This is a first attempt at implemenenting the elastic wave equation as described in:
[1] Jean Virieux (1986). ”P-SV wave propagation in heterogeneous media: Velocity‐stress finite‐difference method.” GEOPHYSICS, 51(4), 889-901. https://doi.org/10.1190/1.1442147
The current version actually attempts to mirror the FDELMODC implementation by Jan Thorbecke:
[2] https://janth.home.xs4all.nl/Software/fdelmodcManual.pdf
We will first attempt to replicate the explosive source test case described in [1], Figure 4. We start by defining the source signature $g(t)$, the derivative of a Gaussian pulse, given by Eq 4:
$$g(t) = -2 \alpha(t - t_0)e^{-\alpha(t-t_0)^2}$$from devito import *
from examples.seismic.source import WaveletSource, RickerSource, GaborSource, TimeAxis
from examples.seismic import plot_image
import numpy as np
from sympy import init_printing, latex
init_printing(use_latex=True)
# Initial grid: 1km x 1km, with spacing 100m
extent = (2000., 2000.)
shape = (201, 201)
x = SpaceDimension(name='x', spacing=Constant(name='h_x', value=extent[0]/(shape[0]-1)))
z = SpaceDimension(name='z', spacing=Constant(name='h_z', value=extent[1]/(shape[1]-1)))
grid = Grid(extent=extent, shape=shape, dimensions=(x, z))
class DGaussSource(WaveletSource):
def wavelet(self, f0, t):
a = 0.004
return -2.*a*(t - 1/f0) * np.exp(-a * (t - 1/f0)**2)
# Timestep size from Eq. 7 with V_p=6000. and dx=100
t0, tn = 0., 600.
dt = (10. / np.sqrt(2.)) / 6.
time_range = TimeAxis(start=t0, stop=tn, step=dt)
src = RickerSource(name='src', grid=grid, f0=0.01, time_range=time_range)
src.coordinates.data[:] = [1000., 1000.]
#NBVAL_SKIP
src.show()
# Now we create the velocity and pressure fields
so = 2
vx= TimeFunction(name='vx', grid=grid, staggered=x, space_order=so)
vz = TimeFunction(name='vz', grid=grid, staggered=z, space_order=so)
txx = TimeFunction(name='txx', grid=grid, staggered=NODE, space_order=so)
tzz = TimeFunction(name='tzz', grid=grid, staggered=NODE, space_order=so)
txz = TimeFunction(name='txz', grid=grid, staggered=(x, z), space_order=so)
# Now let's try and create the staggered updates
t = grid.stepping_dim
time = grid.time_dim
# We need some initial conditions
V_p = 4.0
V_s = 1.0
density = 3.
# The source injection term
src_xx = src.inject(field=txx.forward, expr=src)
src_zz = src.inject(field=tzz.forward, expr=src)
#c1 = 9.0/8.0;
#c2 = -1.0/24.0;
# Thorbecke's parameter notation
cp2 = V_p*V_p
cs2 = V_s*V_s
ro = 1/density
mu = cs2*ro
l = (cp2*ro - 2*mu)
# fdelmodc reference implementation
u_vx = Eq(vx.forward, vx - dt*ro*(txx.dx + txz.dz))
u_vz = Eq(vz.forward, vz - ro*dt*(txz.dx + tzz.dz))
u_txx = Eq(txx.forward, txx - (l+2*mu)*dt * vx.forward.dx - l*dt * vz.forward.dz)
u_tzz = Eq(tzz.forward, tzz - (l+2*mu)*dt * vz.forward.dz - l*dt * vx.forward.dx)
u_txz = Eq(txz.forward, txz - mu*dt * (vx.forward.dz + vz.forward.dx))
op = Operator([u_vx, u_vz, u_txx, u_tzz, u_txz] + src_xx + src_zz)
# Reset the fields
vx.data[:] = 0.
vz.data[:] = 0.
txx.data[:] = 0.
tzz.data[:] = 0.
txz.data[:] = 0.
#NBVAL_IGNORE_OUTPUT
op()
Operator `Kernel` run in 0.43 s
#NBVAL_SKIP
# Let's see what we got....
plot_image(vx.data[0], vmin=-.5*1e-2, vmax=.5*1e-2, cmap="seismic")
plot_image(vz.data[0], vmin=-.5*1e-2, vmax=.5*1e-2, cmap="seismic")
plot_image(txx.data[0], vmin=-.5*1e-2, vmax=.5*1e-2, cmap="seismic")
plot_image(tzz.data[0], vmin=-.5*1e-2, vmax=.5*1e-2, cmap="seismic")
plot_image(txz.data[0], vmin=-.5*1e-2, vmax=.5*1e-2, cmap="seismic")
# Now that looks pretty! But let's do it again with a higher order...
so = 12
vx= TimeFunction(name='vx', grid=grid, staggered=x, space_order=so)
vz = TimeFunction(name='vz', grid=grid, staggered=z, space_order=so)
txx = TimeFunction(name='txx', grid=grid, staggered=NODE, space_order=so)
tzz = TimeFunction(name='tzz', grid=grid, staggered=NODE, space_order=so)
txz = TimeFunction(name='txz', grid=grid, staggered=(x, z), space_order=so)
# fdelmodc reference implementation
u_vx = Eq(vx.forward, vx - dt*ro*(txx.dx + txz.dz))
u_vz = Eq(vz.forward, vz - ro*dt*(txz.dx + tzz.dz))
u_txx = Eq(txx.forward, txx - (l+2*mu)*dt * vx.forward.dx - l*dt * vz.forward.dz)
u_tzz = Eq(tzz.forward, tzz - (l+2*mu)*dt * vz.forward.dz - l*dt * vx.forward.dx)
u_txz = Eq(txz.forward, txz - mu*dt * (vx.forward.dz + vz.forward.dx))
op = Operator([u_vx, u_vz, u_txx, u_tzz, u_txz] + src_xx + src_zz)
# Reset the fields
vx.data[:] = 0.
vz.data[:] = 0.
txx.data[:] = 0.
tzz.data[:] = 0.
txz.data[:] = 0.
#NBVAL_IGNORE_OUTPUT
op()
Operator `Kernel` run in 2.60 s
#NBVAL_SKIP
plot_image(vx.data[0], vmin=-.5*1e-2, vmax=.5*1e-2, cmap="seismic")
plot_image(vz.data[0], vmin=-.5*1e-2, vmax=.5*1e-2, cmap="seismic")
plot_image(txx.data[0], vmin=-.5*1e-2, vmax=.5*1e-2, cmap="seismic")
plot_image(tzz.data[0], vmin=-.5*1e-2, vmax=.5*1e-2, cmap="seismic")
plot_image(txz.data[0], vmin=-.5*1e-2, vmax=.5*1e-2, cmap="seismic")