Waves in strata

Prior RMS velocity

Substituting the theoretical interval velocity $v(\tau)$ from equation (3.40) into the definition of RMS velocity $V(\tau )$ (equation (3.25)) yields:

$$\tau V^2(\tau) = \int_{0}^{\tau} v^2(\tau') d \tau'$$ (47)

$$= v_0^2 \frac {e^{\alpha \tau} - 1} {\alpha} .$$ (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)