Seismic data processing introductory lecture

Embed Size (px)

Citation preview

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

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. Refernces1- 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 5. Illustration of The Problem 6. Part One 7. Mathematical Basis for Fourier transform Complex Numbers VectorsMatrices Linear vector spaces Linear systems 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 OneComplex Numbers Representation Definition OperationsComplex 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 its 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 = 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 denumeratorthen 14. Uses of complex numbers in Seismology Discretizing signals, description with exp(iwt) Poles and zeros for filter descriptions Elastic plane waves Analysis of numerical approximations 15. VectorsLinear 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: 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 17. Comparing Vectors 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)whereA 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 vectorsColumn vectorsMatrix 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 ABBA but (AB)C=A(BC) 22. Matrix Operations TransposeSymmetric MatrixIdentity matrixwith AI=A, Ix=x 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 byA with the following properties:Definition: Norms. 1. 2. 3.||v|| > 0 for any v0 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. 25. Thank youEnd of Lecture