### Light Scattering Reviews 2: Remote Sensing and Inverse Problems

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In all cases, computing the gradient of the objective function often is a key element for the solution of the optimization problem. As mentioned above, information about the spatial distribution of a distributed parameter can be introduced through the parametrization. One can also think of adapting this parametrization during the optimization. Should the objective function be based on a norm other than the Euclidean norm, we have to leave the area of quadratic optimization.

As a result, the optimization problem becomes more difficult. Of course, use of regularization or other kind of prior information reduces the size of the set of almost optimal solutions and, in turn, increases the confidence we can put in the computed solution. We focus here on the recovery of a distributed parameter.

When looking for distributed parameters we have to discretize these unknown functions. Doing so, we reduce the dimension of the problem to something finite. Since a finite number of data does not allow the determination of an infinity of unknowns, the original data misfit functional has to be regularized to ensure uniqueness of the solution.

But many times, regularization has to be integrated explicitly in the objective function. In order to understand what may happen, we have to keep in mind that solving such a linear inverse problem amount to solving a Fredholm integral equation of the first kind:. This holds for a 2D application. Riesz theory states that the set of singular values of such an operator contains zero hence the existence of a null-space , is finite or at most countable, and, in the latter case, they constitute a sequence that goes to zero.

However we can define a solution through the pseudo-inverse of the forward map again up to an arbitrary additive function.

Yet, as in the finite dimension case, we have to question the confidence we can put in the computed solution. Again, basically, the information lies in the eigenvalues of the Hessian operator. The smallest eigenvalue is equal to the weight introduced in Tikhonov regularization.

In such cases, the Hessian is not a bounded operator and the notion of eigenvalue does not make sense any longer. A mathematical analysis is required to make it a bounded operator and design a well-posed problem: an illustration can be found in. Analysis of the spectrum of the Hessian operator is thus a key element to determine how reliable the computed solution is. However, such an analysis is usually a very heavy task.

This has led several authors to investigate alternative approaches in the case where we are not interested in all the components of the unknown function but only in sub-unknowns that are the images of the unknown function by a linear operator. These approaches are referred to as the " Backus and Gilbert method [13] ", Lions 's sentinels approach, [14] and the SOLA method: [15] these approaches turned out to be strongly related with one another as explained in Chavent [16] Finally, the concept of limited resolution , often invoked by physicists, is nothing but a specific view of the fact that some poorly determined components may corrupt the solution.

But, generally speaking, these poorly determined components of the model are not necessarily associated with high frequencies. In these methods we attempt at recovering a distributed parameter, the observation consisting in the measurement of the integrals of this parameter carried out along a family of lines. With such a kernel, the forward map is not compact. Although from a theoretical point of view many linear inverse problems are well understood, problems involving the Radon transform and its generalisations still present many theoretical challenges with questions of sufficiency of data still unresolved.

Such problems include incomplete data for the x-ray transform in three dimensions and problems involving the generalisation of the x-ray transform to tensor fields. Solutions explored include Algebraic Reconstruction Technique , filtered backprojection , and as computing power has increased, iterative reconstruction methods such as iterative Sparse Asymptotic Minimum Variance. In 3D the parameter is not integrated along lines but over surfaces. Should the propagation velocity be constant, such points are distributed on an ellipsoid. The inverse problems consists in retrieving the distribution of diffracting points from the seismograms recorded along the survey, the velocity distribution being known.

Should geometrical optics techniques i. If we consider a rotating stellar object, the spectral lines we can observe on a spectral profile will be shifted due to Doppler effect. Doppler tomography aims at converting the information contained in spectral monitoring of the object into a 2D image of the emission as a function of the radial velocity and of the phase in the periodic rotation movement of the stellar atmosphere.

Non-linear inverse problems constitute an inherently more difficult family of inverse problems. Whereas linear inverse problems were completely solved from the theoretical point of view at the end of the nineteenth century, only one class of nonlinear inverse problems was so before , that of inverse spectral and one space dimension inverse scattering problems , after the seminal work of the Russian mathematical school Krein , Gelfand , Levitan, Marchenko.

A large review of the results has been given by Chadan and Sabatier in their book "Inverse Problems of Quantum Scattering Theory" two editions in English, one in Russian. In this kind of problem, data are properties of the spectrum of a linear operator which describe the scattering.

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The spectrum is made of eigenvalues and eigenfunctions , forming together the "discrete spectrum", and generalizations, called the continuous spectrum. The very remarkable physical point is that scattering experiments give information only on the continuous spectrum, and that knowing its full spectrum is both necessary and sufficient in recovering the scattering operator.

Hence we have invisible parameters, much more interesting than the null space which has a similar property in linear inverse problems. In addition, there are physical motions in which the spectrum of such an operator is conserved as a consequence of such motion. This phenomenon is governed by special nonlinear partial differential evolution equations, for example the Korteweg—de Vries equation.

If the spectrum of the operator is reduced to one single eigenvalue, its corresponding motion is that of a single bump that propagates at constant velocity and without deformation, a solitary wave called a " soliton ". A perfect signal and its generalizations for the Korteweg—de Vries equation or other integrable nonlinear partial differential equations are of great interest, with many possible applications.

## Springer Series in Light Scattering

This area has been studied as a branch of mathematical physics since the s. Nonlinear inverse problems are also currently studied in many fields of applied science acoustics, mechanics, quantum mechanics, electromagnetic scattering - in particular radar soundings, seismic soundings, and nearly all imaging modalities. The goal is to recover the diffusion coefficient in the parabolic partial differential equation that models single phase fluid flows in porous media.

This problem has been the object of many studies since a pioneering work carried out in the early seventies. The goal is to recover the wave-speeds P and S waves and the density distributions from seismograms. Such inverse problems are of prime interest in seismology. These basic hyperbolic equations can be upgraded by incorporating attenuation , anisotropy , The solution of the inverse problem in the 1D wave equation has been the object of many studies.

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It is one of the very few non-linear inverse problems for which we can prove the uniqueness of the solution. Realizing how difficult is the inverse problem in the wave equation, seismologists investigated a simplified approach making use of geometrical optics. In particular they aimed at inverting for the propagation velocity distribution, knowing the arrival times of wave-fronts observed on seismograms. These wave-fronts can be associated with direct arrivals or with reflections associated with reflectors whose geometry is to be determined, jointly with the velocity distribution.

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It is classically solved by shooting rays trajectories about which the arrival time is stationary from the point source. But this tomography like problem is nonlinear, mainly because the unknown ray-path geometry depends upon the velocity or slowness distribution. In spite of its nonlinear character, travel-time tomography turned out to be very effective for determining the propagation velocity in the Earth or in the subsurface, the latter aspect being a key element for seismic imaging, in particular using methods mentioned in Section "Diffraction tomography".

We refer to Chavent [34] for a mathematical analysis of these points. The forward map being nonlinear, the data misfit function is likely to be non-convex, making local minimization techniques inefficient.

Inverse problems, especially in infinite dimension, may be large size, thus requiring important computing time. When the forward map is nonlinear, the computational difficulties increase and minimizing the objective function can be difficult. Much more effective is the evaluation of the gradient of the objective function for some models.

## Light scattering and its inverse problems -- Research Department of Physics Zhejiang University

It is now very widely used. Inverse problem theory is used extensively in weather predictions, oceanography, hydrology, and petroleum engineering. Inverse problems are also found in the field of heat transfer, where a surface heat flux [41] is estimated outgoing from temperature data measured inside a rigid body. The linear inverse problem is also the fundamental of spectral estimation and direction-of-arrival DOA estimation in signal processing.

Inverse, parameter and crack identification problems have been studied, by using optimization and soft computing tools. Many journals on medical imaging, geophysics, non-destructive testing, etc. From Wikipedia, the free encyclopedia. For the use in geodesy , see Inverse geodetic problem. Selected papers of Viktor A. Wave Motion. Inverse problem theory 1st ed. Paris: Masson.