## second order traveltime equations

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Sandralin
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Joined: Sun Aug 23, 2020 4:20 pm

### second order traveltime equations

What is a second-order traveltime equation?
What does a second-order equation mean again?

GuyM
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Joined: Sat Mar 24, 2012 11:35 pm

### Re: second order traveltime equations

It would be useful to have the whole paragraph to give context, but the usual context in reflection seismic data processing is in relation to the Normal Moveout Equation.

This describes how the arrival time of reflected seismic energy from a plane layer varies with offset (x). The reflection shape itself is a hyperbola, but htis is simplified with a Taylor expansion.

The second order term includes x-squared, the next term in the sequence x to the power four, and so on.

We usually use the second order version out to about 45 degrees; with longer offsets you need to include the forth-order term to flatten the reflection.

I'd suggest the SEG wiki for further reading, which includes pages from Oz Yilmaz's book:

https://wiki.seg.org/wiki/NMO_in_a_hori ... fied_earth
https://wiki.seg.org/wiki/Fourth-order_moveout

Sandralin
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Posts: 4
Joined: Sun Aug 23, 2020 4:20 pm

### Re: second order traveltime equations

Thank you for such a good explanation.
One thing was a bit confusing. What is a Taylor expansion in this case you described?
When I searched for it on google it says:
" It is a series that is used to create an estimate (guess) of what a function looks like. "
But how is it simplified with a Taylor expansion?

I also read something about a DSR (Double square root) operator for separation of reflection from diffractions.
They did also use this Taylor expansion.
Are you familiar with the case I described now? Or have you heard about it?
It was really confusing.

GuyM
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Posts: 625
Joined: Sat Mar 24, 2012 11:35 pm

### Re: second order traveltime equations

I'd suggest that you need to get a copy of Oz Yilmaz's book and follow that through. It includes all of the mathematics needed to understand the derivations he is using linked to the equations and in context. That's the source material that underpins the SEG wiki.

The thing with the Taylor expansion (approximation) is that the travel time equation for the reflection hyperbola is not very useful in that form; it's hard to define or use. The Taylor expansion gives us a methodology to unpack the equation into a form that is useful, both at the second and forth order terms. You will find these kind of approximations are used often in seismic processing.

I'm not familiar with the double square-root operator - or rather its something I may have known once, but in these days of full prestack time migration became less important to remember.