# 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 ${\displaystyle {V_{\rm {INT}}}_{i}}$ in the ${\displaystyle i}$-th layer. We label layers from 1 to ${\displaystyle N}$, with the ${\displaystyle N}$-th reflector being at the bottom of the ${\displaystyle N}$-th layer. The two-way traveltime for seismic energy propagating perpendicularly through the ${\displaystyle i}$-th layer is ${\displaystyle \Delta t_{i}}$. The RMS velocity at the ${\displaystyle N}$-th reflector, for travel perpendicular to the layers is

${\displaystyle {V_{\rm {RMS}}}_{N}={\sqrt {\frac {\displaystyle \sum _{i=1}^{N}V_{i}^{2}\;\Delta t_{i}}{\displaystyle \sum _{i=1}^{N}\Delta t_{i}}}}.}$

The average velocity for this path, ${\displaystyle {\bar {V}}}$ at the ${\displaystyle N}$-th reflector, is

${\displaystyle {\bar {V}}_{N}={\frac {\displaystyle \sum _{i=1}^{N}{V_{\rm {INT}}}_{i}\;\Delta t_{i}}{\displaystyle \sum _{i=1}^{N}\Delta t_{i}}}}$

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 ${\displaystyle N}$-th reflector is related to the NMO velocity at the N-th reflector via

${\displaystyle {V_{\rm {NMO}}}_{N}={\frac {{V_{\rm {RMS}}}_{N}}{\cos(\theta _{N})}}}$

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