ABSTRACT. Earthquake perceptibility P(I/m) in the Central and Eastern United States is defined as the joint probability that a point site perceives ground motion ...
Bulletinofthe SeismologicalSocietyofAmerica,Vol.73,No.2, pp. 497-518,April1983
PERCEPTIBLE EARTHQUAKES IN THE CENTRAL AND EASTERN UNITED STATES (EXAMINED USING GUMBEL'S THIRD DISTRIBUTION OF E X T R E M E VALUES) BY PAUL W. BURTON, IAN G. MAIN, AND ROGERE. LONG ABSTRACT Earthquake perceptibility P(I/m) in the Central and Eastern United States is defined as the joint probability that a point site perceives ground motion at least at intensity I in conjunction with an earthquake occurrence specifically of magnitude m. Regional seismicity expressed in the catalogs of Meyers and von Hake (1976) and Nuttli (1979) is examined in terms of magnitude frequency using Gumbel's asymptotic extreme value distribution with an upper bound to magnitude (Gumbel III). The large "rogue" earthquakes associated particularly with the New Madrid seismic zone lead to upper bounded macroseismic body-wave magnitudes with large uncertainties around mb 7¼; but this approximate value is not contradicted by deterministic arguments. Examination of the regional earthquake perceptibility curves then shows a maximum perceptibility at several intensity levels for earthquakes in a relatively narrow band of magnitudes around mb 6½. For the Central and Eastern United States, this most perceptible magnitude is the earthquake which is most likely to be felt at any site, and this might be used in aseismic engineering design of noncritical structures. Extension of the analysis to integrated perceptibility then facilitates calculation of intensity expectation at any site.
INTRODUCTION The probabilistic assessment of earthquake occurrences and associated damaging effects of vibratory ground motions include many problems, among which are: the lack of homogeneous earthquake data catalogs; the choice of statistic to describe earthquake occurrence; the often neglected evaluation of uncertainty in the chosen statistical parameters; and the paucity of regional earthquake strong motion data or attenuation laws. The purpose of this paper is to examine some of these problems in relation to the Central and Eastern United States, where the problems are exacerbated by the juxtaposition of a moderately low seismicity during the available historical documentation, occasional large "rogue" earthquakes which appear inconsistent with the overall regional seismicity history, and low attenuation implying a damage potential from earthquakes which might have been considered inconsequential in areas of higher energy damping. There has been much noteworthy work utilizing (e.g., Karnik, 1968, 1971) and attempting to generalize on the empirical magnitude-frequency relation of Gutenberg and Richter which takes the form log N ( m )
= a -
bm
(1)
where N is the cumulative number of times a magnitude m is exceeded in unit time interval, and a and b are regionally dependent constants. Given any positive values of b, it is obvious that an upper bound to magnitude occurrence is required in order that the rate of release of the earth's strain energy remains finite. This constraint 497
498
PAUL W. BURTON, IAN G. MAIN, AND ROGER E. LONG
can be interpreted physically by limitations on fault dimensions and possible stress drop (Caputo, 1976, 1977) and leads to curvature and the introduction of a third parameter to the cumulative frequency relationship. Such curvature has been observed by Botti et al. (1980) in the western Alps, Burton et al. (1980) in Turkey, Makjani6 (1980) in Yugoslavia, Makropoulos (1978) in Greece, and Cosentino and Luzio (1976) for diverse regions of the world. This list is not exhaustive. In addition, laboratory models by Burridge and Knopoff (1967) and King (1975) show curvature inconsistent with equation (1). A recent theoretical model by Kuznetsova et al. (1981) also produces similar behavior from consideration of inhomogeneities along a fault. From the aforementioned regional studies, curvature does often seem to be apparent on a local scale where the use of historical intensities, or magnitudes below the threshold of instrumental saturation, indicate that such curvature is real. When correction is made to saturated magnitudes on a worldwide basis, the indication is sometimes that a linear fit of the form (1) is more appropriate (Chinnery and North, 1975; but see also Kanamori, 1977). It may be that the superposition of different local seismicities leads to statistics of this form on a global scale. However, when Duda (1965) examined the major global earthquakes of the Aleutians-Alaskan part of the circum-Pacific belt, he found a poor fit to the magnitude-frequency law of equation (1) and concluded that this nonlinearity might itself be caused by the superposition of two earthquake populations. Makropoulos (1978) examined the same but incorporated curvature into the analysis, and also obtained poor fit evidence of two possibly superimposed seismic sources. Earthquake catalogs are often incomplete at the lower magnitudes which can bias the forecasting capability of results based on equation (1) which requires the whole data set (whole process). Additionally, curvature is associated with the larger earthquakes which are of prime importance to us. It is to be expected that statistics which preferentially analyze these larger events will demonstrate the physical phenomenon of curvature better than the methods relying on completeness of the whole process. The method used in this paper is to consider only the largest value of earthquake magnitude in any unit time interval (part process). This extreme value is usually the best-determined one, and, in addition, normally dominates the physical effects such as energy or seismic moment release, or any structural damage that might occur. In practice, the earthquake magnitude set will be divided into discrete magnitude ranges for analysis purposes. If the whole process was required, then partial incompleteness in any magnitude range would adversely influence the outcome, whereas the outcome from any set of extremes will be less susceptible to partial incompleteness of the whole process within a specific magnitude range. EXTREME VALUE THEORY AND EARTHQUAKE PERCEPTIBILITY
E x t r e m e value theory. The Poisson assumption of unrelated events yields the following simple relationship between probabilities of occurrence, P, of the extreme values and the cumulative frequency occurrence, N. P ( M ~ m) = e x p [ - N ( M _->m)).
(2)
A general solution for the cumulative frequency relationship is given by Jenkinson (1955) as
P E R C E P T I B L E E A R T H Q U A K E S I N T H E CENTRAL AND E A S T E R N U.S.
I -ml uj
N(m) = [ w -
499 (3)
which accounts for distributions which are unbounded (type I), bounded from below (type II) and from above (type III), all following from different constraints on the parameters w, u, and h. For example, as the parameter h tends to zero, this equation reduces to the form (1). The parameter ~ relates to curvature. The application of Gumbel's (1958) first distribution of extreme magnitude values to earthquake problems has been criticized by Knopoff and Kagan (1977) on the grounds that the whole process methods in general give more accurate results. This is not generally the case for the type III distribution, where physically realistic curvature at higher magnitudes, which is often apparent in the data, is taken into account by extreme value Gumbel III methods, but not by equations similar to (1). Type III curvature has been observed and analyzed by several authors including: Yegulalp and Kuo (1974) on a global basis; Lilwall (1976) and Burton (1978a) for the low to moderate seismicity of the United Kingdom; Makropoulos (1978) for the high seismicity of Greece; Karnik and Schenkova {1978) for the Balkans region; and Burton (1978b, 1979) for southern Europe through to India. Main (1980) produced preliminary results for the Central and Eastern United States. The third asymptotic distribution of extremes takes the form, from (2) and (3),
P ( m ) = e x p [ -(w-mllp'l[w-uj j
(4)
where w represents the upper bound or limit to the range of extreme values, u is the characteristic extreme value associated with unit time, and ~ allows for curvature asymptotic to the upper bound at low annual probabilities or large return periods. P is a probability of nonexceedence of a magnitude m, and from this equation, P(w)
= 1 and P(u) = lie. It is worth noting that Yegulalp and Kuo (1974), and more recently Howell (1981), have placed emphasis on equation (4) as the means of evaluation of the single parameter w which is the upper bound magnitude of earthquake occurrence. We emphasize our concern with the evaluation of all three parameters of what is one statistical distribution describing earthquake occurrence, the uncertainties and relationships among the statistical parameters, and above all with the forecasting capability of equation (4) for any set of parameters (w, u, and ~). Having obtained the extreme magnitude values mi(i = 1, j ) , where j is the total number of extreme intervals contained by the earthquake history (annual extremes have been used most often in the past), they are ranked in ascending size with a "plotting point" probability of being an extreme. According to Gringorton (1963), this is best approximated for the rarer events at the right of the distribution by
P(mi) = (i - 0.44)/(j + 0.12)
(5)
where i is the rank number. The probability for N-year extremes is related to Pl(m) for annual extremes through
PN(m) = P1N(m).
(6)
500
PAUL W. BURTON, IAN G. MAIN, AND ROGER E. LONG
A complete description of the nonlinear least-squares curve-fitting technique including estimation of uncertainties is given by Burton (1979). The method allows for uncertainties 6m, in each extreme magnitude value mi and incorporates calculation of the complete covariance matrix E, which takes the form
i
Ow
wu
0 2
(7 2
w~ 0 2 0"21 0 2
uw (I 2
u (l S
uX (I 2
hw
Xu
h
(7)
Since P(m) is the probability of a magnitude not being exceeded during the appropriate time interval, the average repeat time or return period T-years associated with an earthquake exceeding m is, for annual extremes, T -
1 1 - P(m)"
(8)
The concept of average repeat time can be overexploited if excessive extrapolations from the time span of the earthquake catalog are invoked; the concept of a contemporary view of annual probability based on an earthquake history known to be short is perhaps preferable. Uncertainty estimates presented later will corroborate this. However, average repeat time is defined as a reciprocal annual probability of exeeedence. Rearranging (4) gives
mT -= W -- (W -- U)[-- In P(m)] x.
(9)
Equations (8) and (9) can then be combined to find the largest earthquake with return period of T-years, say roT, and confidence limits may be placed on this estimate by use of the covariance matrix in the form
(Om~ 2 (Om~ ~ (Om~ 2 (Om)(Om) 02~ ~- °w2 \ O w l + °u2 \ O u ] + ~ \ O w l + 2o~x -~w --~
"'"
(10)
It is important that all of the covariance elements are used, because the negative correlation between w and h is significant. This amounts to a reduction in the error compared to the variance methods (i.e., to using the diagonal elements only). The mathematics of the correlation is considered in Yegulalp and Kuo {1974), but this conclusion results from the fact that a lower maximum earthquake (w) needs a tighter curve (higher h) to reach it, and vice-versa. PERCEPTIBILITY OF GROUND MOTION There are requirements beyond the prediction of probabilities of earthquake occurrence in terms of magnitude, which include the identification of design earthquake parameters and the assessment of the probability of the different levels of ground motion (macroseismic intensity, peak ground acceleration, or velocity, etc.) which may be associated with an earthquake occurrence. To achieve this, probabilities of earthquake occurrence can be combined with regional attenuation laws as in
PERCEPTIBLE
E A R T H Q U A K E S IN THE C E N T R A L A N D E A S T E R N
U.S.
501
Burton (1978a) (also see Willmore and Burton, 1976), which with slight modification leads to Pp(x]m) = Pc(x).~(m).
(11)
Perceptibility Pp is the probability that a point perceives ground motion at least at the level x (x -- intensity, peak acceleration, velocity, etc.) in conjunction with an earthquake occurrence specifically of magnitude m. ~(m) is the probability of a magnitude m earthquake occurrence and is taken as the differential probability d P / dm, the probability density function derived from Gumbel III of equation (4). The ground motion parameter x is taken here to be the Modified Mercalli Intensity IMM. Pc is an estimator of the probability that a point in the Central or Eastern United States is within the area for which an earthquake of magnitude m causes ground motion of at least the level x and, in practice, is taken to be the ratio of the area at an intensity IMM o r greater to the total area covered by the earthquake catalog. In the present work, the area at intensity I or greater is obtained from Chandra's (1979) attenuation law for the Central United States, I ( R ) - Io = 3.534 - 0.00164R - 2.528 logl0(R + D)
(12)
where Io is epicentral intensity, R is surface distance in kilometers, and D is a factor which accounts for finite focal depth and focal volume. This relationship can be linked to magnitude through the epicentral intensity by, mbLg = 0.49/O + 1.66
r = 0.77
V 4.5, in order to reduce problems associated with incompleteness. All data points are assumed to have an uncertainty 8mi = +_0.5 magnitude units• RESULTS
Magnitude recurrence• In order to account for possible instability of the determinations of the statistical parameters of the Gumbel I I I distribution, results were
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FIG• 4. Variation in the Gumbel III parameters w, u, and h, and in the forecast of the 100-yr magnitude, as the extreme interval increases beyond annual extremes of magnitude. In each case, envelopes are drawn at one standard deviation. The data base used is the NOAAcatalog for 1810 to 1976. computed both from different historical time spans, and from data sets constructed using different extreme value time intervals to select the individual sequence of extreme magnitudes. Varying the extreme value time interval, N-years has the effect on parameters (w, u, and X) shown in Figure 4. S t a n d a r d deviations on w and X vary markedly beyond 5-yr extreme intervals; the p a r a m e t e r u is usually well determined but gradually increases with the extreme interval.
PERCEPTIBLE EARTHQUAKES IN THE CENTRAL AND EASTERN U.S.
507
Stability in forecasting capability is a fundamental concern, and each (w, u, and k)N suite for each N-year extreme interval is used to predict the 100-yr magnitude ml00, which is seen in Figure 4d to be stable and reasonably well determined with increasing N-year interval. Clearly, large N-year intervals exclude some available NORA c a t a l o g u e dNORA C o r a l o g u e
For
Presentation
Return
1810-197B
1810-1976
For
Preferred
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(- I nP) ~ I ambda
0'.75
.08
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8,25
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(- I nP) x x I ambda
~.75
.if8
FIG. 5. The Gumbel III asymptotic distribution of extreme values for earthquake magnitudes in the Eastern and Central United States. (Magnitudes less than 4.5 are excluded from the analysis.) (a) The standard geophysical Gumbel III plot with annual extreme magnitudes plotted against the function - l n ( - l n P ) showing curvature toward a maximum magnitude w. P is the probability that a magnitude is an annual extreme event. The data are annual extremes extracted from the NOAA catalog during 1810 to 1976 and the Gumbel III suite (w, u, and k) with standard deviations fitted to these data is [7.73 (1.23), 4.24 (0.16), and 0.253 (0.142)]. The largest events at New Madrid (1811 to 1812) and Charleston (1886) deviate from the fitted curve. (b) The preferred alternative representation to (a) using the function (-lnP) x as abscissa. The data and (w, u, and h) are as in (a). This results in a straight-line fit which is a visually convenient reference: w is now the intercept on the ordinate; curvature k is inco~]~orated into the abscissa parameterization. The h a t c h e d a r e a represents mb 7.0 to 7.5 with average repeat time 500 to 1000 yr which may be more representative of the largest events. (c) The effect of removing the New Madrid events by analyzing the period 1820 to 1976 for the NOAA file. w now reduces to 6.9, a value smaller than that expected from the magnitudes of the New Madrid events, or from knowledge of the dimensions and an appropriate stress drop for the fault area responsible for them. (d) The Central United States from the Nuttli catalog, using annual extremes for the period 1810 to 1975. The (w, u, and ~) suite with standard deviation is [7.63 (2.22), 4.02 (0.24), and 0.198 (0.187)], The h a t c h e d a r e a is as in (b).
508
P/~UL W. BURTON, IAN G. MAIN, AND R O G E R E. LONG
data, and by inspection of Figure 4, annual and occasionally 3-yr extreme intervals were finally selected as being compatible with and representative of the data available to this present work. Examples of the nonlinear least-squares fit to the data are given in Figure 5. In general, the NOAA catalog implies slightly higher magnitudes than the Nuttli catalog for an equivalent average repeat time, primarily because it covers a wider range of seismicity over a larger geographical area. However, these differences are not often statistically significant at larger magnitudes, reflecting the fact that in both catalogs, the extreme value earthquake magnitudes are dominated by those occurring in the New Madrid seismic zone. Differences in forecasting will also arise when using different historical time spans, TABLE 2 EARTHQUAKE BODY-WAVE MAGNITUDES ASSOCIATED WITH AVERAGE REPEAT TIMES T-YEARS (MAGNITUDE UNCERTAINTIES ARE GIVEN IN PARENTHESES) Nuttli Catalog Average Repeat Time ( T - y r }
1983-1975 (Instrumental) rnr(omT)
10 25 50 75 100 150 200
5.12 (0.3) 5.43 (0.5) 5.60 (0.8) 5.69 (1.0) ----
5.21 5.45 5.57 5.63 5.67 5.71 5.74
6.26 (3.9)
5.96 (0.7)
NOAA Catalog
1820-1975 (Mainlyhistorical} mT(amT)
(0.1) (0.1) (0.2) (0.2) (0.2) (0.3) (0.3)
1810-1975 (Mainly historical) mT(amr)
5.32 5.72 5.97 6.09 6.18 6.29 6.37
(0.1) (0.1) (0.1) (0.2) (0.2) (0.3) (0.3)
1810-1976 (Mainlyhistorical) mT(OmT)
5.75 6.17 6.45 6.55 6.64 6.74 6.81
7.63 (2.2)
1820-1976 (Mainly historical) mr(omT)
(0.1) (0.1) (0.2) (0.2) (0.2) (0.3) (0.3)
7.73 (1.2)
5.65 5.98 6.18 6.27 6.33 6.40 6.44
(0.1) (0.1) (0.2) (0.2) {0.2) (0.3) (0.3)
6.92 {0.8}
TABLE 3 GUMBEL PARAMETERS FOR THE CENTRAL AND EASTERN UNITED STATES Data File
Nuttli (Central United States) NOAA (Central and Eastern United States)
Time Period
Extreme Interval (yr)
w
u
1963-1975 1820-1975
1 1
6.26 (3.94) 5.98 (0.68)
3.84 (0.51) 4.10 {0.28)
0.335 (0.892) 0.401 (0.315)
1810-1975 1810-1976 1820-1976
1 1 1
7.62 (2.22) 7.73 (1.23) 6.92 (0.84)
4.02 (0.24) 4.24 (0.16) 4.29 (0.16)
0.198 {0.187) 0.253 {0.142) 0.321 {0.180)
1810-1976
3
7.89 (1.52)
4.96 (0.09)
0.260 (0.183}
not necessarily because the process is nonstationary, although it may be, but primarily because the observed historical epoch available is short compared to the overall seismicity process pertaining in this region. Earthquake magnitudes associated with different average repeat times for some different historical time spans are summarized in Table 2. The upper bound w to earthquake magnitude occurrence corresponds to a notional zero probability earthquake with T = oo. A summary of some representative Gumbel III parameters established in this analysis is given in Table 3. The values of w obtained using samples of different historical time spans taken from both catalogs are seen to be significantly influenced by the seismicity of the New Madrid seismic zone, particularly the New Madrid events during 1811 to 1812. These largest events clearly must be included in the statistical treatment to
PERCEPTIBLE
EARTHQUAKES
IN
THE
CENTRAL
AND
EASTERN
U.S.
509
give all embracing estimates of the magnitude recurrence rates, although these particular data deviate from the best-fitting curve. Assuming that these macroseismically rendered equivalent body-wave magnitudes of these difficult-to-explain "rogue" earthquakes are not too inaccurate, then this deViation is of the "Arm 1" Gumbel III variety described by Burton (1979), implying that the probability of these large magnitudes being an extreme is observationally lower than presently predicted by the theoretical curve. A more realistic plotting point probability for the largest events with magnitude mb = 7.1 to 7.4 might be at the position corresponding to return periods of 600 to 700 yr. This would be compatible with extrastatistical deterministic evidence of Russ (1981) deduced from the liquefaction of trench sediments in the New Madrid zone. Hatched rectangles in Figure 5, b and d, schematically represent this by way of illustration as a range of return periods 500 to 1000 yr with magnitudes 7.0 to 7.5. The large uncertainty in w might also be improved upon deterministically, by using the expected magnitude associated with appropriate capable fault dimensions and potential stress drop. The fault model of Kanamori and Anderson (1975) can be adapted to the New Madrid zone (Main and Burton, 1981). Modeling the zone as a strike-slip fault 20 km deep and 100 km long would give rise to magnitudes mb 7.4 to 7.7 for stress drops of 100 and 400 bars, respectively, compatible with the mean values calculated statistically for w. The effect of removing the New Madrid events from a catalog is interesting in itself (Figure 5c). Generally, this gives a closer fit to the extremes at high magnitudes, but leads to underestimation of w at or around the value of the next highest earthquake at Charleston, South Carolina, 1886, rnb = 6.8. This can be seen by comparing the historically predicted values of w which include the time period 1811 to 1812, and those which use a later historical time "window." It is interesting to speculate on this as evidence of the overall regional seismicity being fundamentally formed by the superposition of two separate seismicities, derived from two unique faulting systems capable of upper bound magnitudes or maximum credible earthquakes around 7 and 7¼, respectively. The covariance matrix associated with the analysis of the NOAA catalog for 1810 to 1976 in Figure 3, a and b, is
• NOAA ~--
1.517 0.126 -0.171
0.126 0.024 -0.017
-0.1717 -0.017| 0.020_]
with the (w, u, and h) NOAA suite being (7.73, 4.24, and 0.253), and the format of the elements of • being as in equation (7). The corresponding values associated with the Nuttli catalog for 1810 to 1975 are
ENuttli ~-~
4.939 0.405 -0.411
0.405 0.057 -0.037
-0.4117 -0.037| 0.035_]
with (w, u, and h) being (7.63, 4.02, and 0.198). These two parameter sets, based on annual extremes starting in 1810, are in good agreement, and are those used in most of the remaining work when either catalog is invoked. When the NOAA catalog for 1820 to 1976 is analyzed, excluding the earlier large New Madrid events, the full parameter set corresponding to Figure 5c is
510
P A U L W. B U R T O N , IAN G. MAIN, A N D ROGER E. LONG
0.709 0.085 -0.0147
=
e
0.085 0.026 -0.022
-0.147~ - 0 . 0 2 2 /| 0.032J
with (w, u, and ~) being (6.92, 4.29, and 0.321). In each case, a large and negative covariance is seen to exist between the NOAR C a t a l o g u e For 1810-1978 Annual
(O)
extremes,and
C h a n d r a's ( t 9 7 9 ) attenuation.
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(1979)
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Fro. 6, Perceptibility of Modified Mercalli Intensity in the Central and Eastern United States. T h e G m n b e l III m a g n i t u d e recurrence statistics are combined with an intensity a t t e n u a t i o n law to give a n n u a l probabilities of perception P,(I) at a point for intensities VI, VII, and VIII arising from a particular m a g n i t u d e event. Solid curves a s s u m e a zero focal d e p t h correction D = 0 kin, broken curves a s s u m e D = 25 km. E a c h perceptibility curve s h o w s a peak perceptibility at a " m o s t perceptible e a r t h q u a k e " m a g n i t u d e mp, which m o v e s slightly to t h e right at higher intensities. T h e range for mp is s h o w n by a horizonal bracket. (a) Perceptibility for t h e Central a n d E a s t e r n U n i t e d States using annual extremes from the N O A A catalog: mp 6.5 to 6.9. (b) perceptibility for t h e Central a n d E a s t e r n United S t a t e s using 3-yr extremes from t h e N O A A catalog: mp 6.55 to 6.95. (c) Perceptibility tbr the Central U n i t e d States using annual extremes from t h e Nuttli catalog: mp 6.1 to 6.5.
P E R C E P T I B L E E A R T H Q U A K E S IN THE CENTRAL AND E A S T E R N U.S.
511
parameters w and h which is important when assessing magnitude forecasts with uncertainties calculated using equation (10). The parameter u is invariably well determined with a relatively small proportional variance. Forecasting based on either the NOAA or Nuttli files back to 1810 leads to the conclusion that a 100-yr event is being seen typically with magnitude mb ~ 6.2 to 6.6, but with an uncertain upper bound to the overall seismicity proce~,s possibly around magnitude 7~. Perceptibility of intensity. The combination of earthquake magnitude recurrence data based on data since 1810 from the NOAA and Nuttli catalogs combined with Chandra's (1979) intensity attenuation law produces the perceptibility diagrams of Figure 6. The most striking feature is the peak to any perceptibility curve which is developed in each case because ~(m) decreases with increasing magnitude whereas Pc(I) increases, hence Pp(IIm) curves based on equation (11) all show the form of Figure 6. The value of magnitude m, at peak perceptibility Ppimax) for different intensities is stable but subject to slight variation from two principal causes, both of which arise within application of the chosen attenuation law rather than the earthquake TABLE 4 MAGNITUDE mp AT PEAK PERCEPTIBILITY mp
Data File
Time Period
Intensity VI (i) (ii)
IntensityVIII (i) (ii)
NOAA catalog (annual extremes)
1810-1976
6.45
6.50
6.70
6.85
NOAA catalog (3-yr extremes)
1810-1976
6.55
6.60
6.90
6.95
Nuttli catalog (annual extremes)
1810-1975
6.10
6.20
6.40
6.55
Columns headed (i) and (ii) correspond to depth corrections of 0 and 25 km, theoretical surface focus, and shallow crustal focus earthquakes, respectively. These magnitude values correspond to the peaks of the graphs at intensity VI and VIII in Figure 6.
magnitude recurrence statistics. Chandra's attenuation law [equation (12)] includes a linear term in distance which displaces the values of mp to higher magnitudes as higher intensities are considered. The effect is about 0.3mb between intensities VI and VIII as seen in Table 4. Other forms of the attenuation law which exclude this linear term in distance would generate exactly equal mp at all intensities. Second, including the quantity D km in equation (12) (broken Curves as opposed to solid curves in Figures 6) to allow for finite focal volume and depth also displaces the values of mp to higher magnitudes. This effect is most noticeable at higher intensities, but the effect is slight, being for intensity VIII displacement of at most 0.15m~ from the mp which is not adjusted for hypocentral depth. Inclusion of finite focal depth obviously also slightly depresses the value of Pp(max) at peak perceptibility. Modifying the earthquake occurrence statistics using 3-yr extremes from the NOAA catalog, rather than annual extremes, has a very small effect on these perceptibility curves (Figure 6b). Overall, these values of mp within intensities VI to VIII for the NOAA file are bracketed by 6.45 to 6.95, and the Nuttli file similarly generates 6.10 to 6.55, for the peak perceptibility magnitude.
512
PAUL
W. BURTON,
IAN
G. MAIN,
AND
ROGER
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PAUL W. BURTON, IAN G. MAIN, AND ROGER E. LONG
T h e upper bound to earthquake magnitude occurrence w is also shown on the plots as the point where the perceptibility curves truncate at the right. It should be borne in mind that w represents the statistically uncertain largest magnitude earthquake which might be associated with a region, whereas the mp range (the most perceptible earthquake) is the one which is most likely to occur and be felt at any intensity in the region analyzed, mp is similar for any level of ground motion, being slightly higher for D = 25 t h a n for the physically unrealizable D = 0 "surface focus" which defines the lower limit to mp, and might be used as a guide to the selection of appropriate time histories for design purposes in the region. Finally, if such earthquakes are chosen from elsewhere to represent this region, then interregional variation among the magnitude values should be accounted for. For instance, if comparable earthquakes are chosen from the Western United States, t h e n it has been suggested by Chung and B e r n r e u t e r (1981) t h a t it is appropriate to select those with body-wave magnitude ½mb smaller t h a n the mb sought in the Eastern United States. Integrated perceptibility of intensity. Perceptibility data for annual extremes in Figure 6 integrated in accord with equation (14) generates the results of Figures 7 and 8, which will be described only briefly. TABLE
5
INTEGRATED INTENSITY PERCEPTIBILITY Pip(I)* P,j,(I). IO ~
MMIntensity NOAACatalogt
Nuttli Catalogt
VI 3.61 2.37 VII 0.915 0.485 VIII 0.180 0.0740 IX 0.0248 0.00746 These data are calculated assuming D = 25 km in equation (12). * Pip(I) is the annual probability of perceiving Modified Mercalli Intensity I at a single point. Geographical extent of the NOAA catalog is taken to have area 10.2 × 10~ km2, similarly the Nuttli catalog is taken as 4.6 × 10~km2. T h e cumulative integrals for Pip (II m
