents for logarithmic (power) regressions f i i bl ll i iof size variables on overall cirque size

_______________________________________________________________Variable expon 95% conf R2 % Northern Fagaras Retezat-GVariable expon. 95% conf. R , % Northern Fagaras Retezat-G

Length 1.095 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 82-0 91 68 | 0 82 0 84 0 79Amplitude 0.861 0.82-0.91 68 | 0.82 0.84 0.79

Height range 0.871 0.83-0.91 72 | 0.86 0.85 0.81

Wall height 0.852 0.78-0.92 50 | 0.81 0.82 0.74_________________________________________________________________

295% 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 faster than size as a whole. Amplitude and other vertical dimensions have exponents significantly below 1 0; vertical growth of cirques lags behindexponents significantly below 1.0; vertical growth of cirques lags behind horizontal growth, their shape changes with size… they are allometric.

L th

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

LengthWesternFagaras

Iezer & North

Width

Iezer & NorthROMANIA

W t

If 95% confidence limits do not overlap, coefficients are significantly diff h 1% l lWestern

FagarasIezer & North

ROMANIA

different at the 1% level

(a rough test…)

Height rangeROMANIA

WesternFFagaras


AmplitudeWesternFagaras

Iezer & NorthIezer & NorthROMANIA

.6 .8 1 1.2isometry

Allometric exponents

Allometry: change in cirque shape with size.L th d idth f t th ti l lit d

length (m)idth ( )

length regressionidth i

length (m)idth ( )

length regressionidth i

Length and width grow faster than vertical amplitude.

width (m)amplitude (m)

width regressionamplitude regression

2000

width (m)amplitude (m)

width regressionamplitude regression

1000

2000

1000

500

1000

500

200

100

200200

260 cirques, Wales 158 Lake District

cirques, England100

200 500 1000Size (m)

200 500 1000Size (m)

Length (m)Width (m)

length regressionwidth regression

length (m )w idth (m )

length regressionwidth regression

Allometric plot

Width (m)Height range (m)

width regressionheight range regression

width (m )am plitu de (m )

width regress ionam plitude re gress ion

Allometric plot2000 2000

1000 1000

500 500

200 200

S i h P

206 cirques, Gallego basin,

200 200

Spanish Pyrenees100200 500 1000

Size (m)198 Cayoosh cirques, B.C.

100

200 500 1000Size (m)Pyrenees data: J.M. Garcia-Ruiz et al.

Length (m) length regression


length regressionwidth regressionamplitude regressionLength (m)

Width (m)Amplitude (m)

length regressionwidth regressionamplitude regression Allometric plot

Amplitude (m) amplitude regression

Allometric plot2000 2000

1000 1000

500 500500 500

222 cirques

200

126 cirques200

B. C. Coast Mtns.

222 cirques, Bendor Range,100 B. C. Coast Mtns.

126 cirques, Shulaps Range,100

200 500 1000Size (m)

200 500 1000Size (m)Provisional data sets – I.S. Evans

British Columbia Coast Mountains

LengthBendor

Shulaps

Width

ShulapsCayoosh

WidthBendor

ShulapsCayoosh

Height range

y

BendorShulapsCayoosh

AmplitudeBendor

ShulapsCayoosh

.6 .8 1isometry

1.2y


Western Europe

LengthPyrenees

N. Scandinavia

Width

N. ScandinaviaLake District

Wales

WidthPyrenees

N. ScandinaviaLake District

Height range

Wales

PyreneesPyreneesN. Scandinavia

Lake DistrictWales

AmplitudeN. Scandinavia

Lake DistrictWales

.6 .8 1isometry

1.2y


ALTERNATIVE RELATIONSHIPS

• Traditionally, allometric relations are logarithmic regressions of one size component, y, treated as dependent, on overall size, x.Thi h th d t th t th t f• This has the advantage that the exponents of

length, width & amplitude sum to 3.0; if one >1 0 another must <1 0if one >1.0, another must <1.0.

• Alternatively, if we give y and x equal status, we may prefer to calculate the standard deviation line (reducedprefer to calculate the standard deviation line (reduced major axis, geometric mean regression).

• This will always be steeper than the regression of y on xThis will always be steeper than the regression of y on x.• Thus there is no constraint on the exponent sum, which

will be greater where scatter is greater (R2 is lower).will be greater where scatter is greater (R is lower).• bootstrap provides 95% confidence limits.

WHAT DIFFERENCE DOES IT MAKE?WHAT DIFFERENCE DOES IT MAKE?

Lake District cirque amplitudeExponents & confidence limits:

Romanian cirque width

Regression 0.736 0.646 - 0.825SDline 0 929 0 830 - 1 028

Exponents & confidence limits:2.70

Exponents & confidence limits:Regression 1.015 0.939 - 1.091

SD line 1.228 1.181 - 1.2763.3

SDline 0.929 0.830 1.028

2.50

e

3.1

th (m

)

2.30

og a

mpl

itud

2 7

2.9

rithm

of w

id

2.10lo

2.5

2.7

Loga

r

log amplitudeSD lineRegression2.3

log of width (m)SD lineRegression

2.30 2.50 2.70 2.90log size

2.3 2.5 2.7 2.9 3.1Logarithm of size (m)

bigger difference where scatter greater…big scatter here for amplitude, 8 cirques < 90 m:

coefficients not well specified, data editing needed.

E t & fid li it

N. Scandinavia cirque lengthExponents & confidence limits:

N. Scandinavia cirque amplitudeExponents & confidence limits:Regression 1.177 1.106 - 1.248

SD line 1.445 1.371 - 1.5203.3

3.5Exponents & confidence limits:

Regression 0.835 0.723 - 0.947SD line 1.561 1.407 - 1.716

3.1

2.9

3.1

h

2.7

2.9

de

2.7

2.9

log

leng

th

2.3

2.5

og a

mpl

itud

2.3

2.5

1.9

2.1lo

1.9

2.1 log lengthSD lineRegression

log amplitudeSD lineRegression

2.3 2.5 2.7 2.9 3.1 3.3logsize

2.3 2.5 2.7 2.9 3.1 3.3logsize

Pyrenees N. Scandinavia Lake District WalesThe

Data:lo

g le

ngth

Data:

L

h

L

log

wid

th

W

mpl

itude

Amp

log

amge

Amp

g he

ight

rang

HR

log

log size log size log size log size

More scatter in N Scandinavia data set - outliers sizeMore scatter in N. Scandinavia data set - outliers with low amplitudes – data problems…

alternative exponents:RMA (geometric regression SD line)RMA (geometric regression, SD line)

exponents for Length, Width, & Amplitude:Pyrenees 1.310 1.204 0.994 (HR)* *N Scandinavia 1 445 1 255 1 561 (unreliable)N. Scandinavia 1.445 1.255 1.561 (unreliable)Lake District 1.231 1.210 0.929 * *Wales 1.207 1.186 1.156 no significant difference

Romania 1.167 1.228 1.044 * *Bendor, B.C. 1.211 1.140 1.113 *Shulaps B C 1 138 1 206 1 009 *Shulaps, B.C. 1.138 1.206 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.

Length

Width

WesternFagaras


Returning to –

95% confidence limits onROMANIAWidth

Height range

WesternFagaras


95% confidence limits on regression coefficients (allometric exponents)

ROMANIA

Height range

Amplitude

WesternFagaras

Iezer & NorthROMANIA Length

PyreneesN. Scandinavia

L k Di t i t

( p )

pWesternFagaras


.6 .8 1isometry

1.2

Width

Lake DistrictWales


Lake DistrictW l

Western Europe

isometry


LengthB d

Height range

Wales


Lake DistrictWales

Width


Bendor

AmplitudeN. Scandinavia

Lake DistrictWales

.6 .8 1isometry

1.2

British Columbia Coast Mountains

Height range


Bendor

isometry


Amplitude



.6 .8 1isometry

1.2

All t i t

SUMMARY TABLEExponents for logarithmic (power) regressions of size variables on overall size

nr. of cirques Length Width Ampl. Height range Data sourceNn Scandinavia * 541 1 18 99 84 65 S Hassinen cf 1998Nn. Scandinavia * 541 1.18 .99 .84 .65 S. Hassinen, cf. 1998Blanca, Rockies 15 1.14 1.20 - .66 Olyphant 1977Lake D * 158 1 17 1 10 74 75 Evans and Cox 1995Lake D. 158 1.17 1.10 .74 .75 Evans and Cox 1995C. Span. Pyrenees 260 1.17 1.06 - .77 J.M. García-Ruiz et al. 2000Cayoosh, B.C. 198 1.10 1.05 .85 .83 Evans & McClean, 1995yShulaps, B.C. 126 1.06 1.07 .87 .83 Evans, unpubl.Maritime Alps 432 1.08 1.08 - .84 Federici and Spagnolo, 2004Bendor, 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 * 260 1 12 98 90 91 E 2006Wales * 260 1.12 .98 .90 .91 Evans 2006Ben 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 t f 1 00 i di t l h i i tExponents of 1.00 indicate equal change, i.e. isometry(found only in Ben Ohau, N.Z.).

Results are consistent in confirming the static allometry of l i l i l i l ti l l d b dglacial cirques: larger cirques are relatively longer and broader, more

than they are deeper. Power coefficients for length and width are ll b 1 0 hil th f d th i ifi tl b l Igenerally above 1.0, while those for depth are significantly below. In

most regions the length exponent exceeds the width exponent: hence th ll t t b l i d b l t l l f ithe allometry cannot be explained by lateral coalescence of cirques.

All length exponents are significantly above 1.0, and all d th t i ifi tl b l h th ti l di idepth exponents are significantly below, whether vertical dimension is expressed as height range, axial amplitude or headwall height.R l ti b t l th d idth h b tRelations between length and width, however, vary between ranges.

It is inferred that cirque headwall retreat is faster than i d i Y t i h d l k th t tt t tcirque deepening. Yet many cirques have deep lakes that attest to

considerable cirque deepening; this means that cirque development i ll th di i i id blin all three dimensions is considerable.

This study shows the importance of considering confidence i t l h ki l i b t l ti t f hintervals when making conclusions about relative rates of change; and of checking consistency between regions.

Headwall recession and floor deepening:DATA BASED MODELS of CIRQUE DEVELOPMENT

(E ans 2006

DATA-BASED MODELS of CIRQUE DEVELOPMENT : means for the 52 cirques in each of five equal size classes in Wales.(Evans 2006,Geomorphology)This data basedThis data-based generalization uses floor and profileuses floor and crest altitudes &max and minmax. and min. gradients.Plan: development ofPlan: development of a hypothetical average mid height

plan

average mid-height contour, using plan closure width andclosure, width, and half the length.

Where glaciation was asymmetric, headward extension of valleys or divide displacement by a few km shows considerable headwall retreat (cirque e s o s co s de ab e ead a e ea (c quelengthening):

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 rangesrapid glacial erosion limiting the height of many mountain ranges.

Instances of complete range truncation are, however, hard to find: coalescent and back-to-back cirques are common but only occasionallycoalescent and back-to-back cirques are common, but only occasionally do intervening ridges seem to have been removed.

Cirques are rarely more than 2 km long or wide. It is interestingCirques are rarely more than 2 km long or wide. It is interesting that cirques in plateau areas, where range truncation has clearly not occurred, are not dissimilar in size to those in more dissected mountains with back-to-back cirques, where the buzzsaw hypothesis might be applicable.

The hypothesis implies that only very small areas can rise high above snowline (firnline, ELA). Yet analysis of the World Glacier Inventory h th t l i i 1000 b ELA i ll i thshows that many glaciers rise >1000 m above ELA, especially in the

Himalayas and Tien Shan but also in other orogenic belts.At the extreme icefields over 3000 km2 in area are found inAt the 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 zoneQuaternary uplift in these areas carried mountains quickly through the zone of rapid glacial erosion, into that of cold ice frozen to its bed.

There is evidence for a glacial buzzsaw, i t f th t USA d th A din parts of the western USA and the Andes:

a parallelism between present and former ELAs and summit altitudes (Mitchell & Montgomery 2006).

But this is by no means universal; many present-day glaciers start well above ELA :day glaciers start well above ELA :

a horizontal buzzsaw would not account f lt f W ld Gl i I tfor many results from World Glacier Inventory.

(& application to Japan limited?)Cirque headwall retreat may limit

any increase in relief,ybut downward erosion of cirque floors and glacial troughs is a major part of the erosionglacial troughs is a major part of the erosion

that affects orogen development (uplift pattern).

INTERIM CONCLUSIONS : cirque allometryq yLarge cirques differ in shape and gradient from small ones.

T ki l i h i d l d f thTaking larger cirques as having developed further,each dimension can be plotted against an overall size

measure to express static allometry or isometry.Vertical dimensions increase more slowly than do horizontalVertical dimensions increase more slowly than do horizontal, and length usually increases faster than width, but length –

width relations vary between areas.(exponents: length > width > height)(exponents: length width height)

The allometric nature of cirque development is supported th b i f t f l i t i f ion the basis of a set of large inventories of cirques.

Acknowledgements to Marcel Mindrescu, Nick Cox, J.M. Garcia-Ruiz and S. Hassinen.

As well as such variation in shape with size, landforms usually show a p ylimited range of size. This is clear for glacial cirques, which vary over only one order of magnitude.

As larger cirques develop in lengthdevelop in length and width more than in depth, th hthey show scalingbehaviour as wellbehaviour as well as scale-specificity.Cl l thClearly these behaviours can be combined.be combined.

Cirque scale-specificity: low standard deviations (StDev).LENGTH log10 scale: metres: geometric arithmetic skewness:Region N mean StDev median mean mean initial log10

Western 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

Shulaps 126 2.85 .204 730 710 797 2.416 .364p

Cayoosh 198 2.85 .206 670 709 798 1.674 .328

Pyrenees 206 2 66 228 450 453 519 1 290 042Pyrenees 206 2.66 .228 450 453 519 1.290 .042

N. Scand. 541 2.86 .243 750 724 842 1.690 -.198

Lake D 158 2 75 190 535 562 620 1 363 184Lake D. 158 2.75 .190 535 562 620 1.363 .184

Wales 260 2.78 .182 610 603 667 2.453 .323

Frequency distributions of drumlin length (& width). Sample sizes: Britain (n = 37,043), Ireland (n = 21,940), and combined (n = 58,983).p ( ) ( ) ( )

Clark, C.D. t l 2009et al. 2009

Quaternary Science Reviews 28 677–692

“the abrupt lower bound andbound and long positiveptail in the frequency histogramshistograms, the 100 m initial bump-scaling”

Evans 2003, in ‘Concepts & modelling in geomorphology’

(see IAG web site)(see IAG web site)

also discussed scale-specificity in aeolian bedforms,

glacial & fluvial bedforms, karst, slopes,

tectonic, volcanic and submarine features

& landslides… & landslides

In earlier papers (e.g. Evans & McClean 1995 ZfG SB 101),

he discussed deviations from the

fractal (~scale-free) model

PikePike,1980

Breaks in the crater depth: diameter scaling relation; the morphologic transition from Again, scaling…simple to complex craters.(a) 230 craters on Mars, showing larger simple craters on plains than on ‘cratered terrain’.

g , g+ scale-specific characteristics

p(b) Based on 203 mare craters and 136 upland craters on the Moon. Simple craters follow a similar relation for maria and for uplands (assimilar relation for maria and for uplands (as for the two divisions of Mars), but complex craters average 12% deeper in uplands.(c) Summary of the relationships on three

n.b. THRESHOLDS(c) Summary of the relationships on three planets and the Moon. The transition sizeincreases as gravity decreases.

(transition sizes)

Length Width

Scale-specific landslides?Logarithmic histograms of length; mean width, and altitude range

H i ht of 3424 landslide masses in south-central Japan

Height

(data: Sugai and Ohmori,cf. 1994 Trans. Japanese

Geomorph. Union).

Length:Length:

arithmetic mean 446 m

geometric mean 409 mgeometric mean 409 m

mode & median 400 m

mean log 2.61

SD log 0.18

(log10 of metres)

Logarithmic quantile-quantile plot of landslide mass length inLogarithmic quantile quantile plot of landslide mass length in south-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 i Li it h th ll t t h f thi l (3424) d t t tsize. Linearity shows the excellent match of this large (3424) data set to

the log-normal frequency distribution.

Next slide shows Dependence of landslide probability densities p on landslide area A for three landslide inventories: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 Umbriai f It l i J 1997 ( i l ) (C di li t l 2000)region of Italy in January 1997 (circles) (Cardinali et al., 2000);

(3) 9594 landslides triggered by heavy rainfall from Hurricane(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).

“there is strong evidence that [these 3 inventories] are substantially completeso that the rollover at small areas is well documentedareas is well documented.The three studies were carried out within hours to a

logarithmic axes (A)

couple of months of the landslide events.Therefore wastingTherefore, wasting processes did not obliterate smaller landslides and the

linear axes

boundaries of even smalllandslides were distinct.”

nearly complete for…nearly complete for landslides with length scales greater than 15 m

2(AL ≈ 225 m2)400 m2 is most frequent

Proposed landslide probability distribution is the best fit to the three landslide Length 10 m 100 m 1 km

p p yinventories 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

Malamud et al. (2004) note that rockfalls / rockslides (left) do notrockfalls / rockslides (left) do not show a rollover: frequency increases down to very smallf ( 1 ³)

1 m³

features (< 1 m³)

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

The cumulative magnitude-frequency L hb h I l t l d lid l tt dThe cumulative magnitude frequencycurve for the landslides from the 370 km²Loughborough Inlet area (50.5 °N, B.C.)

Loughborough Inlet landslides plotted against the double Pareto distribution using maximum likelihood estimation

in the 18 November 2001 storm.The landslides above 10 000 m² are

ll d ib d b l ith

maximum likelihood estimation. The rollover point as determined by Stark and Hovius (2001), t , is 8882 m², however,

well described by a power law with a slope of about −1.24. Several curves would fit the remainder of the data

there is substantial variation around this number and 10 000 m² is an equally good fit.

would fit the remainder of the data.(Guthrie & SG Evans NHESS 2004)

“Th ll ff t i it d f“The rollover effect in magnitude-frequency distributions 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 nearly 1 5 ordersseems to occur at or near 10 000 m , nearly 1.5 orders of magnitude larger than our minimum recorded

l d lid ilandslide size. We note that for total disturbed areas below about

630 m² there may remain a censoring effect.”(Guthrie & SG Evans NHRSS 2004)( )

CONCLUSIONS L d lidCONCLUSIONS: Landslides

• For landslides there is debate about limits to their scaling behaviour, with differing interpretations of the ‘rollover’ in size-behaviour, with differing interpretations of the rollover in sizefrequency plots.

Si di t ib ti h li l ithi li it• Size distributions show power scaling only within limits(between ‘rollover’ size and largest feasible feature)

• Landslides, like cirques, may be scale-specific –especially for single events or particular typessingle events or particular types…

• The lower size limit reflects a threshold ‘critical mass’.

• The upper size limit reflects the frame, the available slope relief.

• More large complete inventories would be welcome• More large, complete inventories would be welcome…

Geomorphology; The Future?Geomorphology; The Future?• Huge numbers of observations – both process and form…

• More precise dating, over long as well as short timescales –

l t f l ti i l t l d d l treplacement of speculation, in long-term landscape development.

• Rapid survey of ground materials, to some depth.p y g p

• 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• No excuse for not testing morphological ideas 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 contextIgnoring spatial & broader context

• Thinking your study area is typical of the whole world…

• Leaving your students behind your technical and analytical progress.y p g

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 allometry also general for them?y g• Scale specificity is important because it relates either to

process thresholds or to the scale of controlling frameworksprocess thresholds or to the scale of controlling frameworks(e.g. whole valley-side, for mass movements)

• I hypothesize that all landforms show some scale-I hypothesize that all landforms show some scalespecificity: there are good process reasons for limits to their scaling behaviour.their scaling behaviour.

• You lucky people! In the digital / GIS / DEM / Lidar / GPS / Google Earth world it is a great time to be a geomorphometristGoogle Earth world, it is a great time to be a geomorphometrist (& a field earth scientist).

ian s. evans, durham university · columbia, britain, romania, scandinavia, and spain. loggp...

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