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
Series
Taylor
Fourier
Delta Function
Fourier integrals
The idea is to illustrate these mathematical tools with examples from
seismology
Complex numbers
) sin
(cos
r i
re ib
a
z
i
Complex numbers
conjugate, etc.
i e
e
e e
r ib
a ib
a zz
z
r ri
r
i r
ib a
z
i i
i i
i
2 / ) (
sin
2 / ) (
cos
) )(
(
*
) sin(
cos
) sin
(cos
*
2 2
Complex numbers
seismological applications
Discretizing signals, description with e
iwt Poles and zeros for filter descriptions
Elastic plane waves
Analysis of numerical approximations
] exp[
) , (
)]
( exp[
) , (
t i
t
ct x
a ik A
t x
u
i j i j j
kx A
x
u
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 additionfrom 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. (x+y)+z = x+(y+z) 2. x+y = y+x
3. There is an element 0 in V such that x+0=x for all x V
4. For each x V there is an element -x V such that x+(-x)=0.
5. a(x+y)= a x+ a y 6. (a + b )x= a x+ bx 7. a(b x)= ab x
8. 1x=x
Matrix Algebra – Linear Systems
Linear system of algebraic equations
n n
nn n
n
n n
n n
b x
a x
a x
a
b x
a x
a x
a
b x
a x
a x
a
...
...
...
...
2 2 1
1
2 2
2 22 1
21
1 1
2 12 1
11
... where the x
1, x
2, ... , x
nare the unknowns ...
in matrix form
b
Ax
Matrix Algebra – Linear Systems
where
nn n
n
n
a a
a
a a
a
a a
a
2 1
11 22
21
1 12
11
a
ijA
b Ax
n i
x x x
x
2 1
x
n i
b b b
b
2 1
b
A is a nxn (square) matrix,
and x and b are column
vectors of dimension n
Matrix Algebra – Vectors
Row vectors Column vectors
v
1v
2v
3
v
3 2 1
w w w w
ij ij
ij
a b
c
A B with
C
Matrix addition and subtraction
ij ij
ij
a b
d
A B with
D
Matrix multiplication
mk
kj ik
ij
a b
c
1
AB with C
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)
Matrix Algebra – Special
Transpose of a matrix Symmetric matrix
T T T
T
A B AB
A A
) (
ji
ij
a
a
Identity matrix
ji
ij
a
a
A
TA
1 0
0
0 1
0
0 0
1
I
with AI=A, Ix=x
Matrix Algebra – Orthogonal
Orthogonal matrices
n
T
Q I
Q
1 1
1 1
2 Q 1
a matrix is Q (nxn) is said to be orthogonal if
... and each column is an
orthonormal vector q
iq
i 1
... examples:
it is easy to show that :
nT
T
Q QQ I
Q
if orthogonal matrices operate on vectors their size (the result of their inner product x.x) does not change -> Rotation
x x Qx
Qx )
T( )
T(
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. ||v|| > 0 for any v0 and ||v|| = 0 implies v=0
2. ||av||=|a| ||v||
3. ||u+v||≤||v||+||u|| (Triangle inequality)
We will only deal with the so-called l
pNorm.
The l
p-Norm
The l
p- Norm for a vector x is defined as (p≥1):
n p
i
p
l
x
ix
p/ 1
1
Examples:
- for p=2 we have the ordinary euclidian norm:
- for p= ∞ the definition is
- a norm for matrices is induced via - for l
2this means :
||A||
2=maximum eigenvalue of A
TA
x x x
l
T2
n i
l i
x
x
max
1x A Ax
x 0
max
Matrix Algebra – Determinants
The determinant of a square matrix A is a scalar number denoted det (A) or |A|, for example
bc d ad
c
b
a
det
or
31 22 13 33
21 12 32
23 11 32
21 13 31
23 12 33
22 11
33 32
31
23 22
21
13 12
11
det
a a a a
a a a
a a a
a a a
a a a
a a
a a
a
a a
a
a a
a
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
I A A
AA
1
-1
For a square and non-
singular matrix A its inverse is defined such as
The cofactor matrix C of
matrix A is given by
C
ij ( 1 )
ijM
ijwhere 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 - 1 - 1
1
A B (AB)
C A
T
A det
1
Matrix Algebra – Solution techniques
... the solution to a linear system of equations is the given by
b A x
-1The 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
... positive definite matrices are non-singular
0 Ax
x
T
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 …
.. the rest on the board …
x
Ax
Some operations on vector fields
Gradient of a vector field
What is the meaning of the gradient?
z z z
y z
x
y z y
y y
x
x z x
y x
x
z y x
z y x
z y x
u u
u u
u
u u
u
u u
u u
u
u
Some operations on vector fields
Divergence of a vector field
When u is the displacement what is ist divergence?
z z y
y x
x z
y x
z y x
z y x
u
u u u u
u u u
Some operations on vector fields
Curl of a vector field
Can we observe it?
x y y
x
z x x
z
y z z
y
z y x
z y x
z y x
u u
u u
u u
u u
u
u
u
Vector product
sin b
a b
a
A
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
The power of series
Many (mildly or wildly nonlinear) physical systems are transformed to linear systems by using Taylor series
1
) (
3 2
! ) (
...
'' 6 '
'' 1 2
' 1 )
( )
(
i
i i
i dx x f
dx f
dx f
dx f
x f dx
x
f
… and Fourier
Let alone the power of Fourier series assuming a periodic function …. (here: symmetric, zero at both ends)
sin 2 2 1 ,
)
(
0n
L x n a
a x
f
nn
L n
L
L dx x x n
L f a
dx x L f
a
0 0 0
sin ) 2 (
) 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
The Delta function
… so weird but so useful …
0 0
) ( , 1
) (
) 0 ( )
( ) (
t für
t t
d t
f dt
t f t
d e
t
a t at
a f a
t t
f
t i
2 ) 1
(
) 1 (
) (
) ( )
(
)
(
Delta function – generating series
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 M
ijthe source moment tensor
As input to any linear system -> response Function, Green’s function
) (
) (
) ,
( x t M t t
0 x x
0s
Fourier Integrals
The basis for the spectral analysis (described in the continuous world) is the transform pair:
dt e
t f F
d e
F t
f
t i
t i
) ( )
(
) 2 (
) 1 (
For actual data analysis it is the discrete version that plays the most
important role.
Complex fourier spectrum
The complex spectrum can be described as
)
)
((
) ( )
( )
(
e
iA
iI R
F
… here A is the amplitude spectrum and is the phase spectrum
The Fourier transform
Seismological applications
