Nonlinear Least Squares for Inverse Problems Theoretical Foundations
This book provides an introduction into the least squares resolution of nonlinear inverse problems. The first goal is to develop a geometrical theory to analyze nonlinear least square (NLS) problems with respect to their quadratic wellposedness, i.e. both
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Least Squares for Inverse Problems: Theoretical Foundations and Step-by-Step Guide for Applications, Scientific Computation, c Springer Science+Business Media B.V. 2009 DOI 10.1007/978-90-481-2785-6 1,
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CHAPTER 1. NONLINEAR INVERSE PROBLEMS
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1.1
Example 1: Inversion of Knott–Zoeppritz Equations
We begin with an example that is intrinsically finite dimensional, where the model consists in a sequence of simple algebraic calculations. The problem occurs in the amplitude versus angle (AVA) processing of seismic data, where the densities ρj , compressional velocity VP,j , and shear velocities VS,j on each side j = 1, 2 of an interface are to be retrieved from the measurement of the compressional (P–P) reflection coefficient Ri at a collection θi , i = 1, . . . , q of given incidence angles. The P-P reflection coefficient R at incidence angle θ is given by the Knott– Zoeppritz equations ([1], pp. 148–151). They are made of quite complicated algebraic formulas involving many trigonometric functions. An in-depth analysis of the formula shows that R depends in fact only on the following four dimensionless combinations of the material parameters [50]: ⎧ ρ1 − ρ2 ⎪ ⎪ eρ = ⎪ ⎪ ρ1 + ρ2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 2 2 ⎪ VP,1 − VP,2 ⎪ ⎪ ⎪ eP = 2 ⎪ 2 ⎪ VP,1 + VP,2 ⎪ ⎪ ⎨ ⎪ ⎪ 2 2 ⎪ VS,1 − VS,2 ⎪ ⎪ eS = 2 ⎪ 2 ⎪ ⎪ VS,1 + VS,2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 2 2 ⎪ VS,1 + VS,2 ⎪ 1 1 ⎪ ⎪ ( 2 + 2 ) ⎩ χ= 2 VP,1 VP,2
(density contrast),
(P-velocity contrast), (1.1) (S-velocity contrast),
(background parameter),
1.1. EXAMPLE 1: INVERSION OF KNOTT–ZOEPPRITZ
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so that R is given by the relatively simple sequence of calculations: ⎧ ⎪ ⎪ e = eS + eρ ⎪ ⎪ f = 1 − e2ρ ⎪ ⎪ ⎪ ⎪ S = χ(1 + eP ) ⎪ ⎪ ⎪ 1 ⎪ S ⎪ 2 = χ(1 − eP ) ⎪ ⎪ ⎪ ⎪ T1 = 2/(1 − eS ) ⎪ ⎪ ⎪ ⎪ T2 = 2/(1 + eS ) ⎪ ⎪ 2 ⎪ ⎪ q 2 = S 1 sin θ ⎪ ⎪ ⎪ ⎪ M1 = S1 − q 2 ⎪ ⎪ ⎪ ⎪ ⎨ M2 = S2 − q 2 (1.2) N1 = T1 − q 2 ⎪ ⎪ 2 ⎪ N2 = T2 − q ⎪ ⎪ ⎪ ⎪ D = eq 2 ⎪ ⎪ ⎪ ⎪ A = eρ − D ⎪ ⎪ ⎪ ⎪ K =D−A ⎪ ⎪ ⎪ ⎪ B =1−K ⎪ ⎪ ⎪ ⎪ C =1+K ⎪ ⎪ ⎪ ⎪ P = M1 (B 2 N1 + f N2 ) + 4eDM1 M2 N1 N2 ⎪ ⎪ ⎪ ⎪ Q = M2 (C 2 N2 + f N1 ) + 4q 2 A2 ⎪ ⎪ ⎩ R = (P − Q)/(P + Q). We call parameter vector the vector x = (eρ , eP , eS , χ) ∈ IR4
(1.3)
of all quantities that are input to the calculation, and state vector the vector y = (e, f, S1 , S2 , . . . , P, Q, R) ∈ IR19
(1.4)
made of all quantities one has to compute to solve the state equations (here at a given incidence angle θ). We have supposed in the above formulas that the incidence angle θ is smaller than the critical angle, so that the reflection coefficient R computed by formula (1.2) is real, but the least squares formulation that follows can be extended without difficulty to postcritical incidence angles with complex reflection coefficients.
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CHAPTER 1. NONLINEAR INVERSE PROBLEMS
If now we are given a sequence Rim , i = 1, . . . , q of “measured” reflection coefficients corresponding to a sequence θ1 , . . . , θq of known (precritical) incidence angles, we can set up a data vector z = (R1m , . . . , Rqm ) ∈ IRq ,
(1.5)
which is to be compared to the output vector v = (R1 , . . . , Rq ) ∈ IRq of reflection
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