models for disc infiltrometers disc infiltrometers are popular devices for determining in situ hydraulic properties of unsaturated soils this paper compares alternative solutions of richards equation for both steady state and time dependent cases the steady state solutions were in general agreement using alternative hydraulic conductivity functions of the same capillary length scale small time solutions for the nonlinear cases were consistent with linear diffusion from a disc source using an average diffusivity value which is simply related to the capillary length this offers a refinement over the one dimensional solution for short times in that the geometric effect of the circular source is included simulations for two examples indicate that the approach to the steady state solution may take considerably longer than what is commonly reported in the literature for field applications

parameter estimation of delay dynamical system from a scalar time series under external noise we introduce an adaptive learning rules for estimating all unknown parameters of delay dynamical system using a scalar time series sufficient condition for synchronization is derived using krasovskii lyapunov theory this scheme is highly applicable in secure communication since multiple messages can be transmitted through multiple parameter modulations one of the advantage of this method is that parameter estimation is also possible even when only one time series of the transmitter is available we present numerical examples for mackey glass system with periodic delay time which are used to illustrate the validity of this scheme the corresponding numerical results and the effect of external noise are also studied

spatiospectral concentration on a sphere we pose and solve the analogue of slepian s time frequency concentration problem on the surface of the unit sphere to determine an orthogonal family of strictly bandlimited functions that are optimally concentrated within a closed region of the sphere or alternatively of strictly spacelimited functions that are optimally concentrated in the spherical harmonic domain such a basis of simultaneously spatially and spectrally concentrated functions should be a useful data analysis and representation tool in a variety of geophysical and planetary applications as well as in medical imaging computer science cosmology and numerical analysis the spherical slepian functions can be found by solving either an algebraic eigenvalue problem in the spectral domain or a fredholm integral equation in the spatial domain the associated eigenvalues are a measure of the spatiospectral concentration when the concentration region is an axisymmetric polar cap the spatiospectral projection operator commutes with a sturm liouville operator this enables the eigenfunctions to be computed extremely accurately and efficiently even when their area bandwidth product or shannon number is large in the asymptotic limit of a small spatial region and a large spherical harmonic bandwidth the spherical concentration problem reduces to its planar equivalent which exhibits self similarity when the shannon number is kept invariant

a general mass conservative numerical solution for the unsaturated flow equation numerical approximations based on different forms of the governing partial differential equation can lead to significantly different results for unsaturated flow problems numerical solution based on the standard h based form of richards equation generally yields poor results characterized by large mass balance errors and erroneous estimates of infiltration depth conversely numerical solutions based on the mixed form of richards equation can be shown to possess the conservative property so that mass is perfectly conserved this leads to significant improvement in numerical solution performance while requiring no additional computational effort however use of the mass conservative method does not guarantee good solutions accurate solution of the unsaturated flow equation also requires use of a diagonal time or mass matrix only when diagonal time matrices are used can the solution be shown to obey a maximum principle which guarantees smooth nonoscillatory infiltration profiles this highlights the fact that proper treatment of the time derivative is critical in the numerical solution of unsaturated flow

dolph chebyshev beampattern design for spherical arrays spherical arrays with array processing based on spherical harmonics have been recently studied for a wide range of applications that require three dimensional beamforming in this correspondence a dolph chebychev beampattern design widely used in array processing due to the direct control over main lobe width and maximum sidelobe level is developed for spherical arrays within the spherical harmonics framework we show that due to the similarity between the legendre polynomials that define the spherical array beampattern and the chebyshev polynomials that define the desired dolph chebyshev beampattern spherical array weights can be computed directly and accurately given desired main lobe width or sidelobe level

slepian spatial spectral concentration problem on the sphere analytical formulation for limited colatitude longitude spatial region in this paper we develop an analytical formulation for the slepian spatial spectral concentration problem on the sphere for a limited colatitude longitude spatial region on the sphere defined as the cartesian product of a range of positive colatitudes and longitudes the solution of the slepian problem is a set of functions that are optimally concentrated and orthogonal within a spatial or spectral region these properties make them useful for applications where measurements are taken within a spatially limited region of the sphere and or a signal is only to be analyzed within a region of the sphere to support localized spectral spatial analysis and estimation and sparse representation of localized data in these applications we exploit the expansion of spherical harmonics in the complex exponential basis to develop an analytical formulation for the slepian concentration problem for a limited colatitude longitude spatial region we also extend the analytical formulation for spatial regions that are comprised of a union of rotated limited colatitude longitude subregions by exploiting various symmetries of the proposed formulation we design a computationally efficient algorithm for the implementation of the proposed analytical formulation such a reduction in computation time is demonstrated through numerical experiments we present illustrations of our results with the help of numerical examples and show that the representation of a spatially concentrated signal is indeed sparse in the slepian basis

markov chain monte carlo methods for high dimensional inversion in remote sensing summary we discuss the inversion of the gas profiles ozone no3 no2 aerosols and neutral density in the upper atmosphere from the spectral occultation measurements the data are produced by the global ozone monitoring of occultation of stars instrument on board the envisat satellite that was launched in march 2002 the instrument measures the attenuation of light spectra at various horizontal paths from about 100 km down to 10 20 km the new feature is that these data allow the inversion of the gas concentration height profiles a short introduction is given to the present operational data management procedure with examples of the first real data inversion several solution options for a more comprehensive statistical inversion are presented a direct inversion leads to a non linear model with hundreds of parameters to be estimated the problem is solved with an adaptive single step markov chain monte carlo algorithm another approach is to divide the problem into several non linear smaller dimensional problems to run parallel adaptive markov chain monte carlo chains for them and to solve the gas profiles in repetitive linear steps the effect of grid size is discussed and we present how the prior regularization takes the grid size into account in a way that effectively leads to a grid independent inversion

estimating unsaturated soil hydraulic properties from tension disc infiltrometer data by numerical inversion tension disc infiltrometers are becoming increasingly popular devices for in situ measurement of the unsaturated hydraulic properties of soil tension infiltration data are generally used to evaluate the parametersks and ain gardner s exponential model of the unsaturated hydraulic conductivity either two measurements using different disc diameters or measurements with a single disc but using multiple pressure heads are then used in this paper we describe a parameter estimation procedure which combines the levenberg marquardt nonlinear parameter optimization method involving weighted least squares with a quasi three dimensional numerical model which solves the variably saturatedflow equation by numerical inversion of richards equation the unknown parameters in van genuchten s model of the unsaturated soil hydraulic properties are estimated from observed cumulative infiltration data during transient waterflow additional measurements of the pressure head or water content as well as a penalty function for constraining the unknown parameters to remain in some feasible region bayesian estimation can be optionally included into the parameter estimation procedure the problem of optimal sampling design that is selecting the best points in space and time for making measurements is addressed by studying the sensitivity of the objective function to changes in the optimized hydraulic parameters we calculate objective functions based on available cumulative infiltration pressure head and water content measurements and also on several combinations of these data the behavior of the objective function in three dimensional parameter space is evaluated by means of a series of two dimensional response surfaces the utility of the parameter estimation procedure is demonstrated using numerically generated data the sensitivity of the procedure to different initial estimates of the model parameters is also discussed

adaptive approximation method for joint parameter estimation and identical synchronization of chaotic systems we introduce a numerical approximation method for estimating an unknown parameter of a primary chaotic system which is partially observed through a scalar time series specifically we show that the recursive minimization of a suitably designed cost function that involves the dynamic state of a fully observed secondary system and the observed time series can lead to the identical synchronization of the two systems and the accurate estimation of the unknown parameter the salient feature of the proposed technique is that the only external input to the secondary system is the unknown parameter which needs to be adjusted we present numerical examples for the lorenz system which show how our algorithm can be considerably faster than some previously proposed methods

markov chain monte carlo method in bayesian reconstruction of dynamical systems from noisy chaotic time series the presence of a noise component in the time series generated by a dynamical system ds results in finite accuracy of reconstruction of the evolution operator eo any model that can be constructed will generally speaking differ from the original system one of the ways to take this fact into consideration is to describe parameters of the model eo as random quantities 1 two limiting cases may be distinguished 2 the perfect model scenario when the form of the operator describing the ds evolution is reliably known and the imperfect model scenario when eo is unknown the perfect model scenario is realized for instance when the problem of hidden information transmission is solved using chaotic time series ts generated by the known ds 3 in this case we are interested in the characteristics of the system perfect model parameters the most probable value mean dispersion distribution in investigations of many natural atmospheric oceanic tectonic biological systems the eo is unknown which corresponds to the imperfect model scenario then information about model parameters is of no value as the physical meaning of the parameters is unknown and we are interested in the properties of the model defined by these parameters and reflecting the properties of the reconstructed ds nevertheless in this case too evaluation of statistical characteristics of model parameters is the key element without which the problem of reconstructing a ds generating the original ts cannot be solved the mathematical body used to reconstruct the ds is determined by a specific application for example for finding the most probable set of model perfect or imperfect parameters it suffices to determine the maximum of posterior probability density of model parameters 4 6 however it should be remembered that in the presence of noise the inverse problem of reconstruction becomes ill posed i e it admits an infinite set of solutions selection of the most correct solutions demands regularization that is a physically justified constraint on admissible values of parameters based solely on a priori information 7 this approach is usually called bayesian within the bayesian approach there are methods which make it possible to estimate dispersions and mathematical expectations of the model parameters 8 a significant advantage of such methods is their relatively low computational resource requirements however due to the model nonlinearity the form of parameter distribution may differ strongly from the normal one in this case estimation of expectation and dispersion is insufficient to construct a correct model and one must use the bayesian approach in full in other words we must construct models which include distributions of parameters and take a priori information about the system into account correctly in this formulation the problem can be solved using the markov chain monte carlo mcmc algorithms 9 however in the case of reconstruction of dss by noisy chaotic time series computational resources required for these algorithms strongly depend on distribution dimensionality this limits applicability of the algorithms to a narrow class of problems e g the case of short ts 10 the method proposed in this paper makes it possible to expand the applicability domain of the mcmc algorithms to the case of arbitrarily long ts

an uncertainty inequality for fourier dunkl series an uncertainty inequality for the fourier dunkl series introduced by the authors in o ciaurri j l varona a whittaker shannon kotel nikov sampling theorem related to the dunkl transform proc amer math soc 135 2007 2939 2947 is proved this result is an extension of the classical uncertainty inequality for the fourier series

error in unsaturated stochastic models parameterized with field data 1 we use monte carlo error analysis to illustrate the impact of measurement errors in field estimated hydraulic properties on predictions made with 1d and 3d unconditional stochastic models of unsaturated flow and transport monte carlo simulations are conducted across a series of simplified artificial realities completely described by the gardner russo parametric model the mean values of properties are varied between simulations to elucidate the relationship between true properties and prediction errors hydraulic properties are reestimated by simulating tension infiltrometer measurements in the presence of small simple errors two types of observation error are considered along with one inversion model error resulting from poor contact between the instrument and the medium errors in the spatial statistics of hydraulic properties cause critical stochastic model assumptions to be violated limiting the usable parameter space for model predictions even where critical assumptions are valid stochastic model predictions show significant error and the magnitude and pattern of error changes with the true property means the flow conditions and the type of measurement error mean velocities may show errors up to an order of magnitude the velocity variance is overestimated by up to three orders of magnitude during 3d flow and eight orders of magnitude during 1d flow the 1d velocity integral scale is underestimated by as much as five orders of magnitude the estimates for 1d longitudinal macrodispersivity are surprisingly robust and show relatively small error across most of the parameter space

parameter equivalence for the brooks corey and van genuchten soil characteristics preserving the effective capillary drive this paper provides a simple way to convert brooks corey bc parameters to van genuchten vg parameters and vice versa for use primarily in situations where saturated conditions are likely to be encountered essential in this conversion is the preservation of the maximum value of a physical characteristic the effective capillary drive h morel seytoux and khanji 1974 defined with a good approximation for a soil water and air system as h jg k dh where k is relative permeability or conductivity to water and ii is capillary pressure head a positive quantity with this conversion infiltration calculations are essentially insensitive to the model used to represent the soil hydraulic properties it is strictly a matter of convenience for the user which expression is used on the other hand the paper shows that other equivalences may lead to great variations in predictions of infiltration capacity consequently the choice of the proper equivalence to use in calculations for rainfall runoff modeling or for low level radioactive waste disposal design is a serious matter

uncertainty principles in harmonic analysis abstract let l 2 r n 2 1 generalizing the heisenberg uncertainty principle lower bounds for the variance of the measure x 2 dx are established under assumptions on the fourier transform tf if e is a subset of r n of codimension one which is suitably uniformly distributed and b is a parameter which measures the maximum distance from a point to e then the estimate var x 2 dx cb 2 holds provided e tf 2 d c b 1 for suitable constants c and c examples of sets e are families of parallel hyperspaces or concentric spheres analogous results are established for normalized l 2 functions on the sphere s n here the lower bound is inf y s n s n sin d x y 2 x 2 dx cb 2 and the hypothesis on the spherical harmonic expansion k 0 k is k a k 2 2 c b 1 where a is a set of integers with gaps at most b for both even and odd numbers

parameter estimation through ignorance dynamical modeling lies at the heart of our understanding of physical systems its role in science is deeper than mere operational forecasting in that it allows us to evaluate the adequacy of the mathematical structure of our models despite the importance of model parameters there is no general method of parameter estimation outside linear systems a relatively simple method of parameter estimation for nonlinear systems is introduced based on variations in the accuracy of probability forecasts it is illustrated on the logistic map the henon map and the 12 dimensional lorenz96 flow and its ability to outperform linear least squares in these systems is explored at various noise levels and sampling rates as expected it is more effective when the forecast error distributions are non gaussian the method selects parameter values by minimizing a proper local skill score for continuous probability forecasts as a function of the parameter values this approach is easier to implement in practice than alternative nonlinear methods based on the geometry of attractors or the ability of the model to shadow the observations direct measures of inadequacy in the model the implied ignorance and the information deficit are introduced

optimum waveform signal sets with amplitude and energy constraints it is desirable to choose the waveforms making up a signaling alphabet so that they are maximally separated one from another this problem is considered in the space of square integrable functions for signals which have finite duration and are constrained in the ranges of their values as well as in energy corresponding to each of the following cases we establish sharp bounds for the minimum distance and for the average distance between elements of a fixed size signal set and construct sets of signals that attain both bounds simultaneously begin list item em case a energy constraint only the average energy of the waveforms in the signal set is at most sigma where 0 leq sigma item em case b energy and peak amplitude constraints the average energy of the waveforms in the signal set is leq sigma 0 leq sigma and the absolute value of each waveform is at most 1 item em case c energy and value constraints the average energy of the waveforms in the signal set is at most b 2 sigma a 2 1 sigma and each waveform takes values in the set a b where 0 leq a and 0 leq sigma leq 1 end list cases a and b are applicable to signal design for communication in channels with additive noise say gaussian and case c is applicable to signal design for optical channels where the signal represents the intensity of a photon stream the general character of the results is that the minimum distance behaves like gamma sigma in cases a and b and like gamma sigma 1 sigma in case c with gamma a suitable constant

fuzzy prediction of chaotic time series based on singular value decomposition for dynamic systems with complex ill conditioned or nonlinear characteristics the modeling method based on fuzzy sets is very effective to describe the properties of the systems in this paper a fuzzy modeling method based on singular value decomposition svd is proposed first a fuzzy clustering method is used to confirm the input space of fuzzy model then the recursive least square algorithm with singular value decomposition is applied to estimate the consequent parameters of fuzzy model in order to avoid error delivery and error accumulation furthermore the parameters of fuzzy model are also optimized by the presented algorithm to demonstrate the performance of this modeling method simulations on mackey glass time series and lorenz chaotic system are performed the results show that this method provides effective and accurate prediction

algebraic models for disc tension permeameters four infiltration models for disc tension permeameters were compared to field and simulated data comparisons were made in order to assess the alternative values of sorptivity and steady state flow rates model 1 was the conventional analysis with assumptions of one dimensional flow at small times and wooding s steady state flow rate at large times model 2 was a modified horton equation defined to fit one dimensional infiltration at small times with an exponential decay to a steady state value for large times model 3 assumed linear diffusion for a soil with constant diffusivity ignoring gravity model 4 was based on linearized flow with gravity the number of unknown parameters to be estimated ranged from one to four sorptivity data using models 2 and 3 generally agreed with the conventional method model 1 both for the field and simulated data for the steady state infiltration rate all of the models gave generally consistent results model 2 with four parameters was the best fitting model to field and simulated data the simulated data allowed comparisons of results based on the total time over which the parameters were evaluated results were sensitive to the time of evaluation up to 0 3 1 3 hours for the sandy loam soil

time to ponding comparison of analytic quasi analytic and approximate predictions an analytic expression for time to ponding is introduced using the nonlinear model of broadbridge and white 1987 the hydraulic properties of this model can encompass properties ranging from those of a highly nonlinear green ampt like soil to those satisfying the weakly nonlinear burgers equation because of its versatility this analytic solution is used as a benchmark against which extant analytic quasi analytic and approximate expressions are compared time to ponding is parameterized here in terms of the readily measured field properties sorptivity and hydraulic conductivity in the limit of green ampt like properties the analytic solution reduces exactly to the parlange and smith 1976 approximation a similar functional dependence of time to ponding on rainfall rate is found from quasi analytic approximations based on this a modified approximation is suggested which should give time to ponding for most soils to within 10 some existing approximations are found to have unacceptable deviations from the analytic solution and their continued use appears unwarranted finally we address the field problem of predicting time to ponding at any antecedent water content given sorptivity measured at only one initial water content

an uncertainty principle for ultraspherical expansions abstract motivated by heisenberg weyl type uncertainty principles for the torustand the spheres2due to breitenberger narowich ward and others we derive an uncertainty relation for radial functions on the spheressn r n 1and more generally for ultraspherical expansions on 0 in this setting the frequency variance of al2 function on 0 is defined by means of the ultraspherical differential operator which plays the role of a laplacian our proof is based on a certain first order differential difference operator on the doubled interval moreover using the densitiesftof gaussian measures on 0 with the timettending to 0 we show that the bound of our uncertainty principle is optimal

inferential framework for nonstationary dynamics i theory a general bayesian framework is introduced for the inference of time varying parameters in nonstationary nonlinear stochastic dynamical systems its convergence is discussed the performance of the method is analyzed in the context of detecting signaling in a system of neurons modeled as fitzhugh nagumo fhn oscillators it is assumed that only fast action potentials for each oscillator mixed by an unknown measurement matrix can be detected it is shown that the proposed approach is able to reconstruct unmeasured hidden variables of the fhn oscillators to determine the model parameters to detect stepwise changes of control parameters for each oscillator and to follow continuous evolution of the control parameters in the adiabatic limit

constant rate rainfall infiltration a versatile nonlinear model 1 analytic solution analytic solutions are presented for a nonlinear diffusion convection model describing constant rate rainfall infiltration in uniform soils and other porous materials the model is based on the darcy buckingham approach to unsaturated water flow and assumes simple functional forms for the soil water diffusivity d and hydraulic conductivity k which depend on a single free parameter c and readily measured soil hydraulic properties these d and k yield physically reasonable analytic moisture characteristics the relation between this model and other models which give analytic solutions is explored as c the model reduces to the weakly nonlinear burgers equation which has been applied in certain field situations at the other end of the range as c 1 the model approaches a green ampt like model a wide range of realistic soil hydraulic properties is encompassed by varying the c parameter the general features of the analytic solutions are illustrated for selected c values gradual and steep wetting profiles develop during rainfall aspects seen in the laboratory and field in addition the time dependent surface water content and surface water pressure potential are presented explicitly a simple traveling wave approximation is given which agrees closely with the exact solution at comparatively early infiltration times

spatial analysis of hydraulic conductivity measured using disc infiltrometers spatial variability of surface hydraulic properties and the extrinsic e g traffic cropping etc and intrinsic e g soil type pore size distribution etc factors associated with these properties are important for infiltration and runoff processes in agricultural fields disc infiltrometers measured infiltration at 296 sites arranged on two parallel transects to examine and differentiate the factors contributing to spatial structure under different field conditions these measurements were made in the corn rows no track interrows and wheel track interrows of the field using four different soil water tensions 0 30 60 and 150 mm unsaturated hydraulic conductivity k and saturated hydraulic conductivity k were maximum in the corn rows and minimum in wheel track inter rows with no track interrows intermediate exponents cz parameters of k and k relationships k k exp av for corn rows and no track interrows were not significantly different from each other but were significantly different from q for the wheel track interrows a t p 0 01 level spatial variability o f k and k values showed some pseudoproportional effect in nugget variance for all three field conditions no track interrows clearly showed an inverse trend for semivariogram o f k with changing tension q values whereas differences were found for corn rows and wheel traffic interrows the spatial structure of y for all three field conditions were mostly white noise under corn rows in addition to random variation a small five row periodic variation at th e p 0 20 level matching the five row traffic configuration was discovered the spatial structure of a was influenced by soil type for the no track interrows spatial structure was absent in wheel track interrows indicating the destruction of pore structure due to compaction

stochastic analysis of unsaturated steady state flow through bounded heterogeneous formations the effect of heterogeneity in saturated hydraulic conductivity ks and in a pore scale distribution parameter on unsaturated steady state flow in bounded domains is studied the properties are assumed to be random space functions having stationary means and covariances an analytic model based on small perturbation approximation in ks and for predicting the spatial moments of the pressure head from the moments of the input variables is developed the solution applies to the entire unsaturated domain without invoking the unit mean hydraulic gradient assumption under conditions of small scale variability the solution simplfies considerably resulting in closed form expressions for the moments of the pressure head these expressions are tested in synthetically generated random fields of and ks having prescribed statistical properties

alternative analyses of hydraulic data from disc tension infiltrometers hydraulic conductivity values were compared based on alternative analyses of data from disc tension infiltrometers the first method was based on a single disc and tension and depends on the estimate of sorptivity and steady state flow a second method used steady state flow measurements for two different disc radii 52 and 118 mm a third method used a single disc with multiple tensions from which steady state flow was obtained at three or more tensions which could be at the same location a fourth method used a single disc with two tensions from which steady state flow was obtained at two tensions finally values based on soil cores were compared the results show reasonable agreement between methods for the hydraulic conductivity with the largest differences for data collected for zero tension for the most part there were no significant differences in hydraulic conductivity due to the disc infiltrometer radius a single disc method with multiple tensions more than 3 points and large disc radius gave results which were the most stable accurate and repeatable

stochastic analysis of unsaturated flow in heterogeneous soils 1 statistically isotropic media steady unsaturated flow with vertical mean infiltration through unbounded heterogeneous porous media is analyzed using a perturbation approximation of the stochastic flow equation which is solved by spectral representation techniques the hydraulic conductivity k is related to the capillary pressure head by k ks exp where ks is the saturated conductivity and is a soil parameter a general formulation is presented for the case with ks and represented as statistically homogeneous spatial random fields in part 1 solutions are developed assuming is constant and representing ks variability by one dimensional and three dimensional isotropic random fields results are obtained for head variances and covariance functions effective hydraulic conductivities variances of the unsaturated hydraulic conductivity flux variances and variance of pressure gradient when the parameter is relatively large corresponding to coarse textured soils the head variance decreases and all of the results demonstrate a trend toward gravitationally dominated one dimensional vertical flow the effective conductivity is dependent on the correlation scale of ln ks and the mean hydraulic gradient

properties of a simple bilinear stochastic model estimation and predictability we analyze the properties of arguably the simplest bilinear stochastic multiplicative process proposed as a model of financial returns and of other complex systems combining both nonlinearity and multiplicative noise by construction it has no linear predictability zero two point correlation but a certain nonlinear predictability non zero three point correlation it can thus be considered as a paradigm for testing the existence of a possible nonlinear predictability in a given time series we present a rather exhaustive study of the process including its ability to produce fat tailed distributions from gaussian innovations the unstable characteristics of the inversion of the key nonlinear parameters and of the two initial conditions necessary for the implementation of a prediction scheme and an analysis of the associated super exponential sensitivity of the inversion of the innovations in the presence of a large impulse our study emphasizes the conditions under which a degree of predictability can be achieved and describe a number of different attempts which overall illuminates the properties of the process in conclusion notwithstanding its remarkable simplicity the bilinear stochastic process exhibits remarkably rich and complex behavior which makes it a serious candidate for the modeling of financial time series among others

linearized unsteady multidimensional infiltration the parameters in the linearized multidimensional unsteady equation for unsaturated flow in homogeneous soils can be chosen so that small time flow rates are given correctly with the quasi linear steady solution as the large time limit matched in this way the linearized equation offers insight into the time course of multidimensional infiltration the laplace transform reduces the equation and its governing conditions into forms similar to those for quasi linear steady flows the problems of unsteady infiltration from buried spherical and cylindrical cavities are solved in many cases the solutions are approximately reducible to the product of the steady solution and a function only of time and radial coordinate theorems show that approximate product solutions apply not only to these configurations but generally to multidimensional cavities of arbitrary shape multidimensional infiltration from finite supply surfaces involves two characteristic lengths and two characteristic times one set deriving from the capillarity gravity interaction and the other from the capillarity geometry interaction we explore the influence of the two characteristic times on the time scales of approach to steady moisture distribution near the supply surface and to steady discharge the analysis provides explanation of the relatively rapid approach to the steady state observed in three dimensional infiltration from small supply surfaces

tensor based cortical surface morphometry via weighted spherical harmonic representation we present a new tensor based morphometric framework that quantifies cortical shape variations using a local area element the local area element is computed from the riemannian metric tensors which are obtained from the smooth functional parametrization of a cortical mesh for the smooth parametrization we have developed a novel weighted spherical harmonic spharm representation which generalizes the traditional spharm as a special case for a specific choice of weights the weighted spharm is shown to be the least squares approximation to the solution of an isotropic heat diffusion on a unit sphere the main aims of this paper are to present the weighted spharm and to show how it can be used in the tensor based morphometry as an illustration the methodology has been applied in the problem of detecting abnormal cortical regions in the group of high functioning autistic subjects

on a recursive method for the estimation of unknown parameters of partially observed chaotic systems we investigate a recently proposed method for on line parameter estimation and synchronization in chaotic systems this novel technique has been shown effective to estimate a single unknown parameter of a primary chaotic system with known functional form that is only partially observed through a scalar time series it works by periodically updating the parameter of interest in a secondary system with the same functional form as the primary one but no explicit coupling between their dynamic variables in order to minimize a suitably defined cost function in this paper we review the basics of the method and investigate its robustness and new extensions in particular we study the performance of the novel technique in the presence of noise either observational i e an additive contamination of the observed time series or dynamical i e a random perturbation of the system dynamics and when there is a mismatch between the primary and secondary systems numerical results including comparisons with other techniques are presented finally we investigate the extension of the original method to perform the estimation of two unknown parameters and illustrate its effectiveness by means of computer simulations

better nonlinear models from noisy data attractors with maximum likelihood a new approach to nonlinear modelling is presented which by incorporating the global behaviour of the model lifts shortcomings of both least squares and total least squares parameter estimates although ubiquitous in practice a least squares approach is fundamentally flawed in that it assumes independent normally distributed ind forecast errors nonlinear models will not yield ind errors even if the noise is ind a new cost function is obtained via the maximum likelihood principle superior results are illustrated both for small data sets and infinitely long data streams

statistical methods of parameter estimation for deterministically chaotic time series we discuss the possibility of applying some standard statistical methods the least square method the maximum likelihood method and the method of statistical moments for estimation of parameters to deterministically chaotic low dimensional dynamic system the logistic map containing an observational noise a segmentation fitting maximum likelihood ml method is suggested to estimate the structural parameter of the logistic map along with the initial value x 1 considered as an additional unknown parameter the segmentation fitting method called piece wise ml is similar in spirit but simpler and has smaller bias than the multiple shooting previously proposed comparisons with different previously proposed techniques on simulated numerical examples give favorable results at least for the investigated combinations of sample size n and noise level besides unlike some suggested techniques our method does not require the a priori knowledge of the noise variance we also clarify the nature of the inherent difficulties in the statistical analysis of deterministically chaotic time series and the status of previously proposed bayesian approaches we note the trade off between the need of using a large number of data points in the ml analysis to decrease the bias to guarantee consistency of the estimation and the unstable nature of dynamical trajectories with exponentially fast loss of memory of the initial condition the method of statistical moments for the estimation of the parameter of the logistic map is discussed this method seems to be the unique method whose consistency for deterministically chaotic time series is proved so far theoretically not only numerically

linear transforms for fourier data on the sphere application to high angular resolution diffusion mri of the brain this paper presents a novel family of linear transforms that can be applied to data collected from the surface of a 2 sphere in three dimensional fourier space this family of transforms generalizes the previously proposed funk radon transform frt which was originally developed for estimating the orientations of white matter fibers in the central nervous system from diffusion magnetic resonance imaging data the new family of transforms is characterized theoretically and efficient numerical implementations of the transforms are presented for the case when the measured data is represented in a basis of spherical harmonics after these general discussions attention is focused on a particular new transform from this family that we name the funk radon and cosine transform fract based on theoretical arguments it is expected that fract based analysis should yield significantly better orientation information e g improved accuracy and higher angular resolution than frt based analysis while maintaining the strong characterizability and computational efficiency of the frt simulations are used to confirm these theoretical characteristics and the practical significance of the proposed approach is illustrated with real diffusion weighted mri brain data these experiments demonstrate that in addition to having strong theoretical characteristics the proposed approach can outperform existing state of the art orientation estimation methods with respect to measures such as angular resolution and robustness to noise and modeling errors

seeker optimization algorithm for parameter estimation of time delay chaotic systems time delay chaotic systems have some very interesting properties and their parameter estimation has received increasing interest in the recent years it is well known that parameter estimation of a chaotic system is a nonlinear multivariable and multimodal optimization problem for which global optimization techniques are required in order to avoid local minima in this work a seeker optimization algorithm soa based method is proposed to address this issue in the soa search direction is based on the empirical gradients by evaluating the response to the position changes and step length is based on uncertainty reasoning by using a simple fuzzy rule the performance of the algorithm is evaluated on two typical test systems moreover two state of the art algorithms i e particle swarm optimization and differential evolution are also considered for comparison the simulation results show that the proposed algorithm is better than or at least as good as the other two algorithms and can effectively solve the parameter estimation problem of time delay chaotic systems

failure of maximum likelihood methods for chaotic dynamical systems the maximum likelihood method is a basic statistical technique for estimating parameters and variables and is the starting point for many more sophisticated methods like bayesian methods this paper shows that maximum likelihood fails to identify the true trajectory of a chaotic dynamical system because there are trajectories that appear to be far more infinitely more likely than truth this failure occurs for unbounded noise and for bounded noise when it is sufficiently large and will almost certainly have consequences for parameter estimation in such systems the reason for the failure is rather simple in chaotic dynamical systems there can be trajectories that are consistently closer to the observations than the true trajectory being observed and hence their likelihood dominates truth the residuals of these truth dominating trajectories are not consistent with the noise distribution they would typically have too small standard deviation and many outliers and hence the situation may be remedied by using methods that examine the distribution of residuals and are not entirely maximum likelihood based

a multigrid model for steady flow in partially saturated porous media a numerical model for steady flow in a partially saturated porous medium is presented the model uses the multigrid solution algorithm for solving the system of equations resulting from the discrete approximation to the partial differential equation the model solutions checked favorably with an existing analytical solution used for a hypothetical problem with realistic soil characteristics the model has been shown to be very efficient for problems with the number of unknowns on the order of 10 000 and more specifically the multigrid scheme was shown to be 22 times faster in terms of cpu time than the line successive overtaxation method for the particular problem chosen in addition the required cpu time is shown to be a linear function of the number of unknowns within the solution domain this indicates that the convergence rate of the multigrid solution algorithm is independent of the grid resolution because of the significant improvement in computational efficiency the multigrid model can be used to simulate field scale partially saturated flow problems with very fine grid resolution as is often required by highly nonlinear soil characteristics further studies and application of the multigrid solution technique are suggested

analyses of infiltration events in relation to determining soil hydraulic properties by inverse problem methodology at the field scale because of the relatively large number of observation points required to estimate the spatial distributions of hydraulic properties application of inversion procedures must be based on a characteristic of the flow that for a given point in the field can be measured with relative ease and can be used as input for the inversion procedure suggesting the use of infiltration data for this purpose and recognizing the fact that a basic requirement for the design of a transient flow experiment is that the resulting inverse problem be sufficiently well posed to allow solution this paper addresses the question under which circumstances the resulting inverse problem is well posed utilizing the concepts of identifiability uniqueness and stability for this purpose three different models of the soil hydraulic properties were analyzed the main conclusion of this study is that when infiltration data measured at the darcy scale are used as input for the inversion procedure the inclusion of prior information on a single measurable parameter the saturated conductivity ks in the estimation criterion will enhance the likelihood of uniqueness and stability of the inverse solution provided that the structure of the hydraulic model is sufficiently simple however since for a particular situation it is impossible to determine a priori whether the resultant inverse problem is well posed or not this must be carried out only a posteriori by solving the problem several times with different initial parameter estimates accompanied by an analysis of the associated estimation errors

determining soil hydraulic properties by parameter estimation on the selection of a model for the hydraulic properties parameter estimation procedures involving solution of the inverse flow problem pertinent to a transient experiment constitute a powerful method of determining the soil hydraulic properties one of the problems associated with these procedures is the selection of an appropriate model to describe the soil hydraulic properties three different models were considered those of van genuchten vg and brooks and corey bc model and the exponential model of gardner for the hydraulic conductivity soil water pressure relationships coupled with a new expression for the soil water content soil water pressure relationship gr model for a given model the soil hydraulic properties were determined from simulated and measured outflow experiments supplemented with water content at a soil water pressure head of 15 000 cm h2o using the parameter estimation procedure of kool et al 1985a and data from two soils a hypothetical sandy loam soil assumed to be described by the vg model and a silt loam soil model validation tests were performed and the most appropriate model was selected from the candidate models by discrimination tests using the akaike information criterion aic in the case of the hypothetical sandy loam the vg model was found to be the most accurate and most consistent with the data the performance of both the bc and the gr models in terms of the aic was less good but very similar to each other in the case of the silt loam soil the vg model with additional parameter m which accounts for the correlation between pores and for the flow path tortuosity m 2 02 rather than a constant m 0 5 was found to be most accurate and most consistent with the data the performance of the bc and the gr models was similar when m was considered as an unknown parameter instead of a constant

differential difference operators associated to reflection groups there is a theory of spherical harmonics for measures invariant under a finite reflection group the measures are products of powers of linear functions whose zero sets are the mirrors of the reflections in the group times the rotation invariant measure on the unit sphere in rn a commutative set of differential difference operators each homogeneous of degree 1 is the analogue of the set of first order partial derivatives in the ordinary theory of spherical harmonics in the case of r2 and dihedral groups there are analogues of the cauchy riemann equations which apply to gegenbauer and jacobi polynomial expansions the analysis of orthogonality structures for polynomials in several variables is a problem of vast dimensions this paper is part of an ongoing program to establish a workable theory for one particular class the underlying structure is based on finite coxeter groups these are finite groups acting on euclidean space generated by reflections in the zero sets of a collection of linear functions the roots the weight functions for the orthogonality are products of powers of these linear functions restricted to the surface of the unit sphere in addition the weight function is required to be invariant under the action of the group the resulting theory has strong similarities to the theory of spherical harmonics this was established in previous papers of the author 3 4 5 most notably a homogeneous polynomial is orthogonal to all polynomials of lower degree if and only if it is annihilated by a certain second order differential difference operator ordinary partial differentiation acts as an endomorphism on ordinary harmonic functions the use of such operators and their adjoints leads to recurrence formulas and orthogonal decompositions for harmonic polynomials in this paper we construct a commutative set of first order differentialdifference operators associated to the second order operator previously mentioned received by the editors june 1 1987 1980 mathematics subject classification 1985 revision primary 33a45 33a65 20h15 secondary 20c30 42c10 51f15

statistical analysis of spatial variability in unsaturated flow parameters core scale estimates of soil parameters of the gardner russo and van gemachten models of the hydraulic conductivity and water retention functions were obtained for 417 undisturbed soil cores taken from a wall of a trench 20 m long 2 5 m deep using a procedure based on inverse problem methodology these estimates were used to evaluate the first two statistical moments of the underlying random space functions rsfs using the restricted maximum likelihood estimation procedure coupled with the weighted least squares procedure to estimate parameters of models of the covariance and the drift functions of the pertinent rsfs the fitted models were used to evaluate the mean and covariance functions of the hydraulic conductivity and water rentention functions for given water saturations covariance functions of log saturated conductivity log ks and the shape parameters of the gardner russo and van genuchten models exhibited statistical anisotropy characterized by aspect ratios that vary between 3 to 4 correlation scales of log ks were larger than those of the shape parameters consequently the product of the variance of log unsaturated conductivity and its correlation scale remained essentially invariant for a considerable range of water saturation the implications of these results regarding stochastic modeling of transport in heterogeneous porous formations and possible applications of the results of this study are discussed briefly

three dimensional analysis of infiltration from the disc infiltrometer 2 physically based infiltration equation in situ measurement of soil hydraulic properties may be achieved by analyzing the unconfined efflux from disc tension infiltrometers once consistent infiltration equations can be derived in this paper an analytical three dimensional infiltration equation is developed based on the use of parameters with sound physical meaning and adjustable for varying initial and boundary conditions the equation is valid over the entire time range for practical purposes a simplified solution is also derived the full and simplified equations give excellent agreement with published experimental results and are particularly useful for determining soil hydraulic properties through application of inverse procedures

degrees of freedom in multiple antenna channels a signal space approach multiple antenna systems that are limited by the area and geometry of antenna arrays are considered given these physical constraints the limit on the available number of spatial degrees of freedom is derived the commonly used statistical multiple input multiple output mimo model is inadequate antenna theory is applied to take into account the area and geometry constraints and to define the spatial signal space so as to interpret experimental channel measurements in an array independent but manageable description of the physical environment based on these modeling strategies for a spherical array of effective aperture a in a physical environment of angular spread spl omega in solid angle the number of spatial degrees of freedom is shown to be a spl omega for uni polarized antennas and 2a spl omega for tri polarized antennas together with the 2wt degrees of freedom for a system of bandwidth w transmitting in an interval t the total degrees of freedom of a multiple antenna channel is therefore 4wta spl omega

nonlinear dynamical system identification with dynamic noise and observational noise in this paper we consider the problem of whether a nonlinear system has dynamic noise and then estimate the level of dynamic noise to add to any model we build the method we propose relies on a nonlinear model and an improved least squares method recently proposed on the assumption that observational noise is not large we do not need any a priori knowledge for systems to be considered and we can apply the method to both maps and flows we demonstrate with applications to artificial and experimental data the results indicate that applying the proposed method can detect presence or absence of dynamic noise from scalar time series and give a reliable level of dynamic noise to add to the model built in some cases

consistent nonlinear dynamics identifying model inadequacy empirical modelling often aims for the simplest model consistent with the data a new technique is presented which quantifies the consistency of the model dynamics as a function of location in state space as is well known traditional statistics of nonlinear models like root mean square rms forecast error can prove misleading testing consistency is shown to overcome some of the deficiencies of rms error both within the perfect model scenario and when applied to data from several physical systems using previously published models in particular testing for consistent nonlinear dynamics provides insight towards i identifying when a delay reconstruction fails to be an embedding ii allowing state dependent model selection and iii optimising local neighbourhood size it also provides a more relevant state dependent threshold for identifying false nearest neighbours

measurement scale network sampling scale and groundwater model parameters the scales at which model parameters are measured with local field tests distributed on a sampling network are examined two scales are defined to characterize the problem 1 the measurement scale associated with the resolution of a single field test and 2 the network scale associated with the separation between samples on a network using a spatial filtering approach it is shown that a network of measurements can only resolve a larger scale component of a parameter field the smaller scale component of the parameter field not seen by measurements here called the subgrid component can only vary on scales larger than the measurement scale and smaller than the network scale these unobserved subgrid scales give rise to the so called closure problem and consequent modeling errors when a significant proportion of the parameter variability is contained in the subgrid scales not only will the closure problem be significant but aliasing errors will also pollute the estimate of the large scale component of the parameter field these concepts are illustrated by performing simple analytical and numerical calculations

plane wave decomposition of the sound field on a sphere by spherical convolution spherical microphone arrays have been recently studied for sound analysis and sound recordings which have the advantage of spherical symmetry facilitating three dimensional analysis this paper complements the recent microphone array design studies by presenting a theoretical analysis of plane wave decomposition given the sound pressure on a sphere the analysis uses the spherical fourier transform and the spherical convolution where it is shown that the amplitudes of the incident plane waves can be calculated as a spherical convolution between the pressure on the sphere and another function which depends on frequency and the sphere radius the spatial resolution of plane wave decomposition given limited bandwidth in the spherical fourier domain is formulated and ways to improve the computation efficiency of plane wave decomposition are introduced the paper concludes with a simulation example of plane wave decomposition

quadratic variational framework for signal design on the 2 sphere this paper introduces a quadratic variational framework for solving a broad class of signal design problems on the 2 sphere the functional to be extremized combines energy concentration measures using a weighting function in the spatial domain multiplicative weights in the spectral domain and a total energy constraint this leads to two formulations of the signal design problem on the 2 sphere one a fredholm integral equation in the spatial domain and the other an infinite matrix equation in the spectral domain the framework is illustrated by deriving the key equations for the two classical spatio spectral concentration problems on the 2 sphere and for an isotropic filter design that maximizes the filtered energy in addition using the proposed framework we formulate a joint 3 d beamforming application which achieves optimal directivity and spatial resolution simultaneously

spatial bias in field estimated unsaturated hydraulic properties hydraulic property measurements often rely on non linear inversion models whose errors vary between samples in non linear physical measurement systems bias can be directly quantified and removed using calibration standards in hydrologic systems field calibration is often infeasible and bias must be quantified indirectly we use a monte carlo error analysis to indirectly quantify spatial bias in the saturated hydraulic conductivity k sub s and the exponential relative permeability parameter alpha estimated using a tension infiltrometer two types of observation error are considered along with one inversion model error resulting from poor contact between the instrument and the medium estimates of spatial statistics including the mean variance and variogram model parameters show significant bias across a parameter space representative of poorly to well sorted silty sand to very coarse sand when only observation errors are present spatial statistics for both parameters are best estimated in materials with high hydraulic conductivity like very coarse sand when simple contact errors are included the nature of the bias changes dramatically spatial statistics are poorly estimated even in highly conductive materials conditions that permit accurate estimation of the statistics for one of the parameters prevent accurate estimation for the other accurate regions for the two parameters do not overlap in parameter space false cross correlation between estimated parameters is created because estimates of k sub s also depend on estimates of alpha and both parameters are estimated from the same data

optimally space localized band limited wavelets on s q 1 the localization of a function can be analyzed with respect to different criteria in this paper we focus on the uncertainty relation on spheres introduced by goh and goodman uncertainty principles and asymptotic behavior appl comput harmon anal 16 2004 69 89 where the localization of a function is measured in terms of the product of two variances the variance in space domain and the variance in frequency domain after deriving an explicit formula for the variance in space domain of a function in the space wn qs of spherical polynomials of degree at most n s which are orthogonal to all spherical polynomials of degree at most n we are able to identify up to rotation and multiplication by a constant the polynomial in wn qs with minimal variance in space domain or in other words to determine the optimally space localized polynomial in wn qs

weighted fourier series representation and its application to quantifying the amount of gray matter we present a novel weighted fourier series wfs representation for cortical surfaces the wfs representation is a data smoothing technique that provides the explicit smooth functional estimation of unknown cortical boundary as a linear combination of basis functions the basic properties of the representation are investigated in connection with a self adjoint partial differential equation and the traditional spherical harmonic spharm representation to reduce steep computational requirements a new iterative residual fitting irf algorithm is developed its computational and numerical implementation issues are discussed in detail the computer codes are also available at http www stat wisc edu mchung softwares weighted spharm weighted spharm html as an illustration the wfs is applied in quantifying the amount of gray matter in a group of high functioning autistic subjects within the wfs framework cortical thickness and gray matter density are computed and compared

optimal filters on the sphere we derive optimal filters on the sphere in the context of detecting compact objects embedded in a stochastic background process the matched filter and the scale adaptive filter are derived on the sphere in the most general setting allowing for directional template profiles and filters the performance and relative merits of the two optimal filters are discussed the application of optimal filter theory on the sphere to the detection of compact objects is demonstrated on simulated data a naive detection strategy is adopted with an initial aim of illustrating the application of the new filter theory derived on the sphere nevertheless this simple object detection strategy is demonstrated to perform well even at low signal to noise ratio

spherical diffusion for 3d surface smoothing a diffusion based approach to surface smoothing is presented surfaces are represented as scalar functions defined on the sphere the approach is equivalent to gaussian smoothing on the sphere and is computationally efficient since it does not require iterative smoothing furthermore it does not suffer from the well known shrinkage problem evolution of important shape features parabolic curves under diffusion is demonstrated a nonlinear modification of the diffusion process is introduced in order to improve smoothing behavior of elongated and poorly centered objects

uncertainty principles for jacobi expansions abstract in this paper an uncertainty principle for jacobi expansions is derived as a generalization of that for ultraspherical expansions by rosler and voit indeed a stronger inequality is proved which is new even for fourier cosine or ultraspherical expansions a complex base of exponential type on the torus z c z 1 related to jacobi polynomials is introduced which are the eigenfunctions both of certain differential difference operators of the first order and the second order an uncertainty principle related to such exponential base is also proved

sorptivity and macroscopic capillary length relationships the relationship of sorptivity s and macroscopic capillary length c is explored sorptivity is the proportionality constant between cumulative intake expressed as a length and the square root of time for sorption into an unsaturated soil the capillary length is defined as the integral of the unsaturated hydraulic conductivity with respect to matric potential normalized by the difference between the conductivity at the two limits of integration the approximation b c wct dry kwct kdry s 2 was found previously to have bounds of 0 5 and 4 with 0 55 as a good overall approximation those results were for two diffusivity functions an exponential d d0 exp b and for d d0 b 2 we now extend the investigation to four additional cases two for which s and c are analytical and two described by the empirical hydraulic functions of brooks and corey 1964 and by van genuchten 1980 the representative value b 0 55 was found to be generally correct for those functions with a finite d value at the wettest value a refinement from the 0 55 value is offered the results are believed of value for future parameter estimation problems including two previous applications related to time of ponding and disk infiltrometers

intrinsic limits of dimensionality and richness in random multipath fields we study the dimensions or degrees of freedom of farfield multipath that is observed in a limited source free region of space the multipath fields are studied as solutions to the wave equation in an infinite dimensional vector space we prove two universal upper bounds on the truncation error of fixed and random multipath fields a direct consequence of the derived bounds is that both fixed and random multipath fields have an effective finite dimension for circular and spherical spatial regions we show that this finite dimension is proportional to the radius and area of the region respectively we use the karhunen loegraveve kl expansion of random multipath fields to quantify the notion of multipath richness the multipath richness is defined as the number of significant eigenvalues in the kl expansion that achieve 99 of the total multipath energy we establish a lower bound on the largest eigenvalue this lower bound quantifies to some extent the well known reduction of multipath richness with reducing the angular power spread of multipath angular power spectrum

efficient computation of slepian functions for arbitrary regions on the sphere in this paper we develop a new method for the fast and memory efficient computation of slepian functions on the sphere slepian functions which arise as the solution of the slepian concentration problem on the sphere have desirable properties for applications where measurements are only available within a spatially limited region on the sphere and or a function is required to be analyzed over the spatially limited region slepian functions are currently not easily computed for large band limits for an arbitrary spatial region due to high computational and large memory storage requirements for the special case of a polar cap the symmetry of the region enables the decomposition of the slepian concentration problem into smaller subproblems and consequently the efficient computation of slepian functions for large band limits by exploiting the efficient computation of slepian functions for the polar cap region on the sphere we develop a formulation supported by a fast algorithm for the approximate computation of slepian functions for an arbitrary spatial region to enable the analysis of modern datasets that support large band limits for the proposed algorithm we carry out accuracy analysis of the approximation computational complexity analysis and review of memory storage requirements we illustrate through numerical experiments that the proposed method enables faster computation and has smaller storage requirements while allowing for sufficiently accurate computation of the slepian functions

the green ampt limit with reference to infiltration coefficients 1 recent progress with an analytic nonlinear model has provided the exact infiltration coefficients for realistic soil behaviors with nonsingular hydraulic functions as well as their exact delta function diffusivity limits after some correction and reinterpretation of the approximate analytical method the exactly solvable model validates some previously obtained approximate infiltration functions the green ampt infiltration function follows from a delta function diffusivity limit with a hydraulic conductivity that may be among other possibilities a linear function of water content just as a linear conductivity function is an overestimate for a realistic soil the second philip infiltration coefficient s1 in the green ampt infiltration function is too large due to conductivity being overestimated better agreement with experiment halving the value of s1 is obtained from the analytic nonlinear model with a limiting delta function diffusivity and a matching gardner exponential hydraulic conductivity function in general infiltration behavior is determined by the limiting forms of the diffusivity and conductivity relative to one another at the saturated water content or alternatively the relationship between the conductivity and soil moisture potential a new infiltration model demonstrates the possible range of s1 for physically valid limiting conductivity functions we show that in the delta function diffusivity limit the solution behaves as if the potential at the wet front were time dependent decreasing in magnitude from an initial value at the traditional green ampt level

cross correlated random field generation with the direct fourier transform method this paper presents a computer algorithm that is capable of cogenerating pairs of three dimensional cross correlated random fields the algorithm produces random fields of real variables by the inverse fourier transform of a randomized discrete three dimensional spectral representations of the variables the randomization is done in the spectral domain in a way that preserves the direct power and cross spectral density structure two types of cross spectra were examined one type specifies a linear relationship between the two fields which produces the same correlation scales for both variables but different variances the second cross spectrum is obtained from a specified transfer function and the two power spectra and it produces fields with different correlation scales for both models the degree of correlation is specified by the coherency a delay vector can also be specified to produce an out of phase correlation between the two fields the algorithm is very efficient computationally is relatively easy to use and does not produce the lineation problems that can be encountered with the turning bands method perhaps most important this random field generator is capable of co generating cross correlated random fields

quantifying uncertainty in geoacoustic inversion i a fast gibbs sampler approach this paper develops a new approach to estimating seabed geoacoustic properties and their uncertainties based on a bayesian formulation of matched field inversion in bayesian inversion the solution is characterized by its posterior probability density ppd which combines prior information about the model with information from an observed data set to interpret the multi dimensional ppd requires calculation of its moments such as the mean covariance and marginal distributions which provide parameter estimates and uncertainties computation of these moments involves estimating multi dimensional integrals of the ppd which is typically carried out using a sampling procedure important goals for an effective bayesian algorithm are to obtain efficient unbiased sampling of these moments and to verify convergence of the sample this is accomplished here using a gibbs sampler gs approach based on the metropolis algorithm which also forms the basis for simulated annealing sa although gs can be computationally slow in its basic form just as modifications to sa have produced much faster optimization algorithms the gs is modified here to produce an efficient algorithm referred to as the fast gibbs sampler fgs an automated convergence criterion is employed based on monitoring the difference between two independent fgs samples collected in parallel comparison of fgs gs and monte carlo integration for noisy synthetic benchmark test cases indicates that fgs provides rigorous estimates of ppd moments while requiring orders of magnitude less computation time

stochastic analysis of steady state unsaturated flow in heterogeneous media comparison of the brooks corey and gardner russo models existing stochastic models of unsaturated flow and transport are usually developed using the simple gardner russo constitutive relationship though it is generally accepted that the more complex van genuchten and brooks corey relationships may perform better in describing experimental data in this paper we develop first order stochastic models for gravity dominated flow in second order stationary media with both the brooks corey and the gardner russo constitutive relationships these models also account for the spatial variability in effective water content while the spatial variability is generally neglected in most existing stochastic models analytical solutions are obtained for the case of one dimensional gravity dominated flow on the basis of the solutions we illustrate the differences between results from these two constitutive models through some one dimensional examples it is found that the impacts of the constitutive models on the statistical moments of suction head effective water content unsaturated hydraulic conductivity and velocity depend on the saturation ranges for example the mean head and the mean effective water content for the brooks corey model differ in a great manner with their counterparts for the gardner russo model near the dry and wet limits while the differences are small at the intermediate range of saturation this finding is confirmed with some two dimensional examples it is also found that the brook corey model has certain advantages over the gardner russo model in analyzing unsaturated flow in heterogeneous media for example the stochastic model developed based on the brooks corey function requires the coefficient of variation of head and soil parameter bc to be small 1 whereas that based on the gardner russo function assumes the one point cross covariance of head and olgr to be small 1 illustrative examples reveal that the latter condition may be violated because the one point covariance is found to increase rapidly to beyond unity as the soil becomes dry whereas the former may be readily satisfied