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
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 application
Plane waves as superposition of harmonic signals using complex notation
Use this „Ansatz“ in the acoustic wave equation and interpret the consequences for wave propagation!
] 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 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. (x+y)= x+ y 6. ( + )x= x+ x 7. ( x)= 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 ( Qx )
T( Qx ) x
Tx
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 2. ||v=0 v||=|| ||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 :
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 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
