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Some basic maths for seismic data processing and inverse problems (Refreshement only!) Complex Numbers Vectors Series Linear vector spaces Linear systems Matrices Determinants Eigenvalue problems Singular values Matrix inversion Mathematical foundations Taylor Fourier Delta Function Fourier integrals The idea is to illustrate these mathematical tools with examples from seismology Computational Geophysics and Data Analysis 1 Complex numbers i z a ib re r (cos i sin ) Mathematical foundations Computational Geophysics and Data Analysis 2 Complex numbers conjugate, etc. z* a ib r (cos i sin ) r cos ri sin( ) r i z 2 zz* ( a ib)(a ib) r 2 cos (e i e i ) / 2 sin (e i e i ) / 2i Mathematical foundations Computational Geophysics and Data Analysis 3 Complex numbers seismological applications Discretizing signals, description with eiwt Poles and zeros for filter descriptions Elastic plane waves Analysis of numerical approximations ui ( x j , t ) Ai exp[ik (a j x j ct)] u(x, t ) A exp[ikx t ] Mathematical foundations Computational Geophysics and Data Analysis 4 Vectors and Matrices 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 Mathematical foundations Computational Geophysics and Data Analysis 5 Matrix Algebra – Linear Systems Linear system of algebraic equations a11 x1 a12 x2 ... a1n xn b1 a21 x1 a22 x2 ... a2 n xn b2 .......... an1 x1 an 2 x2 ... ann xn bn ... where the x1, x2, ... , xn are the unknowns ... in matrix form Ax b Mathematical foundations Computational Geophysics and Data Analysis 6 Matrix Algebra – Linear Systems Ax b a11 a A aij 21 an1 where a1n a22 a11 an 2 ann a12 b1 b b bi 2 bn Mathematical foundations x1 x 2 x xi xn A is a nxn (square) matrix, and x and b are column vectors of dimension n Computational Geophysics and Data Analysis 7 Matrix Algebra – Vectors Row vectors v v1 v2 Column vectors w 1 w w2 w 3 v3 Matrix addition and subtraction C A B D A B with with cij aij bij dij aij bij Matrix multiplication C AB m with cij aik bkj k 1 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) Mathematical foundations Computational Geophysics and Data Analysis 8 Matrix Algebra – Special Transpose of a matrix A aij Symmetric matrix A T a ji A AT aij a ji ( AB) T BT A T Identity matrix 1 0 0 0 1 0 I 0 0 1 with AI=A, Ix=x Mathematical foundations Computational Geophysics and Data Analysis 9 Matrix Algebra – Orthogonal Orthogonal matrices a matrix is Q (nxn) is said to be orthogonal if QT Q I n ... and each column is an orthonormal vector qi qi 1 1 1 1 Q 2 1 1 ... examples: it is easy to show that : QT Q QQT I n if orthogonal matrices operate on vectors their size (the result of their inner product x.x) does not change -> Rotation (Qx)T (Qx) xT x Mathematical foundations Computational Geophysics and Data Analysis 10 Matrix and Vector Norms 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. 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. Mathematical foundations Computational Geophysics and Data Analysis 11 The lp-Norm The lp- Norm for a vector x is defined as (p≥1): x lp 1/ p p xi i 1 n Examples: - for p=2 we have the ordinary euclidian norm: - for p= ∞ the definition is - a norm for matrices is induced via x x l max xi 1 i n A max - for l2 this means : ||A||2=maximum eigenvalue of ATA Mathematical foundations l2 xT x Computational Geophysics and Data Analysis x 0 Ax x 12 Matrix Algebra – Determinants The determinant of a square matrix A is a scalar number denoted det (A) or |A|, for example a b det ad bc c d or a11 a12 a13 det a21 a22 a23 a31 a32 a33 a11a22 a33 a12 a23 a31 a13 a21a32 a11a23 a32 a12 a21a33 a13 a22 a31 Mathematical foundations Computational Geophysics and Data Analysis 13 Matrix Algebra – Inversion A square matrix is singular if det A=0. This usually indicates problems with the system (non-uniqueness, linear dependence, degeneracy ..) Matrix Inversion For a square and non-singular matrix A its inverse is defined such as AA 1 A-1A I The cofactor matrix C of matrix A is given by Cij (1)i j Mij where Mij is the determinant of the matrix obtained by eliminating the i-th row and the j-th column of A. The inverse of A is then given by 1 A CT det A (AB)1 B-1A -1 Mathematical foundations 1 Computational Geophysics and Data Analysis 14 Matrix Algebra – Solution techniques ... the solution to a linear system of equations is the given by x A -1b The main task in solving a linear system of equations is finding the inverse of the coefficient matrix A. Solution techniques are e.g. Gauss elimination methods Iterative methods A square matrix is said to be positive definite if for any non-zero vector x x T Ax 0 ... positive definite matrices are non-singular Mathematical foundations Computational Geophysics and Data Analysis 15 Eigenvalue problems … one of the most important tools in stress, deformation and wave problems! It is a simple geometrical question: find me the directions in which a square matrix does not change the orientation of a vector … and find me the scaling … Ax x .. the rest on the board … Mathematical foundations Computational Geophysics and Data Analysis 16 Some operations on vector fields Gradient of a vector field x x ux x ux u y u y uy x uy u u z z z x z yux yu y yuz z ux zuy z uz What is the meaning of the gradient? Mathematical foundations Computational Geophysics and Data Analysis 17 Some operations on vector fields Divergence of a vector field x x ux u y u y u y x u x yu y z u z u z z z When u is the displacement what is ist divergence? Mathematical foundations Computational Geophysics and Data Analysis 18 Some operations on vector fields Curl of a vector field x x ux yuz z u y u y u y u y z ux x uz u u u y x z z z x y Can we observe it? Mathematical foundations Computational Geophysics and Data Analysis 19 Vector product A a b a b sin Mathematical foundations Computational Geophysics and Data Analysis 20 Matrices –Systems of equations Seismological applications Stress and strain tensors Calculating interpolation or differential operators for finite-difference methods Eigenvectors and eigenvalues for deformation and stress problems (e.g. boreholes) Norm: how to compare data with theory Matrix inversion: solving for tomographic images Measuring strain and rotations Mathematical foundations Computational Geophysics and Data Analysis 21 The power of series Many (mildly or wildly nonlinear) physical systems are transformed to linear systems by using Taylor series 1 1 2 f ( x dx) f ( x) f ' dx f ' ' dx f ' ' ' dx3 ... 2 6 f ( i ) ( x) i dx i! i 1 Mathematical foundations Computational Geophysics and Data Analysis 22 … and Fourier Let alone the power of Fourier series assuming a periodic function …. (here: symmetric, zero at both ends) f ( x) a0 n n an sin 2x 2L n 1, L 1 a0 f ( x)dx L0 2 nx an f ( x) sin dx L0 L L Mathematical foundations Computational Geophysics and Data Analysis 23 Series –Taylor and Fourier Seismological applications Well: any Fouriertransformation, filtering Approximating source input functions (e.g., step functions) Numerical operators (“Taylor operators”) Solutions to wave equations Linearization of strain - deformation Mathematical foundations Computational Geophysics and Data Analysis 24 The Delta function … so weird but so useful … (t ) f (t )dt f (0) (t )dt 1 , (t ) 0 für t 0 f (t ) (t a ) f ( a ) 1 (at) (t ) a 1 (t ) 2 Mathematical foundations it e d Computational Geophysics and Data Analysis 25 Delta function – generating series Mathematical foundations Computational Geophysics and Data Analysis 26 The delta function Seismological applications As input to any system (the Earth, a seismometers …) As description for seismic source signals in time and space, e.g., with Mij the source moment tensor s(x, t ) M (t t0 ) (x x0 ) As input to any linear system -> response Function, Green’s function Mathematical foundations Computational Geophysics and Data Analysis 27 Fourier Integrals The basis for the spectral analysis (described in the continuous world) is the transform pair: 1 f (t ) 2 F ( ) it F ( ) e d f (t )eit dt For actual data analysis it is the discrete version that plays the most important role. Mathematical foundations Computational Geophysics and Data Analysis 28 Complex fourier spectrum The complex spectrum can be described as F ( ) R( ) iI ( ) A( )ei ( ) … here A is the amplitude spectrum and is the phase spectrum Mathematical foundations Computational Geophysics and Data Analysis 29 The Fourier transform Seismological applications • Any filtering … low-, high-, bandpass • Generation of random media • Data analysis for periodic contributions • Tidal forcing • Earth’s rotation • Electromagnetic noise • Day-night variations • Pseudospectral methods for function approximation and derivatives Mathematical foundations Computational Geophysics and Data Analysis 30