Ba y esian Decon v olution of Seismic Arra y Data for Ripple�Fired Explosions Eric A� Suess Advisor� Professor Rob ert Sh um w a y
Outline� �� The Problem �� Mo del �� Ba y esian Statistics �� Ba y esian Decon v olution of Seismic Arra y Data �� Results � Sim ulated Data � Real Data �� Conclusions and F uture W ork
T reaties� �� ���� Limited Nuclear T est Ban T reat y �L TBT� �� ���� Non�Proliferation of Nuclear W eap ons T reat y �NPT� �� ���� Threshold T est Ban T reat y �TTBT� �� ���� P eaceful Nuclear Explosions T reat y �PNET� �� ���� Comprehensiv e T est Ban T reat y �CTBT�
Bac kground� Muc h of the fo cus in the past has b een on distinguishing p ossible n uclear explosions from earthquak es � Curren tly � since the testing treaties ha v e put limitations on the p ermissible sizes of the n uclear explosions� other smaller seismic ev en ts suc h as industrial mining explosions ha v e b ecome of in terest in the discrimination problem� The w ork presen ted here is related to distinguishing lo w�lev el n uclear explosions from ripple��red mining explosions that are on the same seismic lev el�
The Problem� � Monitoring seismic ev en ts at Regional Distances for lo w�lev el n uclear tests� � Other seismic sources need to b e ruled out� Ripple�Fired Mining Explosions �
Ripple�Fired Explosions� This is a mining tec hnique in whic h explosions of single devices �or groups of devices� are detonated in succession�
Monitoring� Arra ys of receiv ers are put in place at Regional Distances and seismic data is con tin ually collected � Seismic disturbances that are ab o v e the baseline noise of the area are in v estigated�
Mo del� m X y � t � � s � t � � a s � t � j � � � � t � k k j k k j �� A mplitudes are distributed according to a random Bernoulli�Gaussian mo del� �ref� Cheng� Chen and Li ����� � p � a j � � � �� � � � I � a � �� � � T N � � � � � I � a � �� j j � j � Signal and p ath e�e cts follo w an AR ��� mo del� �ref� Dargahi�Noubary ����� Tj�stheim ����� s � t � � � s � t � �� � ��� � � s � t � p � � e � t � k � k p k k � � e � t � i�i�d N �� � � � and de�ne the precision � � � k � � � � � t � i�i�d� N �� � c� �� c � � �S N R � c � � is �xed� k
T runcated Normal� � � � � � a � � � j � � � � p p � a � � c � � � � � exp � I � a � �� j j � � � � � � � � � where � � � � � x � � � � Z � � � � c � � � � � � p exp � � � � � � � � � � �
Ba y esian Statistics� Mo del� p � Y j � � Prior distribution� p � � � Join t distribution� p � Y � � � p � Y j � � p � � � � P osterior distribution� p � Y j � � p � � � p � � j Y � � R p � Y j � � p � � � d � � p � Y j � � p � � �
Gibbs Sampler� �ref� Gelfand and Smith ����� � � � � � � � � � p � � j Y � � p �� � � � � j Y � � � ��� ��� Giv en � and � for h � � � ���� R eps � � � h � � h � �� �� Sample � from p � � j Y � � � � � � � h � � h � �� Sample � from p � � j Y � � � � � � �� Set h � h � � and go to �� � B ur nI n ��� � B ur nI n ��� � R eps � � R eps � � � � � � � ���� � � � � � � � � �
��� � R eps � � ���� are realizations of a stationary Mark o v Chain� � � � h � �� � h � with transition probabilit y from � to � � � h � �� � h � � h � �� � h � T � � � � � p � � j Y � � � p � � j Y � � � � � � � � By Ergo dic theory � w e can calculate estimates of sa y � b y � � a�s� � i � X � � � E � � j Y � � � � R eps h R eps � � �
Priors� � � B E T A � � � � � � � � N � � � � � � p � � � � GAM M A � � � � � � �
The Mo del� m X y � t � � s � t � � a s � t � j � � � � t � k k j k k j �� The parameter set� � f � � a � � � � S g � � Hyp erparam ters� � � � � � � � � � � � � � � � � � � � � � � � � Fixed parameters� c� m
Lik eliho o d� � � y � ���� � � Y y � q � � y ��� � ���� y � n �� � y k k k q Y p � Y j � � � p � y j � � k k �� q � � � n� � � � � Y X � n� � � � �� � � exp � � � t � k c � c t k ��
Ov erall Prior� p � � � � p � � � a � � � � S � � � p � � � p � a j � � p � � � p � � � p � S j � � � � Join t Densit y� p � Y � � � p � Y j � � p � � � � Join t P osterior� p � � j Y � � p � Y j � � p � � �
Conditional Marginal P osterior Distributions� � � p � � j Y � r est � � beta � � � � � � � F or �xed j � � � ���� m � p � a j Y est � � �� � � I � a � �� � � � � I � a �� � r � � T N � � � j j j j a j a j j p � � j Y � r est � � N � � � � � p � � � � p � � j Y � r est � � g amma � � � � � � � F or �xed i � � � ���� n and k � � � ���� q � p � s � i � j Y � r est � � N � � � � � k s � i � s � i � k k
� � � j Y � r est � beta � � � � � � � � � � m � n � � � a � � � � � n � � a � �
a j Y � r est � Bernoulli�Gaussian j � � � � � � � X X � � � � � � � � t � � j �� s � t � j � a k a k j � � c j � t k � � � � X X � � � � � � s � t � j � a k � c j � t k � � � j � � � � �� � � c � � �� � � � � � a � � a a � a j � � �� � � � j j exp j � � � � � c � � �� � � � � � � � � a � j
� j Y � r est � N � � � � � p � � � � X X � � � � � � � s � t � � t � � � � s � k k � � � � t k X X � � � � � � � � � � t � � � t � � � s s k k � � t k where � � � t � � � s � t � �� � ���� s � t � p �� s k k k
� � � j Y � r est � g amma � � � � � � � � � � q �� n � l � p � � � � � � � � � X X X X � � � � � � � t � � e � t � � k k � c � t t k k
s � t � j Y � r est � Normal k � � � � � X X � � � � � � a � � t � � i �� � � � e � t � � i �� t � i t � i s � i � k s � i � k c k t t � � � X X � � � � � � a � � � t � i t � i s � i � c k t t
Steps T o P erform The Gibbs Sampler� Giv en the initial v alues� n o ��� ��� ��� ��� ��� � � � � � � a � S for h � � to Reps� � h � �� Sample � from a b eta � � h � �� Sample a � j � � � ���� m � from a Bernoulli�Gaussian� j � h � �� Sample from a p�v ariate normal � � � h � �� Sample � from a gamma � � h � �� Sample s � i � � i � � � ���� n � k � � � ���� q � from a normal � k �� Set h � h � � and go to Step ��
Conclusions and F urther W ork� �� one �� t w o
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