The notions of ‘elementary’ seismic sources and ‘elementary’ earthquakes are introduced, as being associated with ‘elementary’ tensorial point forces with a $$\delta$$-like time dependence (where $$\delta$$ is the Dirac delta function). The tensorial character of these forces, known in Seismology as the dipole (or double-couple) representation, is given by the tensor of the seismic moment. A regular seismic source and a regular earthquake can be represented as a superposition of elementary sources and, respectively, elementary earthquakes, governed by a space-time structure factor of the seismic focal region. All these are new concepts. Elementary seismic sources are considered here for a homogeneous isotropic elastic half-space bounded by a free plane surface, the sources being located at an inner point in the half-space. A transient regime of generation and propagation of seismic waves is identified, as distinct from the stationary regime of elastic vibrations. This is another new concept. It is shown that elementary seismic sources produce (double-shock) spherical-shell waves (in the wave region), which are the well-known $$P$$ and $$S$$ waves associated with the feeble tremor in the recorded seismograms. Their mathematical expression, derived here from the tensorial force, differs from known, particular cases. These waves are called here collectively ‘primary’ waves. It is shown that the primary waves interact with the surface of the half-space, where they give rise to ‘secondary’ wave sources, placed on the surface. The secondary waves generated by the secondary sources (which may be called ‘surface seismic radiation’) are estimated here in a simplified model. It is shown that the secondary waves have a delay time in comparison with the primary waves and give rise to a main shock and a long seismic tail, in qualitative agreement with the seismic records. The secondary wave introduced here is a new concept; the main shock and its long tail derived here are elements of novelty. Similarly, the secondary waves generated by an internal discontinuity in the elastic properties of the half-space (an interface parallel with the free surface) are also estimated; it is shown that the discontinuity reduces appreciably the singular main shock on the free surface of the homogeneous half-space.

We analyze the usefulness of the foreshocks in forecasting seismic mainshocks. The analysis is based on possible correlations which may exist between foreshocks and mainshocks. Such correlations are expressed by a previously established time-magnitude relationship, which indicates the presence of an abrupt magnitude-decreasing sequence of correlated foreshocks in the proximity of a mainshock. By fitting this formula, we are able to derive the occurrence time of a possible mainshock. Also, we can estimate the magnitude of the mainshock, providing we know the parameters of the background seismicity of the seismic region. We report here on the application of this procedure to three Vrancea (Romania) mainshocks, the l’Aquila (Italy), Yangbi (Yunnan, China) and Izmit (Turkey) earthquakes. The limitations of the procedure are discussed. Also, a discussion is included regarding the so-called temporal variability of the Gutenberg-Richter parameter in the proximity of a mainshock, as resulting from time-magnitude and time-time correlations.