536 SEISMIC METHODS [Chap. 9 



normal to the interface between the two upper formations, and Zu and z^ 

 the thicknesses of the second formation (normal to the interface between 

 the second and third formation), 



2Zu , 22u , 5u 1 . 2Zd , 2Z(i Sd . 



tsu = — i 1 and tsd = h 1 ; 



Vi V2 V3„ Vi V2 Vm 



hence, 



Z: = ^.^ (1 - 1) and z'J = ""^ (- + -). (9-59d) 



2 \V2„ V3u/ 2 \V2d Ysd/ 



(h) Wave-front diagrams. In addition to the analj^tical methods dis- 

 cussed in the preceding paragraphs, graphical methods may be employed 

 in the solution of travel-time problems. They involve the construction 

 of "wave-front diagrams" which have the advantage that the advance- 

 ment of the seismic wave through geologic formations, both simple and 

 complex, may be more readily visualized. Their principal application is 

 in indirect interpretation. From a preliminary evaluation of a travel- 

 time curve, an approximate geologic profile may be constructed. Then 

 the wave-front diagrams are drawn; travel times obtained from them are 

 compared with the field data; and the geologic section is changed until 

 complete agreement is obtained. Their construction has been described 

 by Thomburg^ and E. A. Ansel.'^ 



A wave front is defined as the surface which a given phase of a seismic 

 impulse occupies at any particular time. A wave-front diagram is a graph 

 showing a number of such surfaces for many successive instants which for 

 convenience are chosen a given constant time-interval apart. In an in- 

 finite isotropic medium the wave fronts are spherical shells; their inter- 

 section with a vertical plane is represented by circles; their spacing is 

 proportional to the velocit5\ If the time between consecutive wave 

 fronts is At, the spacing is As = v • A^ However, wave fronts are circular 

 only in such portions of a layer in which the propagation is not disturbed 

 by an adjacent formation (see Fig. 9-69). 



The construction of wave-front diagrams for several layers proceeds as 

 follows: Draw a series of concentric circles about the shot point in the 

 upper layer, and calculate their spacing from the above equation. Draw 

 the angles of incidence on the formation boundaries involved (critical and 

 refraction angles). The point of incidence of the critical ray on the first 

 boundary is then determined; wave fronts in the lower layer are drawn 

 about this secondary shot point, their spacing being proportional to the 



" A.A.P.G. Bull., 14(2), 185-200 (Feb., 1930). 

 s"* Gerl. Beit., Erg. Hefte, 1(2), 117-136 (1930). 



