Abstract
The availability of broadband digitally recorded seismic data has led to an increasing number of studies using data from which the instrument transfer function has been deconvolved. In most studies, it is assumed that raw ground motion is the quantity that remains after deconvolution. After deconvolving the instrument transfer function, however, seismograms are usually high-pass filtered to remove low-frequency noise caused by very long-period signals outside the frequency band of interest or instabilities in the instrument response at low frequencies. In some cases, data must also be low-pass filtered to remove high-frequency noise from various sources. Both of these operations are usually performed using either zero-phase (acausal) or minimum-phase (causal) filters. Use of these filters can lead to either bias or increased uncertainty in the results, especially when taking integral measures of the displacement pulse. We present a deconvolution method, based on Backus-Gilbert inverse theory, that regularizes the time-domain deconvolution problem and thus mitigates any low-frequency instabilities. We apply a roughening constraint that minimizes the long-period components of the deconvolved signal along with the misfit to the data, emphasizing the higher frequencies at the expense of low frequencies. Thus, the operator acts like a high-pass filter but is controlled by a trade-off parameter that depends on the ratio of the model variance to the residual variance, rather than an ad hoc selection of a filter corner frequency. The resulting deconvolved signal retains a higher fidelity to the original ground motion than that obtained using a postprocess high-pass filter and eliminates much of the bias introduced by such a filter. A smoothing operator can also be introduced that effectively applies a low-pass filter. This smoothing is useful in the presence of blue noise, or if inferences about source complexity are to be made from the roughness of the deconvolved signal.
A ℘-order R-L high-pass filter modeled by local fractional derivative
