Source code for pyrocko.moment_tensor

# http://pyrocko.org - GPLv3
#
# The Pyrocko Developers, 21st Century
# ---|P------/S----------~Lg----------

'''
This module provides various moment tensor related utility functions.

It can be used to convert between strike-dip-rake and moment tensor
representations and provides different options to produce random moment
tensors.

Moment tensors are represented by :py:class:`MomentTensor` instances. The
internal representation uses a north-east-down (NED) coordinate system, but it
can convert from/to the conventions used by the Global CMT catalog
(up-south-east, USE).

If not otherwise noted, scalar moment is interpreted as the Frobenius norm
based scalar moment (see :py:meth:`MomentTensor.scalar_moment`. The scalar
moment according to the "standard decomposition" can be found in the
output of :py:meth:`MomentTensor.standard_decomposition`.
'''

from __future__ import division, print_function, absolute_import
from builtins import map
from builtins import zip

import math
import numpy as num

from .guts import Object, Float

guts_prefix = 'pf'

dynecm = 1e-7


[docs]def random_axis(rstate=None): ''' Get randomly oriented unit vector. :param rstate: :py:class:`numpy.random.RandomState` object, can be used to create reproducible pseudo-random sequences ''' rstate = rstate or num.random while True: axis = rstate.uniform(size=3) * 2.0 - 1.0 uabs = math.sqrt(num.sum(axis**2)) if 0.001 < uabs < 1.0: return axis / uabs
[docs]def rotation_from_angle_and_axis(angle, axis): ''' Build rotation matrix based on axis and angle. :param angle: rotation angle [degrees] :param axis: orientation of rotation axis, either in spherical coordinates ``(theta, phi)`` [degrees], or as a unit vector ``(ux, uy, uz)``. ''' if len(axis) == 2: theta, phi = axis ux = math.sin(d2r*theta)*math.cos(d2r*phi) uy = math.sin(d2r*theta)*math.sin(d2r*phi) uz = math.cos(d2r*theta) elif len(axis) == 3: axis = num.asarray(axis) uabs = math.sqrt(num.sum(axis**2)) ux, uy, uz = axis / uabs else: assert False ct = math.cos(d2r*angle) st = math.sin(d2r*angle) return num.matrix([ [ct + ux**2*(1.-ct), ux*uy*(1.-ct)-uz*st, ux*uz*(1.-ct)+uy*st], [uy*ux*(1.-ct)+uz*st, ct+uy**2*(1.-ct), uy*uz*(1.-ct)-ux*st], [uz*ux*(1.-ct)-uy*st, uz*uy*(1.-ct)+ux*st, ct+uz**2*(1.-ct)] ])
[docs]def random_rotation(x=None): '''Get random rotation matrix. A random rotation matrix, drawn from a uniform distrubution in the space of rotations is returned, after Avro 1992 - "Fast random rotation matrices". :param x: three (uniform random) numbers in the range [0, 1\[ used as input to the distribution tranformation. If ``None``, random numbers are used. Can be used to create grids of random rotations with uniform density in rotation space. ''' if x is not None: x1, x2, x3 = x else: x1, x2, x3 = num.random.random(3) phi = math.pi*2.0*x1 zrot = num.matrix([ [math.cos(phi), math.sin(phi), 0.], [-math.sin(phi), math.cos(phi), 0.], [0., 0., 1.]]) lam = math.pi*2.0*x2 v = num.matrix([[ math.cos(lam)*math.sqrt(x3), math.sin(lam)*math.sqrt(x3), math.sqrt(1.-x3)]]).T house = num.identity(3) - 2.0 * v * v.T return -house*zrot
def rand(mi, ma): return float(num.random.uniform(mi, ma)) def randdip(mi, ma): mi_ = 0.5*(math.cos(mi * math.pi/180.)+1.) ma_ = 0.5*(math.cos(ma * math.pi/180.)+1.) return math.acos(rand(mi_, ma_)*2.-1.)*180./math.pi
[docs]def random_strike_dip_rake( strikemin=0., strikemax=360., dipmin=0., dipmax=90., rakemin=-180., rakemax=180.): ''' Get random strike, dip, rake triplet. .. note:: Might not produce a homogeneous distribution of mechanisms. Better use :py:meth:`MomentTensor.random_dc` which is based on :py:func:`random_rotation`. ''' strike = rand(strikemin, strikemax) dip = randdip(dipmin, dipmax) rake = rand(rakemin, rakemax) return strike, dip, rake
[docs]def to6(m): '''Get non-redundant components from symmetric 3x3 matrix :returns: 1D NumPy array with entries ordered like ``(a_xx, a_yy, a_zz, a_xy, a_xz, a_yz)`` ''' return num.array([m[0, 0], m[1, 1], m[2, 2], m[0, 1], m[0, 2], m[1, 2]])
[docs]def symmat6(a_xx, a_yy, a_zz, a_xy, a_xz, a_yz): ''' Create symmetric 3x3 matrix from its 6 non-redundant values. ''' return num.matrix([[a_xx, a_xy, a_xz], [a_xy, a_yy, a_yz], [a_xz, a_yz, a_zz]], dtype=num.float)
[docs]def values_to_matrix(values): '''Convert anything to moment tensor represented as a NumPy matrix. Transforms :py:class:`MomentTensor` objects, tuples, lists and NumPy arrays with 3x3 or 3, 4, 6, or 7 elements into NumPy 3x3 matrix objects. The ``values`` argument is interpreted depending on shape and type as follows: * ``(strike, dip, rake)`` * ``(strike, dip, rake, magnitude)`` * ``(mnn, mee, mdd, mne, mnd, med)`` * ``(mnn, mee, mdd, mne, mnd, med, magnitude)`` * ``((mnn, mne, mnd), (mne, mee, med), (mnd, med, mdd))`` * :py:class:`MomentTensor` ''' if isinstance(values, (tuple, list)): values = num.asarray(values, dtype=num.float) if isinstance(values, MomentTensor): return values.m() elif isinstance(values, num.ndarray): if values.shape == (3,): strike, dip, rake = values rotmat1 = euler_to_matrix(d2r*dip, d2r*strike, -d2r*rake) return rotmat1.T * MomentTensor._m_unrot * rotmat1 elif values.shape == (4,): strike, dip, rake, magnitude = values moment = magnitude_to_moment(magnitude) rotmat1 = euler_to_matrix(d2r*dip, d2r*strike, -d2r*rake) return rotmat1.T * MomentTensor._m_unrot * rotmat1 * moment elif values.shape == (6,): return symmat6(*values) elif values.shape == (7,): magnitude = values[6] moment = magnitude_to_moment(magnitude) mt = symmat6(*values[:6]) mt *= moment / (num.linalg.norm(mt) / math.sqrt(2.0)) return mt elif values.shape == (3, 3): return num.asmatrix(values, dtype=num.float) raise Exception('cannot convert object to 3x3 matrix')
[docs]def moment_to_magnitude(moment): ''' Convert scalar moment to moment magnitude Mw. :param moment: scalar moment [Nm] :returns: moment magnitude Mw Moment magnitude is defined as .. math:: M_\\mathrm{w} = {\\frac{2}{3}}\\log_{10}(M_0) - 10.7 where :math:`M_0` is the scalar moment given in [Nm]. .. note:: Global CMT uses 10.7333333 instead of 10.7, based on [Kanamori 1977], 10.7 is from [Hanks and Kanamori 1979]. ''' return num.log10(moment*1.0e7) / 1.5 - 10.7
[docs]def magnitude_to_moment(magnitude): ''' Convert moment magnitude Mw to scalar moment. :param magnitude: moment magnitude :returns: scalar moment [Nm] See :py:func:`moment_to_magnitude`. ''' return 10.0**(1.5*(magnitude+10.7))*1.0e-7
magnitude_1Nm = moment_to_magnitude(1.0)
[docs]def euler_to_matrix(alpha, beta, gamma): '''Given euler angle triplet, create rotation matrix Given coordinate system `(x,y,z)` and rotated system `(xs,ys,zs)` the line of nodes is the intersection between the `x,y` and the `xs,ys` planes. :param alpha: is the angle between the `z`-axis and the `zs`-axis [rad] :param beta: is the angle between the `x`-axis and the line of nodes [rad] :param gamma: is the angle between the line of nodes and the `xs`-axis [rad] Usage for moment tensors:: m_unrot = numpy.matrix([[0,0,-1],[0,0,0],[-1,0,0]]) euler_to_matrix(dip,strike,-rake, rotmat) m = rotmat.T * m_unrot * rotmat ''' ca = math.cos(alpha) cb = math.cos(beta) cg = math.cos(gamma) sa = math.sin(alpha) sb = math.sin(beta) sg = math.sin(gamma) mat = num.matrix([[cb*cg-ca*sb*sg, sb*cg+ca*cb*sg, sa*sg], [-cb*sg-ca*sb*cg, -sb*sg+ca*cb*cg, sa*cg], [sa*sb, -sa*cb, ca]], dtype=num.float) return mat
[docs]def matrix_to_euler(rotmat): '''Get eulerian angle triplet from rotation matrix.''' ex = cvec(1., 0., 0.) ez = cvec(0., 0., 1.) exs = rotmat.T * ex ezs = rotmat.T * ez enodes = num.cross(ez.T, ezs.T).T if num.linalg.norm(enodes) < 1e-10: enodes = exs enodess = rotmat*enodes cos_alpha = float((ez.T*ezs)) if cos_alpha > 1.: cos_alpha = 1. if cos_alpha < -1.: cos_alpha = -1. alpha = math.acos(cos_alpha) beta = num.mod(math.atan2(enodes[1, 0], enodes[0, 0]), math.pi*2.) gamma = num.mod(-math.atan2(enodess[1, 0], enodess[0, 0]), math.pi*2.) return unique_euler(alpha, beta, gamma)
[docs]def unique_euler(alpha, beta, gamma): '''Uniquify eulerian angle triplet. Put eulerian angle triplet into ranges compatible with ``(dip, strike, -rake)`` conventions in seismology:: alpha (dip) : [0, pi/2] beta (strike) : [0, 2*pi) gamma (-rake) : [-pi, pi) If ``alpha1`` is near to zero, ``beta`` is replaced by ``beta+gamma`` and ``gamma`` is set to zero, to prevent this additional ambiguity. If ``alpha`` is near to ``pi/2``, ``beta`` is put into the range ``[0,pi)``. ''' pi = math.pi alpha = num.mod(alpha, 2.0*pi) if 0.5*pi < alpha and alpha <= pi: alpha = pi - alpha beta = beta + pi gamma = 2.0*pi - gamma elif pi < alpha and alpha <= 1.5*pi: alpha = alpha - pi gamma = pi - gamma elif 1.5*pi < alpha and alpha <= 2.0*pi: alpha = 2.0*pi - alpha beta = beta + pi gamma = pi + gamma alpha = num.mod(alpha, 2.0*pi) beta = num.mod(beta, 2.0*pi) gamma = num.mod(gamma+pi, 2.0*pi)-pi # If dip is exactly 90 degrees, one is still # free to choose between looking at the plane from either side. # Choose to look at such that beta is in the range [0,180) # This should prevent some problems, when dip is close to 90 degrees: if abs(alpha - 0.5*pi) < 1e-10: alpha = 0.5*pi if abs(beta - pi) < 1e-10: beta = pi if abs(beta - 2.*pi) < 1e-10: beta = 0. if abs(beta) < 1e-10: beta = 0. if alpha == 0.5*pi and beta >= pi: gamma = - gamma beta = num.mod(beta-pi, 2.0*pi) gamma = num.mod(gamma+pi, 2.0*pi)-pi assert 0. <= beta < pi assert -pi <= gamma < pi if alpha < 1e-7: beta = num.mod(beta + gamma, 2.0*pi) gamma = 0. return (alpha, beta, gamma)
def cvec(x, y, z): return num.matrix([[x, y, z]], dtype=num.float).T def rvec(x, y, z): return num.matrix([[x, y, z]], dtype=num.float) def eigh_check(a): evals, evecs = num.linalg.eigh(a) assert evals[0] <= evals[1] <= evals[2] return evals, evecs r2d = 180. / math.pi d2r = 1. / r2d def random_mt(x=None, scalar_moment=1.0, magnitude=None): if magnitude is not None: scalar_moment = magnitude_to_moment(magnitude) if x is None: x = num.random.random(6) evals = x[:3] * 2. - 1.0 evals /= num.sqrt(num.sum(evals**2)) / math.sqrt(2.0) rotmat = random_rotation(x[3:]) return scalar_moment * rotmat * num.matrix(num.diag(evals)) * rotmat.T def random_m6(*args, **kwargs): return to6(random_mt(*args, **kwargs)) def random_dc(x=None, scalar_moment=1.0, magnitude=None): if magnitude is not None: scalar_moment = magnitude_to_moment(magnitude) rotmat = random_rotation(x) return scalar_moment * (rotmat * MomentTensor._m_unrot * rotmat.T) def sm(m): return "/ %5.2F %5.2F %5.2F \\\n" % (m[0, 0], m[0, 1], m[0, 2]) + \ "| %5.2F %5.2F %5.2F |\n" % (m[1, 0], m[1, 1], m[1, 2]) + \ "\\ %5.2F %5.2F %5.2F /\n" % (m[2, 0], m[2, 1], m[2, 2])
[docs]def as_mt(mt): ''' Convenience function to convert various objects to moment tensor object. Like :py:meth:``MomentTensor.from_values``, but does not create a new :py:class:`MomentTensor` object when ``mt`` already is one. ''' if isinstance(mt, MomentTensor): return mt else: return MomentTensor.from_values(mt)
[docs]class MomentTensor(Object): ''' Moment tensor object :param m: NumPy matrix in north-east-down convention :param m_up_south_east: NumPy matrix in up-south-east convention :param strike,dip,rake: fault plane angles in [degrees] :param scalar_moment: scalar moment in [Nm] :param magnitude: moment magnitude Mw Global CMT catalog moment tensors use the up-south-east (USE) coordinate system convention with :math:`r` (up), :math:`\\theta` (south), and :math:`\\phi` (east). .. math:: :nowrap: \\begin{align*} M_{rr} &= M_{dd}, & M_{ r\\theta} &= M_{nd},\\\\ M_{\\theta\\theta} &= M_{ nn}, & M_{r\\phi} &= -M_{ed},\\\\ M_{\\phi\\phi} &= M_{ee}, & M_{\\theta\\phi} &= -M_{ne} \\end{align*} ''' mnn__ = Float.T(default=0.0) mee__ = Float.T(default=0.0) mdd__ = Float.T(default=0.0) mne__ = Float.T(default=0.0) mnd__ = Float.T(default=-1.0) med__ = Float.T(default=0.0) strike1__ = Float.T(default=None, optional=True) # read-only dip1__ = Float.T(default=None, optional=True) # read-only rake1__ = Float.T(default=None, optional=True) # read-only strike2__ = Float.T(default=None, optional=True) # read-only dip2__ = Float.T(default=None, optional=True) # read-only rake2__ = Float.T(default=None, optional=True) # read-only moment__ = Float.T(default=None, optional=True) # read-only magnitude__ = Float.T(default=None, optional=True) # read-only _flip_dc = num.matrix( [[0., 0., -1.], [0., -1., 0.], [-1., 0., 0.]], dtype=num.float) _to_up_south_east = num.matrix( [[0., 0., -1.], [-1., 0., 0.], [0., 1., 0.]], dtype=num.float).T _m_unrot = num.matrix( [[0., 0., -1.], [0., 0., 0.], [-1., 0., 0.]], dtype=num.float) _u_evals, _u_evecs = eigh_check(_m_unrot)
[docs] @classmethod def random_dc(cls, x=None, scalar_moment=1.0, magnitude=None): ''' Create random oriented double-couple moment tensor The rotations used are uniformly distributed in the space of rotations. ''' return MomentTensor( m=random_dc(x=x, scalar_moment=scalar_moment, magnitude=magnitude))
[docs] @classmethod def random_mt(cls, x=None, scalar_moment=1.0, magnitude=None): ''' Create random moment tensor Moment tensors produced by this function appear uniformly distributed when shown in a Hudson's diagram. The rotations used are unifomly distributed in the space of rotations. ''' return MomentTensor( m=random_mt(x=x, scalar_moment=scalar_moment, magnitude=magnitude))
[docs] @classmethod def from_values(cls, values): ''' Alternative constructor for moment tensor objects This constructor takes a :py:class:`MomentTensor` object, a tuple, list or NumPy array with 3x3 or 3, 4, 6, or 7 elements to build a Moment tensor object. The ``values`` argument is interpreted depending on shape and type as follows: * ``(strike, dip, rake)`` * ``(strike, dip, rake, magnitude)`` * ``(mnn, mee, mdd, mne, mnd, med)`` * ``(mnn, mee, mdd, mne, mnd, med, magnitude)`` * ``((mnn, mne, mnd), (mne, mee, med), (mnd, med, mdd))`` * :py:class:`MomentTensor` object ''' m = values_to_matrix(values) return MomentTensor(m=m)
def __init__( self, m=None, m_up_south_east=None, strike=0., dip=0., rake=0., scalar_moment=1., mnn=None, mee=None, mdd=None, mne=None, mnd=None, med=None, strike1=None, dip1=None, rake1=None, strike2=None, dip2=None, rake2=None, magnitude=None, moment=None): Object.__init__(self, init_props=False) if any(mxx is not None for mxx in (mnn, mee, mdd, mne, mnd, med)): m = symmat6(mnn, mee, mdd, mne, mnd, med) strike = d2r*strike dip = d2r*dip rake = d2r*rake if m_up_south_east is not None: m = self._to_up_south_east * m_up_south_east * \ self._to_up_south_east.T if m is None: if any(x is not None for x in ( strike1, dip1, rake1, strike2, dip2, rake2)): raise Exception( 'strike1, dip1, rake1, strike2, dip2, rake2 are read-only ' 'properties') if moment is not None: scalar_moment = moment if magnitude is not None: scalar_moment = magnitude_to_moment(magnitude) rotmat1 = euler_to_matrix(dip, strike, -rake) m = rotmat1.T * MomentTensor._m_unrot * rotmat1 * scalar_moment self._m = m self._update() def _update(self): m_evals, m_evecs = eigh_check(self._m) if num.linalg.det(m_evecs) < 0.: m_evecs *= -1. rotmat1 = (m_evecs * MomentTensor._u_evecs.T).T if num.linalg.det(rotmat1) < 0.: rotmat1 *= -1. self._m_eigenvals = m_evals self._m_eigenvecs = m_evecs self._rotmats = sorted( [rotmat1, MomentTensor._flip_dc * rotmat1], key=lambda m: num.abs(m.flat).tolist()) @property def mnn(self): return float(self._m[0, 0]) @mnn.setter def mnn(self, value): self._m[0, 0] = value self._update() @property def mee(self): return float(self._m[1, 1]) @mee.setter def mee(self, value): self._m[1, 1] = value self._update() @property def mdd(self): return float(self._m[2, 2]) @mdd.setter def mdd(self, value): self._m[2, 2] = value self._update() @property def mne(self): return float(self._m[0, 1]) @mne.setter def mne(self, value): self._m[0, 1] = value self._m[1, 0] = value self._update() @property def mnd(self): return float(self._m[0, 2]) @mnd.setter def mnd(self, value): self._m[0, 2] = value self._m[2, 0] = value self._update() @property def med(self): return float(self._m[1, 2]) @med.setter def med(self, value): self._m[1, 2] = value self._m[2, 1] = value self._update() @property def strike1(self): return float(self.both_strike_dip_rake()[0][0]) @property def dip1(self): return float(self.both_strike_dip_rake()[0][1]) @property def rake1(self): return float(self.both_strike_dip_rake()[0][2]) @property def strike2(self): return float(self.both_strike_dip_rake()[1][0]) @property def dip2(self): return float(self.both_strike_dip_rake()[1][1]) @property def rake2(self): return float(self.both_strike_dip_rake()[1][2])
[docs] def both_strike_dip_rake(self): '''Get both possible (strike,dip,rake) triplets.''' results = [] for rotmat in self._rotmats: alpha, beta, gamma = [r2d*x for x in matrix_to_euler(rotmat)] results.append((beta, alpha, -gamma)) return results
[docs] def p_axis(self): '''Get direction of p axis.''' return (self._m_eigenvecs.T)[0]
[docs] def t_axis(self): '''Get direction of t axis.''' return (self._m_eigenvecs.T)[2]
[docs] def null_axis(self): '''Get diretion of the null axis.''' return self._m_eigenvecs.T[1]
[docs] def eigenvals(self): ''' Get the eigenvalues of the moment tensor in accending order. :returns: ``(ep, en, et)`` ''' return self._m_eigenvals
[docs] def eigensystem(self): ''' Get the eigenvalues and eigenvectors of the moment tensor. :returns: ``(ep, en, et, vp, vn, vt)``''' vp = self.p_axis().A.flatten() vn = self.null_axis().A.flatten() vt = self.t_axis().A.flatten() ep, en, et = self._m_eigenvals return ep, en, et, vp, vn, vt
[docs] def both_slip_vectors(self): '''Get both possible slip directions.''' return [rotmat*cvec(1., 0., 0.) for rotmat in self._rotmats]
[docs] def m(self): '''Get plain moment tensor as 3x3 matrix.''' return self._m.copy()
[docs] def m6(self): ''' Get the moment tensor as a six-element array. :returns: ``(mnn, mee, mdd, mne, mnd, med)`` ''' return to6(self._m)
[docs] def m_up_south_east(self): '''Get moment tensor in up-south-east convention as 3x3 matrix. .. math:: :nowrap: \\begin{align*} M_{rr} &= M_{dd}, & M_{ r\\theta} &= M_{nd},\\\\ M_{\\theta\\theta} &= M_{ nn}, & M_{r\\phi} &= -M_{ed},\\\\ M_{\\phi\\phi} &= M_{ee}, & M_{\\theta\\phi} &= -M_{ne} \\end{align*} ''' return self._to_up_south_east.T * self._m * self._to_up_south_east
[docs] def m6_up_south_east(self): '''Get moment tensor in up-south-east convention as a six-element array. :returns: ``(muu, mss, mee, mus, mue, mse)`` ''' return to6(self.m_up_south_east())
[docs] def m_plain_double_couple(self): '''Get plain double couple with same scalar moment as moment tensor.''' rotmat1 = self._rotmats[0] m = rotmat1.T * MomentTensor._m_unrot * rotmat1 * self.scalar_moment() return m
[docs] def moment_magnitude(self): '''Get moment magnitude of moment tensor.''' return moment_to_magnitude(self.scalar_moment())
[docs] def scalar_moment(self): ''' Get the scalar moment of the moment tensor (Frobenius norm based) .. math:: M0 = \\frac{1}{\sqrt{2}}\\sqrt{\\sum_{i,j} |M_{ij}|^2} The scalar moment is calculated based on the Euclidean (Frobenius) norm (Silver and Jordan, 1982). The scalar moment returned by this function differs from the standard decomposition based definition of the scalar moment for non-double-couple moment tensors. ''' return num.linalg.norm(self._m_eigenvals) / math.sqrt(2.)
@property def moment(self): return float(self.scalar_moment()) @moment.setter def moment(self, value): self._m *= value / self.moment self._update() @property def magnitude(self): return float(self.moment_magnitude()) @magnitude.setter def magnitude(self, value): self._m *= magnitude_to_moment(value) / self.moment self._update() def __str__(self): mexp = pow(10, math.ceil(num.log10(num.max(num.abs(self._m))))) m = self._m / mexp s = '''Scalar Moment [Nm]: M0 = %g (Mw = %3.1f) Moment Tensor [Nm]: Mnn = %6.3f, Mee = %6.3f, Mdd = %6.3f, Mne = %6.3f, Mnd = %6.3f, Med = %6.3f [ x %g ] ''' % ( self.scalar_moment(), self.moment_magnitude(), m[0, 0], m[1, 1], m[2, 2], m[0, 1], m[0, 2], m[1, 2], mexp) s += self.str_fault_planes() return s def str_fault_planes(self): s = '' for i, sdr in enumerate(self.both_strike_dip_rake()): s += 'Fault plane %i [deg]: ' \ 'strike = %3.0f, dip = %3.0f, slip-rake = %4.0f\n' \ % (i+1, sdr[0], sdr[1], sdr[2]) return s
[docs] def deviatoric(self): ''' Get deviatoric part of moment tensor. Returns a new moment tensor object with zero trace. ''' m = self.m() trace_m = num.trace(m) m_iso = num.diag([trace_m / 3., trace_m / 3., trace_m / 3.]) m_devi = m - m_iso mt = MomentTensor(m=m_devi) return mt
[docs] def standard_decomposition(self): '''Decompose moment tensor into isotropic, DC and CLVD components. Standard decomposition according to e.g. Jost and Herrmann 1989 is returned as:: [ (moment_iso, ratio_iso, m_iso), (moment_dc, ratio_dc, m_dc), (moment_clvd, ratio_clvd, m_clvd), (moment_devi, ratio_devi, m_devi), (moment, 1.0, m) ] ''' epsilon = 1e-6 m = self.m() trace_m = num.trace(m) m_iso = num.diag([trace_m / 3., trace_m / 3., trace_m / 3.]) moment_iso = abs(trace_m / 3.) m_devi = m - m_iso evals, evecs = eigh_check(m_devi) moment_devi = num.max(num.abs(evals)) moment = moment_iso + moment_devi iorder = num.argsort(num.abs(evals)) evals_sorted = evals[iorder] evecs_sorted = (evecs.T[iorder]).T if moment_devi < epsilon * moment_iso: signed_moment_dc = 0. else: assert -epsilon <= -evals_sorted[0] / evals_sorted[2] <= 0.5 signed_moment_dc = evals_sorted[2] * (1.0 + 2.0 * ( min(0.0, evals_sorted[0] / evals_sorted[2]))) moment_dc = abs(signed_moment_dc) m_dc_es = signed_moment_dc * num.diag([0., -1.0, 1.0]) m_dc = num.dot(evecs_sorted, num.dot(m_dc_es, evecs_sorted.T)) m_clvd = m_devi - m_dc moment_clvd = moment_devi - moment_dc ratio_dc = moment_dc / moment ratio_clvd = moment_clvd / moment ratio_iso = moment_iso / moment ratio_devi = moment_devi / moment return [ (moment_iso, ratio_iso, m_iso), (moment_dc, ratio_dc, m_dc), (moment_clvd, ratio_clvd, m_clvd), (moment_devi, ratio_devi, m_devi), (moment, 1.0, m)]
[docs] def rotated(self, rot): ''' Get rotated moment tensor. :param rot: ratation matrix, coordinate system is NED :returns: new :py:class:`MomentTensor` object ''' rotmat = num.matrix(rot) return MomentTensor(m=rotmat * self.m() * rotmat.T)
[docs] def random_rotated(self, angle_std=None, angle=None, rstate=None): ''' Get distorted MT by rotation around random axis and angle. :param angle_std: angles are drawn from a normal distribution with zero mean and given standard deviation [degrees] :param angle: set angle [degrees], only axis will be random :param rstate: :py:class:`numpy.random.RandomState` object, can be used to create reproducible pseudo-random sequences :returns: new :py:class:`MomentTensor` object ''' assert (angle_std is None) != (angle is None), \ 'either angle or angle_std must be given' if angle_std is not None: rstate = rstate or num.random angle = rstate.normal(scale=angle_std) axis = random_axis(rstate=rstate) rot = rotation_from_angle_and_axis(angle, axis) return self.rotated(rot)
[docs]def other_plane(strike, dip, rake): ''' Get the respectively other plane in the double-couple ambiguity. ''' mt = MomentTensor(strike=strike, dip=dip, rake=rake) both_sdr = mt.both_strike_dip_rake() w = [sum([abs(x-y) for x, y in zip(both_sdr[i], (strike, dip, rake))]) for i in (0, 1)] if w[0] < w[1]: return both_sdr[1] else: return both_sdr[0]
def dsdr(sdr1, sdr2): s1, d1, r1 = sdr1 s2, d2, r2 = sdr2 s1 = s1 % 360. s2 = s2 % 360. r1 = r1 % 360. r2 = r2 % 360. ds = abs(s1 - s2) ds = ds if ds <= 180. else 360. - ds dr = abs(r1 - r2) dr = dr if dr <= 180. else 360. - dr dd = abs(d1 - d2) return math.sqrt(ds**2 + dr**2 + dd**2)
[docs]def order_like(sdrs, sdrs_ref): ''' Order strike-dip-rake pair post closely to a given reference pair. :param sdrs: tuple, ``((strike1, dip1, rake1), (strike2, dip2, rake2))`` :param sdrs_ref: as above but with reference pair ''' d1 = min(dsdr(sdrs[0], sdrs_ref[0]), dsdr(sdrs[1], sdrs_ref[1])) d2 = min(dsdr(sdrs[0], sdrs_ref[1]), dsdr(sdrs[1], sdrs_ref[0])) if d1 < d2: return sdrs else: return sdrs[::-1]
def _tpb2q(t, p, b): eps = 0.001 tqw = 1. + t[0] + p[1] + b[2] tqx = 1. + t[0] - p[1] - b[2] tqy = 1. - t[0] + p[1] - b[2] tqz = 1. - t[0] - p[1] + b[2] q = num.zeros(4) if tqw > eps: q[0] = 0.5 * math.sqrt(tqw) q[1:] = p[2] - b[1], b[0] - t[2], t[1] - p[0] elif tqx > eps: q[0] = 0.5 * math.sqrt(tqx) q[1:] = p[2] - b[1], p[0] + t[1], b[0] + t[2] elif tqy > eps: q[0] = 0.5 * math.sqrt(tqy) q[1:] = b[0] - t[2], p[0] + t[1], b[1] + p[2] elif tqz > eps: q[0] = 0.5 * math.sqrt(tqz) q[1:] = t[1] - p[0], b[0] + t[2], b[1] + p[2] else: raise Exception('should not reach this line, check theory!') # q[0] = max(0.5 * math.sqrt(tqx), eps) # q[1:] = p[2] - b[1], p[0] + t[1], b[0] + t[2] q[1:] /= 4.0 * q[0] q /= math.sqrt(num.sum(q**2)) return q _pbt2tpb = num.matrix(((0., 0., 1.), (1., 0., 0.), (0., 1., 0.)))
[docs]def kagan_angle(mt1, mt2): ''' Given two moment tensors, return the Kagan angle in degrees. After Kagan (1991) and Tape & Tape (2012). ''' ai = _pbt2tpb * mt1._m_eigenvecs.T aj = _pbt2tpb * mt2._m_eigenvecs.T u = ai * aj.T tk, pk, bk = u.A qk = _tpb2q(tk, pk, bk) return 2. * r2d * math.acos(num.max(num.abs(qk)))
[docs]def rand_to_gutenberg_richter(rand, b_value, magnitude_min): ''' Draw magnitude from Gutenberg Richter distribution. ''' return magnitude_min + num.log10(1.-rand) / -b_value
if __name__ == '__main__': import sys v = list(map(float, sys.argv[1:])) mt = MomentTensor.from_values(v) print(mt)