Waves in strata

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 (3.18) 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 (3.18), is the same as $V(\tau )$ no matter where you measure $\hat v$ on a hyperbola.

Waves in strata