Introd u ction to statistical seismolog y C ASE STU D IE S IN - - PowerPoint PPT Presentation

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Introd u ction to statistical seismolog y C ASE STU D IE S IN - - PowerPoint PPT Presentation

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Introd u ction to statistical seismolog y C ASE STU D IE S IN STATISTIC AL TH IN K IN G J u stin Bois Lect u rer , Caltech California mo v es and shakes 1 Fa u lt data : USGS Q u aternar y Fa u lt and Fold Database of the United States CASE

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Introduction to statistical seismology

C ASE STU D IE S IN STATISTIC AL TH IN K IN G

Justin Bois

Lecturer, Caltech

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CASE STUDIES IN STATISTICAL THINKING

California moves and shakes

Fault data: USGS Quaternary Fault and Fold Database of the United States

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CASE STUDIES IN STATISTICAL THINKING

The Parkfield region

Fault data: USGS Quaternary Fault and Fold Database of the United States

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CASE STUDIES IN STATISTICAL THINKING

The Parkfield region

Fault data: USGS Quaternary Fault Fault and Fold Database of the United States Earthquake data: USGS ANSS Comprehensive Earthquake Catalog

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CASE STUDIES IN STATISTICAL THINKING

The Parkfield region

Image: Linda Tanner, CC-BY-2.0

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CASE STUDIES IN STATISTICAL THINKING

Seismic Japan

Data source: USGS ANSS Comprehensive Earthquake Catalog (ComCat)

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CASE STUDIES IN STATISTICAL THINKING

ECDF of magnitudes, Japan, 1990-1999

Data source: USGS ANSS Comprehensive Earthquake Catalog (ComCat)

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CASE STUDIES IN STATISTICAL THINKING

Location parameters

m ≡ m − 5 ∼ Exponential m ≡ m − m ∼ Exponential

′ ′ t

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CASE STUDIES IN STATISTICAL THINKING

The Gutenberg-Richter Law

The magnitudes of earthquakes in a given region over a given time period are Exponentially distributed One parameter, given by

− m , describes earthquake

magnitudes for a region

m

t

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CASE STUDIES IN STATISTICAL THINKING

The b-value

b = ( − m ) ⋅ ln 10

# Completeness threshold mt = 5 # b-value b = (np.mean(magnitudes) - mt) * np.log(10) print(b) 0.9729214742632566

m

t

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CASE STUDIES IN STATISTICAL THINKING

ECDF of all magnitudes

Data source: USGS ANSS Comprehensive Earthquake Catalog (ComCat)

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CASE STUDIES IN STATISTICAL THINKING

ECDF of all magnitudes

Data source: USGS ANSS Comprehensive Earthquake Catalog (ComCat)

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CASE STUDIES IN STATISTICAL THINKING

Completeness threshold

The magnitude, m , above which all earthquakes in a region can be detected

t

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Let's practice!

C ASE STU D IE S IN STATISTIC AL TH IN K IN G

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Timing of major earthquakes

C ASE STU D IE S IN STATISTIC AL TH IN K IN G

Justin Bois

Lecturer, Caltech

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CASE STUDIES IN STATISTICAL THINKING

Models for earthquake timing

Exponential: Earthquakes happen like a Poisson process Gaussian: Earthquakes happen with a well-dened period

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CASE STUDIES IN STATISTICAL THINKING

Stable continental region earthquakes

Data source: USGS Earthquake Catalog for Stable Continental Regions

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CASE STUDIES IN STATISTICAL THINKING

The Nankai Trough

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CASE STUDIES IN STATISTICAL THINKING

Earthquakes in the Nankai Trough

Date Magnitude 684-11-24 8.4 887-08-22 8.6 1099-02-16 8.0 1361-07-26 8.4 1498-09-11 8.6 1605-02-03 7.9 1707-10-18 8.6 1854-12-23 8.4

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CASE STUDIES IN STATISTICAL THINKING

ECDF of time between Nankai quakes

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CASE STUDIES IN STATISTICAL THINKING

Formal ECDFs

ECDF(x) = fraction of data points ≤ x

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CASE STUDIES IN STATISTICAL THINKING

Formal ECDFs

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CASE STUDIES IN STATISTICAL THINKING

Formal ECDFs

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CASE STUDIES IN STATISTICAL THINKING # time_gap is an array of interearthquake times _ = plt.plot(*dcst.ecdf(time_gap, formal=True)) _ = plt.xlabel('time between quakes (yr)') _ = plt.ylabel('ECDF')

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CASE STUDIES IN STATISTICAL THINKING

# Compute the mean time gap mean_time_gap = np.mean(time_gap) # Standard deviation of the time gap std_time_gap = np.std(time_gap) # Generate theoretical Exponential distribution of timings time_gap_exp = np.random.exponential(mean_time_gap, size=100000) # Generate theoretical Normal distribution of timings time_gap_norm = np.random.normal( mean_time_gap, std_time_gap, size=100000 ) # Plot theoretical CDFs _ = plt.plot(*dcst.ecdf(time_gap_exp)) _ = plt.plot(*dcst.ecdf(time_gap_norm))

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CASE STUDIES IN STATISTICAL THINKING

Model for Nankai Trough

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Let's practice!

C ASE STU D IE S IN STATISTIC AL TH IN K IN G

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How are the Parkfield interearthquake times distributed?

C ASE STU D IE S IN STATISTIC AL TH IN K IN G

Justin Bois

Lecturer, Caltech

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CASE STUDIES IN STATISTICAL THINKING

The Parkfield Prediction

Adapted from Barkun and Lindh, Science, 229, 619-624, 1985

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Hypothesis test on the Nankai megathrust earthquakes

Hypothesis: The time between Nankai Trough earthquakes is Normally distributed with a mean and standard deviation as calculated from the data Test statistic: ?? At least as extreme as: ??

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CASE STUDIES IN STATISTICAL THINKING

The Kolmogorov-Smirnov statistic

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CASE STUDIES IN STATISTICAL THINKING

The Kolmogorov-Smirnov statistic

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CASE STUDIES IN STATISTICAL THINKING

The Kolmogorov-Smirnov statistic

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CASE STUDIES IN STATISTICAL THINKING

The Kolmogorov-Smirnov statistic

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CASE STUDIES IN STATISTICAL THINKING

The Kolmogorov-Smirnov statistic

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CASE STUDIES IN STATISTICAL THINKING

Kolmogorov-Smirnov test

Hypothesis: The time between Nankai Trough earthquakes is Normally distributed with a mean and standard deviation as calculated from the data Test statistic: Kolmogorov-Smirnov statistic At least as extreme as: ≥ observed K-S statistic

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CASE STUDIES IN STATISTICAL THINKING

Simulating the null hypothesis

Draw and store lots of (say, 10,000) samples out of the theoretical distribution Draw n samples out of the theoretical distribution Compute the K-S statistic from the samples

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CASE STUDIES IN STATISTICAL THINKING

# Generate samples from theoretical distribution x_f = np.random.normal(mean_time_gap, std_time_gap, size=10000) # Initialize K-S replicates reps = np.empty(1000) # Draw replicates for i in range(1000): # Draw samples for comparison x_samp = np.random.normal( mean_time_gap, std_time_gap, size=len(time_gap) ) # Compute K-S statistic reps[i] = ks_stat(x_samp, x_f) # Compute p-value p_val = np.sum(reps >= ks_stat(time_gap, x_f)) / 1000

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Let's practice!

C ASE STU D IE S IN STATISTIC AL TH IN K IN G