The double-elliptic approximation in the group and phase domains

in the impulse-response domain

Muir's trick is to turn the problem on its side and also look at the horizontal paraxial approximating ellipse. This has all the useful properties of the vertical ellipse, with the difference that it fits the true horizontal velocity and the vertical NMO velocity (not the same thing as the true vertical velocity). Figure 2 shows the vertical and horizontal paraxial approximating ellipses, along with Muir's double-elliptic approximation. Muir's approximation fits four parameters, the vertical and horizontal true and NMO velocities. Since it is single-valued, it cannot follow the qSV triplication, but fits well elsewhere.