GEOLOGIJA 30, 391-409 (1987), Ljubljana
UDK 550.34.01:550.343.42(497.12) = 20
A simple macroseismic attenuation model
Enostaven makroseizmičen atenuacijski model
Janez Lapajne
Seizmološki zavod SR Slovenije, Kersnikova 3, 61000 Ljubljana
Abstract
A simple macroseismic attenuation model for an observed intensity I (MSK-78
intensity scale is used) and a "macroseismic" ground acceleration a can be defined
by a system of three linear equations of the type I = f (log r) and log a = f (log r), where
r is the epicentral distance (which can be a distance from the epicenter to an
isoseismal contour in an arbitrary direction, a radius of a circle of equivalent area,
or, generally, a distance from the epicenter to any site). Four constants occurring
in these equations have understandable physical meanings. Two constants deter-
mine the logarithmic degree of attenuation, whereas the two others are characteri-
stic epicentral distances. The model can be easily extended to asymmetrical
macroseismic fields by introducing an additional parameter, which is a function
of azimuth, and can also be a function of the epicentral distance. In the case of an
elliptical field this parameter can be expressed analytically. One extra advantage
of the presented model's equations is that they are dimensionless. An illustration
of the use of the model is given for the region of Slovenia (Yugoslavia).
Kratka vsebina
Na podlagi izbranih kart izoseist preteklih potresov na ozemlju Slovenije
dobimo za opisno potresno stopnjo I (uporabljena je potresna lestvica MSK-78) in
makroseizmičen pospešek tal a enostaven atenuacijski model, ki ga določa sistem
treh linearnih enačb tipa I = f (log r) oz. log a = f (log r), kjer je r epicentralna razdalja
(ta je lahko razdalja od epicentra do izoseiste v katerikoli smeri, polmer kroga, ki
ima enako ploščino kot območje znotraj izoseiste, v splošnem pa je to razdalja od
epicentra do poljubnega mesta). Štiri konstante, ki nastopajo v teh enačbah, imajo
razumljiv fizikalen pomen. Dve konstanti določata logaritemsko stopnjo atenu-
acije, dve pa sta značilni epicentralni razdalji. Model lahko enostavno razširimo
na asimetrična makroseizmična polja tako, da vpeljemo dodaten parameter, ki je
funkcija azimuta, lahko pa je tudi funkcija epicentralne razdalje. V primeru
eliptičnega polja lahko izrazimo ta parameter analitično. Koristna značilnost
predloženih modelnih enačb je tudi njihova brezdimenzijska oblika. Ilustracija
uporabe modela je dana za znani ljubljanski potres iz leta 1895.
392 Janez Lapajne
Attenuation model for intensity
Circular symmetric macroseismic fields
In order-to define the attenuation model, the analysis is based on some marcrosei-
smic data for the territory of Slovenia. In Fig. 1 the territory of Slovenia is presented,
marked with the macroseismic field of the strong earthquake which hit Ljubljana on
April lé''', 1895. The average attenuation of intensity with epicentral distance is
shown, for this earthquake and five others, in Fig. 2. For each earthquake three
straight lines have been drawn through the point values of the average distances of
isoseismals from the epicentre. In this way the attenuation relation I(log r) has been
defined by a group of three linear equations of the type:
Io - I = b log r - c (2)
where r is the average epicentral distance of the isoseismal I. Average epicentral
distance is determined here as a radius of a circle of equivalent area. The coefficients
c can be replaced by the characteristic epicentral distances Го and ri, which determine
the intersections of the straight lines (Fig3).
The system of three linear equations which describes the attenuation relation
I(log r), is:
Introduction
In recent years an ever-increasing number of strong-motion records have become
available for the analysis of seismic hazard. However, in spite of the vague nature of
intensity data, attenuation equations derived from seismic intensity and isoseismal
maps, are still very useful, and sometimes the only practical solution to hazard
estimation.
Sponheuer (1960) conducted an exhaustive survey of macroseismic methods for
determining focal depths, which were directly dependent on attenuation relations-
hips. A more recent extensive survey of commonly used attenuation equations was
given by Campbell (1985). In the latter work, the emphasis is on strong-motion
attenuation relations, and a brief survey of semi-empirical methods with the predic-
tion of ground motion from intensity is also given.
Howell and Schultz (1975) brought together various proposed attenuation
relations into two generalized equations, the more well-known of which is:
Io - I = ai log r + агг - аз (1)
where Io is the epicentral intensity, and r is the hypocentral distance to the intensity
I isoseismal. This equation includes the well-known formula of Kövesligethy, Jánosi
and Gassman (Blake, 1941). In equation (1), outside the near-field region some
authors (e. g. Gupta & Nuttli, 1976, for r>20km) use the epicentral isoseismal
distance for r instead of the hypocentral distance.
Here the use of a particular case of equation (1), when аг = 0, is analysed, r being
the epicentral distance.
A simple macroseismic attenuation model
393
Fig. 1. Isoseismal map relating to the Ljubljana (Slovenia, Yugoslavia) earthquake of April 14«^, 1895; Iq = VIII-IX MSK, m = 6.1, h = 16 km
(macroseismic determinations)
SI. 1. Makroseizmično polje potresa v Ljubljani dne 14. 4. 1895; Io = VIII-IX MSK, m = 6.1, h = 16km (makroseizmične ocene)
394
Janez Lapajne
Fig. 2. Intensity - log-distance plot for 6 earthquakes occurring in Slovenia
(Yugoslavia), with macroseismic values for earthquake magnitude and
focal depth
SI. 2. Atenuacijski grafi opisne potresne stopnje za 6 slovenskih potresov
z makroseizmičnimi ocenami magnitude in globine
A simple macroseismic attenuation model
395
Fig. 3. The parameters of an intensity - log-distance plot
SI. 3. Količine atenuacijskega grafa opisne stopnje
Such an attenuation model might be suitable for several other cases, too. In Fig. 4,
for example, the macroseismic fields of three severe earthquakes from three other
regions have been dealt with in the same way. The Irpinia earthquake of November
23rd, 1980 has also been analysed in a similar, though not identical way (B o 11 a r i et
al., 1986).
If the attenuation relation is to be defined by means of equations (3), (4) and (5), it
is necessary to know the values of the constants bi, b2, го and ri. If we put on one side
all possible mistakes and errors in the preparation of isoseismal maps, and assume
that they are based on fairly objective data, then the parameters bi and b2 represent
logaritmic degree of attenuation (the combination of geometric spreading, rate of
absorption and enhancement due to channeling and path effects). For this reason,
these coefficients will be called "attenuation coefficients". The size of the epicentral
region is determined by the parameter Го, whereas the parameter Гј is the distance at
which the attenuation coefficient changes. Up to distance ri, attenuation is determi-
ned by the constant bi, and beyond that by the constant ba-
Asymmetric macroseismic field
In an approximation when a circular symmetric field is used, considerable errors
can occur, particularly in the case of fields which vary considerably from circular
symmetry. When determining seismic hazard, it would be best to take into account
the actual macroseismic fields. These are not known for all possible earthquakes. For
this reason it is necessary to estimate, more or less accurately, the distribution of the
isoseismals. One way is the calculation of synthetic isoseismals (e. g. S uh a dole et
396
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Fig. 4. Intensity - log-distance plot for the Gediz (Turkey) earthquake of
March 28«h, 1970, for the Skopje (Yugoslavia) earthquake of July 26'\ 1963,
and for the Peloponnesus (Greece) earthquake of April S'*', 1965, with
macroseismically determined magnitudes and depths
SI. 4. Atenuacijski grafi opisne potresne stopnje za potres v Gedizu (Tur-
čija) dne 28. 3. 1970, za potres v Skopju dne 26. 7. 1963 in za potres na
Peloponezu dne 5. 4. 1965
A simple macroseismic attenuation model
397
al., 1987). It is fairly simple to obtain an attenuation model for an asymmetric field by
expanding equations (3), (4) and (5). This is most easily done by introducing a new
parameter k, whose value influences the densification or rarification of isoseismals in
various directions. The influence of this "coefficient of asymmetry" can be seen in
Fig. 5. If k is introduced in the independent variable, then the simplified graph shown
in Fig. 6 is obtained. The attenuation equation (3), (4) and (5) of the corresponding
asymmetric macroseismic field obtain the form:
The coefficient of asymmetry k is a function of azimuth. This dépendance can be
analytically fairly simply expressed in the case of an elliptic field, which is a good
approximation for many practical cases. In Fig. 7 the basic quantities of an elliptical
field are shown, assuming that the epicentre is on the major axis of the ellipse,
between the centre and the focus. For such a field, the coefficient of asymmetry is
given by the equation:
where 0 is the angle between the major axis and the direction in which we are
interested in the attenuation, e = cja^ is the eccentricity of the elipse, and б = d/a^
determines the distance of the epicentre from the centre of the ellipse. If the epicentre
is either at the centre of the ellipse (ó = O) or at the focus (б = e), then equation (9)
becomes very much simplified.
The model of an asymmetric macroseismic field, as defined in Figs. 5 and 6 and by
equations (6), (7) and (8), can if necessary be expanded, e. g. by allowing the
attenuation coefficients bi and b2 to be functions of azimuth. This is easily done by
introducing two new parameters kj and кг, whose meaning can be seen in Fig. 8. In
the case of such an expanded model, the attenuation equations obtain the form:
398
Janez Lapajne
Fig. 5. Intensity - log-distance plot for an asymmetrical macroseismic field;
the influence of values of the parameter k on the attenuation curves
SI. 5. Atenuacijski grafi za nesimetrično makroseizmično polje; vpliv para-
metra k na mesto grafa
Fig. 6. A simplified intensity - log-distance plot for an
asymmetric field
SI. 6. Skupen atenuacijski graf za nesimetrično makrose-
izmično polje
A simple macroseismic attenuation model
399
Fig. 7. The parameters of an elliptic macroseismic field. E is
the epicenter, F is the focus, and r is the radius of a circle
having the same surface area as the ellipse
SI. 7. Količine eliptičnega makroseizmičnega polja. E je epi-
center, F je gorišče, r pa polmer kroga, ki ima enako ploščino
kot elipsa
Fig. 8. Intensity - log-distance plot for an asymmetric macro-
seismic field, showing the influence of parameters kj and кг
on the attenuation curves
SI. 8. Atenuacijski grafi za nesimetrično makroseizmično po-
lje; vpliv parametrov ki in ks na obliko grafa
400 Janez Lapajne
Ground acceleration in the epicentral region is equal to:
ao = lO^^P (17)
In the same way that equations (14), (15) and (16) have been written, it would be
possible to write the equations for ground acceleration for the model which is
expressed for intensity by means of equations (10), (11) and (12).
For tabulated values of the ground acceleration corresponding to the MSK-78
intensity betwen VI and IX (Medvedev, 1978) equation (13) has the following
form:
a = 2^-'' (18)
(a in metres per second squared).
The values of the coefficients p and q are obtained from equations (13) and (18):
In this model the attenuation coefficients are the products kibi and k2b2. The
parameters ki and кг are functions of azimuth, whereas bi and b2 are the attenuation
coefficients of the corresponding circular symmetric field, and thus independent of
azimuth.
Macroseismic attenuation model for ground acceleration
In 1906, Kövesligethy first expressed the dépendance betwen ground acceleration
and the intensity of an earthquake by means of the formula (Sponheuer, 1960):
Io - I = - ploga + q (13)
where p and q are constants.
This form of relation between acceleration and intensity is still in use today (e. g.
Trifunac & Brady, 1975). On the basis of equation (13), it is possible to write
down the attenuation model for ground acceleration, which is expressed for intensity
in equations (6), (7) and (8), as follows:
A simple macroseismic attenuation model 401
Illustration of use of the model
Let us take a look at the use of the attenuation model defined by equations (6), (7)
and (8), on a practical example. For this purpose it is first necessary to estimate
values of the constants bi, b2, Го and ri. This will be done for the seismological
conditions in Slovenia. The starting-point will be the data shown in Figs. 1 and 2.
Due to the small amount of data, it will not be possible to carry out a statistical
evaluation, this not being the aim of this paper, but just a rough estimate for
illustrating the method.
With respect to similarity between individual attenuation curves, the macrosei-
smic fields of the six earthquakes from Slovenia, shown in Fig. 2, can be separated
into two groups: earthquakes with a focal depth of h < 13 km (Fig. 9), and those with
a focal depth of h > 13 km (Fig. 10). For the first group bi « 1.1 and b2 ^ 4.6, and for
the second group bi ~ 2.0 and b2 ~ 3.5.
From the data about the (macroseismically determined) depths h and magnitudes
m, as well as about the values of Го and ri for the earthquakes concerned, the
following two equations for the least square lines have been obtained:
2ro = Mh (24)
m = 2 + 51og^ (25)
^0
The relations are shown graphically in Figs. 11 and 12. From equations (24) and (25)
we obtain:
ri = h lO''"^^-^''^ (26)
26 - Geologija 30
The attenuation model deñned by equations (14), (15) and (16) should, with the
use of equations (18), (19) and (20), give "peak ground accelerations". In order to be
precise, these macroseismically obtained values will be instead called "MSK-78
ground accelerations", since we do not know what these values really are. In this
connection there is an interesting comparison between equation (18) and the correla-
tion of seismic intensities with the peaks of recorded strong ground motion, as was
carried out by Trifunac and Brady (1975). For peak ground acceleration the
following relations were obtanined by these two authors for a Modified Mercalli
intensity Imm between IV and X:
logav = -2-18 + 0-30 Imm (21)
logaH = - 1-99 + 0-30 Imm (22)
where subscripts "V" and "H" designate vertical and horizontal components, respec-
tively. Equations (21) and (22) have been written here in units of metre per second
squared. Let us write equation (18) in the same form:
logaMSK-78 = -2-11 + 0-30 Imsk-78 (23)
The values of амзк-78 are somewhere between the values of ay and ан, which are
determined by equations (21) and (22) for MM intensity. The values of ан are
approximately 30 % greater than the values of ам5к-78, whereas these are about 20 %
greater than ay.
402
Janez Lapajne
Fig. 9. Intensity - log-distance plot for earthquakes occurring in Slovenia
(Yugoslavia), with a focal depth of h < 13 km
SI. 9. Atenuacijski grafi opisne potresne stopnje za 4 slovenske potrese
z žariščno globino manjšo od 13 km
On the basis of the above derivations, using equations (6), (7) and (8) the model
macroseismic field for the earthquake whose macroseismic field is shown in Fig. 1
will be calculated. Since the depth of this earthquake is 16 km, the attenuation
coefficients are bi = 2.0 and b2 = 3.5. From equations (24) and (26) we then obtain:
Го = 8.8 km and ri = 58.1 km. In order to determine the coefficients of asymmetry, an
approximation using an elliptical field has been assumed, estimating the values of
parameters e and б from Fig. 1. Due to the changes in the direction of the ellipse, the
analysis has been limited to the mean epicentral distances, at which the longer axis is
almost in the direction East-West. The eccentricity of the ellipse e is approximately
0.80, and б is approximately 0.24.
The result of the calculation is shown in Fig. 13. The middle line, marked I, defines
the average model attenuation curve for the earthquake of 14. 4. 1985, whereas the
right-hand line, marked Imax, defines the model attenuation curve in the directin of
slowest attenuation (0 = 0, i. e. in the direction East-West), and the left-hand line,
marked Imm, defines the model attenuation curve in the direction of most rapid
attenuation (0 = 104° and 0 = 256°). The model curves for all other directions lie
A simple macroseismic attenuation model
403
Fig. 10. Intensity - log-distance plot for earthquakes occurring in Slovenia
(Yugoslavia), with a focal depth h > 13 km
SI. 10. Atenuacijska grafa opisne potresne stopnje za 2 slovenska potresa
z žariščno globino večjo od 13 km
between the last two mentioned lines. The corresponding attenuation curves for
MSK-78 ground acceleration (á, amin and a^ax) are shown in Fig. 14. The same
relations are shown in a different form in Fig. 15, where the attenuation curve
proposed by Drakopoulos and Makropoulos (1987), marked D&M, has been
drawn in. The latter authors have, for the Balkan area, proposed the following
attenuation equation for peak ground acceleration (written here in units of metre per
second squared):
a = 3-22 e''^^'"(r + 10)-^-^'' (27)
where r is the focal distance in kilometres. In Figs. 13, 14 and 15 the actual values
determined from Fig. 1 are given in addition to the model attenuation curves.
Espinosa (1980) has also defined the depedence of log a on log r for a circular
symmetric field by means of a system of three linear equations. If we use his
attenuation equations for the case being studied here (they are actually valid for the
western United States), then it turns out that in the case of small epicentral distances
they provide larger values, and in the case of large epicentral distances significantly
smaller values then those given by our model in the case of a circular symmetric field.
Fig. 11. Plot of the dependence of the average diameter of the
epicentral region upon the depth of seismic foci, for 6 earth-
quakes occurring in Slovenia (Yugoslavia)
SI. 11. Graf odvisnosti povprečnega premera epicentralnega
območja od žariščne globine za 6 slovenskih potresov
Fig. 12. Plot of the dépendance of macroseismic magni-
tude upon the logarithem of the ratio ri/го, for 6 earth-
quakes occurring in Slovenia (Yugoslavia)
SI. 12. Graf odvisnosti makroseizmične magnitude od
logaritma razmerja ri/го za 6 slovenskih potresov
A simple macroseismic attenuation model
405
Fig. 13. Model intensity - attenuation curves for the Ljubljana (Slovenia,
Yugoslavia) earthquake of April 14'*", 1895; the small empty circules repre-
sent the intensities, which are taken from the isoseismal map of Fig. 1
SI. 13. Modelni atenuacijski grafi opisne potresne stopnje za potres v Ljub-
ljani dne 14. 4. 1895; prazni krogci ponazarjajo vrednosti, ki ustrezajo
makroseizmičnemu polju na si. 1
406
Janez Lapajne
Fig. 14. Model acceleration - attenuation curves for the Ljubljana (Slove-
nia, Yugoslavia) earthquake of April 14'*", 1895; g is the acceleration of
gravity at the earth's surface. The small empty circules represent the
accelerations, which corespond to the intensities in Fig. 1
SI. 14. Modelni atenuacijski grafi makroseizmičnega pospeška za potres
v Ljubljani dne 14. 4. 1895; g je težni pospešek na površju Zemlje. Prazni
krogci ponazarjajo vrednosti makroseizmičnega pospeška, ki ustrezajo
makroseizmičnemu polju na si. 1
A macroseismic field can be very simply presented in the way shown in Fig. 16.
Just one plot is sufficient to define intensity and acceleration values for the asymme-
tric macroseismic field of the chosen earthquake. In order to provide full informa-
tion, values are given for Io and ao, for Го and for the coefficient of asymmetry
k (defined, e. g. in the case of an elliptic field, by means of the values for e and Ô).
Discussion
Although the proposed attenuation rnodel has been verified on only a small
number of macroseismic fields, its applicability appears to be fairly widespread. This
is indicated by the examples in Fig. 4, and several other similar examples from the
literature (e. g. Bottari et al., 1986, and Espinosa, 1980).
For practical purposes we can limit ourselves to the model defined by equations
(6), (7) and (8) for intensity, and by equations (14), (15) and (16) for "macroseismic"
accelaration. If it is assumed that the coefficient of asymmetry changes with epicen-
tral distance (e. g. in sections), then it is possible to deal with macroseismic fields of
various kinds.
A simple macroseismic attenuation model
407
Fig. 15. Model acceleration - attenuation curves for the Ljubljana (Slove-
nia, Yugoslavia) earthquake of April 14"', 1895; g is the acceleration of
gravity at the earth's surface. For comparison, the attenuation curve
according to Drakopoulos and Macropoulos (1986) is given. Empty circules
as in Fig. 14
SI. 15. Modelni atenuacijski grafi makroseizmičnega pospeška za potres
v Ljubljani dne 14. 4. 1895; g je težni pospešek na površju Zemlje. Za
primerjavo je dana atenuacijska krivulja, dobljena z enačbo Drakopoulosa
in Makropoulosa (1986). Prazni krogci kot na si. 14
The strength of the model is in its simplicity (to which the dimensionless form of
the equations contributes, though not essentially) and possibly in the simple relations
between the parameters bi, b2, Го, ri and k, on the one hand, and the geological
structure and focal parameters, on the other. The extent to which the model can be
used depends on how-defined these parameters are. Together with the coefficient of
asymmetry k, the average attenuation coefficients bi and b2 define attenuation at
lesser and greater epicentral distances. In the model, for the epicentral region, i. e. up
to a distance of Го, a constant value of intensity or acceleration, respectively, is
assumed, since usually only one value is available for this area. If a range of values is
given, then it is usually assumed that the spread of values is due to changing local
geological and geotechnical properties of the ground.
408
Janez Lapajne
Fig. 16. Joint model intensity - log-distance and log-acceleration - log-
distance plot for the Ljubljana (Slovenia, Yugoslavia) earthquake of April
14'h, 1895
SI. 16. Skupen poenostavljen atenuacijski graf opisne potresne stopnje in
makroseizmičnega pospeška za potres v Ljubljani dne 14. 4. 1895
For the described earthquakes, occurring in the territory of Slovenia, bi = 1.1 and
b2 = 4.6 (Fig. 9), or bi ~ 2.0 and b2 ~ 3.5 (Fig. 10), respectively. Values of between 1.4
and 2.3 for bi can be read from Fig. 4, and in all three cases b2 is equal to
approximately 4.5. For the case cited from the literature (Bottari et al., 1986), the
value of bi lies between 1.7 and 2.0, whereas b2 is approximately equal to 4.0. It can
be seen that there is a greater scatter of values for bi (in the case of the given data,
these values lie between 1.1 and 2.3), whereas the values for b2 vary less (for the given
data, a fairly good estimate is b2 = 4.0 -f 0.5).
It appears that the parameter bi, and thus attenuation to a distance ri, is
considerably dependent on local seismo-geological conditions, whereas the average
attenuation from a distance ri onwards is fairly similar in the case of earthquakes
from different regions, as the studied macroseismic fields have indicated. It should
also be remembered that errors in determining bi are considerably greater than those
in determining b2.
How well the parameter Го is defined depends upon the determination of the
epicentral region, whose size, as a rule, increases with increasing focal depth and
magnitude. In the cases dealt with here, the parameter ri shows even greater
dependence on these two seismic parameters. Equations (24), (25) and (26) must be
considered as just a temporary aid for determining values of Го and ri, since they have
been derived from a relatively small amount of data.
A simple macroseismic attenuation model 409
References
Blake, A. 1941, On the estimation of focal depth from macroseismic data. Bull. Seism. Soc.
Am., 31, 225-231.
Bottari, A., Corsanego, A., Lo Giudice, E. & Mauger, M. 1986, Some problems in
the MSK scale applications: Use of the quantitative definitions in detailed damage assessment.
Ann. Geophysicae, 4, 191-199.
Campbell, K. W. 1985, Strong motion attenuation relations: A ten - year perspective.
Earthquake Spectra, 1, 759-804.
Drakopoulos, J. & Makropoulos, C. 1987, Uncertainties in hazard assessment due to
attenuation laws. Preprint.
Espinosa, E.F. 1980, Attenuation of strong horizontal ground acceleration in the western
United States and their relation to M. Bull. Seism. Soc. Am., 70, 583-616.
Gupta, I. N. & Nu 11 li, O. W. 1976, Spatial attenuation of intensities for central U. S.
earthquakes. Bull. Seism. Soc. Am., 66, 743-751.
Howell, B. F. & Schultz, T. R. 1975, Attenuation of Modified Mercalli intensity with
distance from the epicenter. Bull. Seism. Soc. Am., 65, 651-665.
Med vede V, S. V., 1978, Opredelenie intensivnosti zemletrjasenij. V: Epicentralna j a zona
zemletrjasenij. Vopr. Inžen. Sejsmol., 19, 108-116, Moskva.
Sponheuer, W. 1960, Methoden zur Herdtiefenbestimmung in der Makroseismik. Fre-
iberger Forschungshefte, C 88, Geophysik, Akademie Verlag, Berlin, 120 pp.
Suhadolc, P., Cernobori, L., Pazzi, G. & Panza, G. F. 1987, Synthetic isoseismals:
Applications to Italian earthquakes. Preprint.
Trifunac, M. D. & Brady, A. G. 1975, On the correlation of seismic intensity scale with
the peaks of recorded strong ground motion. Bull. Seism. Soc. Am., 65, 139-162.
A comparsion of the proposed model with the attenuation equation of Drako-
poulos and Makropoulos (1987), as well as others, not presented in this paper,
has indicated that the proposed macroseismic attenuation model provides, generally,
greater values of acceleration than other attenuation models (which also often add
one standard deviation to the mean values). This is conditioned by the method of
constructing isoseismal maps (the isoseismals are usually drawn on the outer margin
of each isointensity field). Besides that the values of intensities are usually determi-
ned fairly conservatively, and are often over-estimated. A similar situation holds true
for the accelerations derived from these intensities.