Forward Computation
Radial and vertical components of synthetic seismograms are calculated using the Propagator Matrix method of Thomson and Haskell. In the case of offshore setting, we adopt a boundary condition that requires zero traction on the seafloor. Also, we use equations of acoustic wave propagation rather than imitating the acoustic waves by setting Vs ~ 0 on the equations of elastic waves.
We have two options for deconvolution:
Radial component records are deconvolved by the corresponding vertical components using water-level deconvolution. The resulting receiver functions are compared with observed waveforms to measure misfits.
No deconvolution is carried out for this mode. Synthetic radial component seismograms are directly compared with observed waveforms. You may choose this option if you have Green's function estimation (e.g., Kumar et al. 2010; Akuhara et al. 2019).
A Gaussian low-pass filter is applied to the deconvolved seismograms:
where a is a tuning parameter.
The synthetic receiver functions are normalized such that the amplitude of the direct P arrival on the vertical component of becomes unity. The normalization constant, c, is as follows:
where a is the parameter for Gaussian low-pass filter, N is the number of element in the time series, and \Delta t is the sampling interval.