Prior RMS velocity

Substituting the theoretical interval velocity $v(\tau)$from equation (40) into the definition of RMS velocity $V(\tau )$(equation (25)) yields:

$$\begin{eqnarray} \tau \ V^2(\tau) &=& \int_{0}^{\tau} v^2(\tau') \ d \tau' \\ &=& v_0^2 \ \frac {e^{\alpha \tau} - 1} {\alpha} .\end{eqnarray}$$ (47) (48)

Thus the desired expression for RMS velocity as a function of traveltime depth is:  

$$V(\tau) \eq v_0 \ \sqrt{ \frac{e^{\alpha \tau} - 1 }{\alpha \tau} }$$ (49)

For small values of $\alpha \tau$,this can be approximated as

$$V(\tau) \quad\approx \quad v_0\ \sqrt{1 + \alpha \tau / 2} .$$ (50)