def Gyy_shifted(field, cosphi, sinphi, space_order):
"""
3D rotated second order derivative in the direction y as an average of
two non-centered rotated second order derivative in the direction y
:param field: symbolic data whose derivative we are computing
:param cosphi: cosine of the azymuth angle
:param sinphi: sine of the azymuth angle
:param space_order: discretization order
:return: rotated second order derivative wrt y
"""
x, y = field.space_dimensions[:2]
Gyp = (sinphi * field.dx - cosphi * field.dyr)
Gyy = (first_derivative(Gyp * sinphi,
dim=x, side=centered, order=space_order,
matvec=transpose) -
first_derivative(Gyp * cosphi,
dim=y, side=right, order=space_order,
matvec=transpose))
Gyp2 = (sinphi * field.dxr - cosphi * field.dy)
Gyy2 = (first_derivative(Gyp2 * sinphi,
dim=x, side=right, order=space_order,
matvec=transpose) -
first_derivative(Gyp2 * cosphi,
dim=y, side=centered, order=space_order,
matvec=transpose))
return -.5 * (Gyy + Gyy2)
def Gzz_centered_2d(model, field, costheta, sintheta, space_order):
"""
2D rotated second order derivative in the direction z.
Parameters
----------
field : Function
Input for which the derivative is computed.
costheta : Function or float
Cosine of the tilt angle.
sintheta : Function or float
Sine of the tilt angle.
space_order : int
Space discretization order.
Returns
-------
Rotated second order derivative w.r.t. z.
"""
order1 = space_order // 2
x, y = model.space_dimensions[:2]
Gz = -(sintheta * first_derivative(
field, dim=x, side=centered, fd_order=order1) + costheta *
first_derivative(field, dim=y, side=centered, fd_order=order1))
Gzz = (first_derivative(
Gz * sintheta, dim=x, side=centered, fd_order=order1, matvec=transpose
) + first_derivative(
Gz * costheta, dim=y, side=centered, fd_order=order1,
matvec=transpose))
return Gzz
def Gzz_shifted_2d(field, costheta, sintheta, space_order):
"""
2D rotated second order derivative in the direction z as an average of
two non-centered rotated second order derivative in the direction z
:param field: symbolic data whose derivative we are computing
:param costheta: cosine of the tilt
:param sintheta: sine of the tilt
:param space_order: discretization order
:return: rotated second order derivative wrt z
"""
x, y = field.space_dimensions[:2]
Gz1r = (sintheta * field.dxr + costheta * field.dy)
Gzz1 = (first_derivative(Gz1r * sintheta, dim=x,
side=right, order=space_order,
matvec=transpose) +
first_derivative(Gz1r * costheta, dim=y,
side=centered, order=space_order,
matvec=transpose))
Gz2r = (sintheta * field.dx + costheta * field.dyr)
Gzz2 = (first_derivative(Gz2r * sintheta, dim=x,
side=centered, order=space_order,
matvec=transpose) +
first_derivative(Gz2r * costheta, dim=y,
side=right, order=space_order,
matvec=transpose))
return -.5 * (Gzz1 + Gzz2)
def Gxx_shifted_2d(field, costheta, sintheta, space_order):
"""
2D rotated second order derivative in the direction x as an average of
two non-centered rotated second order derivative in the direction x
:param field: symbolic data whose derivative we are computing
:param costheta: cosine of the tilt angle
:param sintheta: sine of the tilt angle
:param space_order: discretization order
:return: rotated second order derivative wrt x
"""
x, y = field.space_dimensions[:2]
Gx1 = (costheta * field.dxr - sintheta * field.dy)
Gxx1 = (first_derivative(Gx1 * costheta, dim=x,
side=right, order=space_order,
matvec=transpose) -
first_derivative(Gx1 * sintheta, dim=y,
side=centered, order=space_order,
matvec=transpose))
Gx2p = (costheta * field.dx - sintheta * field.dyr)
Gxx2 = (first_derivative(Gx2p * costheta, dim=x,
side=centered, order=space_order,
matvec=transpose) -
first_derivative(Gx2p * sintheta, dim=y,
side=right, order=space_order,
matvec=transpose))
return -.5 * (Gxx1 + Gxx2)
def Gzz_centered_2d(field, costheta, sintheta, space_order):
"""
2D rotated second order derivative in the direction z
:param field: symbolic data whose derivative we are computing
:param costheta: cosine of the tilt angle
:param sintheta: sine of the tilt angle
:param space_order: discretization order
:return: rotated second order derivative wrt z
"""
order1 = space_order / 2
x, y = field.space_dimensions[:2]
Gz = -(sintheta * first_derivative(field, dim=x, side=centered, order=order1) +
costheta * first_derivative(field, dim=y, side=centered, order=order1))
Gzz = (first_derivative(Gz * sintheta, dim=x,
side=centered, order=order1,
matvec=transpose) +
first_derivative(Gz * costheta, dim=y,
side=centered, order=order1,
matvec=transpose))
return Gzz
def Gzz_centered_2d(field, costheta, sintheta, space_order):
"""
2D rotated second order derivative in the direction z
:param field: symbolic data whose derivative we are computing
:param costheta: cosine of the tilt angle
:param sintheta: sine of the tilt angle
:param space_order: discretization order
:return: rotated second order derivative wrt z
"""
order1 = space_order / 2
x, y = field.space_dimensions[:2]
Gz = -(sintheta * first_derivative(field, dim=x, side=centered, fd_order=order1) +
costheta * first_derivative(field, dim=y, side=centered, fd_order=order1))
Gzz = (first_derivative(Gz * sintheta, dim=x,
side=centered, fd_order=order1,
matvec=transpose) +
first_derivative(Gz * costheta, dim=y,
side=centered, fd_order=order1,
matvec=transpose))
return Gzz
def Gzz2d(field, costheta, sintheta, rho):
"""
3D rotated second order derivative in the direction z
:param field: symbolic data whose derivative we are computing
:param costheta: cosine of the tilt angle
:param sintheta: sine of the tilt angle
:param cosphi: cosine of the azymuth angle
:param sinphi: sine of the azymuth angle
:param space_order: discretization order
:return: rotated second order derivative wrt ztranspose
"""
order1 = field.space_order // 2
func = list(retrieve_functions(field))[0]
x, z = func.space_dimensions
Gz = -(sintheta * first_derivative(field, dim=x, side=centered, fd_order=order1) +
costheta * first_derivative(field, dim=z, side=centered, fd_order=order1))
Gzz = (first_derivative(Gz * sintheta * rho, dim=x, side=centered,
fd_order=order1, matvec=transpose) +
first_derivative(Gz * costheta * rho, dim=z, side=centered,
fd_order=order1, matvec=transpose))
return Gzz
def Gxx_shifted(field, costheta, sintheta, cosphi, sinphi, space_order):
"""
3D rotated second order derivative in the direction x as an average of
two non-centered rotated second order derivative in the direction x
:param field: symbolic data whose derivative we are computing
:param costheta: cosine of the tilt angle
:param sintheta: sine of the tilt angle
:param cosphi: cosine of the azymuth angle
:param sinphi: sine of the azymuth angle
:param space_order: discretization order
:return: rotated second order derivative wrt x
"""
x, y, z = field.space_dimensions
Gx1 = (costheta * cosphi * field.dx + costheta * sinphi * field.dyr -
sintheta * field.dzr)
Gxx1 = (first_derivative(Gx1 * costheta * cosphi,
dim=x,
side=centered,
fd_order=space_order,
matvec=transpose) +
first_derivative(Gx1 * costheta * sinphi,
dim=y,
side=right,
fd_order=space_order,
matvec=transpose) -
first_derivative(Gx1 * sintheta,
dim=z,
side=right,
fd_order=space_order,
matvec=transpose))
Gx2 = (costheta * cosphi * field.dxr + costheta * sinphi * field.dy -
sintheta * field.dz)
Gxx2 = (first_derivative(Gx2 * costheta * cosphi,
dim=x,
side=right,
fd_order=space_order,
matvec=transpose) +
first_derivative(Gx2 * costheta * sinphi,
dim=y,
side=centered,
fd_order=space_order,
matvec=transpose) -
first_derivative(Gx2 * sintheta,
dim=z,
side=centered,
fd_order=space_order,
matvec=transpose))
return -.5 * (Gxx1 + Gxx2)
def Gzz_shifted(field, costheta, sintheta, cosphi, sinphi, space_order):
"""
3D rotated second order derivative in the direction z as an average of
two non-centered rotated second order derivative in the direction z
:param field: symbolic data whose derivative we are computing
:param costheta: cosine of the tilt angle
:param sintheta: sine of the tilt angle
:param cosphi: cosine of the azymuth angle
:param sinphi: sine of the azymuth angle
:param space_order: discretization order
:return: rotated second order derivative wrt z
"""
x, y, z = field.space_dimensions
Gzr = (sintheta * cosphi * field.dx + sintheta * sinphi * field.dyr +
costheta * field.dzr)
Gzz = (first_derivative(Gzr * sintheta * cosphi,
dim=x, side=centered, order=space_order,
matvec=transpose) +
first_derivative(Gzr * sintheta * sinphi,
dim=y, side=right, order=space_order,
matvec=transpose) +
first_derivative(Gzr * costheta,
dim=z, side=right, order=space_order,
matvec=transpose))
Gzr2 = (sintheta * cosphi * field.dxr + sintheta * sinphi * field.dy +
costheta * field.dz)
Gzz2 = (first_derivative(Gzr2 * sintheta * cosphi,
dim=x, side=right, order=space_order,
matvec=transpose) +
first_derivative(Gzr2 * sintheta * sinphi,
dim=y, side=centered, order=space_order,
matvec=transpose) +
first_derivative(Gzr2 * costheta,
dim=z, side=centered, order=space_order,
matvec=transpose))
return -.5 * (Gzz + Gzz2)
