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Seismic Data ProcessingCode: ZGE 373/4
2013/2014
Dr. Amin E. Khalil
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.
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
Refernces
1- Yilmaz, O., 2001. Seismic Data Processing. Soceity of Exploration Geophysicist (SEG)
2- Mayeda, W., 1993. Digital Signal Processing. Prentice-Hall.
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
Example
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
Lecture One
Complex Numbers●Definition & operations
●Representation●Operations
●Complex Conjugate●Importance
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
Representation of complex numbers
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 formZ1 + 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 dsimplified as:
(a c - b d) + (a d + c b) i
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 isZ* = a - b i. We use * supersccipt to denote complex conjugate. The multiplication Z*Z= a2 - b2
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
Uses of complex numbers in Seismology
•Discretizing signals, description with eiwt
•Poles and zeros for filter descriptions•Elastic plane waves•Analysis of numerical approximations
Vectors
•Linear Vector Spaces.
•Linear Systems.
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 ofscalars is a set of elements V together with a function called addition from VxV into V and a function called scalar multiplication from FxV intoV 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+x3.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 y6.(a + b )x= a x+ bx7.a(b x)= ab x8.1x=x
Comparing Vectors
Linear System of Algebraic Equations
... where the x1, x2, ... , xn are the unknowns ...in matrix form
Ax = b
System of Linear Algebraic Equations (continued)
where
A is a nxn (square) matrix, and x and b are
column vectors of dimension n
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
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)
Matrix Operations
Matrix Operations
Identity matrix
with AI=A, Ix=x
Transpose Symmetric Matrix
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
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
with the following properties:Definition: Norms.1.||v|| > 0 for any v 0 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 lp Norm.
∥ 𝐴∥
Thank you
End of Lecture
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