# How do we convert the seismic trace wavelet to zero phase which is in mixed phase?

As we know the mixed phase data would not give the "real TWTT" through the medium. We need a zero-phase wavelet that can directly correspond to the time taken to travel in a given media. So my question is how do we convert the seismic trace wavelet to zero phase which is in mixed phase? There must be an important processing procedure to do so.

My preferred way of doing this (I've been working in seismic data processing and analysis for over 20 years now) is:

1. Start with theoretical (or notional) source wavelet.
2. Shape the source wavelet to zero phase (or minimum phase, depending on your application).
3. Design a cross-equalization filter that takes the input from step 1 as the source and step 2 as the target. This can then be globally applied to your data.

There are other ways to do this (like with Wiener deconvolution) that make a statistical estimate of what the wavelet should look like, and this was considered acceptable in the industry for many years. However, such statistical processes are not acceptable for all applications, such as 4D (time lapse seismic) analyses, so I prefer processes that are deterministic rather than processes that rely on a stochastic assumption.

It depends on what your starting information about your seismic data is. For example, if you know that your data contains an impulsive (minimum phase) source wavelet (i.e. the acquisition source was dynamite or a big ol' hammer, or some other impulsive source, and has not been modified) then you can apply the workflow of txpaulm or use spiking deconvolution. But realize that you are making an assumption about the shape of your wavelet's phase spectrum, which is a very good assumption most of the time.

If you don't want to make the minimum phase assumption for your wavelet (i.e. you think you have a mixed phase wavelet), you will need to make some other assumption or incorporate some other piece of information.

A very popular method for estimating a mixed phased wavelet is that of Walden and White (1998). They incorporate additional information from a reflectivity series calculated at a well and then find the impulse response filter that best predicts (in the least squares sense) the seismic data extracted along the well path.

The trick here is that you are assuming that you know the velocity function (time-depth curve) along the well to correctly compare the well's calculated reflectivity series (in depth) with the extracted seismic data (in time). Errors in the time-depth curve will propagate into your extracted wavelet. What you can do is to first approximate the real wavelet with a simple zero-phase phase wavelet extracted from the autocorrelation of the seismic data (or you can use a Ricker wavelet of a reasonable dominant frequency if you want) and make a synthetic seismogram at the well. Then modify the time-depth relation along the well path (i.e. stretch and squeeze your synthetic) until your well is kinda-sorta fitting. Then go back and extract a better constant-phase wavelet (no more Ricker) through autocorrelation and try to modify the constant phase while simultaneously tweaking the time-depth curve along the well. You should already be close the right time-depth curve but this will get you pretty damn close to the answer. Once you are happy with the position of your synthetic with respect to the seismic data, then you can apply the method of Walden and White (1998) to get the best fit filter that gets you from the reflectivity at the well to the seismic data along the well path. This filter is your mixed phase wavelet (at that one well location). If you do this at a couple wells and notice that the wavelet is relatively consistent, you can average the wavelets and apply the method of txpaulm to make your entire seismic data zero-phase.

Ok then, sorry for the long block of text...

Now, what if you don't have wells or don't want to use them for some reason?

Well, you can still estimate the phase spectrum of your wavelet directly from the seismic data by using the method Velis and Ulrych (1996) (and optimized by Misra and Sacchi (2007)) based on 4th order cumulant matching between the seismic trace and an all-pass filter. Since this optimization problem is nonlinear, the method is computationally intensive as it requires a global optimization algorithm (Very Fast Simulated Annealing) and can be sensitive to noise. However, since this method does not require a well, it can be used to test the stationarity assumption about the seismic wavelet.

Mixed-phase wavelet extraction is a very popular field of research and you can probably find more information looking through the SEG and EAGE online libraries.

Good luck! -Antonio