2. Syllabus
1- Mathematical Basis for Fourier transform
2- Sampling considerations of seismic time series
3- Main processing sequence
4- Velocity analysis
5- Deconvolution, convolution, filtering and migration in space and time (prestack).
6- Acquisition of seismic data ( land and sea).
7- 3-D seismic data processing
8- Radon transform, tau-p processing, Hilbert transform and AVO
3. Refernces
1- Yilmaz, O., 2001. Seismic Data Processing. Soceity of Exploration
Geophysicist (SEG)
2- Mayeda, W., 1993. Digital Signal Processing. Prentice-Hall.
4. Why Seismic Data Processing is important?
Because reflection seismic energy arrive later, it might be obscured by
another seismic signals like ground roll and direct waves. Hence we apply
Seismic data Processing
1- To remove unwanted Signals and Noises
2- To Enhance Signal to Noise ratio
7. Mathematical Basis for Fourier transform
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Complex Numbers
Vectors
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Matrices
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Linear vector spaces
Linear systems
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Determinants
Eigenvalue problems
Singular values
Matrix inversion
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Series
Taylor
Fourier
Delta Function
Fourier integrals
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The idea is to illustrate these
mathematical tools with
examples from seismology
9. Complex numbers; Definition & operations
Definition:
A combination of a real and an imaginary number in the form a + bi,
where a and b are real, and i is the "unit imaginary number" √(-1),
The values a and b can be zero.
Examples:
1 + i, 2 - 6i, -5.2i, 4
imaginary number is that real number that give negative number
when it’s squared
11. Complex numbers: Basic Operations
Complex numbers are "binomials" of a sort, and are added, subtracted, and multiplied in a similar
way.
◄ Equality:
Two complex numbers
are equal if and only if their
real parts are equals and
their imaginary parts are
Equal.
Ex: 3 – 4i = x + yi
yields that x=3 and y=-4
◄ Addition and subtraction
Addition and subtraction is
done such that real parts are
added (subtracted) together
and same for imaginary parts.
Ex: two complex numbers
Z1=a + bi and Z2=c+di are
added in the form
Z1 + Z2= (a+c) + (b+d)i
◄ Multiplication is done
similar to binomial
multiplication.
Ex: Z1 * Z2 =
ac+adi+cbi-bd
simplified as:
(a c - b d) + (a d + c b) i
12. Complex Numbers: complex Conjugate
A complex conjugate is that number which when multiplied
with original one the result is real number. In this case the
real and imaginary parts for both numbers but the sign of
the imaginary part is reversed in such a wa that if the
complex number Z = a + b i then its complex conjugate is
Z* = a - b i. We use * supersccipt to denote complex
conjugate. The multiplication Z*Z= a2 - b2
13. Complex Numbers: Subdivision
Subdividing two complex numbers Z1 and Z2 is done using the
complex conjugate property, that results in the vanishing of the
imaginary part from the denumerator. Then division is carried out.
Ex.:
Solve
Sol.: First multiply by the
conjugate of the denumerator
then
14. Uses of complex numbers in Seismology
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Discretizing signals, description with exp(iwt)
Poles and zeros for filter descriptions
Elastic plane waves
Analysis of numerical approximations
16. Linear Vector Spaces
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
18. Linear System of Algebraic Equations
... where the x1, x2, ... , xn are the unknowns ...
in matrix form
Ax = b
19. System of Linear Algebraic Equations (continued)
where
A is a nxn (square)
matrix, and x and b are
column vectors of
dimension n
20. Matrix
A matrix is a collection of numbers arranged into a fixed number of rows and columns.
Usually the numbers are real numbers. In general, matrices can contain complex
numbers but we won't see those here. Here is an example of a matrix with three rows
and three columns:
A matrix can be subdivided into column vectors or raw vectors
21. Matrix Operations
Row vectors
Column vectors
Matrix addition and subtraction
Matrix multiplication
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)
23. Orthogonal Matrix
It is such that when multiplied with its transpose the result is the identity
matrix I.e:
AAT=I
Where AT is the transpose of matrix A.
In particular, an orthogonal matrix is always invertible, and
A-1=AT
24. Matrix Norm
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
∥A∥
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.
