def kernel_staggered_2d(model, u, v, space_order, **kwargs):
"""
TTI finite difference. The equation solved is:
vx.dt = - u.dx
vz.dt = - v.dx
m * v.dt = - sqrt(1 + 2 delta) vx.dx - vz.dz + Fh
m * u.dt = - (1 + 2 epsilon) vx.dx - sqrt(1 + 2 delta) vz.dz + Fv
"""
# Forward or backward
forward = kwargs.get('forward', True)
dampl = 1 - model.damp
m, epsilon, delta = model.m, model.epsilon, model.delta
costheta, sintheta = trig_func(model)
epsilon = 1 + 2 * epsilon
delta = sqrt(1 + 2 * delta)
s = model.grid.stepping_dim.spacing
x, z = model.grid.dimensions
# Get source
qu = kwargs.get('qu', 0)
qv = kwargs.get('qv', 0)
# Staggered setup
vx, vz, _ = particle_velocity_fields(model, space_order)
if forward:
# Stencils
phdx = costheta * u.dx - sintheta * u.dy
u_vx = Eq(vx.forward, dampl * vx - dampl * s * phdx)
pvdz = sintheta * v.dx + costheta * v.dy
u_vz = Eq(vz.forward, dampl * vz - dampl * s * pvdz)
dvx = costheta * vx.forward.dx - sintheta * vx.forward.dy
dvz = sintheta * vz.forward.dx + costheta * vz.forward.dy
# u and v equations
pv_eq = Eq(v.forward, dampl * (v - s / m * (delta * dvx + dvz)) + s / m * qv)
ph_eq = Eq(u.forward, dampl * (u - s / m * (epsilon * dvx + delta * dvz)) +
s / m * qu)
else:
# Stencils
phdx = ((costheta*epsilon*u).dx - (sintheta*epsilon*u).dy +
(costheta*delta*v).dx - (sintheta*delta*v).dy)
u_vx = Eq(vx.backward, dampl * vx + dampl * s * phdx)
pvdz = ((sintheta*delta*u).dx + (costheta*delta*u).dy +
(sintheta*v).dx + (costheta*v).dy)
u_vz = Eq(vz.backward, dampl * vz + dampl * s * pvdz)
dvx = (costheta * vx.backward).dx - (sintheta * vx.backward).dy
dvz = (sintheta * vz.backward).dx + (costheta * vz.backward).dy
# u and v equations
pv_eq = Eq(v.backward, dampl * (v + s / m * dvz))
ph_eq = Eq(u.backward, dampl * (u + s / m * dvx))
return [u_vx, u_vz] + [pv_eq, ph_eq]
def kernel_centered_3d(model, u, v, space_order, **kwargs):
"""
TTI finite difference kernel. The equation solved is:
u.dt2 = H0
v.dt2 = Hz
where H0 and Hz are defined as:
H0 = (1+2 *epsilon) (Gxx(u)+Gyy(u)) + sqrt(1+ 2*delta) Gzz(v)
Hz = sqrt(1+ 2*delta) (Gxx(u)+Gyy(u)) + Gzz(v)
and
H0 = (Gxx+Gyy)((1+2 *epsilon)*u + sqrt(1+ 2*delta)*v)
Hz = Gzz(sqrt(1+ 2*delta)*u + v)
for the forward and adjoint cases, respectively. Epsilon and delta are the Thomsen
parameters. This function computes H0 and Hz.
References:
* Zhang, Yu, Houzhu Zhang, and Guanquan Zhang. "A stable TTI reverse
time migration and its implementation." Geophysics 76.3 (2011): WA3-WA11.
* Louboutin, Mathias, Philipp Witte, and Felix J. Herrmann. "Effects of
wrong adjoints for RTM in TTI media." SEG Technical Program Expanded
Abstracts 2018. Society of Exploration Geophysicists, 2018. 331-335.
Parameters
----------
u : TimeFunction
First TTI field.
v : TimeFunction
Second TTI field.
space_order : int
Space discretization order.
Returns
-------
u and v component of the rotated Laplacian in 3D.
"""
# Forward or backward
forward = kwargs.get('forward', True)
costheta, sintheta, cosphi, sinphi = trig_func(model)
delta, epsilon = model.delta, model.epsilon
epsilon = 1 + 2*epsilon
delta = sqrt(1 + 2*delta)
# Get source
qu = kwargs.get('qu', 0)
qv = kwargs.get('qv', 0)
if forward:
Gxx = Gxxyy_centered(model, u, costheta, sintheta, cosphi, sinphi, space_order)
Gzz = Gzz_centered(model, v, costheta, sintheta, cosphi, sinphi, space_order)
H0 = epsilon*Gxx + delta*Gzz
Hz = delta*Gxx + Gzz
return second_order_stencil(model, u, v, H0, Hz, qu, qv)
else:
H0 = Gxxyy_centered(model, (epsilon*u + delta*v), costheta, sintheta,
cosphi, sinphi, space_order)
Hz = Gzz_centered(model, (delta*u + v), costheta, sintheta, cosphi,
sinphi, space_order)
return second_order_stencil(model, u, v, H0, Hz, qu, qv, forward=forward)