Inverse Scattering Problems Chaiwoot Boonyasiriwat October 14, - - PowerPoint PPT Presentation

Inverse Scattering Problems

Chaiwoot Boonyasiriwat

October 14, 2020

Given the velocity model c(x), the source locations xs, 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 xr and is denoted by u(xr,t | xs) where r = 1, 2, …, nr, s = 1, 2, …, ns, and t  [0, T].

2

Direct Scattering Problem

3

▪ Suppose the source function w(t) is known and only one source located at xs is used. ▪ We want to estimate the velocity model c(x) from the wavefields u(xr,t | xs).

Inverse Scattering Problem

Source position Receiver position Medium with unknown property (black box)

4

▪ 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

Optimization Problem

5

▪ 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.

Stochastic Optimization Methods

Local minimum Global minimum Local minimum Objective functional

6

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.

Monte Carlo Methods

Model space M

After discretization, the velocity model becomes n-vector, i.e. where M is a Hilbert space equipped with an inner product and a norm.

7

▪ “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.”

Simulated Annealing

Du and Swamy (2013)

8

▪ Given an initial solution m0 and temperature T0, the solution is iteratively updated by where pi is a random move. ▪ If the move pi leads to a lower energy, the move is accepted. ▪ If the move pi leads to a higher energy, the move will be accepted with probability 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.

Simulated Annealing

Chopard and Tomassini (2018)

Metropolis Rule

9

▪ 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 c0(x), the SD method iteratively updates the velocity model by where subscript i is the iteration number, i is the step length, and gi is the gradient of E with respect to c.

Steepest Descent Method

10

▪ 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

Linear Least-Squares Problem

11

▪ The least-squares solution is equivalent to the solution from the Newton’s method. ▪ Recall the Newton’s method for root finding problem . Let xi be the ith iterate of the estimated root. ▪ Using first-order Taylor’s expansion yields ▪ We want to find x such that f (xi+1) = 0. Thus, ▪ Recall the gradient for the least-squares problem: ▪ At a minimum, g = 0. Let xi be the ith iterate. We seek x such that g(mi+1) = 0. This is a root finding problem.

Newton’s Method

12

▪ 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.

Newton’s Method

13

▪ The steepest descent method can also be used to solve the linear least-squares problem. ▪ Starting from an initial guess m0, 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

Steepest Descent Method

14

▪ 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 –gi to find a good step length.

Nonlinear Least-Squares Problems

15

▪ To use a gradient-based method, the gradient of the

• bjective 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.

Gradient Computation

16

▪ Let X and Y be Hilbert spaces and operator A: X → Y be differentiable at x  X, i.e. where the linear operator Fx: X → Y is called the Fréchet derivative of A at x and is written as Fx = A'(x). ▪ The expression Fx( x) is called Fréchet differential of A(x) at x and is written as ▪ When or , Fx is a linear functional so due to the Riesz representation theorem.

Fréchet Derivative/Differential

Zhdanov (2015, p. 665-666)

17

Let the functional where X is a Hilbert space: . Then, So, the Fréchet differential of is The kernel of the Fréchet derivative is 2x which is also called the functional derivative of

Fréchet Derivative: Example 1

Zhdanov (2015, p. 666), Gelfand and Fomin (2000)

18

Let where . We then have and the gradient is

Fréchet Derivative: Example 2

19

Let where . Fréchet kernel or functional derivative is then

Fréchet Derivative: Example 3

20

▪ 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}.

Method of Lagrange Multiplier

21

Whenever c and u satisfy F(c,u) = 0, The gradient is where

Method of Lagrange Multiplier

22

▪ 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

Method of Lagrange Multiplier

23

E/u

24

K/c

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

As a result, Thus, Thus,

27

K/u and K/

28

▪ In summary, the gradient is ▪ Setting leads to the adjoint equation ▪ Setting leads to the state equation

Adjoint-State Method

▪ 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, 2nd ed., Elsevier.

References