import numpy as np class M_max_models: def __init__(self, data = None, f_name = None, time_win = None, space_win = None, Mc = None, b_method = None, num_bootstraps = None, G = None, Mu = None, dv = None, Mo = None, SER = None, cl = None, ssd = None, C = None, ): self.data = data # Candidate data table: 2darray n x m, for n events and m clonums: x, y, z, t, mag self.f_name = f_name # Feature's name to be calculated, check: def ComputeFeaure(self) self.time_win = time_win # Time window whihc a feature is computed in self.space_win = space_win # Space window ... self.Mc = Mc # Magnitude of completeness for computing b-positive self.b_method = b_method # list of b_methods self.num_bootstraps = num_bootstraps # Num of bootstraps for standard error estimation of b-value self.G = G # Shear modulus self.Mu = Mu # Friction coefficient self.dv = dv # Injected fluid self.SER = SER self.Mo = Mo # Cumulative moment magnitude self.cl = cl # Confidence level self.ssd = ssd # Static stress drop (Shapiro et al. 2013) self.C = C # Geometrical constant (Shapiro et al. 2013) def b_value(self, b_flag): if b_flag == '1': return 1, None # maximum-likelihood estimate (MLE) of b (Deemer & Votaw 1955; Aki 1965; Kagan 2002): elif b_flag == 'b': X = self.data[np.where(self.data[:,-1]>self.Mc)[0],:] if X.shape[0] > 0: b = 1/((np.mean(X[:,-1] - self.Mc))*np.log(10)) std_error = b/np.sqrt(X.shape[0]) else: raise ValueError("All events in the current time window have a magnitude less than 'completeness magnitude'. Use another value either for 'time window', 'minimum number of events' or 'completeness magnitude'.") return b, std_error # B-positive (van der Elst 2021) elif b_flag == 'bp': # Function to perform bootstrap estimation def bootstrap_estimate(data, num_bootstraps): estimates = [] for _ in range(num_bootstraps): # Generate bootstrap sample bootstrap_sample = np.random.choice(data, size=len(data), replace=True) # Perform maximum likelihood estimation on bootstrap sample diff_mat = np.diff(bootstrap_sample) diff_mat = diff_mat[np.where(diff_mat>0)[0]] estimate = 1/((np.mean(diff_mat - np.min(diff_mat)))*np.log(10)) estimates.append(estimate) return np.array(estimates) diff_mat = np.diff(self.data[:,-1]) diff_mat = diff_mat[np.where(diff_mat>0)[0]] bp = 1/((np.mean(diff_mat - np.min(diff_mat)))*np.log(10)) bootstrap_estimates = bootstrap_estimate(diff_mat, self.num_bootstraps) std_error = np.std(bootstrap_estimates, axis=0) return bp, std_error # Tapered Gutenberg_Richter (TGR) distribution (Kagan 2002) elif b_flag == 'TGR': from scipy.optimize import minimize # The logarithm of the likelihood function for the TGR distribution (Kagan 2002) def log_likelihood(params, data): beta, Mcm = params n = len(data) Mt = np.min(data) l = n*beta*np.log(Mt)+1/Mcm*(n*Mt-np.sum(data))-beta*np.sum(np.log(data))+np.sum(np.log([(beta/data[i]+1/Mcm) for i in range(len(data))])) return -l X = self.data[np.where(self.data[:,-1]>self.Mc)[0],:] M = 10**(1.5*X[:,-1]+9.1) initial_guess = [0.5, np.max(M)] bounds = [(0.0, None), (np.max(M), None)] # Minimize the negative likelihood function for beta and maximum moment result = minimize(log_likelihood, initial_guess, args=(M,), bounds=bounds, method='L-BFGS-B', options={'gtol': 1e-12, 'disp': False}) beta_opt, Mcm_opt = result.x eta = 1/Mcm_opt S = M/np.min(M) dldb2 = -np.sum([1/(beta_opt-eta*S[i])**2 for i in range(len(S))]) dldbde = -np.sum([S[i]/(beta_opt-eta*S[i])**2 for i in range(len(S))]) dlde2 = -np.sum([S[i]**2/(beta_opt-eta*S[i])**2 for i in range(len(S))]) std_error_beta = 1/np.sqrt(dldb2*dlde2-dldbde**2)*np.sqrt(-dlde2) return beta_opt*1.5, std_error_beta*1.5 def McGarr(self): b_value, b_stderr = self.b_value(self.b_method) B = 2/3*b_value if B < 1: sigma_m = ((1-B)/B)*(2*self.Mu)*(5*self.G)/3*self.dv Mmax = (np.log10(sigma_m)-9.1)/1.5 if b_stderr: Mmax_stderr = b_stderr/np.abs(np.log(10)*(1.5*b_value-b_value**2)) else: Mmax_stderr = None else: Mmax = None Mmax_stderr = None return b_value, b_stderr, Mmax, Mmax_stderr def Hallo(self): b_value, b_stderr = self.b_value(self.b_method) B = 2/3*b_value if b_value < 1.5: sigma_m = self.SER*((1-B)/B)*(2*self.Mu)*(5*self.G)/3*self.dv Mmax = (np.log10(sigma_m)-9.1)/1.5 if b_stderr: Mmax_stderr = self.SER*b_stderr/np.abs(np.log(10)*(1.5*b_value-b_value**2)) else: Mmax_stderr = None else: Mmax = None Mmax_stderr = None return b_value, b_stderr, Mmax, Mmax_stderr def Li(self): sigma_m = self.SER*2*self.G*self.dv - self.Mo Mmax = (np.log10(sigma_m)-9.1)/1.5 if Mmax < 0: return None else: return Mmax def van_der_Elst(self): b_value, b_stderr = self.b_value(self.b_method) # Seismogenic_Index X = self.data si = np.log10(X.shape[0]) - np.log10(self.dv) + b_value*self.Mc if b_stderr: si_stderr = self.Mc*b_stderr else: si_stderr = None Mmax = (si + np.log10(self.dv))/b_value - np.log10(X.shape[0]*(1-self.cl**(1/X.shape[0])))/b_value if b_stderr: Mmax_stderr = (np.log10(X.shape[0]) + np.log10(X.shape[0]*(1-self.cl**(1/X.shape[0]))))*b_stderr else: Mmax_stderr = None return b_value, b_stderr, si, si_stderr, Mmax, Mmax_stderr def L_Shapiro(self): from scipy.stats import chi2 X = self.data[np.isfinite(self.data[:,1]),1:4] # Parameters STD = 2.0 # 2 standard deviations conf = 2 * chi2.cdf(STD, 2) - 1 # covers around 95% of population scalee = chi2.ppf(conf, 2) # inverse chi-squared with dof=#dimensions # Center the data Mu = np.mean(X, axis=0) X0 = X - Mu # Covariance matrix Cov = np.cov(X0, rowvar=False) * scalee # Eigen decomposition D, V = np.linalg.eigh(Cov) order = np.argsort(D)[::-1] D = D[order] V = V[:, order] # Compute radii VV = V * np.sqrt(D) R1 = np.sqrt(VV[0, 0]**2 + VV[1, 0]**2 + VV[2, 0]**2) R2 = np.sqrt(VV[0, 1]**2 + VV[1, 1]**2 + VV[2, 1]**2) R3 = np.sqrt(VV[0, 2]**2 + VV[1, 2]**2 + VV[2, 2]**2) L = (1/3*(1/R1**3+1/R2**3+1/R3**3))**(-1/3) return R1, R2, R3, L def Shapiro(self): R1, R2, R3, L = self.L_Shapiro() return R1, R2, R3, L, np.log10(R3**2)+(np.log10(self.ssd)-np.log10(self.C)-9.1)/1.5 def All_models(self): return self.McGarr()[2], self.Hallo()[2], self.Li(), self.van_der_Elst()[-2], self.Shapiro()[-1] def ComputeModel(self): if self.f_name == 'max_mcg': return self.McGarr() if self.f_name == 'max_hlo': return self.Hallo() if self.f_name == 'max_li': return self.Li() if self.f_name == 'max_vde': return self.van_der_Elst() if self.f_name == 'max_shp': return self.Shapiro() if self.f_name == 'max_all': return self.All_models()