# Dictionary:Rms velocity

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Root-mean-square (RMS) velocity. Consider an earth model consisting of a sequence of parallel horizontal layers of interval velocity ${V_{\rm {INT}}}_{i}$ in the $i$ -th layer. We label layers from 1 to $N$ , with the $N$ -th reflector being at the bottom of the $N$ -th layer. The two-way traveltime for seismic energy propagating perpendicularly through the $i$ -th layer is $\Delta t_{i}$ . The RMS velocity at the $N$ -th reflector, for travel perpendicular to the layers is

${V_{\rm {RMS}}}_{N}={\sqrt {\frac \sum _{i=1}^{N}V_{i}^{2}\;\Delta t_{i}}\sum _{i=1}^{N}\Delta t_{i}}}}.$ The average velocity for this path, ${\bar {V}}$ at the $N$ -th reflector, is

${\bar {V}}_{N}={\frac \sum _{i=1}^{N}{V_{\rm {INT}}}_{i}\;\Delta t_{i}}\sum _{i=1}^{N}\Delta t_{i}}}$ The stacking velocity, the velocity determined from velocity analysis based on normal-moveout measurements, is often used to approximate RMS velocity; this is valid only in the limit as the offset approaches zero where interfaces are flat and horizontal, and layers are isotropic. The RMS velocity for the $N$ -th reflector is related to the NMO velocity at the N-th reflector via

${V_{\rm {NMO}}}_{N}={\frac {{V_{\rm {RMS}}}_{N}}{\cos(\theta _{N})}}$ where $\theta _{N}$ is the dip of the $N$ -th layer. The NMO and STACK velocities differ if the moveout deviates substantially from being hyperbolic, as would be seen with substantial anisotropy.