In order to estimate the ground motions associated with the 1811 and 1812 earthquakes, one must resort to empirical relations between intensity and ground motion, as obtained from recent earthquakes for which seismometric data are available. Because the attenuation of seismic-wave energy in North America east of the Rocky Mountains is much lower than in the West (Nuttli, 1973), one cannot use the empirical relations of Gutenberg and Richter (1956) or Neumann (1959), which are applicable to the western states.
From a study of recent earthquakes in the Mississippi Valley, Nuttli (1973) has shown that the attenuation of 3- to 6-sec period Rayleigh waves, vertical component, is described by the relation A(DELTA) = K1 DELTA**(-0.5) (sin DELTA)**(-0.5) exp (-0.10 DELTA), where DELTA is the epicentral distance in degrees. For 1-sec period Lg waves, vertical component, the corresponding relation is A(DELTA) = K2 DELTA**(-0.33) (sin DELTA)**(-0.5) exp (-0.07 DELTA). An unpublished formula for 0.2- to 0.3-sec period Lg waves, obtained by an analysis of the data of samller-magnitude earthquakes, is A(DELTA) = K3 DELTA**(-0.5) (sin DELTA)**(-0.5) exp (-0.30 DELTA). The quantities K1, K2, and K3 are related to the amplitude of the source spectrum at the appropriate periods.
Figures 3 and 4 give the maximum particle displacements of the vertical-component surface waves, for the earthquakes of October 21, 1965 and November 9, 1968, respectively. The solid-line curves, whose equations are given above, are based upon a point- source model, which likely will not be appropriate at small epicentral distances. The distance beyond which the point-source model is adequate will depend on the fault length and source volume. Although this distance is not known for the earthquakes in question, the seismographic data of Figures 3 and 4 suggest that it is less than 1 degree. Between 0.1 degree and 1 degree , for which no data are available, the theoretical curves can only be considered as estimates of the motion. They are shown as dotted lines in the figures.
At distances beyond 1 degree, the body waves have appreciably smaller amplitudes than the surface waves, as can be seen if one examines the seismograms. It is conceivable, however, that at very small distances, the amplitudes of the body waves will be greater than those of the surface waves. No amplitude data are available for these earthquakes at distances less than 1 degree, because no strong-motion instruments were in operation at the time, and the standard earthquake-observatory seismograms were unreadable at such distances. For these reasons, the dotted-line estimates for the 1-sec and 3-sec period waves at distances between 0. l degree and l degree may be considered as lower bounds for the maximum value of the vertical component of the ground motion.
On the Benioff seismograms of the WWSSN (World Wide Standardized Seismograph Network) the 0.3-sec period waves are superposed on the 1-sec period waves. The latter have a much larger amplitude on the seismograms for the earthquakes of October 21, 1965 and November 9, 1968. As a result, the amplitudes of the 0.3-sec period waves could not be reliably determined. Rather, they are estimated, assuming that the surface-wave displacement spectrum near the source falls off as the square of the frequency in the range 1 to 3 Hz. Observational data, as yet unpublished, of the surface-wave spectra for these earthquakes in the frequency range 0.3 to 1 Hz indicates such an f**(-2) dependence. Extrapolation to slightly higher frequencies does not seem to be too radical an assumption. The dashed lines in Figures 3 through 8 are estimates of the ground motion for the 0.3-sec period surface waves.
Figures 5 and 6 relate maximum particle velocity to epicentral distance for the vertical- component surface-wave motion of the earthquakes of October 21, 1965 and November 9, 1968. Figures 7 and 8 present similar data for the maximum particle accelerations.
Gordon et al., (1970) have given a detailed description of the damage and intensity data of the November 9, 1968 earthquake, based upon their field studies supplemented by approximately 2,000 questionnaires sent to postmasters. They paid particular attention to the effect of the surficial geology on the intensity of the ground motion; in general, the intensities were greater in the major river valleys and the regions of thick alluvium, compared to other sites at similar epicentral distances. To avoid these effects of anomalous surficial geology on the intensity versus epicentral distance relation, minimum distances from the epicenter to the isoseismal curves were measured. Applying this procedure to the map of Gordon et al., (1970, Figure 3), one then obtains the set of values: Modified Mercalli (MM) intensity VII, 0.2 degrees; MM VI, 0.45 degrees; MM V, 1.1 degrees; MM IV, 2.0 degrees; MM II, 5.0 degrees.
Kisslinger and Nuttli (1965) presented an isoseismal map for the earthquake of October 21, 1965. Of significance to the present study is the fact that there was no damage, even in the immediate epicentral area. Except at isolated points, the maximum intensity can be taken as V. Therefore, the ground motions of Figures 3, 5 and 7 are smaller than those required to produce even minor, nonstructural damage, except at a few scattered places.
For the 1811 and 1812 earthquakes, ground motion must be estimated from intensity values. To do this, it is necessary to know which ground-motion parameter best correlates with intensity, and, furthermore, to know the relationship between that parameter and intensity. Gutenberg and Richter (1956, p. 128) stated that "values of acceleration are of importance, especially for correlation with intensities." In the same paper (Table 16, p. 131), they tabulate values of the logarithm of the ground acceleration versus MM intensity for California earthquakes. Crandell (1949), using data from explosions, and Wiggins (1964), using accelerograph data from California earthquakes, found that intensity is more directly related to particle velocity. The question becomes important in the present study, because the choice of parameter to correlate with intensity will affect the calculated magnitudes of the 1811 and 1812 earthquakes.
Fortunately, Gutenberg and Richter (1956) published all of their intensity and ground motion data, so that it is possible to compare their California earthquake data with that of the Illinois earthquake of November 9, 1968. Gutenberg and Richter used only accelerograph data obtained from instruments located in the basement or ground floor, and they corrected the data for the effects of surficial geology. Thus, their data should be directly comparable to those of the 1968 Illinois earthquake, for which ground motion was obtained from measurements at seismograph observatories (piers on consolidated sediments). Figures 9 and 10 present Gutenberg and Richter's (1956, Table 5) data of intensity versus the logarithm of particle acceleration and intensity versus the logarithm of particle velocity, respectively. The curve in Figure 9 represents Gutenberg and Richter's tabulated relation (1956, Table 16) between intensity and ground acceleration. The large circles represent maximum particle acceleration-intensity values for the 1968 Illinois earthquake, as taken from Figure 8, after the accelerations were multiplied by a factor of 3 (The factor of 3 is approximate. However, in typical cases, the amplitudes of each of the horizontal components of Lg usually are larger than that of the vertical component by about a factor of 2. Lg consists of both higher-mode Love and Rayleigh waves of about the same group velocity, which thus arrive simultaneously.) to go from the vertical component to total motion. From the figure, it can be seen that the Illinois earthquake data lie about 1.4 units below the curve, indicating that the average of the accelerations for California earthquakes at a given intensity is about 25 times that for the Illinois earthquake at the same intensity.
In Figure 10, the particle velocities of the 1968 Illinois earthquake, as obtained from Figure 6 and multiplied by a factor of 3 to convert from vertical component to resultant motion, are seen to fall near the mean values of the California data. Although there is a fair amount of scatter in the California data, the close agreement between their average values and the Illinois data can be taken as justification for the correlation of intensity with particle velocity.
If MM intensity VII is taken as the threshold of structural damage, then, from Figure 6, it corresponds to a particle velocity of about 10 mm/sec (vertical component of motion at a hard-rock recording site). Engineers, from their study of ground vibrations produced by blasting, find that a resultant particle velocity (vertical and horizontal components) of 2 in/sec is a reliable index of the threshold of damage (see, e.g., Nicholls et al., 1971). Allowing a factor of 3 in going from the vertical component of motion to the total motion, and a factor of 2 for the motion of the structure compared to that of its base (or the ground), the observed damage for the November 1968 earthquake can be satisfactorily explained on the basis of the 2 in/sec particle-velocity criterion. Moreover, it implies that for this earthquake the damage was caused principally by waves of period 3 sec and greater.