Chapter 2 



METHOD AND TECHNIQUES OF USING STEREOGRAPHIC 



PROJECTIONS FOR SOLVING SPATIAL PROBLEMS IN 



GEOMETRICAL SEISMICS 



E. I. Gal'perin, G. A. Krasil'shchikova, V. I. Mironova and 



A. V. Frolova 



In seismic prospecting, as in all geophysical methods of prospecting, the 

 solution of linear problems is of great importance for the purpose of analysing 

 data and is an essential step in the working out of methods for interpreting 

 field observations. The solution of ray problems makes it possible to study 

 the shape of surface hodographs for media of various structures, to compare 

 the surface hodographs of different types of waves and discover the possible 

 regions in which they can interfere, to check on the correctness of the con- 

 structions made and estimate the degree of error introduced, to confirm 

 approximate methods of interpretation, to verify the permissibility of any 

 simplified assumptions which have been used in interpreting the seismic 

 data and so forth. Yet it is precisely in seismic prospecting that the solution 

 of linear problems has received comparatively little attention. Until recently 

 opportunities for solving linear problems have been confined to cases where 

 the structure of the medium is very simple, and in the main to two dimen- 

 sional problems, although all the problems in seismic prospecting are by 

 their very nature spatial ones. The reason for this is largely the difficulty 

 of solving spatial problems in geometrical seismics. In those instances when 

 spatial problems have been examined, the examination has been confined 

 as a rule to one interface. The methods used have been both graphical and 

 analytical (I'^'S). 



An earlier paper <^) describes a method for solving linear spatial problems 

 in geometric seismics for multi-layered media with interfaces of arbitrary 

 shape. The method is based on using stereographic projections which make 

 it possible to determine the direction of rays in space after they have struck 

 the interface. 



The method is applicable principally to multi-layered media with a cons- 

 tant velocity in each layer, where there is any number of interfaces of arbi- 

 trary shape, and can be used equally well for calculating the seismic fields 



