Lamb pulse observed in nature

Seismograms observed at Longmire, Washington, for four eruptions of Mt. St. Helens (May 18, June 13, August 7, and August 8, 1980), can be interpreted as Lamb pulses excited by a nearly vertical single force that represents the counter force of the eruption. These data provide reliable estimates of the impulse of the force K (time integral of the force) from which the total momentum and the kinetic energy, E, of the ejecta associated with the eruption can be estimated. The values of K are estimated to be 1.4 x 1019, 1.4 x 1016, 3.7 x 1015, and 2.8 x 1015 dyne-s for the four eruptions (chronological order), respectively. The corresponding values of E are estimated to range from 0.70 to 2.6 x 1023, 0.70 to 2.6 x 1020, 1.9 to 6.9 x 1019, and 1.4 to 5.3 x 1019 ergs using values of ejecta velocity. ranging from 100 to 375 m/s. The ratio of K to the amplitude of the 'air wave excited by the eruption is 20 to 40 times larger for the main event on May 18 than for the other events suggesting a significant difference in the eruptive mechanism. Our results demonstrate that a digital seismograph in the vicinity af volcanoes provides a simple means for quantification of the explosive power of a volcanic eruption. Lamb (1904) computed the transient response of a homogeneous elastic half space to a single force with various geometries, and showed that a pronounced pulse propagates along the free surface with the Rayleigh-wave velocity. This pulse is preceded by minor pulses which propagate with Pand S-wave velocities. Here, we collectively call these pulses the Lamb pulse. Many theoretical studies on this problem have been made since then (e.g. Nakano, 1925; Lapwood, 1949; Pekeris, 1955; Chao, 1960; Burridge, 1966; Johnson, 1974; Kind, 1978; Richards, 1979). Observationally, however, the Lamb pulse is seldom se.en in its simplest form. Usually, observed seismograms are far more complex than predicted' by Lamb (1904), because of the heterogeneity of the propagation medium and the complexity of the source. The source of the most naturally occurring events is represented by force couples rather than single forces. One notable exception is the observation by Benioff (1964). Benioff observed an unusual waveform from a deep earthquake in South America and argued that it was a Lamb pulse produced by a vertical single' force with a step-function time history. He suggested that this single force represents a sudden collapse of a small volume of rock at the focus. Here we present examples of Lamb pulses observed in nature, and discuss its possible use for quantification of volcanic eruptions. After the major eruption on May 18, 1980, Mt. St. Helens, Washington, erupted several more times, one of the largest being on June 13, 1980. Seismic waves excited by this eruption were clearly recorded by a DWWSSN (Digital World-Wide Standardized Seismographic Network) station at Longmire (LON), a distance of about 67 km from the volcano. Figure 1 shows the vertical and radial component records from the long-period channel. The amplitude on the transverse component (not shown) is very small, about one-fourth of the radial component. The major pulse in the beginning of the records exhibits a phase shift of about 90 ø between the vertical and horizontal components, and the particle motion is retrograde. These observations suggest that this pulse is a Rayleigh wave excited' by the erupt.ion. The wavelength of this pulse is about 60 km which is comparable to the distance. Copyright 1983 by the American Geophysical Union. Paper number 3L0352. 0094-8276/83 / 003L-03 52503.00 Kanamori and Given (1982) demonstrate that the massive landslide associated with the May 18, Mt. St. Helens eruption can be modeled by a nearly horizontal single force, and is responsible for the excitation of long-period (•200 s) surface waves. On the other hand, Kanamori et al. (1983) show that the relatively shortperiod (5 30 s) waves observed for the May 18 eruption are due to nearly vertical forces. In general, a volcanic eruption may be modeled by a combination of a single force representing the thrust of the eruption and an isotropic source corresponding to the sudden pressure release in the magma chamber. It can be shown (see Appendix to Kanamori et al., 1983) that, for a simple model of eruption, the amplitude ratio of seismic waves excited by the single force to those excited by the isotropic source is approximately equal to the ratio of the seismic-wave velocity to the particle velocity of the fluid (or gas) in the magma chamber. Since this ratio is about 10, the source can be adequately modeled by a single force. On these grounds, the most straightforward interpretation of the LON records is that the counter force of the eruption excited a Lamb pulse that was recorded by the seismograph after travelling the distance omparable to its wavelen. gth. In orde r to investigate this problem in some detail, we compute Lamb pulses in a homogeneous half space using the expressions given by Richards (1979). A Poisson solid with a P velocity of 5 km/s is used. We take a fight-handed Cartesian coordinate system with the X3 axis pointing vertically upward. The free surface is at X3 -0. We assume that the force is applied at the origin, and consider waves propagating in the positive X1 direction. Following Richards (1979), we write the j-th component of displacement due to a unit step single force in the i-th direction as Gi•. Then the relevant responses are: G• -I1/•r•zr, G•---G•--I4/•r•zr, and G•3 -I3/•r•zr, where •z is the rigidity and r is the distance. I1, I3, and 14 are the functions of time given by equations (10), (12), (13), (14) and (16) of Richards (1979). 13 and 14 correspond to the vertical and radial components of the displacement due to a vertical single force. Figure 2 shows 13 and 14 after they are convalved with the time derivative of the source time function of the form s(t) -p/(p2+t2) that was used by Lamb (1904). Here p -0.15 s is used. These functions correspond to Wo and qo in Lamb's paper which are shown in the figure for comparison. Similar calculations can be made for a horizontal single force. LON DWWSSN June 13, 1980 Mr. St Helens