Pressure Formulation of Acoustic Wave Equation
Without going into great detail, we can also derive a pressure formulation of the acoustic wave equation (see Keiiti Aki and Paul G. Richards). This equation takes the form shown inEquation 2-17.
In contrast to the particle motion described by Equation 2-16,Equation 2-17 measures pressure changes at any given position. I like to call this the reflection formulation because of the presence of the acoustic impedance term, ππ£. This equation can be put in the very compact form of Equation 2-18, where β (the vector differential operator, pronounced del) is given by Equation 2-19.
(2-18) πτ·‘π
Ultimately, we are interested in deriving equations for more complex fully elastic
wavefields, including anisotropic wavefields. These wavefields require more parameters to describe the multiplicity of wavefields that exist in such media. Before increasing the complexity of the discussion, we will focus on a graphical description of how discrete modeling works, and then turn our attention to algorithms for implementing the actual modeling exercise.
Stress-Strain Equations
As we know, isotropic elastic models are described by three familiar parameters: density, compression velocity, and shear velocity. To model elastic wavefields in such three-parameter media, we need to derive a suitable equation or set of equations that describe the wavefield propagation at any given instant. Unfortunately, these three parameters are not the most useful for this purpose. On the other hand, the required parameters are much more directly related to actual rock properties. Moreover, relating the required parameters to more useful quantities is fairly straightforward.
We begin (Equation 2-8and Equation 2-7) by observing thatEquation 2-20 is true, so thatEquation 2-21is also true.
Panorama Technologies Stress-Strain Equations
In continuous terms, Equation 2-21can be restated asEquation 2-22.
(2-22) ππ
ππ₯ = ππτ·‘π’
ππ‘τ·‘
Equation 2-22 is a first order partial-differential equation relating a second order change in time to a first order change in force per unit area. Force per unit area is generally referred to as stress, so our equation relates particle acceleration to stress. In this setting, the stress is one-dimensional and acts along or parallel to the layout of the string. There is also only one compressional wavefield described by this equation.
Stepping up to the simple isotropic elastic models described by the three familiar parameters above, means that it is necessary to include one additional wavefield in the mix, namely shear. While including just two wavefields is certainly an option, there isnβt any reason not to move up to full anisotropic elasticity by incorporating two shear waves for a total of three wavefields. In three dimensions, Equation 2-22takes the form ofEquation 2-23, where π = 1, 2, 3 and we have arbitrarily chosen to set π₯ = π₯τ· , π¦ = π₯τ·‘,
Clearly, this is a three-dimensional equation with nine stress factors, πππ, one for each of the three dimensions and wavefields. To make this into a system of equations governing the three wavefields, we must find a way to relate the stresses, πππ to the π’π. As before, Hookeβs law comes to the rescue. What it says in this anisotropic case is:
Each component of stress is linearly proportional to every component of strain.
Stress-Strain Equations Panorama Technologies Strain, which measures the deformation (compression, extension, ...) of a solid, is
defined in the notation of the previous equation asEquation 2-24.
(2-24) πΈππ = 1
2τΏΆππ’π
ππ₯π + ππ’π
ππ₯πτΏΉ
The mathematical expression of Hookeβs law then takes the form ofEquation 2-25.
(2-25) πππ= τΎ
π,π
ππππππΈππ
InsertingEquation 2-25 into Equation 2-23, we finally get the complex system (π = 1, 2, 3) of fully anisotropic equations of motion,Equation 2-26.
(2-26) πτ·‘π’π
Because Equation 2-26is three-dimensional, each of the πππππ coefficients is actually a three-dimensional volume. Even todayβs massive supercomputers may not have sufficient memory to handle a problem of this size.
We could easily throw up our hands at this point and give up, but, before we panic too much, we might want to analyze the situation a bit. As it turns out there are at least two things we can do to simplify the situation considerably. First, we can simplify the mathematical notation to put us into a setting where we can make some sense of the parameters, and second, we can reformulate the πππππ coefficients in a way that will make a great deal more physical sense.
We are not really interested as much in the math as we are in understanding the kinds of Earth models these πππππ coefficients define for us. We need to know how the various velocities of the wavefields that propagate in the medium are defined. We also want to see if we can understand how direction changes the speed of propagation, and then see if we can find ways to estimate parameters that can be converted into πππππ volumes so we can both synthesize data and image data we have recorded over fully elastic models.
The first simplification to the complexity ofEquation 2-26 is based on the symmetry of stress and strain. Here, the ππ indices representing stress can be switched so that πππππ = πππππ. Similarly, the strain based indices can also be switched so that πππππ = πππππ. Finally, it is also true that πππππ = πππππ. This triple symmetry means that the total number of πππππ volumes has been reduced to only 21! Thus, defining the most general model
Panorama Technologies Stress-Strain Equations we can imagine requires only 21 independent parameters (volumes), instead of the 81 parameters we would need without symmetry.
By applying the indexing scheme (known as the Voigt scheme) in Equation 2-27, we arrive at the 6x6 matrix shown in Equation 2-28.
(2-27) πππππ₯ ππ = 11 22 33 23 13 12
This matrix completely describes the unique set of 21 coefficients fully defining anisotropic elasticity. In this case, the πΆ matrix is the most complicated form of anisotropy we can encounter. For the interested reader, this case is called βtriclinicβ
symmetry and is probably something we will not be able to investigate computationally until computers have advance significantly beyond where they are today. Moreover, we may never be able to measure sufficient data to be able to estimate all of these parameters. Consequently, we will focus on what we consider reasonable today.