Keywords: earthquake, source, free vibration, mean square error, impulse, excitation.

Earthquake source determination is a classical problem in seismology. It is common practice to determine the focus, epicentre and related parameters from wave propagation principles, with the help of teleseismic records. Engineers on the other hand, are more familiar with finite medium and concepts of vibration. Thus, it would be interesting to ask whether strong motion array (SMA) data can be processed as arising out of a structural dynamics problem. The SMA data i.e. acceleration and displacement time histories can be treated as response of intervening medium due to the earthquake forces of unknown magnitude acting at unknown places and an inverse structural dynamics problem can be solved to estimate the source using the responses. Before solving the actual earthquake source determination problem using SMA data, a new numerical scheme for 2D finite elastic media where the source is determined making use of known impulses will be desirable and presented in the paper. The methodology proposes a time marching scheme using normal mode approach for source estimation by minimizing the mean square error.

Let a 2D layered elastic medium be subjected to a sequence of impulses , applied at known instants and at locations . The responses and are measured at a few points in the time interval in and directions respectively. In order to estimate the source, it is required to find ; ; ... and the corresponding values of impulses ; ; ... . The eigen functions , in and directions respectively for the 2D medium are taken to be known.

The displacement at any point can be readily written as

is impulse response function.

There are four unknowns namely and in any time interval . Since the left-hand sides of equations (51.1) and (51.2) being measured response are known for all time , the m.m.s.e. criterion can be adopted for finding the unknowns. The above representation could be used for recursive estimation of and in terms of previous values.

The work presented will be mainly through numerical experimentation. Well-defined 2-D elastic medium is subjected to known forces at known locations. The responses at a few locations are measured (computed) through finite element codes and also response in turn used again along with eigenvalue solution to demonstrate the application of the proposed source identification method. The principle of minimization of the mean square error leads to a robust approach for estimating the location and magnitude of the buried point sources in finite elastic media, for which surface displacements are available. The buried point sources in finite elastic medium can be identified accurately using known surface displacements and free vibration results of the medium. It is seen that earthquake source estimation problem is similar to the 2-D problem solved here. The proposed normal mode approach and principle of minimization of mean square error can be used for the estimation of epicentre of strong earthquakes [1].

1

S.K. Agrawal, "Strong Motion Data as Response of Layered Elastic Medium", Ph. D. Thesis, Department of Earthquake Engineering, University of Roorkee, 2000.

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