Can anybody tell me the theory and methodology for how source time function (moment tensor rate functions) is arrived at for an earthquake?

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Arjun Kumar

This can be estimated by moment tensor inversion of recorded earthquake waveform.

You can get these values from CMT Harvard site. They report these for earthquakes with magnitudes >4.0 occurred anywhere over the globe.

Lei yi

this could be a rupture inversion of this earthquake

Sreekala R.

Other than Harward CMT, there are other methods like USGS MT method and ERI CMT methods.

Your article on slip rate evaluation is very interesting.