Earthquakes
and earthquake effects are controlled by many unknown factors, not always
allowing applications of deterministic methods of analysis. Some of these can be
treated statistically. The probability of an area being struck by an earthquake
is often referred to in engineering applications. The concept is illustrated
below.
Suppose an earthquake occurs in an area A. Consider a small area a
in A. Let the area a
be, further, made of very small areas Da
in it, so that SDa
= a. Let us
call a specified location hit by an earthquake if the peak value of the ground
acceleration there exceeds a certain specified value.
Let the probability of none of the areas Da
being hit by the earthquake be written as p (0,a). The probability of
another smaller area Da outside not being hit by the earthquake may
be written as:
p (0,
a+Da)
= p(0,a).p(0, Da)
= p(0,a).(1- Da/A)
or
(1/ Da
).[p(0,a+Da)-p(0,a)] +(1/A).p(0,a)=0
Letting
Da
-> 0 this equation becomes
dp/da
+ (1/A).p =0
The
solution of this equation is given by:
p(0,a) = e –a/A
The
probability that the area a will be hit by the earthquake is, then,
given by
p(1,a)
= 1 - e –a/A
This
is the expression for probability in space. A similar expression for probability
can be derived, by analogy, in the time domain. The probability of the
earthquake hitting during a time interval of D years (where D years may be the
life of the engineering structures in the area) may be written as:
p(1,D)=1-e-D/T
Equating the two expressions of probability one gets the return period
of the event, having a probability of occurence, p(1,a), during an interval of
D years :
T
= -D/[Loge {1 – p (1,a)}]
In statistical analysis it is often assumed that the number of
earthquakes in year is a Poisson variable and the earthquake magnitude, x, is a
random variable with cumulative density function F (x) = Pr (X £ x) =1 –e -bx;
x ³0. In this model the probability, RD
of an earthquake of magnitude M in D years is given by :
RD
= 1 –Exp. (-aDe-bM)
Here a
and b are constants related to the seismicity
parameters specified by the earthquake magnitude frequency relationship,
namely:Log 10 N (M) = a-bM
Where N (M) is the number of earthquakes occurring annually in the
region and having magnitudes equal to or greater than M, and a = Log 10
N (0) and b the constant b determines the distribution of magnitudes in the
earthquake population. (see
Frequencies of Earthquake occurrence 4.3
).
a
= Exp. [a Log e10] and b = b Log e 10
References:
1.Housner, G.W.(1975). Strong Ground Motion, Chapter 4 in Earthquake Engineering, R.L. Weigel (Ed.), Perentic Hall, N.J.
2.Lomnitz, C.(1976). Global Tectonics And Earthquake Risk, Developments in Geosciences, Elsevier.
3. Epstein,B. and Lomnitz,C. (1966). A model for the occurrence of large earthquakes, Nature 211,954-955.
An earthquake is
detected at a location through the movement of the ground, and the detection
capability depends on the maximum amplitude of the ground motion and the
sensitivity of the detecting agency (a living being, a structure or an
instrument). When the amplitude exceeds this level of sensitivity (which may be
called detection threshold) the earthquake may be considered as “felt”
at that point, and all such points where the ground motion exceeded the
detection threshold lie in the felt area of the earthquake. This perception of
the felt area can be enlarged to include a variable detection threshold, so that
for any specified level of ground motion amplitude (for acceleration, velocity
or displacement) the ‘felt area’ can be defined for an earthquake. A ‘felt
area’ table can then be constructed in which variation of the ‘felt area’
is shown as a function of the earthquake magnitude for a fixed value of ground
motion amplitude, and for a fixed magnitude as a function of different levels of
ground motion. One such ‘felt area’ table is illustrated below.
|
PGA Value Radius of Felt
area of the earthquake kms (PGA exceeded) Magnitude of the
earthquake (g) 4.0
4.5 5.0
5.5 6.0
6.5 7.0
7.5 8.0
8.5 |
|
0.01
50.7 77.1
114.2 168.9
250.0 365.7
533.9 781.1 1140.3
1668.3 |
|
0.03
7.8 23.8
40.6 63.4
95.0 140.6
208.5 304.4
445.4 650.2 |
|
0.05
0.0 2.0
21.4 38.1
59.3 90.2
133.5 195.3
289.0 421.9 |
|
0.07
0.0 0.0
9.4 25.4
42.8 66.7
100.1 146.5
216.7 316.4 |
|
0.09
0.0 0.0
0.0 16.7
32.5 52.1
79.1 118.7
173.6 253.4 |
|
0.11
0.0 0.0
0.0 8.8
25.0 42.2 66.0
99.0 146.5
213.8 |
|
0.13
0.0 0.0
0.0 0.0
19.3 35.2
56.3 84.5
125.0 185.4 |
|
0.15
0.0 0.0
0.0 0.0
14.1 30.1
48.8 75.1
111.3 164.8 |
|
0.17
0.0 0.0
0.0 0.0
9.9 25.4
42.8 66.7
100.1 146.5 |
|
0.19
0.0 0.0
0.0 0.0
2.0 21.4
38.1 59.3
90.2 133.5 |
|
0.21
0.0 0.0
0.0 0.0
0.0 18.1
33.8 54.2
82.4 121.9 |
|
0.23
0.0 0.0
0.0 0.0
0.0 14.8
30.5 50.1
76.1 112.6 |
|
0.25
0.0 0.0
0.0 0.0
0.0 11.7
27.5 45.7
70.3 105.5 |
Such ‘felt area ‘tables can be generated for any chosen empirical relationships described above for any set of PGA values, and come very handy in probability calculations.
Zonation
is a frequently used term in the context of earthquake hazard. This is used for
demarcation of geographically contiguous area on the Earth’s surface
for which certain specified values of a selected set of geological and/or
physical and/or socio-economic parameters applies. Zonation. to identify the
exposure to earthquake hazard, so that a specified hazard mitigation strategy
can be adopted within each zone, is the primary objective of the earthquake
hazard zonation. Depending on the detailing of the information (geological,
geotechnical and seismological data on the one hand and information on cultural
and economic development on the other) the nomenclature (macro, micro or site
specific) will vary. On global scale the plate boundaries provide a
macrozonation of the Earth’s surface. Zonation based on detailed information
of a region leads to microzonation. Concerns of a particular site location are
covered under more site specific investigations.
Microzonation
is considered an important tool in mitigating the disastrous effects o earthquakes, cyclones, floods,
landslides and avalanches. An international conference on this subject was held
in 1972. Since then microzonation has been a subject of several regional
and internationa conferences either
arising from or giving rise to many research programs. Microzonation has found
an important place during the International Decade of Natural Disaster Reduction
(IDNDR) :1990-2000. This is considered a thrust area of research by
seismologists and geologists. Microzonation
was defined far back in 1980 as follows:
“
Microzonation is a process for identifying relevant geological, seismological,
hydrological and geotechnical site characteristics in a specific region and
incorporating them into land use planning and the design of safe structures in
order to reduce damage to human life and property resulting from earthquakes.”
Detailing
the information contained in this definition requires that the geological and
seismotectonic regimes are understood sufficiently well to forecast possible
future earthquake occurrences, and to quantify their effects in the area under
microzonation. For a meaningful assessment of the earthquake hazard at a
location site-specific information is required on the factors, which affect the
ground motion parameters (amplitudes of ground acceleration, velocity and
displacement), e.g. information on locations, magnitudes and source mechanisms
of earthquakes (which could affect the site in future), local geology and
topography and those determining the potential of ground failure due to surface
faulting, liquefaction, subsidence, collapse and landslides. A zonation map is
expected to provide up-to-date information for assessment and mitigation of the
impending earthquake hazard at a specified location. By now it is obvious that:
1.
Zonation is a multidisciplinary exercise covering geology, geophysics,
seismology, geotechnical engineering and landuse planning.
2.
Zonaton is knowledge based (both information and state of the art) making
it a dynamic process requiring revision and upgradation as the knowledge base
advances.
3.
The interpretations and deductions, which enter the zoning maps and are
derived therefrom, need to be unambiguous and traceable in the information base.
4.
Reliability of the information base has to be high.
Constucting reliable zonation maps may be possible only when
sufficient information to identify the locations of future earthquakes has been
collected, geological mapping of the area on sufficiently large scale to deliver
the required geological properties of the are has been carried out and surface
and subsurface investigations on soil cover and the underlying rock beds has
been completed. (Though it has been well recognized that earthquakes occur along
geological faults, and it is believed that faults can be discovered through
planned field investigations, faults are often discovered
only after the occurrence of a major earthquake, making one to believe
that the area was not properly investigated at earlier occassions. Existence of
a single major fault capable of producing a major earthquake would, thus, render
any zonation exercise redundant)