SYSTEM IDENTIFICATION FROM VERTICAL DOWNHOLE ARRAY DATA

Abstract eng:
The paper presents an approach to identify the characteristics of layered soil media from the records of vertical downhole arrays. It is assumed that the behavior of soil is linear and it is subjected to vertically propagating shear waves. The approach is based on the discrete-time formulation of wave propagation. The waves in each layer are represented by up-going and down-going waves at the top of the layer. We develop the layer transfer matrices that relate the up-going and down-going waves in one layer to those in the next layer. We divide the soil media into fictitious layers, and assume that each layer has a thickness such that the two-way wave travel time in the layer is equal to one sampling interval of the data. With this assumption, it is shown that the relationship between the bedrock motion and the surface motion can be represented by an IIR (Infinite Impulse Response) filter. We model the filter as an AR (Auto-Regressive) process subjected to Gaussian white-noise input and convert the filter into a cascade of lattice filters defined by the reflection coefficients at layer interfaces. By using the layer transfer matrices and the reflection coefficients, and starting from the top layer, we can calculate the up-going and down-going waves at the interfaces of the fictitious layers. The motion (acceleration, velocity, or displacement) at the interface would be the sum of the up-going and down-going waves. When the calculated interface motion matches the recorded motion by one of the downhole sensors, we conclude that the layers above the matching interface represent the layers between the surface and the downhole sensor.

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