- In this Knowledgette we will discuss wavelets and polarity in some detail. As we noted in the last Knowledgette, if you have a wavelet and an impedance log, simply convolving the two will generate a synthetic seismogram. So it is imperative that we deepen our understanding of wavelets. Note that we are talking about "seismic wavelets", not to be confused with "mathematical wavelets". Wavelet theory in mathematics, while of great utility in Geophysics, is unrelated to seismic wavelets. It is just an unfortunate and confusing use of the same name. A typical source wavelet, from an air gun or dynamite, is causal. That is, there is no energy before time zero the dynamite is quiet until it explodes. As it turns out, these causal wavelets are also minimum phase. What does that mean? What is wavelet phase? Well, our little wavelet squiggle can be completely represented through three other quantities, obtained via a Fourier transform: the frequency spectrum, the phase spectrum, and the amplitude scale. From a seismic interpreter's standpoint, there are two important facts about these minimum phase source wavelets. First, the zero-time point of the wavelet, where the energy first bumps above zero, is where the reflection should be posted. This is a difficult spot to pick, and so can cause practical difficulties. Secondly, a dynamite or air gun source wavelet tends to be very long and loopy, so that each reflective interface will produce a long series of loops, making it difficult if not impossible to distinguish geology from acquisition artifacts. Note how the energy in the wavelet shown is all crowded towards the front. So what is to be done? In the last Knowledgette we discussed convolution, combining a wavelet with geology to produce seismic. Our resourceful seismic processors can do the inverse process for us, deconvolution, solving for the input wavelet and then replacing it with a shorter, better behaved wavelet. If done properly, deconvolution should remove any shot-to-shot wavelet variation, often a problem onshore, and also remove most of the extra loops from the source wavelet. Additionally, the new wavelet should always be zero-phase, which means that the wavelet is symmetric, and time zero is centered on the peak, or trough, that begins the wavelet. So this is no longer a causal wavelet, stuff happens before time zero, but that is okay. It is a much easier wavelet to interpret with. Let's dig a little deeper into wavelet phase, and look into rotating the wavelet phase. If we take a sin wave, and we add a constant angle to it, we "rotate" the sin wave. The peaks and troughs move in time. We can do exactly the same thing to a seismic wavelet. In the slide, we start with a nice zero phase wavelet, and then start adding phase to it in constant 30 degree increments. In green we show the envelope, within which the wavelet always stays, regardless of the rotation. As an aside, thirty degrees is roughly the limit of what one can see with the naked eye. Rotations less than that are very hard to see. Here we see four identical wavelets, with their phases rotated by 30, 60, and 90 degrees. A brief discussion of analytical wavelets. There are several formulations that yield things that look like wavelets, in particular the Ricker and the Cosine Bell wavelets. Personally, I am not a fan of these, especially the Ricker. In my experience, they do not really reproduce true wavelets very well, and so I don't use them. They typically exaggerate the side lobes. However, there are those who swear by them. As they say, your mileage may vary. The most important, and most difficult topic around wavelets is polarity. When presented with a nice seismic section, what does a positive wavelet peak represent? Does it represent an increase in the reflection coefficient? Or a decrease? This is, of course, a fundamental issue, and absolutely key to doing a proper interpretation of anything but the gross structure. The SEG has defined that for a zero-phase wavelet, a positive reflection coefficient is represented by a central peak, that is, a positive number. However, in many companies and in many basins, this convention is actually reversed, so it cannot be depended on. Additionally, since the only difference between a positive polarity and a negative polarity seismic section is a multiplication by minus one, it is very easy for the processing to get it wrong. So from a practical standpoint, an interpreter can never trust that the polarity is as advertised, but must check it by looking at the water bottom reflector, doing a well tie, or some other method. So why do we care about the wavelet and its phase? For structural well ties, if we have the wrong wavelet polarity, our tie can easily be off by a loop, which equates to roughly 30 meters, or 100 feet. That can certainly introduce significant error into a structural interpretation, as well as put the interpreter onto the wrong loop, making tying loops and the entire interpretation process difficult, and suspect. Certainly for quantitative interpretation, where the interpreter is trying to link the seismic response to rock and fluid properties, having the wavelet phase wrong can make it nearly impossible to develop a coherent story, since the seismic impedances will all be incorrect. Similarly, a phase problem will cripple stratigraphic interpretation, where subtle changes in loop shape often signal important stratigraphic changes. If the loop is not correct to begin with, then the changes become very difficult to interpret. So what are the most common wavelet pitfalls? The most common is that the phase is not actually zero. This is much more common than you might think. After many years of generating high quality well-ties, I have actually rarely seen zero phase wavelets on seismic data. It seems that usually the wavelet has a residual phase of around 20-30 degrees. I'm not actually sure why this is, but it is what I have observed. Since this is at the limit of detectability by eye, it's not too serious for normal work, but it can cause issues when looking at subtle features. Sometimes, the phase error is much larger, often being off by around 90 degrees, or 180 degrees, a polarity flip. These sorts of errors can be disastrous to an interpretation, and usually are caused either by a processing error, or by a miscommunication in the processing or the data loading process. These illustrate why it is critical for the interpreter to be involved in the processing, to ensure that there is a common understanding. More rarely, but I have seen it happen, are simply poorly done deconvolutions. We will not discuss the details here, but especially adaptive deconvolution, if not done thoughtfully, can corrupt the phase of the wavelet in strange and unpredictable ways. Finally, I have, rarely, seen data where the phase of the wavelet is not constant, but rotates as a function of depth. This can be especially pernicious, since the phase could be correct at the water bottom, and incorrect at the objective. In summary, we have expanded the meaning of a wavelet, to mean much more than the signal from our seismic source. In effect, we are saying the wavelet is whatever little loopy thing we have convolved with our earth model to create our seismic section. Wavelets are defined by their frequency spectrum and their phase spectrum. A wavelet may have constant phase, or the phase spectrum may vary with frequency. A special case of varying phase is the minimum phase, or causal wavelet. A constant phase wavelet with phase zero is the goal for seismic processing, as this yields a nice symmetric wavelet with the peak centered on the reflecting horizon, making for easy interpretation. There is an SEG standard for wavelet polarity, however it is not always followed, and claims about the polarity of a seismic dataset should always, always be confirmed, and never taken for granted. Errors in wavelet phase can easily propagate into interpretation errors of all sorts, from structural to stratigraphic and quantitative. The wavelet is our lens through which we view our geology, and we need to make certain that our lens is as clear and clean as possible.