trans dimensional geoacoustic inversion this paper develops a general trans dimensional bayesian methodology for geoacoustic inversion trans dimensional inverse problems are a generalization of fixed dimensional inversion that includes the number and type of model parameters as unknowns in the problem by extending the inversion state space to multiple subspaces of different dimensions the posterior probability density quantifies the state of knowledge regarding inversion parameters including effects due to limited knowledge about appropriate parametrization of the environment and error processes the inversion is implemented here using a reversible jump markov chain monte carlo algorithm and the seabed is parametrized using a partition model unknown data errors are addressed by including a data error model jumps between dimensions are implemented with a birth death methodology that allows transitions between dimensions by adding or removing interfaces while maintaining detailed balance in the markov chain trans dimensional inversion results in an inherently parsimonious solution while partition modeling provides a naturally self regularizing algorithm based on data information content not on subjective regularization functions together this results in environmental estimates that quantify appropriate seabed structure as supported by the data allowing sharp discontinuities while approximating smooth transitions where needed this approach applies generally to geoacoustic inversion and is illustrated here with seabed reflection coefficient data

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

stochastic inversion for electromagnetic geophysics practical challenges and improving convergence efficiency traditional deterministic geophysical inversion algorithms are not designed to provide a robust evaluation of uncertainty that reflects the limitations of the geophysical technique stochastic inversions which do provide a sampling based measure of uncertainty are computationally expensive and not straightforward to implement for nonexperts nonstatisticians our results include stochastic inversion for magnetotelluric and controlled source electromagnetic data two markov chain sampling algorithms metropolis hastings and slice sampler can significantly decrease the computational expense compared to using either sampler alone the statistics of the stochastic inversion allow for 1 variances that better reveal the measurement sensitivities of the two different electromagnetic techniques than traditional techniques and 2 models defined by the median and modes of parameter probability density functions which produce amplitude and phase data that are consistent with the observed data in general pa

hierarchical bayesian inference for ill posed problems via variational method this paper investigates a novel approximate bayesian inference procedure for numerically solving inverse problems a hierarchical formulation which determines automatically the regularization parameter and the noise level together with the inverse solution is adopted the framework is of variational type and it can deliver the inverse solution and regularization parameter together with their uncertainties calibrated it approximates the posteriori probability distribution by separable distributions based on kullback leibler divergence two approximations are derived within the framework and some theoretical properties e g variance estimate and consistency are also provided algorithms for their efficient numerical realization are described and their convergence properties are also discussed extensions to nonquadratic regularization nonlinear forward models are also briefly studied numerical results for linear and nonlinear cauchy type problems arising in heat conduction with both smooth and nonsmooth solutions are presented for the proposed method and compared with that by markov chain monte carlo the results illustrate that the variational method can faithfully capture the posteriori distribution in a computationally efficient way

bayesian evidence computation for model selection in non linear geoacoustic inference problems this paper applies a general bayesian inference approach based on bayesian evidence computation to geoacoustic inversion of interface wave dispersion data quantitative model selection is carried out by computing the evidence normalizing constants for several model parameterizations using annealed importance sampling the resulting posterior probability density estimate is compared to estimates obtained from metropolis hastings sampling to ensure consistent results the approach is applied to invert interface wave dispersion data collected on the scotian shelf off the east coast of canada for the sediment shear wave velocity profile results are consistent with previous work on these data but extend the analysis to a rigorous approach including model selection and uncertainty analysis the results are also consistent with core samples and seismic reflection measurements carried out in the area

level set and total variation regularization for elliptic inverse problems with discontinuous coefficients we propose a level set approach for elliptic inverse problems with piecewise constant coefficients the geometry of the discontinuity of the coefficient is represented implicitly by level set functions the inverse problem is solved using a variational augmented lagrangian formulation with total variation regularization of the coefficient the corresponding euler lagrange equation gives the evolution equation for the level set functions and the constant values of the coefficients we use a multiple level set representation which allows the coefficient to have multiple constant regions knowledge of the exact number of regions is not required only an upper bound is needed numerical experiments show that the method can recover coefficients with rather complicated geometries of discontinuities under moderate amount of noise in the observation data the method is also robust with respect to the initial guess for the geometry of the coefficient discontinuities

multidimensional scaling for the evaluation of a geostatistical seismic elastic inversion methodology due to the nature of seismic inversion problems there are multiple possible solutions that can equally fit the observed seismic data while diverging from the real subsurface model consequently it is important to assess how inverse impedance models are converging toward the real subsurface model for this purpose we evaluated a new methodology to combine the multidimensional scaling mds technique with an iterative geostatistical elastic seismic inversion algorithm the geostatistical inversion algorithm inverted partial angle stacks directly for acoustic and elastic impedance ai and ei models it was based on a genetic algorithm in which the model perturbation at each iteration was performed recurring to stochastic sequential simulation to assess the reliability and convergence of the inverted models at each step the simulated models can be projected in a metric space computed by mds this projection allowed distinguishing similar from variable models and assessing the convergence of inverted models toward the real impedance ones the geostatistical inversion results of a synthetic data set in which the real ai and ei models are known were plotted in this metric space along with the known impedance models we applied the same principle to a real data set using a crossvalidation technique these examples revealed that the mds is a valuable tool to evaluate the convergence of the inverse methodology and the impedance model variability among each iteration of the inversion process particularly for the geostatistical inversion algorithm we evaluated it retrieves reliable impedance models while still producing a set of simulated models with considerable variability

expanded uncertainty quantification in inverse problems hierarchical bayes and empirical bayes a common way to account for uncertainty in inverse problems is to apply bayes rule and obtain a posterior distribution of the quantities of interest given a set of measurements a conventional bayesian treatment however requires assuming specific values for parameters of the prior distribution and of the distribution of the measurement errors e g the standard deviation of the errors in practice these parameters are often poorly known a priori and choosing a particular value is often problematic moreover the posterior uncertainty is computed assuming that these parameters are fixed if they are not well known a priori the posterior uncertainties have dubious value this paper describes extensions to the conventional bayesian treatment that assign uncertainty to the parameters defining the prior distribution and the distribution of the measurement errors these extensions are known in the statistical literature as empirical bayes and hierarchical bayes we demonstrate the practical applicati

high resolution prestack seismic inversion using a hybrid fista least squares strategy abstracta new inversion method to estimate high resolution amplitude versus angle attributes ava attributes such as intercept and gradient from prestack data is presented the proposed technique promotes sparse spike reflectivities that when convolved with the source wavelet fit the observed data the inversion is carried out using a hybrid two step strategy that combines fast iterative shrinkage thresholding algorithm fista and a standard least squares ls inversion fista which can be viewed as an extension of the classical gradient algorithm provides sparse solutions by minimizing the misfit between the modeled and the observed data and the l1 norm of the solution fista is used to estimate the location in time of the main reflectors then ls is used to retrieve the appropriate reflectivity amplitudes that honor the data fista like other iterative solvers for l1 norm regularization does not require matrices in explicit form making it easy to apply economic in computational terms and ad

every real ellipsoid in mathbb c 2 admits cr umbilical points we prove that every real ellipsoid m c c 2 admits at least four umbilical points which can be compared to the result of webster that a generic real ellipsoid in c with n 3 does not admit any umbilical point

markov chain monte carlo convergence diagnostics a comparative review abstract a critical issue for users of markov chain monte carlo mcmc methods in applications is how to determine when it is safe to stop sampling and use the samples to estimate characteristics of the distribution of interest research into methods of computing theoretical convergence bounds holds promise for the future but to date has yielded relatively little of practical use in applied work consequently most mcmc users address the convergence problem by applying diagnostic tools to the output produced by running their samplers after giving a brief overview of the area we provide an expository review of 13 convergence diagnostics describing the theoretical basis and practical implementation of each we then compare their performance in two simple models and conclude that all of the methods can fail to detect the sorts of convergence failure that they were designed to identify we thus recommend a combination of strategies aimed at evaluating and accelerating mcmc sampler convergence including ap

quasi linear geostatistical theory for inversing a quasi linear theory is presented for the geostatistical solution to the inverse problem the archetypal problem is to estimate the log transmissivity function from observations of head and log transmissivity at selected locations the unknown is parameterized as a realization of a random field and the estimation problem is solved in two phases structural analysis where the random field is characterized followed by estimation of the log transmissivity conditional on all observations the proposed method generalizes the linear approach of kitanidis and vomvoris 1983 the generalized method is superior to the linear method in cases of large contrast in formation properties but informative measurements i e there are enough observations that the variance of estimation error of the log transmissivity is small the methodology deals rigorously with unknown drift coefficients and yields estimates of covariance parameters that are unbiased and grid independent the applicability of the methodology is demonstrated through an example that includes structural analysis determination of best estimates and conditional simulations

umbilical points on three dimensional strictly pseudoconvex cr manifolds i manifolds with u 1 action the question of existence of umbilical points in the cr sense on compact three dimensional strictly pseudoconvex cr manifolds was raised in the seminal paper by chern and moser acta math 133 219 271 1974 in the present paper we consider compact three dimensional strictly pseudoconvex cr manifolds that possess a free transverse action by the circle group u 1 we show that every such cr manifold m has at least one orbit of umbilical points provided that the riemann surface x m u 1 is not a torus in particular every compact circular and strictly pseudoconvex hypersurface in mathbb c 2 has at least one circle of umbilical points the existence of umbilical points in the case where x is a torus is left open in general but it is shown that if such an m has additional symmetries in a certain sense then it must possess umbilical points as well

bayesian closed skew gaussian inversion of seismic avo data for elastic material properties bayesian closed skew gaussian inversion is defined as a generalization of traditional bayesian gaussian inversion which is used frequently in seismic amplitude versus offset avo inversion the new model captures skewness in the variables of interest hence the posterior model for log transformed elastic material properties given seismic avo data might be a skew probability density function the model is analytically tractable and this makes it applicable in high dimensional 3d inversion problems assessment of the posterior models in high dimensions requires numerical approximations however the bayesian closed skew gaussian inversion approach has been applied on real elastic material properties from a well in the sleipner field in the north sea a comparison with results from traditional bayesian gaussian inversion shows that the mean square error of predictions of p wave and s wave velocities are reduced by a factor of two although somewhat less for density predictions

simultaneous estimation of transmissivity values and zonation the extended kalmanfilter ekf has long been recognized as a powerful yet computationally intensive methodology for stochastic parameter estimation three improvements to traditional algorithms are presented and applied to heterogeneous transmissivity estimation first the costly ekf covariance updates are replaced by more efficient approximations second the zonation structure of the distributed parameterfield being estimated is dynamically determined and refined using a partitional clustering algorithm third a new method of mergingfirst and second moments of randomfields that have heterogeneous statistics is introduced we apply this method called random field union as an alternative to conventional randomfield averaging for the systematic shrinking of covariance matrices as the dimensionality of the parameter space is reduced the effects of these three improvements are examined in applications to steady state groundwaterflow test problems we show that thefirst and second improvements reduce the computational time requirements dramatically while the second and third can improve the accuracy and stability of the results the resulting integrated method is successfully applied to a larger more realistic calibration test case under steady and cyclostationary flow conditions similar to regular seasonalfluctuations whenflow is steady the method can be viewed as iterative whenflow is transient the method is fully recursive

fast bayesian approach for parameter estimation this paper presents two techniques i e the proper orthogonal decomposition pod and the stochastic collocation method scm for constructing surrogate models to accelerate the bayesian inference approach for parameter estimation problems associated with partial differential equations pod is a model reduction technique that derives reduced order models using an optimal problem adapted basis to effect significant reduction of the problem size and hence computational cost scm is an uncertainty propagation technique that approximates the parameterized solution and reduces further forward solves to function evaluations the utility of the techniques is assessed on the non linear inverse problem of probabilistically calibrating scalar robin coefficients from boundary measurements arising in the quenching process and non destructive evaluation a hierarchical bayesian model that handles flexibly the regularization parameter and the noise level is employed and the posterior state space is explored by the markov chain monte carlo the numerical results indicate that significant computational gains can be realized without sacrificing the accuracy

on the ambiguity of elasticity measurements in layered rocks different boundary conditions applied to finite size heterogeneous rocks result in different average or apparent elastic stiffnesses in statics the apparent stiffness tensors that correspond to the homogeneous stress and homogeneous strain boundary conditions provide the lower and upper bounds for the true or effective stiffness tensor it appears that similar bounds cannot be established for experiments utilizing wave propagation we present static and dynamic computations of the apparent properties of stacks of horizontal isotropic layers the effective stiffness tensors of such media are known to be vertically transversely isotropic vti and are given by backus theory for statics in the limit of zero thicknesses of the constituent layers we verify the theoretical static bounds for these effective stiffnesses with the finite element method the existing stiffness bounds however do not lead to bounds for the effective thomsen anisotropic coefficients in particular we show that any static measure

estimation of aquifer parameters under transient and steady state conditions 1 maximum likelihood method incorporating prior information in this series of three papers a method is presented to estimate the parameters of groundwater flow models under steady and nonsteady state conditions the parameters include values and directions of principal hydraulic conductivities or transmissivities in anisotropic media specific storage or storativity interior and boundary recharge or leakage rates coefficients of head dependent interior and boundary sources and boundary heads in transient situations the initial head distribution can also be estimated if the system is originally at a steady state paper 1 of the series discusses some of the advantage in treating the inverse problem statistically and in regularizing its solution by means of penalty criteria based on prior estimates of the parameters the inverse problem is posed in the framework of maximum likelihood theory cast in a manner that accounts for prior information about the parameters since not all the factors which contribute to the prior errors can be quantified statistically at the outset the covariance matrices of these errors are expressed in terms of several parameters which if unknown can be estimated jointly with the hydraulic parameters by a stagewise optimization process when transient head data are separated by a fixed time interval the temporal structure of these data is approximated by a lag one autoregressive model with a correlation coefficient that can be treated as another unknown parameter estimation errors are analyzed by examining the lower bound of their covariance matrix in the eigenspace paper 1 concludes by suggesting that certain model identification criteria developed in the time series literature all of which are based on the maximum likelihood concept might be useful for selecting the best groundwater model or the best method of parameterizing a particular model among a number of given alternatives

target oriented linear least squares and nonlinear trust region newton inversions of plane waves using ava and pva data for elastic model parameters abstractvariations in elastic properties directly impact observed seismic amplitude and phase as a function of incident angle amplitude variation with angle ava and with phase pva conventional simultaneous prestack inversion uses precritical pp reflections or joint pp and ps reflections data to invert for elastic properties via linearized least squares ls methods incorporating measured phase information as additional constraints for joint pp ps and joint ava pva inversions helps to stabilize the convergence path and improves the accuracy of the inverted parameters combining these dual joint constraints in a nonlinear trust region reflective newton trn solver produces a robust inversion with improved convergence rates partial derivatives for the jacobian and hessian matrices are computed numerically two weighting functions one for pp ps and one for ava pva data are developed for minimizing the global objective function during inversion numerical results for data from a salt interface r

characterization and identification of aquifer heterogeneity with generalized parameterization and bayesian estimation 1 this research proposes a generalized parameterization method referred to as gp and bayesian estimation for parameter heterogeneity characterization and identification in groundwater modeling gp unifies zonation and interpolation through a set of weighting coefficients and is capable of creating a zonation structure a continuous distribution or a mixed structure gp shows greater flexibility not only in manipulating the highly complex spatial distribution but also in identifying the parameter structure with gp parameter structure identification seeks to identify the parameter dimension parameter pattern parameter values as well as the values of the weighting coefficients simultaneously through a set of basis points additionally this study develops an embedded genetic algorithm ga for solving the structure identification problem a bayesian estimator that estimates the basis point values as well as the values of the weighing coefficients is embedded in the ga which searches for the best basis point locations we demonstrate the inverse methodology by a numerical example in which the distributed transmissivity in a two dimensional confined aquifer is identified we calculate the jacobian matrix by the adjoint state method with gp we have successfully identified the transmissivity structure with four basis points that results in a good fitting in groundwater heads and captures the nonsmooth characteristic as well as the trend of the true transmissivity field we compare gp with voronoi tessellation zonation and natural neighbor interpolation results show that gp outperforms the other two parameterization methods in that gp identified the transmissivity field with a smaller parameter uncertainty along with a sufficiently small fitting residual and without over parameterization

bayesian linearized rock physics inversion abstractthe estimation of rock and fluid properties from seismic attributes is an inverse problem rock physics modeling provides physical relations to link elastic and petrophysical variables most of these models are nonlinear therefore the inversion generally requires complex iterative optimization algorithms to estimate the reservoir model of petrophysical properties we have developed a new approach based on the linearization of the rock physics forward model using first order taylor series approximations the mathematical method adopted for the inversion is the bayesian approach previously applied successfully to amplitude variation with offset linearized inversion we developed the analytical formulation of the linearized rock physics relations for three different models empirical granular media and inclusion models and we derived the formulation of the bayesian rock physics inversion under gaussian assumptions for the prior distribution of the model the application of the inversion to real

prediction of reservoir variables based on seismic data and well observations offshore petroleum reservoirs are usually explored by acquisition of vast amounts of seismic reflection data and observations along a small number of wells drilled through the reservoirs the seismic data are indirect measurements of reservoir characteristics and have good spatial coverage but low precision the well observations are sparse but precise information from these sources of data are integrated to characterize the reservoir variables of interest reservoir porosity in the current study critical parameters in the acquisition procedure are unknown and are simultaneously estimated this process termed seismic inversion constitutes a spatial multivariate ill posed inverse problem the problem is cast in a bayesian framework with a focus on sampling from the posterior model high dimensionality nonlinear components and unknown parameters in the likelihood model along with a complex design of the conditioning data complicate the problem a fast sequential sampling algorithm containing severa

effective media a forward modeling view the existing effective media theories such as backus averaging can be only used in media that possess certain characteristics regarding to concentration shape or geometry of their heterogeneities these limitations originate from necessity of having analytical description of complex stress and strain fields that normally arise in microheterogeneous solids the need for explicit solutions can be eliminated by computing the stresses and strains numerically as a result effective media can be in principle constructed for solids of arbitrary complexity this simple idea is tested on two 2d models where conventional analytical effective media theories are likely to break down the first model is an isotropic layered solid with cracks that intersect the layer interfaces the second is a layered medium containing random inclusions in both cases some differences are observed between the effective stiffness coefficients obtained numerically and those derived using the existing effective media theories for

bayesian inversion for facies detection an extensible level set framework 1 in many cases it has been assumed that the variability in hydrologic parameters can be adequately described through a simple geostatistical model with a given variogram in other cases variability may be best described as a series of jumps in parameter behavior e g those that occur at geologic facies contacts when using indirect measurements such as pump tests to try to map such heterogeneity during inverse modeling the resulting images of the subsurface are always affected by the assumptions invoked in this paper we discuss inversion for parameter fields where prior information has suggested that major variability can be described by boundaries between geologic units or facies in order to identify such parameter fields we propose a bayesian level set inversion protocol framework which allows for flexible zones of any shape size and number we review formulas for defining facies locations using the level set method and for moving the boundaries between zones using a gradient based technique that improves fit through iterative deformation of the boundaries we describe the optimization algorithm employed when multiple level set functions are used to represent a field with more than two facies we extend these formulas to the investigation of the inverse problem in a bayesian context in which prior information is taken into account and through which measures of uncertainty can be derived we also demonstrate that the level set method is well suited for joint inversion problems and present a strategy for integrating different data types such as hydrologic and geophysical without assuming strict petrophysical relations our framework for joint inversion also contrasts with many previous methods in that all data sources e g both hydrologic and geophysical contribute to boundary delineation at once

a geostatistical approach to the inverse problem in groundwater modeling steady state and one dimensional simulations the problem of estimating hydrogeologic parameters in particular permeability from input output measurements is reexamined in a geostatistical framework the field of the unknown parameters is represented as a random field and the estimation procedure consists of two main steps first the structure of the parameter field is identified i e mathematical representations of the variogram and the trend are selected and their parameters are established by using all available information including measurements of hydraulic head and permeability second linear estimation theory is applied to provide minimum variance and unbiased point estimates of hydrogeologic parameters kriging structure identification is achieved iteratively in three substeps structure selection maximum likelihood estimation and model validation and diagnostic checking the methodology was extensively tested through simulations on a simple one dimensional case the results are remarkably stable and well behaved the estimated field is smooth while small scale variability is statistically described as the quality of measurements improves the procedure reproduces more features of the original field the results are also shown to be rather insensitive to deviations from assumptions about the geostatistical structure of the field

a numerical homogenization method for heterogeneous anisotropic elastic media based on multiscale theory the development of reliable methods for upscaling finescale models of elastic media has long been an important topic for rock physics and applied seismology several effective medium theories have been developed to provide elastic parameters for materials such as finely layered media or randomly oriented or aligned fractures in such cases the analytic solutions for upscaled properties can be used for accurate prediction of wave propagation however such theories cannot be applied directly to homogenize elastic media with more complex arbitrary spatial heterogeneity therefore we have proposed a numerical homogenization algorithm based on multiscale finite element methods for simulating elastic wave propagation in heterogeneous anisotropic elastic media specifically our method used multiscale basis functions obtained from a local linear elasticity problem with appropriately defined boundary conditions homogenized effective medium parameters were then computed using these basis functions and the approach applied a numerical discretization that was similar to the rotated staggered grid finite difference scheme comparisons of the results from our method and from conventional analytical approaches for finely layered media showed that the homogenization reliably estimated elastic parameters for this simple geometry additional tests examined anisotropic models with arbitrary spatial heterogeneity in which the average size of the heterogeneities ranged from several centimeters to several meters and the ratio between the dominant wavelength and the average size of the arbitrary heterogeneities ranged from 10 to 100 comparisons to finite difference simulations proved that the numerical homogenization was equally accurate for these complex cases

on the riemann mapping theorem we prove a generalization of the riemann mapping theorem if a bounded simply connected domain q with connected smooth boundary has the spherical boundary then it is biholomorphic to the unit ball

an iterative geostatistical inverse method for steady flow in the vadose zone an iterative geostatistical inverse approach is developed to estimate conditional effective unsaturated hydraulic conductivity parameters soil water pressure head and degree of saturation in heterogeneous vadose zones this approach is similar to the classical cokriging technique and it uses a linear estimator that depends on covariances and cross covariances of unsaturated hydraulic parameters soil water pressure head and degree of saturation the linear estimator is however improved successively by solving the governing flow equation and by updating the residual covariance and cross covariance functions in an iterative manner as a result the nonlinear relationship between unsaturated hydraulic conductivity parameters and head is incorporated in the estimation and the estimated fields are approximate conditional means the ability of the iterative approach is demonstrated through some numerical examples

velocity log upscaling based on reversible jump markov chain monte carlo simulated annealing abstractupscaling log measurements acquired at high frequencies and correlating them with corresponding low frequency values from surface seismic and vertical seismic profile data is a challenging task we have applied a sampling technique called the reversible jump markov chain monte carlo rjmcmc method to this problem a key property of our approach is that it treats the number of unknowns itself as a parameter to be determined specifically we have considered upscaling as an inverse problem in which we considered the number of coarse layers layer boundary depths and material properties as the unknowns the method applies bayesian inversion with rjmcmc sampling and uses simulated annealing to guide the optimization at each iteration the algorithm will randomly move a boundary in the current model add a new boundary or delete an existing boundary in each case a random perturbation is applied to backus average values we have developed examples showing that the mismatch between seismograms com

simultaneous joint inversion of amplitude versus offset and controlled source electromagnetic data by implicit representation of common parameter structure abstracti evaluated a simultaneous joint inversion of seismic amplitude versus offset avo and controlled source electromagnetic csem data for fluid flow monitoring a new approach for structure coupled joint inversion was presented in which the coupling of the two data types was obtained by allowing for the direct identification of parameter structure that is shared by the different geophysical model parameters the main idea was to use a composite parameter representation which enables inversion with respect to the parameter magnitude and parameter structure in the current application parameter structure refers to transitions between dominating property values and is represented by the position and shape of the flooding front hence with this approach the position and shape of the flooding front are inverted for directly and the coupling between the different data sets is obtained without the inclusion of an additional penalizing term in the objective function regularization of the inverse pr

preperiodic points of polynomials over global fields given a global field k and a polynomial defined over k of degree at least two morton and silverman conjectured in 1994 that the number of k rational preperiodic points of is bounded in terms of only the degree of k and the degree of in 1997 for quadratic polynomials over k q call and goldstine proved a bound which was exponential in s the number of primes of bad reduction of by careful analysis of the filled julia sets at each prime we present an improved bound on the order of s log s our bound applies to polynomials of any degree at least two over any global field k let k be a field and let k z be a rational function let denote the n iterate of under composition that is is the identity function and for n 1 n 1 we will study the dynamics on the projective line p k in particular we say a point x is preperiodic under if there are integers n m 0 such that x x the point y x satisfies n m y y and is said to be periodic of period n m note that x p k is preperiodic if and only if its orbit x n 0 is finite for example let k q and z z 29 16 then 5 4 1 4 7 4 forms a periodic cycle of period 3 and 5 4 1 4 7 4 and 3 4 each land on this cycle after one or two iterations in addition the point is of course fixed these nine q rational points are all preperiodic under meanwhile it is not difficult to see that no other point in p q is preperiodic by showing that the denominator of a rational preperiodic point must be 4 and that the absolute value must be less than 2 in general for any global field k any dimension n 1 and any morphism p p over k of degree at least two northcott proved in 1950 that the number of k rational preperiodic points of is finite 25 more precisely he showed that the preperiodic points form a set of bounded arithmetic height years later by analogy with the theorems of mazur 19 and merel 20 on k rational torsion of elliptic curves morton and silverman proposed the following conjecture 23 uniform boundedness conjecture morton and silverman 1994 given integers d n 1 and d 2 there is a constant d n d with the following property let k be a number field with k q d and let p p be a morphism of degree d defined over k then has at most preperiodic points in p k the analogy between preperiodic points and torsion comes from the fact that the torsion points of an elliptic curve e are precisely the preperiodic points of the multiplication bytwo map 2 e e in fact taking x coordinates the map 2 induces a rational function known to dynamicists as a lattes map p p of degree 4 whose preperiodic points date june 21 2005 revised december 21 2005 2000 mathematics subject classification primary 11g99 secondary 11d45 37f10

aquifer parameter identification with optimum dimension in parameterization this paper presents a systematic procedure whereby the functional coefficients imbedded in a two dimensional partial differential equation which governs unsteady groundwater flow are optimally identified the coefficients to be identified are transmissivities which vary spatially finite elements are used to represent the unknown transmissivity function parametrically in terms of nodal values over a suitable discretization of a flow region a modified gauss newton algorithm is used for parameter optimization covariance analysis is used to estimate the reliability of the estimated parameters as the dimension of the unknown parameter increases the modeling error represented by a least squares criterion will generally decrease but errors in data would be propagated to a greater degree into the estimated parameters thus reducing the reliability of estimation the reliability of the estimated parameters is characterized by a norm of the covariance matrix this information is used for the determination of the optimum dimension in parameterization

global local optimization for parameter structure identification in three dimensional groundwater modeling 1 this paper develops two global local optimization methods for three dimensional parameter structure identification in groundwater modeling parameter structure identification is formulated in terms of solving a generalized inverse problem which allows for a determination of an appropriate level of parameter structure complexity and the identification of its pattern and the associated parameter values the structure complexity is determined by calculating the parameter structure error measured in the prediction space or management space while the structure pattern and the associated parameter values are identified simultaneously by minimizing the fitting residual measured in the observation space the former requires solving a continuous maximum minimum max min optimization problem and the latter requires solving a combinatorial optimization problem in this paper voronoi zonation is used to parameterize the unknown distributed parameter for each given level of structure complexity a sequential global local optimization method which consists of a genetic algorithm a quasi newton method and local search is developed to solve the combinatorial problem in addition the continuous max min problem is transformed to a hierarchical optimization problem and solved by a hybrid global local method sensitivities of state variables to the unknown parameters are calculated by the sensitivity equations the validity and applicability of the proposed methodology are demonstrated by numerical experiments we have shown that the choice of an objective function in model application impacts the determination of the parameter structure complexity

a comparison between gauss newton and markov chain monte carlo based methods for inverting spectral induced polarization data for cole cole parameters we have developed a bayesian model to invert spectral induced polarization sip data for cole cole parameters using markov chain monte carlo mcmc sampling methods we comparedtheperformanceofthemcmc basedstochasticmethod with an iterative gauss newton based deterministic method for cole cole parameter estimation through inversion of syntheticandlaboratorysipdata thegauss newton basedmethod canprovideanoptimalsolutionforgivenobjectivefunctionsunder constraints but the obtained optimal solution generally dependsonthechoiceofinitialvaluesandtheestimateduncertainty information often is inaccurate or insufficient in contrast the mcmc based inversion method provides extensive global informationonunknownparameters suchasthemarginalprobability distribution functions from which we can obtain better estimates and tighter uncertainty bounds of the parameters than with the deterministic method in addition the results obtained with the mcmc method are independent of the choice of initial values becausethemcmc basedmethoddoesnotexplicitlyofferasingleoptimalsolutionforgivenobjectivefunctions thedeterministic and stochastic methods can complement each other forexample thestochasticmethodcanbeusedfirsttoobtainthe mediansofunknownparametersbystartingfromanarbitraryset of initial values the deterministic method then can be initiated using the medians as starting values to obtain the optimal estimatesofthecole coleparameters

transdimensional seismic inversion using the reversible jump hamiltonian monte carlo algorithm abstractprestack or angle stack gathers are inverted to estimate pseudologs at every surface location for building reservoir models recently several methods have been proposed to increase the resolution of the inverted models all of these methods however require that the total number of model parameters be fixed a priori we have investigated an alternate approach in which we allow the data themselves to choose model parameterization in other words in addition to the layer properties the number of layers is also treated as a variable in our formulation such transdimensional inverse problems are generally solved by using the reversible jump markov chain monte carlo rjmcmc approach which is a tool for model exploration and uncertainty quantification this method however has very low acceptance we have developed a two step method by combining rjmcmc with a fixed dimensional mcmc called hamiltonian monte carlo which makes use of gradient information to take large steps acceptance probability

the selection of prior distributions by formal rules abstract subjectivism has become the dominant philosophical foundation for bayesian inference yet in practice most bayesian analyses are performed with so called noninformative priors that is priors constructed by some formal rule we review the plethora of techniques for constructing such priors and discuss some of the practical and philosophical issues that arise when they are used we give special emphasis to jeffreys s rules and discuss the evolution of his viewpoint about the interpretation of priors away from unique representation of ignorance toward the notion that they should be chosen by convention we conclude that the problems raised by the research on priors chosen by formal rules are serious and may not be dismissed lightly when sample sizes are small relative to the number of parameters being estimated it is dangerous to put faith in any default solution but when asymptotics take over jeffreys s rules and their variants remain reasonable choices we also provide an annotated b

a calculus for finely layered anisotropic media matrix algebra and group theory combine to offer a formalism for the simple calculation of the elastic anisotropic homogeneous medium which is equivalent in the long wavelength limit to a heterogeneous distribution of fine layers each layer itself an elastic anisotropic medium the properties of each anisotropic constituent in a set of fine layers map to an element of a commutative group a reverse mapping returns the material properties of the constituent adding group elements gives the group element for the homogeneous medium equivalent to a heterogeneous set of layers addition of an inverse element that is subtraction provides the means to remove a set of layers from an anisotropic medium then if the remaining layer is a stable anisotropic medium a valid decomposition of the original medium into anisotropic constituents is obtained within the group structure eight subgroups corresponding to eight types of elastic symmetry systems may be identified immediately yielding the symmetry of the e

a multiresolution method for distributed parameter estimation a multiresolution method for distributed parameter estimation or inverse problems is studied numerically the identification of the coefficient of an elliptic equation in one dimension is considered as our model problem first multiscale bases are used to analyze the degree of ill posedness of the inverse problem second based on some numerical results it is shown that the method of scale by scale multiresolution yields robust and fast convergence finally it is shown how the method gives a natural regularization approach which is complementary to tikhonov s regularization

bayesian linearized avo inversion a new linearized avo inversion technique is developed in a bayesian framework the objective is to obtain posterior distributions for p wave velocity s wave velocity and density distributions for other elastic parameters can also be assessed for example acoustic impedance shear impedance and p wave to s wave velocity ratio the inversion algorithm is based on the convolutional model and a linearized weak contrast approximation of the zoeppritz equation the solution is represented by a gaussian posterior distribution with explicit expressions for the posterior expectation and covariance hence exact prediction intervals for the inverted parameters can be computed under the specified model the explicit analytical form of the posterior distribution provides a computationally fast inversion method tests on synthetic data show that all inverted parameters were almost perfectly retrieved when the noise approached zero with realistic noise levels acoustic impedance was the best determined parameter

linear equations in variables which lie in a multiplicative group let k be a field of characteristic 0 and let n be a natural number let r be a subgroup of the multiplicative group k n of finite rank r given a 1 a n e k write a a 1 a n for the number of solutions x x 1 x n of the equation a 1 x 1 a n x n 1 such that no proper subsum of a 1 x 1 a n x n vanishes we derive an explicit upper bound for a a 1 a n r which depends only on the dimension n and on the rank r

the number of solutions of decomposable form equations has only finitely many solutions the diophantine approximation techniques of thue and mahler and improvements by siegel dyson roth and bombieri made it possible to derive good explicit upper bounds for the number of solutions of 1 1 the best such upper bound to date due to bombieri 1 is 2 12r bombieri assumed that f is irreducible and r 6 which was not essential in his proof for t 0 i e f x y 1 in x y z bombieri and schmidt 2 derived the upper bound constant r which is best possible in terms of r

reservoir parameter identification using minimum relative entropy based bayesian inversion of seismic ava and marine csem data a stochastic joint inversion approach for estimating reservoir fluid saturations and porosity is proposed the approach couples seismic amplitude variation with angle ava and marine controlled source electromagnetic csem forward models into a bayesian framework which allows for integration of complementary information to obtain minimally subjective prior probabilities required for the bayesian approach the principle of minimum relative entropy mre is employed instead of single value estimates provided by deterministic methods the approach gives a probability distribution for any unknown parameter of interest such as reservoir fluid saturations or porosity at various locations the distribution means modes and confidence intervals can be calculated providing a more complete understanding of the uncertainty in the parameter estimates the approach is demonstrated using synthetic and field data sets results show that joint inversion using seismic and em data gives better estimates of reservoir parameters than estimates from either geophysical data set used in isolation moreover a more informative prior leads to much narrower predictive intervals of the target parameters with mean values of the posterior distributions closer to logged values

two ways to quantify uncertainty in geophysical inverse problems we present two approaches to invert geophysical measurements and estimate subsurface properties and their uncertainties when little is known a priori about the size of the errors associated with the data we illustrate these approaches by inverting first arrival traveltimes of seismic waves measured in a vertical well to infer the variation of compressional slowness in depth first we describe a bayesian formulation based on probability distributions that define prior knowledge about the slowness and the data errors we use an empirical bayes approach where hyperparameters are not well known ahead of time e g the variance of the data errors and are estimated from their most probable value given the data the second approach is a non bayesian formulation that we call spectral in the sense that it uses the power spectral density of the traveltime data to constrain the inversion e g to estimate the variance of the data errors in the spectral approach we vary assumptions made about the characteristics of the slowness signal and evaluate the resulting slowness estimates and their uncertainties this approach is computationally simple and starts from a few assumptions these assumptions can be checked during the analysis on the other hand it requires evenly spaced traveltime measurements and it cannot be extended easily e g to data that have gaps in contrast the bayesian framework is based on a general theory that can be generalized immediately but it is more involved computationally despite the conceptual and practical differences we find that the two approaches give the same results when they start from the same assumptions the allegiance to a bayesian or non bayesian formulation matters less than what one is willing to assume when solving the inverse problem

the influence of crack shape on the effective elasticity of fractured rocks a circle is the basic fracture shape adopted by conventional effective media theories to describe the overall elasticity of cracked solids fractures in rocks do not resemble circles so it is important to find out to what extent the available theoretical results apply to realistic fracture shapes to address this issue we conduct 3d numerical experiments on the so called digital rocks containing irregular cracks that might be partially closed and might intersect each other despite profound deviations of our fracture geometries from circles we find that basic theoretical results originally developed for penny shaped cracks remain valid for arbitrary fracture shapes based on a series of finite element computations we show that as far as the effective elasticity is concerned flat fractures with random in plane irregularities are represented accurately by circular ones we also show that approximate effective elliptical orthotropy established for multiple sets of dry penny shaped cracks embedded in is

amplitude variation with offset inversion using the reflectivity method abstractmost existing amplitude variation with offset avo inversion methods are based on the zoeppritz s equation or its approximations these methods assume that the amplitude of seismic data depends only on the reflection coefficients which means that the wave propagation effects such as geometric spreading attenuation transmission loss and multiples have been fully corrected or attenuated before inversion however these requirements are very strict and can hardly be satisfied under a 1d assumption reflectivity method based inversions are able to handle transmission losses and internal multiples applications of these inversions however are still time consuming and complex in computation of differential seismograms we have evaluated an inversion methodology based on the vectorized reflectivity method in which the differential seismograms can be calculated from analytical expressions it is computationally efficient a modification is implemented to transform the inversion from the interce

a bayesian inference approach to identify a robin coefficient in one dimensional parabolic problems this paper investigates a nonlinear inverse problem associated with the heat conduction problem of identifying a robin coefficient from boundary temperature measurement a bayesian inference approach is presented for the solution of this problem the prior modeling is achieved via the markov random field mrf the use of a hierarchical bayesian method for automatic selection of the regularization parameter in the function estimation inverse problem is discussed the markov chain monte carlo mcmc algorithm is used to explore the posterior state space numerical results indicate that mrf provides an effective prior regularization and the bayesian inference approach can provide accurate estimates as well as uncertainty quantification to the solution of the inverse problem

prior information and uncertainty in inverse problems solving any inverse problem requires understanding the uncertainties in the data to know what it means to fit the data we also need methods to incorporate data independent prior information to eliminate unreasonable models that fit the data both of these issues involve subtle choices that may significantly influence the results of inverse calculations the specification of prior information is especially controversial how does one quantify information what does it mean to know something about a parameter a priori in this tutorial we discuss bayesian and frequentist methodologies that can be used to incorporate information into inverse calculations in particular we show that apparently conservative bayesian choices such as representing interval constraints by uniform probabilities as is commonly done when using genetic algorithms for example may lead to artificially small uncertainties we also describe tools from statistical decision theory that can be used to characterize the performance of inv

identifying arbitrary parameter zonation using multiple level set functions abstract in this paper we extended the analytical level set method 1 2 for identifying a piece wisely heterogeneous zonation binary system to the case with an arbitrary number of materials with unknown material properties in the developed level set approach starting from an initial guess the material interfaces are propagated through iterations such that the residuals between the simulated and observed state variables hydraulic head is minimized we derived an expression for the propagation velocity of the interface between any two materials which is related to the permeability contrast between the materials on two sides of the interface the sensitivity of the head to permeability and the head residual we also formulated an expression for updating the permeability of all materials which is consistent with the steepest descent of the objective function the developed approach has been demonstrated through many examples ranging from totally synthetic cases to a case where the flow conditions are representative of a groundwater contaminant site at the los alamos national laboratory these examples indicate that the level set method can successfully identify zonation structures even if the number of materials in the model domain is not exactly known in advance although the evolution of the material zonation depends on the initial guess field inverse modeling runs starting with different initial guesses fields may converge to the similar final zonation structure these examples also suggest that identifying interfaces of spatially distributed heterogeneities is more important than estimating their permeability values

a note on higher order corrections of the head covariances in steady aquifer flow average uniform flow takes place in a heterogeneous aquifer of infinite extent the input to the problem is the hydraulic conductivity which is regarded as a random space function that is lognormal stationary and statistically isotropic the output is the water head field which is a random function satisfying the equation of steady flow the first order approximation of the head in an asymptotic expansion for small log conductivity variance is a normal function characterized completely by the head log conductivity cross covariance and the head covariance or variogram these covariances are proportional to the log conductivity variance by using spectral methods second order corrections of the head covariances proportional to the log conductivity variance squared are derived explicitly detailed calculations are carried out for an exponential log conductivity covariance the main finding of the note is that the first order approximation is very robust and even for a log conductivity variance equal to unity the second order correction of the head variances is smaller than 10 of the first order approximation

electrical impedance tomography using level set representation and total variational regularization in this paper we propose a numerical scheme for the identification of piecewise constant conductivity coefficient for a problem arising from electrical impedance tomography the key feature of the scheme is the use of level set method for the representation of interface between domains with different values of coefficients numerical tests show that our method can recover sharp interfaces and can tolerate a relatively high level of noise in the observation data results concerning the effects of number of measurements noise level in the data as well as the regularization parameters on the accuracy of the scheme are also given

iterative estimation of reflectivity and image texture least squares migration with an empirical bayes approach in many geophysical inverse problems smoothness assumptions on the underlying geology are used to mitigate the effects of nonuniqueness poor data coverage and noise in the data and to improve the quality of the inferred model parameters within a bayesian inference framework a priori assumptions about the probabilistic structure of the model parameters can impose such a smoothness constraint analogous to regularization in a deterministic inverse problem we have considered anempirical bayes generalization of the kirchhoff based least squares migration lsm problem we have developed a novel methodology for estimation ofthe reflectivitymodel andregularization parameters using a bayesian statistical framework that treats both of these as random variables to be inferred from the data hence rather than fixing the regularization parameters prior to inverting for the image we allow the data to dictate where to regularize estimating these regularization parameters givesusinformation about the degree of conditional correlation or lack thereof between neighboring image parameters and subsequently incorporating this information in the final model produces more clearly visible discontinuities in the estimated image the inference framework is verified on 2d synthetic data sets in which the empirical bayes imaging results significantly outperform standard lsm images we note that although we evaluated this method within the context of seismic imaging it is in fact a general methodology that can be applied to any linear inverse problem in which there are spatially varying correlations in the model parameter space

identification of parameter structure in groundwater inverse problem this paper develops an optimization method that can be used to identify parameter structure as well as its values in connection with the inverse problem in groundwater modeling the complexity of the parameter structure is determined by the quantity and quality of observation data output least square error is used as the criterion of identification sensitivity coefficients at finite element nodes are evaluated by the variational method the problem of identifying the parameter structure is formulated as a problem of combinatorial optimization a new parameterization technique is presented for describing the parameter structure it can represent various distributions of parameters from piecewise constant to continuous variability based on this representation the combinatorial optimization problem for determining the parameter structure is then solved by a simple one at a time searching procedure numerical experimentations are used to demonstrate the efficiency and practicality of the proposed method

amplitude variation with angle inversion using the exact zoeppritz equations theory and methodology to overcome the weaknesses of conventional prestack amplitude variation with angle inversion based on various linear or quasi linear approximations we have conducted a nonlinear inversion method using the exact zoeppritz matrix ezai however the inversion using the exact zoeppritz matrix was highly nonlinear and often unstable if not properly treated to tackle these issues we have used an iteratively regularizing levenberg marquardt scheme irlm which regularizes the inversion problem within an algorithm that minimizes the misfit between the observed and the modeled data at the same time by incorporating the tikhonov regularization method as a result the new ezai method solved using the irlm scheme is feasible for seismic data sets with large incidence angles even up to or beyond the critical angle as well as strong parameter contrasts single and multilayered synthetic examples were used to test these features these tests also showed that ezai is robust on noisy gathers for parameter extraction and has weak dependence on the initial model for the influence of inaccurate amplitudes dominant frequencies and phase angles we found that ezai is less sensitive to the variation in amplitude and phase shifts than to the dominant frequencies specifically the inversion results of ezai for p and s wave velocities and density were reliable if the inaccurate range for the amplitude was within 20 or the angle of the phase shift was no more than 20 the superiority of ezai makes it a very promising method for the estimation of subsurface elastic parameters