SPATIAL PROBLEMS IN GEOMETRICAL SEISMICS 53 



The problems we have considered are sufficient to enable us to proceed 

 to a description of the methods and techniques used in solving three- 

 dimensional ray problems in geometrical seismics. 



GENERAL SCHEME OF SOLUTION 



The solution of three-dimensional ray problems resolves itself into 

 studying seismic fields in space. Seismic fronts are traced along the seismic 

 rays. For this purpose, instead of using individual rays, we find it more 

 convenient to use certain aggregations of rays, for example rays emerging 

 from a source and maldng a known angle with the vertical or intersecting 

 an interface along certain definite lines (isohypses). In the first case these 

 rays will form, in the first medium a conic ray surface, which after striking 

 the very first interface can get deformed. If we take a definite number of 

 such surfaces which differ only in the apical angle of the cone, we can use 

 these to fill in the whole portion of the space which interests us. The behaviour 

 of each such conic ray surface is traced in space along the individual rays 

 forming it, which in the first medium differ from one another only with 

 respect to the azimuth. By tracing consecutively the behaviour of the conic 

 ray surface on all the given interfaces we can find the trace of intersection 

 of the ray surface with the surface or plane of observation. 



All the constructions are produced on a plan where trace projections 

 of the ray surface intersections with each of the interfaces in turn are drawn. 



The intersection trace of a ray surface with an interface if the latter has 

 a simple shape can be found by ordinary geometrical constructions, the 

 trace being subsequently projected onto the horizontal observation plane. 



In cases where the interface is not distinguished by a simple form, the 

 projection of the intersection trace of the ray surface with the interface 

 can be constructed as the geometrical position of the trace projections of 

 the intersection with the interface of the rays forming the ray surface. The 

 point of intersection of the ray with the interface is determined as the point 

 where the interface and the ray have exactly the same depth. Each point 

 of the projection of the ray on the plan corresponds to a particular depth 

 of the ray which can be calculated from the known angle made by the ray 

 with the vertical (the inclination of the ray). Projection points of rays for which 

 the depth along the rays coincide with the interface depths (isohypses) 

 are projections of traces of intersection with the interface of the corresponding 

 rays. The travel times of waves are calculated from the rays separately for 

 each section of the wave path. The length of the section of a ray which is 

 included between interfaces is determined from the angle of inclination of 



