SPATIAL PROBLEMS IN GEOMETRICAL SEISMICS 51 



In Fig. 3 the point A with the co-ordinates 54° and 31° corresponds to the 

 direction of an incident ray, while the point iV^ (112° and 14°) corresponds 

 to the normal to the interface at the point of incidence. 



To find the plane of the ray by rotating the tracing paper, we plot both 

 points on the same meridian (points A', iV^) and draw it on the tracing 

 paper (lines produced on the tracing paper are shown on the drawing by 

 a dotted line). We mark on the tracing paper the normal to the plane of the 

 circle (the point N' i). Next we move the tracing paper into its original 

 position and we can determine the position of the plane of the rays in space ; 

 in our case the co-ordinates of the normal to the plane of the rays are 

 determined as 300° and 76°*. 



If one of the directions is above the plane of drawing and the other is below 

 it then the points will be on symmetrical (equidistant from zero) meridians. 

 This is correct since the lower half of the meridian is projected from the 

 zenith in a symmetrical arc. 



Problem 2. To determine the angle between two directions in space. 

 Both directions are plotted by means of the net on tracing paper (points A 

 and B Fig. 3), and by rotating the tracing paper the plane in which both 

 points lie is found. The points under consideration, A and B, lake up positions 

 A' and B' respectively. The angle between points A' and B' in this plane 

 are counted off on the net; in our case this is equal to 54°. This angle is also 

 the angle between the given directions in space. 



Problem 3. Given the direction of an incident ray and of the normal to 

 the interface at the point of incidence, to determine the direction of a reflected, 

 a refracted and a grazing ray. In order to determine the directions of these 

 rays in space, we must first find the plane of the rays, that is we must first 

 solve Problem 1, after which we can construct the directions of the rays 

 in which we are interested in the plane of the rays. Let a point iVj, (see 

 Fig. 3) (112° and 14°) correspond to the direction of the normal to the 

 interface, and a point A (54° and 31°) to the direction of the incident ray 

 (with inverse azimuth). Then the plane of the rays is determined by the 

 points A' and 7V^. 



We shall consider separately how to determine the direction of each of 

 the rays which interests us (the reflected, the refracted and the grazing rays) 

 in this plane. 



(a) To determine the direction of the reflected ray. Since the angle of 

 reflection is equal to the angle of incidence, it follows that when we have 

 determined the angle between the incident ray and the normal by means 



* The Russian original states mistakenly 86°. 



