Reciprocity Mode

This mode excites the seismic wave at a specified station location and exports the velocity and/or strain velocity of multiple virtual source locations. Based on the reciprocity theorem, this result corresponds to the body force and/or moment tensor response from virtual source locations observed at specified stations. If the time duration of the source time function is sufficiently short, they can be treated as Green's functions.

If we denote the Green's tensor, from the virtual source $\mathbf{\xi}$ to the receiver $\mathbf{r}$, as $G_{ij}\left( \mathbf{r}, t; \mathbf{\xi}\right)$, this mode simulates the convolution of the spatial derivatives of Green's tensor with the source time function $s(t)$ as

$$\begin{align} \begin{split} G^{M1}_{i}\left( \mathbf{r}, t; \mathbf{\xi}\right) &\equiv \frac{\partial G_{ix} \left( \mathbf{r}, t; \mathbf{\xi}\right) }{\partial \xi_x} \ast s(t) = \frac{\partial G_{ix} \left( \mathbf{\xi}, t; \mathbf{r}\right) }{\partial \xi_x} \ast s(t) \\ G^{M2}_{i}\left( \mathbf{r}, t; \mathbf{\xi}\right) &\equiv \frac{\partial G_{iy} \left( \mathbf{r}, t; \mathbf{\xi}\right) }{\partial \xi_y} \ast s(t) = \frac{\partial G_{iy} \left( \mathbf{\xi}, t; \mathbf{r}\right) }{\partial \xi_y} \ast s(t) \\ G^{M3}_{i}\left( \mathbf{r}, t; \mathbf{\xi}\right) &\equiv \frac{\partial G_{iz} \left( \mathbf{r}, t; \mathbf{\xi}\right) }{\partial \xi_z} \ast s(t) = \frac{\partial G_{iz} \left( \mathbf{\xi}, t; \mathbf{r}\right) }{\partial \xi_z} \ast s(t) \\ G^{M4}_{i}\left( \mathbf{r}, t; \mathbf{\xi}\right) &\equiv \left( \frac{\partial G_{iy} \left( \mathbf{r}, t; \mathbf{\xi}\right) }{\partial \xi_z} + \frac{\partial G_{iz} \left( \mathbf{r}, t; \mathbf{\xi}\right) }{\partial \xi_y} \right) \ast s(t) \\ &= \left( \frac{\partial G_{iy} \left( \mathbf{\xi}, t; \mathbf{r}\right) }{\partial \xi_z} + \frac{\partial G_{iz} \left( \mathbf{\xi}, t; \mathbf{r}\right) }{\partial \xi_y} \right) \ast s(t) \\ G^{M5}_{i}\left( \mathbf{r}, t; \mathbf{\xi}\right) &\equiv \left( \frac{\partial G_{ix} \left( \mathbf{r}, t; \mathbf{\xi}\right) }{\partial \xi_z} + \frac{\partial G_{iz} \left( \mathbf{r}, t; \mathbf{\xi}\right) }{\partial \xi_x}\right) \ast s(t) \\& = \left( \frac{\partial G_{ix} \left( \mathbf{\xi}, t; \mathbf{r}\right) }{\partial \xi_z} + \frac{\partial G_{iz} \left( \mathbf{\xi}, t; \mathbf{r}\right) }{\partial \xi_x}\right) \ast s(t) \\ G^{M6}_{i} \left( \mathbf{r}, t; \mathbf{\xi}\right) &\equiv \left( \frac{\partial G_{ix} \left( \mathbf{r}, t; \mathbf{\xi}\right) }{\partial \xi_y} + \frac{\partial G_{iy} \left( \mathbf{\xi}, t; \mathbf{r}\right) }{\partial \xi_x} \right) \ast s(t) \\ &= \left( \frac{\partial G_{ix} \left( \mathbf{r}, t; \mathbf{\xi}\right) }{\partial \xi_y} + \frac{\partial G_{iy} \left( \mathbf{\xi}, t; \mathbf{r}\right) }{\partial \xi_x} \right) \ast s(t) \end{split} \end{align}$$

which corresponds to the moment tensor response. Optionally, the body-force response

$$\begin{align} \begin{split} G^{B1}_{i}\left( \mathbf{r}, t; \mathbf{\xi}\right) &\equiv G_{ix} \left( \mathbf{r}, t; \mathbf{\xi}\right) \ast s(t) = G_{ix} \left( \mathbf{\xi}, t; \mathbf{r}\right) \ast s(t) \\ G^{B2}_{i}\left( \mathbf{r}, t; \mathbf{\xi}\right) &\equiv G_{iy} \left( \mathbf{r}, t; \mathbf{\xi}\right) \ast s(t) = G_{iy} \left( \mathbf{\xi}, t; \mathbf{r}\right) \ast s(t) \\ G^{B3}_{i}\left( \mathbf{r}, t; \mathbf{\xi}\right) &\equiv G_{iz} \left( \mathbf{r}, t; \mathbf{\xi}\right)\ast s(t) = G_{iz} \left( \mathbf{\xi}, t; \mathbf{r}\right)\ast s(t) \end{split} \end{align}$$

can be calculated.

To use this mode, the users should specify the station name green_stnm of the receiver. This station name should be contained in the station list file. OpenSWPC radiates the seismic wave by an excitation force with a direction specified by the green_cmp parameter and a source time function of the rise time, green_trise. To obtain the full response of all components, three independent simulations with green_cmp=’x’, ’y’, and ’z’ are necessary.

The virtual source location should be given in the Cartesian or geographical coordinates and depth (the format is described in Table [table:greenf]{reference-type="ref" reference="table:greenf"}) with unique integer ID numbers (gid). Multiple virtual source locations can be specified in the simulation. The gids do not need to be sequential.

The output file is stored in the directory (odir)/green/(gid) in the SAC format with the name convention (title)__(green_cmp)__mij__.sac (for the moment tensor response) or (title)__(green_cmp)__fi__.sac (for the body force response).

The amplitudes of the output files are multiplied by $10^9$ to compare the SAC-formatted files in nm or nm/s units. The vertical component of the output file is changed to be positive upward. However, the derivative with respect to depth is performed according to the original definition of positive downward.

Parameters

green_mode :Flags to turn the reciprocity mode on. If this is .true., the other earthquake source parameters will be ignored.

green_stnm

Name of the virtual station. This name must be included in the station list.

green_cmp

Component at the virtual receiver. Choose from ’x’, ’y’, or ’z’.

green_trise

Rise time of the source time function convolved with the simulated Green's function.

green_bforce

If .true., calculate the body force response as well as the moment tensor response. The default setting is .false..

green_fmt

Format specification of the virtual source location. Choose from ’xyz’ (Cartesian coordinate; default) or ’llz’ (longitude, latitude, and depth).

green_maxdist

The reciprocity wave will only be calculated if the horizontal distance is shorter than this parameter. Specify in units of km.

fn_glst

Name of the virtual source location file.

stftype

Source time function type. Same as the case with the moment tensor source.

ntdec_w

Temporal decimation factor of the output waveforms. Same as the case with the normal waveform output.

The virtual source location file takes the following format according to the settings of green_fmt:

green_fmt	format
’xyz’	x	y	z	gid
’llz’	lon	lat	z	gid