"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

http://quebec.hwr.arizona.edu/classes/hwr696t/gosse01-cosmogenic-nuclide-review.pdf

Terrestrial in situ cosmogenic nuclides: theory and application

http://hess.ess.washington.edu/math/docs/al_be_v2/al_be_calc_2007.pdf

A complete and easily accessible means of calculating surface exposure ages or erosion rates from 10Be and 26Al measurements

Magnetotelluric (MT) data are inverted for smooth 2-D models using an extension of the existing 1-D algorithm, Occam’s inversion. Since an MT data set consists of a finite number of imprecise data, an infinity of solutions to the inverse problem exists. Fitting field or synthetic electromagnetic data as closely as possible results in theoretical models with a maximum amount of roughness, or structure. However, by relaxing the misfit criterion only a small amount, models which are maximally smooth may be generated. Smooth models are less likely to result in overinterpretation of the data and reflect the true resolving power of the MT method. The models are composed of a large number of rectangular prisms, each having a constant conductivity. Apriori information, in the form of boundary locations only or both boundary locations and conductivity, may be included, providing a powerful tool for improving the resolving power of the data. Joint inversion of TE and TM synthetic data generated from known models al...

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Occam's inversion to generate smooth, two-dimensional models from magnetotelluric data

We investigate a new algorithm for computing regularized solutions of the 2-D magnetotelluric inverse problem. The algorithm employs a nonlinear conjugate gradients (NLCG) scheme to minimize an objective function that penalizes data residuals and second spatial derivatives of resistivity. We compare this algorithm theoretically and numerically to two previous algorithms for constructing such “minimum‐structure” models: the Gauss‐Newton method, which solves a sequence of linearized inverse problems and has been the standard approach to nonlinear inversion in geophysics, and an algorithm due to Mackie and Madden, which solves a sequence of linearized inverse problems incompletely using a (linear) conjugate gradients technique. Numerical experiments involving synthetic and field data indicate that the two algorithms based on conjugate gradients (NLCG and Mackie‐Madden) are more efficient than the Gauss‐Newton algorithm in terms of both computer memory requirements and CPU time needed to find accurate solutio...

/pdf/nonlinear-conjugate-gradients-algorithm-for-2-d-ktm8gicvyc.pdf

Nonlinear conjugate gradients algorithm for 2-D magnetotelluric inversion

The inversion of electromagnetic sounding data does not yield a unique solution, but inevitably a single model to interpret the observations is sought. We recommend that this model be as simple, or smooth, as possible, in order to reduce the temptation to overinterpret the data and to eliminate arbitrary discontinuities in simple layered models.To obtain smooth models, the nonlinear forward problem is linearized about a starting model in the usual way, but it is then solved explicitly for the desired model rather than for a model correction. By parameterizing the model in terms of its first or second derivative with depth, the minimum norm solution yields the smoothest possible model.Rather than fitting the experimental data as well as possible (which maximizes the roughness of the model), the smoothest model which fits the data to within an expected tolerance is sought. A practical scheme is developed which optimizes the step size at each iteration and retains the computational efficiency of layered models, resulting in a stable and rapidly convergent algorithm. The inversion of both magnetotelluric and Schlumberger sounding field data, and a joint magnetotelluric-resistivity inversion, demonstrate the method and show it to have practical application.

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Occam's inversion; a practical algorithm for generating smooth models from electromagnetic sounding data

[1] We present two continuous global geomagnetic field models for recent millennia: CALS3K.2, covering the past 3000 years, and CALS7K.2, covering 7000 years from 5000 BC to 1950 AD. The models were determined by regularized least squares inversion of archeomagnetic and paleomagnetic data using spherical harmonics in space and cubic B splines in time. They are derived from a greatly increased number of paleomagnetic directional data, compared to previous efforts, and for the first time a significant amount of archeointensity data is used in this kind of global model, allowing the determination of evolution of geomagnetic dipole strength. While data accuracy and dating uncertainties remain a limitation, reliable low-resolution global models can be obtained. The results agree well with previous results from virtual axial dipole moment (VADM) studies from archeomagnetic intensity data apart from a systematic offset in strength. A comparison of model predictions with the previous 3000 year model, CALS3K.1, gives general agreement but also some significant differences particularly for the early epochs. The new models suggest that the prominent two northern hemisphere flux lobes are more stationary than CALS3K.1 implied, extending considerably the time span of stationary flux lobes observed in historical models. Between 5000 BC and 2000 BC there are time intervals of weak dipole moment where dipole power is exceeded by low-degree nondipole power at the core-mantle boundary. Model coefficients and evaluation code can be obtained from the EarthRef Digital Archive (ERDA) together with animations and snapshots plots for every 100 years at http://www.earthref.org. Detailed URLs for the different material are listed in Appendix A.

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Continuous geomagnetic field models for the past 7 millennia: 2. CALS7K

Preface 1. General facts about the main field 2. Classical electrodynamics 3. Spherical harmonics 4. Gauss' theory of the main field 5. The Mie representation 6. Hydromagnetics of the core 7. Appendix: mathematical background Bibliography Index.

/pdf/foundations-of-geomagnetism-53hcut2jaz.pdf

Foundations of geomagnetism

Reconstructing the Holocene geomagnetic field

[1] Steadily increasing numbers of archeomagnetic and paleomagnetic data for the Holocene have allowed development of temporally continuous global spherical harmonic models of the geomagnetic field extending present and historical global descriptions of magnetic field evolution. The current work uses various subsets of improved data compilations, details of which are given in a companion paper by Donadini et al. (2009), and minor modifications of standard modeling strategies (using temporally and spatially regularized inversion of the data and cubic spline parameterizations for temporal variations) to produce five models with enhanced spatial and temporal resolution for 0–3 ka. Spurious end effects present in earlier models are eliminated by enforcing large-scale agreement with the gufm1 historical model for 1650–1990 A.D. and by extending the model range to accommodate data older than 3 ka. Age errors are not considered as a contribution to data uncertainties but are included along with data uncertainties in an investigation of statistical uncertainty estimates for the models using parametric bootstrap resampling techniques. We find common features but also significant differences among the various models, indicating intrinsic uncertainties in global models based on the currently available Holocene data. Model CALS3k.3 based on all available archeomagnetic and sediment data, without a priori quality selection, currently constitutes the best global representation of the past field. The new models have slightly higher dipole moments than our previous models. Virtual axial dipole moments (VADMs) calculated directly from the data are in good agreement with all corresponding model predictions of VADMs. These are always higher than the spherical harmonic dipole moment, indicating the limitations of using VADMs as a measure of geomagnetic dipole moments.

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Catherine Constable

Papers

Occam's inversion; a practical algorithm for generating smooth models from electromagnetic sounding data

Continuous geomagnetic field models for the past 7 millennia: 2. CALS7K

Foundations of geomagnetism

Reconstructing the Holocene geomagnetic field

Geomagnetic field for 0–3 ka: 2. A new series of time‐varying global models