SPATIAL PROBLEMS IN GEOMETRICAL SEISMICS 55 



CONSTRUCTING A SURFACE HODOGRAPH FOR REFLECTED 

 RAYS IN THE CASE OF A TWO LAYERED MEDIUM 



By way of example we shall consider the construction of a surface hodograph 

 or waves reflected from an interface which has a cylindrical form. 



Formulation of the problem— To solve a linear problem in geometrical 

 seismics (to construct a chart of isochronous lines) for waves reflected from 

 a cylindrical surface with a horizontal axis of infinite extent. The cylindrical 

 surface is joined to planes. A section of the interface across the extent of its 

 vertical plane is shown in Fig. 4, h. The source of vibrations (shot point) is 

 situated above the axis of the cylinder. 



Description of the constructions —The reflecting boundary has two 

 mutually perpendicular axes of symmetry, the source being above the 

 point of intersection of these axes. In this case we can limit our consideration 

 of the constructions to one -quarter alone, for the field in the remaining 

 quarters will be synunetrical. Taking the direction in which the structure 

 spreads as the initial azimuth reading, we shall make the constructions 

 within the limits to 90°. 



To solve this problem we shall trace several ray surfaces formed by rays 

 emerging from the source at arbitrarily determined angles (p. The section 

 of these surfaces in the vertical plane is shown in Fig. 8, 6. By way of example 

 we shall consider in detail constructions for any one of these surfaces alone 

 (for example the surface formed by rays which make an angle 99* equal 

 to 20°). For this purpose we shall consider the rays of which the azimuths 

 differ from one another by a definite quantity for example 10°. The projections 

 of the incident rays in the first medium are shown in Fig. 4, a by the lines 5. 



The constructions for eacla ray surface can be broken down into separate 

 stages : 



(a) construction of the trace of intersection of the ray surface with the 

 interface on the plane of the projection; 



(b) determination by Wulff"'s net of the directions (azimuth and angle 

 with vertical) of the reflected rays; 



(c) construction of the trace of intersection of the ray surface with the 

 observation plane after reflection. 



Let us examine these stages for each ray surface separately. 



(a) Construction of the projection of the trace of intersection of the ray 

 surface intersection with the interface. Since the interface is not simple 

 in form we shall find the projection of the trace of intersection of the ray 



* For the sake of brevity we shall henceforward call such a surface the "9?° ray surface" . 



