The Amplitudes of Waves to Be Expected in Seismic Prospecting

A method of calculating the amplitudes of bodily waves in earthquakes is applied to a study of the amplitudes of longitudinal waves produced by an artificial explosion. Formulae are given for calculating the energy of the reflected longitudinal wave arriving at the surface. The percentage of energy reflected at a discontinuity increases rapidly after the angle of incidence exceeds the critical refraction angle. However, no corresponding large amplitudes are recorded by the im;truments. Instead, the maximum amplitudes of reflected waves are found near the shot point. This is because the ground movement is due both to the arriving wave and to the wave reflected downward from the ground. The amplitude of this movement depends, besides, on the rate of change of the angle of incidence with distance. It is pointed out that the same methods can be applied to a dipping bed, and that the amplitudes of the reflected waves are in general slightly larger in the up-dip than in the down-dip direction. A method for calculating the amplitudes of bodily waves in earthquakes was suggested by K. Zoeppritz in rgo8. The formulae to be used have been worked out by the author in the years following the death of Zoeppritz, and the first application to earthquake waves was made by L. Geiger and B. Gutenberg.2 The method has since been used successfully in several investigations. The following is an application of this method in the study of amplitudes of longitudinal waves produced by an artificial explosion. We suppose that the energy is produced at a point A (Fig. r) on the surface of the earth, that the same amount of energy is radiated in all directions, and that there is no change in properties in horizontal directions. K o assumptions are made concerning the vertical direction; the velocity may increase or decrease either gradually or suddenly with depth. We consider two rays which leave the point A (Fig. r) under the angles of incidence i 1 and i 2 which differ by the small amount di. After refractions and reflections the rays arrive fmally at the distances~~ and ~ 2 at the surface which differ by the small amount d~. According to our assumptions, through each unit area of a sphere around A having the very small radius r, the same amount e of energy 1 Balch Graduate School of the Geological Sciences, California Institute of Technology, Pasadena, California; Contribution No. 202. 2 Karl Zoeppritz, Ludwig Geiger unci Beno Gutenberg, Ueber Enlbebenwellcn V. :-.!achrichten der K. Ges. der Wiss. Gottingen, math.-phys. Kl., 121~2o6 (1912). WAVE AMPLITUDES IN SEISMIC PROSPECTING 253 is passing. The rays which leave A between the cones given by the angles i1 and i2 intersect the sphere in the zone (I) I= 21rrh = 21rr 2di sin i. The total energy flowing between the two cones just mentioned is el. At each reflection and refraction of the rays, the energy is split into the longitudinal and transverse reflected and refracted waves. If F