- Convolutions and Synthetic Seismic. In the last Knowledgette, we discussed acoustic impedance and reflection coefficients. When we look at a seismic section, however, what we see are a bunch of squiggles, that resemble sine waves. What we are seeing is the interaction between the sound pulse we generated in the near surface, with the reflective horizons in the subsurface. In this Knowledgette, we will explore the basic theory of how these wiggles get formed. In Science, and indeed in much of life, we work with models of reality. We construct simplified versions of the real world so that we can more easily understand, make predictions, and manipulate. A good model is one that captures the behaviors we are interested in, but is also as simple as possible. One key to using models is to fully understand the assumptions built into the model, the limitations of the model, and the potential pitfalls. We will examine each of these areas for the Convolutional Model. To briefly review the previous Knowledgette, Acoustic Impedance is simply the product of density and velocity. Changes in acoustic impedance create reflections, which we record and which generate our seismic sections. We imagine the earth, so here is the first part of our model, being a series of homogeneous layers, each layer having a near constant acoustic impedance. Here is a bit of a deepwater turbidite in southern France. We have a thick, sandy facies sitting on a thin-bedded, shaly section. Imagine that each layer in this roadcut is nearly homogeneous, so we can draw a simple, layered model as shown. If we look at each interface between different layers, we can calculate the change in acoustic impedance to find the reflection coefficients. If we then plot those as a spike log, we have a graphical representation of the reflectivity series, showing reflection coefficients between plus one and minus one. If we have a well that has a density and a sonic log in it, we can do a similar thing for the entire well, by blocking off sections of similar impedance, as shown in the illustration. Briefly, let's discuss a different topic, wavelets. These will be covered in more detail in the next Knowledgette, but we need to introduce the concept now. Imagine tossing a pebble into a pond. As we all know, a series of ripples will expand outwards from where the pebble hits. If we look closely, we will see that we have a crest, followed by a trough, followed by a smaller crest, then a smaller trough, and so forth. A completely analogous situation occurs when we set off a seismic source, such as dynamite or an airgun. If we look at a cross section of the ripple pattern, we see something like this. We call this our geophysical wavelet. This is the signal we have pumped into the earth, and it is what we will see echoing back to us, from each interface it encounters in its travels. So we have a spike log representing our reflection coefficients, and we have our input signal, our wavelet. What will we see when we record the echoes coming back to us? For each spike in our spike log, we will see a copy of the input wavelet, flipped so that the tail points down, since it is now traveling upwards, its amplitude shrunk by the reflection coefficient, and having its peaks and troughs swapped whenever the reflection coefficient is negative. Where the returning wavelets overlap, we simply add up the amplitudes of each point in time to get the final answer as to what we will see. We have just done a convolution! It's a bit tedious, but I think the basic concept is pretty simple. Symbolically, we represent the convolution with an asterisk, and it is, in fact, a form of multiplication, but a special way to multiply signals together. As you can see I have added an extra term, "noise", to the equation. That just recognizes that there is some level of random noise in all of our measurements. They are never perfect. As an aside, if one takes the Fourier transform of the input signals, the convolution in the transformed space is really just a simple, everyday multiply. In practice, in software, usually the signals are Fourier transformed, multiplied, and then inverse transformed. So let's review what we have just covered. We can take a sonic and density log from a well, and we can use those to create an acoustic impedance log. By taking the differences at each interface we end up with a spike log, which represents the reflection coefficient at each interface. If we then convolve that with our input wavelet, we will get a synthetic seismic trace. There are a few assumptions, both implicit and explicit, that we have made in constructing this model. We assume that the earth can be fairly represented as a constant acoustic impedance layerS. We assume that our wavelet does not change as it moves through the earth. We also assume that we are dealing with a plane wave travelling only vertically. How good are these assumptions? Usually, not too bad. When I started out 30 plus years ago as a young Geophysicist, the process I have outlined is exactly how I created synthetics for doing our well ties, convolving a wavelet with a spike log. And it usually worked pretty well. But there are some pitfalls. Many times the acoustic impedance of a layer is not, in fact, a constant value. This is not as bad as it sounds, however, as a thick, heterogeneous layer can often be replaced with a mathematically equivalent homogeneous layer using effective medium theory. That topic will be covered in a Rock Physics Knowledgette. A more serious issue is that wavelets are not, in fact, constant. As a seismic signal propagates through the earth, there is a frequency dependent absorption. The wavelet loses energy at the higher frequencies faster than at the lower, so it changes shape. Usually this is a relatively slow process. The change in a wavelet going through relatively benign geology can be assumed to be negligible over about one second of data. Finally, we assume a vertically traveling wave. We know that seismic is acquired with a range of reflection angles, and stacked together, so this assumption is clearly violated. And, in a later Knowledgette, we will discuss Amplitude versus Offset, and the Zoeppritz equation. We will see how to relax this assumption. This is probably the largest source of mismatches between a single convolutional synthetic and what we will see on seismic data. All of these issues will be addressed in some detail in succeeding Knowledgettes. We have discussed the basic theory of how our seismic gets formed, by seeing how a wavelet convolved with a series of reflection coefficients will yield a pretty decent approximation of our seismic. We have noted some of the assumptions and pitfalls in this approach. All that said, this basic approach is behind much of how we think about seismic data, as a series of reflectors that are convolved with a near-constant wavelet. As we explore further, we will relax some of our assumptions, and look more closely at the basic physics, but the framework in this Knowledgette will remain at the core of what we do.