a moving average approach for spatial statistical models of stream networks in this article we use moving averages to develop new classes of models in a flexible modeling framework for stream networks streams and rivers are among our most important resources yet models with autocorrelated errors for spatially continuous stream networks have been described only recently we develop models based on stream distance rather than on euclidean distance spatial autocovariance models developed for euclidean distance may not be valid when using stream distance we begin by describing a stream topology we then use moving averages to build several classes of valid models for streams various models are derived depending on whether the moving average has a tail up stream a tail down stream or a two tail construction these models also can account for the volume and direction of flowing water the data for this article come from the ecosystem health monitoring program in southeast queensland australia an important national program aimed at monitoring water quality we model two w

objective analyses and mapping techniques for rainfall fields an objective comparison many methods have been proposed and applied individually to rainfall fields in order to estimate point or average values at ungauged sites this paper reviews some of the usual as well as some of the more recent techniques they are classified according to their fundamental principles optimality criteria and practical aspects some theoretical developments mostly for the statistically based techniques are offered and the practical aspects are mainly directed to the particular case of rainfall fields a case study is presented for a region with particularly high rainfall variability providing a good sample of events for testing the sensitivity of the estimation to the different methods used the identification of statistical structure functions required by certain methods using regionalized variable concepts is discussed in detail for the case considered method evaluation is based on a classical validation technique that divides the available stations into two separate networks i e a base network is used to perform estimations at the stations of a test network for a set of independent storm events careful attention is paid to result interpretation and visualisation finally conclusions are drawn as to which method should be implemented taking into account cost efficiency considerations

a front tracking method for multiphase free boundary problems a front tracking method for the solution of multiphase free boundary problems is presented based on the method of lines with invariant imbedding this method is sufficiently general to allow the solution of problems with three or more phases and problems in which phases appear and disappear a discussion of the method for the three phase stefan problem and for the moving contact problem as well as a proof of convergence for the latter is included in addition the method is demonstrated for a free boundary problem with a combination of stefan and ablation boundary conditions and for the two dimensional contact problem numerical results indicate that this front tracking method is fast enough to solve a wide range of problems on a personal computer

the method of lines and invariant imbedding for elliptic and parabolic free boundary problems a combination of the method of lines and invariant imbedding is suggested as a general purpose numerical algorithm for free boundary problems its effectiveness is illustrated by computing the solidification of a binary alloy in one dimension electrochemical machining and hele shaw flow in two dimensions and a stefan and ablation problem in three dimensions

a bayesian analysis of kriging this article is concerned with predicting for gaussian random fields in a way that appropriately deals with uncertainty in the covariance function to this end we analyze the best linear unbiased prediction procedure within a bayesian framework particular attention is paid to the treatment of parameters in the covariance structure and their effect on the quality both real and perceived of the prediction these ideas are implemented using topographical data from davis

spectral methods for nonstationary spatial processes summary we propose a nonstationary periodogram and various parametric approaches for estimating the spectral density of a nonstationary spatial process we also study the asymptotic properties of the proposed estimators via shrinking asymptotics assuming the distance between neighbouring observations tends to zero as the size of the observation region grows without bound with this type of asymptotic model we can uniquely determine the spectral density avoiding the aliasing problem we also present a new class of nonstationary processes based on a convolution of local stationary processes this model has the advantage that the model is simultaneously defined everywhere unlike moving window approaches but it retains the attractive property that locally in small regions it behaves like a stationary spatial process applications include the spatial analysis and modelling of air pollution data provided by the us environmental protection agency

on the spectra of so 3 scalars in the enveloping algebra of su 3 formulas are given that make it possible to calculate the eigenvalues of the two independent so 3 scalars o0l and q0l in the su 3 enveloping algebra

bayesian estimation and experimental design in linear regression models estimation and design as a bayesian decision problem choice of a prior distribution conjugate prior distributions bayes estimation of the regression parameter optimality and robustness of the bayes estimator bayesian interpretation of estimators using non bayesian prior knowledge bayes estimation in case of prior ignorance further problems the design problem for the linear bayes estimator characterization of optimal designs construction of optimal continuous designs construction of exact optimal designs

hele shaw flow with a cusping free boundary abstract hele shaw flow with suction for an incompressible fluid in a porous medium is introduced as a nontrivial model problem for two dimensional transient free boundary problems in this model an initially smooth free boundary can move with unbounded velocities and develop cusps two analytic solutions are worked out for a special geometry with which numerical results can be compared it is then shown that the method of lines can reasonably well reproduce the analytic solutions although at the expense of long computer times some strategies for improving the efficiency of the method of lines including a multi grid method are described

on multimodality of the likelihood in the spatial linear model summary a popular covariance scheme used for the spatial linear model in geostatistics has spherical form however the likelihood is not twice differentiable with respect to the range parameter and this raises some questions regarding the unimodality of the likelihood we compare the likelihoods of the spatial linear model for small samples under this scheme and the doubly geometric scheme also a power covariance with range parameter is proposed in view of potential multimodal likelihoods for small samples for this model a convenient profile likelihood is introduced and studied

statistical estimation of polynomial generalized covariance functions and hydrologic applications minimum variance unbiased linear estimation theory has found many applications in the analysis of spatial rainfall and hydrogeologic data estimation of point values or areal averages calculation of variances of estimation error and network design however the important problem of inferring the spatial structure or interdependence of the function of interest has received relatively less attention three methods for the estimation of the parameters of matheron s polynomial generalized covariance function a useful model for nonstationary functions are presented and evaluated maximum likelihood ml minimum variance unbiased quadratic mvuq and minimum norm least squares the estimation error covariance matrix of the parameters is calculated with each of these methods ml and mvuq estimation have very good properties and for reasonably large samples give very similar results minimum norm estimation is appropriate for preliminary data analysis the application of these methodologies is illustrated with a set of rainfall data and problems of implementation are discussed it is shown that the estimation of the parameters of the covariance function involves sizeable errors it is suggested that in practice it may be rather difficult to identify the parameters of all but simple models it is important that future research evaluate the effects of parameter uncertainty in the circulation of estimation error variances of point or areal averages and in the design of hydrologic data collection networks

a general approach to the systematic derivation of so 3 shift operator relations ii applications the technique reported on in the preceding paper is applied to construct shift operator relations in the su 3 and so 5 lie algebras comments are made concerning the integrity basis for so 3 scalar operators appearing in their respective enveloping algebras

numerical solution of a free boundary problem in hypersonic flow theory nonequilibrium viscous shock layers abstract non equilibrium viscous shock layer flows produced near the stagnation point of a blunt body constitute a standard problem in hypersonic flow theory this flow configuration is usually modeled with the viscous shock layer vsl equations derived by writing the full navier stokes equations in boundary layer coordinates and performing an order of magnitude analysis the terms retained in this process assure that for moderate reynolds numbers the resulting equations are uniformly valid between the body surface and the shock which may be treated as a thin discontinuity the vsl equations are generally solved by starting with the thin layer approximation tvsl boundary conditions are specified at the body surface and the rankine hugoniot relations are imposed at the shock because the position of the shock is not known one has to solve a free boundary problem this paper presents a novel solution procedure for this situation a reduced coordinate is introduced and the free boundary problem is transformed into a nonlinear eigenvalue problem the new problem for an augmented set of variables is then solved with newton iterations and adaptive gridding the method is illustrated in the paper with a solution of the thin shock layer equations at the stagnation streamline the solution obtained on this line is then used as an initial condition for a two dimensional marching procedure the complete axisymmetric problem is solved by performing fully coupled newton iterations moreover we consider a new enlarged unknown function including the shock location the model includes detailed transport properties and complex kinetics for air dissociation and ionization however in order to focus on the numerical method and for the sake of simplicity thermodynamic equilibrium is assumed

nonscalar extension of shift operator techniques for su 3 in an o 3 basis iii shift operators of second degree in the tensor components shift operators qkl 2 k 2 of second degree in the tensor components q 2 2 are constructed relations connecting quadratic shift operator products of the type ojl k qkl or qkl j ojl and of the type qjl k qkl are derived the usefulness of these relations is demonstrated by the example of the ool and qol eigenvalue calculation for various irreducible respresentations p q of su 3

problems with likelihood estimation of covariance functions of spatial gaussian processes summary maximum likelihood has frequently been suggested as a way to estimate covariance parameters in spatial gaussian processes the results of some simple examples based on simulated data are discussed it is shown by an example with real data that the profile likelihood can be multimodal and that the global maximum may not correspond to a sensible value of the parameters mardia marshall 1984 worked in the following framework we have a gaussian process z x i x e x with mean given by a regression equation e z x f x t for a p dimensional vector the covariance function cov z x z y c x y 0 is also specified by a q dimensional vector 0 this process is observed at n points xi xn and from these observations we wish to estimate 3 and 0 let zn z xi z xn t be the vector of observations and k c xi xj 0 be its covariance matrix finally let f be the matrix with rows f xi t then zn has a multivariate normal distribution with mean f18 dispersion matrix k and the log likelihood is up to an additive constant

the shift operator technique for so 7 in an su 2 3 basis i theory shift operators for the chain of lie groups so 7 su 2 3 are constructed and several properties are examined quite a number of relations connecting quadratic products of shift operators are built up finally the connection between matrix elements of the shift operators and reduced matrix elements of the tensor operator which forms part of the so 7 generator basis is explained

modeling spatial covariance using the limiting distribution of spatio temporal random walks abstractwe present an approach for modeling areal spatial covariance in observed genetic allele data by considering the stationary limiting distribution of a spatio temporal markov random walk model for gene flow this stationary distribution corresponds to an intrinsic simultaneous autoregressive sar model for spatial correlation and provides a principled approach to specifying areal spatial models when a spatio temporal generating process can be assumed we apply the approach to a study of spatial genetic variation of trout in a stream network in connecticut usa

nonscalar extension of shift operator techniques for su 3 in an o 3 basis ii applications in a preceding paper relations have been derived which connect nonscalar quadratic shift operator products here the extreme usefulness of these relations is demonstrated by the example of the o0l eigenvalue calculation for the cases l p i i 0 1 2 3 and 4 where p q is any su 3 representation for the first time a case of threefold l degeneracy is completely solved in a pure analytical way

free boundary problems with nonlinear source terms the method of lines is used to semi discretize the non linear poisson equation over a domain with a free boundary the resulting multipoint free boundary problem is solved with a line gauss seidel method which is shown to converge monotonically the method of lines solution is then shown to converge to the continuous solution of the variational inequality form of the obstacle problem some numerical results for the diffusion reaction equation indicate that the method is applicable to more general free boundary problems for nonlinear elliptic equations

analysis of the spatial structure of properties of selected aquifers data from 31 regional aquifers are analyzed to identify the horizontal spatial correlation structure of transmissivity hydraulic conductivity and storage coefficient three parameters are estimated the variance of small scale variability or nugget and the variance and integral scale range of an exponential covariance function the assumption of normality of the aforementioned geohydrologic parameters and their logarithmic transformations is also examined results are obtained which indicate that the logarithms of these properties generally pass normality tests

a pair of commuting scalars for g 2 su 2 su 2 g 2 su 2 su 2 is a two missing labels problem and therefore in order to give a complete and orthogonal specification of states of irreducible representations of g 2 in an su 2 su 2 basis one needs to find a pair of commuting hermitian operators which are scalar with respect to the su 2 su 2 subalgebra a theorem due to peccia and sharp states that there are apart from the lie algebra invariants twice as many functionally independent scalars as missing labels here two commuting su 2 su 2 scalars are obtained both of sixth order in the g 2 basis elements they are in fact combinations of five scalars of different tensorial types indicating that the functionally independent ones are in general insufficient to provide the lowest order commuting scalars an expression for the sixth order invariant of g 2 is also obtained

application of invariant imbedding to the estimation of process duration abstract this work deals with the application of invariant imbedding to solve a particularly important design problem namely the duration of the process a numerical example is used to illustrate the approach the advantage of this approach is its straightforward nature and uses only the usual design data it avoids any iterations and thus no convergence problems need to be considered

nonscalar extension of shift operator techniques for su 3 in an o 3 basis i theory a set of relations is set up which connect quadratic products of the shift operators okl k 0 1 2 which are nonscalar with respect to the o 3 subgroup of su 3 the usefulness of these relations is illustrated by the calculation of the eigenvalues of the scalar shift operator o0l for various irreducible representations p q of su 3

totally symmetric irreducible representations of the group so 6 in the principal so 3 subgroup basis explicit matrix elements are found for the generators of the group so 6 in an arbitrary totally symmetric irreducible representation using the physical principal so 3 subgroup in the chain so 6 so 5 so 3 the internal one missing label problem is solved through the definition of intrinsic states associated to the su 2 su 2 subgroup in the chain so 6 so 5 su 2 su 2 and out of which is projected a complete set of states in the physical basis by integrations over the physical rotation group manifold the matrix elements of the so 6 generators in the su 2 su 2 basis are themselves obtained by the intermediate use of an su 2 su 2 u 1 basis the latter group being a subgroup of so 6 but not of so 5

maximum likelihood estimation of models for residual covariance in spatial regression we describe the maximum likelihood method for fitting the linear model when residuals are correlated and when the covariance among the residuals is determined by a parametric model containing unknown parameters observations are assumed to be gaussian we give conditions which ensure consistency and asymptotic normality of the estimators our main concern is with the analysis of spatial data and in this context we describe some simulation experiments to assess the small sample behaviour of estimators we also discuss an application of the spectral approximation to the likelihood for processes on a lattice

an eigenlength problem arising from a model of patch formation abstract the patch formation model of del grosso gerardi and marchetti 1 is reformulated as an eigenlength problem the latter is then solved numerically by using recent results of elder 2 3 in which the method of invariant imbedding is extended to linear problems having a singularity of the first kind the results agree very well with those previously obtained 4 for the eigenvalue problem by means of the frobenius series sturmian theory and a certain monotone iteration