Inverse Scattering Problems Chaiwoot Boonyasiriwat October 14, 2020
Direct Scattering Problem Given the velocity model c ( x ), the source locations x s , and the source function w ( t ), find the wave field u ( x , t ) that satisfies the initial value problem Typically, the wavefield u ( x , t ) is recorded by receiver at x r and is denoted by u ( x r , t | x s ) where r = 1, 2, …, n r , s = 1, 2, …, n s , and t [0, T]. 2
Inverse Scattering Problem ▪ Suppose the source function w ( t ) is known and only one source located at x s is used. ▪ We want to estimate the velocity model c ( x ) from the wavefields u ( x r , t | x s ). Medium with Source position unknown Receiver position property (black box) 3
Optimization Problem ▪ To solve the inverse scattering problem, we turn it into an optimization problem in which the objective functional (a function of function) is minimized with respect to the velocity model c ( x ). ▪ To solve the optimization problem, we can use • Deterministic methods such as steepest descent, conjugate gradient, Newton, quasi-Newton methods • Stochastic methods (make use of random variables) such as Monte Carlo methods, simulated annealing, genetic algorithms, swarm algorithms 4
Stochastic Optimization Methods ▪ Gradient-based methods are typically faster than stochastic methods but only converge to the local minimum closest to the initial guess. ▪ On the other hand, stochastic optimization methods are slower but can theoretically reach the global minimum. Objective functional Local minimum Global minimum Local minimum 5
Monte Carlo Methods In a Monte Carlo method, models are randomly drawn from the model space and the best model is iteratively updated based on the tested models. After discretization, the velocity model becomes n -vector, i.e. where M is a Hilbert space equipped with an inner product and a norm. Model space M 6
Simulated Annealing ▪ “Annealing is a process where a solid is heated above its melting point, and slowly cooled until it solidifies into a perfect crystalline structure corresponding to the global minimum energy configuration.” ▪ Simulated annealing is a variant of the Metropolis method which is composed of 2 stochastic processes: (1) generation of solutions and (2) acceptance of solution. ▪ “SA is a descent algorithm modified by random ascent moves to escape local minima.” Du and Swamy (2013) 7
Simulated Annealing ▪ Given an initial solution m 0 and temperature T 0 , the solution is iteratively updated by where p i is a random move. ▪ If the move p i leads to a lower energy, the move is accepted. ▪ If the move p i leads to a higher energy, the move will be accepted with probability Metropolis Rule where E is the energy change. ▪ The probability of uphill move is large at high T and is low at low T . Temperature T is decreased with iteration. Chopard and Tomassini (2018) 8
Steepest Descent Method ▪ Deterministic methods are typically based on the gradient of the objective functional with respect to the model parameters – so called gradient-based methods. ▪ The simplest deterministic method is the steepest descent (SD) method which iteratively updates the model parameter by searching in the negative gradient direction. ▪ Given an initial velocity model c 0 ( x ), the SD method iteratively updates the velocity model by where subscript i is the iteration number, i is the step length, and g i is the gradient of E with respect to c . 9
Linear Least-Squares Problem ▪ Consider a linear system Am = d where the m n matrix A and the RHS vector d are known. ▪ An optimization problem corresponding to solving the linear system consists of the objective functional ▪ It can be shown that the gradient of E with respect to m is given by ▪ At a minimum, which leads to the normal equation whose solution is called the least-squares solution 10
Newton’s Method ▪ The least-squares solution is equivalent to the solution from the Newton’s method. ▪ Recall the Newton’s method for root finding problem . Let x i be the i th iterate of the estimated root. ▪ Using first- order Taylor’s expansion yields ▪ We want to find x such that f ( x i +1 ) = 0. Thus, ▪ Recall the gradient for the least-squares problem: ▪ At a minimum, g = 0 . Let x i be the i th iterate. We seek x such that g ( m i +1 ) = 0 . This is a root finding problem. 11
Newton’s Method ▪ Since , then we have ▪ Rearranging this yields ▪ Therefore, the formula for the least-squares solution is equivalent to the Newton’s method. ▪ For a large- scale problem, the Newton’s method requires a large amount of memory and computing time. ▪ A quasi-Newton method is often the method of choice. 12
Steepest Descent Method ▪ The steepest descent method can also be used to solve the linear least-squares problem. ▪ Starting from an initial guess m 0 , the solution is iteratively updated using the where ▪ For this linear problem, the step length can be computed using the formula which is obtained by minimizing the objective function 13
Nonlinear Least-Squares Problems ▪ The nonlinear least-squares problem is different from the linear problem in that the relationship between the data d and model m are nonlinear, i.e., d = A ( m ) where A is a nonlinear operator, not a matrix. ▪ In this case, the objective function becomes ▪ If we can compute the gradient g , we can also use the steepest descent method but the step length formula no longer works in this case. ▪ So, we have to perform a line search along the direction – g i to find a good step length. 14
Gradient Computation ▪ To use a gradient-based method, the gradient of the objective functional with respect to the model parameter is required. ▪ The most efficient method for computing the gradient is the adjoint-state method. ▪ In this case, we turn the inverse scattering into the constrained minimization problem with constraint where c M , u U , and M and U are Hilbert spaces. 15
Fréchet Derivative/Differential ▪ Let X and Y be Hilbert spaces and operator A : X → Y be differentiable at x X , i.e. where the linear operator F x : X → Y is called the Fréchet derivative of A at x and is written as F x = A '( x ). ▪ The expression F x ( x ) is called Fréchet differential of A ( x ) at x and is written as ▪ When or , F x is a linear functional so due to the Riesz representation theorem. Zhdanov (2015, p. 665-666) 16
Fréchet Derivative: Example 1 Let the functional where X is a Hilbert space: . Then, So, the Fréchet differential of is The kernel of the Fréchet derivative is 2 x which is also called the functional derivative of Zhdanov (2015, p. 666), Gelfand and Fomin (2000) 17
Fréchet Derivative: Example 2 Let where . We then have and the gradient is 18
Fréchet Derivative: Example 3 Let where . Fréchet kernel or functional derivative is then 19
Method of Lagrange Multiplier ▪ The method of Lagrange multiplier can be used to turn a constrained problem into an unconstrained problem. ▪ First, the Lagrangian is defined as where is the Lagrange multiplier and Here, is the spatial domain and d is the dimension. So, d {1, 2, 3}. 20
Method of Lagrange Multiplier Whenever c and u satisfy F ( c,u ) = 0, The gradient is where 21
Method of Lagrange Multiplier ▪ By setting , the gradient becomes since E does not explicitly depend on c . ▪ We will later find out the effect of setting ▪ First, let’s rewrite E ( c , u ) as 22
E / u 23
K / c 24
K / u Using integration by part twice and the initial and final conditions , it can be shown that 25
K / u Using the identities we obtain Suppose . We then obtain 26
K / u and K / As a result, Thus, Thus, 27
Adjoint-State Method ▪ In summary, the gradient is ▪ Setting leads to the adjoint equation ▪ Setting leads to the state equation 28
References ▪ K. L. Du and M. N. S. Swamy, 2016, Search and Optimization by Metaheuristics: Techniques and Algorithms Inspired by Nature, Birkhauser. ▪ I. M. Gelfand and S. V. Fomin, 2000, Calculus of Variations, Dover. ▪ M. S. Zhdanov, 2015, Inverse Theory and Applications in Geophysics, 2 nd ed., Elsevier.
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