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Some basic maths for seismic data processing and inverse problems (Refreshement only!) • Complex Numbers • Vectors – Linear vector spaces • Matrices – – – – Determinants Eigenvalue problems Singular values Matrix inversion Mathematical foundations The idea is to illustrate these mathematical tools with examples from seismology Modern Seismology – Data processing and inversion 1 Complex numbers i z a ib re r (cos i sin ) Mathematical foundations Modern Seismology – Data processing and inversion 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 Modern Seismology – Data processing and inversion 3 Complex numbers seismological application Plane waves as superposition of harmonic signals using complex notation ui ( x j , t ) Ai exp[ ik (a j x j ct )] u(x, t ) A exp[ ikx t ] Use this „Ansatz“ in the acoustic wave equation and interpret the consequences for wave propagation! Mathematical foundations Modern Seismology – Data processing and inversion 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 Modern Seismology – Data processing and inversion 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 Modern Seismology – Data processing and inversion 6 Matrix Algebra – Linear Systems Ax b where a11 a12 a a22 21 A aij an1 an 2 b1 b 2 b bi bn Mathematical foundations a1n a11 ann x1 x x xi 2 xn A is a nxn (square) matrix, and x and b are column vectors of dimension n Modern Seismology – Data processing and inversion 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 d ij aij bij Matrix multiplication C AB with m 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 Modern Seismology – Data processing and inversion 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 Modern Seismology – Data processing and inversion 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 Q 2 ... examples: 1 1 1 1 it is easy to show that : Q T Q QQ T 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 Modern Seismology – Data processing and inversion 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 Modern Seismology – Data processing and inversion 11 The lp-Norm The lp- Norm for a vector x is defined as (p≥1): xl p 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 1i n A max x0 - for l2 this means : ||A||2=maximum eigenvalue of ATA Mathematical foundations l2 xT x Modern Seismology – Data processing and inversion 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 a11a22a33 a12a23a31 a13a21a32 a11a23a32 a12a21a33 a13a22a31 Mathematical foundations Modern Seismology – Data processing and inversion 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 Modern Seismology – Data processing and inversion 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 nonzero vector x x T Ax 0 ... positive definite matrices are non-singular Mathematical foundations Modern Seismology – Data processing and inversion 15 Matrices –Systems of equations Seismological applications • Stress and strain tensors • Tomographic forward and inverse problems • Calculating interpolation or differential operators for finitedifference methods Mathematical foundations Modern Seismology – Data processing and inversion 16