# -*- coding: utf-8 -*-
# vim: tabstop=4 shiftwidth=4 softtabstop=4
#
# LICENSE
#
# Copyright (C) 2015-2021 GEM Foundation, G. Weatherill, M. Pagani
#
# 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.
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# The software Hazard Modeller's Toolkit (openquake.hmtk) provided herein
# is released as a prototype implementation on behalf of
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# Earthquake Model).
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# It is distributed for the purpose of open collaboration and in the
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# The software is NOT distributed as part of GEM’s OpenQuake suite
# (https://www.globalquakemodel.org/tools-products) and must be considered as a
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# developers, as GEM’s OpenQuake software suite.
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# Feedback and contribution to the software is welcome, and can be
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import numpy as np
from openquake.hmtk.seismicity.occurrence.base import (
SeismicityOccurrence, OCCURRENCE_METHODS)
[docs]@OCCURRENCE_METHODS.add(
'calculate',
completeness=True,
reference_magnitude=0.0,
mmax=None,
b_prior=1.0,
b_prior_weight=1.0,
area=0.0,
a_prior_weight=1.0)
class PenalizedMLE(SeismicityOccurrence):
"""
Test Implementation of the Penalized Maximum Likelihood function
"""
IERR = {1: "No events in catalogue - returning prior",
2: "Failure of convergence - returning prior"}
def _null_outcome(self, config, ierr):
"""
Method to handle failures and return information on their causes
:param dict config:
Configuration of method
:param int ierr:
Error code
"""
print(self.IERR[ierr])
a_4 = 0.05 * config["area"]
return config["b_prior"], 0.0, (
10.0 ** (np.log10(a_4) + (4. - config["reference_magnitude"])
* config["b_prior"]), 0.0)
[docs] def calculate(self, catalogue, config, completeness):
"""
Calculates the b-value and rates (and their corresponding standard
deviations) using the Penalized MLE approach
:param dict config:
Configuration parameters
:param catalogue:
Earthquake catalogue as instance of :class:
openquake.hmtk.seismicity.catalogue.Catalogue
:param completeness:
Completeness table
:returns:
b-value, standard deviation on b, rate (or a-value), standard
deviation on a
"""
# Setup
if config["b_prior"]:
betap = config["b_prior"] * np.log(10.)
beta = np.copy(betap)
has_prior = True
wbu = config["b_prior_weight"] / np.log(10.)
wau = config["a_prior_weight"]
if config["a_prior_weight"]:
apu = config["a_prior_weight"]
else:
apu = 1.0
else:
# No prior assumed. Take initial b-value of 1.0 for beta
betap = 0.0
beta = np.log(10.)
wbu = 1.0E-5
wau = 0.0
apu = 1.0
has_prior = False
# Get the counts of earthquakes in their completeness windows
delta, kval, tval, lamda, cum_rate, cum_count, nmx, nmt =\
self._get_rate_counts(catalogue, config, completeness)
n_val = np.sum(kval)
if not n_val:
return self._null_outcome(config, 1)
assert n_val == cum_count[0]
# Get the penalized MLE value (also returns correlation coefficient
# rho - but not used!)
bval, sigmab, rate, sigma_rate, rho = self._run_penalized_mle(
config, delta, kval, tval, cum_count, betap, beta, wbu, wau)
aval = np.log10(rate) + bval * completeness[0, 1]
if "reference_magnitude" in config.keys() and\
config["reference_magnitude"]:
dm = config["reference_magnitude"] - completeness[0, 1]
rate = 10.0 ** (np.log10(rate) - bval * dm)
sigma_rate = 10.0 ** (np.log10(rate + sigma_rate) - bval * dm) -\
rate
else:
dm = -completeness[0, 1]
rate = np.log10(rate) - bval * dm
sigma_rate = np.log10(rate + sigma_rate) - np.log10(rate)
rate = np.log10(rate)
return bval, sigmab, rate, sigma_rate
def _run_penalized_mle(self, config, delta, kval, tval, cum_count,
betap, beta, wbu, wau):
"""
Implements the core of the penalised maximum likelihood method for
the b-value
"""
nrloop = 0
while nrloop <= 10:
e_b = np.exp(beta * delta)
deb = e_b[:-1] - e_b[1:]
skmeb = np.sum(kval * ((delta[:-1] * e_b[:-1]) -
(delta[1:] * e_b[1:])) / deb)
skm2eb = np.sum(kval * (
((((delta[:-1] ** 2.) * e_b[:-1]) -
((delta[1:] ** 2.) * e_b[1:])) / deb) -
(((delta[:-1] * e_b[:-1]) - (delta[1:] * e_b[1:])) / deb)
** 2.))
sateb = np.sum(config["area"] * tval * deb)
satmeb = np.sum(config["area"] * tval *
((delta[:-1] * e_b[:-1]) - (delta[1:] * e_b[1:])))
satm2eb = np.sum(config["area"] * tval *
(((delta[:-1] ** 2.) * e_b[:-1]) -
((delta[1:] ** 2.) * e_b[1:])))
dldb = skmeb - cum_count[0] * (satmeb / sateb) -\
(wbu * (beta - betap))
d2ldb2 = skm2eb - cum_count[0] * (satm2eb / sateb -
(satmeb / sateb) ** 2.) - wbu
beta0 = np.copy(beta)
am0 = cum_count[0] * (1.0 - e_b[-1]) / sateb
if cum_count[1]:
# More than one interval
beta -= (dldb / d2ldb2)
if beta < 0.0:
if nrloop > 10:
# Total failure of convergence - return prior
return self._null_outcome(config, 2)
nrloop += 1
beta = np.log(10.) / (1.0 + float(nrloop))
continue
if np.abs(beta - beta0) < 1.0E-5:
break
else:
break
bval = beta / np.log(10.0)
v11 = (cum_count[0] / ((am0 * config["area"]) ** 2.)) + (wau / am0)
v22 = -d2ldb2 * (np.log(10.) ** 2.)
v12 = np.log(10.) * ((satmeb / (1.0 - e_b[-1])) +
delta[-1] * e_b[-1] * sateb /
((1.0 - e_b[-1]) ** 2.)) / config["area"]
vmat = np.array([[v11, v12], [v12, v22]])
error_mat = np.linalg.inv(vmat)
sigmab = np.sqrt(error_mat[1, 1])
sigma_rate = np.sqrt(error_mat[0, 0])
rho = error_mat[0, 1] / np.sqrt(error_mat[0, 0] * error_mat[1, 1])
return bval, sigmab, am0 * config["area"], sigma_rate, rho
def _get_rate_counts(self, catalogue, config, completeness):
"""
Using the earthquake catalogue and the completeness table determine
the number of complete earthquakes in each time and magnitude bin
:returns:
delta: Mmin - Mi
kval: Number of earthquakes in magnitude bin
tval: Effective duration of completeness for magnitude bin
lamda: Rate of earthquake in bin
cum_count: Number of earthquakes >= Mi
nmx: Number of magnitude bins
nmt: number of time bins
"""
# If the observed mmax is greater than the specified mmax then replace
# with observed Mmax
mmax_inp = np.max([config["mmax"],
np.max(catalogue.data["magnitude"])])
# Stack a maximum magnitude on the completeness magnitudes
if mmax_inp > np.max(completeness[:, 1]):
cmag = np.hstack([completeness[:, 1], mmax_inp + 1.0E-10])
high_event = True
else:
cmag = np.hstack([completeness[:, 1],
completeness[-1, 1] + 1.0E-10])
high_event = False
# Pre-pend the last year of the catalogue as the completeness year
cyear = np.hstack([catalogue.end_year + 1, completeness[:, 0]])
count_table = np.zeros([len(cmag) - 1, len(cyear) - 1])
nmx, nmt = count_table.shape
count_years = np.zeros_like(count_table)
for i in range(len(cyear) - 1):
time_idx = np.logical_and(catalogue.data["dtime"] < cyear[i],
catalogue.data["dtime"] >= cyear[i + 1])
nyrs = cyear[i] - cyear[i + 1]
sel_mags = catalogue.data["magnitude"][time_idx]
for j in range(i, len(cmag) - 1):
mag_idx = np.logical_and(sel_mags >= cmag[j],
sel_mags < cmag[j + 1])
count_table[j, i] += float(np.sum(mag_idx))
count_years[j, i] += float(nyrs)
delta = cmag[0] - cmag
if not high_event:
# Remove last row
delta = delta[:-1]
count_table = count_table[:-1, :]
count_years = count_years[:-1, :]
nmx -= 1
nmt -= 1
kval = np.sum(count_table, axis=1)
tval = np.sum(count_years, axis=1)
lamda = kval / tval
cum_rates = np.zeros_like(count_table)
cum_count = np.zeros_like(count_table)
# Get cumulative values
for i in range(nmt):
cum_rates[:, i] = np.sum(cum_rates[:, i:], axis=1)
cum_count[:, i] = np.sum(cum_count[:, i:], axis=1)
cum_rate = np.array([np.sum(lamda[i:]) for i in range(nmx)])
cum_count = np.array([np.sum(kval[i:]) for i in range(nmx)])
return delta, kval, tval, lamda, cum_rate, cum_count, nmx, nmt