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Some properties of earthquake frequency distributions
Tectonophysics, 51 (1978) T63-T69 o Elsevier Scientific Publishing Company, Amsterdam - Printed in The Netherlands
T63
Letter Section Some properties of earthquake frequency distributions
MARKUS BATH Seismological Institute,
Uppsala (Sweden)
(Submitted July 24, 1978; accepted for publication September 19, 1978)
ABSTRACT B&h, M., 1978. Some properties of earthquake-frequency 51: T63-T69
distributions. Tectonophysics,
Various statistical parameters are calculated for frequency-magnitude distributions of earthquakes. Theoretical expressions are given for the mean, median, quartiles, variance and skewness and applied to four different earthquake sequences.
INTRODUCTION
Statistics has introduced a number of parameters to characterize frequency distributions, such as mode, mean, median, quartiles, variance, skewness. When a distribution is not available in mathematical form, the parameters have to be calculated directly from original data. But if it is possible to give a simple mathematical formula for the distribution, this will provide a suitable means to develop mathematical expressions for the various statistical parameters. These expressions will then replace the original data for the statistical calculations. Using this procedure, the task of this paper is to derive some statistical properties of frequency-magnitude distributions of earthquakes. Besides being an exercise in statistical mathematics, the problem will throw some light on a much used relation in seismology. To some extent, the paper is supplementary to another paper (B&h, 1978c), where relations between single and cumulative frequency are investigated. THEORETICAL
FORMULAS
We assume the following log N = a-bM
frequency-magnitude
relation: (1)
where the frequency N represents data from M = Ml and upwards for a given area, a given interval of time and a certain interval AM of magnitude M (log = decadic logarithm). The kind of magnitude M must also be given. These
T64
specifications are necessary and sufficient in order to define all properties of this positively J-shaped frequency distribution. We restrict our attention to the magnitude range MI 5 M < Mz where M2 = a/b is the upper bound, corresponding to N = 1 (Fig. 1). Equation 1 has been much discussed in literature, for example by K&&k (1968,197l).
Fig. 1. Schematic log N-M regressions with one observed curve (full line) and one “saturated” curve for a longer interval of time and with higher station detectability (dashed line).
Starting from eq. 1, we deduce formulas for a number of statistical parameters, which characterize the given distribution in terms of the independent variable (M). For definitions of the statistical concepts, see any standard textbook in statistics, e.g., Aitken (1962, ch. 2). The derivations include summations of geometric and combined arithmetic-geometric progressions as well as integrations. Details will be omitted in the following summary. Mode (maximum N) = MI. Mean (arithmetic mean, first moment, or mathematical expectation) ii%
(B, MI-B,
M2)/(BI---&
) + b’-’ log e
(2)
where log B, = --bM, ; log B2 = -bMz ; log B; = -bM;; M; = M2 + AM; D = B: AM/(B2--B;) = AM/(10 bAM-l). The expression after the arrow is the special case of AM = 0 (also obtained simpler by integration; cf., Hamilton, 1967). Note that it may be simpler to derive the sum of a finite progression
T65
as the difference
between
M??l
M*
c
NAM = (l/2)
c
= MI--b-’
Median M,
:
NAM
log [(B, + Bz )/2 B1 ]
Lower quartile 41 c
progressions.
MI
MI M,
the sums of two infinite
NAM=
(3)
M, ,:
(114);
NAM MI
M* MV = MI--b-’
log [(3 B1 + Bz )/4 B1 ]
(4)
Upper quartile M, 3: MC c
NAM=(3,4);
NAM
MI
MI
M, 3 = MI--b-’
log [(B,
+ 3 B, )/4 B, ]
(5)
Equations 3, 4 and 5 remain the same for AM = 0, as verified by integration. Note that in the summations leading to eqs. 3, 4 and 5, only half the areas NAM shall be included at the lower and upper limits. These equations can easily be generalized into one formula, corresponding to any subdivision of the range M, to Mz. Semi-interquartile range (dispersion, spread or ‘probable’ error of distribution) Q:
Q = (M,,--M,
1 l/2 = (2 W’ log [(3 BI + & MB,
Magnitude
M+
MI/* = M,
+ b-’ log 2
for which the initial N (for M = MI ) is reduced
Variance (second u2 = C (M-@)‘N/XN = D(D + AM)-BI + be2 log’e-B,
+ 3 & )I
(6) to one-half: (7)
moment
with respect to the mean) u2 :
= ~M2N/~N-(~)” B; (M,-M;)*
B2 (MI-M2
=
/(B,-B;)’
)’ /(B1---B2 )’
(8)
where the expression after the arrow is the special case for AM = 0 (also obtained simpler by integration; cf., Hamilton, 1967). Skewness (asymmetry; cf., Aitken, 1962, p. 37): SI = (M,,-2
M,
or S, = (M-M,
+ M, 1 )/o = -(ba)-’
log [3/4 + Br B2 /(B, + B2 )2 ] (9)
)/u
T66
As an alternative to the upper bound M2 = a/b, we could proceed to infinite M, both in the summations and the integrations. Results are obtained immediately from the formulas above by inserting B,’ = B2 = 0. The formulas are thus applicable to any magnitude range, as long as eq. 1 remains valid. In case a distribution is better represented by two or more linear segments, each corresponding to eq. 1, the equations are still applicable by adding contributions from each segment. Moreover, by appropriate substi~tions the formulas are valid generally for any exponential frequency distribution. SEISMOLOGICAL
APPLICATION
Table I summarizes the results of a numerical application of our equations to four different sequences: rockbursts at GrZngesberg in central Sweden (B&h, 1977), earthquakes in Sweden (B&h, 1978a), in Turkey (Alsan et al., 1975) and world-wide (Duda, 1965, B&h, 1973). Calculations are based on the summation formulas (AM finite), unless otherwise menTABLE I Statistical parameters for frequency-magnitude
distributions The Earth ____ _
Parameter
Grangesberg
Sweden
Turkey _____~_.
Observation period
1974-1977 (1159 days)
1967-1976 (10 years)
1964-1970 (reduced to 10 years)
1918-1964 (47 years)
a b M
3.10 1.13
3.22 0.84
ML
ML
6.00 0.78 M&S (surface wave) 0.5 4.1 7.69 4.44 4.49 4.26 4.87 0.31 4.49 0.28 0.28 0.64
10.40 1.15 KS (surface wave) O..l 7.0 9.04 7.32 7.26 7.11 7.52 0.21 7.26 0.12 0.31 0.91
(regional. 1 (regional f AM 0.1 0.1 1.1 2.3 M, M 2.74 3.83 -2 M 1.42 2.70 1.36 2.63 M, 1.21 2.44 Mqi M q3 1.62 2.94 0.21 0.25 Q M 112 1.37 2.66 cl2 0.11 0.14 0.33 0.32 Sl 0.97 1.08 s, From integration formulas (AM = 0): ‘@ 1.46 2.73 a2 0.11 0.13 Directly from original data: fi 1.45 2.68 a= 0.12 0.14
4.65 0.29
7.37 0.12
4.57 0.21
7.30 0.15
T67
tioned. For comparison, mean and variance are calculated also from integration formulas (AM = 0) and directly from original data. As seen from Table I, there is no essential difference between the results of the three approaches, even though they can be listed as follows in order of decreasing reliability: (1) directly from given data, (2) by summation formulas (with the same AM as in the given data), 13) by integration formulas. The three magnitudes M, M1jZ and M, agree well with each other for each sequence. In average, they exceed Ml by 0.31 + 0.05 units after reduction of ML-to MS by B&h et al. (1976, p. 24). Excepting Turkey, the ratio (MmM,)/(M-M,) averages4.4.Moreover, M,,-M, = 0.12,M,,--M, = 0.58and = 0.058 in average (reduced to MS).The Turkey sequence (Ms 1 + M, d/2-M, deviates by larger AM and a larger range M,---Ml, more than twice as large as the average range of the other three sequences, with clear effects on calculated parameters. The mean magnitude G has found application in several studies of earthquake sequences to demonstrate a certain magnitude stability (see for example Lomnitz, 1966, and Olsson, 1974). The b-coefficient has often attracted interest because of its tectonic significance (Wyss, 1973). Also in our expressions we can trace the significance of b. This becomes even more pronounced when we observe that a is generally well correlated with b for groups of equivalent events. For example, for 26 fracture zones in Sweden (Bath, 1978b), we find a correlation coefficient of +0.95 f 0.02 between a and b for the interval 1951-1976 and the following regressions: b = (0.32 a-0.03)
f 0.10
a = (2.77 b + 0.42) f 0.30
(10)
Likewise, for eight different regions all over the world for the interval 1918 1963 (Duda, 1965), we calculate an even larger correlation coefficient of +0.99 f 0.01 between a and b and the following regressions: b = (0.13 u-0.12)
f 0.06
a = (7.72 b + 1.16) f 0.30
(11)
In eqs. 10 and 11, a and b refer to cumulative frequency CN. Relations like 10 and 11 imply the existence of a nearly common intersection at M, + (M,-Ml)/3 of all regressions log EN-M for groups of equivalent events. Similar results are discussed by Tsuboi (1958), Duda (1965), Kaila and Madhava Rao (1975), Bdth (1979) and others. Even if relations like 10 and 11 suggest close connection between a and b,they are still too inaccurate to justify any substitution of b for a or vice versa. DISCUSSION
It should be emphasized that no complete frequency distribution curve is available. The limit towards higher magnitudes is due to limited observation
T68
periods, and the limit towards lower magnitudes is due to insufficient detectability of seismograph networks for small events. By longer observation periods and by higher detectability, we will ultimately arrive at more complete frequency curves (cf., Tsuboi, 1952, Alsan et al., 1975). Lacking these, it is customary to extrapolate limited frequency distributions towards both higher and lower magnitudes. However, such extrapolations cannot be done without limit, as they will lead to unrealistic results. For example, extrapolating the frequency distribution for Sweden (Table I) beyond M2, we would get one earthquake of magnitude ML = 8.6 (MS = 7.6) per 100.000 years. Likewise, extrapolating below Ml we would get one earthquake of magnitude ML = -2.0 (MS = -1.7) per hour. None of these extrapolations appears reasonable (cf., B&h et al., 1976). On the basis of rock dynamics, we have to assume an upper bound as well as a lower bound for the magnitude, as suggested by the dashed curve in Fig. 1. The difficulty is that we do not know exactly when extrapolations cease to yield realistic results in either direction. As regards the upper bound, compare with observational findings of Gutenberg and Richter (1954). Historical data on large earthquakes have got renewed interest in recent attempts to improve the upper end of the distribution. Recordings of microearthquakes by dense seismograph networks help to extrapolate towards lower magnitudes, but have to be handled with caution when extrapolated upwards. Detailed investigations of rock behaviour under stress (Scholz, 1968) as well as statistical theories of extreme values represent promising attacks on this problem.
REFERENCES Aitken, A.C., 1962. Statistical Mathematics. Oliver and Boyd, Edinburgh and London, 153 pp. Aisan, E., Tezuvan, L. and B&h, M., 1975. An earthquake catalogue for Turkey for the interval 1913-1970. Seismol. Inst., Uppsala, Rep. No. 7-75, 166 pp. Also: Tectonophysics, 31: T13-T19. B&h, M., 1973. Introduction to Seismology. BirkhEuser, Basel, 395 pp. B%th, M., 1977. A rockburst sequence at the Grlngesberg iron ore mines in central Sweden - Part II. Seismol. Inst., UppsaIa, Rep. No. 5-77, 21 pp. B&h, M., 1978a. Earthquakes in Sweden in 1951 to 1976. Sver. Geol. Unders., in press. Blth, M., 1978b. Deep-seated fracture.zones in the Swedish crust. Tectonophysics, 51: T47-T51. B&h, M., 1978c. A note on recurrence relations for earthquakes. Tectonophysics, 51: T23-T30. B&h, M., 1979. Seismic risk in Fennoscandia. Tectonophysics, in press. B&h, M., Kulhgnek, O., Van Eck, T. and Wahlstram, R., 1976. Engineering analysis of ground motion in Sweden. Seismol. Inst., UppsaIa, Rep. No. 5-76, 37 pp. Duda, S.J., 1965. Secular seismic energy release in the circum-Pacific belt. Tectonophysics, 2: 409-452. Gutenberg, B. and Richter, C.F., 1954. Seismicity of the Earth and Associated Phenomena. Princeton Univ. Press, Princeton, 310 pp. Hamilton, R.M., 1967. Mean magnitude of an earthquake sequence. Bull. Seismol. Sot. Am., 57: 1115-1116. Kaila, K.L. and Madhava Rao, N., 1975. Seismotectonic maps of the European area. Bull. Seismol. Sot. Am.. 65: 1721-1732.
T69 K&n;k, V., 1968, 1971. Seismicity of the European Area, Part 1 and 2. Academia, Prague, 364 and 218 pp. Lomnitz, C., 1966. Magnitude stability in earthquake sequences. Bull. Seismol. Sot. Am., 56: 247-249. Olsson, R., 1974. Earthquake activity in the Kamchatka-Kurile Islands-Japan region November 6, 1958-November 30, 1970. Riv. Ital. Geofis., 23: 43-56. Scholz, C.H., 1968. The frequency-magnitude relation of microfracturing in rock and its relation to earthquakes. Bull. Seismol. Sot. Am., 58: 399-415. Tsuboi, Ch., 1952. Magnitude-frequency relation for earthquakes in and near Japan. J. Phys. Earth (Tokyo), 1: 47-54. Tsuboi, Ch., 1958. A new formula connecting magnitude and number of earthquakes. J. Phys. Earth (Tokyo), 6: 51-55. Wyss, M., 1973. Towards a physical understanding of the earthquake frequency distribution. Geophys. J. R. Astron. Sot., 31: 341-359.
T63
Letter Section Some properties of earthquake frequency distributions
MARKUS BATH Seismological Institute,
Uppsala (Sweden)
(Submitted July 24, 1978; accepted for publication September 19, 1978)
ABSTRACT B&h, M., 1978. Some properties of earthquake-frequency 51: T63-T69
distributions. Tectonophysics,
Various statistical parameters are calculated for frequency-magnitude distributions of earthquakes. Theoretical expressions are given for the mean, median, quartiles, variance and skewness and applied to four different earthquake sequences.
INTRODUCTION
Statistics has introduced a number of parameters to characterize frequency distributions, such as mode, mean, median, quartiles, variance, skewness. When a distribution is not available in mathematical form, the parameters have to be calculated directly from original data. But if it is possible to give a simple mathematical formula for the distribution, this will provide a suitable means to develop mathematical expressions for the various statistical parameters. These expressions will then replace the original data for the statistical calculations. Using this procedure, the task of this paper is to derive some statistical properties of frequency-magnitude distributions of earthquakes. Besides being an exercise in statistical mathematics, the problem will throw some light on a much used relation in seismology. To some extent, the paper is supplementary to another paper (B&h, 1978c), where relations between single and cumulative frequency are investigated. THEORETICAL
FORMULAS
We assume the following log N = a-bM
frequency-magnitude
relation: (1)
where the frequency N represents data from M = Ml and upwards for a given area, a given interval of time and a certain interval AM of magnitude M (log = decadic logarithm). The kind of magnitude M must also be given. These
T64
specifications are necessary and sufficient in order to define all properties of this positively J-shaped frequency distribution. We restrict our attention to the magnitude range MI 5 M < Mz where M2 = a/b is the upper bound, corresponding to N = 1 (Fig. 1). Equation 1 has been much discussed in literature, for example by K&&k (1968,197l).
Fig. 1. Schematic log N-M regressions with one observed curve (full line) and one “saturated” curve for a longer interval of time and with higher station detectability (dashed line).
Starting from eq. 1, we deduce formulas for a number of statistical parameters, which characterize the given distribution in terms of the independent variable (M). For definitions of the statistical concepts, see any standard textbook in statistics, e.g., Aitken (1962, ch. 2). The derivations include summations of geometric and combined arithmetic-geometric progressions as well as integrations. Details will be omitted in the following summary. Mode (maximum N) = MI. Mean (arithmetic mean, first moment, or mathematical expectation) ii%
(B, MI-B,
M2)/(BI---&
) + b’-’ log e
(2)
where log B, = --bM, ; log B2 = -bMz ; log B; = -bM;; M; = M2 + AM; D = B: AM/(B2--B;) = AM/(10 bAM-l). The expression after the arrow is the special case of AM = 0 (also obtained simpler by integration; cf., Hamilton, 1967). Note that it may be simpler to derive the sum of a finite progression
T65
as the difference
between
M??l
M*
c
NAM = (l/2)
c
= MI--b-’
Median M,
:
NAM
log [(B, + Bz )/2 B1 ]
Lower quartile 41 c
progressions.
MI
MI M,
the sums of two infinite
NAM=
(3)
M, ,:
(114);
NAM MI
M* MV = MI--b-’
log [(3 B1 + Bz )/4 B1 ]
(4)
Upper quartile M, 3: MC c
NAM=(3,4);
NAM
MI
MI
M, 3 = MI--b-’
log [(B,
+ 3 B, )/4 B, ]
(5)
Equations 3, 4 and 5 remain the same for AM = 0, as verified by integration. Note that in the summations leading to eqs. 3, 4 and 5, only half the areas NAM shall be included at the lower and upper limits. These equations can easily be generalized into one formula, corresponding to any subdivision of the range M, to Mz. Semi-interquartile range (dispersion, spread or ‘probable’ error of distribution) Q:
Q = (M,,--M,
1 l/2 = (2 W’ log [(3 BI + & MB,
Magnitude
M+
MI/* = M,
+ b-’ log 2
for which the initial N (for M = MI ) is reduced
Variance (second u2 = C (M-@)‘N/XN = D(D + AM)-BI + be2 log’e-B,
+ 3 & )I
(6) to one-half: (7)
moment
with respect to the mean) u2 :
= ~M2N/~N-(~)” B; (M,-M;)*
B2 (MI-M2
=
/(B,-B;)’
)’ /(B1---B2 )’
(8)
where the expression after the arrow is the special case for AM = 0 (also obtained simpler by integration; cf., Hamilton, 1967). Skewness (asymmetry; cf., Aitken, 1962, p. 37): SI = (M,,-2
M,
or S, = (M-M,
+ M, 1 )/o = -(ba)-’
log [3/4 + Br B2 /(B, + B2 )2 ] (9)
)/u
T66
As an alternative to the upper bound M2 = a/b, we could proceed to infinite M, both in the summations and the integrations. Results are obtained immediately from the formulas above by inserting B,’ = B2 = 0. The formulas are thus applicable to any magnitude range, as long as eq. 1 remains valid. In case a distribution is better represented by two or more linear segments, each corresponding to eq. 1, the equations are still applicable by adding contributions from each segment. Moreover, by appropriate substi~tions the formulas are valid generally for any exponential frequency distribution. SEISMOLOGICAL
APPLICATION
Table I summarizes the results of a numerical application of our equations to four different sequences: rockbursts at GrZngesberg in central Sweden (B&h, 1977), earthquakes in Sweden (B&h, 1978a), in Turkey (Alsan et al., 1975) and world-wide (Duda, 1965, B&h, 1973). Calculations are based on the summation formulas (AM finite), unless otherwise menTABLE I Statistical parameters for frequency-magnitude
distributions The Earth ____ _
Parameter
Grangesberg
Sweden
Turkey _____~_.
Observation period
1974-1977 (1159 days)
1967-1976 (10 years)
1964-1970 (reduced to 10 years)
1918-1964 (47 years)
a b M
3.10 1.13
3.22 0.84
ML
ML
6.00 0.78 M&S (surface wave) 0.5 4.1 7.69 4.44 4.49 4.26 4.87 0.31 4.49 0.28 0.28 0.64
10.40 1.15 KS (surface wave) O..l 7.0 9.04 7.32 7.26 7.11 7.52 0.21 7.26 0.12 0.31 0.91
(regional. 1 (regional f AM 0.1 0.1 1.1 2.3 M, M 2.74 3.83 -2 M 1.42 2.70 1.36 2.63 M, 1.21 2.44 Mqi M q3 1.62 2.94 0.21 0.25 Q M 112 1.37 2.66 cl2 0.11 0.14 0.33 0.32 Sl 0.97 1.08 s, From integration formulas (AM = 0): ‘@ 1.46 2.73 a2 0.11 0.13 Directly from original data: fi 1.45 2.68 a= 0.12 0.14
4.65 0.29
7.37 0.12
4.57 0.21
7.30 0.15
T67
tioned. For comparison, mean and variance are calculated also from integration formulas (AM = 0) and directly from original data. As seen from Table I, there is no essential difference between the results of the three approaches, even though they can be listed as follows in order of decreasing reliability: (1) directly from given data, (2) by summation formulas (with the same AM as in the given data), 13) by integration formulas. The three magnitudes M, M1jZ and M, agree well with each other for each sequence. In average, they exceed Ml by 0.31 + 0.05 units after reduction of ML-to MS by B&h et al. (1976, p. 24). Excepting Turkey, the ratio (MmM,)/(M-M,) averages4.4.Moreover, M,,-M, = 0.12,M,,--M, = 0.58and = 0.058 in average (reduced to MS).The Turkey sequence (Ms 1 + M, d/2-M, deviates by larger AM and a larger range M,---Ml, more than twice as large as the average range of the other three sequences, with clear effects on calculated parameters. The mean magnitude G has found application in several studies of earthquake sequences to demonstrate a certain magnitude stability (see for example Lomnitz, 1966, and Olsson, 1974). The b-coefficient has often attracted interest because of its tectonic significance (Wyss, 1973). Also in our expressions we can trace the significance of b. This becomes even more pronounced when we observe that a is generally well correlated with b for groups of equivalent events. For example, for 26 fracture zones in Sweden (Bath, 1978b), we find a correlation coefficient of +0.95 f 0.02 between a and b for the interval 1951-1976 and the following regressions: b = (0.32 a-0.03)
f 0.10
a = (2.77 b + 0.42) f 0.30
(10)
Likewise, for eight different regions all over the world for the interval 1918 1963 (Duda, 1965), we calculate an even larger correlation coefficient of +0.99 f 0.01 between a and b and the following regressions: b = (0.13 u-0.12)
f 0.06
a = (7.72 b + 1.16) f 0.30
(11)
In eqs. 10 and 11, a and b refer to cumulative frequency CN. Relations like 10 and 11 imply the existence of a nearly common intersection at M, + (M,-Ml)/3 of all regressions log EN-M for groups of equivalent events. Similar results are discussed by Tsuboi (1958), Duda (1965), Kaila and Madhava Rao (1975), Bdth (1979) and others. Even if relations like 10 and 11 suggest close connection between a and b,they are still too inaccurate to justify any substitution of b for a or vice versa. DISCUSSION
It should be emphasized that no complete frequency distribution curve is available. The limit towards higher magnitudes is due to limited observation
T68
periods, and the limit towards lower magnitudes is due to insufficient detectability of seismograph networks for small events. By longer observation periods and by higher detectability, we will ultimately arrive at more complete frequency curves (cf., Tsuboi, 1952, Alsan et al., 1975). Lacking these, it is customary to extrapolate limited frequency distributions towards both higher and lower magnitudes. However, such extrapolations cannot be done without limit, as they will lead to unrealistic results. For example, extrapolating the frequency distribution for Sweden (Table I) beyond M2, we would get one earthquake of magnitude ML = 8.6 (MS = 7.6) per 100.000 years. Likewise, extrapolating below Ml we would get one earthquake of magnitude ML = -2.0 (MS = -1.7) per hour. None of these extrapolations appears reasonable (cf., B&h et al., 1976). On the basis of rock dynamics, we have to assume an upper bound as well as a lower bound for the magnitude, as suggested by the dashed curve in Fig. 1. The difficulty is that we do not know exactly when extrapolations cease to yield realistic results in either direction. As regards the upper bound, compare with observational findings of Gutenberg and Richter (1954). Historical data on large earthquakes have got renewed interest in recent attempts to improve the upper end of the distribution. Recordings of microearthquakes by dense seismograph networks help to extrapolate towards lower magnitudes, but have to be handled with caution when extrapolated upwards. Detailed investigations of rock behaviour under stress (Scholz, 1968) as well as statistical theories of extreme values represent promising attacks on this problem.
REFERENCES Aitken, A.C., 1962. Statistical Mathematics. Oliver and Boyd, Edinburgh and London, 153 pp. Aisan, E., Tezuvan, L. and B&h, M., 1975. An earthquake catalogue for Turkey for the interval 1913-1970. Seismol. Inst., Uppsala, Rep. No. 7-75, 166 pp. Also: Tectonophysics, 31: T13-T19. B&h, M., 1973. Introduction to Seismology. BirkhEuser, Basel, 395 pp. B%th, M., 1977. A rockburst sequence at the Grlngesberg iron ore mines in central Sweden - Part II. Seismol. Inst., UppsaIa, Rep. No. 5-77, 21 pp. B&h, M., 1978a. Earthquakes in Sweden in 1951 to 1976. Sver. Geol. Unders., in press. Blth, M., 1978b. Deep-seated fracture.zones in the Swedish crust. Tectonophysics, 51: T47-T51. B&h, M., 1978c. A note on recurrence relations for earthquakes. Tectonophysics, 51: T23-T30. B&h, M., 1979. Seismic risk in Fennoscandia. Tectonophysics, in press. B&h, M., Kulhgnek, O., Van Eck, T. and Wahlstram, R., 1976. Engineering analysis of ground motion in Sweden. Seismol. Inst., UppsaIa, Rep. No. 5-76, 37 pp. Duda, S.J., 1965. Secular seismic energy release in the circum-Pacific belt. Tectonophysics, 2: 409-452. Gutenberg, B. and Richter, C.F., 1954. Seismicity of the Earth and Associated Phenomena. Princeton Univ. Press, Princeton, 310 pp. Hamilton, R.M., 1967. Mean magnitude of an earthquake sequence. Bull. Seismol. Sot. Am., 57: 1115-1116. Kaila, K.L. and Madhava Rao, N., 1975. Seismotectonic maps of the European area. Bull. Seismol. Sot. Am.. 65: 1721-1732.
T69 K&n;k, V., 1968, 1971. Seismicity of the European Area, Part 1 and 2. Academia, Prague, 364 and 218 pp. Lomnitz, C., 1966. Magnitude stability in earthquake sequences. Bull. Seismol. Sot. Am., 56: 247-249. Olsson, R., 1974. Earthquake activity in the Kamchatka-Kurile Islands-Japan region November 6, 1958-November 30, 1970. Riv. Ital. Geofis., 23: 43-56. Scholz, C.H., 1968. The frequency-magnitude relation of microfracturing in rock and its relation to earthquakes. Bull. Seismol. Sot. Am., 58: 399-415. Tsuboi, Ch., 1952. Magnitude-frequency relation for earthquakes in and near Japan. J. Phys. Earth (Tokyo), 1: 47-54. Tsuboi, Ch., 1958. A new formula connecting magnitude and number of earthquakes. J. Phys. Earth (Tokyo), 6: 51-55. Wyss, M., 1973. Towards a physical understanding of the earthquake frequency distribution. Geophys. J. R. Astron. Sot., 31: 341-359.