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A general landslide distribution applied to a small inventory in Todi, Italy

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A general landslide distribution applied to a small inventoryin Todi, Italy
DONALD L. TURCOTTE
1
, BRUCE D. MALAMUD
2
, FAUSTO GUZZETTI
3
&PAOLA REICHENBACH
31
Department of Geology, University of California, Davis, CA, 95616, USA(e-mail: turcotte@geology.ucdavis.edu)
2
Environmental Monitoring and Modelling Research Group, Department of Geography,King’s College London, Strand, London WC2R 2LS, UK (e-mail: bruce@malamud.com)
3
CNR-IRPI Perugia, via della Madonna Alta 126, Perugia 06128, Italy(e-mail: fausto.guzzetti@irpi.cnr.it; paola.reichenbach@irpi.cnr.it)
Abstract:
Large numbers of landslides can be associated with a trigger, for example, an earth-quake or a large storm. We have previously hypothesized that the frequency–area statistics of landslides triggered in an event are well approximated by a three-parameter inverse-gamma dis-tribution, irrespective of the trigger type. The use of this general distribution was established usingthree substantially complete and well-documented landslide event inventories: 11,000 landslidestriggered by the Northridge California Earthquake, 4000 landslides triggered by rapidly meltingsnow cover in the Umbria region of Italy, and 9000 landslides triggered by heavy rainfall associ-ated with Hurricane Mitch in Guatemala. In this paper, we examine further this general landslidedistribution by using an inventory of 165 landslides triggered by heavy rainfall in the region of Todi, Central Italy. Our previous studies have shownthe applicability of our general landslide dis-tribution to events with 4000–11,000 landslides. This smaller inventory provides a critical step inexaminingtheapplicabilityofthegenerallandslidedistribution.WeﬁndverygoodagreementoftheTodi event with our general distribution. This also provides support for our further hypothesis thatthe mean area of landslides triggered by an event is approximately independent of the event size.
Landslides are complex phenomena inﬂuenced bymany factors, including soil and rock types,bedding planes, topography, and moisture content.Landslide events consist of one to many thousandsof landslides, generally associated with a triggersuch as an earthquake, a large storm, a rapid snow-melt, or a volcanic eruption. A landslide event maybe quantiﬁed by the frequency–area distribution of the triggered landslides. We have recently shown(Malamud
et al.
2004
a
) that the frequency–areastatistics of three substantially complete landslideinventories are well approximated by the sameprobability density function, a three-parameterinverse-gamma distribution. We also introduced alandslide-event magnitude scale
m
L
¼
log(
N
LT
),with
N
LT
the total number of landslides associatedwith the landslide event, in analogy to the Richterearthquake magnitude scale. We argue that thestatistics of triggered-landslide events under awide variety of conditions follow the samegeneral statistical behaviour to a good approxi-mation. Such a ‘general’ statistical behaviour iswidely accepted for the frequency–magnitude stat-istics of earthquakes, which are also complex,but generally follow a power-law relationshipbetween the number of earthquakes and theearthquake rupture area, the Gutenberg–Richterrelation.In this paper, we will ﬁrst discuss our ‘general’probability distribution of landslide areas fortriggered landslide events, and then several impli-cations of having a ‘general’ distribution, including(1) a magnitude scale for landslide events, (2) thesame theoretical average area of landslides associ-ated with any given landslide event, and (3) theability to extrapolate for incomplete landslideevents or historical inventories. We will alsopresent a fourth, much smaller, landslide inventoryfrom Todi, central Italy.
Frequency–area distributions
In order to give the statistical distribution of land-slide areas, a probability density function
p
(
A
L
) isdeﬁned according to
p
(
A
L
)
¼
1
N
LT
d
N
L
d
A
L
(1)
From
: C
ELLO
, G. & M
ALAMUD
, B. D. (eds) 2006.
Fractal Analysis for Natural Hazards
.Geological Society, London, Special Publications,
261
, 105–111.0305-8719/06/$15.00
#
The Geological Society of London 2006.
with the normalization condition
ð
1
0
p
(
A
L
)
dA
L
¼
1 (2)where
A
L
is landslide area,
N
LT
is the total numberof landslides in the inventory, and
d
N
L
is thenumber of landslides with areas between
A
L
and
A
L
þ
d
A
L
. In Figure 1 we present the probabilitydensities
p
(
A
L
) for three substantially completelandslide inventories, from the USA, Italy andGuatemala. A detailed discussion of each inventoryis found in Malamud
et al.
(2004
a
).The three sets of probability densities given inFigure 1 exhibit a characteristic shape (Guzzetti
et al.
2002; Malamud
et al.
2004
a
), with densitiesincreasing to a maximum value (most abundantlandslide size) and then decreasing with a power-law tail. The inventories were estimated to be nearlycomplete (Harp & Jibson 1995, 1996; Cardinali
et al.
2000; Bucknam
et al.
2001; Guzzetti
et al.
2002) for landslides with length scales greater than5–15 m (
A
L
25–225 m
2
), therefore the ‘rollover’in Figure 1 is regarded as real. Based on the goodagreement between these three sets of probabilitydensities, we proposed (Malamud
et al.
2004
a
) ageneral probability distribution for landslides, athree-parameter inverse-gamma distribution, givenby (Johnson & Kotz 1970; Evans
et al.
2000)
p
(
A
L
;
r
,
a
,
s
)
¼
1
a
G
(
r
)
a A
L
s
r
þ
1
exp
a A
L
s
(3)
600
Northridge earthquakeUmbria snowmeltGuatemala rainfall
I
nverse Gamma
P r o b a b i l i t y d e n s i t y ,
p
( k m
- 2
) P r o b a b i l i t y d e n s i t y ,
p
( m
- 2
)
0.001 0.002 0.003 0.004 0.00500200400800
10
2
10
3
10
1
10
0
10
-1
Landslide area,
A
L
(km
2
)Landslide area,
A
L
(m
2
)
10
-5
10
-4
10
-3
10
-2
10
-1
10
0
10
1
10
2
10
3
10
4
10
5
10
6
10
-7
10
-4
10
-3
10
-5
10
-6
10
-11
10
-8
10
-9
10
-10
10
-2
10
-3
10
-4
10
-5
(
a
)(
b
)
Fig. 1.
Dependence of landslide probability densities
p
on landslide area
A
L
, for three landslide inventories(ﬁgure after Malamud
et al.
2004
a
): (1) 11,111 landslides triggered by the 17 January 1994 Northridge earthquakein California (Harp & Jibson 1995, 1996); (2) 4233 landslides triggered by rapid snowmelt in the Umbria region of Italy in January 1997 (Cardinali
et al.
2000; Guzzetti
et al.
2002); (3) 9594 landslides triggered by heavy rainfallfrom Hurricane Mitch in Guatemala in late October and early November 1998 (Bucknam
et al.
2001). Probabilitydensities are given on logarithmic axes in (
a
) and linear axes in (
b
). Also included is our proposed general landslideprobability distribution. This is the best-ﬁt to the three landslide inventories of the three-parameterinverse-gamma distribution of Eq. (3), with
r
¼
1.40,
a
¼
1.28
10
2
3
km
2
, and
s
¼
1.32
10
2
4
km
2
(coefﬁcient of determination
r
2
¼
0.965).D. L. TURCOTTE
ET AL.
106
where
G
(
r
) is the gamma function of
r
. The inverse-gammadistributionhasapower-lawdecaywithexpo-nent –(
r
þ
1) for medium and large areas and anexponential rollover for small areas. The maximumlikelihood ﬁt of Eq. (3) to the three data sets inFigure 1 yields
r
¼
1.40,
a
¼
1.28
10
2
3
km
2
,and
s
¼
1.32
10
2
4
km
2
, with coefﬁcient of deter-mination
r
2
¼
0.965;thepower-lawtailhasexponent
r
þ
1
¼
2.40. Many authors (see Malamud
et al.
2004
a
for a review) have also noted that the fre-quency–area distributions of large landslides corre-late with a power-law tail. This common behaviouris observed despite large differences in landslidetypes, topography, soil types, and triggeringmechanisms.On the basis of the good agreement between thethree landslide inventories and the inverse-gammadistribution illustrated in Figure 1, Malamud
et al.
(2004
a
) hypothesized that the distribution given inEq. (3) is general for landslide events. It is notexpected that all landslide-event inventories will bein as good agreement as the three considered, butwe do argue that the quantiﬁcation, if only approxi-mate, is valuable in assessing the landslide hazard(Guzzetti
et al
. 2005, 2006).In this paper we present the probability densities
p
(
A
L
) for a fourth landslide inventory consisting of
N
LT
¼
165 rainfall-triggered landslides in the vicin-ity of Todi, central Italy, with landslides occurringin the period March to May 2004. The inventorywas compiled through reconnaissance ﬁeldsurveys, and is reasonably complete. Probabilitydensities for these landslides are given inFigure 2, along with the inverse-gamma distributionfrom Eq. (3) with
r
¼
1.40,
a
¼
1.28
10
2
3
km
2
,and
s
¼
1.32
10
2
4
km
2
, the best-ﬁt to the threeinventories in Figure 1. A reasonable agreement isobtained between this fourth set of data (Fig. 2)from Todi, Italy, and the inverse-gamma distri-bution. It should be emphasized that the totalnumber of landslides in the Todi event,
N
LT
¼
165, is a factor of thirty to eighty less thanthe number of landslides in the three substantiallycomplete inventories given in Figure 1.Before discussing implications of a ‘general’landslide distribution, we brieﬂy discuss rockfallinventories. It has been shown (Dussauge
et al.
2003; Malamud
et al.
2004
a
) that the frequency–size statistics of rockfalls are very different thanthe statistics for other types of landslides as
Fig. 2.
Dependence of landslide probability densities
p
on landslide area
A
L
, for 165 rainfall-triggered landslides inthe vicinity of Todi, central Italy, occurring March to May 2004. Probability densities are given on logarithmicaxes. Also included is the three-parameter inverse-gamma distribution of Eq. (3) with
r
¼
1.40,
a
¼
1.28
10
2
3
km
2
,and
s
¼
1.32
10
2
4
km
2
, the best-ﬁt to the three inventories in Figure 1.A GENERAL LANDSLIDE DISTRIBUTION 107
discussed above. Malamud
et al.
(2004
a
) examinedthree rockfall inventories, and found that the non-cumulative frequency–volume relationship is bestdescribed by a power-law with exponent –1.93,with no ‘rollover’ for smaller rockfall volumes;the equivalent tail for the medium and large land-slides in our ‘general’ distribution Eq. (3) with
r
¼
1.40, with areas converted to volumes, wouldbe a power-law exponent of
2
1.07. There is a sig-niﬁcant difference in slope for the medium andlarge landslides, compared to rockfalls, and therockfalls have no ‘rollover’ for the smaller land-slides. This difference has been attributed to differ-ent applicable physics, with rockfalls controlled byprocesses of fragmentation, compared to landslidesthat are primarily controlled by the process of slopestability. We now discuss the implication of ourgeneral landslide distribution for the ‘average’ land-slide area in the landslide event.
Average landslide area
Assuming the validity of Eq. (3) for the probabilitydistributionoflandslideareasinindividualtriggeredlandslide events, Malamud
et al.
(2004
a
) used thedistribution to derive a theoretical mean landslidearea
A
L
. This is the mean of all landslide areas thatoccur during a landslide event. The theoreticalmean area is obtained by taking the ﬁrst momentof the probability distribution function, giving
A
L
¼
ð
1
0
A
L
p
(
A
L
)
dA
L
:
(4)Substitution of the three-parameter inverse-gammadistribution from Eq. (3) into Eq. (4) and integratinggives
A
L
¼
a
r
1
þ
s
:
(5)For the landslide probability distribution givenin Figure 1 we have
r
¼
1.40,
a
¼
1.28
10
2
3
km
2
, and
s
¼
1.32
10
2
4
km
2
, so that
A
L
¼
3.07
10
2
3
km
2
. One implication of our landslidedistribution is that because the probability distri-bution always has the same mean, all landslideevents should have the same mean landslide area
A
L
¼
3
:
07
10
3
km
2
¼
3070 m
2
. This followsdirectly from the applicability of our proposed land-slide distribution, and is independent of the numberof landslides associated with a landslide event.Malamud
et al.
(2004
a
) found that the measuredmean landslide areas
A
L
for the three event inven-tories, Northridge, Umbria, and Guatemala, are
A
L
¼
3.01
10
2
3
km
2
, 2.14
10
2
3
km
2
, and3.07
10
2
3
km
2
, in good agreement with thevalue predicted by our general landslidedistribution. The mean landslide area for the fourthinventory from Todi, Italy, given in Figure 2, is
A
L
¼
4.05
10
2
3
km
2
. This is also in reasonablygood agreement with the theoretical
A
L
¼
3.07
10
2
3
km
2
, particularly for a triggered land-slide event with so few landslides, and is potentialfurther conﬁrmation of our general landslidedistribution.
Landslide magnitude scale
A second implication of having a ‘general’ landslideprobability distribution is the ability to create a land-slide event magnitude scale. Measures of event sizesare useful for natural hazards. For example, theRichter magnitude scale for earthquakes is univer-sally used and the general public has some under-standing of the implications of an
M
¼
7.0earthquake. Over a dozen magnitude scales areavailable for other natural hazards, including theSafﬁr–Simpson scale (hurricanes), the Fujita scale(tornadoes), and the Volcanic Explosivity Index.Malamud
et al.
(2004
a
) proposed a magnitudescale
m
L
for a landslide event based on the logarithmto the base 10 of the total number of landslidesassociated with the landslide event:
m
L
¼
log
N
LT
:
(6)Keefer (1984) and later Rodrı´guez
et al.
(1999) usedasimilarscaletoquantifythenumberoflandslidesinearthquake-triggered landslide events: 100–1000landslides were classiﬁed as a two, 1000–10,000landslides a three, and so on. The landslideevent magnitudes for the three event inventoriesconsidered by Malamud
et al.
(2004
a
) are (1) North-ridge earthquake-triggered event,
m
L
¼
4.05; (2)Umbria snowmelt-triggered event,
m
L
¼
3.63;(3) Guatemala rainfall-triggered event,
m
L
¼
3.98.These are in the range of
m
L
¼
3.6–4.0.Although observed earthquakes span a widerange on the Richter scale, available landslide-event inventories are often restricted to a relativelynarrow magnitude range. There are several reasonsfor this. Accurate inventories are restricted to popu-lated areas and have been carried out only duringthe last ten years or so. Thus, very few large land-slide events with
m
L
.
4.0 have occurred inregions where studies have been carried out. Inaddition, there has been little incentive to carryout studies of small-magnitude landslide events,
m
L
,
3.6. A limited range was also the case forearthquakes in the 1940s, when instrumental limit-ations limited studies to large earthquakes andthe timespan for studies was short, so few largeearthquakes had occurred. Therefore, one of thepurposes of this paper was to introduce a fourth
D. L. TURCOTTE
ET AL.
108
‘substantially’ complete landslide inventory, butwith a low magnitude. The Todi rainfall-triggeredevent has
m
L
¼
2.22.Given several hundred landslide events and theirmagnitudes in a given region and time period, wehypothesize that there will be many more‘smaller’ magnitude events compared to the largerones. In analogy to earthquakes, we further hypoth-esize that these will follow a relationship such thatlog
N
C
¼
2
bm
L
þ
a
, where
N
C
is the number of landslide events with magnitudes greater than orequal to
m
L
, and
b
and
a
are constants. To partiallyconﬁrm (or disprove) this hypothesis, we will needto assign landslide event magnitudes using substan-tially complete inventories, or making extrapol-ations based on ‘incomplete’ inventories, whichwe now discuss.
Historical and incomplete inventories
An historical landslide inventory includes the sumof many landslide events that have occurred overtime. Assuming that our landslide probability distri-bution is applicable to all landslide events, the sumof events over time (the historical inventory) willalso satisfy this distribution (Malamud
et al.
2004
a
). However, in historical inventories, the evi-dence for the existence of many smaller andmedium landslides will have been lost due towasting processes over time. Therefore, for the his-torical inventories, we attribute the deviation fromour general landslide distribution to the incomplete-ness of the inventories. Using the general landslidedistribution of Eq. (3), we can extrapolate an inven-tory that contains just the largest landslides to givethe total number and total volume of all landslidesin the region.Malamud
etal.
(2004
b
)usedthisextrapolationbyconsidering two examples. Frequency densitieswere used because the inventories are incompleteand the normalization given in Eq. (2) no longerholds. From Eq. (1), the frequency density
f
(
A
L
) is
f
(
A
L
)
¼
d
N
L
d
A
L
¼
N
LT
p
(
A
L
)
:
(7)Theoretical curves of
f
(
A
L
) for various landslide-event magnitudes
m
L
are obtained by multiplyingthe probability distribution
p
(
A
L
) given in Eq. (3)by the total number of landslides in the event
N
LT
.Curves are given in Figure 3 for
m
L
¼
1(
N
LT
¼
10) to
m
L
¼
8 (
N
LT
¼
10
8
).The same method can be used for a single incom-plete landslide event inventory where only thelargest landslides have been measured (e.g. thoselargest landslides that comprise just 1–2% of thetotal inventory) and the medium and smallerlandslide sizes are not known. Using the generallandslide distribution Eq. (3), we can extrapolatethe frequency densities of the largest landslides togive the total number of all landslides in theevent, and estimate the equivalent landslide eventmagnitude.Figure 3 includes frequency densities for two his-torical landslide inventories, from Italy and Japan.Also included, for reference, are frequency den-sities for the snowmelt-triggered Umbria landslideevent (Fig. 1). The ﬁrst historical inventory includes44,724 landslides in Umbria, Italy (Guzzetti
et al.
2003), estimated to have occurred in the last 5–10 ka (thousand years). The power-law tail of thefrequency densities is in good agreement with thelandslide-event probability distribution of Eq. (3).With a landslide magnitude of
m
L
¼
5.8
+
0.1,we estimate that over the last 5–10 ka the totalnumber of landslides that have occurred is
N
LT
¼
650,000
+
150,000. The second historicalinventory in Figure 3 includes 3424 landslides inthe Akaishi Ranges of central Japan (Ohmori &Sugai 1995) estimated to have occurred in the last10 ka. The power-law tail of the frequency densitiesgives
m
L
¼
6.0
+
0.2, corresponding to
N
LT
¼
1,100,000
+
500,000.Malamud
et al.
(2004
b
) related the landslide-event magnitude for individual events to the totalarea and volume of associated landslides, as wellas the area and volume of the maximum landslides.They then used the historical landslide inventories just discussed (Fig. 3) from Italy and Japan, andmade estimates of total area and volumes of land-slides involved over time for each of the regions,and from these a lower bound estimate on regionalerosion rates due to landslides. They inferred long-term erosion rates due to landslides in these tworegions of Italy and Japan as 0.4 and 2.2 mmyear
2
1
, respectively.
Conclusions
Landslide events display large variations in landslidetypes, sizes, distributions, patterns, and triggeringmechanisms. Many would question whether suchcomplex phenomena can be quantiﬁed. Malamud
et al.
(2004
a
) showed that three well-documentedand substantially complete landslide-event inven-tories from different parts of the world, each withdifferent triggering mechanisms, have frequency–area statistics that are well approximated by a three-parameter inverse-gamma distribution (Eq. 3). Inthispaper,wehaveshownthatafourth,muchsmallerlandslide inventory, is also well-approximated byour proposed ‘general’ landslide distribution. It isclearly desirable to test this distribution using othersubstantially complete landslides inventories.
A GENERAL LANDSLIDE DISTRIBUTION 109

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