Non-linear Vibroseis models for generating harmonics

Michael P. Lamoureux

ABSTRACT

Vibroseis devices are a convenient seismic source for generating energy to propagate into the earth, driving a spot on surface of the earth with a controlled force that sets up a seismic wave used in the seismic imaging experiment. For a variety of reasons, harmonics are generated when a pure frequency drives the Vibroseis device - these harmonics may be considered as noise, or as extra correlated data that might be used in the imaging algorithms.

In this short project, we build several simple mathematical models for the Vibroseis device, to help to understand where these harmonics come from. The simplest models are one-dimensional non-linear oscillators, where the response of the oscillator includes non-linear effects of the earth, limiting devices on the motion, and other mechanisms that may introduce harmonics. Better models include more physical details of the Vibroseis device - the motion of the reaction mass, the baseplate, the control machinery, and so on. These require a system of ordinary differential equations to represent the mathematics. Not all models are able to capture all the harmonic details of the Vibroseis device.

We make use of the Gabor transform to view the time-frequency characteristics of the seismic signal generated in these Vibroseis models. This provides a simple test to verify the presence and behavior of the harmonics in these signals.

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