after the NMO application we have stretch for far offset , i read a lot of articles but i can't have a complete idea how this phenomena is really created ! any one have a rich explanation ?


My understanding is the following (NB: it could be wrong!).

  1. The assumption is that on a common shot gather, your travel time follows a hyperbolic curve: $$ f(x)=t^2=t_0^2 + \frac{x^2}{v^2}, $$ where $t_0$ is the zero-offset travel time, $x$ the offset and $v$ the speed of the medium above the interface. The hyperbolic travel-times, for example, create gathers like the one shown in https://wiki.seg.org/wiki/NMO_stretching#/media/File:Ch03_fig1-9.png (which is figure '(a)' below).

  2. Furthermore, the recorded wavelet has the same shape along the time ($t$) direction, regardless of which offset ($x$) you recorded it. Hence, on the figure below, check that there is no discernible change in the wavelet shape in figure '(a)' below as you move along the hyperbolic curve.

  3. Then, NMO correction essentially comes down to stretching the original data. In the figure '(a)' below, I put a red line close to the zero offset point, and a purple line at far offset. In figure '(b)', which is an NMO-corrected picture, I highlight that same lines, with identical colors. You see that the zero offset (red line) hardly requires a change, but the far offset (purple line) needs to be stretched considerably to create a flat curve in the diagram! A neat feature is that at at ~90% of the offsets, you still see 2 events cross in both figures '(a)' and '(b)', to indeed confirm that we just stretch the traces with no real further magic behind it...

So, the wavelet shape is constant in the vertical direction everywhere in figure '(a)', but is seriously stretched out in figure '(b)'.

From the page of the linked figure: "Note that stretching is confined mainly to large offsets and shallow times. For example, a waveform with a 30-Hz dominant frequency at 2000-m offset and t0 = 0.25 s shifts to nearly 10 Hz after NMO correction. Because of the stretched waveform at large offsets, stacking the NMO-corrected CMP gather will severely damage the shallow events. This problem can be circumvented by muting the stretched zones in the gather."

NMO example

tl;dr: because the red and purple line in figure '(a)' must become the red and purple line in figure '(b)', the wavelets of particularly the purple line are stretched.

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