Discrete Slant Stack on seismic data

In the formula for slant stacking (tau-p transform) a substitution t=tau+d/v is made.

If we consider the substitution as a vector equation then how can vector d/v have the same length as vector t given that in most surveys the number of data points for offset(d) is less than for time (t) .

The $\tau$–$p$ time domain $t$ is a function of offset $d$, so the result is an array of size $n_t \times n_d$, where $n$ is the number of samples.

The inputs are a vector $\tau$ of travel time at zero offset, or $t(0)$, an array $p$ of ray parameter, which is in turn a function of velocity $V$ and angle, and a vector $d$ (usually called $x$) of offset.

Here's an example of some visualizations of the $\tau$–$p$ domain from a paper by Sergey Fomel; they might help explain what I mean by an 'array' — you can think of them as images:

The resulting travel-time $t(d)$ is an array of size $n_t \times n_d$, where $n$ is the number of samples. For a given offset at a given time sample, the result is a scalar — a single value of $t$.

By the way, stevej's answer is a great explanation of how exactly to calculate the ray parameter term, given limited information. Do follow that link to Yilmaz, and maybe also read the Radon transform article.

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:

1. Sample ray parameter $p$ such $np = nx$; i.e. number of ray parameter samples equals number of offset samples (Yilmaz, 2001).
2. 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.

• This is a really nice contribution but I think it answers a slightly different question, so I had a go at my interpretation of what the OP wants. Jul 14, 2015 at 14:37
• Yeah good call; in your answer the portion in bold answers the OP nicely. Jul 15, 2015 at 1:39