Some basic maths for seismic data processing and inverse problems
(Refreshement only!)
 Complex Numbers
 Vectors
 Series
 Linear vector spaces
 Linear systems
 Matrices




Determinants
Eigenvalue problems
Singular values
Matrix inversion
Mathematical foundations
 Taylor
 Fourier
 Delta Function
 Fourier integrals
The idea is to illustrate
these mathematical tools
with examples from
seismology
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Complex numbers
i
z  a  ib  re  r (cos  i sin  )
Mathematical foundations
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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
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Complex numbers
seismological applications




Discretizing signals, description with eiwt
Poles and zeros for filter descriptions
Elastic plane waves
Analysis of numerical approximations
ui ( x j , t )  Ai exp[ik (a j x j  ct)]
u(x, t )  A exp[ikx t ]
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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
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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
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Matrix Algebra – Linear Systems
Ax  b
 a11
a
A  aij   21


an1
 
where
 a1n 
a22  a11 
  

an 2  ann 
a12
 b1 
b 
 
b  bi    2 

bn 
Mathematical foundations
 x1 
x 
 2
x  xi    

 xn 
A is a nxn (square) matrix, and
x and b are column vectors of
dimension n
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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
dij  aij  bij
Matrix multiplication
C  AB
m
with
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)
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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
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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 1  1
Q


2 1 1 
... examples:
it is easy to show that :
QT Q  QQT  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
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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.
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The lp-Norm
The lp- Norm for a vector x is defined as (p≥1):
x
lp
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
- for l2 this means :
||A||2=maximum eigenvalue of ATA
Mathematical foundations
l2
 xT x
Computational Geophysics and Data Analysis
x 0
Ax
x
12
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 
 a11a22 a33  a12 a23 a31  a13 a21a32  a11a23 a32  a12 a21a33  a13 a22 a31
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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
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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 non-zero
vector x
x T  Ax  0
... positive definite matrices are non-singular
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Eigenvalue problems
… one of the most important tools in stress, deformation and wave
problems!
It is a simple geometrical question: find me the directions in which a
square matrix does not change the orientation of a vector … and
find me the scaling …
Ax  x
.. the rest on the board …
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Some operations on vector fields
Gradient of a vector field
 x 
  x  ux    x ux
 
   
u    y u    y  uy     x uy
 
   u    u
 z
 z  z   x z
 yux
 yu y
 yuz
 z ux 

 zuy 

 z uz 
What is the meaning of the gradient?
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Some operations on vector fields
Divergence of a vector field
x 
  x   ux 
 
   
  u    y   u    y    u y    x u x   yu y   z u z
 
  u 
 z
 z  z
When u is the displacement what is ist divergence?
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Some operations on vector fields
Curl of a vector field
 x 
  x   ux    yuz   z u y 

 
    
  u    y   u    y    u y     z ux   x uz 
 
  u   u   u 
y x
 z
 z  z  x y
Can we observe it?
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Vector product
A  a  b  a b sin 
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Matrices –Systems of equations
Seismological applications
 Stress and strain tensors
 Calculating interpolation or differential
operators for finite-difference methods
 Eigenvectors and eigenvalues for
deformation and stress problems (e.g.
boreholes)
 Norm: how to compare data with theory
 Matrix inversion: solving for tomographic
images
 Measuring strain and rotations
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The power of series
Many (mildly or wildly nonlinear) physical systems
are transformed to linear systems by using Taylor
series
1
1
2
f ( x  dx)  f ( x)  f ' dx  f ' ' dx  f ' ' ' dx3  ...
2
6

f ( i ) ( x) i

dx
i!
i 1
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… and Fourier
Let alone the power of Fourier series assuming a
periodic function …. (here: symmetric, zero at
both ends)
f ( x)  a0  
n
n 

an sin 2x

2L 

n  1, 
L
1
a0   f ( x)dx
L0
2
nx
an   f ( x) sin
dx
L0
L
L
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Series –Taylor and Fourier
Seismological applications
 Well: any Fouriertransformation, filtering
 Approximating source input functions (e.g.,
step functions)
 Numerical operators (“Taylor operators”)
 Solutions to wave equations
 Linearization of strain - deformation
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The Delta function
… so weird but so useful …






 (t ) f (t )dt  f (0)
 (t )dt  1
,  (t )  0
für t  0
f (t ) (t  a )  f ( a )
1
 (at)   (t )
a
1
 (t ) 
2
Mathematical foundations

it
e
 d

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Delta function – generating series
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The delta function
Seismological applications
 As input to any system (the Earth, a
seismometers …)
 As description for seismic source signals in
time and space, e.g., with Mij the source
moment tensor
s(x, t )  M (t  t0 ) (x  x0 )
 As input to any linear system -> response
Function, Green’s function
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Fourier Integrals
The basis for the spectral analysis (described in the continuous
world) is the transform pair:
1
f (t ) 
2
F ( ) 



it
F
(

)
e
d


f (t )eit dt

For actual data analysis it is the discrete version that plays the most
important role.
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Complex fourier spectrum
The complex spectrum can be described as
F ( )  R( )  iI ( )
 A( )ei ( )
… here A is the amplitude spectrum and  is the phase spectrum
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The Fourier transform
Seismological applications
• Any filtering … low-, high-, bandpass
• Generation of random media
• Data analysis for periodic contributions
• Tidal forcing
• Earth’s rotation
• Electromagnetic noise
• Day-night variations
• Pseudospectral methods for function
approximation and derivatives
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