# -*- 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 :mod:`openquake.hazardlib.gsim.base` defines base classes for
different kinds of :class:`ground shaking intensity models
<GroundShakingIntensityModel>`.
"""
import abc
import math
import warnings
import functools
import numpy
from scipy.special import ndtr
from openquake.baselib.general import DeprecationWarning
from openquake.hazardlib import imt as imt_module
from openquake.hazardlib import const
from openquake.hazardlib.contexts import KNOWN_DISTANCES
from openquake.hazardlib.contexts import * # for backward compatibility
ADMITTED_STR_PARAMETERS = ['DEFINED_FOR_TECTONIC_REGION_TYPE',
'DEFINED_FOR_INTENSITY_MEASURE_COMPONENT']
ADMITTED_FLOAT_PARAMETERS = ['DEFINED_FOR_REFERENCE_VELOCITY']
ADMITTED_TABLE_PARAMETERS = ['COEFFS_STRESS', 'COEFFS_HARD_ROCK',
'COEFFS_SITE_RESPONSE']
ADMITTED_SET_PARAMETERS = ['DEFINED_FOR_INTENSITY_MEASURE_TYPES',
'DEFINED_FOR_STANDARD_DEVIATION_TYPES',
'REQUIRES_DISTANCES',
'REQUIRES_SITES_PARAMETERS',
'REQUIRES_RUPTURE_PARAMETERS']
registry = {} # GSIM name -> GSIM class
[docs]class NotVerifiedWarning(UserWarning):
"""
Raised when a non verified GSIM is instantiated
"""
[docs]class ExperimentalWarning(UserWarning):
"""
Raised for GMPEs that are intended for experimental use or maybe subject
to changes in future version.
"""
[docs]def gsim_imt_dt(sorted_gsims, sorted_imts):
"""
Build a numpy dtype as a nested record with keys 'idx' and nested
(gsim, imt).
:param sorted_gsims: a list of GSIM instances, sorted lexicographically
:param sorted_imts: a list of intensity measure type strings
"""
dtlist = [(imt, numpy.float32) for imt in sorted_imts]
imt_dt = numpy.dtype(dtlist)
return numpy.dtype([(str(gsim), imt_dt) for gsim in sorted_gsims])
[docs]@functools.total_ordering
class GroundShakingIntensityModel(metaclass=MetaGSIM):
"""
Base class for all the ground shaking intensity models.
A Ground Shaking Intensity Model (GSIM) defines a set of equations
for computing mean and standard deviation of a Normal distribution
representing the variability of an intensity measure (or of its logarithm)
at a site given an earthquake rupture.
This class is not intended to be subclassed directly, instead
the actual GSIMs should subclass either :class:`GMPE` or :class:`IPE`.
Subclasses of both must implement :meth:`get_mean_and_stddevs`
and all the class attributes with names starting from ``DEFINED_FOR``
and ``REQUIRES``.
"""
#: Reference to a
#: :class:`tectonic region type <openquake.hazardlib.const.TRT>` this GSIM
#: is defined for. One GSIM can implement only one tectonic region type.
DEFINED_FOR_TECTONIC_REGION_TYPE = abc.abstractproperty()
#: Set of :mod:`intensity measure types <openquake.hazardlib.imt>`
#: this GSIM can
#: calculate. A set should contain classes from module
#: :mod:`openquake.hazardlib.imt`.
DEFINED_FOR_INTENSITY_MEASURE_TYPES = abc.abstractproperty()
#: Reference to a :class:`intensity measure component type
#: <openquake.hazardlib.const.IMC>` this GSIM can calculate mean
#: and standard
#: deviation for.
DEFINED_FOR_INTENSITY_MEASURE_COMPONENT = abc.abstractproperty()
#: Set of
#: :class:`standard deviation types <openquake.hazardlib.const.StdDev>`
#: this GSIM can calculate.
DEFINED_FOR_STANDARD_DEVIATION_TYPES = abc.abstractproperty()
#: Set of site parameters names this GSIM needs. The set should include
#: strings that match names of the attributes of a :class:`site
#: <openquake.hazardlib.site.Site>` object.
#: Those attributes are then available in the
#: :class:`SitesContext` object with the same names.
REQUIRES_SITES_PARAMETERS = abc.abstractproperty()
#: Set of rupture parameters (excluding distance information) required
#: by GSIM. Supported parameters are:
#:
#: ``mag``
#: Magnitude of the rupture.
#: ``dip``
#: Rupture's surface dip angle in decimal degrees.
#: ``rake``
#: Angle describing the slip propagation on the rupture surface,
#: in decimal degrees. See :mod:`~openquake.hazardlib.geo.nodalplane`
#: for more detailed description of dip and rake.
#: ``ztor``
#: Depth of rupture's top edge in km. See
#: :meth:`~openquake.hazardlib.geo.surface.base.BaseSurface.get_top_edge_depth`.
#:
#: These parameters are available from the :class:`RuptureContext` object
#: attributes with same names.
REQUIRES_RUPTURE_PARAMETERS = abc.abstractproperty()
#: Set of types of distance measures between rupture and sites. Possible
#: values are:
#:
#: ``rrup``
#: Closest distance to rupture surface. See
#: :meth:`~openquake.hazardlib.geo.surface.base.BaseSurface.get_min_distance`.
#: ``rjb``
#: Distance to rupture's surface projection. See
#: :meth:`~openquake.hazardlib.geo.surface.base.BaseSurface.get_joyner_boore_distance`.
#: ``rx``
#: Perpendicular distance to rupture top edge projection.
#: See :meth:`~openquake.hazardlib.geo.surface.base.BaseSurface.get_rx_distance`.
#: ``ry0``
#: Horizontal distance off the end of the rupture measured parallel to
# strike. See:
#: See :meth:`~openquake.hazardlib.geo.surface.base.BaseSurface.get_ry0_distance`.
#: ``rcdpp``
#: Direct point parameter for directivity effect centered on the site- and earthquake-specific
# average DPP used. See:
#: See :meth:`~openquake.hazardlib.source.rupture.ParametricProbabilisticRupture.get_dppvalue`.
#: ``rvolc``
#: Source to site distance passing through surface projection of volcanic zone
#:
#: All the distances are available from the :class:`DistancesContext`
#: object attributes with same names. Values are in kilometers.
REQUIRES_DISTANCES = abc.abstractproperty()
_toml = '' # set by valid.gsim
minimum_distance = 0 # set by valid.gsim
superseded_by = None
non_verified = False
experimental = False
@classmethod
def __init_subclass__(cls):
registry[cls.__name__] = cls
def __init__(self, **kwargs):
self.kwargs = kwargs
cls = self.__class__
if cls.superseded_by:
msg = '%s is deprecated - use %s instead' % (
cls.__name__, cls.superseded_by.__name__)
warnings.warn(msg, DeprecationWarning)
if cls.non_verified:
msg = ('%s is not independently verified - the user is liable '
'for their application') % cls.__name__
warnings.warn(msg, NotVerifiedWarning)
if cls.experimental:
msg = ('%s is experimental and may change in future versions - '
'the user is liable for their application') % cls.__name__
warnings.warn(msg, ExperimentalWarning)
[docs] def init(self):
"""
Override this method if you want to further initialize the GSIM
"""
[docs] @abc.abstractmethod
def get_mean_and_stddevs(self, sites, rup, dists, imt, stddev_types):
"""
Calculate and return mean value of intensity distribution and it's
standard deviation.
Method must be implemented by subclasses.
:param sites:
Instance of :class:`openquake.hazardlib.site.SiteCollection`
with parameters of sites
collection assigned to respective values as numpy arrays.
Only those attributes that are listed in class'
:attr:`REQUIRES_SITES_PARAMETERS` set are available.
:param rup:
Instance of :class:`openquake.hazardlib.source.rupture.BaseRupture`
with parameters of a rupture
assigned to respective values. Only those attributes that are
listed in class' :attr:`REQUIRES_RUPTURE_PARAMETERS` set are
available.
:param dists:
Instance of :class:`DistancesContext` with values of distance
measures between the rupture and each site of the collection
assigned to respective values as numpy arrays. Only those
attributes that are listed in class' :attr:`REQUIRES_DISTANCES`
set are available.
:param imt:
An instance (not a class) of intensity measure type.
See :mod:`openquake.hazardlib.imt`.
:param stddev_types:
List of standard deviation types, constants from
:class:`openquake.hazardlib.const.StdDev`.
Method result value should include
standard deviation values for each of types in this list.
:returns:
Method should return a tuple of two items. First item should be
a numpy array of floats -- mean values of respective component
of a chosen intensity measure type, and the second should be
a list of numpy arrays of standard deviation values for the same
single component of the same single intensity measure type, one
array for each type in ``stddev_types`` parameter, preserving
the order.
Combining interface to mean and standard deviation values in a single
method allows to avoid redoing the same intermediate calculations
if there are some shared between stddev and mean formulae without
resorting to keeping any sort of internal state (and effectively
making GSIM not reenterable).
However it is advised to split calculation of mean and stddev values
and make ``get_mean_and_stddevs()`` just combine both (and possibly
compute interim steps).
"""
[docs] def get_poes(self, sctx, rctx, dctx, imt, imls, truncation_level):
"""
Calculate and return probabilities of exceedance (PoEs) of one or more
intensity measure levels (IMLs) of one intensity measure type (IMT)
for one or more pairs "site -- rupture".
:param sctx:
An instance of :class:`SitesContext` with sites information
to calculate PoEs on.
:param rctx:
An instance of :class:`RuptureContext` with a single rupture
information.
:param dctx:
An instance of :class:`DistancesContext` with information about
the distances between sites and a rupture.
All three contexts (``sctx``, ``rctx`` and ``dctx``) must conform
to each other. The easiest way to get them is to call
ContextMaker.make_contexts.
:param imt:
An intensity measure type object (that is, an instance of one
of classes from :mod:`openquake.hazardlib.imt`).
:param imls:
List of interested intensity measure levels (of type ``imt``).
:param truncation_level:
Can be ``None``, which means that the distribution of intensity
is treated as Gaussian distribution with possible values ranging
from minus infinity to plus infinity.
When set to zero, the mean intensity is treated as an exact
value (standard deviation is not even computed for that case)
and resulting array contains 0 in places where IMT is strictly
lower than the mean value of intensity and 1.0 where IMT is equal
or greater.
When truncation level is positive number, the intensity
distribution is processed as symmetric truncated Gaussian with
range borders being ``mean - truncation_level * stddev`` and
``mean + truncation_level * stddev``. That is, the truncation
level expresses how far the range borders are from the mean
value and is defined in units of sigmas. The resulting PoEs
for that mode are values of complementary cumulative distribution
function of that truncated Gaussian applied to IMLs.
:returns:
A dictionary of the same structure as parameter ``imts`` (see
above). Instead of lists of IMLs values of the dictionaries
have 2d numpy arrays of corresponding PoEs, first dimension
represents sites and the second represents IMLs.
:raises ValueError:
If truncation level is not ``None`` and neither non-negative
float number, and if ``imts`` dictionary contain wrong or
unsupported IMTs (see :attr:`DEFINED_FOR_INTENSITY_MEASURE_TYPES`).
"""
if truncation_level is not None and truncation_level < 0:
raise ValueError('truncation level must be zero, positive number '
'or None')
self._check_imt(imt)
if truncation_level == 0:
# zero truncation mode, just compare imls to mean
imls = self.to_distribution_values(imls)
mean, _ = self.get_mean_and_stddevs(sctx, rctx, dctx, imt, [])
mean = mean.reshape(mean.shape + (1, ))
return (imls <= mean).astype(float)
else:
# use real normal distribution
assert (const.StdDev.TOTAL
in self.DEFINED_FOR_STANDARD_DEVIATION_TYPES)
imls = self.to_distribution_values(imls)
mean, [stddev] = self.get_mean_and_stddevs(sctx, rctx, dctx, imt,
[const.StdDev.TOTAL])
mean = mean.reshape(mean.shape + (1, ))
stddev = stddev.reshape(stddev.shape + (1, ))
values = (imls - mean) / stddev
if truncation_level is None:
return _norm_sf(values)
else:
return _truncnorm_sf(truncation_level, values)
[docs] @abc.abstractmethod
def to_distribution_values(self, values):
"""
Convert a list or array of values in units of IMT to a numpy array
of values of intensity measure distribution (like taking the natural
logarithm for :class:`GMPE`).
This method is implemented by both :class:`GMPE` and :class:`IPE`
so there is no need to override it in actual GSIM implementations.
"""
[docs] @abc.abstractmethod
def to_imt_unit_values(self, values):
"""
Convert a list or array of values of intensity measure distribution
(like ones returned from :meth:`get_mean_and_stddevs`) to values
in units of IMT. This is the opposite operation
to :meth:`to_distribution_values`.
This method is implemented by both :class:`GMPE` and :class:`IPE`
so there is no need to override it in actual GSIM implementations.
"""
def _check_imt(self, imt):
"""
Make sure that ``imt`` is valid and is supported by this GSIM.
"""
names = set(f.__name__
for f in self.DEFINED_FOR_INTENSITY_MEASURE_TYPES)
if imt.name not in names:
raise ValueError('imt %s is not supported by %s' %
(imt.name, type(self).__name__))
def __lt__(self, other):
"""
The GSIMs are ordered according to string representation
"""
return str(self) < str(other)
def __eq__(self, other):
"""
The GSIMs are equal if their string representations are equal
"""
return str(self) == str(other)
def __hash__(self):
"""
We use the __str__ representation as hash: it means that we can
use equivalently GSIM instances or strings as dictionary keys.
"""
return hash(str(self))
def __repr__(self):
"""
String representation for GSIM instances in TOML format.
"""
if self._toml:
return self._toml
return '[%s]' % self.__class__.__name__
def _truncnorm_sf(truncation_level, values):
"""
Survival function for truncated normal distribution.
Assumes zero mean, standard deviation equal to one and symmetric
truncation.
:param truncation_level:
Positive float number representing the truncation on both sides
around the mean, in units of sigma.
:param values:
Numpy array of values as input to a survival function for the given
distribution.
:returns:
Numpy array of survival function results in a range between 0 and 1.
>>> from scipy.stats import truncnorm
>>> truncnorm(-3, 3).sf(0.12345) == _truncnorm_sf(3, 0.12345)
True
"""
# notation from http://en.wikipedia.org/wiki/Truncated_normal_distribution.
# given that mu = 0 and sigma = 1, we have alpha = a and beta = b.
# "CDF" in comments refers to cumulative distribution function
# of non-truncated distribution with that mu and sigma values.
# assume symmetric truncation, that is ``a = - truncation_level``
# and ``b = + truncation_level``.
# calculate CDF of b
phi_b = ndtr(truncation_level)
# calculate Z as ``Z = CDF(b) - CDF(a)``, here we assume that
# ``CDF(a) == CDF(- truncation_level) == 1 - CDF(b)``
z = phi_b * 2 - 1
# calculate the result of survival function of ``values``,
# and restrict it to the interval where probability is defined --
# 0..1. here we use some transformations of the original formula
# that is ``SF(x) = 1 - (CDF(x) - CDF(a)) / Z`` in order to minimize
# number of arithmetic operations and function calls:
# ``SF(x) = (Z - CDF(x) + CDF(a)) / Z``,
# ``SF(x) = (CDF(b) - CDF(a) - CDF(x) + CDF(a)) / Z``,
# ``SF(x) = (CDF(b) - CDF(x)) / Z``.
return ((phi_b - ndtr(values)) / z).clip(0.0, 1.0)
def _norm_sf(values):
"""
Survival function for normal distribution.
Assumes zero mean and standard deviation equal to one.
``values`` parameter and the return value are the same
as in :func:`_truncnorm_sf`.
>>> from scipy.stats import norm
>>> norm.sf(0.12345) == _norm_sf(0.12345)
True
"""
# survival function by definition is ``SF(x) = 1 - CDF(x)``,
# which is equivalent to ``SF(x) = CDF(- x)``, since (given
# that the normal distribution is symmetric with respect to 0)
# the integral between ``[x, +infinity]`` (that is the survival
# function) is equal to the integral between ``[-infinity, -x]``
# (that is the CDF at ``- x``).
return ndtr(- values)
[docs]class GMPE(GroundShakingIntensityModel):
"""
Ground-Motion Prediction Equation is a subclass of generic
:class:`GroundShakingIntensityModel` with a distinct feature
that the intensity values are log-normally distributed.
Method :meth:`~GroundShakingIntensityModel.get_mean_and_stddevs`
of actual GMPE implementations is supposed to return the mean
value as a natural logarithm of intensity.
"""
[docs] def to_distribution_values(self, values):
"""
Returns numpy array of natural logarithms of ``values``.
"""
with warnings.catch_warnings():
warnings.simplefilter("ignore")
# avoid RuntimeWarning: divide by zero encountered in log
return numpy.log(values)
[docs] def to_imt_unit_values(self, values):
"""
Returns numpy array of exponents of ``values``.
"""
return numpy.exp(values)
[docs] def set_parameters(self):
"""
Combines the parameters of the GMPE provided at the construction level
with the ones originally assigned to the backbone modified GMPE.
"""
for key in (ADMITTED_STR_PARAMETERS + ADMITTED_FLOAT_PARAMETERS +
ADMITTED_SET_PARAMETERS):
try:
val = getattr(self.gmpe, key)
except AttributeError:
pass
else:
setattr(self, key, val)
[docs]class IPE(GroundShakingIntensityModel):
"""
Intensity Prediction Equation is a subclass of generic
:class:`GroundShakingIntensityModel` which is suitable for
intensity measures that are normally distributed. In particular,
for :class:`~openquake.hazardlib.imt.MMI`.
"""
[docs] def to_distribution_values(self, values):
"""
Returns numpy array of ``values`` without any conversion.
"""
return numpy.array(values, dtype=float)
[docs] def to_imt_unit_values(self, values):
"""
Returns numpy array of ``values`` without any conversion.
"""
return numpy.array(values, dtype=float)
[docs]class CoeffsTable(object):
r"""
Instances of :class:`CoeffsTable` encapsulate tables of coefficients
corresponding to different IMTs.
Tables are defined in a space-separated tabular form in a simple string
literal (heading and trailing whitespace does not matter). The first column
in the table must be named "IMT" (or "imt") and thus should represent IMTs:
>>> CoeffsTable(table='''imf z
... pga 1''')
Traceback (most recent call last):
...
ValueError: first column in a table must be IMT
Names of other columns are used as coefficients dicts keys. The values
in the first column should correspond to real intensity measure types,
see :mod:`openquake.hazardlib.imt`:
>>> CoeffsTable(table='''imt z
... pgx 2''')
Traceback (most recent call last):
...
ValueError: unknown IMT 'PGX'
Note that :class:`CoeffsTable` only accepts keyword argumets:
>>> CoeffsTable()
Traceback (most recent call last):
...
TypeError: CoeffsTable requires "table" kwarg
>>> CoeffsTable(table='', foo=1)
Traceback (most recent call last):
...
TypeError: CoeffsTable got unexpected kwargs: {'foo': 1}
If there are :class:`~openquake.hazardlib.imt.SA` IMTs in the table, they
are not referenced by name, because they require parametrization:
>>> CoeffsTable(table='''imt x
... sa 15''')
Traceback (most recent call last):
...
ValueError: specify period as float value to declare SA IMT
>>> CoeffsTable(table='''imt x
... 0.1 20''')
Traceback (most recent call last):
...
TypeError: attribute "sa_damping" is required for tables defining SA
So proper table defining SA looks like this:
>>> ct = CoeffsTable(sa_damping=5, table='''
... imt a b c d
... pga 1 2.4 -5 0.01
... pgd 7.6 12 0 44.1
... 0.1 10 20 30 40
... 1.0 1 2 3 4
... 10 2 4 6 8
... ''')
Table objects could be indexed by IMT objects (this returns a dictionary
of coefficients):
>>> from openquake.hazardlib import imt
>>> ct[imt.PGA()] == dict(a=1, b=2.4, c=-5, d=0.01)
True
>>> ct[imt.PGD()] == dict(a=7.6, b=12, c=0, d=44.1)
True
>>> ct[imt.SA(damping=5, period=0.1)] == dict(a=10, b=20, c=30, d=40)
True
>>> ct[imt.PGV()]
Traceback (most recent call last):
...
KeyError: PGV
>>> ct[imt.SA(1.0, 4)]
Traceback (most recent call last):
...
KeyError: SA(1.0, 4)
Table of coefficients for spectral acceleration could be indexed
by instances of :class:`openquake.hazardlib.imt.SA` with period
value that is not specified in the table. The coefficients then
get interpolated between the ones for closest higher and closest
lower period. That scaling of coefficients works in a logarithmic
scale of periods and only within the same damping:
>>> '%.5f' % ct[imt.SA(period=0.2, damping=5)]['a']
'7.29073'
>>> '%.5f' % ct[imt.SA(period=0.9, damping=5)]['c']
'4.23545'
>>> '%.5f' % ct[imt.SA(period=5, damping=5)]['c']
'5.09691'
>>> ct[imt.SA(period=0.9, damping=15)]
Traceback (most recent call last):
...
KeyError: SA(0.9, 15)
Extrapolation is not possible:
>>> ct[imt.SA(period=0.01, damping=5)]
Traceback (most recent call last):
...
KeyError: SA(0.01)
It is also possible to instantiate a table from a tuple of dictionaries,
corresponding to the SA coefficients and non-SA coefficients:
>>> coeffs = {imt.SA(0.1): {"a": 1.0, "b": 2.0},
... imt.SA(1.0): {"a": 3.0, "b": 4.0},
... imt.PGA(): {"a": 0.1, "b": 1.0},
... imt.PGV(): {"a": 0.5, "b": 10.0}}
>>> ct = CoeffsTable(sa_damping=5, table=coeffs)
"""
def __init__(self, **kwargs):
if 'table' not in kwargs:
raise TypeError('CoeffsTable requires "table" kwarg')
table = kwargs.pop('table')
self.sa_coeffs = {}
self.non_sa_coeffs = {}
sa_damping = kwargs.pop('sa_damping', None)
if kwargs:
raise TypeError('CoeffsTable got unexpected kwargs: %r' % kwargs)
if isinstance(table, str):
self._setup_table_from_str(table, sa_damping)
elif isinstance(table, dict):
for imt in table:
if imt.name == 'SA':
self.sa_coeffs[imt] = table[imt]
else:
self.non_sa_coeffs[imt] = table[imt]
else:
raise TypeError("CoeffsTable cannot be constructed with inputs "
"of the form '%s'" % table.__class__.__name__)
def _setup_table_from_str(self, table, sa_damping):
"""
Builds the input tables from a string definition
"""
table = table.strip().splitlines()
header = table.pop(0).split()
if not header[0].upper() == "IMT":
raise ValueError('first column in a table must be IMT')
coeff_names = header[1:]
for row in table:
row = row.split()
imt_name = row[0].upper()
if imt_name == 'SA':
raise ValueError('specify period as float value '
'to declare SA IMT')
imt_coeffs = dict(zip(coeff_names, map(float, row[1:])))
try:
sa_period = float(imt_name)
except Exception:
if imt_name not in imt_module.registry:
raise ValueError('unknown IMT %r' % imt_name)
imt = imt_module.registry[imt_name]()
self.non_sa_coeffs[imt] = imt_coeffs
else:
if sa_damping is None:
raise TypeError('attribute "sa_damping" is required '
'for tables defining SA')
imt = imt_module.SA(sa_period, sa_damping)
self.sa_coeffs[imt] = imt_coeffs
def __getitem__(self, imt):
"""
Return a dictionary of coefficients corresponding to ``imt``
from this table (if there is a line for requested IMT in it),
or the dictionary of interpolated coefficients, if ``imt`` is
of type :class:`~openquake.hazardlib.imt.SA` and interpolation
is possible.
:raises KeyError:
If ``imt`` is not available in the table and no interpolation
can be done.
"""
if imt.name != 'SA':
return self.non_sa_coeffs[imt]
try:
return self.sa_coeffs[imt]
except KeyError:
pass
max_below = min_above = None
for unscaled_imt in list(self.sa_coeffs):
if unscaled_imt.damping != imt.damping:
continue
if unscaled_imt.period > imt.period:
if min_above is None or unscaled_imt.period < min_above.period:
min_above = unscaled_imt
elif unscaled_imt.period < imt.period:
if max_below is None or unscaled_imt.period > max_below.period:
max_below = unscaled_imt
if max_below is None or min_above is None:
raise KeyError(imt)
# ratio tends to 1 when target period tends to a minimum
# known period above and to 0 if target period is close
# to maximum period below.
ratio = ((math.log(imt.period) - math.log(max_below.period))
/ (math.log(min_above.period) - math.log(max_below.period)))
max_below = self.sa_coeffs[max_below]
min_above = self.sa_coeffs[min_above]
return dict(
(co, (min_above[co] - max_below[co]) * ratio + max_below[co])
for co in max_below)