Root-mean-square velocity

When a ray travels in a depth-stratified medium, Snell's parameter $p=v^{-1}\sin\theta$ is constant along the ray. If the ray emerges at the surface, we can measure the distance x that it has traveled, the time t it took, and its apparent speed dx/dt=1/p. A well-known estimate $\hat v$for the earth velocity contains this apparent speed.  

$$\hat v \eq \sqrt{ {x\over t} \ {dx\over dt} }$$ (18)

To see where this velocity estimate comes from, first notice that the stratified velocity v(z) can be parameterized as a function of time and take-off angle of a ray from the surface.

$$v(z) \eq v(x,z) \eq v'(p,t)$$ (19)

The x coordinate of the tip of a ray with Snell parameter p is the horizontal component of velocity integrated over time.

$$x(p,t) \eq \int_0^t \ v'(p,t) \ \sin\theta(p,t)\ dt \eq p\ \int_0^t v'(p,t)^2\ dt \$$ (20)

Inserting this into equation (18) and canceling p=dt/dx we have  

$$\hat v \eq v_{\rm RMS}\eq \sqrt{ {1\over t} \ \int_0^t v'(p,t)^2\ dt\ \ }$$ (21)

which shows that the observed velocity is the ``root-mean-square'' velocity.

When velocity varies with depth, the traveltime curve is only roughly a hyperbola. If we break the event into many short line segments where the i-th segment has a slope p_i and a midpoint (t_i,x_i) each segment gives a different $\hat v(p_i,t_i)$and we have the unwelcome chore of assembling the best model. Instead, we can fit the observational data to the best fitting hyperbola using a different velocity hyperbola for each apex, in other words, find $V(\tau )$ so this equation will best flatten the data in $(\tau,x)$-space.

$$t^2 \eq \tau^2 + x^2/V(\tau)^2$$ (22)

Differentiate with respect to x at constant $\tau$ getting

$$2t\, dt/dx \eq 2x/V(\tau)^2$$ (23)

which confirms that the observed velocity $\hat v$ in equation (18), is the same as $V(\tau )$ no matter where you measure $\hat v$ on a hyperbola.