To be consistent with the references referred to in this post, I will re-write your equation of the $\tau-p$ transform as:
$t = \tau + px$
Where $t$ is two-way time, $x$ is offset, $p$ is ray parameter ($\Delta x / \Delta t$ or $\sin(\theta) / velocity$), $\tau$ is the intercept time when $p=0$ (Yilmaz, 2001).
There seem to be at least two methods for overcoming the finite aperture, or limited offset and ray parameter range, problem when transforming from time-offset ($t-x$) domain to $\tau-p$ domain:
- Sample ray parameter $p$ such $np = nx$; i.e. number of ray parameter samples equals number of offset samples (Yilmaz, 2001).
- Formulate the the computation of the transform as a least-squares estimation problem (Kostov, 1990; Thorson and Claerbout, 1985).
Kostov, C., 1990. Multi-channel seismic experiment with a drill-bit source. Phd Thesis, Standford University. See Appendix C for details on least-squares solution.
Thorson, J. and Claerbout, J., 1985. Velocity-stack and slant-stack stochastic inversion. Geophysics, Vol. 50, N. 12, P. 2727-2741.
Yilmaz, Öz (2001). Seismic data analysis. Society of Exploration Geophysicists. ISBN 1-56080-094-1. See SEG Wiki page on the problem.