This document provides an overview of the course "Seismic Data Processing" including the protocol, syllabus, references, and importance of seismic data processing. The syllabus covers topics like Fourier transforms, sampling considerations, various processing sequences, acquisition methods, and more. Seismic data processing is important because it helps remove unwanted signals and noise to enhance the signal-to-noise ratio and make seismic reflections clearer.

1. Seismic Data Processing
Code: ZGE 373/4
2013/2014
Dr. Amin E. Khalil

2. Protocol
The protocol of the lecture is as follow:
1- There nothing called stupid question so don’t hesitate to ask a
question because it is the shortest way to learn something.
2- There will be two periods during the lecture for questions and
discussion.

3. 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

4. Refernces
1- Yilmaz, O., 2001. Seismic Data Processing. Soceity of Exploration
Geophysicist (SEG)
2- Mayeda, W., 1993. Digital Signal Processing. Prentice-Hall.

5. 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

6. Example

7. Part I: Mathematical Basis for Fourier transform
•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

8. Lecture One
Complex Numbers
●Definition & operations
●Representation
●Operations
●Complex Conjugate
●Importance

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

10. Representation of complex numbers

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 =
a c + a d i + c b i - b d
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; such that:
Solve Z1/Z2 such that Z1= a + b i and Z2 = c + d i
To be solved on the whiteboard

14. Uses of complex numbers in Seismology
•Discretizing signals, description with eiwt
•Poles and zeros for filter descriptions
•Elastic plane waves
•Analysis of numerical approximations

15. Vectors
•Linear Vector Spaces.
•Linear Systems.

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:LinearVectorSpace.A linearvectorspaceoverafieldFof
scalarsisa setofelementsVtogetherwithafunctioncalledaddition
fromVxV intoVand afunctioncalledscalarmultiplicationfromFxVinto
Vsatisfyingthefollowingconditionsforallx,y,z ∈V andalla,b ∈ F
1.(x+y)+z = x+(y+z)
2.x+y =y+x
3.Thereisan element0inV suchthatx+0=xforallx ∈ V
4.Foreachx ∈ Vthereisan element-x∈ Vsuchthatx+(-x)=0.
5. a(x+y)=a x+a y
6.(a +b )x=a x+ bx
7.a(bx)= ab x

17. Comparing Vectors

18. Linear System of Algebraic Equations
... wherethex1,x2, ... , xnare theunknowns...
inmatrixform
Ax = b

19. System of Linear Algebraic Equations (continued)
where
A is anxn(square) matrix,andx
andbare columnvectors of
dimensionn

20. Matrix
A matrix is a collection of numbers arranged into a fixed number of rows and columns. Usually the
A matrix can be subdivided into column vectors or raw vectors

21. Rowvectors Columnvectors
Matrixadditionandsubtraction
Matrixmultiplication
whereA (size lxm) andB (sizemxn) andi=1,2,...,land j=1,2,...,n.
Note thatingeneral AB≠BA but(AB)C=A(BC)
Matrix Operations

22. Matrix Operations
Identitymatrix
withAI=A,Ix=x
Transpose Symmetric Matrix

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
Howcan wecomparethesizeofvectors,matrices(andfunctions!)?
For scalars it is easy (absolute value). The generalization of this concept to vectors, matrices and functions is called a norm. Formally the
normis afunctionfromthespaceofvectorsintothespaceof scalarsdenoted by
withthefollowingproperties:
Definition:Norms.
1.||v||>0foranyv∈0and||v||=0impliesv=0
2.||av||=|a|||v||
3.||u+v||≤||v||+||u|| (Triangleinequality)
Wewillonlydealwiththeso-called lpNorm.
∥ 𝐴 ∥

25. Thank you
End of Lecture