# Generalized Dix equation and analytic treatment of normal-moveout velocity for anisotropic media

## TL;DR

This paper derives exact solutions for normal-moveout velocity in arbitrary anisotropic media, providing a generalized Dix equation that enhances moveout inversion accuracy in complex geological settings.

## Contribution

It introduces a comprehensive analytic framework for NMO velocity in anisotropic media of any symmetry, extending previous elliptical models to arbitrary orientations and stratifications.

## Key findings

- Explicit NMO expression for single-layer models
- Effective NMO velocity via Dix-type averaging in layered media
- Analytic basis for moveout inversion in complex anisotropic structures

## Abstract

Despite the complexity of wave propagation in anisotropic media, reflection moveout on conventional common-midpoint (CMP) spreads is usually well described by the normal-moveout (NMO) velocity defined in the zero-spread limit. In their recent work, Grechka and Tsvankin showed that the azimuthal dependence of NMO velocity generally has an elliptical shape and is determined by the spatial derivatives of the slowness vector evaluated at the CMP location. This formalism is used here to develop exact solutions for normal-moveout velocity in anisotropic media of arbitrary symmetry. For the model of a single homogeneous layer above a dipping reflector, we obtain an explicit NMO expression valid for all pure modes and any orientation of the CMP line with respect to the reflector strike. The influence of anisotropy on normal-moveout velocity is absorbed by the slowness components of the zero-offset ray. If the medium above a dipping reflector is horizontally stratified, the effective NMO velocity is determined through a Dix-type average of the matrices responsible for the interval NMO ellipses in the individual layers. This generalized Dix equation provides an analytic basis for moveout inversion in vertically inhomogeneous, arbitrary anisotropic media.