Conditional co-simulation of continuous and categorical variables for geostatistical applications

The modeling of uncertainty in continuous and categorical regionalized variables is a common issue in the geosciences. We present a hybrid continuous/categorical model, in which the continuous variable is represented by the transform of a Gaussian random field, while the categorical variable is obtained by truncating one or more Gaussian random fields. The dependencies between the continuous and categorical variables are reproduced by assuming that all the Gaussian random fields are spatially cross-correlated. Algorithms and computer programs are proposed to infer the model parameters and to co-simulate the variables, and illustrated through a case study on a mining data set.     

Generalized perturbation-based stochastic finite element method in elastostatics

Generalised nth order stochastic perturbation technique that can be applied to solve some boundary value or boundary initial problems in computational physics and/or engineering with random coefficients is presented here. This technique is implemented in conjunction with the finite element method (FEM) to model 1D linear elastostatics problem with a single random variable. Main motivation of this work is to improve essentially the accuracy of the stochastic perturbation technique, which in its second order realization was ineffective for large variations of the input random fields. The nth order approach makes it possible to specify the accuracy of the computations a priori for the expected values and variances separately. The symbolic computer program is employed to perform computational studies on convergence of the first two probabilistic moments for simple unidirectional tension of the bar. These numerical studies verify the influence of coefficient of variation of the random input and, in the same time, of the perturbation parameter on the first four probabilistic moments of the final solution vector.     

Experimental evaluation of solution approaches for the K-route maximum flow problem

We describe the implementation and testing of approaches for the solution of the maximum K-route flow problem. More specifically, we focus on Kishimoto's algorithm, the binary search algorithm and the K-route flow algorithm. With the aim to compare the performances of the aforementioned methods, we report results of computational experiments carried out on a large set of randomly generated problems with varying topology and arc capacities. The numerical results show that Kishimoto's algorithm is generally the fastest for solving fully random networks and 3-D grid random networks. The binary search algorithm provides the best performance in solving grid random networks. The K-route flow algorithm is generally the fastest for solving fully random networks and 3-D grid random networks with capacity range of [1,1000] and certain values of K.     

Scalable processing of continuous K-nearest neighbor queries with uncertain velocity

Continuous K-nearest neighbor (CKNN) query is an important type of spatio-temporal queries. Given a time interval [t_s, t_e] and a moving query object q, a CKNN query is to find the K-nearest neighbors (KNNs) of q at each time instant within [t_s, t_e]. In this paper, we focus on the issue of scalable processing of CKNN queries over moving objects with uncertain velocity. Due to the large amount of CKNN queries that need to be evaluated concurrently, efficiently processing such queries inevitably becomes more complicated. We propose an index structure, namely the CI-tree, to predetermine and organize the candidates for each query issued by the user from anywhere and anytime. When the CKNN queries are evaluated, their corresponding candidates can be rapidly retrieved by traversing the CI-tree so that the processing time is greatly reduced. A comprehensive set of experiments is performed to demonstrate the effectiveness and the efficiency of the CI-tree.     

An approximation of random field with a bounded discrete parameter space

The model of n-dimensional random field with a bounded discrete parameter space is presented. To obtain simple method for generating random field with a computer it is assumed the field is Gaussian and Markovian. The special cases of two- and three-dimensional random fields are considered exactly. A numerical example is presented.     

Stable polyhedra in parameter space

A typical uncertainty structure of a characteristic polynomial is P(s)=A(s)Q(s)+B(s) with A(s) and B(s) fixed and Q(s) uncertain. In robust controller design Q(s) may be a controller numerator or denominator polynomial; an example is the PID controller with Q(s)=K_I+K_Ps+K_Ds². In robustness analysis Q(s) may describe a plant uncertainty. For fixed imaginary part of Q(jω), it is shown that Hurwitz stability boundaries in the parameter space of the even part of Q(jω) are hyperplanes and the stability regions are convex polyhedra. A dual result holds for fixed real part of Q(jω). Also σ-stability with the real parts of all roots of P(s) smaller than σ is treated.Under the above conditions, the roots of P(s) can cross the imaginary axis only at a finite number of discrete “singular” frequencies. Each singular frequency generates a hyperplane as stability boundary. An application is robust controller design by simultaneous stabilization of several representatives of A(s) and B(s) by a PID controller. Geometrically, the intersection of convex polygons must be calculated and represented tomographically for a grid on K_P.     

TBSIM: A computer program for conditional simulation of three-dimensional Gaussian random fields via the turning bands method

The simulation of spatially correlated Gaussian random fields is widespread in geologic, hydrologic and environmental applications for characterizing the uncertainty about the unsampled values of regionalized attributes. In this respect, the turning bands method has received attention among practitioners, for it allows multidimensional simulations to be generated at the CPU cost of one-dimensional simulations.This work provides and documents a set of computer programs for (i) constructing three-dimensional realizations of stationary and intrinsic Gaussian random fields, (ii) conditioning these realizations to a set of data and (iii) back-transforming the Gaussian values to the original attribute units. Such programs can deal with simulations over large domains and handle anisotropic and nested covariance models.The quality of the proposed programs is examined through an example consisting of a non-conditional simulation of a spherical covariance model. The artifact banding in the simulated maps is shown to be negligible when thousands of lines are used. The main parameters of the univariate and bivariate distributions, as well as their expected ergodic fluctuations, also prove to be accurately reproduced.     

Computer implementation of the constraint method

A computer program based on the constraint method approach to finite elements is described. The program permits the user to select arbitrarily polynomial orders for the various fields to be approximated. A set of nodal variables enforcing C¹ continuity for arbitrary polynomial order is presented, and those aspects of the program algorithm which differ from conventional finite element programs are described, including the numerical determination of a set of independent variables. An example is presented and the results obtained by the constraint method program compared with results from a conventional program. The constraint method approach is seen to be competitive with the conventional approach.     

Holographic Method for Reconstruction of Random Rough Surface

A holographic principle for computer reconstruction of random rough surface (RRS) and determination of the surface parameters is used. The random function describing the surface is expanded in a Fourier series. The scattering from the RRS light field is approximated as a field, obtained as result of diffraction from sinusoidal gratings. The mixed field (obtained from the mixing of the diffracted field and the field of the laser beam) carries information about the phase of each sinusoidal grating. Sections of the diffracted and mixing fields are correspondingly registered by photodiode array. The first, the second and the third diffraction orders from each grating are taken into account for the surface reconstruction. The statistical distribution and the autocorrelation function of the surface height fluctuation are calculated. The surface parameters of mean roughness, root mean square and correlation length are determined. The studied surface, having a Gaussian statistic, is measured [from the work of Daintyet al.Talystep profilometer. The values of the above-mentioned surface parameters calculated in Daintyet al.are in good agreement with those obtained in the present work.     

Simulation of constituent transport using a reduced 3D constituent transport model (CTM) driven by HF Radar: Model application and error analysis

Data-driven constituent transport models (CTM), which take surface current measurements from High Frequency (HF) Radar as input can be applied within the context of real-time environmental monitoring, oceanographic assessment, response to episodic events, as well as search and rescue in surface waters.This paper discusses a numerical method that allows for the evaluation of diffusion coefficients in anisotropic flow fields from surface current measurements using HF Radar. The numerical scheme developed was incorporated into a data-driven CTM and through model error analyses, the effects of using spatially variable transport coefficients on model predictions were examined. The error analyses were performed on the model by varying the cell Reynolds number, Re = f(u,K,Δx) between 0.15 and 100, where u is the velocity vector within the flow field, K is a diffusivity tensor and Δx is the computational grid cell size.Two instantaneous releases of conservative material were then modeled, the model being initialized at two different locations within the domain. From the two simulation runs, marked differences in the predicted spatial extent of the conservative material resulting from the spatially varying diffusivity values within the study area were observed. Model predictions in terms of variance or size estimates of a diffusing patch were found to be more affected from using inaccurate diffusivity estimates, and less affected from using inaccurate current measurements. The largest errors occurred at Re > 2 associated with changing diffusivity values, going up to as much as a 25-fold difference in variance estimates at Re = 100. Very little effect on variance estimates due to varying velocity values were observed even at Re > 2. This model was applied within the framework of constituent tracking to Corpus Christi Bay in the Texas Gulf of Mexico region.     

Texture classification using features derived from random field models

This paper presents a new feature extraction method for classifying a texture image into one of the l possible classes C_i, i=1,…,l. It is assumed that the given M × M image characterized by a set of intensity levels, {y(s₁,S₂), 0≤s_s,s₂≤M?1}, is a realization of an underlying random field model, known as the Simultaneous Autoregressive Model (SAR). This model is characterized by a set of parameters φ whose probability density function p_i(φ), depends on the class to which the image belongs. First it is shown that the maximum likelihood estimate (M.L.E.) $\text{φ}{}^1$, of φ is an appropriate feature vector for classification purposes. The optimum Bayes classifier which minimizes the average probability of classification error, is then designed using $\text{φ}{}^1$. Finally the efficiency of the feature vector is demonstrated through experimental results obtained with some natural texture data and a simpler quadratic mean classifier.     

Column sizing design aid

Presented in this paper is an interactive computer program which will aid the engineer in selecting the trial section properties of a column for the iterative design and analysis process. An algorithm is developed around the effective length factor (K) and the end restraint factor (ψ) for an unbraced frame. The output is a table of values of the ratio of the column moment of inertia to the beam moment of inertia (IR) as a function of the effective length factor (K). The program is written in standard Fortran.     

First Arrival Seismic Tomography (FAST) vs. PStomo_eq applied to crooked line seismic data from the Siljan ring area

When seismic profiles deviate significantly from straight lines, the results from 2D traveltime inversion programs will be in error due to the inherent 3D component present in the data. Thus, it is necessary to use a program that can handle the 3D aspects of the acquisition geometry. This study compares the performance and results from two computer programs for 3D seismic tomography. These algorithms are the package for First Arrival Seismic Tomography (FAST) and a Local Earthquake tomography program, PStomo_eq. Although both codes invert for the velocity field using the conjugate gradient solver LSQR, the common smoothness constraint is handled differently. In addition, the programs do not incorporate the same options for user-specified constraints. These differences in implementation are clearly observed in the inverted velocity fields obtained in this study. Both FAST and PStomo_eq are applied to synthetic and real data sets with crooked line geometry. First arrival traveltimes from seismic data acquired in the Siljan ring impact area are used for the real data set test. The results show that FAST gives smoother models than PStomo_eq. On the real data set PStomo_eq showed a better correlation to the information at hand. Different criteria exist for what is desirable in a model; thus, the choice of which program to use will mostly depend on the particular goals of the study.     

Deriving maximal light use efficiency from coordinated flux measurements and satellite data for regional gross primary production modeling

Remote sensing models based on light use efficiency (LUE) provide promising tools for monitoring spatial and temporal variation of gross primary production (GPP) at regional scale. In most of current LUE-based models, maximal LUE (ε_max) heavily relies on land cover types and is considered as a constant, rather than a variable for a certain vegetation type or even entire eco-region. However, species composition and plant functional types are often highly heterogeneous in a given land cover class; therefore, spatial heterogeneity of ε_max must be fully considered in GPP modeling, so that a single cover type does not equate to a single ε_max value. A spatial dataset of ε_max accurately represents the spatial heterogeneity of maximal light use would be of significant beneficial to regional GPP models. Here, we developed a spatial dataset of ε_max by integrating eddy covariance flux measurements from 14 field sites in a network of coordinated observation across northern China and satellite derived indices such as enhanced vegetation index (EVI) and visible albedo to simulate regional distribution of GPP. This dynamic modeling method recognizes the spatial heterogeneity of ε_max and reduces the uncertainties in mixed pixels. Further, we simulated GPP with the spatial dataset of ε_max generated above. Both ε_max and growing season GPP show complex patterns over northern China that reflect influences of humidity, green vegetation fractions, and land use intensity. “Green spots” such as oasis meadow and alpine forests in dryland and “brown spots” such as build-up and heavily degraded vegetation in the east are clearly captured by the simulation. The correlation between simulated GPP and EC measured GPP indicate that the simulated GPP from this new approach is well matched with flux-measured GPP. Those results have demonstrated the importance of considering ε_max as both a spatially and temporally variable values in GPP modeling.     

Parameter uncertainty and temporal dynamics of sensitivity for hydrologic models: A hybrid sequential data assimilation and probabilistic collocation method

In this study, a hybrid sequential data assimilation and probabilistic collocation (HSDAPC) approach is proposed for analyzing uncertainty propagation and parameter sensitivity of hydrologic models. In HSDAPC, the posterior probability distributions of model parameters are first estimated through a particle filter method based on streamflow discharge data. A probabilistic collocation method (PCM) is further employed to show uncertainty propagation from model parameters to model outputs. The temporal dynamics of parameter sensitivities are then generated based on the polynomial chaos expansion (PCE) generated by PCM, which can reveal the dominant model components for different catchment conditions. The maximal information coefficient (MIC) is finally employed to characterize the correlation/association between model parameter sensitivity and catchment precipitation, potential evapotranspiration and observed discharge. The proposed method is applied to the Xiangxi River located in the Three Gorges Reservoir area. The results show that: (i) the proposed HSDAPC approach can generate effective 2nd and 3rd PCE models which provide accuracy predictions; (ii) 2^nd-order PCE, which can run nearly ten time faster than the hydrologic model, can capably represent the original hydrological model to show the uncertainty propagation in a hydrologic simulation; (iii) the slow (R_s) and quick flows (R_q) in Hymod show significant sensitivities during the simulation periods but the distribution factor (α) shows a least sensitivity to model performance; (iv) the model parameter sensitivities show significant correlation with the catchment hydro-meteorological conditions, especially during the rainy period with MIC values larger than 0.5. Overall, the results in this paper indicate that uncertainty propagation and temporal sensitivities of parameters can be effectively characterized through the proposed HSDAPC approach.     

A new empirical model to retrieve soil moisture and roughness from C-band radar data

A new empirical model for the retrieval, at a field scale, of the bare soil moisture content and the surface roughness characteristics from radar measurements is proposed. The derivation of the algorithm is based on the results of three experimental radar campaigns conducted under natural conditions over agricultural areas. Radar data were acquired by means of several C-band space borne (SIR-C, RADARSAT) or helicopter borne (ERASME) sensors, operating in different configurations of polarization (HH or VV) and incidence angle. Simultaneously to radar acquisitions, a complete ground truth data base was built up with different surface condition measurements of the mean standard deviation (rms) height s, the correlation length l, and the volumetric surface moisture M_v. This algorithm is more specifically developed using the radar cross-section σ⁰ (HH polarization and 39° incidence angle off nadir), namely, σ_0HH,39, and the differential (HH polarization) radar cross-section Δσ₀=σ_0,23°−σ_0,39° in terms of an original roughness parameter, Z_s, namely Z_s=s²/l, and M_v. A good agreement is observed between model outputs and backscattering measurements over different test fields. Eventually, an inversion technique is proposed to retrieve Z_s and M_v from radar measurements.     

ASM: An adaptive simplification method for 3D point-based models

Due to the popularity of computer games and computer-animated movies, 3D models are fast becoming an important element in multimedia applications. In addition to the conventional polygonal representation for these models, the direct adoption of the original scanned 3D point set for model representation is recently gaining more and more attention due to the possibility of bypassing the time consuming mesh construction stage, and various approaches have been proposed for directly processing point-based models. In particular, the design of a simplification approach which can be directly applied to 3D point-based models to reduce their size is important for applications such as 3D model transmission and archival. Given a point-based 3D model which is defined by a point set P () and a desired reduced number of output samples n^s, the simplification approach finds a point set P_s which (i) satisfies |P_s|=n^s (|P_s| being the cardinality of P_s) and (ii) minimizes the difference of the corresponding surface S_s (defined by P_s) and the original surface S (defined by P). Although a number of previous approaches has been proposed for simplification, most of them (i) do not focus on point-based 3D models, (ii) do not consider efficiency, quality and generality together and (iii) do not consider the distribution of the output samples. In this paper, we propose an Adaptive Simplification Method (ASM) which is an efficient technique for simplifying point-based complex 3D models. Specifically, the ASM consists of three parts: a hierarchical cluster tree structure, the specification of simplification criteria and an optimization process. The ASM achieves a low computation time by clustering the points locally based on the preservation of geometric characteristics. We analyze the performance of the ASM and show that it outperforms most of the current state-of-the-art methods in terms of efficiency, quality and generality.     

Finite elements for elliptic problems with stochastic coefficients

We describe a deterministic finite element (FE) solution algorithm for a stochastic elliptic boundary value problem (sbvp), whose coefficients are assumed to be random fields with finite second moments and known, piecewise smooth two-point spatial correlation function. Separation of random and deterministic variables (parametrization of the uncertainty) is achieved via a Karhunen–Loève (KL) expansion. An O(N log N) algorithm for the computation of the KL eigenvalues is presented, based on a kernel independent fast multipole method (FMM). Truncation of the KL expansion gives an (M, 1) Wiener polynomial chaos (PC) expansion of the stochastic coefficient and is shown to lead to a high dimensional, deterministic boundary value problem (dbvp). Analyticity of its solution in the stochastic variables with sharp bounds for the domain of analyticity are used to prescribe variable stochastic polynomial degree r = (r₁, …, r_M) in an (M, r) Wiener PC expansion for the approximate solution. Pointwise error bounds for the FEM approximations of KL eigenpairs, the truncation of the KL expansion and the FE solution to the dbvp are given. Numerical examples show that M depends on the spatial correlation length of the random diffusion coefficient. The variable polynomial degree r in PC-stochastic Galerkin FEM allows to handle KL expansions with M up to 30 and r₁ up to 10 in moderate time.     

Effects of soil spatial variability on rainfall-induced landslides

This paper presents a probabilistic framework to assess the stability of unsaturated slope under rainfall. The effects of soil spatial variability on the probability of rainfall-induced slope failure (landslides) are investigated. Soil spatial variability is considered by modeling the saturated hydraulic conductivity of the soil (k_s) as a stationary lognormal random field. Subset simulation with a modified Metropolis–Hastings algorithm is used to estimate the probability of slope failure. It is demonstrated numerically that probabilistic analysis accounting for spatial variability of k_s can reproduce a shallow failure mechanism widely observed in real rainfall-induced landslides. This shallow failure is attributed to positive pore-water pressures developed in layers near the ground surface. In contrast, analysis assuming a homogeneous profile cannot reproduce a shallow failure except for the extreme case of infiltration flux being almost equal to k_s. Therefore, ignoring spatial variability leads to unconservative estimates of failure probability. The correlation length of k_s affects the probability of slope failure significantly. The applicability of subset simulation with a modified Metropolis–Hastings algorithm to assess the reliability of problems involving spatial variability is highlighted.