Earth models, as we understand them, have the following three basic formulations:
Earth models, as we understand them, have three basic formulations. The first is what we usually call purely isotropic or acoustic. Acoustic models are based on the assumption that the only physical parameters defining wave propagation are density, ρ ( x;y;z ), and interval or instantaneous velocity v ( x;y;z ). Only fluids can be described by these two properties, but because propagation in such environments can be simulated efficiently, they are the most prevalent at this writing. Empirical evidence also seems to suggest that in many geologic settings the real Earth does not vary much from this assumption.
Isotropic elastic models are described by density, compressional velocity, and shear velocity. The notation for these parameters is ρ ( x;y;z ), v p ( x;y;z ), and v s ( x;y;z ). Isotropic elastic models support two wavefields, one of which is a compressional wave and the other is a shear wave. Compressional waves in such models are identical to those in acoustic models. They are characterized by particle motions consistent with what might be called compression and rarefaction where the particle vibrations are normal to the direction of propagation. In contrast, we tend to think of waves where the the particle motion is tangential to the direction of propagation as shear waves. As a point of fact, the truth is probably somewhat different. Simulations tend to support the conclusion that the compressional wave is what we would record if we were to measure purely vertical particle motion and the shear wave is the one characterized by purely horizontal particle motions. The speed of shear waves is frequently much slower then the velocity of the compressional wave. Nevertheless, shear and compressional waves continually interact and convert from one to the other as the propagation progresses. Thus, if we are to successfully handle isotropic elastic data, we must acquire something at least directly related to the vertical and horizontal particle motions. In other words, we have to acquire vector data.
Anisotropic models represent the Earth at its most complex. For our purposes, a model is said to be anisotropic whenever the sound speed is a function of the angle of propagation. In models of this type, not only does the velocity of sound vary with propagation angle, but there are three possible propagating modes at any given instant. One is our familiar compressional wave and the other two are shear wave, each of which propagates with its own local angle dependent velocity profile.
Over the last 20 or so years we have come to specify what we might call the first realistic anisotropic models by density ρ ( x;y;z ), vertical velocity v p ( x;y;z ), shear horizontal velocity v s ( x;y;z ) and three additional parameters, δ ( z;y;z ), ε ( x;y;z ), and γ ( x;y;z ) . Models described by these "Thomsen" parameters are the so called vertically transverse isotropic or VTI models. Anisotropic VTI models have a very convenient form of symmetry that makes using them somewhat easier and less computationally complex then more complex versions of anisotropy. It is reasonable to expect anisotropic models to become the norm in future exploration exercises. In this case true anisotropic processing will also require the acquisition of vector data. The difference between this and isotropic elastic acquisition is that here each vector has three components.
Regardless of the source we use, the Earth's response always contains compressional and at least one, but most probably, two shear wavefields. Thus, the expected Earth model is quite complex. While we may not have the ability to estimate the necessary parameters to image the recorded multicomponent data, there are many algorithms for doing so. It makes sense to at least understand the kinds of data we might expect to record and what it might look like. The basic idea of using modeling to help us understand the recordings has a significant history in the exploration for hydrocarbons. As this book progresses, we will attempt to define the various models currently in vogue and make some additional comments about how the required parameters might be estimated from our acoustical recordings.
