8. Wheeler Ridge, California. 30. 4 25 1954. 1233PST. 36 48 00. -121 48 00. 5.3. 7. Central California. Soil Dynamics and Earthquake Engineering, 1989, Vol.
Empirical models for scaling Fourier amplitude spectra of strong ground acceleration in terms of earthquake magnitude source to station distance, site intensity and recording site conditions M. D. Trifunac and V. W. Lee Department of Civil Engineering, University of Southern California, University Park, Los Angeles, CA 90089-0242, USA
In this paper, we present the current improvements in empirical scaling of Fourier spectrum amplitudes of strong earthquake accelerations by introducing the frequency dependent attenuation function which has been developed" from the same data base. This function replaces the Richter's empirical attenuation function which we previously used together with a linear term in R, the epicentral distance. By using the new attenuation function, the scaling model has the additional flexibility for estimating the Fourier spectral amplitudes from earthquakes of given source dimensions and focal depths.
INTRODUCTION The idea to scale Fourier amplitude spectra of strong earthquake ground motion directly in terms of earthquake magnitude or Modified Mercalli Intensity at a site is not new and has been considered by a number of investigators during the past ten years. The basic ideas and equations employed here are essentially the same as those we presented in 1976'. With recent significant increase in the number of uniformly processed strong motion accelerograms, however, it has been possible to detect frequency and source size dependent trends in the attenuation of spectrum amplitudes with distance. Because such refinement of attenuation laws should lead to smaller scatter of observed Fourier spectrum amplitudes about the empirical scaling models and thus to more reliable estimates of the amplitudes of strong ground motion, the aim of this paper is to present the second generation of `preliminary empirical scaling models ...'' by including this new description of amplitude attenuation with distance. In this analysis we continue to employ the `published' magnitude scale to describe the size of earthquakes in our data base". With the increasing number of well studied earthquakes for which strong motion data are available, it should be possible, in the near future, to develop similar empirical scaling relations, but in terms of more `physical' earthquake source parameters, for example, seismic moment and stress drop. Such scaling parameters are expected to decrease the overall fluctuations of the recorded amplitudes about the average empirical estimates. However, one of the principal uses of the scaling models presented here will continue to be for the calculation of Uniform Risk Spectra". Such probabilistic estimates of strong ground motion are still, in many parts of the world, based on old seismicity records where in some cases even the estimates of earthquakes magnitude
have to be derived from old and often incomplete data on reported intensities of shaking. While the empirical relations between the seismic moment, magnitude and the reported intensities are available, it is not clear, at present, how much could be gained by converting all scaling relationships from magnitude to seismic moment, for example. In defining the `distance' between the earthquake source and the recording stations, in this work we consider approximately the effects of source depth and source size, but we continue to employ the epicentral distance to define the principal horizontal distance component. One could consider instead, the closest distance to a fault or distance perpendicular to the fault projection on ground surface. Such distance definitions would imply that some information on the distribution of energy release along the fault is available. Since this is available only for a small subset of earthquakes contributing to the data base considered here, we chose to continue with the simplified distance definition in terms of the epicentral distance. PART 1: SCALING OF FOURIER SPECTRA IN TERMS OF M, R, H, S, h AND v 1.1 Previous analysis During the regression analyses of earthquake strongmotion parameters in the 1970's Trifunac' suggested that the Fourier amplitude spectra (FS) of strong motion acceleration at a selected set of discrete periods, T, can be scaled in terms of the definition of the earthquake magnitude scale and a `correction' function in the following form: log, 0[FS(T),n] = M + log, 0A0(R)
Paper accepted April 1987. Discussion closes November 1989. where © Computational Mechanics Publications 1989
110
Soil Dynamics and Earthquake Engineering, 1989, Vol. 8, No. 3
-logs o {FSQ(T, M, p, s, v, R)} M
is the local earthquake magnitude,
(1) ML;
Empirical models for scaling Fourier amplitude : M. D. Trifunac and V. W. Lee should be just a linear function of the depth of sediments. Before choosing the final form for equation (2) then, studies were carried out to find whether there is any significant dependence of spectral amplitudes on h2, h3, ... etc. It was found that with the database available then, the least-squares coefficients associated with these higher order term of h are indistinguishable from zero at the 95 % confidence level. It should be noted, also, that there is no physical justification for the chosen parabolic dependence on M. This choice is motivated by the simplicity of its mathematical form and the apparent trend of data indicated in earlier analyses8. Note that if the log10Ao(R) in (2) were to represent the geometric spreading, the term g(T)R would model the equivalent anelastic attenuation. However, log10A,,(R) was derived empirically from data on actual peak amplitudes in Southern California, and thus represents an average combination of geometric spreading and anelastic attenuation. The term g(T)R then only represents a correction to the average attenuation given by log,,A0(R)•
log10Ao (R) represents the amplitude attenuation functions versus distance . The term logs 0{FS0(T, M, p, s, v, R)} represents a `correction ' function which incorporates the effects of: (1) distribution of observations with respect to the assumed empirical model, as represented by the confidence level p selected for the approximate bound of spectral amplitudes FS(T),p, (2) geologic site conditions , s, (s = 0 for alluvium , s = 2 for basement rock , s =1 for intermediate sites ), (3) horizontal versus vertical ground motion differences , (v = 0 for horizontal and v =1 for vertical ), and (4) the frequency dependent attenuation effects of amplitudes versus distances , R. The term log10 FSO (T, M, p, s, v, R) was then determined by regression analysis. The same empirical model was also used for scaling of pseudo relative velocity spectra, PSV10 and relative velocity spectra, SV11 Trifunac and Lee12 refined the above analyses by introducing a measure of the depth of sedimentary deposits beneath the recording station , h, as a site characteristic to replace the scaling parameter s mentioned above . The new scaling equation then became (equation ( 1) of Trifunac and Lee12):
1.2 The new database The above regression analysis was carried out for 186 free-field records corresponding to a total of 558 components of data from 57 earthquakes starting with the Long Beach earthquake in 1933 and ending with the San Fernando earthquake in 1971. Through the years new earthquake acceleration data have been added to the original database. The list of 57 earthquakes has now grown to 104, most of which occurred in the regions of northern and southern California. Table 1 is the list of earthquakes now used in our database. Each line contains information on the date and time of the earthquake,
1og10 [FS(T)] =M+ log10Ao (R)-b(T)M-c(T)-d(T)h -e(T)v-f(T)M2-g(T)R (2) with all the parameters defined as above. The functions b(T), c(T), ..., and g(T) have been estimated by regression analysis at 91 periods T between 0.04 sec and 15 sec. Note that in the regression equation (2) the second and higher order terms of h, R, and the third and higher order terms of M are neglected. It was pointed out then that there is really no physical basis to assume that log10FS(T)
Table 1.
Month/day/year
Time code
Longitude Latitude Degree, minute and second
Depth (km)
Magnitude
1 2 3
3 10 1933 10 2 1933 7 6 1934
1754PST O11OPST 1449PST
33 37 00 33 47 00 41 42 00
-117 58 00 -its 08 00 -124 36 00
16.0 16.0
6.3 5.4
4 5 6
12 30 1934 10 31 1935 10 31 1935
0552PST 1I38MST 1218MST
32 15 00 46 37 00 46 37 00
-115 30 00 -111 58 00 -111 58 00
16.0
7 8
11 21 1935 11 28 1935
2058MST 0742MST
46 36 00 46 37 00
-112 00 00 -111 58 00
9 10 11 12
2 6 1937 4 12 1938 6 5 1938 6 6 1938
2042PST 0825PST 1842PST 0435PST
40 32 32 32
-125 -115 -115 -115
13 14
9 11 1938 5 18 1940
221OPST 2037PST
15 16 17
2 9 1941 6 30 1941 10 3 1941
18 19
Equation no.
30 53 54 15
00 00 00 00
15 35 13 10
Maximum MMI
Name
6.5
9 6 5 9
Long Beach, California Southern California Eureka, California Lower California
6.0
8
Helena, MT
3 6 6
Helena, MT Helena, MT Helena, MT
5
Humboldt Bay, California Imperial Valley, California Imperial Valley, California Imperial Valley, California
00 00 00 00
16.0 16.0 16.0
3.0 5.0 4.0
40 18 00 32 44 00
-124 48 00 -115 30 00
16.0
6 10
0145PST 2351PST 0813PST
40 42 00 34 22 00 40 36 00
-125 24 00 -119 35 00 -124 36 00
5.5 6.7 6.4
NW California Imperial Valley, California NW California
16.0
5.9 6.4
8 7
Santa Barbara, California Northern California
11 14 1941 10 21 1942
0042PST 0822PST
33 47 00 32 58 00
-118 15 00 -116 00 00
16.0 16.0
20 21 22
3 9 1949 4 13 1949 1 23 1951
0429PST 1156PST 2317PST
37 06 00 47 06 00 32 59 00
-121 18 00 -122 42 00 -115 44 00
5.4 6.5 5.3
8 7 7
Torrance-Gardena, California Borrego Valley, California Northern California
16.0
23 24
10 7 1951 7 21 1952
2011 PST 0453PDT
40 17 00 35 00 00
-124 48 00 -119 Ol 00
7.1 5.6 5.8
8 7 7
Western Washington Imperial Valley, California NW California
16.0
7.7
11
25 26 27
7 23 1952 9 22 1952 11 21 1952
0441 PDT 2346PST
35 17 00 40 12 00 35 50 00
-118 39 00 -124 25 00 -121 10 00
5.5 6.0
7 7
Northern California Southern California
28
6 13 1953
2017PST
32 57 00
-115 43 00
16.0
5.5
7
Imperial Valley, California
29
1 12 1954
1534PST
35 00 00
-119 01 00
16.0
5.9
8
Wheeler Ridge, California
30
4 25 1954
1233PST
36 48 00
-121 48 00
5.3
7
Central California
Kern County, California Kern County, California
Soil Dynamics and Earthquake Engineering , 1989, Vol. 8, No. 3
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Empirical models for scaling Fourier amplitude : M. D. Trifunac and V. W. Lee Table 1 (continued) Equation no. Month /day/year
Time code
Latitude Longitude Degree, minute and second
Depth (km)
31 30 00 40 47 00 37 22 00
-116 00 00 -123 52 00 -121 47 00
16.0
6.3 6.5 5.8
Magnitude
Maximum MMI
Name
31 32 33
11 12 1954 12 21 1954 9 4 1955
0427PST 1156PST 1801PST
34
12 16 1955
2117PST
33 00 00
-115 30 00
16.0
4.3
35 36 37 38 39 40 41 42 43
12 12 2 2 3 3 3 3 3
1955 1955 1956 1956 1957 1957 1957 1957 1957
2142PST 2207PST 0633PST 0725PST 1056PST 1048PST 1144PST 1515PST 1627PST
Imperial County, California
33 00 33 00 31 42 31 42 34 07 37 40 37 40 37 39 37 39
-115 -115 -115 -115 -119 - 122 -122 -122 -122
16.0 16.0 16.0
3.9 5.4 6.8 6.4 4.7 3.8 5.3 4.4 4.0
6 5 7 5 5
1 19 1960 6 5 1960
Imperial County, California Imperial County, California El Alamo, Baja, California El Alamo, Baja , California Southern California San Francisco, California San Francisco, California San Francisco, California San Francisco, California
1926PST 1718PST
36 47 00 40 49 00
-121 26 00 -124 53 00
5.0 5.7
6 6
Central California Northern California
2323PST 0917PST 0729PST 2346PST 2026PST 0936PST 0841 PST 0407PST 0925PST
36 30 40 58 47 24 34 29 35 57 31 48 39 24 40 30 37 00
-121 -124 -122 -118 -120 -114 -120 -124 -121
7 6 8 6 7 6 7 6 6
Hollister , California Northern California Puget Sound, Washington Southern California Parkfield, California Gulf of California Northern California Northern California Northern California
44 45
46 47 48 49 50 51 52 53 54
4 9 4 7 6 8 9 12 12
55
4
16 16 9 9 18 22 22 22 22
8 4 29 15 27 7 12 10 18
1961 1962 1965 1965 1966 1966 1966 1967 1967
00 00 00 00 06 00 00 00 00
00 00 00 06 18 00 00 00 36
30 30 54 54 13 28 29 27 29
18 12 18 31 29 30 06 36 47
00 00 00 00 12 00 00 00 00
13.8
00 00 00 18 54 00 00 00 18
11.0
15.1 6.0 16.0
5.7 5.0 6.5 4.0 5.6 6.3 6.3 5.8 5.2
5 7 7
7
8 1968
1830PST
33 11 24
-116 07 42
11.1
6.4
7
56 57 58 59
9 12 1970 2 9 1971 10 15 1979 8 6 1979
0630PST 0600PST 1417PST 0805PST
34 16 34 24 32 37 37 06
-117 -118 -115 -121
32 24 24 00 19 59 31 59
8.0 13.0 12.0 9.6
5.4 6.4 6.6 5.9
7 11
60
8 13 1978
12 42 59 43
2254GMT
34 21 04
-119 42 00
61 62 63 64
1 1 08 08
1980 1980 1975 1975
1100PST 1833PST 2022GMT 2059GMT
37 49 37 45 39 26 39 26
-121 -121 -121 -121
65
08 03 1975
0103GMT
66 67 68 69
08 08 08 08
1975 1975 1975 1975
0247GMT 0228GMT 0350GMT 1641GMT
70 71
08 08 1975 08 11 1975
0700GMT 0611GMT
72 73 74
08 11 1975 08 16 1975 08 16 1975
75 76 77
09 27 1975 11 28 1974 1 11 1975
24 26 02 02 03 05 06 06
Lower California Eureka, California San Jose, California
Borrego MTN, California
Lytle Creek, California San Fernando, California Imperial Valley, California Coyote Lake, California
12.5
5.5
Santa Barbara , California
13 47 25 31
5.9 7.3 4.1 5.1
5.9 5.2 5.2 5.2
MT Diablo, Livermore MT Diablo, Livermore Oroville Aftershock Oroville Aftershock
39 29 19
-121 30 59
8.8
4.6
Oroville Aftershock
39 28 39 24 39 29 39 29
-121 -121 -121 -121
21 43 49 45
7.4 6.2 9.2 9.7
4.1 3.2 4.7 3.9
Oroville Aftershock Oroville Aftershock Oroville Aftershock Oroville Aftershock
39 29 50 39 27 29
-121 30 41 -121 28 59
7.7 3.1
4.8 4.4
Oroville Aftershock Oroville Aftershock
1559GMT 0548GMT 1223GMT
39 30 20 39 28 12 39 29 52
-121 31 35 -121 31 42 -121 30 16
9.8 8.5 7.1
3.8 4.1 3.1
Oroville Aftershock Oroville Aftershock Oroville Aftershock
2234GMT 2301GMT 1737PST
39 31 12 36 54 0 40 13 12
-121 31 56 -121 30 0 -124 15 36
10.4 9.0 2.0
4.6 0.0 4.7
6 6
Oroville Aftershock Hollister, California Northern California
37 00 58 00 52 18 46 31
47 42 28 28 30 29 31 31
78 79
5 6
6 1975 7 1975
1835PST 0846GMT
40 16 48 40 34 12
-124 40 12 -124 08 24
0.0 21.0
4.0 5.7
7
Northern California Northern California
80 81 82
3 8 1971 5 2 1971 9 12 1971
1508PST 0608GMT 1132PST
35 40 0 51 24 00 41 17 54
-118 24 12 -177 12 00 -123 40 24
6.0 43.0 20.0
4.7 7.1 4.6
5 6 5
Central California Andreanof, Alaska Northern California
83 84 85
7 30 1972 9 4 1972 5 26 1980
2145GMT 1804GMT 1857GMT
56 49 12 36 38 13 37 32 37
-135 40 48 -121 17 13 -118 51 41
25.0 2.0 2.8
7.1 4.8 4.9
7 6
Southeast Alaska Central California Mammoth Aftershock
86 87
5 27 1980 5 27 1980
1450GMT 1901GMT
37 27 49 37 36 15
-118 49 24 -118 46 11
2.4 3.8
6.3 5.0
Mammoth Aftershcok Mammoth Aftershock
88 89 90
5 28 1980 5 31 1980 6 11 1980
0516GMT 1516GMT 0441GMT
37 34 49 37 32 22 37 30 24
-118 53 09 -118 54 22 -119 02 34
3.3 8.2 14.1
4.8 5.1 5.0
Mammoth Aftershock Mammoth Aftershock Mammoth Aftershock
91 92 93
6 28 1980 10 16 1979 10 16 1979
0058GMT 1616PDT 1445PDT
37 33 23 33 4 29 33 2 44
-118 51 45 -115 33 16 -115 29 24
5.1 5.0 3.9
4.1 4.9 4.6
Mammoth Aftershock Imperial Valley Aftershock Imperial Valley Aftershock
94 95
10 16 1979 10 15 1979
1114PDT 2319GMT
32 58 19 32 46 00
-115 36 22 -115 26 29
4.7 9.5
4.2 5.0
Imperial Valley Aftershock Imperial Valley Aftershock
96 97
4 26 1981 1 24 1980
1209GMT 1900GMT
33 7 48 37 50 24
-115 39 00 -121 48 00
8.0 5.9
5.6 5.9
Westmoreland, California Livermore, California
98 99 100
1 26 1980 5 25 1980 5 25 1980
0233GMT 0934PDT 0949PDT
37 45 36 37 36 32 37 37 41
-121 42 00 -118 50 49 -118 55 37
7.3 9.0 14.0
5.2 6.1 6.0
Livermore, California Mammoth Aftershock Mammoth Aftershock
101 102 103
5 25 1980 5 25 1980 5 26 1980
1245PDT 1336PDT 1158PDT
37 33 40 37 37 30 37 32 35
-118 49 52 -118 51 32 -118 53 17
16.0 2.0 5.0
6.1 5.7 5.7
Mammoth Aftershock Mammoth Aftershock Mammoth Aftershock
104
5 27 1980
0751 PDT
37 30 22
-118 49 34
14.0
6.2
Mammoth Aftershock
112
Soil Dynamics and Earthquake Engineering, 1989, Vol. 8, No. 3
Empirical models for scaling Fourier amplitude : M. D. Trifunac and V. W. Lee latitude and longitude of the epicentre, focal depth, local earthquake magnitude and maximum intensity, if available, and the name of the earthquake. The original list of 186 free-field records corresponding to 57 earthquakes has now grown to 438 free-field records from these 104 earthquakes. With 3 components available for each record, this amounts to a total of 1314 acceleration components, of which there are 876 horizontal and 438 vertical components. 1.3 The attenuation function The advantage in using the attenuation function, log10Ao(R) (Ref. 5), in the previous regression analyses has been that it contains information on the average properties of wave propagation through the crust in southern California, where virtually all strong motion data have been recorded up to and during the 70's. The disadvantages and limitations have been that its shape does not depend on the magnitude, source dimension and focal depth of an earthquake, on the geological environment of the recording station, or on the amplitudes of the recorded motions. That log10Ao(R) or its analogue should depend on the geometric size of the fault has been discussed in some detail previously'. Up to the 1970's only a few of the 186 records had epicentral distances less than 10 km and the empirical derivation of different shapes of log10Aa(R), or its equivalent, to reflect different magnitudes or source dimensions then was not feasible. With the new database now available, Trifunac and Lee13 have developed an iteration procedure for determining a new frequency dependent attenuation function, a complete description of which is given in the above reference. A brief summary and description of this new attenuation function is given here. To take into account that the attenuation function should depend on the epicentral distance, R, on the focal depth, H, of the earthquake, and the `size' of the fault, S, a parameter, denoted by A, is introduced to replace the epicentral distance, R, and is defined as follows:
M=3 S(M,T)=0.2km M=6.5 S(M,T)=S6.5(T)km (5) and S6.5(T) is an empirically determined function. From (5) S(M,T) takes the form (also see Appendix A, equation (A.4))
S(M, T)=0.2+(M 53) (S6.5 (T) - 0.2) (6) This definition of fault size `felt' or `experienced' at the site is independent of how close the site is to the epicentre of the earthquake. The new frequency dependent attenuation function then takes the form13: sltt(A,M,T)=
./o(T) log, 0A R ^< Ro do(T) log10A0- (R-R0)/200 R>Ro
with R3+H2 A =S(1n R2
+S2
+H2
+Sz
+H2
+S2
-1/2
and Ao=S
Ro
-1/2
1nR2+H2+So (7)
where do(T) (see Appendix A) is an empirically determined parabolic function of T. It is used to calculate the attenuation function at distances R less than R0. For distances R> Ro, the attenuation function is a linear function of distance with slope - 1/200. The transition distance Ro is given by (Model III13): 1 (- 200.d0(T)(1-So /S2)
S 2 + R 2 + H 2 1/2
A=S In I ( (So + RZ + HZ
where S(M, T) is the size of the fault `felt' at the period T, and is assumed to be a linear function of magnitude, M, so that for
(3)
A can be thought of as a `representative distance' from the earthquake source of size S, at depth H and at distance R from the recording site. S,, is the coherence radius of the source. The definition of A used here in equation (3) is identical to that used in Model III (equation (4.8)) presented in Trifunac and Lee13 . It has been proposed by Gusev2 in his descriptive statistical model of earthquake source radiation for the description of short-period strong ground motion . The coherence radius So is taken to be a half of the wavelength , ), for radiation of frequency f (or period T), namely, coherence radius, So = ti/2 = CS /2f = C5T/2, where C. is the velocity of the radiation (in this work CS is taken to be I km/sec). Since the fault size, S, of the earthquake is not available for most of the earthquakes used in the database, an empirical formula for the size , as a function of magnitude , epicentral distance R and period of the spectral amplitudes has been introduced as follows,
R° 2 In 10 +(200 0(T)(1-So/ S2))2-4H2 In 10
(8)
which is a function of H,S (hence M , R, T), S, and sdo(T). Detailed description and plots of S6.5(T), slo(T) and Ro and of the attenuation function s/tt(A, M, T) are all given in Trifunac and Lee13 1.4 The new scaling relation With the new attenuation function defined, the regression equation of Fourier amplitudes now takes the form: log10 FS (T) = M + ,d tt(A, M, T)+ b, (T )M +b2(T)h+b3(T)v+b4(T)A/100 +b5(T)+b6(T)M2
(9)
Equation (9) is of the same form as equation (2), the old scaling equation, with the old attenuation function S = S(M, R, T) (4) log 1 o A0(T) replaced by new attenuation function
Soil Dynamics and Earthquake Engineering , 1989, Vol. 8, No. 3
113
Empirical models for scaling Fourier amplitude : M. D. Trifunac and V. W. Lee -4tt(A, M, T). The regression analysis is performed on the new database of 1314 components of Fourier amplitude data FS ( T), at 91 discrete periods T ranging from 0.04 to 15.0 sec. This is in fact Step 2 of the iteration procedure described in Trifunac and Lee13 for the determination of the new attenuation function 4tt(A, M, T), and is identical to the regression analysis procedure used with the old database6 , 1,12. For completeness , the details of this step are repeated here. The data are screened to minimize possible bias in the model that could result from possible uneven distribtuion of data among the different magnitudes and from excessive contribution to the database from several abundantly recorded earthquakes . To carry out this screening the data are partitioned into six groups corresponding to magnitude ranges: 2.0 -2.9, 3.0-3.9, 4.04.9, 5.0-5.9, 6.0-6.9 and 7.0-7.9. The data in each of these magnitude ranges are next subdivided according to the site classifications s = 0, 1 and 2. The data within each of these subgroups were then divided into 2 sets corresponding to horizontal (v = 0) and vertical (v = 1) components . The resulting data in each of the groups correspond to the Fourier spectral amplitudes from a specified earthquake magnitude range for a specified site classification and with specified component orientation. To properly balance the effects of attenuation at small and large distances , the data in each of the subgroups are subdivided further into 2 sets : one for epicentral distances < 100 km and the other for distances > 100 km. The data in each of these two final subsets are then arranged in increasing order in terms of their amplitudes. If the number of data points in the first set (R < 100 km) is less than 19 , all the data points are taken . If there are more than 19 points in this first set, at most 19 points are selected from among the ordered set of data so that they correspond uniformly , as close as possible , to the 5 %, 10%, ..., 90 % and 95 % percentiles at distances R 100 km . Similarly , at most 5 points are selected from the second set (R> 100km ) of data so that they correspond uniformly to around 16 2/3 %, 33 1 /3 %, 50 %, 66 2/3 and 83 1/3 % percentiles at distances R> 100 km. This approximate scheme has the effect of reducing the biases described above. Note that this selection process is repeated for each of the 91 periods in the range 0.04 sec to 15 sec. At the long period end, the Fourier data whose amplitudes are below that of the average digitization noise i.e., those with signal -to-noise ratio less than one, are automatically eliminated before the above selection process. This will be the case for many of the data from earthquake of smaller magnitudes and/or recorded at sites of larger epicentral distances . The number of data points used in the regression analysis at the long period end are thus comparatively smaller than those at the rest of the period ranges. The resulting fitted coefficients at each period T resulting from linear regression will be denoted by b1(T), b2(T), b3(T), b4(T), b,(T) and b6(T), (equation (9)) respectively.
log10 FS(T)=M+dtt(A, M, T) +b1(T)M+b2(T)h+b3(T)v +bs(T)+b6(T)M2
(10)
Fig. 1 shows bl(T), b2 (T), b3(T), bs(T) and b6 ( T) (solid lines ) and the estimates of their 80 %, 90% and 95 % confidence intervals 15 , represented by the corresponding dashed lines. Substituting these coefficients in equation (10) gives FS(T), where:
loglo FS(T)= M+dtt(A, M, T)
+b1(T)M+b2(T)h+b3(T)v
+bs(T)+b6(T)M2 FS(T) then represents the least squares estimate of the Fourier amplitude spectrum at period T. For given values of T, h, v and A, log10 represents a parabola when plotted versus M. Following the previous work, it is also assumed in the present analysis that equation (11) applies only in the range Mm,n 1.8 sec as follows: 0.732025 T > 1.8 sec a+blogto T+c (log10 T)2 T
