^{1}

^{*}

^{1}

^{1}

^{1}

^{1}

A software (EQK_SRC_PARA) has been developed to estimate spectral parameters of earthquake source spectrum, namely: low frequency displacement spectral level (Ω
_{0}), corner frequency above which spectrum decays with a rate of 2 (
f_{c}), the cut-off frequency above which the spectrum again decays (
f_{max}) and the rate of decay above
f_{max} (N). A Brune’s source model [1,2] that yield a fall-off of 2 beyond corner frequency is considered with high cut-off frequency factor presented by Boore [3] that fits well for frequencies greater than
f_{max}. The software EQK_SRC_PARA is written in MATLAB and uses input data in Sesame ASCII Format (SAF) format. The obtained spectral parameters have been used to estimate source parameters (e.g., seismic moment, source dimension and stress drop etc.) and to develop scaling laws for the study region. The cut-off frequency “
f_{max}” can also be studied and interpreted to confirm about its origin.

The seismic design of engineered structures depends upon a quantitative estimate of the characteristics of strong ground motion at desired site. In case recorded data are not available at that site, then simulation of ground motion is the only way for engineers to rely upon. The stochastic approach is the most popular way, which requires the knowledge of spectral and source parameters (e.g., corner frequency and stress drop) of study region and has very good results recently in both regions with and without recordings. The shape of the seismic spectrum and how it scales with earthquake size has been a topic of importance as a way of gaining insight into the character of earthquake source processes and a guide to the simulation of strong ground motion for engineering purposes [^{2} model. Satisfactory agreement has been found with observations based on the assumption of similarity. A constant stress drop has been considered, however, it is pointed out that if the stress drop differs, the scaling law will not apply and if the stress drop varies systematically with respect to environmental factors as focal depth, orientation of the fault plane and crust-mantle structure, different scaling laws can be constructed for different environments.

Brune [1,2] modeled an earthquake source as a tangential stress pulse applied instantaneously to the interior of a dislocation surface. This model employs three independent parameters (moment, source dimension and fractional stress drop) those determine the shape of the farfield displacement spectrum of body waves. The relationship of the corner frequency to the fault radius has been constrained by assuming that the effective stress was equal to the average static stress drop. This model has been extensively used to estimate source parameters from the observational data of seismic waves [6-12].

In these models, the displacement source spectrum has a simple ω^{2} shape, i.e., it has a flat level for frequencies below a source-corner frequency (f_{c}) and a decay of ω^{−2} for frequencies greater than source-corner frequency. Similarly, the acceleration the acceleration spectrum has ω^{2} shape for frequencies below a source-corner frequency (f_{c}) and a flat level for frequencies greater than source-corner frequency. Hanks [_{max}, above which acceleration spectral amplitudes diminish abruptly. This cut-off frequency, f_{max}, is an important parameter from earthquake engineering point as it controls the peak ground acceleration. There is a controversy about its origin. Hanks [_{max} is a recording site effect. However, most of the studies attribute this due to source [15-24].

A software has been developed employing Brune’s source model [1,2] and a high-cut filter presented by Boore [_{0}), corner frequency above which spectrum decays with a rate of 2 (f_{c}), the frequency above which the spectrum again decays (f_{max}) and the rate of decay above f_{max} (N). These spectral parameters are used to estimate source parameters and to develop scaling laws. The source parameters of an earthquake of 21st September 2009 (Mw 4.7) that occurred near Uttarkashi have been estimated by this software as an example.

The time histories are first corrected for instrument response and then rotated to obtained SH-component of ground motion. The SH-spectrum is corrected for attenuation due to path. In this study Brune’s source model [1, 2] that yield a fall-off of 2 beyond corner frequency is considered with high frequency dimunition factor, a Butterworth high-cut filter presented by Boore [_{max} is fitted in observed acceleration spectrum as

And for displacement spectrum

The software automatically picks the spectral parameters:

1) low frequency displacement spectral level, Ω_{0}2) corner frequency above which spectrum decays with a rate of 2, f_{c}3) the frequency above which the spectrum again decays, f_{max} and 4) the rate of decay above f_{max}, N.

A brief explanation to the technique has been presented here with an example data of Srikot (SRIK) station for 21st September 2009 earthquake (Mw = 4.7) that occurred near Uttarkashi. The time histories are first corrected for instrument response using transfer function estimated from zero-poles values and then rotated about azimuth to obtained SH-component of ground motion. A typical example of selected SH-component of time history obtained after applying instrument response and rotation about azimuth is shown in

The Fast Fourier Transform (FFT) of selected SHcomponent of time history is performed to obtain spectrum of SH-component. A frequency dependent attenua-

tion correction, Q_{c} = 110f^{1.02} [_{c} is obtained from velocity spectrum; it is a value of frequency where the velocity spectrum has a peak. This is shown in log-log plot in

Then a value of f_{max} is estimated from snap (double differential of acceleration, snap (f) = ω^{2} A (f)), it is approximately a value of frequency where spectral snap has a peak. This is shown in

Glassmoyer and Borcherdt [_{c}” is to the constant spectral levels of displacement “” and acceleration “” as given below:

Glassmoyer and Borcherdt [_{c}. However in this study, “” shown in _{c} and f_{max}. This leads to the approximation of “” shown in

The source model for acceleration spectrum given by Equation (1) is fitted to observed spectrum with different values of f_{c} between f_{1} and f_{max}, and for f_{max} between f_{c} and f_{Ny} (Nyquest frequency) and its value is obtained from observed and modeled spectra based on root mean square error (rmse). Plot in _{c} between f_{1} and f_{max} and the difference in observed and modeled spectra, the value for f_{c} is considered where the root mean square error (rmse) is minimum. Similary _{max} between f_{c} and f_{Nq} and the difference in observed and modeled spectra; the value for f_{max} is considered where the root mean square error (rmse) is least.

Then the corrected value of Ω_{0} is obtained from relation (4). Finally, values of N between 2 to 10 are given and

the value of N having least difference between the considered model and observed spectrum is obtained (

A flowchart of the procedure is given in

The seismic moment is estimated from the value of Ω_{0} following Kellis-Borok [

Here is the average density (=2.67 g/cm^{3}), is shear wave velocity in the source zone (=3.2 km/s), is the hypocentral distance, is the average radiation pattern (=0.63), is free surface amplification (=2).

The moment magnitude is obtained following Hanks and Kanamori [

Following Brune [1,2] the source radius and stress drop can be estimated as:

The data collected from two networks in the Garhwal Himalaya (

ponent used for analysis from seismogram (Vinakhal) and strong motion instrument (Dhanolti). The acceleration and displacement spectra along with the fitted model are also shown below in respective figures.

The spectral parameters obtained from the velocity and acceleration records at various sites are given along with

estimated source parameters in Appendix 1.

The seismic moment for this event has been found to be of the order of (107 ± 0.19) × 10^{23} dyne.cm and the moment magnitude has been calculated 4.7 ± 0.09 at different stations. The stress drop is found to be 76.3 ± 11.5 bars, while source radius for the earthquake is estimated to be (850.0 ± 38.0) m. The value of f_{max} for this earthquake is 9.1 ± 1.7 Hz obtained from records at various stations of different site conditions. A change in spectral fall-off above f_{max} has been observed in short period instruments while strong motion instruments has same value. This may be due to different band-width of recording instruments.

The software EQK_SRC_PARA has been written in MATLAB and is based on Brune’s source model (1970) and high cut-off frequency factor of Boore (1983). It uses input data in Sesame ASCII Format (SAF) format. The software automatically picks the spectral parameters: low frequency displacement spectral level (Ω_{0}), corner frequency above which spectrum decays with a rate of 2 (f_{c}), the cut-off frequency above which the spectrum again decays (f_{max}) and the rate of decay above f_{max} (N). The obtained spectral parameters have been used to estimate source parameters (e.g., seismic moment, source dimension and stress drop etc.) and to develop scaling laws. The cut-off frequency “f_{max}” can also be studied and interpreted to find clues about its origin.

The estimated stress drop for this event is 76.3 ± 11.5 bars that is higher than the stress drop of 52.6 ± 5.9 bars for Uttarkashi earthquake (Mw 6.7) of 1991 and an average of about 60 bars for Garhwal-Kumaon Himalayan earthquakes (Kumar, 2011). The difference in focal depth may be the cause of higher stress drop in this earthquake.

The author (Arjun Kumar) is profusely thankful to Ministry of Human Resources Development (MHRD) for providing fellowship. The authors are also thankful to Ministry of Earth Sciences (MoES) and Tehri Hydropower Development Corporation (THDC) for funding projects under which data was collected.