SPATIAL PROBLEMS IN GEOMETRICAL SEISMICS 45 



of reflected and of refracted waves. As the graphical constructions are cum- 

 bersome, however, the practical use of the method is confined to three- or 

 four -layered media. 



It should be stressed that further development both of the graphical and 

 of the analytical methods of solving linear problems in geometrical seisinics 

 is essential. Only by combining both methods of construction is it possible, 

 where necessary, to increase the degree of accuracy by very simple 

 methods. 



In seismology a Wulfif net is used in processing earthquake records. 

 The construction of spatial fields by means of a Wulff net as apphed to 

 problems in seismology has been described in a thesis by N. Bessonova. 



The present paper is devoted to a detailed exposition of problems in 

 methodology and technique for the solution of spatial problems in geo- 

 metrical seismics by means of stereographic projections as applied to problems 

 in seismic prospecting. The first part explains the principal properties of 

 stereographic projections, without demonstration, and describes the tech- 

 nique of working with a Wulfif net together with methods for solving problems 

 in geometrical seismics by means of these grids. In the second part we des- 

 cribe the technique for solving linear problems in geometric seismics for 

 multi-layered media with interfaces of arbitrary shape. 



STEREOGRAPHIC PROJECTIONS 



Stereographic projections were used in astronomy as far back as over two 

 thousand years ago to represent the surface of the heavenly vault on a plane. 

 Later the method began to be used for the same purpose in map making. 

 At the end of last century stereographic projections began to be used suc- 

 cessfully for studying the angles between lines and planes in space. This 

 use of stereographic projections is of major interest for geometric seismics 

 since it can be used to solve problems connected with the propagation of 

 seismic waves, which by their very nature are spatial. Here we shall not 

 dwell on the theory of stereographic projection, which has been expounded 

 in a number of works <'>, but shall merely describe their main properties 

 which enable them to be used for studying on a drawing the mutual incli- 

 nations of rays in space by first projecting them on to a sphere. 



1. The entire upper hemisphere can be represented by a circle. 



2. The angles between the rays of the great circles in the sphere are 

 equal to the angles between the arcs of their projections. 



3. The arcs of the circumferences of both small and great circles are re- 

 presented in the projection by arcs of circles or in a particular case by straight 



