Wave Phase Velocity Measurements Research Articles

Reply [to “Comments on ‘The correction of great circular surface wave phase velocity measurements from the rotation and ellipticity of the Earth’ by F. A. Dahlen”

Reply [to “Comments on ‘The correction of great circular surface wave phase velocity measurements from the rotation and ellipticity of the Earth’ by F. A. Dahlen”

The correction of great circular surface wave phase velocity measurements for the rotation and ellipticity of the Earth

Normal mode perturbation theory is combined with a geometrical optics approximation in order to establish a procedure for correcting great circular Love and Rayleigh surface wave phase velocity measurements for the effects of the rotation and the hydrostatic ellipsoidal shape of the earth. The necessary correction can be made by utilizing in the measurement process an apparent path length Lapp(T, Θ) for great circular propagation of surface waves of period T around any path with a positive pole inclined at an angle Θ to the rotation axis of the earth. The apparent great circular path length Lapp(T, Θ) is given by Lapp(T, Θ) = 2πa[1 - χ1(T) cos Θ - χ2(T)(1 - 3 cos2 Θ)], where a is the mean radius of the earth and χ1(T) and χ2(T) can be expressed, respectively, in terms of the normal mode multiplet rotational and elliptical splitting parameters of the earth. For some models of the earth, especially those having sharp discontinuities or steep gradients in radial structure in the upper mantle, the apparent path length Lapp(T, Θ) appropriate to fundamental long-period Love and, especially, Rayleigh waves can differ from the actual path length by several tenths of a percent, in which case the need for correction becomes real.