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  4. Ray Theory

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- Thus far, we've been focused on acoustic impedance, reflections, and wavelets, basically these small-scale seismic features. In this Knowledgette, we want to step back and look at a much larger scale. How do seismic waves propagate at the large scale and what are the important effects? In the literature, this area of geophysics is often called kinematics. As we've noted before, seismic waves look much like the ripples on a pond when a stone is tossed into the water, having a nearly circular appearance. For developing our understanding and for doing simple calculations, however, Ray Theory is usually much more useful. Rays are in an isotropic medium, orthogonal to the wavefront, and are the basis for much of geophysical theory. From freshman physics, you probably remember Snell's Law as applied to light rays. Simply stated, it says that light will refract or bend when it crosses a boundary between materials with different refractive indices and the ratio of the sines of the angles equals the ratio of the refractive indices or, alternatively, the velocity ratios. Exactly the same thing occurs with seismic waves. Our rays get bent according to Snell's Law, purely based on velocity changes in the earth. So, this is interesting. Reflections depend on acoustic changes in acoustic impedance, that is, velocity times density. But refraction of rays depends only on changes in velocity, not density. Notably, Snell's Law is a direct consequence of Fermat's Principle, which states, roughly speaking, that the energy will follow the path of least time. Sometimes, in very complex geology, there are other allowed solutions, like maximum time paths, but we're not gonna worry about those here, as that is quite an advanced topic. Seismic wave velocities in the earth are not constant. In general, although there are exceptions, velocities increase as a function of depth. Because of this often gradual change in velocity, our raypaths will not be straight lines, but will curve. So, why do the raypaths curve? This is easily understood as a consequence of Fermat's Principle. The ray is trying to minimize his travel time between the surface and the reflector. Think of the shallow, slow section of your geology being like surface streets in a city. The deeper, faster parts of the earth are like super highways. If you want to minimize your travel time, you will jump onto the highway as soon as possible and stay there for most of your journey. As can be seen in the illustration, the curved raypaths try to go straight down in the shallow section and then flatten out a bit at depth so they can spend more of their journey in the deeper section, getting to the destination faster than the straight rays. Don't be confused by the fact that the curved raypaths are longer in distance. Like taking a freeway, this is simply a reflection of the fact that the shortest distance is not necessarily the quickest route. As an aside, I've used a generic velocity function that is reflective of young, unconsolidated sediments like one would find in the Gulf of Mexico or West Africa. So, all of this is very interesting, but why do we care? How is this actually useful? Well, let's use this simple ray-tracing model to explore a number of practical issues. Perfect analogy to the way a lens can focus rays of light, the same phenomenon can occur with seismic, especially when we are trying to see beneath, or near, high velocity layers of salt, carbonates or igneous rocks. The wave field, as represented by our raypaths, can be distorted in ways that will cause both shadow zones and very confused images. For clarity, the illustration shows zero offset rays, where the source and receiver are at the same position on the surface. As can be seen, parts of the reflector below the high velocity zone are in shadow and parts are over-illuminated with the rays actually being scrambled a bit. This can obviously cause issues with an interpretation, especially if amplitudes are an important consideration. Let's talk a bit about migration. When we create a seismic section, we post the data we record at the common mid-point, that is, at the halfway point between our shot and our receiver. For a dipping reflector, the reflection is not actually coming from the mid-point, but for an isotropic velocity field, from a point up-dip of the mid-point. So, when we post the data, we will post it in the wrong place. This is exactly analogous to the distortion we see when we view an object in a pond. It is not actually where it appears to be. Migration is the process which we use to correct for this problem, and some others, and place the data in the correct position spatially. As you can see in the illustration, the reflector has moved substantially. This is all fine and good, but ever since the late 1980s, the industry standard for interpretation has been fully migrated 3D seismic data, so, most of these mispositioning problems are resolved for us, right? Not quite. Fracking, as I'm sure you are aware, has completely altered the industry, generated a huge push back onshore, and, consequently, generated a tremendous need to acquire and interpret 2D seismic data, since onshore 3D acquisition tends to be rather pricey. This has reintroduced many interpreters to the fact that 2D seismic does not actually work unless the 2D line is a perfect dip line. This is easy to see by considering a strike line. Along strike, a dipping horizon will appear flat and so, will not be moved by migration. However, the reflection that is seen is actually coming from off to one side of the 2D line and is, in fact, incorrect. However, migrating a dip line will position the reflector correctly. What this means, from a practical standpoint, is that the interpreter cannot use migrated lines to tie lines and construct loops. Unmigrated lines must be used for this key interpretive process. Then once the lines are tied, mapping must be done using only the dip lines, as those are the only lines correctly positioned. Alternatively, an unmigrated map can be made and then migrated with a map migration tool. Let's summarize what we've covered in this Knowledgette. We can model seismic wave propagation with wave theory, a process exactly analogous to how light rays are treated in freshman physics. Snell's Law applied, when a seismic ray crosses an interface, the ratio of the velocities is equal to the sines of the angles. Seismic rays obey Fermat's Principle, that is, they generally take the minimum time path between any two points, which, for the earth, where velocity generally increases with depth, results in curved raypaths. From a practical standpoint, energy can be focused or shadow zones can be created by high or low velocity zones distorting the raypaths, creating interpretational challenges. Ray Theory also allows us to start looking at the migration process and understand how dipping horizons are misplaced before migration. Additionally, it helps us to understand the special challenges of interpreting 2D seismic. Tying 2D data must be done on the stack data, not the migrated data, and mapping must be done using only the migrated dip lines. Ray Theory, like all our models, is only an approximation, but it is a very useful approximation and, at least for me, helps me to visualize what is happening with my data.