Source%20parameters%20determination%20using%20moment%20tensor%20inversion

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• Source parameters determination using
  moment tensor inversion

Panagiotis Papadimitriou, Alexandra Moshou,
Konstantinos Makropoulos Department of
Geophysics and Geothermics National and
Kapodistrian University of Athens
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Contents
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Mathematical Methods in Science and Technology

• Determination of seismic parameters of an
  earthquake
• Methodology
• Bodywave modeling by calculation synthetic
  seismograms and fitting with observed
• Non linear inverse problem

• Seismic moment tensor, Mij
• Seismic source
• Depth

• Forward problem
• Inverse problem

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Bodywave propagation
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Mathematical Methods in Science and Technology
4
Seismic Faults
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• motion between two blocks
• the two sides of a block slippage relative to
  each other with a standard geometry

footwall
Slip vector
Hanging wall
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Seismic fault plane
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Slip 3 D representation
f azimuth of the fault plane with respect to
North, 0 lt f lt 360 d dip of the fault plane
from horizontal, 0 lt d lt 90 ? rake of the
direction of slip in the fault plane with
respect to the horizontal -180 lt ? lt 180

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Calculation of Synthetic Seismograms
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Mathematical Methods in Science and Technology
U(t) displacement seismogram s(t) seismic
source operator m(t) structure response
operator Q(t) attenuation operator I(t)
instrument response operator
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Bodywave synthetic seismograms
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Mathematical Methods in Science and Technology
8
Equivalent Body forces double couple
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earthquake
Double couple
Focal mechanism
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Equivalent body forces representing seismic
sources
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Mathematical Methods in Science and Technology
Single Force
Single couple
Double couple
10

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General representation of seismic source using 9
force couples
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Forward problem
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Mathematical Methods in Science and Technology
Given initial
parameters of the model
Calculate synthetic seismograms and
comparison with the corresponding

observed Determine
the parameters of the model
by trial and error
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Forward problem
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Mathematical Methods in Science and Technology

• G is a non square matrix (n x m) whose
  elements are a set of five elementary Greens
  functions.
• m a vector of length
• m 6 the six elements of the moment
  tensor
• m 7 six elements of the moment
  tensor
• depth
• m 9 six elements of the moment
  tensor source location (x, z, t)
• d a vector of length n observations

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Inverse problem
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Mathematical Methods in Science and Technology
Given field
observations
Determine parameters of the
model
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Inverse problem
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Mathematical Methods in Science and Technology

• d n x 1 matrix," lives in data space of n
  components, represents the observed data set
• G non - square n x m matrix, greens function
• m m x 1 vector, the model vector m lives in
  model space of m components

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Moment tensor inversion - methodology
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Mathematical Methods in Science and Technology

u synthetics seismograms G Greens function m
model of parameters
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Moment tensor inversion
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Mathematical Methods in Science and Technology
Where n is the number of waveforms, when n gt 6,
the system is overdetermined and it is should be
possible to resolve the system. That means to
found the inverse of G.
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Moment tensor inversion
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Mathematical Methods in Science and Technology

• To determine the model m must be inversed the
  matrix G
• G non square matrix
• The system is overdetermined
• Process 1) determination the transpose of the
• matrix G, GT
• 2) multiplication GTG
• 3) determination G-1
• The non square matrix G lives in work
  space n x m components. The matrix G maps a
  vector from model - space to a vector in data
  space, is composed by a set of five elementary
  Greens function

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Methods for overcomplete system
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Mathematical Methods in Science and Technology

• Normal equations
• QR factorization
• SVD decomposition

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Normal equations
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Mathematical Methods in Science and Technology
The system of normal equations
and the solution is
pseudo inverse
where

• G non square, n x m

Square matrices
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QR factorization
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Mathematical Methods in Science and Technology
where

• Q orthogonal matrix, Gram Schmidt
  orthogonalization of G
• R upper triangular,

QR factorization transform the least square
problem into a triangular least - square
where
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Singular value decomposition
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Mathematical Methods in Science and Technology

• Any real matrix G (n x m) can be decomposed in
  three parts

where

• U,V n x n, m x m orthogonal matrices
  respectively
• The columns of U,V are the eigenvectors
  , respectively
• ? unique n x m diagonal matrix, with real and
  non negative
• elements ?i ,(singular values of G) i 1,2, ,
  min (m, n) gt 0, in order

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Singular value decomposition Determination G-1
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Mathematical Methods in Science and Technology
The matrices U,V are orthogonal, that means
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Singular value decomposition Determination G-1
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where
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Singular value decomposition
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Mathematical Methods in Science and Technology
25
Singular value decomposition
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Mathematical Methods in Science and Technology
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SVD decomposition
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Mathematical Methods in Science and Technology

• We do not need to know if the problem is
  overdetermined or undetermined. The singular
  values in S will show what kind of problem we are
  trying to solve. SVD always produces a solution.
• In the case that the problem is undetermined, SVD
  finds the solution in the complement null
  space. This is the minimum solution.

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Normal equations QR factorization Singular
value decomposition
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Mathematical Methods in Science and Technology

• Normal equations and QR factorization only work
  for fully ranked matrices (i.e. r(A)n). If A
  is rank deficient, there are infinite number of
  solutions to the least squares problems.
• The algorithm of normal equations is the fastest
  and the least accurate among the three.
• Singular value decomposition is the slowest and
  the most accurate.

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SVD Least squares
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Mathematical Methods in Science and Technology
If the model fit exactly the data, then E is be a
vector with all elements approximate zero. Since
this will not usually be the case, the inverse
problem is designed to find a model that
minimizes E.
be minimum
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Moment tensor inversion
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Mathematical Methods in Science and Technology

• Conditions for double couple solution
• det Mij 0
• trace M 0

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General moment tensor
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Earthquake Aigio 15/06/1995
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Mathematical Methods in Science and Technology
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Non linear problem
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• the problem becomes non linear

Because the derivative of the envelope misfit
function is non linear
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Example of non linear problem
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Non linear problem
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Mathematical Methods in Science and Technology

• Methodology for non linear inversion
  conditions

where
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Non linear problem
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Mathematical Methods in Science and Technology

• where
• µ, 1-µ relative weights of the waveforms
  misfit function E1 (m) and the envelope misfit
  function E2 (m)
• ?1 , ?2 Lagrange multipliers associated with
  the constraints C1 (m), C2 (m)

• Conditions for non linear problem
• ? ? 0 or µ ? 0

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Non linear problem
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Mathematical Methods in Science and Technology

• Linearization problem
• The source parameters mnew may be determined by
  solving the set of n equations

• Subject the constraint
• C2 (m) 0, when ?2 ? 0
• and
• C1 (m) 0, when ?1 ? 0

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Non linear problem
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Mathematical Methods in Science and Technology

• In any event, an initial solution m0 is required
  to solve the equation (10). In our case we use
  point source parameters inverted using
  alternative methods as the starting solution. The
  synthetics can be linearize with respect to the
  initial source parameters

then (11) becomes
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Non linear problem
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Mathematical Methods in Science and Technology
where
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Non linear problem methodology
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Mathematical Methods in Science and Technology

• Newtons method
• We define a vector
• and an initial solution vector

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Non linear problem methodology
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Mathematical Methods in Science and Technology

• The (i 1) th iteration can be expressed as
• The source model parameters m, can be determine
  by iterating the following linear system with
  respect to x

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Non linear problem methodology
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Mathematical Methods in Science and Technology
where
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