RUSSIAN JOURNAL OF EARTH SCIENCES VOL. 10, ES1004, doi:10.2205/2007ES000267, 2008
[5] The study of natural disaster pattern was based on the authors' compiled database of damage from earthquakes [Rodkin and Pisarenko, 2000]. Such a choice was caused by the fact, that data allowing to reliably track the character in change of the death toll from natural disasters over the last century are available only for earthquakes, moreover starting from the 1950-1960s the data on the economic damage are fairly comprehensive. As to the other natural disasters, the information is not abundant. However, the data available convincingly imply a qualitative uniformity of damage patterns from different types of natural and natural-technogenic disasters and make one assume, that the obtained conclusions may be used not only in case of earthquakes but may be applied to other types of natural disasters. The presented results of the damage pattern from earthquakes were obtained by Pisarenko and Rodkin [2003a, 2003b], Rodkin and Pisarenko [2000, 2004, 2006], etc.
[6] The above mentioned general tendency of nonlinear growth of cumulative damage is well exemplified by earthquakes. A number of the 1900-1999 earthquakes resulted in casualties growth with time as t1.4 and the total death toll as t1.6. This might result in the conclusion about non-stationary growth of damage from earthquakes. However, such a conclusion is not correct.
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Figure 2 |
![]() |
Figure 3 |
[9] Figure 3a shows that sequence of values for the death toll in events of scale-range III does not change systematically. The curve of cumulative sum of logarithms in fact does not differ from straight line suggesting stability of the death toll logarithm mean. Hence a conclusion about stationary of a number of strongest seismic events and distribution of the death toll from such earthquakes. A sequence of values of cumulative death toll in events of scale-range II (Figure 3b) also does not show important changes, whereas growth of cumulative sum shifts slightly from a straight line. Only the event pattern of scale-range I reveals a strong non-stationarity. A number of such events grows fast (Figure 2), while an average death toll in case of unit event tends to decrease (Figure 3c). However, such changes is naturally to attribute to a better record system and not to a change in recurrence pattern of seismic events.
[10] Thus, a strong non-stationarity of seismic event pattern is reported only for weak events totaling less then 1% of the total death toll. Such a non-stationarity cannot explain an observed tendency of a strong growth of the death toll with time.
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Figure 4 |
![]() | (1) |
[12] Another condition (typical of damage cases) that the power distribution starts to be fulfilled with some minimal damage quantity a is entered into equation (1). The condition is caused by a fairly incomplete statistic on minor disasters. Noteworthy, that an actual law of distribution for a number of weak disasters is not of real interest, because they contribute slightly to the total damage amount.
![]() |
Figure 5 |
[14] Atypical (imitating the presence of non-stationarity) behavior of cumulative damage values is due to the
fact, that mean value and dispersion are infinite for power distribution of eq, (1) type with an exponent of
power of distribution
b1 (such distribution are called the heavy tale distribution).
A unit maximal event and the total effect for such distributions turn to be of the same order. Probability of
realization of an extremely strong disaster increases with time of observations, and, correspondingly, accumulative
damage value increases nonlinearly. As a result, a tendency of nonlinear growth cumulative damage with time is
observed in the context of a well-defined stationary model.
[15] In case of heavy tail power-law distributions, medians, i.e. values accounting for not more than 50% of
elements of a sample and no less than other 50% of elements, are used instead of mean values that are formally
infinite. In our case of special interest is an estimate of
median of expected total damage in time
t,
t, or resulted from
n events,
n. Analytically,
this problem can be solved using the relation between the total damage median
n and that of distribution of M
max of sample
mn
![]() | (2) |
![]() | (3) |
[16] Equations (2) and (3) show that the quantities
mn and
n increase with number of events or
with time nonlinearly as
n1/b or
t1/b, respectively. Relations (2) and (3) are convenient to use for evaluation
of specific total damage in these cases, when the empirical distribution of damage is described by the Pareto law
at
b<1. Quantitatively close estimates of damage can be obtained by means of numerical simulation.
[17] Let us sum up. The analysis of the power pattern of the damage value distribution (1) implies, that nonlinear growth of cumulative damage can occur within the framework of the stationary model, i.e. empirically observed nonlinear growth of damage is not necessarily assumes the non-stationary disaster pattern. In case of earthquakes, it turns out that the observed tendency of damage growth with time might be easily attributed to the presence of the heavy tail of the damage distribution (a number of casualties and economic losses) caused by occasional earthquakes. Actually, according to relations (2) and (3) an expected nonlinear growth of the death toll is given by exponent 1/b = 1/0.7 =1.4; whereas an empirically revealed law of average growth amounted to 1.6. It is obvious that an observed growth of casualties caused by earthquakes accounted for an important scatter in behavior of different statistical achievements is not in conflict with the stationary model (1). Actual non-stationarity (discussed below) is of secondary importance.
[18] Naturally, in case of other types of natural disasters, for example, hurricanes, whose recurrence pattern depends on climate changes, a real non-stationarity may be of great importance. However, at least partly the observed effect of nonlinear growth of damage amounts with time, will be due to the power distribution law of damage, caused by individual hurricanes. Noteworthy, for the case of the number of people, who became homeless during floods the readers are referred to the paper by Pisarenko [1998] showing a key role of this factor.
[19] The presented schematic diagram shows the growth of damage with time for the case of the heavy tale distribution. However, two important questions remain unsolved. First, within which range simulation of damage distribution can be described by the power law, and hence when is nonlinear growth of total damage with time to be expected. Second, what is the relation between damage amount and such important social-economic processes as the growth the population and the development of technosphere. Let us discuss these issues.
Citation: 2008), Damage from natural disasters: Fast growth of losses or stable ratio?, Russ. J. Earth Sci., 10, ES1004, doi:10.2205/2007ES000267.
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