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"""
Module :mod: mfd.anderson_luco_1_mmax implements :class:
AndersonLucoType1Mmax. This calculates the magnitude occurrence rate on a fault
given a known slip value using the exponential models described by
Anderson & Luco (1983) referring to the whole fault area.
Anderson, J. G., and Luco, J. E. (1983) "Consequences of slip rate constraints
on earthquake recurrence relations". Bull. Seis. Soc. Am. 73(2) 471 - 496
"""
import abc
import numpy as np
from openquake.hmtk.faults.mfd.base import _scale_moment, BaseMFDfromSlip
C_VALUE = 16.05
D_VALUE = 1.5
[docs]class BaseRecurrenceModel(object):
"""
Abstract base class to implement cumulative value formula
"""
[docs] @abc.abstractmethod
def cumulative_value(self, slip_moment, mmax, mag_value, bbar, dbar):
"""
Returns the rate of earthquakes with M > mag_value
:param float slip_moment:
Product of slip (cm/yr) * Area (cm ^ 2) * shear_modulus (dyne-cm)
:param float mmax:
Maximum magnitude
:param float mag_value:
Magnitude value
:param float bbar:
\bar{b} parameter (effectively = b * log(10.))
:param float dbar:
\bar{d} parameter
"""
raise NotImplementedError
[docs] @abc.abstractmethod
def incremental_value(self, slip_moment, mmax, mag_value, bbar, dbar):
"""
Returns the incremental rate of earthquakes with M = mag_value
"""
raise NotImplementedError
[docs]class Type1RecurrenceModel(BaseRecurrenceModel):
"""
Calculate N(M > mag_value) using Anderson & Luco Type 1 formula as
inverse of formula I.5 of Table 2 in Anderson & Luco (1993).
"""
[docs] def cumulative_value(self, slip_moment, mmax, mag_value, bbar, dbar):
"""
Returns the rate of events with M > mag_value
:param float slip_moment:
:param float slip_moment:
Product of slip (cm/yr) * Area (cm ^ 2) * shear_modulus (dyne-cm)
:param float mmax:
Maximum magnitude
:param float mag_value:
Magnitude value
:param float bbar:
\bar{b} parameter (effectively = b * log(10.))
:param float dbar:
\bar{d} parameter
"""
delta_m = mmax - mag_value
a_1 = self._get_a1(bbar, dbar, slip_moment, mmax)
return a_1 * np.exp(bbar * (delta_m)) * (delta_m > 0.0)
@staticmethod
def _get_a1(bbar, dbar, slip_moment, mmax):
"""
Returns the A1 term (I.4 of Table 2 in Anderson & Luco)
"""
return ((dbar - bbar) / dbar) * (slip_moment / _scale_moment(mmax))
[docs] def incremental_value(self, slip_moment, mmax, mag_value, bbar, dbar):
"""
Returns the incremental rate of earthquakes with M = mag_value
"""
delta_m = mmax - mag_value
dirac_term = np.zeros_like(mag_value)
dirac_term[np.fabs(delta_m) < 1.0e-12] = 1.0
a_1 = self._get_a1(bbar, dbar, slip_moment, mmax)
return (
a_1 * (bbar * np.exp(bbar * delta_m) * (delta_m > 0.0))
+ a_1 * dirac_term
)
[docs]class Type2RecurrenceModel(BaseRecurrenceModel):
"""
Calculate N(M > mag_value) using Anderson & Luco Type 1 formula as
inverse of formula II.5 of Table 3 in Anderson & Luco (1993).
"""
[docs] def cumulative_value(self, slip_moment, mmax, mag_value, bbar, dbar):
"""
Returns the rate of events with M > mag_value
:param float slip_moment:
Product of slip (cm/yr) * Area (cm ^ 2) * shear_modulus (dyne-cm)
:param float mmax:
Maximum magnitude
:param float mag_value:
Magnitude value
:param float bbar:
\bar{b} parameter (effectively = b * log(10.))
:param float dbar:
\bar{d} parameter
"""
delta_m = mmax - mag_value
a_2 = self._get_a2(bbar, dbar, slip_moment, mmax)
return a_2 * (np.exp(bbar * delta_m) - 1.0) * (delta_m > 0.0)
@staticmethod
def _get_a2(bbar, dbar, slip_moment, mmax):
"""
Returns the A2 value defined in II.4 of Table 2
"""
return ((dbar - bbar) / bbar) * (slip_moment / _scale_moment(mmax))
[docs] def incremental_value(self, slip_moment, mmax, mag_value, bbar, dbar):
"""
Returns the incremental rate with Mmax = Mag_value
"""
delta_m = mmax - mag_value
a_2 = self._get_a2(bbar, dbar, slip_moment, mmax)
return a_2 * bbar * np.exp(bbar * delta_m) * (delta_m > 0.0)
[docs]class Type3RecurrenceModel(BaseRecurrenceModel):
"""
Calculate N(M > mag_value) using Anderson & Luco Type 1 formula as
inverse of formula III.5 of Table 4 in Anderson & Luco (1993).
"""
[docs] def cumulative_value(self, slip_moment, mmax, mag_value, bbar, dbar):
"""
Returns the rate of events with M > mag_value
:param float slip_moment:
Product of slip (cm/yr) * Area (cm ^ 2) * shear_modulus (dyne-cm)
:param float mmax:
Maximum magnitude
:param float mag_value:
Magnitude value
:param float bbar:
\bar{b} parameter (effectively = b * log(10.))
:param float dbar:
\bar{d} parameter
"""
delta_m = mmax - mag_value
a_3 = self._get_a3(bbar, dbar, slip_moment, mmax)
central_term = np.exp(bbar * delta_m) - 1.0 - (bbar * delta_m)
return a_3 * central_term * (delta_m > 0.0)
@staticmethod
def _get_a3(bbar, dbar, slip_moment, mmax):
"""
Returns the A3 term (III.4 in Table 4)
"""
return ((dbar * (dbar - bbar)) / (bbar**2.0)) * (
slip_moment / _scale_moment(mmax)
)
[docs] def incremental_value(self, slip_moment, mmax, mag_value, bbar, dbar):
"""
Returns the incremental rate with Mmax = Mag_value
"""
delta_m = mmax - mag_value
a_3 = self._get_a3(bbar, dbar, slip_moment, mmax)
return a_3 * bbar * (np.exp(bbar * delta_m) - 1.0) * (delta_m > 0.0)
RECURRENCE_MAP = {
"First": Type1RecurrenceModel(),
"Second": Type2RecurrenceModel(),
"Third": Type3RecurrenceModel(),
}
[docs]class AndersonLucoArbitrary(BaseMFDfromSlip):
"""
Class to implement the fault activity rate calculators of Anderson & Luco
(1983) referring to the whole fault
:param str mfd_type:
Type of magnitude frequency distribution
:param float mfd_weight:
Weight of the mfd distribution (for subsequent logic tree processing)
:param float bin_width:
Width of the magnitude bin (rates are given for the centre point)
:param float mmin:
Minimum magnitude
:param float mmax:
Maximum magnitude
:param float mmax_sigma:
Uncertainty on maximum magnitude
:param float b_value:
Exponent (b-value) for the magnitude frequency distribution
:param numpy.ndarray occurrence_rate:
Activity rates for magnitude in the range mmin to mmax in steps of
bin_width
"""
[docs] def setUp(self, mfd_conf):
"""
Input core configuration parameters as specified in the
configuration file
:param dict mfd_conf:
Configuration file containing the following attributes:
* 'Type' - Choose between the 1st, 2nd or 3rd type of recurrence
model {'First' | 'Second' | 'Third'}
* 'Model_Weight' - Logic tree weight of model type (float)
* 'MFD_spacing' - Width of MFD bin (float)
* 'Minimum_Magnitude' - Minimum magnitude of activity rates (float)
* 'b_value' - Tuple of (b-value, b-value uncertainty)
* 'Maximum_Magnitude' - Maximum magnitude on fault (if not defined
will use scaling relation)
* 'Maximum_Magnitude_Uncertainty' - Uncertainty on maximum
magnitude (If not defined and the MSR has a sigma term then this
will be taken from sigma)
"""
self.mfd_type = mfd_conf["Model_Type"]
self.mfd_model = "Anderson & Luco (Arbitrary) " + self.mfd_type
self.mfd_weight = mfd_conf["Model_Weight"]
self.bin_width = mfd_conf["MFD_spacing"]
self.mmin = mfd_conf["Minimum_Magnitude"]
self.mmax = None
self.mmax_sigma = None
self.b_value = mfd_conf["b_value"][0]
self.b_value_sigma = mfd_conf["b_value"][1]
self.occurrence_rate = None
[docs] def get_mmax(self, mfd_conf, msr, rake, area):
"""
Gets the mmax for the fault - reading directly from the config file
or using the msr otherwise
:param dict mfd_config:
Configuration file (see setUp for paramters)
:param msr:
Instance of :class:`nhlib.scalerel`
:param float rake:
Rake of the fault (in range -180 to 180)
:param float area:
Area of the fault surface (km^2)
"""
if mfd_conf["Maximum_Magnitude"]:
self.mmax = mfd_conf["Maximum_Magnitude"]
else:
self.mmax = msr.get_median_mag(area, rake)
if (
"Maximum_Magnitude_Uncertainty" in mfd_conf
and mfd_conf["Maximum_Magnitude_Uncertainty"]
):
self.mmax_sigma = mfd_conf["Maximum_Magnitude_Uncertainty"]
else:
self.mmax_sigma = msr.get_std_dev_mag(None, rake)
[docs] def get_mfd(self, slip, area, shear_modulus=30.0):
"""
Calculates activity rate on the fault
:param float slip:
Slip rate in mm/yr
:param fault_area:
Width of the fault (km)
:param float shear_modulus:
Shear modulus of the fault (GPa)
:returns:
* Minimum Magnitude (float)
* Bin width (float)
* Occurrence Rates (numpy.ndarray)
"""
# Convert shear modulus GPa -> dyne-cm, area km ** 2 -> cm ** 2 and
# slip mm/yr -> cm/yr
slip_moment = (shear_modulus * 1e10) * (area * 1e10) * (slip / 10.0)
dbar = D_VALUE * np.log(10.0)
bbar = self.b_value * np.log(10.0)
mags = np.arange(
self.mmin - (self.bin_width / 2.0),
self.mmax + self.bin_width,
self.bin_width,
)
if bbar >= dbar:
print(
"b-value larger than 1.5 will produce invalid results in "
"Anderson & Luco models"
)
self.occurrence_rate = np.nan * np.ones(len(mags) - 1)
return self.mmin, self.bin_width, self.occurrence_rate
self.occurrence_rate = np.zeros(len(mags) - 1, dtype=float)
for ival in range(0, len(mags) - 1):
self.occurrence_rate[ival] = RECURRENCE_MAP[
self.mfd_type
].cumulative_value(
slip_moment, self.mmax, mags[ival], bbar, dbar
) - RECURRENCE_MAP[self.mfd_type].cumulative_value(
slip_moment, self.mmax, mags[ival + 1], bbar, dbar
)
return self.mmin, self.bin_width, self.occurrence_rate
