Geophysical Center RASGeophysical Center RAS

Department of Complex Data Department of Complex Data Studies Studies

Laboratory of GeoinformaticsLaboratory of Geoinformatics

Current State of Art and Review

(some results were obtained in co-authorship with scientists from other bodies of the Russian Academy of Sci.)

• Geophysical objects could be rather complex objects with a memory and with a complex dependence from a number of parameters and variables; both the number and the character of a part of these dependences can be unknown. Moreover, the occurrence of unstable avalanche-like regimes in the behavior of the complex geophysical objects is possible.

• Rather frequently, for example in the case of monitoring of dangerous geophysical processes (seismic activity, volcanic eruptions, landslides, …), the prognosis of the behavior of the complex dynamic system should be presented in the case of the deficiency both in data amount and in our understanding of the underlying geophysical processes.

• The deficiency in understanding and in data could be tried to be partly compensated by the use of fuzzy ideas of the experts.

In a number of cases the task is to simulate the expert work because of an enormous volume of data that should be analyzed in the near real time regime.

In this situation it seems reasonably to try to use : - the fuzzy logics based algorithms of data processing , - the general models of unstable and avalanche-like behavior, - other similar approaches.

.

Fuzzy Logics:Discrete

Mathematical Analysis (DMA):

Theory

Discrete Mathematical

Analysis: GeophysicalApplications

Cluster Analysis,

Clearing of Geological Structure

Morphologic Analysis

of 1-D (Time)Data Rows

Application toRegime ofNaturalHazards

Monitoring of Hazardous

GeophysicalPhenomena

Power LawHeavy Tail

Distributions

Does an IncreaseIn Loss ValuesContradicts to

SustainableDevelopment?

DMA definition

Discrete mathematical analysis (DMA) is an approach to studying of multidimensional massifs and time series, based

on modeling of limit in a finite situation, realized in a series of algorithms.

The basis of the finite limit was formed on a more stable character, compared to a mathematic character, of human idea of

discontinuity and stochasticity. Fuzzy mathematics and fuzzy logic are sufficient for modeling of human ideas

and judgments. That was reason why they became technical foundation of DMA.

Construction scheme of DMA

Fussy comparisons onpositive numbers

Nearness in finite metrical space

Limit in finite metrical space

Density as measure oflimitness

Smoothtime series.Equilibrium

Monotonous time series

Fussy logicand geometry on

time series:geometrymeasures

Separation of dense subset.

Crystal. Monolith.

Clasterization.Rodin

Predicationof time series.

Forecast

Anomalies ontime series.

DRAS. FLARS

Extremums on

time series.

Convextime series

Search of linearstructure.Tracing

Fuzzy comparisons

Definition. A fuzzy comparison ( , )f a b of real numbers a and b measures in the alternative

scale of [ 1,1] segment the rate of superiority of “b” over “a ”:

( , ) ( ) [ 1,1]n a b es a b .

Example. For ( 1,1) we define

0( , ) ( ( , ))n a b n a b , где

, [ ,1]1

( )

, [ 1, ]1

tt

tt

t

, 0( , )max( , )

b an a ba b

Remark. For given pairs (5,10) and (70,75)

0

5 1(5 10) (5,10)

10 2es n and 0

5 1(70 75) (70,75)

75 15es n

Extension. Let us extend ( , )n a b to some arbitrary non-negative finite sets A and B . I f

1 20 NA a a a , 1 20 MB b b b , then

, ( , )( ) ( , ) [ 1,1]i j i jn a b

es A B n A BNM

Cluster Analysis, Clearing of Geological Structure

(2-D Data Analysis)

Algorithm “Choosing of foundation”Let ( , )X d be FMS and ( )XP x – chosen model of density, x X , ( ) { ( ), }XP X P x x X .

Definition: Let us call point x strong (weak) n – foundation in X , if for measure of

superiority ( )XP x over ( )P X , induced by fuzzy comparison n , inequality is corrected

μes( ( ) ( )) ( ( ), ( )) 0.5X XP X P x n P X P x ( 0) ( )

“Choosing of foundation” – algorithm of recognition in dense areas

in X by correlation ( )

Model examples

=2.5 =4.0

blue – foundations; red – crystallised points

Geophysical applications:

region Hoggar (Algeria), aeromagnetic data

Географич. к-ты региона

70ВД, 210СШ

TN 38 500 нТл

DN -3.50

IN 250

Region Hoggar (Algeria), cluster and classification analysis of potential field anomaly data (the Euler solutions clustering ).

Initial geomagnetic field (left), the routinely adjusted Euler solutions of the local geomagnetic field (centre) and the result of the use of CRYSTAL algorithm (right)

T

Algorithm of clustering RODIN: overview

• The cluster definition:

cluster in X, if

A – cluster in X, if

RODIN: model examples

C artographied area

L a n n io n

G u in g a m p

Golfe deSaint-Malo

J ersey

MCCPC

Zone

Shear Armorican

North 0 10 20 km

S a i n t B r i e u c U n i t

ortho-gneisses

brioverian meta-volcanics

brioveriansediments

gabbroic & tonaliticintrusions

Guingamp Unit

metagabbrosmigmatitesleuco-granites

Saint Malo Unit

migmatites granite(Lamballe)

sediments

granitoid (Ploufragan-Saint brieuc)

post cadomian intrusions

paleozoic and mezosoic sediments

variscan granitic intrusions

doleriticdikes

Geophysical Applications of RODIN:Saint Malo Region, regional geomagnetic data

Geophysical applicationsSaint Malo Region

Geophysical applicationsSaint Malo Region

Algorithm “Monolith”

X – multi-dimensional massif, xX, A – subset in X

Crossing AA (1) =MonA X – “multi-dimensional topological” smoothing A in X

MonA X – subsep points in X with large A-monolithness (A-foundations)

radius of monolithness r weight of monolithness parameter of choosing number of iterations i

Algorithm “Monolith”: parameters

monA x – monolithness A in x measure of limitness A in x

Algorithm “Monolith”: block-scheme

A m o n A (. ) A (1 ) m o n A (1 )(. ) A (2 ) A ( i)m o n A ( i-1 )(. )...

r (X )

...

Etna Volcano. Typical SAR – imaging data

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Smooth cells determination

Let be a set of pixels, ( ) п - a shift of pixel п measured by satellite, ( )D п - its

neighborhood (it could be neighboring points or a sphere of radius r around pixel п).

There are different variants of smoothness ( )G п of shift in point п. For example:

1. Mean value;

2. Linear regression value;

3. Similar approaches with excluding of evident error values

A palette was used in our calculations ( )D п - 3х3 and

ï

ï

( ) ( )( )

8G

п п

п

Etna. Smooth points, without special processing

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Etna. Monolith (4-th iteration)

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Etna. General solution

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Current Section Conclusions:

1. Discrete Mathematical Analysis (DMA) was elaborated and the collection of DMA based algorithms was realized in a program form.

2. Algorithms of clustering CRYSTAL and RODIN based on fuzzy logics were verified at the model examples and were shown to be effective to make the features of geological structure essentially more clear.

Analysis of 1-D (Time) Data Rows and

Monitoring of Hazardous Geological Phenomena

La Fournaise volcano (Reunion Island)

Application to the detection of electrotelluric signals associated with the volcanic activity of La Fournaise volcano (Reunion Island)

DRAS and FLARS

Algorithms DRAS (Difference Recognition Algorithm for Signals ) and FLARS (Fuzzy Logic Algorithm for Recognition of Signals) are results of “smooth” modeling (in fuzzy mathematics sense) of interpreter’s logic, which searches for anomalies at the record.

DRAS and FLARS are based on fuzzy mathematic principals.

The Used Assumption on the Interpreter’s Logic

Local level:

Interpreter glances along the record and estimates activity of its sufficiently small fragments by positive numbers. At the same time, he puts some numeric marks to the centers of the fragments. In this way, from initial record interpreter necessarily proceeds to a non-negative function. It is naturally to call this function as a “rectification” of the initial time series. Indeed, greater values of this function correspond to more anomalous points (centers of fragments).

Global level:

The anomalies on the record are the uplifts on its rectification. And the real anomalies should be searched among the uplifts of the rectification function.

Interpreter’s Logic. Illustration

Record

Local level - rectification of the record

Global level - searching the uplifts on rectification

Examples of rectifications

1 Length of the fragment:

2 Energy of the fragment:

3 Difference of the fragment from its regression of order n:

here as usual is an optimal mean squares approximation of order n of the fragment . If n=0 we get the previous functional “energy of the fragment”:

4 Oscillation of the fragment:

1

1

kh

kj j

j kh

L y y y

2k

hk

j k

j kh

E y y y

2

kh

k j

j kh

hy y

h

2( ) [ Regr ( )]k

kh

k nn j y

j kh

R y y jk

0Regr2

k

kh

j ky

j kh

hy y

h

2 200 ( ) Regr ( ) ( )k

k kh h

k kj j ky

j k j kh h

R y y jh y y E y

( ) max mink k

h hk

j jj kj k

hh

O y y y

Illustration of rectification

Record

Rectification “Energy”

DRAS: block-scheme of the algorithm

Background measures

Record rectification

Record fragmentation

Potential anomaly on the record

Anomaly on the record

,

,y yL R

y

( )ky y kh

Record

DRAS: application to electric signals associated with the volcanic activity of La Fournaise volcano (Reunion Island)

Station – DON, direction - EW

FLARS: block-scheme of the algorithm

FLARS: recognition of anomaly on the record

α[0,1] – vertical level of how extreme are the current measure values

No anomaly on the record y NA = { kh Y : μ(k)<α}

Anomaly on the record y A = { kh Y : μ(k)α}

FLARS: application to the Superconducting Gravimeters data preprocessing (Strasbourg, France)

Morphological Analysis of 1-D (Time) Data Rows and

the Monitoring of Hazardous Geological Phenomena

Geometrical measures

T - any fuzzy conjunction ( minT , T )

(1,1,1,1)

( 1,1,1,1)

( 1, 1, 1, 1)

(1,1,1,1)

( 1,1,1,1)

( 1, 1, 1, 1)

( ) ( ( ), ( ), ( ), ( ))

( ) ( ( ), ( ), ( ), ( ))

( ) ( ( ), ( ), ( ), ( ))

li l s ri rs

li l s ri rs

li l s ri rs

t T t t t t

t T t t t t

t T t t t t

Examples:

1) (1,1,1,1)( )t - measure “background”

2) (1,1, 1,1)( )t - measure “beginning of mountain”

3) ( 1,1,1,1)( )t - measure “end of mountain”

4) (1, 1, 1,1)( )t - measure “left slope”

5) ( 1,1,1, 1)( )t - measure “right slope”

6) ( 1,1, 1,1)( )t - measure “hollow”

7) (1, 1,1, 1)( )t - measure “peak”

Geometrical measure “background”

5800 5900 6000 6100 6200 6300 6400 6500 6600 6700 6800700

720

740

760

780

мл

Га

л

5800 5900 6000 6100 6200 6300 6400 6500 6600 6700 6800

200

400

600

5800 5900 6000 6100 6200 6300 6400 6500 6600 6700 6800-1

-0.5

0

0.5

1

Geometrical measure “Beginning of mountain”

5800 5900 6000 6100 6200 6300 6400 6500 6600 6700 6800700

720

740

760

780

мл

Га

л

5800 5900 6000 6100 6200 6300 6400 6500 6600 6700 6800

200

400

600

5800 5900 6000 6100 6200 6300 6400 6500 6600 6700 6800-1

-0.5

0

0.5

1

Geometrical measure “peak”

5800 5900 6000 6100 6200 6300 6400 6500 6600 6700 6800700

720

740

760

780

мл

Га

л

5800 5900 6000 6100 6200 6300 6400 6500 6600 6700 6800

200

400

600

5800 5900 6000 6100 6200 6300 6400 6500 6600 6700 6800-1

-0.5

0

0.5

1

• .

The Other Scheme of Morphologic Analysis: The monotonous character of change (decrease, increase

or stable level of data row) is evaluated by comparison with the model of random deviations using Einstein relation, where S – an expected random change for n steps, d – error. Mean random displacement in result of n step:

The distance R from the needed model anomaly and the

current segment of data row can be evaluated as where the first and the second parts are connected with monotonous and oscillatory character of the record

,1 1

; , ; , 0m o

N N

m im im o io io m o m oi i

R x y x y

.

S d n

Morphological analysis:The signal under examination, upper panel;The trend characteristic: increase, stable level or decrease ( coded as “1”,

“0” or “-1”), second panel;The oscillation existence ( “1” or “0”), third panel;

The “distance” between the chosen part of the record and the current segment of the data row, lower panel.

The case of the model 1-D rowThe case of the model 1-D row

The use of the morphological analysis for the case of geolectrical monitoring data at la The use of the morphological analysis for the case of geolectrical monitoring data at la Fournaise volcano (Ile de la Réunion).Fournaise volcano (Ile de la Réunion).

Anomalies connected :Anomalies connected :1)1) with heavy rains (of high amplitude and mainly monotonous), and with heavy rains (of high amplitude and mainly monotonous), and2)2) with an increase in geothermal activity (strongly oscillating but lower amplitude with an increase in geothermal activity (strongly oscillating but lower amplitude

variations) variations) are determined and subdivided into the different classes.are determined and subdivided into the different classes.

The geoelectric monitoring data from La Fournaise volcano (Réunion Island).The geoelectric monitoring data from La Fournaise volcano (Réunion Island). Heavy rain type anomaly is searched.

The segment of the record under examination (above);The segment of the record under examination (above);The change in the “distance” between the model segment of the record and the The change in the “distance” between the model segment of the record and the

current segment of the record (lower panel).current segment of the record (lower panel).

The use for the real data examination: geoelectric monitoring data from La The use for the real data examination: geoelectric monitoring data from La Fournaise volcano (Réunion Island).Fournaise volcano (Réunion Island).

The hydrothermal type anomaly is searched.

The segment of the record under examination (above);The segment of the record under examination (above);The change in the “distance” between the model segment of the record and the current The change in the “distance” between the model segment of the record and the current

segment of the record (lower panel).segment of the record (lower panel).

.

Recognition of intervals of activation Recognition of intervals of activation with the use of the morphologic with the use of the morphologic

analysis for the case of the regime analysis for the case of the regime of heavy social incidents (norm of heavy social incidents (norm sum of accidents, unexpected sum of accidents, unexpected

death, and suicides) in Yaroslavldeath, and suicides) in Yaroslavl

The same morphological analysis procedure was applied to the analysis of the statistics The same morphological analysis procedure was applied to the analysis of the statistics of crimes and of the heavy social incidents in Yaroslavl, typical Central Russia regional of crimes and of the heavy social incidents in Yaroslavl, typical Central Russia regional

center with 650 thousands habitants. center with 650 thousands habitants. Below the change in “non-equilibrium activity” of the crimes and incidents numbers is Below the change in “non-equilibrium activity” of the crimes and incidents numbers is

shownshown

Recognition of time intervals of activation with the use of the morphologic analysis for the case of the regime of heavy crimes in Yaroslavl. The maximums in instability of regime of heavy crimes correlate with crises of middle 1990-s and after the default of 1998 year

Current Section Conclusions:

1. Discrete Mathematical Analysis (DMA) was elaborated and the collection of DMA based algorithms was realized in a program form.

2. Algorithms based on DMA and fuzzy logics (DRAS and FLARS) and the morphologic analysis approach were shown to be effective to reveal the anomalies of different form in 1- D data rows.

2. The results of the algorithms’ application showed the possibility of automatic detection of anomalous parts in the records and for the diversification of anomalies into a few different types, related to different morphology and (presumably) of different physical nature. The proposed algorithms can be used either in the functional monitoring systems for automation of the expert work, or for research purposes for recognition of characteristic morphological successions in the examined time series.

Fuzzy Logics:Discrete

Mathematical Analysis (DMA):

Theory

Discrete Mathematical

Analysis: GeophysicalApplications

Cluster Analysis,

Clearing of Geological Structure

Morphologic Analysis

of 1-D (Time)Data Rows

Application toRegime ofNaturalHazards

Monitoring of Hazardous

GeophysicalPhenomena

Power LawHeavy Tail

Distributions

Does an IncreaseIn Loss ValuesContradicts to

SustainableDevelopment?

.

The numbers of fatalities from natural (earthquakes, floods, hurricanes etc.) and man-made disasters as well as economic losses from the disasters have a tendency to an essential increase with time. This effect is commonly related to the growth of the Earth’s population, to spreading of potentially dangerous technologies and to environment degradation.

In this interpretation the process is assumed to be non-stationary that evidently interferes with the idea of the sustainable development of society.

The alternative interpretation is presented and examined below.

The problems: The problems: 1. Does the tendency of the loss values increase exist really?1. Does the tendency of the loss values increase exist really?2. How long this tendency can be extrapolated?2. How long this tendency can be extrapolated?

If this tendency is valid for a long time interval it would contradict with the very idea of sustainable development

Examples of increase of yearly numbers of events (a) and loss values (b) from

natural catastrophes world-wide (from www.em-dat.net/documents )

. Mean increase rates of looses from disasters in Russia are following:Mean increase rates of looses from disasters in Russia are following:

Number of victims – 4.3%Number of victims – 4.3%Number of suffering - 8.6%Number of suffering - 8.6%

Economic losses - 10.4%Economic losses - 10.4%

An example of the loss increase with time:An example of the loss increase with time: Number of disasters (N) and number of suffering people (S) Number of disasters (N) and number of suffering people (S)

in Russia from all disasters and from natural disasters only (N’, S’)in Russia from all disasters and from natural disasters only (N’, S’)

1993 1994 1995 1996 1997

0

500

1000

1500

N S , thoupeople

50

100

N

S

N '

S '

Examples: Power-law distributions of loss values in USA from

earthquakes (E), hurricanes (H), and floods (F), (from

coastal.er.usgs.gov/hurricane_forecast/barton4.htm)

=-0.74 (F), =-0.98 (H), and =-0.41 (E) .

Note, all values <1.

• .

An alternative approach: The effect of an increase in loss values could be connected with the

specific character of distribution of loss values in the case if the distribution of losses obeys the power distribution law with power index <1. In this case this effect could be an “apparent” effect.

The power distribution law with <1 is rather typical of losses from different types of disasters.

The specific feature of this distribution law is a highly increased probability of occurrence of huge events. This causes a statistical tendency of increase of loss value with time.

. We have: We have: The total loss values from earthquakes have a clear tendency to increase with The total loss values from earthquakes have a clear tendency to increase with

time.time.The distributions of loss values obey the power law with power index The distributions of loss values obey the power law with power index <1.<1.

The case of earthquake disasters much better supplied with data is examined. Distributions of yearly numbers of victims (a) and losses (c) and

the numbers of victims from the individual strong earthquake disasters (b)

1 100 10,000

V , victim s

1

10

100

N а

1 100 10,000

L , thou .$

1

10

100

N c

1 100 10,000

V , victim s

1

10

100

1000

N b

.

All seismic disasters were subdivided into three classes according to the number of All seismic disasters were subdivided into three classes according to the number of fatalities.fatalities.

• III V>100III V>100 (more 99% of total number of fataliries); (more 99% of total number of fataliries);• II 100II 100V>10 (about 1% of total number);V>10 (about 1% of total number);• I VI V10 ( < 0.1% of total number)10 ( < 0.1% of total number)

The apparent contradiction: The apparent contradiction: Only the strongest disasters (III class) are important in total number of fatalities, but Only the strongest disasters (III class) are important in total number of fatalities, but

the flow rate of these events seams to be stable with timethe flow rate of these events seams to be stable with time

Does the regime of the seismic disasters is unstable?Does the regime of the seismic disasters is unstable?

1900 1940 1980

t, years

0

100

200

300

N

III -

II-

- I

But maybe the character of distribution of events in the chosen intervals could But maybe the character of distribution of events in the chosen intervals could change.change.

The sequences of number of victims Vi and cumulative numbers The sequences of number of victims Vi and cumulative numbers Vi for the Vi for the different used ranges of disasters are given.different used ranges of disasters are given.

It can be seen that the regime of strongest III class disasters (It can be seen that the regime of strongest III class disasters (more more 100 victims, causing more than 99% of total number of victims 100 victims, causing more than 99% of total number of victims) is stable) is stable

1900 1920 1940 1960 1980 years

0

10

Vi

0

400

800

1200

V i

1900 1920 1940 1960 1980 2000

0

4000

8000

12000

V i

0

100

V i

1900 1920 1940 1960 1980 2000

0

200

400

600

800

1000

ln (V i) V i

102

104

а b

с

V>100 100V>10

V<1 0

.

Thus, we have apparent contradiction:

The numbers of victims and economic losses from earthquakes (as well as from most of the others natural and man-made disasters) have a clear tendency to a non-linear increase with time.

But the flow rate of the strong seismic disasters that cause more than 99% of the total number of victims is shown to be stable.

This disagreement can be explained if the specific character of the distribution of loss values from the strong seismic disasters is taken into account.

Specific properties of the power-law distribution with power index 1

• .

The power Pareto law with distribution function F(x) is used : F(x) = 1 (A/x); xA, (1)

When 1 the Pareto law (1) has a “heavy tail” with infinite mean value and dispersion. Using routine statistic procedures is incorrect in this case, and the order statistics has been used.Medians of the cumulative effect n and the maximum event n value increase

with the number of events n in a non-linear manner: n ~n1/ and n ~n1/ (2)

and in the case of a stable flow of events with time t we have: t ~t1/ and t~t1/ , where 1/ >1 (2’)

Important !Effect (2) obtained here in the strictly stable model can be incorrectly treated as the evidence of the unstable growth of loss values from disasters with time.

Analytical and numerical methods were used for the calculations of the cumulative effect values in the case of the power law distribution with 1

• .

1. Analytical approach. The Pareto law was used as a model and the following results were obtained for the expected values of the cumulative effect values:

E ln 1 +{ln(ci(}. (1)

Where - incomplete Gamma-function, c – constant; i - integral exponential function. T – time interval, - mean flow of the cases. In the case n= a more simple variant of (1) has place: E ln 1 Г + {ln(c}; (2)

lnlnln c; . where Г– Gamma function. The upper and lower limits of with probability

(1-2) can be evaluated from the following relations : () <an1/ -1/. (3)

and

() > an1/ {ln(1/ )}- 1/ . (3а)

The very useful approximate relation between the median value of the cumulative effect and the median maximum individual value VT for the case n= was

obtained and used also: VT/(1-) (4)

2. Numerical modeling. The other variant of the calculation is based on the use of

the numerical modeling with use of the bootstrap-method.

Naturally, both the analytical and numerical approaches give similar results.

• The clue parameter is the magnitude C and the recurrence time Tc of the critical event corresponding to the boundary between the power distribution law with <1 and the unknown distribution law of rarest huge events

• The model distribution law (left) and the corresponding law of increase of typical cumulative loss values (right). Change in the distribution law (left) causes the change in the increase of cumulative loss value with time (right)

1 0 1 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 0 0n u m b e r o f v ic t im s , N

0.001

0.01

0.1

1

10

n/ye

ar

Zone of the power-law d istribution w ith "heavy ta il",non-linear increase in loss va lue w ith tim e

Zone of d istribution w ithout "heavy ta il",linear increase w ith tim e (right fromthe boundary)

boundaryzone

1 10 100

T, "years"

100

1000

10000

100000

C um ulative Loss Value

Tc, recurrence tim e

C , critica l event m agnitude

The power distribution law (1) with power index <1 has an infinite mean value that is nonsense in practical sense. It means that the real distribution law of rarest huge events should differ from this law. It is the case.The model case of seismic disasters distribution:

Cross-over point of the change in the distribution law corresponding to the typical and rare strongest events (Vc) can be defined as a point corresponding to =1 slope of the frequency-size curve

It means Vc value can be defined from the following:

d{lg(1-F(V)}/d lg(V) = 1. (*)

The described regularity in the growth of cumulative effect values has a rather common character, and the definition (*) is applied in a rather common case also. It can be applied in the every case when three following requirements are fitted: 1) if the distribution of events of moderate size is described by the Pareto law with <1, 2) if the distribution of rare major events can be described by a distribution function G(M) such that there are parameters such that:

{1-G(V)} = ( /V) , where >

and 3) if the density of distribution g(V) is a monotonous function.

More robust and similar approach to the Vc evaluation is based on the change from the non-linear to the linear regimes of increase of the

cumulative loss values (t). The following relation was used

<lg((t))> = a2lg2(t) + a1lg(t) + a0.

Parameters аi (i=0, 1, 2) were evaluated by the least squares method. The duration of a time interval Tc of the change to the linear mode of (t) growth can be determined from the evident condition

d{lg((t))}/d(lg(t))|t=Tc = 1. (*)

This relation was used for Tc value determination. According with the definition given above, characteristic Vc value is a loss value V with recurrence time Tc, i.e. Vc is the maximum single event occurring typically during Tc time interval. The following relation between cumulative effect value (Tc) and the maximum value Vc was used. (Tc) Vc/(1-). Thus, in result, the characteristic Vc value that meets the equations (*) and d{lg(1-F(V)}/d lg(V) = 1, was evaluated with the typical recurrence time Tc.

The analysis was performed for the first (red) and second (blue) half of the XX-century and for the developed (a) and

developing (b) countries.It can be seen that the characteristic disaster size Vc

corresponding to the beginning of the linear regime of V(t) growth had decreased essentially in the developed countries.

The regime of growth of number of victims depends on the social/economic situation

1 10 100

t, years

1 0

1 0 0

1 0 0 0

1 0 0 0 0

1 0 0 0 0 0

V(t)

1 10 100

t, years

1000

10000

100000

1000000V(t)

Vc

Vc

Vc Vc

Vc

Vc

Vc <Vc

Examples of C and Tc values for different types of disasters (numbers of fatalities)

Disaster type Regions,countries

Recurrence time

Tc, years

Characteristic boundary

event magnitude,Vc, victims

number

Really observed

maximum disaster, fatalities number

Data source

earthquakes Developed countries1900 - 19591960 - 1999

3330

9500024000

11000017000

Significant earthquakes

Data base(corrected)

earthquakes Developing countries,1900 - 1959,1960 - 1999

4065

270000260000

200000240000

Significant earthquakes

Data base(corrected)

floods North America and European Union

1950-19801980-2005

1510

1500500

650200

Em-dat

hurricanes Atlantic Ocean1850-1950

30 20000 11000 Disaster center

floods SE and S Asia1984-2006

20 10000 6000 Em-dat

In all cases the recurrence time Tc is a few dozens years only, it means the loss increase in this model is restricted to the similar time intervals.

The similar approach was used in analysis of the seismic regime (the recurrence time of

earthquakes with the different seismic moment M values

. + - data from the + - data from the Harvard world catalog of the seismic moments: Harvard world catalog of the seismic moments: BlueBlue dashed linedashed line - Gutenberg-Richter law with - Gutenberg-Richter law with 0.65; 0.65; red linered line - - =1.=1.

The distribution of the rare strongest earthquakes clearly deviates The distribution of the rare strongest earthquakes clearly deviates from the Gutenberg-Richter relation that corresponds to the from the Gutenberg-Richter relation that corresponds to the power law distribution with power law distribution with <1<1

The difference in the distribution law of the typical and rare major earthquakes

1e+024 1e+025 1e+026 1e+027 1e+028 1e+029

M , d y n -с м

1

10

100

1000

N

Events Events with depth less than 70 km were used. The different regions correspond to the with depth less than 70 km were used. The different regions correspond to the

subduction zones, shear (transform fault) zones, middle ocean ridge (MOR) zones, subduction zones, shear (transform fault) zones, middle ocean ridge (MOR) zones, and zones of continental collision.and zones of continental collision.

Harvard seismic moments catalog was used; a number of homogeneous geodynamic regions were chosen for the study

• Seismotectonic parameters of different regions

Regional parameters of seismic regime * Region number

1 2 3 4 5 6 7 8

1 -1.24 4.3 50 1 3.43 0.3 - -

2 -0.64 12 50 4 4.88 0.31 0.15 23

3 -0.85 8.8 52 1 4.43 0.44 0.16 16

4 -1.0 8.8 90 5 2.98 0.88 - -

5 -0.58 8.7 119 5 3.33 0.88 1.45 63

6 -0.62 8.2 88 5 3.38 0.58 0.4 42.5

7 -0.69 7.5 117 1 2.94 0.88 0.68 41

8 -0.61 4.6 80 3 3.64 0.32 0.12 17

9 -0.7 0.9 49 1 2.45 0.017 0.02 53

10 -0.52 10. 38 7 3.62 1.28 0.62 22

11 -0.45 7.2 17 6 4.27 0.87 0.36 16

12 -0.84 7.6 94 4 2.52 0.12 - -

13 -0.71 9.9 67 6 3.87 0.28 0.2 37

14 -0.62 6.3 49 6 3.57 0.41 0.86 90

15 -0.62 3.11 0.17 0.08 25

16 -0.77 2.38 0.05 0.02 26

17 -0.61 3.55 0.06 0.04 30

18 -0.92 3.31 0.01 0.004 22

19 -0.8 3.3 0.02 0.01 14

20 -0.45 5.67 0.33 1.0 84

21 -1.1 3.23 0.014 0.003 9

22 -0.69 5.62 0.44 0.23 21

23 -0.61 6.46 0.26 0.09 14

24 -0.58 5.5 0.48 5.1 330?

25 -0.65 3.3 3.1 1.5 5

* 1 – slope of the seismic moment–frequency relation in the linear part; 2 – subduction velocity, sm/year; 3 – plate age, 106 years; 4 – «stress class»; 5 – median seismic moment value (М1024 dyn.cm)1024; 6 – Mmax, 1028 dyn.cm; 7 - Mc, 1028 dyn.cm; 8 - Tc, years.

Coefficients of correlation of characteristics oftectonic and seismic regime in different regions*

Seismic and tectonic characteristicsSeismic and

tectoniccharacteristics 1 2 3 4 5 6 7 8

1 in linear partof the seismic

moment-frequencyrelation

1

2 Subductionvelocity

0.2 1

3 Plate age -0.12 -0.1 1

4 Stress class 0.52 0.52 -0.16 1

5 Median Mvalue

0.38 0.56 -0.51 0.25 1

6 Mmax 0.4 0.63 0.15 0.46 0.38 1

7 Mc 0.72 0.54 0.36 0.53 0.43 0.87 1

8 Tc 0.44 -0.27 0.41 0.1 0.11 -0.2 0.97 1

* - logarithms are used if the range of change of variable is too large.

Current Section on Earthquake Size Distribution Conclusions Power-law with <1 is typical of the distributions of strong events in the

regimes of losses from natural and man-made disasters, earthquake seismic moment and energy values and in other fields. But, the distribution of rare strongest events differs from the power law with 1.

The non-linear and linear regimes of the growth of cumulative effects are shown to be typical for the mentioned cases and correspond to the difference in the distribution law of the typical and the rare strongest events.

The case of the distribution the of seismic moments is examined. The problem of parameterisation of the regime of the rare strongest events is treated in terms of the change from the non-linear to the linear mode of the growth of cumulative seismic moment values. This approach due to the inherent averaging of cumulative values ensures more robust results than the alternative approaches.

Using the presented parameterisation it was shown that the parameters of the seismic regime depend on the tectonic situation in different regions. Thus, the frequency-seismic moment distribution of earthquakes appears to be not universal but tectonic dependent.

The regime of loss from earthquakes was shown to depend from the social and economic situation in

different regions.Lets examine these connection in a more details

.

The connection of loss regime with social and economic situation

The flow rate of the strong (>100 fatalities) seismic disasters increases in the developing countries (1) and decreases in the developed countries (2)

(in comparison with the linear prognosis performed using the data for 1900-1940 years)

1900 1920 1940 1960 1980 t, years

0

100

200

N

1

2

V>100 people

The distribution of values of (economic loss)/(number of victims) ratio differs essentially in the developing (1) and in the developed countries (2)

0.001 0.1 10

L , 106 $/victim

1

10

100

N

2

1

A number of correlation exist between economical and loss characteristics,an example:

loss/victims number ratio value and year per capita income correlation

0.1 1 10 100

Р , thou . $ /m an

0.1

1

10

100

L,10 6

$ / v ic t im

The loss value/victim number ratio L increases with the increase of the year per capita income value P as (approximately) L~ P2.

Data for the strong earthquakes occurring at the territory of major cities were used.

Social and Economic Characteristics of Damaging Earthquakes for 1900-1999 in Different Regions

Region

(1)

Mean Number of casualties

(2)

Mean loss in

$

million

(3)

Mean ratio loss/casu-alties in $ million per

victim

(4)

Annual per capita product,

$ 1000 (as of 1970)

(5)

Ratio of values in columns

(4)/(5) 1000

(6)

Ratio of values in columns

(3)/(5) 1000

man-years (7)

North America

45

800

32 4.5 7.1

180

South Europe

800

340

8 1.5

5.3

230

Japan 2500 430 5.5 1.6 3.4 270 Latin America

560

130

1.3 0.5 2.6 260

Asia 2800 50 1. 0.2 5. 250 Indo-China

480

18

1.2 0.15 8. 120

Max/min ratio in columns

70

45

32 30 3 2.

Thus, being norm the loss values appear to be rather stable, and even have a tendency to decrease.

Different sets of data testify that the regime of disasters depends essentially from the current change in economic and social situation. Thus, the more exact results of a near real time prognosis of a number of victims and of a damage value can be obtained if the current social and economic situation would be taken into account.

1900 1920 1940 1960 1980 2000

T , y ea rs

0

0.01

0.02

0.03

0.04

N

Number of strong catastrophes in the Russian Federation. Points - number of events per million habitants, line - mean 5-year number of catastrophes. It can be seen an increase in a number of disasters during economical and social crisis in 1990-s years.

The Economic Disaster in Russia in the End of XX-century: the View from the Space

The current information on the night lights intensity and distribution can be used for the monitoring of current change in the population density and in the land use in the damaged areas.

Below an example of essential change in land use and population density in Russia in connection with the crisis of 1990-s years is given. The Cola Peninsular region (heavy suffered from the crisis)

and the Moscow region (essentially less suffered) are compared

A, Cola Peninsula, central region, night lights in 1993 B, the same region, night lights in

2000

A, die out in night lights (given in blue color) predominates in the suffered region

B, change in the less suffered Moscow region

Change in night lights in the Central part of the Cola peninsular (A) and in the Moscow region (B) in the interval from 1993 until 2000 year.Red color corresponds to origin of new lights, and blue color means that lights die out.

Current Section Conclusions:

1. The alternative explanation to the opinion on the non-stationary increase in loss values from natural disasters with time is presented. The non-linear growth in cumulative numbers of casualties and losses from disasters can be explained in terms of the stable model in the case when the loss values distribution has a “heavy tail”.

2. Nonlinearity in the growth of earthquake-caused casualties and losses occurs (at the planetary scale) over time interval about 20-30 years. When longer time intervals are considered, the size of the maximum disaster stops increasing owing to the natural restrictions on the amount of maximum loss. The total loss from earthquakes increases approximately linearly with time at intervals 50 years and longer.

3. Relations of the numbers of casualties and loss values from earthquake with social and economic parameters were examined, and a number of correlations were revealed. This results give ground to expect a decrease in a number of disasters with large numbers of fatalities with economic and social development despite the increase in population and urbanization. This prognosis is much more optimistic than those presented before in result of a formal extrapolation of observed numbers of fatalities and loss values.

4. Being norm to the year per capita income the typical loss characteristics from earthquakes appear to be rather stable in the countries with different level of economic development.

5. This result does can not contradict with the idea of sustainable development.

Thank you for attentionThank you for attention