Download seismological application

Some basic maths for seismic data
processing and inverse problems
(Refreshement only!)
• Complex Numbers
• Vectors
– Linear vector spaces
• Matrices
–
–
–
–
Determinants
Eigenvalue problems
Singular values
Matrix inversion
Mathematical foundations
The idea is to illustrate these
mathematical tools with
examples from seismology
Modern Seismology – Data processing and inversion
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Complex numbers
i
z  a  ib  re  r (cos   i sin  )
Mathematical foundations
Modern Seismology – Data processing and inversion
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Complex numbers
conjugate, etc.
z*  a  ib  r (cos   i sin  )
 r cos    ri sin(  )  r i
z 2  zz*  ( a  ib )( a  ib )  r 2
cos   (e i  e i ) / 2
sin   (e i  e i ) / 2i
Mathematical foundations
Modern Seismology – Data processing and inversion
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Complex numbers
seismological application
Plane waves as superposition of harmonic signals
using complex notation
ui ( x j , t )  Ai exp[ ik (a j x j  ct )]
u(x, t )  A exp[ ikx  t ]
Use this „Ansatz“ in the acoustic wave equation and
interpret the consequences for wave propagation!
Mathematical foundations
Modern Seismology – Data processing and inversion
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Vectors and Matrices
For discrete linear inverse problems we will need the concept of
linear vector spaces. The generalization of the concept of size of a vector
to matrices and function will be extremely useful for inverse problems.
Definition: Linear Vector Space.
A linear vector space over a field F of
scalars is a set of elements V together with a function called addition
from VxV into V and a function called scalar multiplication from FxV into
V satisfying the following conditions for all x,y,z  V and all a,b  F
1.
2.
3.
4.
5.
6.
7.
8.
(x+y)+z = x+(y+z)
x+y = y+x
There is an element 0 in V such that x+0=x for all x  V
For each x  V there is an element -x  V such that x+(-x)=0.
a(x+y)= a x+ a y
(a + b )x= a x+ bx
a(b x)= ab x
1x=x
Mathematical foundations
Modern Seismology – Data processing and inversion
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Matrix Algebra – Linear Systems
Linear system of algebraic equations
a11 x1  a12 x2  ...  a1n xn  b1
a21 x1  a22 x2  ...  a2 n xn  b2
..........
an1 x1  an 2 x2  ...  ann xn  bn
... where the x1, x2, ... , xn are the unknowns ...
in matrix form
Ax  b
Mathematical foundations
Modern Seismology – Data processing and inversion
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Matrix Algebra – Linear Systems
Ax  b
where
 a11 a12
a
a22
21

A  aij 
 

an1 an 2
 
b1 
b 
 2
b  bi    

bn 
Mathematical foundations




a1n 

a11 


ann 
 x1 
x 
 
x  xi    2 

 xn 
A is a nxn (square) matrix,
and x and b are column
vectors of dimension n
Modern Seismology – Data processing and inversion
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Matrix Algebra – Vectors
Row vectors
v  v1 v2
Column vectors
w 1 
 
w  w2 
w 
 3
v3 
Matrix addition and subtraction
C  A B
D  A B
with
with
cij  aij  bij
d ij  aij  bij
Matrix multiplication
C  AB
with
m
cij   aik bkj
k 1
where A (size lxm) and B (size mxn) and i=1,2,...,l and j=1,2,...,n.
Note that in general AB≠BA but (AB)C=A(BC)
Mathematical foundations
Modern Seismology – Data processing and inversion
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Matrix Algebra – Special
Transpose of a matrix
 
A  aij
Symmetric matrix
 
A T  a ji
A  AT
aij  a ji
( AB) T  BT A T
Identity matrix
1 0  0
0 1  0

I
   


0 0  1
with AI=A, Ix=x
Mathematical foundations
Modern Seismology – Data processing and inversion
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Matrix Algebra – Orthogonal
Orthogonal matrices
a matrix is Q (nxn) is said to be
orthogonal if
QT Q  I n
... and each column is an orthonormal
vector
qi qi  1
1
Q
2
... examples:
1  1
1 1 


it is easy to show that :
Q T Q  QQ T  I n
if orthogonal matrices operate on
vectors their size (the result of
their inner product x.x) does not
change -> Rotation
(Qx )T (Qx )  xT x
Mathematical foundations
Modern Seismology – Data processing and inversion
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Matrix and Vector Norms
How can we compare the size of vectors, matrices (and functions!)?
For scalars it is easy (absolute value). The generalization of this
concept to vectors, matrices and functions is called a norm. Formally
the norm is a function from the space of vectors into the space of
scalars denoted by
(.)
with the following properties:
Definition: Norms.
1.
2.
3.
||v|| > 0 for any v0 and ||v|| = 0 implies
v=0
||av||=|a| ||v||
||u+v||≤||v||+||u|| (Triangle inequality)
We will only deal with the so-called lp Norm.
Mathematical foundations
Modern Seismology – Data processing and inversion
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The lp-Norm
The lp- Norm for a vector x is defined as (p≥1):
xl
p
1/ p

p
   xi 
 i 1

n
Examples:
- for p=2 we have the ordinary euclidian norm:
- for p= ∞ the definition is
- a norm for matrices is induced via
x
x l  max xi

1i  n
A  max
x0
- for l2 this means :
||A||2=maximum eigenvalue of ATA
Mathematical foundations
l2
 xT x
Modern Seismology – Data processing and inversion
Ax
x
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Matrix Algebra – Determinants
The determinant of a square matrix A is a scalar number
denoted det (A) or |A|, for example
a b 
det 
 ad  bc

c d 
or
 a11 a12 a13 
det a21 a22 a23 
a31 a32 a33 
 a11a22a33  a12a23a31  a13a21a32  a11a23a32  a12a21a33  a13a22a31
Mathematical foundations
Modern Seismology – Data processing and inversion
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Matrix Algebra – Inversion
A square matrix is singular if det A=0. This usually indicates
problems with the system (non-uniqueness, linear dependence,
degeneracy ..)
Matrix Inversion
For a square and non-singular
matrix A its inverse is defined
such as
AA 1  A-1A  I
The cofactor matrix C of
matrix A is given by
Cij  (1)i j Mij
where Mij is the determinant of
the matrix obtained by
eliminating the i-th row and the
j-th column of A.
The inverse of A is then given
by
1
A 
CT
det A
(AB)1  B-1A -1
Mathematical foundations
1
Modern Seismology – Data processing and inversion
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Matrix Algebra – Solution techniques
... the solution to a linear system of equations is the given by
x  A -1b
The main task in solving a linear system of equations is
finding the inverse of the coefficient matrix A.
Solution techniques are e.g.
Gauss elimination methods
Iterative methods
A square matrix is said to be positive definite if for any nonzero vector x
x T  Ax  0
... positive definite matrices are non-singular
Mathematical foundations
Modern Seismology – Data processing and inversion
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Matrices –Systems of equations
Seismological applications
• Stress and strain tensors
• Tomographic forward and
inverse problems
• Calculating interpolation or
differential operators for finitedifference methods
Mathematical foundations
Modern Seismology – Data processing and inversion
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