1# https://pyrocko.org - GPLv3
2#
3# The Pyrocko Developers, 21st Century
4# ---|P------/S----------~Lg----------
6import numpy as num
7import logging
9from pyrocko import moment_tensor as mt
10from pyrocko.guts import Float, String, Timestamp, Int
11from pyrocko.model import Location
12from pyrocko.modelling import okada_ext
13from pyrocko.util import get_threadpool_limits
15guts_prefix = 'modelling'
17logger = logging.getLogger(__name__)
19d2r = num.pi/180.
20r2d = 180./num.pi
21km = 1e3
24class AnalyticalSource(Location):
25 '''
26 Base class for analytical source models.
27 '''
29 name = String.T(
30 optional=True,
31 default='')
33 time = Timestamp.T(
34 default=0.,
35 help='Source origin time',
36 optional=True)
38 vr = Float.T(
39 default=0.,
40 help='Rupture velocity [m/s]',
41 optional=True)
43 @property
44 def northing(self):
45 return self.north_shift
47 @property
48 def easting(self):
49 return self.east_shift
52class AnalyticalRectangularSource(AnalyticalSource):
53 '''
54 Rectangular analytical source model.
56 Coordinates on the source plane are with respect to the origin point given
57 by `(lat, lon, east_shift, north_shift, depth)`.
58 '''
60 strike = Float.T(
61 default=0.0,
62 help='Strike direction in [deg], measured clockwise from north.')
64 dip = Float.T(
65 default=90.0,
66 help='Dip angle in [deg], measured downward from horizontal.')
68 rake = Float.T(
69 default=0.0,
70 help='Rake angle in [deg], measured counter-clockwise from '
71 'right-horizontal in on-plane view.')
73 al1 = Float.T(
74 default=0.,
75 help='Left edge source plane coordinate [m].')
77 al2 = Float.T(
78 default=0.,
79 help='Right edge source plane coordinate [m].')
81 aw1 = Float.T(
82 default=0.,
83 help='Lower edge source plane coordinate [m].')
85 aw2 = Float.T(
86 default=0.,
87 help='Upper edge source plane coordinate [m].')
89 slip = Float.T(
90 default=0.,
91 help='Slip on the rectangular source area [m].',
92 optional=True)
94 @property
95 def length(self):
96 return abs(-self.al1 + self.al2)
98 @property
99 def width(self):
100 return abs(-self.aw1 + self.aw2)
102 @property
103 def area(self):
104 return self.width * self.length
107class OkadaSource(AnalyticalRectangularSource):
108 '''
109 Rectangular Okada source model.
110 '''
112 opening = Float.T(
113 default=0.,
114 help='Opening of the plane in [m].',
115 optional=True)
117 poisson__ = Float.T(
118 default=0.25,
119 help='Poisson\'s ratio :math:`\\nu`.',
120 optional=True)
122 lamb__ = Float.T(
123 help='First Lame\' s parameter :math:`\\lambda` [Pa].',
124 optional=True)
126 shearmod__ = Float.T(
127 default=32.0e9,
128 help='Shear modulus along the plane :math:`\\mu` [Pa].',
129 optional=True)
131 @property
132 def poisson(self):
133 '''
134 Poisson\' s ratio :math:`\\nu` (if not given).
136 The Poisson\' s ratio :math:`\\nu` can be calculated from the Lame\'
137 parameters :math:`\\lambda` and :math:`\\mu` using :math:`\\nu =
138 \\frac{\\lambda}{2(\\lambda + \\mu)}` (e.g. Mueller 2007).
139 '''
141 if self.poisson__ is not None:
142 return self.poisson__
144 if self.shearmod__ is None or self.lamb__ is None:
145 raise ValueError('Shearmod and lambda are needed')
147 return (self.lamb__) / (2. * (self.lamb__ + self.shearmod__))
149 @poisson.setter
150 def poisson(self, poisson):
151 self.poisson__ = poisson
153 @property
154 def lamb(self):
155 '''
156 First Lame\' s parameter :math:`\\lambda` (if not given).
158 Poisson\' s ratio :math:`\\nu` and shear modulus :math:`\\mu` must be
159 available to calculate the first Lame\' s parameter :math:`\\lambda`.
161 .. important ::
163 We assume a perfect elastic solid with :math:`K=\\frac{5}{3}\\mu`.
165 Through :math:`\\nu = \\frac{\\lambda}{2(\\lambda + \\mu)}` this
166 leads to :math:`\\lambda = \\frac{2 \\mu \\nu}{1-2\\nu}`.
168 '''
170 if self.lamb__ is not None:
171 return self.lamb__
173 if self.shearmod__ is None or self.poisson__ is None:
174 raise ValueError('Shearmod and poisson ratio are needed')
176 return (
177 2. * self.poisson__ * self.shearmod__) / (1. - 2. * self.poisson__)
179 @lamb.setter
180 def lamb(self, lamb):
181 self.lamb__ = lamb
183 @property
184 def shearmod(self):
185 '''
186 Shear modulus :math:`\\mu` (if not given).
188 Poisson ratio\' s :math:`\\nu` must be available.
190 .. important ::
192 We assume a perfect elastic solid with :math:`K=\\frac{5}{3}\\mu`.
194 Through :math:`\\mu = \\frac{3K(1-2\\nu)}{2(1+\\nu)}` this leads to
195 :math:`\\mu = \\frac{8(1+\\nu)}{1-2\\nu}`.
197 '''
199 if self.shearmod__ is not None:
200 return self.shearmod__
202 if self.poisson__ is None:
203 raise ValueError('Poisson ratio is needed')
205 return (8. * (1. + self.poisson__)) / (1. - 2. * self.poisson__)
207 @shearmod.setter
208 def shearmod(self, shearmod):
209 self.shearmod__ = shearmod
211 @property
212 def seismic_moment(self):
213 '''
214 Scalar Seismic moment :math:`M_0`.
216 Code copied from Kite. It disregards the opening (as for now).
217 We assume :math:`M_0 = mu A D`.
219 .. important ::
221 We assume a perfect elastic solid with :math:`K=\\frac{5}{3}\\mu`.
223 Through :math:`\\mu = \\frac{3K(1-2\\nu)}{2(1+\\nu)}` this leads to
224 :math:`\\mu = \\frac{8(1+\\nu)}{1-2\\nu}`.
226 :return:
227 Seismic moment release.
228 :rtype:
229 float
230 '''
232 mu = self.shearmod
234 disl = 0.
235 if self.slip:
236 disl = self.slip
237 if self.opening:
238 disl = (disl**2 + self.opening**2)**.5
240 return mu * self.area * disl
242 @property
243 def moment_magnitude(self):
244 '''
245 Moment magnitude :math:`M_\\mathrm{w}` from seismic moment.
247 We assume :math:`M_\\mathrm{w} = {\\frac{2}{3}}\\log_{10}(M_0) - 10.7`.
249 :returns:
250 Moment magnitude.
251 :rtype:
252 float
253 '''
254 return mt.moment_to_magnitude(self.seismic_moment)
256 def source_patch(self):
257 '''
258 Get source location and geometry array for okada_ext.okada input.
260 The values are defined according to Okada (1992).
262 :return:
263 Source data as input for okada_ext.okada. The order is
264 northing [m], easting [m], depth [m], strike [deg], dip [deg],
265 al1 [m], al2 [m], aw1 [m], aw2 [m].
266 :rtype:
267 :py:class:`~numpy.ndarray`: ``(9, )``
268 '''
269 return num.array([
270 self.northing,
271 self.easting,
272 self.depth,
273 self.strike,
274 self.dip,
275 self.al1,
276 self.al2,
277 self.aw1,
278 self.aw2])
280 def source_disloc(self):
281 '''
282 Get source dislocation array for okada_ext.okada input.
284 The given slip is splitted into a strike and an updip part based on the
285 source rake.
287 :return:
288 Source dislocation data as input for okada_ext.okada. The order is
289 dislocation in strike [m], dislocation updip [m], opening [m].
290 :rtype:
291 :py:class:`~numpy.ndarray`: ``(3, )``
292 '''
293 return num.array([
294 num.cos(self.rake * d2r) * self.slip,
295 num.sin(self.rake * d2r) * self.slip,
296 self.opening])
298 def discretize(self, nlength, nwidth, *args, **kwargs):
299 '''
300 Discretize fault into rectilinear grid of fault patches.
302 Fault orientation, slip and elastic parameters are passed to the
303 sub-faults unchanged.
305 :param nlength:
306 Number of patches in strike direction.
307 :type nlength:
308 int
310 :param nwidth:
311 Number of patches in down-dip direction.
312 :type nwidth:
313 int
315 :return:
316 Discrete fault patches.
317 :rtype:
318 list of :py:class:`~pyrocko.modelling.okada.OkadaPatch`
319 '''
320 assert nlength > 0
321 assert nwidth > 0
323 il = num.repeat(num.arange(nlength), nwidth)
324 iw = num.tile(num.arange(nwidth), nlength)
326 patch_length = self.length / nlength
327 patch_width = self.width / nwidth
329 al1 = -patch_length / 2.
330 al2 = patch_length / 2.
331 aw1 = -patch_width / 2.
332 aw2 = patch_width / 2.
334 source_points = num.zeros((nlength * nwidth, 3))
335 source_points[:, 0] = il * patch_length + patch_length / 2.
336 source_points[:, 1] = iw * patch_width + patch_width / 2.
338 source_points[:, 0] += self.al1
339 source_points[:, 1] -= self.aw2
341 rotmat = mt.euler_to_matrix(self.dip*d2r, self.strike*d2r, 0.)
343 source_points_rot = num.dot(rotmat.T, source_points.T).T
344 source_points_rot[:, 0] += self.northing
345 source_points_rot[:, 1] += self.easting
346 source_points_rot[:, 2] += self.depth
348 kwargs = {
349 prop: getattr(self, prop) for prop in self.T.propnames
350 if prop not in [
351 'north_shift', 'east_shift', 'depth',
352 'al1', 'al2', 'aw1', 'aw2']}
354 return (
355 [OkadaPatch(
356 parent=self,
357 ix=src_point[0],
358 iy=src_point[1],
359 north_shift=coord[0],
360 east_shift=coord[1],
361 depth=coord[2],
362 al1=al1, al2=al2, aw1=aw1, aw2=aw2, **kwargs)
363 for src_point, coord in zip(source_points, source_points_rot)],
364 source_points)
367class OkadaPatch(OkadaSource):
369 '''
370 Okada source with additional 2D indexes for bookkeeping.
371 '''
373 ix = Int.T(help='Relative index of the patch in x')
374 iy = Int.T(help='Relative index of the patch in y')
376 def __init__(self, parent=None, *args, **kwargs):
377 OkadaSource.__init__(self, *args, **kwargs)
378 self.parent = parent
381def make_okada_coefficient_matrix(
382 source_patches_list,
383 pure_shear=False,
384 rotate_sdn=True,
385 nthreads=1, variant='normal'):
387 '''
388 Build coefficient matrix for given fault patches.
390 The boundary element method (BEM) for a discretized fault and the
391 determination of the slip distribution :math:`\\Delta u` from stress drop
392 :math:`\\Delta \\sigma` is based on
393 :math:`\\Delta \\sigma = \\mathbf{C} \\cdot \\Delta u`. Here the
394 coefficient matrix :math:`\\mathbf{C}` is built, based on the displacements
395 from Okada's solution (Okada, 1992) and their partial derivatives.
397 :param source_patches_list:
398 Source patches, to be used in BEM.
399 :type source_patches_list:
400 list of :py:class:`~pyrocko.modelling.okada.OkadaSource`.
402 :param pure_shear:
403 If ``True``, only shear forces are taken into account.
404 :type pure_shear:
405 optional, bool
407 :param rotate_sdn:
408 If ``True``, rotate to strike, dip, normal.
409 :type rotate_sdn:
410 optional, bool
412 :param nthreads:
413 Number of threads.
414 :type nthreads:
415 optional, int
417 :return:
418 Coefficient matrix for all source combinations.
419 :rtype:
420 :py:class:`~numpy.ndarray`:
421 ``(len(source_patches_list) * 3, len(source_patches_list) * 3)``
422 '''
424 if variant == 'slow':
425 return _make_okada_coefficient_matrix_slow(
426 source_patches_list, pure_shear, rotate_sdn, nthreads)
428 source_patches = num.array([
429 src.source_patch() for src in source_patches_list])
430 receiver_coords = source_patches[:, :3].copy()
432 npoints = len(source_patches_list)
434 if pure_shear:
435 n_eq = 2
436 else:
437 n_eq = 3
439 coefmat = num.zeros((npoints * 3, npoints * 3))
441 lambda_mean = num.mean([src.lamb for src in source_patches_list])
442 mu_mean = num.mean([src.shearmod for src in source_patches_list])
444 unit_disl = 1.
445 disl_cases = {
446 'strikeslip': {
447 'slip': unit_disl,
448 'opening': 0.,
449 'rake': 0.},
450 'dipslip': {
451 'slip': unit_disl,
452 'opening': 0.,
453 'rake': 90.},
454 'tensileslip': {
455 'slip': 0.,
456 'opening': unit_disl,
457 'rake': 0.}
458 }
460 diag_ind = [0, 4, 8]
461 kron = num.zeros(9)
462 kron[diag_ind] = 1.
464 if variant == 'normal':
465 kron = kron[num.newaxis, num.newaxis, :]
466 else:
467 kron = kron[num.newaxis, :]
469 for idisl, case_type in enumerate([
470 'strikeslip', 'dipslip', 'tensileslip'][:n_eq]):
471 case = disl_cases[case_type]
472 source_disl = num.array([
473 case['slip'] * num.cos(case['rake'] * d2r),
474 case['slip'] * num.sin(case['rake'] * d2r),
475 case['opening']])
477 if variant == 'normal':
478 results = okada_ext.okada(
479 source_patches,
480 num.tile(source_disl, npoints).reshape(-1, 3),
481 receiver_coords,
482 lambda_mean,
483 mu_mean,
484 nthreads=nthreads,
485 rotate_sdn=int(rotate_sdn),
486 stack_sources=int(variant != 'normal'))
488 eps = 0.5 * (
489 results[:, :, 3:] +
490 results[:, :, (3, 6, 9, 4, 7, 10, 5, 8, 11)])
492 dilatation \
493 = eps[:, :, diag_ind].sum(axis=-1)[:, :, num.newaxis]
495 stress_sdn = kron*lambda_mean*dilatation + 2.*mu_mean*eps
496 coefmat[:, idisl::3] = stress_sdn[:, :, (2, 5, 8)]\
497 .reshape(-1, npoints*3).T
498 else:
499 for isrc, source in enumerate(source_patches):
500 results = okada_ext.okada(
501 source.reshape(1, -1),
502 source_disl.reshape(1, -1),
503 receiver_coords,
504 lambda_mean,
505 mu_mean,
506 nthreads=nthreads,
507 rotate_sdn=int(rotate_sdn))
509 eps = 0.5 * (
510 results[:, 3:] +
511 results[:, (3, 6, 9, 4, 7, 10, 5, 8, 11)])
513 dilatation \
514 = num.sum(eps[:, diag_ind], axis=1)[:, num.newaxis]
515 stress_sdn \
516 = kron * lambda_mean * dilatation+2. * mu_mean * eps
518 coefmat[:, isrc*3 + idisl] \
519 = stress_sdn[:, (2, 5, 8)].ravel()
521 if pure_shear:
522 coefmat[2::3, :] = 0.
524 return -coefmat / unit_disl
527def _make_okada_coefficient_matrix_slow(
528 source_patches_list, pure_shear=False, rotate_sdn=True, nthreads=1):
530 source_patches = num.array([
531 src.source_patch() for src in source_patches_list])
532 receiver_coords = source_patches[:, :3].copy()
534 npoints = len(source_patches_list)
536 if pure_shear:
537 n_eq = 2
538 else:
539 n_eq = 3
541 coefmat = num.zeros((npoints * 3, npoints * 3))
543 def ned2sdn_rotmat(strike, dip):
544 rotmat = mt.euler_to_matrix(
545 (dip + 180.) * d2r, strike * d2r, 0.)
546 return rotmat
548 lambda_mean = num.mean([src.lamb for src in source_patches_list])
549 shearmod_mean = num.mean([src.shearmod for src in source_patches_list])
551 unit_disl = 1.
552 disl_cases = {
553 'strikeslip': {
554 'slip': unit_disl,
555 'opening': 0.,
556 'rake': 0.},
557 'dipslip': {
558 'slip': unit_disl,
559 'opening': 0.,
560 'rake': 90.},
561 'tensileslip': {
562 'slip': 0.,
563 'opening': unit_disl,
564 'rake': 0.}
565 }
566 for idisl, case_type in enumerate([
567 'strikeslip', 'dipslip', 'tensileslip'][:n_eq]):
568 case = disl_cases[case_type]
569 source_disl = num.array([
570 case['slip'] * num.cos(case['rake'] * d2r),
571 case['slip'] * num.sin(case['rake'] * d2r),
572 case['opening']])
574 for isource, source in enumerate(source_patches):
575 results = okada_ext.okada(
576 source[num.newaxis, :].copy(),
577 source_disl[num.newaxis, :].copy(),
578 receiver_coords,
579 lambda_mean,
580 shearmod_mean,
581 nthreads=nthreads,
582 rotate_sdn=int(rotate_sdn))
584 for irec in range(receiver_coords.shape[0]):
585 eps = num.zeros((3, 3))
586 for m in range(3):
587 for n in range(3):
588 eps[m, n] = 0.5 * (
589 results[irec][m * 3 + n + 3] +
590 results[irec][n * 3 + m + 3])
592 stress = num.zeros((3, 3))
593 dilatation = num.sum([eps[i, i] for i in range(3)])
595 for m, n in zip([0, 0, 0, 1, 1, 2], [0, 1, 2, 1, 2, 2]):
596 if m == n:
597 stress[m, n] = \
598 lambda_mean * \
599 dilatation + \
600 2. * shearmod_mean * \
601 eps[m, n]
603 else:
604 stress[m, n] = \
605 2. * shearmod_mean * \
606 eps[m, n]
607 stress[n, m] = stress[m, n]
609 normal = num.array([0., 0., -1.])
610 for isig in range(3):
611 tension = num.sum(stress[isig, :] * normal)
612 coefmat[irec * n_eq + isig, isource * n_eq + idisl] = \
613 tension / unit_disl
615 return coefmat
618def invert_fault_dislocations_bem(
619 stress_field,
620 coef_mat=None,
621 source_list=None,
622 pure_shear=False,
623 epsilon=None,
624 nthreads=1,
625 **kwargs):
626 '''
627 BEM least squares inversion to get fault dislocations given stress field.
629 Follows least squares inversion approach by Menke (1989) to calculate
630 dislocations on a fault with several segments from a given stress field.
631 The coefficient matrix connecting stresses and displacements of the fault
632 patches can either be specified by the user (``coef_mat``) or it is
633 calculated using the solution of Okada (1992) for a rectangular fault in a
634 homogeneous half space (``source_list``).
636 :param stress_field:
637 Stress change [Pa] for each source patch (as
638 ``stress_field[isource, icomponent]`` where isource indexes the source
639 patch and ``icomponent`` indexes component, ordered (strike, dip,
640 tensile).
641 :type stress_field:
642 :py:class:`~numpy.ndarray`: ``(nsources, 3)``
644 :param coef_mat:
645 Coefficient matrix connecting source patch dislocations and the stress
646 field.
647 :type coef_mat:
648 optional, :py:class:`~numpy.ndarray`:
649 ``(len(source_list) * 3, len(source_list) * 3)``
651 :param source_list:
652 Source patches to be used for BEM.
653 :type source_list:
654 optional, list of
655 :py:class:`~pyrocko.modelling.okada.OkadaSource`
657 :param epsilon:
658 If given, values in ``coef_mat`` smaller than ``epsilon`` are set to
659 zero.
660 :type epsilon:
661 optional, float
663 :param nthreads:
664 Number of threads allowed.
665 :type nthreads:
666 int
668 :return:
669 Inverted displacements as ``displacements[isource, icomponent]``
670 where isource indexes the source patch and ``icomponent`` indexes
671 component, ordered (strike, dip, tensile).
672 :rtype:
673 :py:class:`~numpy.ndarray`: ``(nsources, 3)``
674 '''
676 if source_list is not None and coef_mat is None:
677 coef_mat = make_okada_coefficient_matrix(
678 source_list, pure_shear=pure_shear, nthreads=nthreads,
679 **kwargs)
681 if epsilon is not None:
682 coef_mat[coef_mat < epsilon] = 0.
684 idx = num.arange(0, coef_mat.shape[0])
685 if pure_shear:
686 idx = idx[idx % 3 != 2]
688 coef_mat_in = coef_mat[idx, :][:, idx]
689 disloc_est = num.zeros(coef_mat.shape[0])
691 if stress_field.ndim == 2:
692 stress_field = stress_field.ravel()
694 threadpool_limits = get_threadpool_limits()
696 with threadpool_limits(limits=nthreads, user_api='blas'):
697 try:
698 disloc_est[idx] = num.linalg.multi_dot([
699 num.linalg.inv(num.dot(coef_mat_in.T, coef_mat_in)),
700 coef_mat_in.T,
701 stress_field[idx]])
702 except num.linalg.LinAlgError as e:
703 logger.warning('Linear inversion failed!')
704 logger.warning(
705 'coef_mat: %s\nstress_field: %s',
706 coef_mat_in, stress_field[idx])
707 raise e
708 return disloc_est.reshape(-1, 3)
711__all__ = [
712 'AnalyticalSource',
713 'AnalyticalRectangularSource',
714 'OkadaSource',
715 'OkadaPatch',
716 'make_okada_coefficient_matrix',
717 'invert_fault_dislocations_bem']