96 Some basic maths for seismic data processing and inverse problems

Some basic maths for seismic data processing and inverse problems
(Refreshement only!)

 Complex Numbers
 Vectors
 Linear vector spaces
 Linear systems

 Matrices





Determinants
Eigenvalue problems
Singular values
Matrix inversion

Mathematical foundations

 Series
 Taylor
 Fourier

 Delta Function
 Fourier integrals
The idea is to illustrate
these mathematical tools
with examples from
seismology

Complex numbers

iφ

z = a + ib = re = r (cos φ + i sin φ )

Mathematical foundations


Complex numbers
conjugate, etc.

z* = a − ib = r (cos φ − i sin φ )
= r cos − φ − ri sin( −φ ) = r −iφ
z 2 = zz* = ( a + ib)(a − ib) = r 2
cos φ = (e iφ + e −iφ ) / 2
sin φ = (e iφ − e −iφ ) / 2i

Mathematical foundations


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 − ωt ]

Mathematical foundations


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 vecto
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


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


Matrix Algebra – Linear Systems

Ax = b

 a11
a
21

A = aij =


an1

[ ]

where

 a1n 

a22  a11 
  

an 2  ann 
a12

 b1 
b 
 2
b = { bi } =  
 
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


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


Matrix Algebra – Special
Transpose of a matrix

[ ]

A = aij

Symmetric matrix

[ ]

A T = 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

A = AT
aij = a ji


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

... examples:

1
Q=
2

1 − 1
1 1 



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

Mathematical foundations


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

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:x
- for p= ∞ the definition is
- a norm for matrices is induced via
- for l2 this means :
||A||2=maximum eigenvalue of ATA
Mathematical foundations

x

l∞

l2

= xT x

= max xi
1≤i ≤ n

A = max
x≠0

Ax
x


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 a23a31 + a13a21a32 − a11a23a32 − a12 a21a33 − a13a22 a31

Mathematical foundations


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 nonsingular 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


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


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 = λx
.. the rest on the board …
Mathematical foundations


Some operations on vector fields
Gradient of a vector field

 ∂x 
 ∂ x  ux   ∂ x ux
 
   
∇u =  ∂ y u =  ∂ y  u y  =  ∂ x u y
∂ 
 ∂  u   ∂ u
 z
 z  z   x z

∂ yux
∂ yu y
∂ yu z

What is the meaning of the gradient?

Mathematical foundations

∂ z ux 

∂ zuy 

∂ z uz 


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?

Mathematical foundations


Some operations on vector fields
Curl of a vector field

 ∂x 
 ∂ x   u x   ∂ yu z − ∂ z u y 


 
    
∇ × u =  ∂ y  × u =  ∂ y  ×  uy  =  ∂ zux − ∂ x uz 
∂ 
∂  u  ∂ u − ∂ u 
y x
 z
 z  z  x y
Can we observe it?

Mathematical foundations


Vector product

A = a × b = a b sin θ
Mathematical foundations


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

Mathematical foundations


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 ' ' ' dx 3 + ...
2
6
∞
f (i ) ( x) i
=∑
dx
i!
i =1

Mathematical foundations


… 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  2πx

2L 

L

1
a0 = ∫ f ( x)dx
L0
2
nπx
an = ∫ f ( x) sin
dx
L0
L
L

Mathematical foundations

n = 1, ∞


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

Mathematical foundations


The Delta function
… so weird but so useful …

∫

∞

∫

∞

−∞

−∞

δ (t ) f (t )dt = f (0)
δ (t )dt = 1

, δ (t ) = 0

f (t )δ (t − a ) = f (a )
1
δ (at ) = δ (t )
a

δ (t ) =

Mathematical foundations

1
2π

∞

i ωt
e
∫ dω

−∞

für t ≠ 0