Introduction

When generating a time dependent forcing function, nonlinear behavior can occur in the electronic amplification part of the system. These nonlinearities can be avoided by having feedback mechanisms. Another nonlinear mechanism is the transfer of the desired signal into the ground. Most of the signal distortion occurs at this instance with high momentum devices, be it by impacting or vibrating. The subsurface volume in which such mechanisms dominate is rather small, yet such nonlinear behavior shows up in an effective alteration of the signal spectrum and depends on radiation direction. It is this interaction of the device with the earth's surface that cannot be described by using infinitesimal displacements or stresses. When describing nonlinearities we need to separate two effects: one is the finiteness of the variables, the other is the possibility of entering a nonlinear region in the stress-strain relationship. In general the function describing the stress-strain dependency can be arbitrary, as long as some underlying conservation principles are satisfied. It is always possible to expand this function around the equilibrium point at which we measure. A conventional elastic method would only use the linear term of such a Taylor series expansion. This works well if the magnitude of the variables are small. If the magnitude exceeds a certain limit we have to use higher order terms in the approximation in order to describe the medium properly. It is seldom the case that these higher order stiffness constants are measured. For this reason, this paper mainly deals with effects introduced by the finiteness of the variables, but not so much with the nonlinear stress-strain relationship. The finiteness of displacements introduces no new material parameters which must be known or estimated. Allowing the displacements to be non-infinitesimal merely means that the volume is deformed by finite displacements. This is equivalent to allowing higher order terms in the Taylor expansion of the displacement field.

This paper does not deal with receiver properties, although the same framework can be applied to modeling and estimating receiver functions. I assume that receiver properties can be effectively estimated as their transfer function. Most of the external effect is assumed to be caused by ground coupling. Stresses and displacements are generally assumed to be small by many orders of magnitudes in the case of receivers.