Revisiting Dix’s RMS approximation for Normal Move-Out Velocity

The RMS velocity approximation continues to be used in seismic data processing and interpretation while its limitations are not well understood because its derivation is opaque to most modern geophysicists. One of the reasons for this is because Dix describes in (Dix, 1955) NMO velocity functions determined by T²-X² analysis of shot records, which very few current geophysicists have ever performed. A clearer understanding of T²-X² analysis and the derivation of the RMS formula will enable NMO-derived velocities to be better understood and so better used in reflection seismic imaging, inversion and interpretation. Dix’s derivation closely follows the process of correcting the apparent depth from say refraction of objects in water when viewed from the air. So it is useful to search for this subject and follow the development which relies on the observer’s eye separation being small relative to the depth of the object and also relies on refraction being towards the normal of any interface so that angles of incidence will reduce and remain small. This is true for light because the speed of light in water is much slower in water than in air. Reflection seismic uses sound which in general bends away from the normal at each interface as sound velocities tend to increase with depth so that angles of incidence increase which causes a problem with the method which we will see later.