GEOLOGIJA 31, 32, 577-580 (1988/89), Ljubljana
UDK 550.34.01:550.343.42 = 20
Generalization of the Kövesligethy equation for non-circular
macroseismic fields
Posplošitev Kövesligethyjeve enačbe za nekrožna makroseizmična polja
Janez Lapajne
Seizmološki zavod SR Slovenije, Kersnikova 3, 61000 Ljubljana
Abstract
In European countries the equation of Kövesligethy for macroseismic determi-
nation of focal depths is often used as the attenuation law for seismic hazard
assessment. For non-circular fields the expression with an additional parameter
- "the coefficient of non-circulatity" is much better model as the original form of
the Kövesligethy equation. This coefficient is a function of azymuth and can also
be a function of epicentral distance. The function of azymuth can be analytically
simply expressed for elliptic macroseismic fields.
Kratka vsebina
Kövesligethyjeva enačba za makroseizmično določanje globine potresnih ža-
rišč se v Evropi pogosto uporablja kot enačba pojemanje učinkov pri oceni
potresne nevarnosti. Za krožna in približno krožna polja je enačba uporabna
v svoji prvotni obliki. Za nekrožna polja pa daje ustreznejši model enačba, v kateri
nastopa dodatna količina - »koeficient nekrožnosti«:
kjer je Io potresna stopnja v epicentru, r epicentralna razdalja, h žariščna globina,
5 povprečni absorpcijski koeficient in k koeficient nekrožnosti. Ta koeficient je
funkcija azimuta, lahko pa se tudi spreminja z oddaljenostjo od epicentra. Odvis-
nost od azimuta analitično enostavno izrazimo za eliptično polje.
The formula of Kövesligethy, Jánosi, and Gassmannn (Blake, 1941) or simply
the equation of Kövesligethy (Sponheuer, 1960) has been often used for the
determination of the focal depth and other macroseismic parameters of historic
earthquakes from the available isoseismal maps. Its vv^ell known form is:
578	Janez Lapajne
in which Io is the epicentral intensity, f is the radius of the area enclosed within the
isoseismal I, h is the focal depth, and â is the average value of the absorption
coefficient.
The procedure of evaluation of macroseismic parameters consists of two steps:
1.	the determination of f for all isoseismals of the given macroseismic field,
2.	the calculation of parameters Iq, h, and à from the system of equations of the type
(1), corresponding to different isoseismals of the given macroseismic field.
In European countries Kövesligethy equation is also very often applied as a mac-
roseismic attenuation law I = I(r, Ф) in the form:
where r is the epicentral distance, a = а(Ф) is the absorption coefficient, and Ф is
azimuth. It is presumed that the formula of Kövesligethy holds also for all and
singular direction from the epicentre.
To use the equation (2) as the attenuation relation we must first evaluate
macroseismic parameters Io, h, and a = а(Ф) for all earthquake sources from the
available isoseismal maps. In the calculation procedure a system of equations of the
type (2) for different directions Ф is used. This procedure works usually quite well for
circular and near-circular macroseismic fields. In the case of non-circular fields the
results of calculations are often great standard deviations. The more the macroseis-
mic field deviates from the circular one the bigger are standard errors.
Obviously, the equation (2) is generally not appropriate for non-circular fields. If
for instance, the fitness of the model to the observed field is good for the direction in
which the field is prolongated, then for the direction in which the isoseismals are
squeezed together is very bad, and vice versa. Farther, if the fitness is good for
medium epicentral distances, then for small and large distances is very bad.
Main deficiency of the equation (2) lies in the fact that basically only the last part
of the equation can be (through a) a function of azimuth. The influence of the last
part of the equation (2) increases with increasing epicentral distance, but is negligi-
able in the vicinity of the epicentre. For this reason the isoseismals of the model field
corresponding to the equation (2) are almost circular near the epicentre and become
more and more non-circular with increasing epicentral distance (Fig. la). The shape
of the isoseismals of an observed field has mostly just the oposite tendency (Fig. Ic).
To get a better expression for non-circular fields we may still rely on the formula
of Kövesligethy. But we must consider its original purpose: the determination of the
average characteristics of the given macroseismic field. Namely, Kövesligethy equa-
tion has been proved a good model for the equivalent circular fields of many non-
circular fields. The radius f is actually the average epicentral distance to the intensity
I isoseimal. It can be written as f = r/k, where r is the epicentral distance to the
intensity I isoseismal in an arbitrary direction Ф. Parameter k is a function of
azimuth (and could be also a function of the epicentral distance). We can call it "the
coefficient of non-circularity". Puting k into equation (1), we get the following
expression:
Generalization of the Kövesligethy equation for non-circular macroseismic fields	579
Fig. 1. Three macroseismic field models
SI. 1. Trije modeli makroseizmičnega polja
Fig. 2. The parameters of an elliptic macroseismic field
SI. 2. Količine eliptičnega makroseizmičnega polja
For many practical cases radially constant non-circularity (k is only a function of
azimuth) is a good approximation (Fig. lb). For circular fields r = f and k = 1; equation
(3) converts to equation (1).
Now the calculation of input parameters consists of three steps (first two were
already mentioned):
1.	the determination of f for all isoseismals of the given macroseismic field,
2.	the calculation of Io, h and ä from the system of equations of the type (1),
corresponding to different isoseismals of the given macroseismic field,
3.	the calculation of k as a function of azimuth from the system of equations кф = Гф/г,
corresponding to different azimuths Ф.
580	Janez Lapajne
The coefficient of non-circularity can be analyticaly fairly simply expressed for
an elliptic field (Fig. 2):
where 0 is the angle between the major axis of the ellipse and the direction in which
we are interested in attenuation, and e is the eccentricity of the ellipse. More general
expression of k is given in the previous paper (Lapajne, 1987), where k is called "the
coefficient of asymmetry". The eccentricity can be constant for the entire macroseis-
mic field as in Fig. lb (k is only a function of azimuth) or different for different
isoseismals as for instance in Fig. la and Ic (k is a function of azimuth and epicentral
distance).
Knowing the maximum value of Iq, the range of h, a, and k = к(Ф) for the given
earthquake source, we can use the equation (3) as the attenuation model for intensity
I = 1(г,Ф), where r is the epicentral distance.
Writing kh as hk and ä/k as a^ the equation (3) becommes:
where hk = hk (Ф) and ak = ak (Ф) for k = (Ф)
Let us substitute the parameter h in the equation (2) with the parameter h beeging
a function of azimuth:
The quantities Iq, h = h (Ф), and a = a (Ф) in equation (6) should be calculated from
the given isoseismal map using the system of equations of the type (6) for different
directions Ф.
Realizing that hk = h and ak = a, it is obvious that equations (3) (or (5)) and (6)
express identic attenuation models. Comparing the corresponding calculation pro-
cedures, the use of the equation (3) is less complicated. Besides, the parameter h in
equation (6) has no visible physical meaning. On the other hand, assuming ä/k = a, the
geometric parameter k has also an understandable physical meaning. Namely, its
reciprocal 1/k = a/a is "relative absorption coefficient" for different directions from
the epicentre.
It should be emphasized that in aplying the Kövesligethy equation as an attenua-
tion law, earthquake parameters determined by the same equation should be used as
input parameters. This is especially important for the focal depth. Macroseismic focal
depths are usually much smaller than the corresponding instrumental values.
References
Blake, A. 1941, On the estimation of focal depth from macroseismic data. Bull. Seism. Soc.
Am., 31, 225-231.
Lapajne, J. 1987, A simple macroseismic attenuation model. Geologija 30, 391-409,
Ljubljana.
Sponheuer, W. 1960, Methoden zur Herdtiefenbestimmung in der Makroseismik.
Freiberger Forschungshefte, C 88, Geophysik, Akademie Verlag, Berlin, 120 p.