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...

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Inversion of seismic reflection data in the acoustic approximation

▪ 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...

GLOBAL GLACIAL ISOSTASY AND THE SURFACE OF THE ICE-AGE EARTH: The ICE-5G (VM2) Model and GRACE

"Parameter Estimation and Inverse Problems, 2/e" provides geoscience students and professionals with answers to common questions like how one can derive a physical model from a finite set of observations containing errors, and how one may determine the quality of such a model. This book takes on these fundamental and challenging problems, introducing students and professionals to the broad range of approaches that lie in the realm of inverse theory. The authors present both the underlying theory and practical algorithms for solving inverse problems. The authors' treatment is appropriate for geoscience graduate students and advanced undergraduates with a basic working knowledge of calculus, linear algebra, and statistics. "Parameter Estimation and Inverse Problems, 2/e" introduces readers to both Classical and Bayesian approaches to linear and nonlinear problems with particular attention paid to computational, mathematical, and statistical issues related to their application to geophysical problems. The textbook includes Appendices covering essential linear algebra, statistics, and notation in the context of the subject. This book includes a companion website that features computational examples (including all examples contained in the textbook) and useful subroutines using MATLAB. It: includes appendices for review of needed concepts in linear, statistics, and vector calculus; features a companion website that contains comprehensive MATLAB code for all examples, which readers can reproduce, experiment with, and modify; offers an online instructor's guide that helps professors teach, customize exercises, and select homework problems; and, is accessible to students and professionals without a highly specialized mathematical background.

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Parameter estimation and inverse problems

GLOBAL GLACIAL ISOSTASY AND THE SURFACE OF THE ICE-AGE EARTH: The ICE-5G (VM2)

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.

Spectral unmixing

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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Generalized Nonlinear Inverse Problems Solved Using the Least Squares Criterion

We examine the general non-linear inverse problem with a nite number of parameters In order to permit the incorporation of any a priori information about parameters and any distribution of data (not only of gaussian type) we propose to formulate the problem not using single quantities (such as bounds, means, etc) but using probability density functions for data and parameters We also want our formulation to allow for the incorporation of theoretical errors, ie non-exact theoretical relationships between data and parameters (due to discretization, or incomplete theoretical knowledge); to do that in a natural way we propose to dene general theoretical relationships also as probability density functions We show then that the inverse problem may be formulated as a problem of combination of information: the experimental information about data, the a priori information about parameters, and the theoretical information With this approach, the general solution of the non-linear inverse problem is unique and consistent (solving the same problem, with the same data, but with a dierent system

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Inverse problems = Quest for information

Summary. We attempt a general definition of the inverse problem of computing the components of the regional stress tensor from a set of field data including the measurements of the strike and dip of several faults, and the directions and senses of relative motion along these faults (as indicated by slickenslides). In previous treatments of this problem, no experimental errors could be taken into account except those in measuring the pitch of slickenslides; thus, errors in the orientation of the fault (strike and dip), which have considerable practical importance, were neglected. In our work, all experimental errors in the field measurements are taken into account, so that the agreement between the computed stress tensor and the set of field measurements can be rigorously checked.

Inversion of field data in fault tectonics to obtain the regional stress — I. Single phase fault populations: a new method of computing the stress tensor

Bernard Valette

Papers

Generalized Nonlinear Inverse Problems Solved Using the Least Squares Criterion

Inverse problems = Quest for information

Inversion of field data in fault tectonics to obtain the regional stress — I. Single phase fault populations: a new method of computing the stress tensor