Normal moveout velocity for pure-mode and converted waves in layered orthorhombic medium

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We study the azimuthally dependent hyperbolic moveout approximation for small angles (or offsets) for quasi-compressional, quasi-shear, and converted waves in one-dimensional multi-layer orthorhombic media. The vertical orthorhombic axis is the same for all layers, but the azimuthal orientation of the horizontal orthorhombic axes at each layer may be different. By starting with the known equation for normal moveout velocity with respect to the surface-offset azimuth and applying our derived relationship between the surface-offset azimuth and phase-velocity azimuth, we obtain the normal moveout velocity versus the phase-velocity azimuth. As the surface offset/azimuth moveout dependence is required for analysing azimuthally dependent moveout parameters directly from time-domain rich azimuth gathers, our phase angle/azimuth formulas are required for analysing azimuthally dependent residual moveout along the migrated local-angle-domain common image gathers. The angle and azimuth parameters of the local-angle-domain gathers represent the opening angle between the incidence and reflection slowness vectors and the azimuth of the phase velocity ψphs at the image points in the specular direction. Our derivation of the effective velocity parameters for a multi-layer structure is based on the fact that, for a one-dimensional model assumption, the horizontal slowness Symbol and the azimuth of the phase velocity ψphs remain constant along the entire ray (wave) path. We introduce a special set of auxiliary parameters that allow us to establish equivalent effective model parameters in a simple summation manner. We then transform this set of parameters into three widely used effective parameters: fast and slow normal moveout velocities and azimuth of the slow one. For completeness, we show that these three effective normal moveout velocity parameters can be equivalently obtained in both surface-offset azimuth and phase-velocity azimuth domains.

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