seismic inversion based on proximal objective function optimization algorithm abstractseismic impedance inversion has become a common approach in reservoir prediction at present the critical issue in the application of seismic inversion is its low computational efficiency especially in 3d to improve the computational efficiency we have developed an inversion method derived from the proximal objective function optimization algorithm our inversion method calculates each unknown parameter in the model vector one by one during iteration compared with routine gradient dependent inversion algorithms such as the iteratively reweighted least squares irls algorithm our inversion method has lower computational complexity as well as higher efficiency in addition to obtain a sparse reflectivity series a long tailed cauchy distribution is used as the a priori constraint the weak nonlinear problem owing to the introduction of cauchy sparse constraint is addressed by taking advantage of reweighting strategy results of synthetic and real data tests illustrate that the proposed inv

simultaneous multifrequency inversion of full waveform seismic data we present a simultaneous multifrequency inversion approach for seismic data interpretation this algorithm inverts all frequency data components simultaneously a data weighting scheme balances the contributions from different frequency data components so the inversion process does not become dominated by high frequency data components which produce a velocity image with many artifacts a gauss newton minimization approach achieves a high convergence rate and an accurate reconstructed velocity image by introducing a modified adjoint formulation we can calculate the jacobian matrix efficiently allowing the material properties in the perfectly matched layers pmls to be updated automatically during the inversion process this feature ensures the correct behavior of the inversion and implies that the algorithm is appropriate for realistic applications where a priori information of the background medium is unavailable two different regularization schemes an l2 norm and a weighted l2 norm function a

algebraic relations among solutions of linear differential equations fano s theorem on formalise la notion de resolution d une equation differentielle lineaire en fonction d equations differentielles lineaires d ordre inferieur on deduit des conditions necessaires et suffisantes en termes d algebre de lie du groupe de galois pour que les solutions d une equation differentielle lineaire homogene s exprime en fonction d equations differentielles lineaires d ordre inferieur

a measure of multivariate phase synchrony using hyperdimensional geometry phase synchrony has been used to investigate the dynamics of subsystems that make up a complex system current measures of phase synchrony are mostly bivariate focusing on the synchrony between pairs of time series bivariate measures do not necessarily lead to a complete picture of the global interactions within a complex system current multivariate synchrony measures are based on either averaging all possible pairwise synchrony values or eigendecomposition of the pairwise bivariate synchrony matrix these approaches are sensitive to the accuracy of the bivariate synchrony indices computationally complex and indirect ways of quantifying the multivariate synchrony recently we had proposed a method to compute the multivariate phase synchrony using a hyperdimensional coordinate system this method referred to as hyperspherical phase synchrony hps has been found to be dependent on the ordering of the phase differences in this paper we propose a more general hyperspherical coordinate system along with a new higher dimensional manifold representation to eliminate the dependency on the ordering of the signals phases this new framework referred to as hyper torus synchrony hts is shown to be equivalent to the root mean square of a sufficient set of squared phase locking values whose phase differences contain information about all oscillators in the network the statistical properties of hts are given analytically and its performance is evaluated thoroughly for both synthetic and real signals

analysis of depth resolution in potential field inversion we study the inversion of potential fields and evaluate the degree of depth resolution achievable for a given problem to this end we introduce a powerful new tool the depth resolution plot drp the drp allows a theoretical study of how much the depth resolution in a potential field inversion is influenced by the way the problem is discretized and regularized the drp also allows a careful study of the influence of various kinds of ambiguities such as those from data errors or of a purely algebraic nature the achievable depth resolution is related to the given discretization regularization and data noise level we compute drp by means of singular value decomposition svd or its generalization gsvd depending on the particular regularization method chosen to illustrate the use of the drp we assume a source volume of specified depth and horizontal extent in which the solution is piecewise constant within a 3d grid of blocks we consider various linear regularization terms in a tikhonov damped least squares formulation some based on using higher order derivatives in the objective function drps are illustrated for both synthetic and real data our analysis shows that if the algebraic ambiguity is not too large and a suitable smoothing norm is used some depth resolution can be obtained without resorting to any subjective choice of depth weighting

quantifying and interpreting collective behavior in financial markets firms having similar business activities are correlated we analyze two different cross correlation matrices c constructed from i 30 min price fluctuations of 1000 us stocks for the two year period 1994 95 and ii one day price fluctuations of 422 us stocks for the 35 year period 1962 96 we find that the eigenvectors of c corresponding to the largest eigenvalues allow us to partition the set of all stocks into distinct subsets these subsets are similar to business sectors and are stable for extended periods of time we find that price fluctuations of these subsets are characterized by power law decaying time correlations reminiscent of strongly interacting systems the internal structure of a complex system manifests itself in correlations among its constituents in complex physical systems interactions between constituents cause collective modes having special statistical properties which reflect the underlying dynamics can we quantify collective movement of stock prices in analogous terms to address this question we analyze the equal time correlation matrix c constructed from the price fluctuations of a large number of stocks first we find that the collective modes for the stock market problem partition the set of all stocks studied into distinct subsets typically these subsets are formed by combinations of related industries and in some cases they go beyond grouping by industry due to company diversification the traditional partitioning of firms into subsets by products and services is difficult and sometimes arbitrary and thus our results could be viewed as a statistical alternative to traditional industry classification 1 furthermore we find that the price fluctuations of the collective modes display long range power law time correla

systematic analysis of group identification in stock markets we propose improved methods to identify stock groups using the correlation matrix of stock price changes by filtering out the marketwide effect and the random noise we construct the correlation matrix of stock groups in which nontrivial high correlations between stocks are found using the filtered correlation matrix we successfully identify the multiple stock groups without any extra knowledge of the stocks by the optimization of the matrix representation and the percolation approach to the correlation based network of stocks these methods drastically reduce the ambiguities while finding stock groups using the eigenvectors of the correlation matrix

a multiple level set method for 3d inversion of magnetic data abstractwe have developed a multiple level set method for inverting magnetic data produced by weak induced magnetization only the method is designed to deal with a specific class of 3d magnetic inverse problems in which the magnetic susceptibility is known and the objective of the inversion is to find the boundary or geometric shape of the causative bodies we adopt the conceptual representation of the subsurface geologic structure by a set of magnetic bodies each having a uniform magnetic susceptibility embedded in a nonmagnetic background this representation enables us to reformulate the magnetic inverse problem into a domain inverse problem for those unknown domains defining the supports of the magnetic causative bodies because each body may take on a variety of shapes and we may not know the number of bodies a priori either we use multiple level set functions to parameterize these domains so that the domain inverse problem can be further reduced to an optimization problem of multiple level set f

constrained least squares spectral analysis application to seismic data abstractan inversion based algorithm for computing the time frequency analysis of reflection seismograms using constrained least squares spectral analysis is formulated and applied to modeled seismic waveforms and real seismic data the fourier series coefficients are computed as a function of time directly by inverting a basis of truncated sinusoidal kernels for a moving time window the method resulted in spectra that have reduced window smearing for a given window length relative to the discrete fourier transform irrespective of window shape and a time frequency analysis with a combination of time and frequency resolution that is superior to the short time fourier transform and the continuous wavelet transform the reduction in spectral smoothing enables better determination of the spectral characteristics of interfering reflections within a short window the degree of resolution improvement relative to the short time fourier transform increases as window length decreases as compared with the continu

distributions of singular values for some random matrices the singular value decomposition is a matrix decomposition technique widely used in the analysis of multivariate data such as complex space time images obtained in both physical and biological systems in this paper we examine the distribution of singular values of low rank matrices corrupted by additive noise past studies have been limited to uniform uncorrelated noise using diagrammatic and saddle point integration techniques we extend these results to heterogeneous and correlated noise sources we also provide perturbative estimates of error bars on the reconstructed low rank matrix obtained by truncating a singular value decomposition 1999 the american physical society

uniform approach to linear and nonlinear interrelation patterns in multivariate time series currently a variety of linear and nonlinear measures is in use to investigate spatiotemporal interrelation patterns of multivariate time series whereas the former are by definition insensitive to nonlinear effects the latter detect both nonlinear and linear interrelation in the present contribution we employ a uniform surrogate based approach which is capable of disentangling interrelations that significantly exceed random effects and interrelations that significantly exceed linear correlation the bivariate version of the proposed framework is explored using a simple model allowing for separate tuning of coupling and nonlinearity of interrelation to demonstrate applicability of the approach to multivariate real world time series we investigate resting state functional magnetic resonance imaging rsfmri data of two healthy subjects as well as intracranial electroencephalograms ieeg of two epilepsy patients with focal onset seizures the main findings are that for our rsfmri data interrelations can be described by linear cross correlation rejection of the null hypothesis of linear ieeg interrelation occurs predominantly for epileptogenic tissue as well as during epileptic seizures

measuring synchronization in coupled model systems a comparison of different approaches the investigation of synchronization phenomena on measured experimental data such as biological time series has recently become an increasing focus of interest different approaches for measuring synchronization have been proposed that rely on certain characteristic features of the dynamical system under investigation for experimental data the underlying dynamics are usually not completely known therefore it is difficult to decide a priori which synchronization measure is most suitable for an analysis in this study we use three different coupled model systems to create a controlled setting for a comparison of six different measures of synchronization all measures are compared to each other with respect to their ability to distinguish between different levels of coupling and their robustness against noise results show that the measure to be applied to a certain task can not be chosen according to a fixed criterion but rather pragmatically as the measure which most reliably yields plausible information in test applications although certain dynamical features of a system under investigation e g power spectra dimension may render certain measures more suitable than others c 2006 elsevier b v all rights reserved

a local level set method for 3d inversion of gravity gradient data abstractwe have developed a local level set method for inverting 3d gravity gradient data to alleviate the inherent nonuniqueness of the inverse gradiometry problem we assumed that a homogeneous density contrast distribution with the value of the density contrast specified a priori was supported on an unknown bounded domain d so that we may convert the original inverse problem into a domain inverse problem because the unknown domain d may take a variety of shapes we parametrized the domain d by a level set function implicitly so that the domain inverse problem was reduced to a nonlinear optimization problem for the level set function because the convergence of the level set algorithm relied heavily on initializing the level set function to enclose the gravity center of a source body we applied a weighted l1 regularization method to locate such a gravity center so that the level set function can be properly initialized to rapidly compute the gradient of the nonlinear functional arising in the level

joint analysis of refractions with surface waves an inverse solution to the refraction traveltime problem we describe a possible solution to the inverse refraction traveltime problem irtp that reduces the range of possible solutions nonuniqueness this approach uses a reference model derived from surface wave shear wave velocity estimates as a constraint the application of the joint analysis of refractions with surface waves jars method provided a more realistic solution than the conventional refraction tomography methods which did not benefit from a reference model derived from real data this confirmed our conclusion that the proposed method is an advancement in the irtp analysis the unique basic principles of the jars method might be applicable to other inverse geophysical problems

bivariate surrogate techniques necessity strengths and caveats the concept of surrogates allows testing results from time series analysis against specified null hypotheses in application to bivariate model dynamics we here compare different types of surrogates each designed to test against a different null hypothesis e g an underlying bivariate linear stochastic process two measures that aim at a characterization of interdependence between nonlinear deterministic dynamics were used as discriminating statistics we analyze eight different stochastic and deterministic models not only to demonstrate the power of the surrogates but also to reveal some pitfalls and limitations

random matrix approach to cross correlations in financial data we analyze cross correlations between price fluctuations of different stocks using methods of random matrix theory rmt using two large databases we calculate cross correlation matrices c of returns constructed from i 30 min returns of 1000 us stocks for the 2 yr period 1994 1995 ii 30 min returns of 881 us stocks for the 2 yr period 1996 1997 and iii 1 day returns of 422 us stocks for the 35 yr period 1962 1996 we test the statistics of the eigenvalues lambda i of c against a null hypothesis a random correlation matrix constructed from mutually uncorrelated time series we find that a majority of the eigenvalues of c fall within the rmt bounds lambda lambda for the eigenvalues of random correlation matrices we test the eigenvalues of c within the rmt bound for universal properties of random matrices and find good agreement with the results for the gaussian orthogonal ensemble of random matrices implying a large degree of randomness in the measured cross correlation coefficients further we find that the distribution of eigenvector components for the eigenvectors corresponding to the eigenvalues outside the rmt bound display systematic deviations from the rmt prediction in addition we find that these deviating eigenvectors are stable in time we analyze the components of the deviating eigenvectors and find that the largest eigenvalue corresponds to an influence common to all stocks our analysis of the remaining deviating eigenvectors shows distinct groups whose identities correspond to conventionally identified business sectors finally we discuss applications to the construction of portfolios of stocks that have a stable ratio of risk to return less

applying compactness constraints to differential traveltime tomography tomographic imaging problems are typically ill posed and often require the use of regularization techniques to guarantee a stable solution minimization of a weighted norm of model length is one commonly used secondary constraint tikhonov methods exploit low order differential operators to select for solutions that are small flat or smooth in one or more dimensions this class of regularizing functionals may not always be appropriate particularly in cases where the anomaly being imaged is generated by a nonsmooth spatial process time lapse imaging of flow induced velocity anomalies is one such case flow features are often characterized by spatial compactness or connectivity by performing inversions on differenced arrival time data the properties of the time lapse feature can be directly constrained we develop a differential traveltime tomography algorithm whichselects for compact solutions i e models with a minimum area of support through application of model space iteratively reweighted leas

compact gravity inversion we present a new criterion for the inversion of gravity data the principle employed is to minimize the volume of the causative body which is equivalent to maximizing its compactness the anomalous density distribution is obtained using an iterative technique which is numerically stable and rapidly convergent the principle can also be adapted to include modeling of gravity anomalies by single density sources the advantage of this approach is that desirable geologic characteristics are automatically incorporated into the model with a minimum of subjective judgments on the part of the interpreter the treatment of noise in the data fits naturally into the formulation of the inversion procedure the method is illustrated by the inversion of noise free and noisy data generated from a two dimensional model consisting of a regular array of identical rectangular blocks whose densities can be individually specified in every case the algorithm successfully recovers the correct density distribution from the dat

data driven estimates of the number of clusters in multivariate time series an important problem in unsupervised data clustering is how to determine the number of clusters here we investigate how this can be achieved in an automated way by using interrelation matrices of multivariate time series two nonparametric and purely data driven algorithms are expounded and compared the first exploits the eigenvalue spectra of surrogate data while the second employs the eigenvector components of the interrelation matrix compared to the first algorithm the second approach is computationally faster and not limited to linear interrelation measures

detecting synchronization clusters in multivariate time series via coarse graining of markov chains synchronization cluster analysis is an approach to the detection of underlying structures in data sets of multivariate time series starting from a matrix r of bivariate synchronization indices a previous method utilized the eigenvectors of r for cluster identification analogous to several recent attempts at group identification using eigenvectors of the correlation matrix all of these approaches assumed a one to one correspondence of dominant eigenvectors and clusters which has however been shown to be wrong in important cases we clarify the usefulness of eigenvalue decomposition for synchronization cluster analysis by translating the problem into the language of stochastic processes and derive an enhanced clustering method harnessing recent insights from the coarse graining of finite state markov processes we illustrate the operation of our method using a simulated system of coupled lorenz oscillators and we demonstrate its superior performance over the previous approach finally we investigate the question of robustness of the algorithm against small sample size which is important with regard to field applications

nonuniform state space reconstruction and coupling detection we investigate the state space reconstruction from multiple time series derived from continuous and discrete systems and propose a method for building embedding vectors progressively using information measure criteria regarding past current and future states the embedding scheme can be adapted for different purposes such as mixed modeling cross prediction and granger causality in particular we apply this method in order to detect and evaluate information transfer in coupled systems as a practical application we investigate in records of scalp epileptic eeg the information flow across brain areas

three dimensional gravity inversion using graph theory to delineate the skeleton of homogeneous sources abstractwe developed a new strategy based on graph theory concepts to invert gravity data using an ensemble of simple point masses our method consisted of a genetic algorithm with elitism to generate a set of possible solutions each estimate was associated to a graph to solve the minimum spanning tree mst problem to produce unique and stable estimates we restricted the position of the point masses by minimizing the statistical variance of the distances of an mst jointly with the data misfit function during the iterations of the genetic algorithm hence the 3d spatial distribution of the point masses identified the skeleton of homogeneous gravity sources in addition our method also gave an estimation of the anomalous mass of the source so together with the anomalous mass the skeleton could aid other 3d methods with promising geometric a priori parameters several tests with different values of regularizing parameter were made to bespeak this new regularizing strategy the inversion results ap

a generalized kolmogorov sinai like entropy under markov shifts in symbolic dynamics abstract in this paper a generalized kolmogorov sinai like entropy ks q entropy in the sense of tsallis is proposed with a nonextensive parameter q under markov shifts which contains the classical kolmogorov sinai ks entropy and the renyi entropy as well as bernoulli shifts as special cases to verify the formula of this ks q entropy a one dimensional iterative system is chosen as an example of markov shifts and its ks q entropy is evaluated by a new refinement method of symbolic dynamics called symbolic refinement which differs from the conventional numerical method the numerical results show that this ks q entropy is monotonically decreasing as q increases

fuzzy constrained lp norm inversion of direct current resistivity data abstractwe evaluate the use of a fuzzy c means clustering procedure to improve an inverted 2d resistivity model within the iterative error minimization procedure the algorithm is coded in matlab language for the lp norm inversion of 2d direct current resistivity data and is referred to as fuzzy constrained inversion fci two additional input parameters are required to be provided by the interpreter 1 the number of geologic units in the model i e the number of clusters and 2 the mean resistivity values of each geologic unit i e cluster center values of the geologic units the efficacy of our approach is evaluated by tests carried on the synthetic and field electrical resistivity tomography ert data inversion results from the fci algorithm are presented for conventional l1 and l2 norm minimization techniques fci indicates improvement over conventional inversion approaches in differentiating the geologic units if a proper number of the geologic units is provided to the algorithm inappro

a hybrid regularized inversion methodology for highly parameterized environmental models 1 a hybrid approach to the regularized inversion of highly parameterized environmental models is described the method is based on constructing a highly parameterized base model calculating base parameter sensitivities and decomposing the base parameter normal matrix into eigenvectors representing principal orthogonal directions in parameter space the decomposition is used to construct super parameters super parameters are factors by which principal eigenvectors of the base parameter normal matrix are multiplied in order to minimize a composite least squares objective function these eigenvectors define orthogonal axes of a parameter subspace for which information is available from the calibration data the coordinates of the solution are sought within this subspace super parameters are estimated using a regularized nonlinear gauss marquardt levenberg scheme though super parameters are estimated tikhonov regularization constraints are imposed on base parameters tikhonov regularization mitigates over fitting and promotes the estimation of reasonable base parameters use of a large number of base parameters enables the inversion process to be receptive to the information content of the calibration data including aspects pertaining to small scale parameter variations because the number of super parameters sustainable by the calibration data may be far less than the number of base parameters used to define the original problem the computational burden for solution of the inverse problem is reduced the hybrid methodology is described and applied to a simple synthetic groundwater flow model it is then applied to a real world groundwater flow and contaminant transport model the approach and programs described are applicable to a range of modeling disciplines copyright 2005 by the american geophysical union

imaging geologic surfaces by inverting gravity gradient data with depth horizons the nonuniqueness problem that occurs when inverting potential field data is well known it can however be surmounted by jointly inverting these data with independent data sets incorporating depth information and regularizing the solution the goal is to produce a geologic model that is compatible with all measured quantities does not exceed any prescribed limits and is geologically plausible to achieve this we have developed a spatially based surface inversion algorithm that solves for the geometric interface between geologic bodies the bodies are constructed from grids of rectangular prisms that have their bottom depths adjusted by the algorithm to form the inverted surface to solve large scale inversions approximations are used in the potential field calculations that allow internal matrices to be stored in sparse format with minimal loss of accuracy the impetus for the work came from the need to combine airborne gravity gradient data with depth horizons estimated from interpreted 2d seismic

harmonic and biharmonic biases in potential field inversion abstractwe found that minimum l2 norm and smoothness constrained continuous solutions of the linear inverse problem of potential field data are harmonic and biharmonic respectively in the case of a discrete distribution the minimum l2 norm and smoothness constrained solutions become biased toward being harmonic or biharmonic respectively as a result the estimated discrete distribution of density or magnetization contrast tends to be smooth and to satisfy the maximum principle which forces the solution maxima and minima to lie on any boundary of the discretized region the above findings were illustrated with 2d numerical examples the harmonic or biharmonic bias is brought forth when the strengths of the minimum l2 norm or the smoothness constraint are enhanced relative to all other constraints by approximating the continuous case a large number of discretizing cells relative to the number of independent observations and or by using a regularizing parameter we discovered that by inspecting th

estimating mutual information we present two classes of improved estimators for mutual information m x y from samples of random points distributed according to some joint probability density mu x y in contrast to conventional estimators based on binnings they are based on entropy estimates from k nearest neighbor distances this means that they are data efficient with k 1 we resolve structures down to the smallest possible scales adaptive the resolution is higher where data are more numerous and have minimal bias indeed the bias of the underlying entropy estimates is mainly due to nonuniformity of the density at the smallest resolved scale giving typically systematic errors which scale as functions of k n for n points numerically we find that both families become exact for independent distributions i e the estimator m x y vanishes up to statistical fluctuations if mu x y mu x mu y this holds for all tested marginal distributions and for all dimensions of x and y in addition we give estimators for redundancies between more than two random variables we compare our algorithms in detail with existing algorithms finally we demonstrate the usefulness of our estimators for assessing the actual independence of components obtained from independent component analysis ica for improving ica and for estimating the reliability of blind source separation

enhancing the resolving power of least squares inversion with active constraint balancing most geophysical inverse problems are solved using least squares inversion schemes with damping or smoothness constraints to improve stability and convergence rate since the lagrangian multiplier controls resolution and stability of the inverse problem we always want to use the optimum multiplier which is not easy to get and is usually obtained by experience or a time consuming optimization process we present a new regularization approach in which the lagrangian multiplier is set as a spatial variable at each parameterized block and automatically determined via the parameter resolution matrix and spread function analysis for highly resolvable parameters a small value of the lagrangian multiplier is assigned and vice versa this approach named active constraint balancing acb tries to balance the constraints of the least squares inversion according to sensitivity for a given problem so that it enhances the resolution as well as the stability of the inversion process we demonstrate the performa

3 d inversion of gravity data we present two methods for inverting surface gravity data to recover a 3 d distribution of density contrast in the first method we transform the gravity data into pseudomagnetic data via poisson s relation and carry out the inversion using a 3 d magnetic inversion algorithm in the second we invert the gravity data directly to recover a minimum structure model in both approaches the earth is modeled by using a large number of rectangular cells of constant density and the final density distribution is obtained by minimizing a model objective function subject to fitting the observed data the model objective function has the flexibility to incorporate prior information and thus the constructed model not only fits the data but also agrees with additional geophysical and geological constraints we apply a depth weighting in the objective function to counteract the natural decay of the kernels so that the inversion yields depth information applications of the algorithms to synthetic and field data produce density models representative of true structures our results have shown that the inversion of gravity data with a properly designed objective function can yield geologically meaningful information

markov chain monte carlo estimation of distributed debye relaxations in spectral induced polarization there is growing interest in the link between electrical polarization and physical properties of geologic porous media in particular spectral characteristics may be controlled by the same pore geometric properties that influence fluid permeability of such media various models have been proposed to describe the spectral induced polarization sip response of permeable rocks and the links between these models and hydraulic properties have been explored albeit empirically computation of the uncertainties in the parameters of such electrical models is essential for effective use of these relationships the formulation of an electrical dispersion model in terms of a distribution of relaxation times and associated chargeabilities has been demonstrated to be an effective generalized approach however thus far such an approach has only been considered in a deterministic framework here we formulate a spectral model based on a distribution of polarizations by using a simple polynomial descriptor of such a distribution we are able to cast the model in a stochastic manner and solve it using a markov chain monte carlo mcmc sampler thus allowing the computation of model parameter uncertainties we apply the model to synthetic data and demonstrate that the stochastic method can provide posterior distributions of model parameters with narrow bounds around the true values when little or no noise is added to the synthetic data with posterior distributions that broaden with increasing noise we also apply our model to experimental measurements of six sandstone samples and compare physical properties of a number of samples of porous media with stochastic estimates of characteristic relaxation times we demonstrate the utility of our method on electrical spectra with different response characteristics and show that a single metric of relaxation time for the sip response is not sufficient to provide clear insight into the physical characteristics of a sample

universal and nonuniversal properties of cross correlations in financial time series we use methods of random matrix theory to analyze the cross correlation matrix c of stock price changes of the largest 1000 u s companies for the 2 year period 1994 1995 we find that the statistics of most of the eigenvalues in the spectrum of c agree with the predictions of random matrix theory but there are deviations for a few of the largest eigenvalues we find that c has the universal properties of the gaussian orthogonal ensemble of random matrices furthermore we analyze the eigenvectors of c through their inverse participation ratio and find eigenvectors with large ratios at both edges of the eigenvalue spectrum a situation reminiscent of localization theory results

reliable detection of directional couplings using rank statistics to detect directional couplings from time series various measures based on distances in reconstructed state spaces were introduced these measures can however be biased by asymmetries in the dynamics structure noise color or noise level which are ubiquitous in experimental signals using theoretical reasoning and results from model systems we identify the various sources of bias and show that most of them can be eliminated by an appropriate normalization we furthermore diminish the remaining biases by introducing a measure based on ranks of distances this rank based measure outperforms existing distance based measures concerning both sensitivity and specificity for directional couplings therefore our findings are relevant for a reliable detection of directional couplings from experimental signals

3d magnetic sparse inversion using an interior point method rock susceptibility measurements are sometimes taken on outcrop and borehole rocks and provide valuable information for constraining magnetic data inversion we present two approaches for 3d magnetic sparse inversion that effectively take advantage of the rock susceptibility information both approaches minimize a total objective function subject to bound constraints using an interior point method the first approach directly minimizes an l1 norm of the susceptibility model by keeping the bounds positive in which case the objective function is differentiable in the feasible region the second approach minimizes a more generalized lp like norm 0 p 1 of the susceptibility model by approximating the lp like norm inversion as an iteratively reweighed least squares problem moreover this approach allows the model values to be either positive or negative we also demonstrate the equivalence of the proposed approaches and the binary inversion the recovered models of both approaches are characterized by

relative performance of mutual information estimation methods for quantifying the dependence among short and noisy data commonly used dependence measures such as linear correlation cross correlogram or kendall s tau cannot capture the complete dependence structure in data unless the structure is restricted to linear periodic or monotonic mutual information mi has been frequently utilized for capturing the complete dependence structure including nonlinear dependence recently several methods have been proposed for the mi estimation such as kernel density estimators kde k nearest neighbors knn edgeworth approximation of differential entropy and adaptive partitioning of the xy plane however outstanding gaps in the current literature have precluded the ability to effectively automate these methods which in turn have caused limited adoptions by the application communities this study attempts to address a key gap in the literature specifically the evaluation of the above methods to choose the best method particularly in terms of their robustness for short and noisy data based on comparisons with the theoretical mi estimates which can be computed analytically as well with linear correlation and kendall s tau here we consider smaller data sizes such as 50 100 and 1 000 where this study considers 50 and 100 data points as very short and 1 000 as short we consider a broader class of functions specifically linear quadratic periodicmore and chaotic contaminated with artificial noise with varying noise to signal ratios the case studies presented here are motivated by domain consideration in the earth sciences where the data are short and noisy our results indicate kde as the best choice for very short data at relatively high noise to signal levels whereas the performance of knn is the best for short data at relatively low noise levels as well as for short data consistently across noise levels in addition the optimal smoothing parameter of a gaussian kernel appears to be the best choice for kde while three nearest neighbors appear optimal for knn thus in situations where the approximate data sizes are known in advance and exploratory data analysis and or domain knowledge can be used to provide a priori insights on the noise to signal ratios the results in the paper point to a way forward for automating the process of mi estimation less

adaptive sampling of potential field data a direct approach to compressive inversion abstractcompressive inversion uses computational algorithms that decrease the time and storage needs of a traditional inverse problem most compression approaches focus on the model domain and very few other than traditional downsampling focus on the data domain for potential field applications to further the compression in the data domain a direct and practical approach to the adaptive downsampling of potential field data for large inversion problems has been developed the approach is formulated to significantly reduce the quantity of data in relatively smooth or quiet regions of the data set while preserving the signal anomalies that contain the relevant target information two major benefits arise from this form of compressive inversion first because the approach compresses the problem in the data domain it can be applied immediately without the addition of or modification to existing inversion software second as most industry software use some form of model or sensitivity compression the

inversion and uncertainty estimation of gravity data using simulated annealing an application over lake vostok east antarctica interpretation of gravity data warrants uncertainty estimation because of its inherent nonuniqueness although the uncertainties in model parameters cannot be completely reduced they can aid in the meaningful interpretation of results here we have employed a simulated annealing sa based technique in the inversion of gravity data to derive multilayered earth models consisting of two and three dimensional bodies in our approach we assume that the density contrast is known and we solve for the coordinates or shapes of the causative bodies resulting in a nonlinear inverse problem we attempt to sample the model space extensively so as to estimate several equally likely models we then use all the models sampled by sa to construct an approximate marginal posterior probability density function ppd in model space and several orders of moments the correlation matrix clearly shows the interdependence of different model parameters and the corresponding trade offs such correlation plots are used to stud

3d magnetic data space inversion with sparseness constraints i have developed an inversion approach that determines a 3d susceptibility distribution that produces a given magnetic anomaly the subsurface model consists of a 3d equally spaced array of dipoles the inversion incorporates a model norm that enforces sparseness and depth weighting of the solution sparseness is imposed by using the cauchy norm on model parameters the inverse problem is posed in the data space leading to a linear system of equations with dimensions based on the number of data n this contrasts with the standard least squares solution derived through operations within the m dimensional model space m being the number of model parameters hence the data space method combined with a conjugate gradient algorithm leads to computational efficiency by dealing with an n n system versus an m m one where n m tests on synthetic data show that sparse inversion produces a much more focused solution compared with a standard model space least squares inversion the inversion of aeromagnet

mean phase coherence as a measure for phase synchronization and its application to the eeg of epilepsy patients abstract we apply the concept of phase synchronization of chaotic and or noisy systems and the statistical distribution of the relative instantaneous phases to electroencephalograms eegs recorded from patients with temporal lobe epilepsy using the mean phase coherence as a statistical measure for phase synchronization we observe characteristic spatial and temporal shifts in synchronization that appear to be strongly related to pathological activity in particular we observe distinct differences in the degree of synchronization between recordings from seizure free intervals and those before an impending seizure indicating an altered state of brain dynamics prior to seizure activity

laws of composition of backlund transformations and the universal form of completely integrable systems in dimensions two and three abstract backlund transformations are defined as operations on solutions of a riemann boundary value problem vector bundles over p1 that add apparent singularities for solutions of difference and differential linear spectral problems backlund transformations are presented in explicit form through the christoffel formula and its generalizations identities satisfied by iterations of elementary backlund transformations are represented in the form of the law of addition or as the three dimensional difference equation of hirota s type matrix two dimensional isospectral deformation equations are imbedded into three dimensional scalar systems of kadomtzev petviashvili law of addition form two dimensional matrix systems correspond to reductions of kadomtzev petviashvili equations with pseudodifferential operators satisfying algebraic equations

cross gradients joint 3d inversion with applications to gravity and magnetic data we extend the cross gradient methodology for joint inversion to three dimensional environments and introduce a solution procedure based on a statistical formulation and equality constraints for structural similarity resemblance we apply the proposed solution to the joint 3d inversion of gravity and magnetic data and gauge the advantages of this new formulation on test and field data experiments combining singular value decomposition svd and other conventional regularizing constraints we determine 3d distributions of the density and magnetization with enhanced structural similarity the algorithm reduces some misleading features of the models which are introduced commonly by conventional separate inversions of gravity and magnetic data and facilitates an integrated interpretation of the models

3d stochastic gravity inversion using nonstationary covariances abstractthe flexibility of geostatistical inversions in geophysics is limited by the use of stationary covariances which implicitly and mostly for mathematical convenience assumes statistical homogeneity of the studied field for fields showing sharp contrasts due for example to faults or folds an approach based on the use of nonstationary covariances for cokriging inversion was developed the approach was tested on two synthetic cases and one real data set inversion results based on the nonstationary covariance were compared to the results from the stationary covariance for two synthetic models the nonstationary covariance better recovered the known synthetic models with the real data set the nonstationary assumption resulted in a better match with the known surface geology

3 d magnetic inversion with data compression and image focusing we develop a method of 3 d magnetic anomaly inversion based on traditional tikhonov regularization theory we use a minimum support stabilizing functional to generate a sharp focused inverse image an iterative inversion process is constructed in the space of weighted model parameters that accelerates the convergence and robustness of the method the weighting functions are selected based on sensitivity analysis to speed up the computations and to decrease the size of memory required we use a compression technique based on cubic interpolation our method is designed for inversion of total magnetic anomalies assuming the anomalous field is caused by induced magnetization only the method is applied to synthetic data for typical models of magnetic anomalies and is tested on real airborne data provided by exxonmobil upstream research company

a fast algorithm for regularized focused 3d inversion of gravity data using randomized singular value decomposition a fast algorithm for solving the under determined 3 d linear gravity inverse problem based on the randomized singular value decomposition rsvd is developed the algorithm combines an iteratively reweighted approach for l1 norm regularization with the rsvd methodology in which the large scale linear system at each iteration is replaced with a much smaller linear system although the optimal choice for the low rank approximation of the system matrix with m rows is q m acceptable results are achievable with q m in contrast to the use of the iterative lsqr algorithm for the solution of the linear systems at each iteration the singular values generated using the rsvd yield a good approximation of the dominant singular values of the large scale system matrix the regularization parameter found for the small system at each iteration is thus dependent on the dominant singular values of the large scale system matrix and appropriately regularizes the dominant singular space of the large scale problem the

2d time lapse seismic tomography using an active time constraint atc approach a 2d seismic time lapse inversion approach can image the evolution of seismic velocities over time and space the forward modeling is based on solving the eikonal equation using a second order fast marching method wave paths are represented by fresnel volumes rather than by conventional rays this approach accounts for complex velocity models and has the advantage of considering the effects of wave frequency on velocity resolution the aim of time lapse inversion is to find changes in velocities of each cell in the model as a function of time each model along the time axis is called a reference space model this approach can be simplified into an inverse problem that seeks the optimum of several reference space models taken together using the approximation that the change in seismic velocity varies linearly in time between two subsequent reference models the method is demonstrated on a synthetic example that includes regularization in time in the cost function and reduces inversion artifacts associated with noise in the data by comparison with independent inversions at each time

practical applications of uniqueness theorems in gravimetry part i constructing sound interpretation methods to obtain a unique and stable solution to the gravity inverse problem a priori information reflecting geological attributes of the gravity source must be used mathematical conditions to obtain stable solutions are established in tikhonov s regularization method where the a priori information is introduced via a stabilizing functional which may be suitably designed to incorporate some relevant geological information however there is no unifying approach establishing general uniqueness conditions for a gravity inverse problem rather there are many theorems usually establishing just abstract mathematical conditions and making it difficult to devise the type of geological information needed to guarantee a unique solution in part i of these companion papers we show that translating the mathematical uniqueness conditions into geological constraints is an important step not only in establishing the type of geological setting where a particular method may be applied but also in designing new gravity inv

derivation of forward and adjoint operators for least squares shot profile split step migration the forward and adjoint operators for shot profile least squares migration are derived the forward operator is demigration and the adjoint operator is migration the demigration operator is derived from the born approximation the process begins with a green s function that allows for a laterally varying migration velocity model using the split step approximation next the earth is divided into horizontal layers and within each layer the migration velocity model is made to be constant with respect to depth for a given layer 1 the source side wavefield is propagated down to its top using the background wavefield this gives a background wavefield incident at the layer s upper boundary 2 the layer s contribution to the scattered wavefield is computed using the born approximation to the scattered wavefield and the background wavefield 3 next its scattered wavefield is propagated back up to the measurement surface using again the background wavefield the measured wavefield is approximated by the sum of scattered wavefields from each layer in the derivation of the measured wavefield the shot profile migration geometry is used for each shot the resulting wavefield modeling operator takes the form of a fredholm integral equation of the first kind and this is used to write down its adjoint the shot profile migration operator this forward adjoint pair is used for shot profile least squares migration shot profile least squares migration is illustrated with two synthetic examples the first uses data collected over a four layer acoustic model and the second uses data from the sigsbee 2a model

identifying phase synchronization clusters in spatially extended dynamical systems we investigate two recently proposed multivariate time series analysis techniques that aim at detecting phase synchronization clusters in spatially extended nonstationary systems with regard to field applications the starting point of both techniques is a matrix whose entries are the mean phase coherence values measured between pairs of time series the first method is a mean field approach which allows one to define the strength of participation of a subsystem in a single synchronization cluster the second method is based on an eigenvalue decomposition from which a participation index is derived that characterizes the degree of involvement of a subsystem within multiple synchronization clusters simulating multiple clusters within a lattice of coupled lorenz oscillators we explore the limitations and pitfalls of both methods and demonstrate a that the mean field approach is relatively robust even in configurations where the single cluster assumption is not entirely fulfilled and b that the eigenvalue decomposition approach correctly identifies the simulated clusters even for low coupling strengths using the eigenvalue decomposition approach we studied spatiotemporal synchronization clusters in long lasting multichannel eeg recordings from epilepsy patients and obtained results that fully confirm findings from well established neurophysiological examination techniques multivariate time series analysis methods such as synchronization cluster analysis which account for nonlinearities in the data are expected to provide complementary information which allows one to gain deeper insights into the collective dynamics of spatially extended complex systems

a new sensitivity controlled focusing regularization scheme for the inversion of induced polarization data based on the minimum gradient support we present a new application of a focusing regularization scheme for the inversion of resistivity and induced polarization ip data that supports large resistivity magnitude and phase contrasts similar approaches so far have only been used for the interpretation of gravity magnetic or seismic data sets unlike methods based on smoothness constraints the approach is able to resolve sharp boundaries of bodies and layers and it allows slight parameter variations within them therefore it can be used in hydrogeologic applications where we need focused images to resolve high contrast aquifer boundaries our approach is based on the minimum gradient support which seeks to minimize the occurrence of parameter contrasts independent of their magnitude we study the effects of a variable control parameter on the reweighting optimization allowing a continuous transition from smooth to sharp images we also take the spatially varying sensitivity into account to allow focusing even where sensitivities are small the implemented weighting leads to increased smoothing in well resolved areas and a decrease in regions of lower sensitivity the opposite approach is examined as well this gradient dependent sensitivity weighting is basically an extension of depth dependent sensitivity weighting we demonstrate the effectiveness and limitations of the approach and the influence of the control parameter using different synthetic models and field data from a hydrogeophysical test site the technique has proven particularly suitable for revealing sharp parameter contrasts

magnetotelluric inversion for minimum structure structure can be measured in terms of a norm of the derivative of a model with respect to a function of depth f z where the model m z is either the conductivity or log an iterative linearized algorithm can find models that minimize norms of this form for chosen levels of chi squared misfit the models found may very well be global minima of these norms since they are not observed to depend on the starting model overfitting data causes extraneous structure some choices of the depth function result in systematic overfitting of high frequencies a blue fit and extraneous shallow structure others result in systematic overfitting of low frequencies a red fit and extraneous deep structure a robust statistic is used to test for whiteness the fit can be made acceptably white by varying the depth function f z which defines the norm an optimum norm produces an inversion which does not introduce false structure and which approaches the true structure in a reasonable way as data errors decrease

the effects of noise on occam s inversion of resistivity tomography data an occam s inversion algorithm for crosshole resistivity data that uses a finite element method forward solution is discussed for the inverse algorithm the earth is discretized into a series of parameter blocks each containing one or more elements the occam s inversion finds the smoothest 2 d model for which the chi squared statistic equals an a priori value synthetic model data are used to show the effects of noise and noise estimates on the resulting 2 d resistivity images resolution of the images decreases with increasing noise the reconstructions are underdetermined so that at low noise levels the images converge to an asymptotic image not the true geoelectrical section if the estimated standard deviation is too low the algorithm cannot achieve an adequate data fit the resulting image becomes rough and irregular artifacts start to appear when the estimated standard deviation is larger than the correct value the resolution decreases substantially the image is too smooth the same effects are demonstrated for field data from a site near livermore california however when the correct noise values are known the occam s results are independent of the discretization used a case history of monitoring at an enhanced oil recovery site is used to illustrate problems in comparing successive images over time from a site where the noise level changes in this case changes in image resolution can be misinterpreted as actual geoelectrical changes one solution to this problem is to perform smoothest but non occam s inversion on later data sets using parameters found from the background data set

practical applications of uniqueness theorems in gravimetry part ii pragmatic incorporation of concrete geologic information we illustrate the importance of establishing solution uniqueness through mathematical restrictions reflecting a source attribute we also illustrate the validity and utility of a guideline derived in an accompanying paper for constructing sound gravity inversion methods for the class of sources presenting either homogeneous or depth independent density distributions the two part guideline is 1 to introduce a priori information favoring uniqueness either by assuming that a nonnull density distribution depending only on x and y is confined to the interior of a horizontal slab with known position or by limiting the class of possible solutions to homogeneous simply connected polygons or polyhedra with known density displaying no fancy shapes and no curling apophyses at their borders and 2 to introduce information favoring solution stability by estimating only the features of the source which may be resolved by the data following the guideline we apply different methods to gravity data using interp

the wronskian formalism for linear differential equations and pade approximations on demontre des theoremes generaux sur la determination des meilleures approximations rationnelles des solutions d equations differentielles lineaires a coefficients fonction rationnelle

dynamic smoothing in crosswell traveltime tomography variable size dynamic smoothing operator constraints are applied in crosswell traveltime tomography to reconstruct both the smooth and fine scale details of the tomogram in mixed and underdetermined problems a large number of iterations may be necessary to introduce the slowly varying slowness features into the tomogram to speed up convergence the dynamic smoothing operator applies adaptive regularization to the traveltime prediction error function with the help of the model covariance matrix by so doing the regularization term has a larger weight at initial iterations and the prediction error term dominates the final iterations with a small regularization term weight in addition it is shown that adaptive regularization acts by reweighting the adjoint modeling operator preconditioning and by providing additional damping comparisons of two dynamic smoothing operators the low pass filter smoothing and the multigrid technique with the fixed size static smoothing operators show that the dynamic smoothing operator yields more accurate velocity distributions with greater stability for larger velocity contrasts consequently it is a preferred choice for regularization

detection and characterization of changes of the correlation structure in multivariate time series we propose a method based on the equal time correlation matrix as a sensitive detector for phase shape correlations in multivariate data sets the key point of the method is that changes of the degree of synchronization between time series provoke level repulsions between eigenstates at both edges of the spectrum of the correlation matrix consequently detailed information about the correlation structure of the multivariate data set is imprinted into the dynamics of the eigenvalues and into the structure of the corresponding eigenvectors the performance of the technique is demonstrated by application to n f tori autoregressive models and coupled chaotic systems the high sensitivity the comparatively small computational effort and the excellent time resolution of the method recommend it for application to the analysis of complex spatially extended nonstationary systems

robust perron cluster analysis in conformation dynamics the key to molecular conformation dynamics is the direct identification of metastable conformations which are almost invariant sets of molecular dynamical systems once some reversible markov operator has been discretized a generalized symmetric stochastic matrix arises this matrix can be treated by perron cluster analysis a rather recent method involving a perron cluster eigenproblem the paper presents an improved perron cluster analysis algorithm which is more robust than earlier suggestions numerical examples are included

statistically optimal almost invariant sets abstract chaotic dynamical systems are often transitive although this transitivity is sometimes very weak it is of interest to divide the phase space into large regions between which there is relatively little communication of trajectories we present fast simple algorithms to find such divisions the present work builds on the results of froyland and dellnitz g froyland m dellnitz detecting and locating near optimal almost invariant sets and cycles siam j sci comput 24 6 2003 1839 1863 focussing on a statistical description of transitivity that takes into account the fact that trajectories tend to visit different regions of phase space with different frequencies the new work takes advantage of theoretical results from the theory of reversible markov chains a new adaptive algorithm is put forward to efficiently deal with situations where the boundaries of the weakly communicating regions are complicated this algorithm is illustrated with the standard map relevant convergence results are proven

2 d and 3 d interpretation of electrical tomography measurements part 2 the inverse problem the interpretation of borehole to borehole electrical measurements requires solving an inverse problem for a given class of model geometries the usual approach to an inverse problem includes a model dependent task i e forward modeling and a generic task i e an optimization scheme we have developed an optimization algorithm using a nonlinear inversion technique this algorithm allows recovery of a possible resistivity distribution in an investigated zone between two boreholes or in the vicinity of them this resistivity distribution is defined as a set of 2 d or 3 d volumes of constant resistivity the inversion procedure minimizes a least squares term plus a damping term this latter term seeks to minimize the roughness of the solution an improved form of this smoothness term may enhance the spatial resolution of the resistivity image assuming that the resistivity contrast is known a priori this reconstruction algorithm has been tested for both 2 d and 3 d geometries these inversion tests we

gravity and magnetic inversion with minimization of a specific functional the nonuniqueness of gravity or magnetic data inversion is well known in order to remove ambiguity some authors have sought solutions minimizing a functional describing geometrical or physical properties last and kubik 1983 in particular developed a method explaining the observed anomaly by structures of minimum volume in this method the domain where anomalous sources are searched is divided into elementary prisms of a constant density or susceptibility contrast each elementary contrast is allowed to vary individually thus a contrast distribution is computed the search for this kind of solution leads in general to geologically more appropriate bodies but exceptions do occur in this paper the technique is broadened to include the search for solutions minimizing the moment of inertia with respect to the center of gravity or with respect to a given dip line passing through it the resulting structures are both deeper and more compact precisely as is required in specific cases theoretical and

statistics of atmospheric correlations for a large class of quantum systems the statistical properties of their spectrum show remarkable agreement with random matrix predictions recent advances show that the scope of random matrix theory is much wider in this work we show that the random matrix approach can be beneficially applied to a completely different classical domain namely to the empirical correlation matrices obtained from the analysis of the basic atmospheric parameters that characterize the state of atmosphere we show that the spectrum of atmospheric correlation matrices satisfy the random matrix prescription in particular the eigenmodes of the atmospheric empirical correlation matrices that have physical significance are marked by deviations from the eigenvector distribution

random matrix theory analysis of cross correlations in financial markets we confirm universal behaviors such as eigenvalue distribution and spacings predicted by random matrix theory rmt for the cross correlation matrix of the daily stock prices of tokyo stock exchange from 1993 to 2001 which have been reported for new york stock exchange in previous studies it is shown that the random part of the eigenvalue distribution of the cross correlation matrix is stable even when deterministic correlations are present some deviations in the small eigenvalue statistics outside the bounds of the universality class of rmt are not completely explained with the deterministic correlations as proposed in previous studies we study the effect of randomness on deterministic correlations and find that randomness causes a repulsion between deterministic eigenvalues and the random eigenvalues this is interpreted as a reminiscent of level repulsion in rmt and explains some deviations from the previous studies observed in the market data we also study correlated groups of issues in these markets and propose a refined method to identify correlated groups based on rmt some characteristic differences between properties of tokyo stock exchange and new york stock exchange are found

structural inversion of gravity data using linear programming structural inversion of gravity data deriving robust images of the subsurface by delineating lithotype boundaries using density anomalies is an important goal in a range of exploration settings e g ore bodies salt flanks application of conventional inversion techniques in such cases using l2 norms and regularization produces smooth results and is thus suboptimal we investigate an l1 norm based approach which yields structural images without the need for explicit regularization the density distribution of the subsurface is modeled with a uniform grid of cells the density of each cell is inverted by minimizing the l1 norm of the data misfit using linear programming lp while satisfying a priori density constraints the estimate of the noise level in a given data set is used to qualitatively determine an appropriate parameterization the 2 5d and 3d synthetic tests adequately reconstruct the structure of the test models the quality of the inversion depends upon a good prior estimation of

testing for nonlinearity in time series the method of surrogate data we describe a statistical approach for identifying nonlinearity in time series the method first specifies some linear process as a null hypothesis then generates surrogate data sets which are consistent with this null hypothesis and finally computes a discriminating statistic for the original and for each of the surrogate data sets if the value computed for the original data is significantly different than the ensemble of values computed for the surrogate data then the null hypothesis is rejected and nonlinearity is detected we discuss various null hypotheses and discriminating statistics the method is demonstrated for numerical data generated by known chaotic systems and applied to a number of experimental time series which arise in the measurement of superfluids brain waves and sunspots we evaluate the statistical significance of the evidence for nonlinear structure in each case and illustrate aspects of the data which this approach identifies

surrogate time series before we apply nonlinear techniques e g those inspired by chaos theory to dynamical phenomena occurring in nature it is necessary to first ask if the use of such advanced techniques is justified by the data while many processes in nature seem very unlikely a priori to be linear the possible nonlinear nature might not be evident in specific aspects of their dynamics the method of surrogate data has become a very popular tool to address such a question however while it was meant to provide a statistically rigorous foolproof framework some limitations and caveats have shown up in its practical use in this paper recent efforts to understand the caveats avoid the pitfalls and to overcome some of the limitations are reviewed and augmented by new material in particular we will discuss specific as well as more general approaches to constrained randomisation providing a full range of examples new algorithms will be introduced for unevenly sampled and multivariate data and for surrogate spike trains the main limitation which lies in the interpretability of the test results will be illustrated through instructive case studies we will also discuss some implementational aspects of the realisation of these methods in the tisean software package

improved surrogate data for nonlinearity tests current tests for nonlinearity compare a time series to the null hypothesis of a gaussian linear stochastic process for this restricted null assumption random surrogates can be constructed which are constrained by the linear properties of the data we propose a more general null hypothesis allowing for nonlinear rescalings of a gaussian linear process we show that such rescalings cannot be accounted for by a simple amplitude adjustment of the surrogates which leads to spurious detection of nonlinearity an iterative algorithm is proposed to make appropriate surrogates which have the same autocorrelations as the data and the same probability distribution

occam s inversion a practical algorithm for generating smooth models from electromagnetic sounding data 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

a rapid method for three dimensional modeling of magnetic anomalies a computer program has been developed for three dimensional analysis of total field magnetic anomalies due to arbitrary polarization suitable for present day personal computers a vertical sided prism with arbitrary polarization is used as a basic model a nonlinear optimization technique based on marquardt s algorithm is used to estimate all parameters of the model a combination of prisms is used to analyze more complex magnetic fields analytical methods are used to estimate the derivatives required in the simultaneous solutions of the normal equations methods have been developed to minimize the computing time in forward calculation as well as in inversion approximate equations have been derived for rapid calculation of magnetic anomalies and partial derivatives of anomalies of prismatic bodies which are valid beyond short distances from the sources the algorithm has been developed in such a way that the use of the exact and approximate equations may be efficiently monitored as a trade off between

resolution and uncertainty in 1d csem inversion a bayesian approach and open source implementation we show that resolution and uncertainty in csem inversion is most naturally approached using a bayesian framework resolution can be inferred by either hierarchical models with free parameters for effective correlation lengths bayesian smoothing or model choice frameworks applied to variable resolution spatial models bayesian splitting merging we find that typical 1d csem data can be modelled with quite parsimonious models typically o 10 parameters per common midpoint efficient opt imizations for the csem problem must address the challenges of poor scaling strong nonlinearity multimodality and the necessity of bound constraints the posterior parameter uncertainties are frequently controlled by the nonlinearity and linearised approaches to uncertainty are usually very poor in markov chain monte carlo mcmc approaches the nonlinearity and poor scaling make good mixing hard to achieve a novel approximate frequentist method we call the bayesianized parametric bootstrap sometimes called randomized maximum likelihood is much more efficient than mcmc in this problem is considerably better than linearized analysis but tends to modestly overstate uncertainties the software that implements these ideas for the 1d csem problem is made available under an open source license agreement

occam s inversion to generate smooth two dimensional models from magnetotelluric data 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

algebraic relations among solutions of linear differential equations using power series methods harris and sibuya 3 4 recently showed that if a is an ordinary differential field of characteristic zero and y 5 0 is an element of a differential extension of fc such that y and l y satisfy linear differential equations with coefficients in fc then y iy is algebraic over fc using differential galois theory we generalize this and characterize those polynomial relations among solutions of linear differential equations that force these solutions to have algebraic logarithmic derivatives we also show that if is an algebraic function of genus 1 and if y and f y or y and ei v satisfy linear differential equations then y is an algebraic function proposition 1 a is obtained by letting n 3 m2 m 1 and yi 1 proposition 1 b is obtained by letting n 2 and m2 m sperber s techniques also allow him to handle solutions of certain nonlinear diferential equations in this paper we prove results that imply