SPATIAL PROBLEMS IN GEOMETRICAL SEISMICS 49 



diameter with the mark 90° (point Oq') and let us plot the angle 9? = 54° 

 (point b') from the centre of the projections to the circumference. 



When the tracing paper is rotated into the original position (the index 

 coinciding with 0°) the point b' takes up the position b which is also a pro- 

 jection of the trace of the intersection of a direction determined by the given 

 angles a = 162°, (p = 54° with the sphere. 



It must be noted that all the directions for which q> is equal to or less than 

 90° will be above the plane of drawing, and in this case they will be marked 

 on the grid by points. If 90° < 93 < 180°, then such directions will be 

 marked on the drawing by means of crosses. 



If the angle given is not from the vertical but from the horizontal then 

 it will be calculated not from the centre of the projection but from the 

 outer circumference of the grid towards its centre. 



2. To determine the co-ordinates of points given on the net (the problem 

 in reverse). To solve this problem we draw a straight Hne from the centre 

 through the given point up to its intersection with the circumference of the 

 net, and we count off on the circumference the azimuth a. We then transfer 

 the point to the equator and count off the angle 9? from the centre. 



We can now pass immediately to the consideration of problems encountered 

 in geometrical seismics. 



Problem 1. To determine on the net the plane which includes the direction 

 of the ray and the normal to the boundary. 



It is known that incident, reflected and refracted rays and the normal 

 to an interface Ue in one plane, which is also the plane of the rays. We shall 

 use this property to find the plane of the rays. Imagine the centre of a stereo- 

 graphic net at the point of incidence of a ray; we now plot on the tracing 

 paper, using the net, the direction of the incident ray and the normal to the 

 interface. We must bear in mind that since the centre of the net is set at 

 the point of incidence of the ray, it will always be essential to take its inverse 

 azimuth when we plot the direction of the incident ray onto the net. 



Two directions in space have thus been plotted on the tracing paper 

 and the problem is reduced to finding the plane in which both directions he. 

 The meridians of the net correspond to an assembly of circles (planes), 

 differently incHned to the plane of drawing. Consequently, if we rotate 

 the tracing paper until both given directions fall on one and the same meridian 

 of the grid, we shall thereby find the plane in which both given directions 

 He. We produce this meridian and find the pole of arc of the great circle, 

 which will also determine the direction of the normal to the plane of the 

 rays. For this purpose it is sufficient if we count off 90° along the diameter 

 from the arc. Rotating the tracing paper again until it reaches its original 



Applied geophysics 4 



