Seismic Travel-time Power-series Expansions Involving Terms in ?2, etc.

Abstract

Summary Velocity distributions corresponding to seismic travel-time relations of the form where B≠ 0, have been examined analytically, and a number of properties derived. In applications of the theory special reference is made to the particular cases T=AΔ+BΔ2 and p-1=E+FΔ, where p=dT/dΔ. It is shown that when B≠ 0, a previously obtained expression for dΔ/dp takes the specially simple form v is the velocity at distance r from the centre, η=r/v, ξ= 2 d ln r/d ln η, and the subscript p denotes values at the lowest point of the ray whose parameter is p. The use of (ii) in connecting (T, Δ) relations of the form (i) with (v, r) relations is illustrated in particular cases. A new formula where the subscript zero denotes values at the surface of the medium, is derived for determining velocity distributions from (T, Δ) relations in general. The formula (iii) is not limited to (T, Δ) relations of the form (i). It is specially useful in generating distributions for which the (T, Δ) and (v, r) relations both take simple forms. Various applications of interest to the present context are given. It is further shown for cases, more general than (i), of the form T=AΔ+BΔk+…, that the velocity gradient is infinite at the surface when 1 < k < 3. Further, dΔ/dp approaches zero, a finite non-zero value or infinity, according as 1< k < 2, k= 2 or 2 < k < 3, respectively. The earlier part of the paper is concerned with the case k= 2. Reference is made to the practical relevance of the theoretical results obtained.