# The Hazard Library
# Copyright (C) 2012-2014, GEM Foundation
#
# This program is free software: you can redistribute it and/or modify
# it under the terms of the GNU Affero General Public License as
# published by the Free Software Foundation, either version 3 of the
# License, or (at your option) any later version.
#
# This program is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
# GNU Affero General Public License for more details.
#
# You should have received a copy of the GNU Affero General Public License
# along with this program. If not, see <http://www.gnu.org/licenses/>.
"""
Module exports :class:`BooreAtkinson2008`.
"""
from __future__ import division
import numpy as np
from openquake.hazardlib.gsim.base import GMPE, CoeffsTable
from openquake.hazardlib import const
from openquake.hazardlib.imt import PGA, PGV, SA
[docs]class BooreAtkinson2008(GMPE):
"""
Implements GMPE developed by David M. Boore and Gail M. Atkinson
and published as "Ground-Motion Prediction Equations for the
Average Horizontal Component of PGA, PGV, and 5%-Damped PSA
at Spectral Periods between 0.01 and 10.0 s" (2008, Earthquake Spectra,
Volume 24, No. 1, pages 99-138).
"""
#: Supported tectonic region type is active shallow crust, see
#: paragraph 'Introduction', page 99.
DEFINED_FOR_TECTONIC_REGION_TYPE = const.TRT.ACTIVE_SHALLOW_CRUST
#: Supported intensity measure types are spectral acceleration,
#: peak ground velocity and peak ground acceleration, see table 3
#: pag. 110
DEFINED_FOR_INTENSITY_MEASURE_TYPES = set([
PGA,
PGV,
SA
])
#: Supported intensity measure component is orientation-independent
#: measure :attr:`~openquake.hazardlib.const.IMC.GMRotI50`, see paragraph
#: 'Response Variables', page 100 and table 8, pag 121.
DEFINED_FOR_INTENSITY_MEASURE_COMPONENT = const.IMC.GMRotI50
#: Supported standard deviation types are inter-event, intra-event
#: and total, see equation 2, pag 106.
DEFINED_FOR_STANDARD_DEVIATION_TYPES = set([
const.StdDev.TOTAL,
const.StdDev.INTER_EVENT,
const.StdDev.INTRA_EVENT
])
#: Required site parameters is Vs30.
#: See paragraph 'Predictor Variables', pag 103
REQUIRES_SITES_PARAMETERS = set(('vs30', ))
#: Required rupture parameters are magnitude, and rake.
#: See paragraph 'Predictor Variables', pag 103
REQUIRES_RUPTURE_PARAMETERS = set(('mag', 'rake'))
#: Required distance measure is Rjb.
#: See paragraph 'Predictor Variables', pag 103
REQUIRES_DISTANCES = set(('rjb', ))
[docs] def get_mean_and_stddevs(self, sites, rup, dists, imt, stddev_types):
"""
See :meth:`superclass method
<.base.GroundShakingIntensityModel.get_mean_and_stddevs>`
for spec of input and result values.
"""
# extracting dictionary of coefficients specific to required
# intensity measure type.
C = self.COEFFS[imt]
C_SR = self.COEFFS_SOIL_RESPONSE[imt]
# compute PGA on rock conditions - needed to compute non-linear
# site amplification term
pga4nl = self._get_pga_on_rock(rup, dists, C)
# equation 1, pag 106, without sigma term, that is only the first 3
# terms. The third term (site amplification) is computed as given in
# equation (6), that is the sum of a linear term - equation (7) - and
# a non-linear one - equations (8a) to (8c).
# Mref, Rref values are given in the caption to table 6, pag 119.
if imt == PGA():
# avoid recomputing PGA on rock, just add site terms
mean = np.log(pga4nl) + \
self._get_site_amplification_linear(sites.vs30, C_SR) + \
self._get_site_amplification_non_linear(sites.vs30, pga4nl, C_SR)
else:
mean = self._compute_magnitude_scaling(rup, C) + \
self._compute_distance_scaling(rup, dists, C) + \
self._get_site_amplification_linear(sites.vs30, C_SR) + \
self._get_site_amplification_non_linear(sites.vs30, pga4nl, C_SR)
stddevs = self._get_stddevs(C, stddev_types, num_sites=len(sites.vs30))
return mean, stddevs
def _get_stddevs(self, C, stddev_types, num_sites):
"""
Return standard deviations as defined in table 8, pag 121.
"""
stddevs = []
for stddev_type in stddev_types:
assert stddev_type in self.DEFINED_FOR_STANDARD_DEVIATION_TYPES
if stddev_type == const.StdDev.TOTAL:
stddevs.append(C['std'] + np.zeros(num_sites))
elif stddev_type == const.StdDev.INTRA_EVENT:
stddevs.append(C['sigma'] + np.zeros(num_sites))
elif stddev_type == const.StdDev.INTER_EVENT:
stddevs.append(C['tau'] + np.zeros(num_sites))
return stddevs
def _compute_distance_scaling(self, rup, dists, C):
"""
Compute distance-scaling term, equations (3) and (4), pag 107.
"""
Mref = 4.5
Rref = 1.0
R = np.sqrt(dists.rjb ** 2 + C['h'] ** 2)
return (C['c1'] + C['c2'] * (rup.mag - Mref)) * np.log(R / Rref) + \
C['c3'] * (R - Rref)
def _compute_magnitude_scaling(self, rup, C):
"""
Compute magnitude-scaling term, equations (5a) and (5b), pag 107.
"""
U, SS, NS, RS = self._get_fault_type_dummy_variables(rup)
if rup.mag <= C['Mh']:
return C['e1'] * U + C['e2'] * SS + C['e3'] * NS + C['e4'] * RS + \
C['e5'] * (rup.mag - C['Mh']) + \
C['e6'] * (rup.mag - C['Mh']) ** 2
else:
return C['e1'] * U + C['e2'] * SS + C['e3'] * NS + C['e4'] * RS + \
C['e7'] * (rup.mag - C['Mh'])
def _get_fault_type_dummy_variables(self, rup):
"""
Get fault type dummy variables, see Table 2, pag 107.
Fault type (Strike-slip, Normal, Thrust/reverse) is
derived from rake angle.
Rakes angles within 30 of horizontal are strike-slip,
angles from 30 to 150 are reverse, and angles from
-30 to -150 are normal. See paragraph 'Predictor Variables'
pag 103.
Note that the 'Unspecified' case is not considered,
because rake is always given.
"""
U, SS, NS, RS = 0, 0, 0, 0
if np.abs(rup.rake) <= 30.0 or (180.0 - np.abs(rup.rake)) <= 30.0:
# strike-slip
SS = 1
elif rup.rake > 30.0 and rup.rake < 150.0:
# reverse
RS = 1
else:
# normal
NS = 1
return U, SS, NS, RS
def _get_site_amplification_linear(self, vs30, C):
"""
Compute site amplification linear term,
equation (7), pag 107.
"""
return C['blin'] * np.log(vs30 / 760.0)
def _get_pga_on_rock(self, rup, dists, _C):
"""
Compute and return PGA on rock conditions (that is vs30 = 760.0 m/s).
This is needed to compute non-linear site amplification term
"""
# Median PGA in g for Vref = 760.0, without site amplification,
# that is equation (1) pag 106, without the third and fourth terms
# Mref and Rref values are given in the caption to table 6, pag 119
# Note that in the original paper, the caption reads:
# "Distance-scaling coefficients (Mref=4.5 and Rref=1.0 km for all
# periods, except Rref=5.0 km for pga4nl)". However this is a mistake
# as reported in http://www.daveboore.com/pubs_online.php:
# ERRATUM: 27 August 2008. Tom Blake pointed out that the caption to
# Table 6 should read "Distance-scaling coefficients (Mref=4.5 and
# Rref=1.0 km for all periods)".
C_pga = self.COEFFS[PGA()]
pga4nl = np.exp(self._compute_magnitude_scaling(rup, C_pga) +
self._compute_distance_scaling(rup, dists, C_pga))
return pga4nl
def _get_site_amplification_non_linear(self, vs30, pga4nl, C):
"""
Compute site amplification non-linear term,
equations (8a) to (13d), pag 108-109.
"""
# non linear slope
bnl = self._compute_non_linear_slope(vs30, C)
# compute the actual non-linear term
return self._compute_non_linear_term(pga4nl, bnl)
def _compute_non_linear_slope(self, vs30, C):
"""
Compute non-linear slope factor,
equations (13a) to (13d), pag 108-109.
"""
V1 = 180.0
V2 = 300.0
Vref = 760.0
# equation (13d), values are zero for vs30 >= Vref = 760.0
bnl = np.zeros(vs30.shape)
# equation (13a)
idx = vs30 <= V1
bnl[idx] = C['b1']
# equation (13b)
idx = np.where((vs30 > V1) & (vs30 <= V2))
bnl[idx] = (C['b1'] - C['b2']) * \
np.log(vs30[idx] / V2) / np.log(V1 / V2) + C['b2']
# equation (13c)
idx = np.where((vs30 > V2) & (vs30 < Vref))
bnl[idx] = C['b2'] * np.log(vs30[idx] / Vref) / np.log(V2 / Vref)
return bnl
def _compute_non_linear_term(self, pga4nl, bnl):
"""
Compute non-linear term,
equation (8a) to (8c), pag 108.
"""
fnl = np.zeros(len(pga4nl))
a1 = 0.03
a2 = 0.09
pga_low = 0.06
# equation (8a)
idx = pga4nl <= a1
fnl[idx] = bnl[idx] * np.log(pga_low / 0.1)
# equation (8b)
idx = np.where((pga4nl > a1) & (pga4nl <= a2))
delta_x = np.log(a2 / a1)
delta_y = bnl[idx] * np.log(a2 / pga_low)
c = (3 * delta_y - bnl[idx] * delta_x) / delta_x ** 2
d = -(2 * delta_y - bnl[idx] * delta_x) / delta_x ** 3
fnl[idx] = bnl[idx] * np.log(pga_low / 0.1) +\
c * (np.log(pga4nl[idx] / a1) ** 2) + \
d * (np.log(pga4nl[idx] / a1) ** 3)
# equation (8c)
idx = pga4nl > a2
fnl[idx] = bnl[idx] * np.log(pga4nl[idx] / 0.1)
return fnl
#: Coefficient table is constructed from values in tables 6, 7 and 8
#: (pages 119, 120, 121). Spectral acceleration is defined for damping
#: of 5%, see 'Response Variables' page 100.
#: c1, c2, c3, h are the period-dependent distance scaling coefficients.
#: e1, e2, e3, e4, e5, e6, e7, Mh are the period-dependent magnitude-
# scaling coefficients.
#: sigma, tau, std are the intra-event uncertainty, inter-event
#: uncertainty, and total standard deviation, respectively.
#: Note that only the inter-event and total standard deviation for
#: 'specified' fault type are considered (because rake angle is always
#: specified)
COEFFS = CoeffsTable(sa_damping=5, table="""\
IMT c1 c2 c3 h e1 e2 e3 e4 e5 e6 e7 Mh sigma tau std
pgv -0.87370 0.10060 -0.00334 2.54 5.00121 5.04727 4.63188 5.08210 0.18322 -0.12736 0.00000 8.50 0.500 0.256 0.560
pga -0.66050 0.11970 -0.01151 1.35 -0.53804 -0.50350 -0.75472 -0.50970 0.28805 -0.10164 0.00000 6.75 0.502 0.260 0.564
0.010 -0.66220 0.12000 -0.01151 1.35 -0.52883 -0.49429 -0.74551 -0.49966 0.28897 -0.10019 0.00000 6.75 0.502 0.262 0.566
0.020 -0.66600 0.12280 -0.01151 1.35 -0.52192 -0.48508 -0.73906 -0.48895 0.25144 -0.11006 0.00000 6.75 0.502 0.262 0.566
0.030 -0.69010 0.12830 -0.01151 1.35 -0.45285 -0.41831 -0.66722 -0.42229 0.17976 -0.12858 0.00000 6.75 0.507 0.274 0.576
0.050 -0.71700 0.13170 -0.01151 1.35 -0.28476 -0.25022 -0.48462 -0.26092 0.06369 -0.15752 0.00000 6.75 0.516 0.286 0.589
0.075 -0.72050 0.12370 -0.01151 1.55 0.00767 0.04912 -0.20578 0.02706 0.01170 -0.17051 0.00000 6.75 0.513 0.320 0.606
0.10 -0.70810 0.11170 -0.01151 1.68 0.20109 0.23102 0.03058 0.22193 0.04697 -0.15948 0.00000 6.75 0.520 0.318 0.608
0.15 -0.69610 0.09884 -0.01113 1.86 0.46128 0.48661 0.30185 0.49328 0.17990 -0.14539 0.00000 6.75 0.518 0.290 0.594
0.20 -0.58300 0.04273 -0.00952 1.98 0.57180 0.59253 0.40860 0.61472 0.52729 -0.12964 0.00102 6.75 0.523 0.288 0.596
0.25 -0.57260 0.02977 -0.00837 2.07 0.51884 0.53496 0.33880 0.57747 0.60880 -0.13843 0.08607 6.75 0.527 0.267 0.592
0.30 -0.55430 0.01955 -0.00750 2.14 0.43825 0.44516 0.25356 0.51990 0.64472 -0.15694 0.10601 6.75 0.546 0.269 0.608
0.40 -0.64430 0.04394 -0.00626 2.24 0.39220 0.40602 0.21398 0.46080 0.78610 -0.07843 0.02262 6.75 0.541 0.267 0.603
0.50 -0.69140 0.06080 -0.00540 2.32 0.18957 0.19878 0.00967 0.26337 0.76837 -0.09054 0.00000 6.75 0.555 0.265 0.615
0.75 -0.74080 0.07518 -0.00409 2.46 -0.21338 -0.19496 -0.49176 -0.10813 0.75179 -0.14053 0.10302 6.75 0.571 0.299 0.645
1.0 -0.81830 0.10270 -0.00334 2.54 -0.46896 -0.43443 -0.78465 -0.39330 0.67880 -0.18257 0.05393 6.75 0.573 0.302 0.647
1.5 -0.83030 0.09793 -0.00255 2.66 -0.86271 -0.79593 -1.20902 -0.88085 0.70689 -0.25950 0.19082 6.75 0.566 0.373 0.679
2.0 -0.82850 0.09432 -0.00217 2.73 -1.22652 -1.15514 -1.57697 -1.27669 0.77989 -0.29657 0.29888 6.75 0.580 0.389 0.700
3.0 -0.78440 0.07282 -0.00191 2.83 -1.82979 -1.74690 -2.22584 -1.91814 0.77966 -0.45384 0.67466 6.75 0.566 0.401 0.695
4.0 -0.68540 0.03758 -0.00191 2.89 -2.24656 -2.15906 -2.58228 -2.38168 1.24961 -0.35874 0.79508 6.75 0.583 0.385 0.698
5.0 -0.50960 -0.02391 -0.00191 2.93 -1.28408 -1.21270 -1.50904 -1.41093 0.14271 -0.39006 0.00000 8.50 0.601 0.437 0.744
7.5 -0.37240 -0.06568 -0.00191 3.00 -1.43145 -1.31632 -1.81022 -1.59217 0.52407 -0.37578 0.00000 8.50 0.626 0.477 0.787
10.0 -0.09824 -0.13800 -0.00191 3.04 -2.15446 -2.16137 -2.53323 -2.14635 0.40387 -0.48492 0.00000 8.50 0.645 0.477 0.801
""")
#: Table 3, pag. 110. + coefficient values for additional frequencies
#: extracted from Fortran code implementing soil response function
#: developed by the original author (ab06_fmrvs_evaluate_gmpes.for
#: available at http://www.daveboore.com/pubs_online.html - see code
#: available for Atkinson, G. M. and D. M. Boore (2006). Earthquake ground
#: -motion prediction equations for eastern North America)
COEFFS_SOIL_RESPONSE = CoeffsTable(sa_damping=5, table="""\
IMT blin b1 b2
pgv -0.60 -0.50 -0.06
pga -0.36 -0.64 -0.14
0.010 -0.36 -0.64 -0.14
0.020 -0.34 -0.63 -0.12
0.030 -0.33 -0.62 -0.11
0.040 -0.31 -0.61 -0.11
0.050 -0.29 -0.64 -0.11
0.060 -0.25 -0.64 -0.11
0.075 -0.23 -0.64 -0.11
0.090 -0.23 -0.64 -0.12
0.100 -0.25 -0.60 -0.13
0.120 -0.26 -0.56 -0.14
0.150 -0.28 -0.53 -0.18
0.170 -0.29 -0.53 -0.19
0.200 -0.31 -0.52 -0.19
0.240 -0.38 -0.52 -0.16
0.250 -0.39 -0.52 -0.16
0.300 -0.44 -0.52 -0.14
0.360 -0.48 -0.51 -0.11
0.400 -0.50 -0.51 -0.10
0.460 -0.55 -0.50 -0.08
0.500 -0.60 -0.50 -0.06
0.600 -0.66 -0.49 -0.03
0.750 -0.69 -0.47 -0.00
0.850 -0.69 -0.46 -0.00
1.000 -0.70 -0.44 -0.00
1.500 -0.72 -0.40 -0.00
2.000 -0.73 -0.38 -0.00
3.000 -0.74 -0.34 -0.00
4.000 -0.75 -0.31 -0.00
5.000 -0.75 -0.291 -0.00
7.500 -0.692 -0.247 -0.00
10.00 -0.650 -0.215 -0.00
""")