# https://pyrocko.org - GPLv3
#
# The Pyrocko Developers, 21st Century
# ---|P------/S----------~Lg----------
'''
Elastostatic solutions and boundary element modelling for rectangular
dislocation sources.
'''
import numpy as num
import logging
from pyrocko import moment_tensor as mt
from pyrocko.guts import Float, String, Timestamp, Int
from pyrocko.model import Location
from pyrocko.modelling import okada_ext
from pyrocko.util import get_threadpool_limits
guts_prefix = 'modelling'
logger = logging.getLogger(__name__)
d2r = num.pi/180.
r2d = 180./num.pi
km = 1e3
[docs]class AnalyticalSource(Location):
'''
Base class for analytical source models.
'''
name = String.T(
optional=True,
default='')
time = Timestamp.T(
default=0.,
help='Source origin time',
optional=True)
vr = Float.T(
default=0.,
help='Rupture velocity [m/s]',
optional=True)
@property
def northing(self):
return self.north_shift
@property
def easting(self):
return self.east_shift
[docs]class AnalyticalRectangularSource(AnalyticalSource):
'''
Rectangular analytical source model.
Coordinates on the source plane are with respect to the origin point given
by `(lat, lon, east_shift, north_shift, depth)`.
'''
strike = Float.T(
default=0.0,
help='Strike direction in [deg], measured clockwise from north.')
dip = Float.T(
default=90.0,
help='Dip angle in [deg], measured downward from horizontal.')
rake = Float.T(
default=0.0,
help='Rake angle in [deg], measured counter-clockwise from '
'right-horizontal in on-plane view.')
al1 = Float.T(
default=0.,
help='Left edge source plane coordinate [m].')
al2 = Float.T(
default=0.,
help='Right edge source plane coordinate [m].')
aw1 = Float.T(
default=0.,
help='Lower edge source plane coordinate [m].')
aw2 = Float.T(
default=0.,
help='Upper edge source plane coordinate [m].')
slip = Float.T(
default=0.,
help='Slip on the rectangular source area [m].',
optional=True)
@property
def length(self):
return abs(-self.al1 + self.al2)
@property
def width(self):
return abs(-self.aw1 + self.aw2)
@property
def area(self):
return self.width * self.length
[docs]class OkadaSource(AnalyticalRectangularSource):
'''
Rectangular Okada source model.
'''
opening = Float.T(
default=0.,
help='Opening of the plane in [m].',
optional=True)
poisson__ = Float.T(
default=0.25,
help='Poisson ratio :math:`\\nu`. '
'The Poisson ratio :math:`\\nu`. If set to ``None``, calculated '
"from the Lame' parameters :math:`\\lambda` and :math:`\\mu` "
'using :math:`\\nu = \\frac{\\lambda}{2(\\lambda + \\mu)}` (e.g. '
'Mueller 2007).',
optional=True)
lamb__ = Float.T(
help='First Lame parameter :math:`\\lambda` [Pa]. '
'If set to ``None``, it is computed from Poisson ratio '
':math:`\\nu` and shear modulus :math:`\\mu`. **Important:** We '
'assume a perfect elastic solid with :math:`K=\\frac{5}{3}\\mu`. '
'Through :math:`\\nu = \\frac{\\lambda}{2(\\lambda + \\mu)}` '
'this leads to :math:`\\lambda = \\frac{2 \\mu \\nu}{1-2\\nu}`.',
optional=True)
shearmod__ = Float.T(
default=32.0e9,
help='Shear modulus :math:`\\mu` [Pa]. '
'If set to ``None``, it is computed from poisson ratio. '
'**Important:** We assume a perfect elastic solid with '
':math:`K=\\frac{5}{3}\\mu`. Through '
':math:`\\mu = \\frac{3K(1-2\\nu)}{2(1+\\nu)}` this leads to '
':math:`\\mu = \\frac{8(1+\\nu)}{1-2\\nu}`.',
optional=True)
@property
def poisson(self):
if self.poisson__ is not None:
return self.poisson__
if self.shearmod__ is None or self.lamb__ is None:
raise ValueError('Shearmod and lambda are needed')
return (self.lamb__) / (2. * (self.lamb__ + self.shearmod__))
@poisson.setter
def poisson(self, poisson):
self.poisson__ = poisson
@property
def lamb(self):
if self.lamb__ is not None:
return self.lamb__
if self.shearmod__ is None or self.poisson__ is None:
raise ValueError('Shearmod and poisson ratio are needed')
return (
2. * self.poisson__ * self.shearmod__) / (1. - 2. * self.poisson__)
@lamb.setter
def lamb(self, lamb):
self.lamb__ = lamb
@property
def shearmod(self):
if self.shearmod__ is not None:
return self.shearmod__
if self.poisson__ is None:
raise ValueError('Poisson ratio is needed')
return (8. * (1. + self.poisson__)) / (1. - 2. * self.poisson__)
@shearmod.setter
def shearmod(self, shearmod):
self.shearmod__ = shearmod
@property
def seismic_moment(self):
'''
Scalar Seismic moment :math:`M_0`.
Code copied from Kite. It disregards the opening (as for now).
We assume :math:`M_0 = mu A D`.
.. important ::
We assume a perfect elastic solid with :math:`K=\\frac{5}{3}\\mu`.
Through :math:`\\mu = \\frac{3K(1-2\\nu)}{2(1+\\nu)}` this leads to
:math:`\\mu = \\frac{8(1+\\nu)}{1-2\\nu}`.
:return:
Seismic moment release.
:rtype:
float
'''
mu = self.shearmod
disl = 0.
if self.slip:
disl = self.slip
if self.opening:
disl = (disl**2 + self.opening**2)**.5
return mu * self.area * disl
@property
def moment_magnitude(self):
'''
Moment magnitude :math:`M_\\mathrm{w}` from seismic moment.
We assume :math:`M_\\mathrm{w} = {\\frac{2}{3}}\\log_{10}(M_0) - 10.7`.
:returns:
Moment magnitude.
:rtype:
float
'''
return mt.moment_to_magnitude(self.seismic_moment)
[docs] def source_patch(self):
'''
Get source location and geometry array for okada_ext.okada input.
The values are defined according to Okada (1992).
:return:
Source data as input for okada_ext.okada. The order is
northing [m], easting [m], depth [m], strike [deg], dip [deg],
al1 [m], al2 [m], aw1 [m], aw2 [m].
:rtype:
:py:class:`~numpy.ndarray`: ``(9, )``
'''
return num.array([
self.northing,
self.easting,
self.depth,
self.strike,
self.dip,
self.al1,
self.al2,
self.aw1,
self.aw2])
[docs] def source_disloc(self):
'''
Get source dislocation array for okada_ext.okada input.
The given slip is splitted into a strike and an updip part based on the
source rake.
:return:
Source dislocation data as input for okada_ext.okada. The order is
dislocation in strike [m], dislocation updip [m], opening [m].
:rtype:
:py:class:`~numpy.ndarray`: ``(3, )``
'''
return num.array([
num.cos(self.rake * d2r) * self.slip,
num.sin(self.rake * d2r) * self.slip,
self.opening])
[docs] def discretize(self, nlength, nwidth, *args, **kwargs):
'''
Discretize fault into rectilinear grid of fault patches.
Fault orientation, slip and elastic parameters are passed to the
sub-faults unchanged.
:param nlength:
Number of patches in strike direction.
:type nlength:
int
:param nwidth:
Number of patches in down-dip direction.
:type nwidth:
int
:return:
Discrete fault patches.
:rtype:
list of :py:class:`~pyrocko.modelling.okada.OkadaPatch`
'''
assert nlength > 0
assert nwidth > 0
il = num.repeat(num.arange(nlength), nwidth)
iw = num.tile(num.arange(nwidth), nlength)
patch_length = self.length / nlength
patch_width = self.width / nwidth
al1 = -patch_length / 2.
al2 = patch_length / 2.
aw1 = -patch_width / 2.
aw2 = patch_width / 2.
source_points = num.zeros((nlength * nwidth, 3))
source_points[:, 0] = il * patch_length + patch_length / 2.
source_points[:, 1] = iw * patch_width + patch_width / 2.
source_points[:, 0] += self.al1
source_points[:, 1] -= self.aw2
rotmat = mt.euler_to_matrix(self.dip*d2r, self.strike*d2r, 0.)
source_points_rot = num.dot(rotmat.T, source_points.T).T
source_points_rot[:, 0] += self.northing
source_points_rot[:, 1] += self.easting
source_points_rot[:, 2] += self.depth
kwargs = {
prop: getattr(self, prop) for prop in self.T.propnames
if prop not in [
'north_shift', 'east_shift', 'depth',
'al1', 'al2', 'aw1', 'aw2']}
return (
[OkadaPatch(
parent=self,
ix=src_point[0],
iy=src_point[1],
north_shift=coord[0],
east_shift=coord[1],
depth=coord[2],
al1=al1, al2=al2, aw1=aw1, aw2=aw2, **kwargs)
for src_point, coord in zip(source_points, source_points_rot)],
source_points)
[docs]class OkadaPatch(OkadaSource):
'''
Okada source with additional 2D indexes for bookkeeping.
'''
ix = Int.T(help='Relative index of the patch in x')
iy = Int.T(help='Relative index of the patch in y')
def __init__(self, parent=None, *args, **kwargs):
OkadaSource.__init__(self, *args, **kwargs)
self.parent = parent
[docs]def make_okada_coefficient_matrix(
source_patches_list,
pure_shear=False,
rotate_sdn=True,
nthreads=1, variant='normal'):
'''
Build coefficient matrix for given fault patches.
The boundary element method (BEM) for a discretized fault and the
determination of the slip distribution :math:`\\Delta u` from stress drop
:math:`\\Delta \\sigma` is based on
:math:`\\Delta \\sigma = \\mathbf{C} \\cdot \\Delta u`. Here the
coefficient matrix :math:`\\mathbf{C}` is built, based on the displacements
from Okada's solution (Okada, 1992) and their partial derivatives.
:param source_patches_list:
Source patches, to be used in BEM.
:type source_patches_list:
list of :py:class:`~pyrocko.modelling.okada.OkadaSource`.
:param pure_shear:
If ``True``, only shear forces are taken into account.
:type pure_shear:
bool
:param rotate_sdn:
If ``True``, rotate to strike, dip, normal.
:type rotate_sdn:
bool
:param nthreads:
Number of threads.
:type nthreads:
int
:return:
Coefficient matrix for all source combinations.
:rtype:
:py:class:`~numpy.ndarray`:
``(len(source_patches_list) * 3, len(source_patches_list) * 3)``
'''
if variant == 'slow':
return _make_okada_coefficient_matrix_slow(
source_patches_list, pure_shear, rotate_sdn, nthreads)
source_patches = num.array([
src.source_patch() for src in source_patches_list])
receiver_coords = source_patches[:, :3].copy()
npoints = len(source_patches_list)
if pure_shear:
n_eq = 2
else:
n_eq = 3
coefmat = num.zeros((npoints * 3, npoints * 3))
lambda_mean = num.mean([src.lamb for src in source_patches_list])
mu_mean = num.mean([src.shearmod for src in source_patches_list])
unit_disl = 1.
disl_cases = {
'strikeslip': {
'slip': unit_disl,
'opening': 0.,
'rake': 0.},
'dipslip': {
'slip': unit_disl,
'opening': 0.,
'rake': 90.},
'tensileslip': {
'slip': 0.,
'opening': unit_disl,
'rake': 0.}
}
diag_ind = [0, 4, 8]
kron = num.zeros(9)
kron[diag_ind] = 1.
if variant == 'normal':
kron = kron[num.newaxis, num.newaxis, :]
else:
kron = kron[num.newaxis, :]
for idisl, case_type in enumerate([
'strikeslip', 'dipslip', 'tensileslip'][:n_eq]):
case = disl_cases[case_type]
source_disl = num.array([
case['slip'] * num.cos(case['rake'] * d2r),
case['slip'] * num.sin(case['rake'] * d2r),
case['opening']])
if variant == 'normal':
results = okada_ext.okada(
source_patches,
num.tile(source_disl, npoints).reshape(-1, 3),
receiver_coords,
lambda_mean,
mu_mean,
nthreads=nthreads,
rotate_sdn=int(rotate_sdn),
stack_sources=int(variant != 'normal'))
eps = 0.5 * (
results[:, :, 3:] +
results[:, :, (3, 6, 9, 4, 7, 10, 5, 8, 11)])
dilatation \
= eps[:, :, diag_ind].sum(axis=-1)[:, :, num.newaxis]
stress_sdn = kron*lambda_mean*dilatation + 2.*mu_mean*eps
coefmat[:, idisl::3] = stress_sdn[:, :, (2, 5, 8)]\
.reshape(-1, npoints*3).T
else:
for isrc, source in enumerate(source_patches):
results = okada_ext.okada(
source.reshape(1, -1),
source_disl.reshape(1, -1),
receiver_coords,
lambda_mean,
mu_mean,
nthreads=nthreads,
rotate_sdn=int(rotate_sdn))
eps = 0.5 * (
results[:, 3:] +
results[:, (3, 6, 9, 4, 7, 10, 5, 8, 11)])
dilatation \
= num.sum(eps[:, diag_ind], axis=1)[:, num.newaxis]
stress_sdn \
= kron * lambda_mean * dilatation+2. * mu_mean * eps
coefmat[:, isrc*3 + idisl] \
= stress_sdn[:, (2, 5, 8)].ravel()
if pure_shear:
coefmat[2::3, :] = 0.
return -coefmat / unit_disl
def _make_okada_coefficient_matrix_slow(
source_patches_list, pure_shear=False, rotate_sdn=True, nthreads=1):
source_patches = num.array([
src.source_patch() for src in source_patches_list])
receiver_coords = source_patches[:, :3].copy()
npoints = len(source_patches_list)
if pure_shear:
n_eq = 2
else:
n_eq = 3
coefmat = num.zeros((npoints * 3, npoints * 3))
def ned2sdn_rotmat(strike, dip):
rotmat = mt.euler_to_matrix(
(dip + 180.) * d2r, strike * d2r, 0.)
return rotmat
lambda_mean = num.mean([src.lamb for src in source_patches_list])
shearmod_mean = num.mean([src.shearmod for src in source_patches_list])
unit_disl = 1.
disl_cases = {
'strikeslip': {
'slip': unit_disl,
'opening': 0.,
'rake': 0.},
'dipslip': {
'slip': unit_disl,
'opening': 0.,
'rake': 90.},
'tensileslip': {
'slip': 0.,
'opening': unit_disl,
'rake': 0.}
}
for idisl, case_type in enumerate([
'strikeslip', 'dipslip', 'tensileslip'][:n_eq]):
case = disl_cases[case_type]
source_disl = num.array([
case['slip'] * num.cos(case['rake'] * d2r),
case['slip'] * num.sin(case['rake'] * d2r),
case['opening']])
for isource, source in enumerate(source_patches):
results = okada_ext.okada(
source[num.newaxis, :].copy(),
source_disl[num.newaxis, :].copy(),
receiver_coords,
lambda_mean,
shearmod_mean,
nthreads=nthreads,
rotate_sdn=int(rotate_sdn))
for irec in range(receiver_coords.shape[0]):
eps = num.zeros((3, 3))
for m in range(3):
for n in range(3):
eps[m, n] = 0.5 * (
results[irec][m * 3 + n + 3] +
results[irec][n * 3 + m + 3])
stress = num.zeros((3, 3))
dilatation = num.sum([eps[i, i] for i in range(3)])
for m, n in zip([0, 0, 0, 1, 1, 2], [0, 1, 2, 1, 2, 2]):
if m == n:
stress[m, n] = \
lambda_mean * \
dilatation + \
2. * shearmod_mean * \
eps[m, n]
else:
stress[m, n] = \
2. * shearmod_mean * \
eps[m, n]
stress[n, m] = stress[m, n]
normal = num.array([0., 0., -1.])
for isig in range(3):
tension = num.sum(stress[isig, :] * normal)
coefmat[irec * n_eq + isig, isource * n_eq + idisl] = \
tension / unit_disl
return coefmat
[docs]def invert_fault_dislocations_bem(
stress_field,
coef_mat=None,
source_list=None,
pure_shear=False,
epsilon=None,
nthreads=1,
**kwargs):
'''
BEM least squares inversion to get fault dislocations given stress field.
Follows least squares inversion approach by Menke (1989) to calculate
dislocations on a fault with several segments from a given stress field.
The coefficient matrix connecting stresses and displacements of the fault
patches can either be specified by the user (``coef_mat``) or it is
calculated using the solution of Okada (1992) for a rectangular fault in a
homogeneous half space (``source_list``).
:param stress_field:
Stress change [Pa] for each source patch (as
``stress_field[isource, icomponent]`` where isource indexes the source
patch and ``icomponent`` indexes component, ordered (strike, dip,
tensile).
:type stress_field:
:py:class:`~numpy.ndarray`: ``(nsources, 3)``
:param coef_mat:
Coefficient matrix connecting source patch dislocations and the stress
field.
:type coef_mat:
:py:class:`~numpy.ndarray`:
``(len(source_list) * 3, len(source_list) * 3)``
:param source_list:
Source patches to be used for BEM.
:type source_list:
list of
:py:class:`~pyrocko.modelling.okada.OkadaSource`
:param epsilon:
If given, values in ``coef_mat`` smaller than ``epsilon`` are set to
zero.
:type epsilon:
float
:param nthreads:
Number of threads allowed.
:type nthreads:
int
:return:
Inverted displacements as ``displacements[isource, icomponent]``
where isource indexes the source patch and ``icomponent`` indexes
component, ordered (strike, dip, tensile).
:rtype:
:py:class:`~numpy.ndarray`: ``(nsources, 3)``
'''
if source_list is not None and coef_mat is None:
coef_mat = make_okada_coefficient_matrix(
source_list, pure_shear=pure_shear, nthreads=nthreads,
**kwargs)
if epsilon is not None:
coef_mat[coef_mat < epsilon] = 0.
idx = num.arange(0, coef_mat.shape[0])
if pure_shear:
idx = idx[idx % 3 != 2]
coef_mat_in = coef_mat[idx, :][:, idx]
disloc_est = num.zeros(coef_mat.shape[0])
if stress_field.ndim == 2:
stress_field = stress_field.ravel()
threadpool_limits = get_threadpool_limits()
with threadpool_limits(limits=nthreads, user_api='blas'):
try:
disloc_est[idx] = num.linalg.multi_dot([
num.linalg.inv(num.dot(coef_mat_in.T, coef_mat_in)),
coef_mat_in.T,
stress_field[idx]])
except num.linalg.LinAlgError as e:
logger.warning('Linear inversion failed!')
logger.warning(
'coef_mat: %s\nstress_field: %s',
coef_mat_in, stress_field[idx])
raise e
return disloc_est.reshape(-1, 3)
__all__ = [
'AnalyticalSource',
'AnalyticalRectangularSource',
'OkadaSource',
'OkadaPatch',
'make_okada_coefficient_matrix',
'invert_fault_dislocations_bem']
