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Old physics for new images

Old physics for new images

At its core, seismology is concerned with how objects move when forces act on them. Over 300 years ago, two gentlemen outlined everything we need to know: Robert Hooke, with his law describing elasticity, and Isaac Newton with his second law describing inertia. Anyone working with seismic data should try to develop an intuitive understanding of their ideas and the equations that manifest them.

For rocks, a rudimentary but useful analogy is to imagine a mass suspended by a spring. Hooke discovered that when the spring is stretched, stress is proportional to strain. In other words, the force vector F exerted by the spring is proportional to the magnitude of the displacement vector u. The proportionality constant k is called the stiffness coefficient, also known as the spring constant:

$$ \mathbf{F} = - k \mathbf{u} $$

This is the simplest form of Hooke’s law of elasticity. Importantly, it implies that the stiffness coefficient is the defining property of elastic materials.

Newton’s second law says that a body of mass m, has a resistance to acceleration ü (that is, the second derivative of displacement with respect to time) under an applied force F:

$$ \mathbf{F} = m \ddot{\mathbf{u}} $$

If displaced from equilibrium, a mass attached to the end of a spring will feel two forces: a tensional force described by Hooke’s law, and an inertial force from its motion, described by Newton’s second law. The system of a mass and a single spring yields simple harmonic motion, characterized by acceleration being proportional to displacement but opposite in direction:

$$ m \ddot{\mathbf{u}} = - k \mathbf{u} $$

Simple harmonic motion has many applications in physics, but doesn’t quite fit the behaviour of rocks and seismic waves. A rock is bounded, like a mass held under the opposing tension of two springs. In this case, there are two tensional forces acting in the line along which the mass can oscillate. Writing out the forces in this system and doing a bit of calculus yields the well-known wave equation:

$$ \ddot{\mathbf{u}} = \frac{k}{m} \nabla^2\mathbf{u} $$

The wave equation says the acceleration of the mass with respect to time is proportional to the acceleration of the mass with respect to space, a tricky concept described by the Laplacian \(\nabla^2\). The point is, the only properties that control the propagation of waves through time and through space are the elasticity of the springs and the inertia of the mass.

Some vector calculus can move our spring–mass–spring system to three dimensions and unpack \(k\), \(m\), and \(\nabla^2\) into more familiar earth properties:

$$ \ddot{\mathbf{u}} = \frac{\mathbf{F}}{\rho} + \left [ \frac{\lambda + 2\mu}{\rho} \right ] \nabla (\nabla \cdot \mathbf{u} ) - \left [ \frac{\mu}{\rho} \right ] \nabla \times (\nabla \times \mathbf{u} ) $$

Here, \(\lambda\) and \(\mu\) are the Lamé parameters, representing Hooke’s elasticity, and ρ is the density of the medium, representing Newton’s inertia. You don’t need to fully comprehend the vector calculus to see the link between wave mechanics, as described by the displacement terms, and rock properties. I have deliberately written this equation this way to group all the earth parameters in the square brackets. These terms are equal to the squares of P-wave velocity \(V_\mathrm{P}\) and S-wave velocity \(V_\mathrm{S}\), which are therefore nothing but simple ratios of tensional (\(\lambda\) and \(\mu\)) to inertial properties (\(\rho\)).

To sum up, the Lamé parameters and density are the coefficients that scale the P-wave and S-wave terms in the wave equation. When rock properties change throughout space, the travelling waveform reacts accordingly. We have a direct link between intrinsic properties and extrinsic dynamic behaviours. The implication is that propagating waves in the earth carry information about the medium’s intrinsic parameters. Rock properties dance upon the crests of travelling waves, and they dance to the tune of seismic rock physics.

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