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# https://pyrocko.org - GPLv3 

# 

# The Pyrocko Developers, 21st Century 

# ---|P------/S----------~Lg---------- 

import numpy as num 

import logging 

 

from pyrocko import moment_tensor as mt 

import pyrocko.guts as guts 

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('pyrocko.modelling.okada') 

 

d2r = num.pi/180. 

r2d = 180./num.pi 

km = 1e3 

 

 

class AnalyticalSource(Location): 

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', 

optional=True) 

 

@property 

def northing(self): 

return self.north_shift 

 

@property 

def easting(self): 

return self.east_shift 

 

clone = guts.clone 

 

 

class AnalyticalRectangularSource(AnalyticalSource): 

''' 

Rectangular analytical source model 

''' 

 

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='Distance "left" side to source point [m]') 

 

al2 = Float.T( 

default=0., 

help='Distance "right" side to source point [m]') 

 

aw1 = Float.T( 

default=0., 

help='Distance "lower" side to source point [m]') 

 

aw2 = Float.T( 

default=0., 

help='Distance "upper" side to source point [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 

 

 

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\'s ratio, typically 0.25', 

optional=True) 

 

shearmod = Float.T( 

default=32e9, 

help='Shear modulus along the plane [Pa]', 

optional=True) 

 

@property 

def lamb(self): 

''' 

Calculation of first Lame's parameter 

 

According to Mueller (2007), the first Lame parameter lambda can be 

determined from the formulation for the poisson ration nu: 

nu = lambda / (2 * (lambda + mu)) 

with the shear modulus mu 

''' 

 

return (2. * self.poisson * self.shearmod) / (1. - 2*self.poisson) 

 

@property 

def seismic_moment(self): 

''' 

Scalar Seismic moment 

 

Code copied from Kite 

Disregarding the opening (as for now) 

We assume a shear modulus of :math:`mu = 36 mathrm{GPa}` 

and :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 

''' 

 

if self.shearmod: 

mu = self.shearmod 

elif self.poisson: 

self.shearmod = (8. * (1. + self.poisson)) / (1. - 2*self.poisson) 

mu = self.shearmod 

else: 

raise ValueError( 

'Shear modulus or poisson ratio needed for moment calculation') 

 

disl = 0. 

if self.slip: 

disl = (disl**2 + self.slip**2)**.5 

if self.opening: 

disl = (disl**2 + self.opening**2)**.5 

 

return mu * self.area * disl 

 

@property 

def moment_magnitude(self): 

''' Moment magnitude 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) 

 

def source_patch(self): 

''' Build source information array for okada_ext.okada input 

 

:return: array of the source data as input for okada_ext.okada 

:rtype: :py:class:`numpy.ndarray`, ``(1, 9)`` 

''' 

return num.array([ 

self.northing, 

self.easting, 

self.depth, 

self.strike, 

self.dip, 

self.al1, 

self.al2, 

self.aw1, 

self.aw2]) 

 

def source_disloc(self): 

''' Build source dislocation for okada_ext.okada input 

 

:return: array of the source dislocation data as input for 

okada_ext.okada 

:rtype: :py:class:`numpy.ndarray`, ``(1, 3)`` 

''' 

return num.array([ 

num.cos(self.rake * d2r) * self.slip, 

num.sin(self.rake * d2r) * self.slip, 

self.opening]) 

 

def discretize(self, nlength, nwidth, *args, **kwargs): 

''' Discretize the given fault by ``nlength * nwidth`` fault patches 

 

Discretizing the fault into several sub faults. ``nlength`` is 

number of points in strike direction, ``nwidth`` in down dip direction 

along the fault. Fault orientation, slip and elastic parameters are 

kept. 

 

:param nlength: Number of discrete points in faults strike direction 

:type nlength: int 

:param nwidth: Number of discrete points in faults down-dip direction 

:type nwidth: int 

:return: Discrete fault patches 

:rtype: list of :py:class:`pyrocko.modelling.OkadaSource` objects 

''' 

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 = num.asarray( 

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) 

 

 

class OkadaPatch(OkadaSource): 

 

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, *args, parent=None, **kwargs): 

super().__init__(*args, **kwargs) 

self.parent = parent 

 

 

class DislocationInverter(object): 

''' 

Toolbox for Boundary Element Method (BEM) and dislocation inversion based 

on okada_ext.okada 

''' 

 

@staticmethod 

def get_coef_mat(source_patches_list, pure_shear=False, 

rotate_sdn=True, nthreads=1): 

''' 

Build coefficient matrix for given source_patches 

 

The BEM for a fault and the determination of the slip distribution from 

the stress drop is based on the relation stress = coef_mat * displ. 

Here the coefficient matrix is build and filled based on the 

okada_ext.okada displacements and partial displacement 

differentiations. 

 

:param source_patches_list: list of all OkadaSources, which shall be 

used for BEM 

:type source_patches_list: list of 

:py:class:`pyrocko.modelling.OkadaSource` 

:param pure_shear: Shall only shear forces be taken into account, 

or additionally include opening, default False. 

:type pure_shear: optional, bool 

:param rotate_sdn: Rotation towards strike, dip, normal, default True. 

:type rotate_sdn: optional, bool 

:param nthreads: Number of threads, default 1 

:type nthreads: optional, int 

 

:return: coefficient matrix for all sources 

:rtype: :py:class:`numpy.ndarray`, 

``(len(source_patches_list) * 3, len(source_patches_list) * 3)`` 

''' 

 

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. 

kron = kron[num.newaxis, 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']]) 

 

results = okada_ext.okada( 

source_patches, 

num.tile(source_disl, npoints).reshape(-1, 3), 

receiver_coords, 

lambda_mean, 

mu_mean, 

nthreads=nthreads, 

rotate_sdn=rotate_sdn, 

stack_sources=False) 

 

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 

 

if pure_shear: 

coefmat[2::3, :] = 0. 

 

return -coefmat / unit_disl 

 

@staticmethod 

def get_coef_mat_single(source_patches_list, pure_shear=False, 

rotate_sdn=True, nthreads=1): 

''' 

Build coefficient matrix for given source_patches 

 

The BEM for a fault and the determination of the slip distribution from 

the stress drop is based on the relation stress = coef_mat * displ. 

Here the coefficient matrix is build and filled based on the 

okada_ext.okada displacements and partial displacement 

differentiations. 

 

:param source_patches_list: list of all OkadaSources, which shall be 

used for BEM 

:type source_patches_list: list of 

:py:class:`pyrocko.modelling.OkadaSource` 

:param pure_shear: Flag, if also opening mode shall be taken into 

account (False) or the fault is described as pure shear (True). 

:type pure_shear: optional, bool 

:param rotate_sdn: Rotation towards strike, dip, normal, default True. 

:type rotate_sdn: optional, bool 

:param nthreads: Number of threads, default 1 

:type nthreads: optional, int 

 

:return: coefficient matrix for all sources 

:rtype: :py:class:`numpy.ndarray`, 

``(len(source_patches_list) * 3, len(source_patches_list) * 3)`` 

''' 

 

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. 

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']]) 

 

for isrc, source in enumerate(source_patches): 

results = okada_ext.okada( 

source[num.newaxis, :], 

source_disl[num.newaxis, :], 

receiver_coords, 

lambda_mean, 

mu_mean, 

nthreads=nthreads, 

rotate_sdn=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, isrc * 3 + idisl] = 0. 

 

return -coefmat / unit_disl 

 

@staticmethod 

def get_coef_mat_slow(source_patches_list, pure_shear=False, 

rotate_sdn=True, nthreads=1): 

''' 

Build coefficient matrix for given source_patches (Slow version) 

 

The BEM for a fault and the determination of the slip distribution from 

the stress drop is based on the relation stress = coef_mat * displ. 

Here the coefficient matrix is build and filled based on the 

okada_ext.okada displacements and partial displacement 

differentiations. 

 

:param source_patches_list: list of all OkadaSources, which shall be 

used for BEM 

:type source_patches_list: list of 

:py:class:`pyrocko.modelling.OkadaSource` 

:param pure_shear: Flag, if also opening mode shall be taken into 

account (False) or the fault is described as pure shear (True). 

:type pure_shear: optional, bool 

:param rotate_sdn: Rotation towards strike, dip, normal, default True. 

:type rotate_sdn: optional, bool 

:param nthreads: Number of threads, default 1 

:type nthreads: optional, int 

 

:return: coefficient matrix for all sources 

:rtype: :py:class:`numpy.ndarray`, 

``(source_patches_list.shape[0] * 3, 

source_patches_list.shape[0] * 3(2))`` 

''' 

 

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.).A 

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=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 

 

@staticmethod 

def get_disloc_lsq( 

stress_field, 

coef_mat=None, 

source_list=None, 

pure_shear=False, 

epsilon=None, 

nthreads=1, 

**kwargs): 

''' 

Least square inversion to get displacement from stress 

 

Follows approach for Least-Square Inversion published in Menke (1989) 

to calculate displacements on a fault with several segments from a 

given stress field. If not done, the coefficient matrix is determined 

within the code. 

 

:param stress_field: Array containing the stress change [Pa] for each 

source patch (order: [ 

src1 dstress_Strike, src1 dstress_Dip, src1 dstress_Tensile, 

src2 dstress_Strike, ...]) 

:type stress_field: :py:class:`numpy.ndarray`, ``(n_sources * 3, )`` 

:param coef_mat: Coefficient matrix to connect source patches 

displacement and the resulting stress field 

:type coef_mat: optional, :py:class:`numpy.ndarray`, 

``(source_patches_list.shape[0] * 3, 

source_patches.shape[] * 3(2)`` 

:param source_list: list of all OkadaSources, which shall be 

used for BEM 

:type source_list: optional, list of 

:py:class:`pyrocko.modelling.OkadaSource` 

:param epsilon: regularize small values from the coefficient matrix, 

default None. 

:type epsilon: optional, float 

 

:param nthreads: number of threads for the least squares inversion, 

default 1 

:type nthreads: optional, int 

 

:return: inverted displacements (u_strike, u_dip , u_tensile) for each 

source patch. order: [ 

[patch1 u_Strike, patch1 u_Dip, patch1 u_Tensile], 

[patch2 u_Strike, patch2 u_Dip, patch2 u_Tensile], 

...] 

:rtype: :py:class:`numpy.ndarray`, ``(n_sources, 3)`` 

''' 

 

if source_list is not None and coef_mat is None: 

coef_mat = DislocationInverter.get_coef_mat( 

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', 

'DislocationInverter']