Second-Order Calculation of Transit Time in a Medium with a Velocity Variation in One Dimension

Abstract

The transit time along a ray path in a medium with a one-dimensional sound-velocity variation has been calculated to the second order. The calculation makes use of the first and second moments of the velocity variation. Define the velocity (a function of the Z coordinate only) as C = C0/1 + μ(Z). Define μl¯ = 1Z2 − Z1 ∫ Z1Z2 μldZ. The transit time from a point P1(ρ1 = 0, Z1) to a second point P2(ρ2,Z2) is found to be t = (R/C0)[1 + μ̄ − 12(μ2¯ − μ̄2) ctn2β], where R2 = ρ22 + (Z2 − Z1)2; ctn2β = ρ2/(Z2 − Z1). The solution is useful for situations where β is large enough so that 12(μ2¯ − μ̄2) ctn2β≪μ̄ holds and has been verified to second order with known eikonal solutions.