magnitude completeness script added
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src/CompletenessMagnitudeEstimation/MCcalc.m
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src/CompletenessMagnitudeEstimation/MCcalc.m
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% function [OUTstruct,maxc,gft10,gft5,mgft]=MCcalc(m,t,s,iop)
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% ----------------------------------------------------------------------------------------------
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% Program that calculates the completess magnitude of a given
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% catalog by application of different methods, ASSUMING THAT
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% FREQUENCY-MANGITUDE DISTRIBUTION IS EXPONENTIAL
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% The program calls the following functions which are appended in the scipt:
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% function aa=FM_Param(m,t,iop) - GR-law parameters as a function of mmin
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% function [b]=bmle(x,y,ml,dm) - Maximum likelihood b-value estimation
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% function Rs=rescalc(No,m,a,b) - Residuals calculation
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% ----------------------------------------------------------------------------------------------
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% DESCRIPTION:
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% The completeness magnitude is estimated by the application of Goodness–of–Fit Test (GFT),
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% proposed by Wiemer and Wyss (2000). The procedure they followed is that a power law,
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% as a function of minimum magnitude, Mi, is fitted for events with M?Mi, by application
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% of maximum likelihood estimation (Aki, 1965). The synthetic data, i.e. the distribution of
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% magnitudes which represent a perfect fit to the power law, is constructed in this way. Then
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% the normalized absolute difference, R, between the cumulative number of observed events
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% (No) and the simulated ones (Ns) in each magnitude bin is computed. A model is found at
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% an R-value at which a predefined percentage (90% and 95%) of the observed data is modeled
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% by a straight line, which means that 90% and 95% of the observed data can be simulated by
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% the specific power law. In addition to this approach, synthetic catalogs are constructed following
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% the defined G-R, law, and then the number of events for each magnitude bin of the synthetic
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% catalogs is compared with the corresponding observed one. In this way a discrete minimum
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% residual point is calculated, and considered as the completeness magnitude for a given dataset.
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% The Maximum curvature method is also applied for comparison, which corresponds to the
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% magnitude bin with the highest frequency of events in the non-cumulative F-M distribution.
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% NOTE: Recall that the assumption of self similarity of the magnitude distribution may be invalid
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% in some cases. In such cases, alternative (e.g. network based) methods should be applied.
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% THE USER SHOULD COMPREHEND THE PROGRAM RESULTS TOGETHER WITH THEIR
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% VISUALIZATION IN ORDER TO MAKE HIS/HERS FINAL DECISION.
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% ----------------------------------------------------------------------------------------------
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% COMPLIED BY:
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% Leptokaropoulos Konstantinos, last updated December 2015, (kleptoka@igf.edu.pl)
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% within IS-EPOS Project,
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% ----------------------------------------------------------------------------------------------
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% INPUT:
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% m: magnitude of the events in the catalog
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% t: occurrence time of the events
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% s: number of synthetic catalogs following the same distribution with
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% the observed data (for modified goodness of fit test)-default is 500
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% iop: unit time selection (default is 'day'):
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% 0: day
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% 1: month
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% 2: year
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%--------------------------------------------------------------------------------------------
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% OUTPUT
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% aa: output (not extracted) is an 9-column array with seismicity parameters as
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% a function of minimum magnitude. This array is also saved as
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% 'parameters.txt'. Its columns correspond to:
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% aa(1): magnitude bin (the step is fixed to 0.1)
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% aa(2): number of events in the respective magnitude bin
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% aa(3): cumulative number of events in the respective magnitude bin
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% aa(4): b-value estimated by Maximum Likelihood (Aki, 1965)
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% function bmle, appended
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% aa(5): b-value uncertainty (Aki, 1965)
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% aa(6): a-value calculated from GR law logN=a-bM, given N,M from catalog
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% and b from MLE (aa(4)).
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% aa(7): at-value, normalized per unit time (at=a-log10(DT))
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% aa(8): Residual between observed data and power law,
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% as a function of minimum magnitude
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% aa(9): Average residual between synthetic catalogs (s) and power law,
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% as a function of minimum magnitude
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%
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% OUTstruct: Structure containing all the aforementioned parameters in matlab format
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% maxc: Mc estimation by Maximum Curvature Method(Wiemer & Wyss, 2000)
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% gft10: Mc estimation by Goodness of fit test at 90% confidence bounds (Wiemer & Wyss, 2000)
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% gft5: Mc estimation by Goodness of fit test at 95% confidence bounds (Wiemer & Wyss, 2000)
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% mgft: Mc estimation by Modified goodness of fit test (Leptokaropoulos et al., 2013)
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%----------------------------------------------------------------------------------------------
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% REFERENCES
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% - Aki, K. (1965), Maximum likelihood estimate of b in the formula log N=a-bM and its
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% confidence limits, Bull. Earthq. Res. Inst., Univ. Tokyo, 43, 237-239
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% - Wiemer, S. and M. Wyss (2000), Minimum magnitude of completeness in earthquake catalogs:
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% Examples from Alaska, the Western United States, and Japan, Bull. Seismol. Soc. Am., 90, 859-869.
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% - Leptokaropoulos et al., (2013), A homogeneous earthquake catalogue compilation for western
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% Turkey and magnitude of completeness determination, Bull. Seismol. Soc. Am., 103, 5, 2739-2751.
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%------------------------------------------------------------------------------------------------
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function [OUTstruct,maxc,gft10,gft5,mgft]=MCcalc(m,t,s,iop)
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if nargin==2;s=500;iop=0;end
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if nargin==3;iop=0;end
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% -------------- VALIDATION RULES -------------
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if s<10 || s>10000;error('Number of synthetic catalogs should be between 10 and 10000');end
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%----------------------------------------------------------
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%call FM_Param function
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aa=FM_Param(m,t,iop);
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no=aa(:,2);
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% Nobs-Npl residuals
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No=aa(:,3);m=aa(:,1);b=aa(:,4);a=aa(:,6);
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n=length(No);ml=min(m);mu=max(m);
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% call rescalc function
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Rs=rescalc(No,m,a,b);
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% Nsynth-Npl residuals
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for i=1:s
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ra=rand(No(1),1);
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nN=No/No(1);
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te=histc(ra,flipud(nN));
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te(1)=numel(find(ra<min(nN)));
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te=flipud(cumsum(te));
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Sy=te(2:n);Sy(n)=numel(find(ra<min(nN)));
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Rsyn=rescalc(Sy,m,a,b);
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R(i,:)=Rsyn;
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Rf=mean(R(:,1:n))';
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end
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Rf(Rf==inf)=100; Rf(Rf>100)=100; % K 17APR2014
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maxc=max(m(no==max(no)));
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gft10=min(m(Rs<10)); gft5=min(m(Rs<5));
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mgft=m(Rf==min(Rf)); rmgft=Rf(Rf==min(Rf));
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if gft10==max(m);gft10=NaN;end %K4NOE2014 %K 12NOV 2015- replace [] with NaNs
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if gft5==max(m);gft5=NaN;end %K4NOE2014 %K 12NOV 2015- replace [] with NaNs
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aa(:,8)=Rs;aa(:,9)=Rf;
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[OUTstruct M n N b berr a at Rs Rf]=ar2vec(aa);
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end
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% function aa=FM_Param(m,t,iop)
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% Function that calculates and saves seismicity parameters as a
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% function of minimum magnitude for given catalog
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% -------------------------------------------------------------------------------------------
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% -------------------------------------------------------------------------------------------
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function aa=FM_Param(m,t,iop)
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if nargin==2;iop=0;end
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dt=range(t);
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if iop==0 % unit time
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dt=dt;un='day';
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elseif iop==1;
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dt=dt/30.4;un='month';
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elseif iop==2
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dt=dt/365;un='year';
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end
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dm=0.1;
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m=round(m*10)*dm; %rounds magnitides to 1 decimal digit
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mu=max(m);ml=min(m); % maximum catalog magnitude
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mb=(ml:dm:mu)'; % magnitude bins
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n1=hist(m,min(m):0.1:max(m))'; % K-15MAY2015 number of events belonging to each magnitude bin
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nx=length(n1);
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cn1=flipud(cumsum(n1(nx:-1:1))); % cumulative number of events in each magnitude bin
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for i=1:nx % calculation of b-value as a function of magnitude
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b1=bmle(mb(i:nx),n1(i:nx),mb(i),dm);
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sb(i)=b1/sqrt(cn1(i)); % b-value error
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a(i)=log10(cn1(i))+b1*mb(i); % a-value
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b(i)=b1;
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end
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b=b';sb=sb';a=a';
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a1=a-log10(dt); % equivalent a-value for unit-time period
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aa=[mb n1 cn1 b sb a a1]; % output parameters
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end
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%------------------------------------------------------------
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%------------------------------------------------------------
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function [b]=bmle(x,y,ml,dm)
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% Function calcutates the b-value of frequency-magnitude
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% distribution of earthquakes following the Maximum
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% Likelihood Estimate (Aki, 1965).
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% INPUT:
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% x: magnitude
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% y: number of events with magnitude x
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% ml: minimum magnitude (cuttoff)
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% dm: the binning width of the catalog (default dm=0.1)
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me=sum(x.*y)/sum(y); % arithmetic mean of sample
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b=1/((me-ml+dm/2)*log(10));
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end
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%------------------------------------------------------------
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%------------------------------------------------------------
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function Rs=rescalc(No,m,a,b)
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% Residuals Calculation
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% Input:
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% No: number of observed data per magnitude bin
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% m: magnitude vector
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% a: GR law a-value
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% b: GR law b-value
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% OUTPUT
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% Rs: Fitting residuals as a function of minimum magnitude
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n=length(No);
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for i=1:n
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Ns=10.^(a(i)-m*b(i));
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Rs(i)=sum(abs(Ns(i:n)-No(i:n)))/sum(No(i:n));
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end
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Rs=Rs'*100;
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end
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%------------------------------------------------------------
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%------------------------------------------------------------ Added 17DEC2015
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function [OUTstruct M n N b berr a at Rs Rf]=ar2vec(aa)
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% Separates the output result 'aa' of the 'MCcalc'
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% function to 9 vectors and 1 structure, corresponding to:
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% M: Magnitude bins, from the minimum to the maximum
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% magnitude of the data set.
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% n: Number of events in the respective magnitude bin
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% N: Cumulative number of events in the respective magnitude bin
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% b: b-value of the GR law
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% berr: b-value uncertainty
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% a: a-value of the GR law
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% at: at-value (a-value normalized per unit time)
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% Rs: Residual between observed data and power law
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% Rf: Average residual between synthetic catalogs (s) and power law
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%
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%OUTstruct: is a structure containing the aforementioned fields
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% -----------------------------------------------------------------
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% Input is the 'aa' array as it is derived from 'MCcalc' program
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%------------------------------------------------------------------
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M=aa(:,1);
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n= aa(:,2);
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N=aa(:,3);
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b= aa(:,4);
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berr= aa(:,5);
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a =aa(:,6);
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at=aa(:,7);
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Rs= aa(:,8);
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Rf=aa(:,9);
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OUTstruct.M=M;OUTstruct.n=n;OUTstruct.N=N;
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OUTstruct.b=b;OUTstruct.berr=berr;OUTstruct.a=a;
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OUTstruct.at=at;OUTstruct.Rs=Rs;OUTstruct.Rf=Rf;
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end
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