# -*- coding: utf-8 -*-
# vim: tabstop=4 shiftwidth=4 softtabstop=4
#
# Copyright (C) 2012-2019 GEM Foundation
#
# OpenQuake 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.
#
# OpenQuake 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 OpenQuake. If not, see <http://www.gnu.org/licenses/>.
"""
Module exports :class:`Bradley2013`, :class:`Bradley2013Volc`.
"""
import numpy as np
from openquake.hazardlib.gsim.base import GMPE, CoeffsTable
from openquake.hazardlib import const
from openquake.hazardlib.imt import PGA, SA
[docs]class Bradley2013(GMPE):
"""
Implements GMPE developed by Brendan Bradley for Active Shallow Crust
Earthquakes for New Zealand, and published as "A New Zealand-Specific
Pseudospectral Acceleration Ground-Motion Prediction Equation for Active
Shallow Crustal Earthquakes Based on Foreign Models" (2013, Bulletin of
the Seismological Society of America, Volume 103, No. 3, pages 1801-1822).
This model is modified from Chiou and Youngs, 2008 and has been adapted
for New Zealand conditions. Specifically, the modifications are related to:
1) small magnitude scaling;
2) scaling of short period ground motion from normal faulting events in
volcanic crust;
3) scaling of ground motions on very hard rock sites;
4) anelastic attenuation in the New Zealand crust;
5) consideration of the increates anelastic attenuation in the Taupo
Volcanic Zone (not implemented in this model, use Bradley2013Volc)
"""
#: Supported tectonic region type is active shallow crust, see page 1801
DEFINED_FOR_TECTONIC_REGION_TYPE = const.TRT.ACTIVE_SHALLOW_CRUST
#: Supported intensity measure types are spectral acceleration,
#: peak ground velocity and peak ground acceleration. Note that PGV is
#: the Chiou & Youngs PGV and has not been modified for New Zealand.
DEFINED_FOR_INTENSITY_MEASURE_TYPES = set([
PGA,
SA
])
#: Supported intensity measure component is geometric mean
#: of two horizontal components
#: attr:`~openquake.hazardlib.const.IMC.AVERAGE_HORIZONTAL`,
#: see abstract page 1801.
DEFINED_FOR_INTENSITY_MEASURE_COMPONENT = const.IMC.AVERAGE_HORIZONTAL
#: Supported standard deviation types are inter-event, intra-event
#: and total, see chapter "Variance model".
DEFINED_FOR_STANDARD_DEVIATION_TYPES = set([
const.StdDev.TOTAL,
const.StdDev.INTER_EVENT,
const.StdDev.INTRA_EVENT
])
#: Required site parameters are Vs30 (eq. 13b), Vs30 measured flag (eq. 20)
#: and Z1.0 (eq. 13b).
REQUIRES_SITES_PARAMETERS = set(('vs30', 'vs30measured', 'z1pt0'))
#: Required rupture parameters are magnitude, rake (eq. 13a and 13b),
#: dip (eq. 13a) and ztor (eq. 13a).
REQUIRES_RUPTURE_PARAMETERS = set(('dip', 'rake', 'mag', 'ztor'))
#: Required distance measures are RRup, Rjb and Rx (all are in eq. 13a).
REQUIRES_DISTANCES = set(('rrup', 'rjb', 'rx'))
[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]
# intensity on a reference soil is used for both mean
# and stddev calculations.
ln_y_ref = self._get_ln_y_ref(rup, dists, C)
# exp1 and exp2 are parts of eq. 7
exp1 = np.exp(C['phi3'] * (sites.vs30.clip(-np.inf, 1130) - 360))
exp2 = np.exp(C['phi3'] * (1130 - 360))
# v1 is the period dependent site term. The Vs30 above which, the
# amplification is constant
v1 = self._get_v1(imt)
mean = self._get_mean(sites, C, ln_y_ref, exp1, exp2, v1)
stddevs = self._get_stddevs(sites, rup, C, stddev_types,
ln_y_ref, exp1, exp2)
return mean, stddevs
def _get_mean(self, sites, C, ln_y_ref, exp1, exp2, v1):
"""
Add site effects to an intensity.
Implements eq. 5
"""
# we do not support estimating of basin depth and instead
# rely on it being available (since we require it).
z1pt0 = sites.z1pt0
# we consider random variables being zero since we want
# to find the exact mean value.
eta = epsilon = 0
ln_y = (
# first line of eq. 13b
ln_y_ref + C['phi1'] *
np.log(np.clip(sites.vs30, -np.inf, v1) / 1130)
# second line
+ C['phi2'] * (exp1 - exp2)
* np.log((np.exp(ln_y_ref) + C['phi4']) / C['phi4'])
# third line
+ C['phi5']
* (1.0 - 1.0 / np.cosh(
C['phi6'] * (z1pt0 - C['phi7']).clip(0, np.inf)))
+ C['phi8'] / np.cosh(0.15 * (z1pt0 - 15).clip(0, np.inf))
# fourth line
+ eta + epsilon
)
return ln_y
def _get_stddevs(self, sites, rup, C, stddev_types, ln_y_ref, exp1, exp2):
"""
Get standard deviation for a given intensity on reference soil.
Implements equations 19, 20 and 21 of Chiou & Youngs, 2008 for
inter-event, intra-event and total standard deviations respectively.
This has not been modified for NZ conditions.
"""
# aftershock flag is zero, we consider only main shock.
AS = 0
Fmeasured = sites.vs30measured
Finferred = 1 - sites.vs30measured
# eq. 19 to calculate inter-event standard error
mag_test = min(max(rup.mag, 5.0), 7.0) - 5.0
tau = C['tau1'] + (C['tau2'] - C['tau1']) / 2 * mag_test
# b and c coeffs from eq. 10
b = C['phi2'] * (exp1 - exp2)
c = C['phi4']
y_ref = np.exp(ln_y_ref)
# eq. 20
NL = b * y_ref / (y_ref + c)
sigma = (
# first line of eq. 20
(C['sig1']
+ 0.5 * (C['sig2'] - C['sig1']) * mag_test
+ C['sig4'] * AS)
# second line
* np.sqrt((C['sig3'] * Finferred + 0.7 * Fmeasured)
+ (1 + NL) ** 2)
)
ret = []
for stddev_type in stddev_types:
assert stddev_type in self.DEFINED_FOR_STANDARD_DEVIATION_TYPES
if stddev_type == const.StdDev.TOTAL:
# eq. 21
ret += [np.sqrt(((1 + NL) ** 2) * (tau ** 2) + (sigma ** 2))]
elif stddev_type == const.StdDev.INTRA_EVENT:
ret.append(sigma)
elif stddev_type == const.StdDev.INTER_EVENT:
# this is implied in eq. 21
ret.append(np.abs((1 + NL) * tau))
return ret
def _get_ln_y_ref(self, rup, dists, C):
"""
Get an intensity on a reference soil.
Implements eq. 4 in Bradley 2013. This is the same as Chiou and
Youngs 2008, with addition of TVZ attentuation term, and addition of
c8 which constains the ZTOR. Note that the TVZ scaling is set to 1
(i.e. no TVZ attenuation)
"""
# Taupo Volcanic Zone Path Distance. Set to zero.
rtvz = self._get_tvz_path_distance(dists.rrup)
# reverse faulting flag
Frv = 1 if 30 <= rup.rake <= 150 else 0
# normal faulting flag
Fnm = 1 if -120 <= rup.rake <= -60 else 0
# hanging wall flag
Fhw = (dists.rx >= 0)
# aftershock flag. always zero since we only consider main shock
AS = 0
ln_y_ref = (
# first line of eq. 4 in Bradley 2013
C['c1']
+ (C['c1a'] * Frv
+ C['c1b'] * Fnm
+ C['c7'] * (np.clip(rup.ztor, -np.inf, C['c8']) - 4))
* (1 - AS)
+ (C['c10'] + C['c7a'] * (rup.ztor - 4)) * AS
# second line
+ C['c2'] * (rup.mag - 6)
+ ((C['c2'] - C['c3']) / C['cn'])
* np.log(1 + np.exp(C['cn'] * (C['cm'] - rup.mag)))
# third line
+ C['c4']
* np.log(dists.rrup
+ C['c5']
* np.cosh(C['c6'] * max(rup.mag - C['chm'], 0)))
# fourth line
+ (C['c4a'] - C['c4'])
* np.log(np.sqrt(dists.rrup ** 2 + C['crb'] ** 2))
# fifth line
+ (C['cg1'] + C['cg2'] / (np.cosh(max(rup.mag - C['cg3'], 0))))
# sixth line
* ((1 + C['ctvz'] * (rtvz / dists.rrup)) * dists.rrup)
# seventh line
+ C['c9'] * Fhw
* np.tanh(dists.rx
* (np.cos(np.radians(rup.dip)) ** 2)
/ C['c9a'])
* (1 - np.sqrt(dists.rjb ** 2 + rup.ztor ** 2)
/ (dists.rrup + 0.001))
)
return ln_y_ref
def _get_v1(self, imt):
"""
Calculates Bradley's V1 term. Equation 2 (page 1814) and 6 (page 1816)
based on SA period
"""
if imt == PGA():
v1 = 1800.
else:
T = imt.period
v1a = np.clip((1130 * (T / 0.75)**-0.11), 1130, np.inf)
v1 = np.clip(v1a, -np.inf, 1800.)
return v1
def _get_tvz_path_distance(self, rrup):
"""
Returns Taupo Volcanic Zone (TVZ) path distance.
Set to zero.
"""
return 0
#: Coefficient tables are constructed from values in tables 1, 2 and 3
#: (pages 197, 198 and 199) in Chiou & Youngs,2008. Only Coefficients c1,
#: c1b, c3, cm, c8, cg1, cg2, ctvz are modified by Bradley 2013.
#: Spectral acceleration is defined for damping of 5%, see page 208 (CY08).
COEFFS = CoeffsTable(sa_damping=5, table="""\
IMT c2 c3 c4 c4a crb chm cg3 c1 c1a c1b cn cm c5 c6 c7 c7a c8 c9 c9a c10 cg1 cg2 ctvz phi1 phi2 phi3 phi4 phi5 phi6 phi7 phi8 tau1 tau2 sig1 sig2 sig3 sig4
pga 1.06 1.50000 -2.1 -0.5 50.0 3.0 4.0 -1.1985 0.1000 -0.4550 2.996 5.85000 6.1600 0.4893 0.0512 0.0860 10.00 0.7900 1.5005 -0.3218 -0.00960 -0.00480 2.000 -0.4417 -0.1417 -0.007010 0.102151 0.2289 0.014996 580.0 0.0700 0.3437 0.2637 0.4458 0.3459 0.8000 0.0663
0.010 1.06 1.50299 -2.1 -0.5 50.0 3.0 4.0 -1.1958 0.1000 -0.4550 2.996 5.81711 6.1600 0.4893 0.0512 0.0860 10.00 0.7900 1.5005 -0.3218 -0.00960 -0.00481 2.000 -0.4417 -0.1417 -0.007010 0.102151 0.2289 0.014996 580.0 0.0700 0.3437 0.2637 0.4458 0.3459 0.8000 0.0663
0.020 1.06 1.50845 -2.1 -0.5 50.0 3.0 4.0 -1.1756 0.1000 -0.4550 3.292 5.80023 6.1580 0.4892 0.0512 0.0860 10.00 0.8129 1.5028 -0.3323 -0.00970 -0.00486 2.000 -0.4340 -0.1364 -0.007279 0.108360 0.2289 0.014996 580.0 0.0699 0.3471 0.2671 0.4458 0.3459 0.8000 0.0663
0.030 1.06 1.51549 -2.1 -0.5 50.0 3.0 4.0 -1.0909 0.1000 -0.4550 3.514 5.78659 6.1550 0.4890 0.0511 0.0860 10.00 0.8439 1.5071 -0.3394 -0.01010 -0.00503 2.000 -0.4177 -0.1403 -0.007354 0.119888 0.2289 0.014996 580.0 0.0701 0.3603 0.2803 0.4535 0.3537 0.8000 0.0663
0.040 1.06 1.52380 -2.1 -0.5 50.0 3.0 4.0 -0.9793 0.1000 -0.4550 3.563 5.77472 6.1508 0.4888 0.0508 0.0860 10.00 0.8740 1.5138 -0.3453 -0.01050 -0.00526 2.000 -0.4000 -0.1591 -0.006977 0.133641 0.2289 0.014996 579.9 0.0702 0.3718 0.2918 0.4589 0.3592 0.8000 0.0663
0.050 1.06 1.53319 -2.1 -0.5 50.0 3.0 4.0 -0.8549 0.1000 -0.4550 3.547 5.76402 6.1441 0.4884 0.0504 0.0860 10.00 0.8996 1.5230 -0.3502 -0.01090 -0.00549 2.000 -0.3903 -0.1862 -0.006467 0.148927 0.2290 0.014996 579.9 0.0701 0.3848 0.3048 0.4630 0.3635 0.8000 0.0663
0.075 1.06 1.56053 -2.1 -0.5 50.0 3.0 4.0 -0.6008 0.1000 -0.4540 3.448 5.74056 6.1200 0.4872 0.0495 0.0860 10.00 0.9442 1.5597 -0.3579 -0.01170 -0.00588 2.000 -0.4040 -0.2538 -0.005734 0.190596 0.2292 0.014996 579.6 0.0686 0.3878 0.3129 0.4702 0.3713 0.8000 0.0663
0.10 1.06 1.59241 -2.1 -0.5 50.0 3.0 4.0 -0.4700 0.1000 -0.4530 3.312 5.72017 6.0850 0.4854 0.0489 0.0860 10.00 0.9677 1.6104 -0.3604 -0.01170 -0.00591 2.000 -0.4423 -0.2943 -0.005604 0.230662 0.2297 0.014996 579.2 0.0646 0.3835 0.3152 0.4747 0.3769 0.8000 0.0663
0.15 1.06 1.66640 -2.1 -0.5 50.0 3.0 4.0 -0.4139 0.1000 -0.4500 3.044 5.68493 5.9871 0.4808 0.0479 0.0860 10.00 0.9660 1.7549 -0.3565 -0.01110 -0.00540 2.000 -0.5162 -0.3113 -0.005845 0.266468 0.2326 0.014988 577.2 0.0494 0.3719 0.3128 0.4798 0.3847 0.8000 0.0612
0.20 1.06 1.75021 -2.1 -0.5 50.0 3.0 4.0 -0.5237 0.1000 -0.4149 2.831 5.65435 5.8699 0.4755 0.0471 0.0860 10.00 0.9334 1.9157 -0.3470 -0.01000 -0.00479 2.000 -0.5697 -0.2927 -0.006141 0.255253 0.2386 0.014964 573.9 -0.0019 0.3601 0.3076 0.4816 0.3902 0.8000 0.0530
0.25 1.06 1.84052 -2.1 -0.5 50.0 3.0 4.0 -0.6678 0.1000 -0.3582 2.658 5.62686 5.7547 0.4706 0.0464 0.0860 10.50 0.8946 2.0709 -0.3379 -0.00910 -0.00427 2.000 -0.6109 -0.2662 -0.006439 0.231541 0.2497 0.014881 568.5 -0.0479 0.3522 0.3047 0.4815 0.3946 0.7999 0.0457
0.30 1.06 1.93480 -2.1 -0.5 50.0 3.0 4.0 -0.8277 0.0999 -0.3113 2.505 5.60162 5.6527 0.4665 0.0458 0.0860 11.00 0.8590 2.2005 -0.3314 -0.00820 -0.00384 2.500 -0.6444 -0.2405 -0.006704 0.207277 0.2674 0.014639 560.5 -0.0756 0.3438 0.3005 0.4801 0.3981 0.7997 0.0398
0.40 1.06 2.12764 -2.1 -0.5 50.0 3.0 4.0 -1.1284 0.0997 -0.2646 2.261 5.55602 5.4997 0.4607 0.0445 0.0850 12.00 0.8019 2.3886 -0.3256 -0.00690 -0.00317 3.200 -0.6931 -0.1975 -0.007125 0.165464 0.3120 0.013493 540.0 -0.0960 0.3351 0.2984 0.4758 0.4036 0.7988 0.0312
0.50 1.06 2.31684 -2.1 -0.5 50.0 3.0 4.0 -1.3926 0.0991 -0.2272 2.087 5.51513 5.4029 0.4571 0.0429 0.0830 13.00 0.7578 2.5000 -0.3189 -0.00590 -0.00272 3.500 -0.7246 -0.1633 -0.007435 0.133828 0.3610 0.011133 512.9 -0.0998 0.3353 0.3036 0.4710 0.4079 0.7966 0.0255
0.75 1.06 2.73064 -2.1 -0.5 50.0 3.0 4.0 -1.8664 0.0936 -0.1620 1.812 5.38632 5.2900 0.4531 0.0387 0.0690 14.00 0.6788 2.6224 -0.2702 -0.00450 -0.00209 4.500 -0.7708 -0.1028 -0.008120 0.085153 0.4353 0.006739 441.9 -0.0765 0.3429 0.3205 0.4621 0.4157 0.7792 0.0175
1.0 1.06 3.03000 -2.1 -0.5 50.0 3.0 4.0 -2.1935 0.0766 -0.1400 1.648 5.31000 5.2480 0.4517 0.0350 0.0450 15.00 0.6196 2.6690 -0.2059 -0.00370 -0.00175 5.000 -0.7990 -0.0699 -0.008444 0.058595 0.4629 0.005749 391.8 -0.0412 0.3577 0.3419 0.4581 0.4213 0.7504 0.0133
1.5 1.06 3.43384 -2.1 -0.5 50.0 3.0 4.0 -2.6883 0.0022 -0.1184 1.511 5.29995 5.2194 0.4507 0.0280 0.0134 16.00 0.5101 2.6985 -0.0852 -0.00280 -0.00142 5.400 -0.8382 -0.0425 -0.007707 0.031787 0.4756 0.005544 348.1 0.0140 0.3769 0.3703 0.4493 0.4213 0.7136 0.0090
2.0 1.06 3.67464 -2.1 -0.5 50.0 3.0 4.0 -3.1040 -0.0591 -0.1100 1.470 5.32730 5.2099 0.4504 0.0213 0.0040 18.00 0.3917 2.7085 0.0160 -0.00230 -0.00143 5.800 -0.8663 -0.0302 -0.004792 0.019716 0.4785 0.005521 332.5 0.0544 0.4023 0.4023 0.4459 0.4213 0.7035 0.0068
3.0 1.06 3.64933 -2.1 -0.5 50.0 3.0 4.0 -3.7085 -0.0931 -0.1040 1.456 5.43850 5.2040 0.4501 0.0106 0.0010 19.00 0.1244 2.7145 0.1876 -0.00190 -0.00115 6.000 -0.9032 -0.0129 -0.001828 0.009643 0.4796 0.005517 324.1 0.1232 0.4406 0.4406 0.4433 0.4213 0.7006 0.0045
4.0 1.06 3.60999 -2.1 -0.5 50.0 3.0 4.0 -4.1486 -0.0982 -0.1020 1.465 5.59770 5.2020 0.4501 0.0041 0.0000 19.75 0.0086 2.7164 0.3378 -0.00180 -0.00104 6.150 -0.9231 -0.0016 -0.001523 0.005379 0.4799 0.005517 321.7 0.1859 0.4784 0.4784 0.4424 0.4213 0.7001 0.0034
5.0 1.06 3.50000 -2.1 -0.5 50.0 3.0 4.0 -4.4881 -0.0994 -0.1010 1.478 5.72760 5.2010 0.4500 0.0010 0.0000 20.00 0.0000 2.7172 0.4579 -0.00170 -0.00099 6.300 -0.9222 0.0000 -0.001440 0.003223 0.4799 0.005517 320.9 0.2295 0.5074 0.5074 0.4420 0.4213 0.7000 0.0027
7.5 1.06 3.45000 -2.1 -0.5 50.0 3.0 4.0 -5.0891 -0.0999 -0.1010 1.498 5.98910 5.2000 0.4500 0.0000 0.0000 20.00 0.0000 2.7177 0.7514 -0.00170 -0.00094 6.425 -0.8346 0.0000 -0.001369 0.001134 0.4800 0.005517 320.3 0.2660 0.5328 0.5328 0.4416 0.4213 0.7000 0.0018
10.0 1.06 3.45000 -2.1 -0.5 50.0 3.0 4.0 -5.5530 -0.1000 -0.1000 1.502 6.19300 5.2000 0.4500 0.0000 0.0000 20.00 0.0000 2.7180 1.1856 -0.00170 -0.00091 6.550 -0.7332 0.0000 -0.001361 0.000515 0.4800 0.005517 320.1 0.2682 0.5542 0.5542 0.4414 0.4213 0.7000 0.0014
""")
[docs]class Bradley2013Volc(Bradley2013):
"""
Extend :class:`Bradley2013` for earthquakes with paths across the Taupo
Volcanic Zone (rtvz) that have increased anelastic attenuation.
Implements GMPE developed by Brendan Bradley for Active Shallow Crust
Earthquakes for New Zealand, and published as "A New Zealand-Specific
Pseudospectral Acceleration Ground-Motion Prediction Equation for Active
Shallow Crustal Earthquakes Based on Foreign Models" (2013, Bulletin of
the Seismological Society of America, Volume 103, No. 3, pages 1801-1822).
This model is modified from Chiou and Youngs, 2008 and has been adapted
for New Zealand conditions. Specifically, the modifications are related to:
1) small magnitude scaling;
2) scaling of short period ground motion from normal faulting events in
volcanic crust;
3) scaling of ground motions on very hard rock sites;
4) anelastic attenuation in the New Zealand crust;
5) consideration of the increates anelastic attenuation in the Taupo
Volcanic Zone (rtvz is equal to rrup)
"""
DEFINED_FOR_TECTONIC_REGION_TYPE = const.TRT.VOLCANIC
def _get_tvz_path_distance(self, rrup):
"""
Returns Taupo Volcanic Zone (TVZ) path distance.
rtvz = rrup as implemented for New Zealand seismic hazard model
"""
return rrup
[docs]class Bradley2013LHC(Bradley2013):
"""
Extend :class:`Bradley2013` to provide the model in terms of the larger
of two as-recorded horizontal components. This definition is required
by New Zealand building design standards.
"""
DEFINED_FOR_INTENSITY_MEASURE_COMPONENT = \
const.IMC.GREATER_OF_TWO_HORIZONTAL
#: This implementation is non-verified because this version of the
#: model has not been published, nor is independent code available.
non_verified = True
[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]
# intensity on a reference soil is used for both mean
# and stddev calculations.
ln_y_ref = self._get_ln_y_ref(rup, dists, C)
# exp1 and exp2 are parts of eq. 7
exp1 = np.exp(C['phi3'] * (sites.vs30.clip(-np.inf, 1130) - 360))
exp2 = np.exp(C['phi3'] * (1130 - 360))
# v1 is the period dependent site term. The Vs30 above which, the
# amplification is constant
v1 = self._get_v1(imt)
mean = self._get_mean(sites, C, ln_y_ref, exp1, exp2, v1)
mean += convert_to_LHC(imt)
stddevs = self._get_stddevs(sites, rup, C, stddev_types,
ln_y_ref, exp1, exp2)
return mean, stddevs
[docs]class Bradley2013VolcLHC(Bradley2013LHC):
"""
Extend :class:`Bradley2013LHC` for earthquakes with paths across the Taupo
Volcanic Zone (rtvz) that have increased anelastic attenuation.
"""
DEFINED_FOR_TECTONIC_REGION_TYPE = const.TRT.VOLCANIC
def _get_tvz_path_distance(self, rrup):
"""
Returns Taupo Volcanic Zone (TVZ) path distance.
rtvz = rrup as implemented for New Zealand seismic hazard model
"""
return rrup
[docs]def convert_to_LHC(imt):
"""
Converts from GMRotI50 to Larger of two horizontal components using
global equation of:
Boore, D and Kishida, T (2016). Relations between some horizontal-
component ground-motion intensity measures used in practice.
Bulletin of the Seismological Society of America, 107(1), 334-343.
doi:10.1785/0120160250
No standard deviation modification required.
"""
# get period t
if isinstance(imt, SA):
t = imt.period
else:
t = 0.01
T1 = 0.08
T2 = 0.56
T3 = 4.40
T4 = 8.70
R1 = 1.106
R2 = 1.158
R3 = 1.178
R4 = 1.241
R5 = 1.241
Ratio = max(R1,
max(min(R1+(R2-R1)/np.log(T2/T1)*np.log(t/T1),
R2+(R3-R2)/np.log(T3/T2)*np.log(t/T2)),
min(R3+(R4-R3)/np.log(T4/T3)*np.log(t/T3), R5)))
SF = np.log(Ratio)
return SF