# Source code for openquake.hmtk.seismicity.occurrence.aki_maximum_likelihood

```
# -*- coding: utf-8 -*-
# vim: tabstop=4 shiftwidth=4 softtabstop=4
#
# LICENSE
#
# Copyright (C) 2010-2019 GEM Foundation, G. Weatherill, M. Pagani,
# D. Monelli.
#
# The Hazard Modeller's Toolkit 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.
#
# You should have received a copy of the GNU Affero General Public License
# along with OpenQuake. If not, see <http://www.gnu.org/licenses/>
#
# DISCLAIMER
#
# The software Hazard Modeller's Toolkit (openquake.hmtk) provided herein
# is released as a prototype implementation on behalf of
# scientists and engineers working within the GEM Foundation (Global
# Earthquake Model).
#
# It is distributed for the purpose of open collaboration and in the
# hope that it will be useful to the scientific, engineering, disaster
# risk and software design communities.
#
# The software is NOT distributed as part of GEM’s OpenQuake suite
# (https://www.globalquakemodel.org/tools-products) and must be considered as a
# separate entity. The software provided herein is designed and implemented
# by scientific staff. It is not developed to the design standards, nor
# subject to same level of critical review by professional software
# developers, as GEM’s OpenQuake software suite.
#
# Feedback and contribution to the software is welcome, and can be
# directed to the hazard scientific staff of the GEM Model Facility
# (hazard@globalquakemodel.org).
#
# The Hazard Modeller's Toolkit (openquake.hmtk) is therefore distributed WITHOUT
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# FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License
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# liability for use of the software.
import warnings
import numpy as np
from openquake.hmtk.seismicity.occurrence.base import (
SeismicityOccurrence, OCCURRENCE_METHODS)
from openquake.hmtk.seismicity.occurrence.utils import recurrence_table, input_checks
[docs]@OCCURRENCE_METHODS.add('calculate', completeness=True)
class AkiMaxLikelihood(SeismicityOccurrence):
[docs] def calculate(self, catalogue, config=None, completeness=None):
"""
Calculation of b-value and its uncertainty for a given
catalogue, using the maximum likelihood method of Aki (1965),
with a correction for discrete bin width (Bender, 1983).
:param catalogue:
See :class:`openquake.hmtk.seismicity.occurrence.base.py`
for further explanation
:param config:
The configuration in this case do not contains specific
information
:keyword float completeness:
Completeness magnitude
:return float bval:
b-value of the Gutenberg-Richter relationship
:return float sigma_b:
Standard deviation of the GR b-value
"""
# Input checks
_cmag, _ctime, _ref_mag, dmag, config = input_checks(catalogue, config,
completeness)
rt = recurrence_table(
catalogue.data['magnitude'], dmag, catalogue.data['year'])
bval, sigma_b = self._aki_ml(rt[:, 0], rt[:, 1])
return bval, sigma_b
def _aki_ml(self, mval, number_obs, dmag=0.1, m_c=0.0):
"""
:param numpy.ndarray mval:
array of reference magnitudes (column 0 from recurrence
table)
:param numpy.ndarray number_obs:
number of observations in magnitude bin (column 1 from
recurrence table)
:keyword float dmag:
magnitude interval
:keyword float m_c:
completeness magnitude
:return float bval:
b-value of the Gutenberg-Richter relationship
:return float sigma_b:
Standard deviation of the GR b-value
"""
# Exclude data below Mc
id0 = mval >= m_c
mval = mval[id0]
number_obs = number_obs[id0]
# Get Number of events, minimum magnitude and mean magnitude
neq = np.sum(number_obs)
if neq <= 1:
# Cannot determine b-value (too few event) return NaNs
warnings.warn('Too few events (<= 1) to calculate b-value')
return np.nan, np.nan
m_min = np.min(mval)
m_ave = np.sum(mval * number_obs) / neq
# Calculate b-value
bval = np.log10(np.exp(1.0)) / (m_ave - m_min + (dmag / 2.))
# Calculate sigma b from Bender estimator
sigma_b = np.sum(number_obs * ((mval - m_ave) ** 2.0)) /\
(neq * (neq - 1))
sigma_b = np.log(10.) * (bval ** 2.0) * np.sqrt(sigma_b)
return bval, sigma_b
```