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