15-DEGREE PARTIAL NORMAL MOVEOUT

Let us repeat basic definitions from Claerbout's paper. A truncated Taylor series of the normal moveout equation is given by the equation:

$$\tau = t-{x^2/v^2 \over 2t}.$$ (1)

The N^th partial normal moveout is given by the equation:  

$$\tau_N(t) = t-{x^2/v^2 \over 2tN}.$$ (2)

The N^th partial normal moveout can be applied N times:  

$$T_N(t) = \tau_N \tau_N \cdots \tau_N(t)$$ (3)

The equation to be proved is

$$T_\infty = \sqrt{t^2-x^2/v^2};$$ (4)

i.e., normal moveout can be expressed as a superposition of N N^th partial normal moveouts with arbitrary precision as $N \to \infty$.