Generalized Nonlinear Inverse Problems Solved Using the Least Squares Criterion
TL;DR: In this article, a general definition of the nonlinear least squares inverse problem is given, where the form of the theoretical relationship between data and unknowns may be general (in particular, nonlinear integrodierentia l equations).
Abstract: We attempt to give a general definition of the nonlinear least squares inverse problem. First, we examine the discrete problem (finite number of data and unknowns), setting the problem in its fully nonlinear form. Second, we examine the general case where some data and/or unknowns may be functions of a continuous variable and where the form of the theoretical relationship between data and unknowns may be general (in particular, nonlinear integrodierentia l equations). As particular cases of our nonlinear algorithm we find linear solutions well known in geophysics, like Jackson’s (1979) solution for discrete problems or Backus and Gilbert’s (1970) a solution for continuous problems.
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TL;DR: In this paper, the nonlinear inverse problem for seismic reflection data is solved in the acoustic approximation, which is based on the generalized least squares criterion, and it can handle errors in the data set and a priori information on the model.
Abstract: The nonlinear inverse problem for seismic reflection data is solved in the acoustic approximation. The method is based on the generalized least‐squares criterion, and it can handle errors in the data set and a priori information on the model. Multiply reflected energy is naturally taken into account, as well as refracted energy or surface waves. The inverse problem can be solved using an iterative algorithm which gives, at each iteration, updated values of bulk modulus, density, and time source function. Each step of the iterative algorithm essentially consists of a forward propagation of the actual sources in the current model and a forward propagation (backward in time) of the data residuals. The correlation at each point of the space of the two fields thus obtained yields the corrections of the bulk modulus and density models. This shows, in particular, that the general solution of the inverse problem can be attained by methods strongly related to the methods of migration of unstacked data, and commerc...
3,198 citations
TL;DR: The impact of the changing surface ice load upon both Earth's shape and gravitational field, as well as upon sea-level history, have come to be measurable using a variety of geological and geophysical techniques.
Abstract: ▪ Abstract The 100 kyr quasiperiodic variation of continental ice cover, which has been a persistent feature of climate system evolution throughout the most recent 900 kyr of Earth history, has occurred as a consequence of changes in the seasonal insolation regime forced by the influence of gravitational n-body effects in the Solar System on the geometry of Earth's orbit around the Sun. The impacts of the changing surface ice load upon both Earth's shape and gravitational field, as well as upon sea-level history, have come to be measurable using a variety of geological and geophysical techniques. These observations are invertible to obtain useful information on both the internal viscoelastic structure of the solid Earth and on the detailed spatiotemporal characteristics of glaciation history. This review focuses upon the most recent advances that have been achieved in each of these areas, advances that have proven to be central to the construction of the refined model of the global process of glacial isos...
2,333 citations
01 Jan 2004
2,160 citations
Journal Article•
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TL;DR: The outputs of spectral unmixing, endmember, and abundance estimates are important for identifying the material composition of mixtures and the applicability of models and techniques is highly dependent on the variety of circumstances and factors that give rise to mixed pixels.
Abstract: Spectral unmixing using hyperspectral data represents a significant step in the evolution of remote decompositional analysis that began with multispectral sensing. It is a consequence of collecting data in greater and greater quantities and the desire to extract more detailed information about the material composition of surfaces. Linear mixing is the key assumption that has permitted well-known algorithms to be adapted to the unmixing problem. In fact, the resemblance of the linear mixing model to system models in other areas has permitted a significant legacy of algorithms from a wide range of applications to be adapted to unmixing. However, it is still unclear whether the assumption of linearity is sufficient to model the mixing process in every application of interest. It is clear, however, that the applicability of models and techniques is highly dependent on the variety of circumstances and factors that give rise to mixed pixels. The outputs of spectral unmixing, endmember, and abundance estimates are important for identifying the material composition of mixtures.
1,917 citations
TL;DR: In this paper, the frequency-domain inversion (FDI) method was proposed to solve the non-linear problem of extracting a smooth background velocity model from surface seismic-reuse data.
Abstract: SUMMARY By specifying a discrete matrix formulation for the frequency^space modelling problem for linear partial diierential equations (‘FDM’ methods), it is possible to derive a matrix formalism for standard iterative non-linear inverse methods, such as the gradient (steepest descent) method, the Gauss^Newton method and the full Newton method We obtain expressions for each of these methods directly from the discrete FDM method, and we refer to this approach as frequency-domain inversion (FDI)The FDI methods are based on simple notions of matrix algebra, but are nevertheless very general The FDI methods only require that the original partial diierential equations can be expressed as a discrete boundary-value problem (that is as a matrix problem) Simple algebraic manipulation of the FDI expressions allows us to compute the gradient of the mis¢t function using only three forward modelling steps (one to compute the residuals, one to backpropagate the residuals, and a ¢nal computation to compute a step length) This result is exactly analogous to earlier backpropagation methods derived using methods of functional analysis for continuous problems Following from the simplicity of this result, we give FDI expressions for the approximate Hessian matrix used in the Gauss^Newton method, and the full Hessian matrix used in the full Newton method In a new development, we show that the additional term in the exact Hessian, ignored in the Gauss^Newton method, can be e⁄ciently computed using a backpropagation approach similar to that used to compute the gradient vector The additional term in the Hessian predicts the degradation of linearized inversions due to the presence of ¢rst-order multiples (such as free-surface multiples in seismic data) Another interpretation is that this term predicts changes in the gradient vector due to second-order non-linear eiects In a numerical test, the Gauss^Newton and full Newton methods prove eiective in helping to solve the di⁄cult non-linear problem of extracting a smooth background velocity model from surface seismic-re£ection data
1,432 citations
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Book•
01 Jan 1965TL;DR: Algebra of Vectors and Matrices, Probability Theory, Tools and Techniques, and Continuous Probability Models.
Abstract: Algebra of Vectors and Matrices. Probability Theory, Tools and Techniques. Continuous Probability Models. The Theory of Least Squares and Analysis of Variance. Criteria and Methods of Estimation. Large Sample Theory and Methods. Theory of Statistical Inference. Multivariate Analysis. Publications of the Author. Author Index. Subject Index.
8,300 citations
4,122 citations
TL;DR: In this article, it was shown that a given set G of measured gross Earth data permits such a construction of localized averages, and if so, how to find the shortest length scale over which G gives a local average structure at a particular depth if the variance of the error in computing that local average from G is to be less than a specified amount.
Abstract: A gross Earth datum is a single measurable number describing some property of the whole Earth, such as mass, moment of inertia, or the frequency of oscillation of some identified elastic-gravitational normal mode. We suppose that a finite set G of gross Earth data has been measured, that the measurements are inaccurate, and that the variance matrix of the errors of measurement can be estimated. We show that some such sets G of measurements determine the structure of the Earth within certain limits of error except for fine-scale detail. That is, from some setsG it is possible to compute localized averages of the Earth structure at various depths. These localized averages will be slightly in error, and their errors will be larger as their resolving lengths are shortened. We show how to determine whether a given set G of measured gross Earth data permits such a construction of localized averages, and, if so, how to find the shortest length scale over which G gives a local average structure at a particular depth if the variance of the error in computing that local average from G is to be less than a specified amount. We apply the general theory to the linear problem of finding the depth variation of a frequency-independent local elastic dissipation ( Q ) from the observed damping rates of a finite number of normal modes. We also apply the theory to the nonlinear problem of finding density against depth from the total mass, moment and normal-mode frequencies, in case the compressional and shear velocities are known.
1,291 citations
TL;DR: In this paper, the discrete general linear inverse problem is reduced to a set of m equations in n unknowns and a linear combination of the eigenvectors of the coefficient matrix can be used to determine parameter resolution and information distribution among the observations.
Abstract: The discrete general linear inverse problem reduces to a set of m equations in n unknowns. There is generally no unique solution, but we can find k linear combinations of parameters for which restraints are determined. The parameter combinations are given by the eigenvectors of the coefficient matrix. The number k is determined by the ratio of the standard deviations of the observations to the allowable standard deviations in the resulting solution. Various linear combinations of the eigenvectors can be used to determine parameter resolution and information distribution among the observations. Thus we can determine where information comes from among the observations and exactly how it constraints the set of possible models. The application of such analyses to surface-wave and free-oscillation observations indicates that (1) phase, group, and amplitude observations for any particular mode provide basically the same type of information about the model; (2) observations of overtones can enhance the resolution considerably; and (3) the degree of resolution has generally been overestimated for many model determinations made from surface waves.
738 citations
547 citations