Stochastic Realization Research Articles

Schloegl’s second model for autocatalysis with particle diffusion: Lattice-gas realization exhibiting generic two-phase coexistence

We analyze a discontinuous nonequilibrium phase transition between an active (or reactive) state and a poisoned (or extinguished) state occurring in a stochastic lattice-gas realization of Schloegl's second model for autocatalysis. This realization, also known as the quadratic contact process, involves spontaneous annihilation, autocatalytic creation, and diffusion of particles on a square lattice, where creation at empty sites requires a suitable nearby pair of particles. The poisoned state exists for all annihilation rates p>0 and is an absorbing particle-free "vacuum" state. The populated active steady state exists only for p below a critical value, p(e). If p(f) denotes the critical value below which a finite population can survive, then we show that p(f)<p(e). This strict inequality contrasts a postulate of Durrett, and is a direct consequence of the occurrence of coexisting stable active and poisoned states for a finite range p(f)<or=p<or=p(e) (which shrinks with increasing diffusivity). This so-called generic two-phase coexistence markedly contrasts behavior in thermodynamic systems. However, one still finds metastability and nucleation phenomena similar to those in discontinuous equilibrium transitions.

Notched concrete beams under bending -- calculations of size effects within stochastic elasto-plasticity with non-local softening

Numerical FE investigations of a deterministic and statistical size effect in notched concrete beams of a similar geometry under three-point bending were performed. The FE analyses were carried out with four different beam sizes. Deterministic calculations were performed assuming constant values of tensile strength. In turn, in statistical calculations, the tensile strength took the form of random spatial fields described by a truncated Gaussian random distribution. In order to reduce the number of stochastic realizations without loosing the accuracy of the calculations, Latin hypercube sampling was applied. The numerical results were compared with the corresponding laboratory tests. The numerical outcomes show that the bearing capacity of beams and their ductility increase with decreasing specimen size. If the distribution of the tensile strength is stochastically distributed, the mean beam strength is always smaller than the deterministic value.

An “Indefinite Realization” Algorithm via Riccati Difference Equation

This paper studies a realization algorithm for modeling dynamical systems subject to a bounded error. An iterative algorithm is developed based on “indefinite realization”, which is a generalization of stochastic realization. A Riccati difference equation for indefinite realization is derived, and it is shown that the Riccati recursion converges to the stabilizing solution to the steady state Riccati equation under certain assumptions. Numerical simulation results are also included.

Continuous-Time Model Identification and State Estimation Using Non-Uniformly Sampled Data

This paper presents theory, algorithms and validation results for system identification of continuous-time state-space models from finite non-uniformly sampled input-output sequences. The algorithms developed are methods of model identification and stochastic realization adapted to the continuous-time model context using non-uniformly sampled input-output data. The resulting model can be decomposed into an input-output model and a stochastic innovations model. For state estimation dynamics, we have designed a procedure to provide separate continuous-time temporal update and error-feedback update based on non-uniformly sampled input-output data. Stochastic convergence analysis is provided.

A Simple Stochastic Model for Generating Broken Cloud Optical Depth and Cloud-Top Height Fields

Abstract A simple and fast algorithm for generating two correlated stochastic two-dimensional (2D) cloud fields is described. The algorithm is illustrated with two broken cumulus cloud fields: cloud optical depth and cloud-top height retrieved from the Moderate Resolution Imaging Spectroradiometer (MODIS). Only two 2D fields are required as an input. The algorithm output is statistical realizations of these two fields with approximately the same correlation and joint distribution functions as the original ones. The major assumption of the algorithm is statistical isotropy of the fields. In contrast to fractals and the Fourier filtering methods frequently used for stochastic cloud modeling, the proposed method is based on spectral models of homogeneous random fields. To retain the same probability density function as the (first) original field, the method of inverse distribution function is used. When the spatial distribution of the first field has been generated, a realization of the correlated second field is simulated using a conditional distribution matrix. This paper serves as a theoretical justification of the publicly available software “Simulation of a two-component cloud field,” which has been recently released. Although 2D rather than full 3D, the stochastic realizations of two correlated cloud fields that mimic statistics of given fields have proven to be very useful to study 3D radiative transfer features of broken cumulus clouds for a better understanding of shortwave radiation and the interpretation of remote sensing retrievals.

Nearest-neighbour modelling of reciprocal chains

This paper focuses on the class of finite-state, discrete-index, reciprocal processes (reciprocal chains). Such a class of processes seems to be a suitable setup in many applications and, in particular, it appears well-suited for image-processing. While addressing this issue, the aim is 2-fold: theoretic and practical. As to the theoretic purpose, some new results are provided: first, a general stochastic realization result is provided for reciprocal chains endowed with a known, arbitrary, distribution. Such a model has the form of a fixed-degree, nearest-neighbour polynomial model. Next, the polynomial model is shown to be exactly linearizable, which means it is equivalent to a nearest-neighbour linear model in a different set of variables. The latter model turns out to be formally identical to the Levi–Frezza–Krener linear model of a Gaussian reciprocal process, although actually non-linear with respect to the chain's values. As far as the practical purpose is concerned, in order to yield an example of application an estimation issue is addressed: a suboptimal (polynomial-optimal) solution is derived for the smoothing problem of a reciprocal chain partially observed under non-Gaussian noise. To this purpose, two kinds of boundary conditions (Dirichlet and Cyclic), specifying the reciprocal chain on a finite interval, are considered, and in both cases the model is shown to be well-posed, in a ‘wide-sense’. Under this view, some well-known representation results about Gaussian reciprocal processes extend, in a sense, to a ‘non-Gaussian’ case.

Information Gathering Externalities in Product Markets

Goods and services vary along a number of dimensions independently. Customers can choose to acquire information to assess the quality of some dimensions and not others. Their choices affect firms incentives to invest in quality and so lead to indirect externalities in consumers choices. We illustrate these ideas in a simple model with a monopolist selling a product with two characteristics, investment in quality with stochastic realizations, and heterogeneousconsumers. Consumers in choosing which information to acquire do not consider the effects on firm investment incentives and so there are indirect externalities in information gathering. Therefore, a fall in the cost of acquiring information, by changing the pattern of consumers information gathering and thereby firm investment, can paradoxically reduce consumer surplus, profits, and welfare. We briefly consider a number of potential extensions and in particular, highlight a benefit of diversity in tastes.

Simulated-annealing-based conditional simulation for the local-scale characterization of heterogeneous aquifers

Simulated-annealing-based conditional simulations provide a flexible means of quantitatively integrating diverse types of subsurface data. Although such techniques are being increasingly used in hydrocarbon reservoir characterization studies, their potential in environmental, engineering and hydrological investigations is still largely unexploited. Here, we introduce a novel simulated annealing (SA) algorithm geared towards the integration of high-resolution geophysical and hydrological data which, compared to more conventional approaches, provides significant advancements in the way that large-scale structural information in the geophysical data is accounted for. Model perturbations in the annealing procedure are made by drawing from a probability distribution for the target parameter conditioned to the geophysical data. This is the only place where geophysical information is utilized in our algorithm, which is in marked contrast to other approaches where model perturbations are made through the swapping of values in the simulation grid and agreement with soft data is enforced through a correlation coefficient constraint. Another major feature of our algorithm is the way in which available geostatistical information is utilized. Instead of constraining realizations to match a parametric target covariance model over a wide range of spatial lags, we constrain the realizations only at smaller lags where the available geophysical data cannot provide enough information. Thus we allow the larger-scale subsurface features resolved by the geophysical data to have much more due control on the output realizations. Further, since the only component of the SA objective function required in our approach is a covariance constraint at small lags, our method has improved convergence and computational efficiency over more traditional methods. Here, we present the results of applying our algorithm to the integration of porosity log and tomographic crosshole georadar data to generate stochastic realizations of the local-scale porosity structure. Our procedure is first tested on a synthetic data set, and then applied to data collected at the Boise Hydrogeophysical Research Site.

Real‐time groundwater flow modeling with the Ensemble Kalman Filter: Joint estimation of states and parameters and the filter inbreeding problem

Real‐time groundwater flow modeling with filter methods is interesting for dynamical groundwater flow systems, for which measurement data in real‐time are available. The Ensemble Kalman Filter (EnKF) approach is used here to update states together with parameters by adopting an augmented state vector approach. The performance of EnKF is investigated in a synthetic study with a two‐dimensional transient groundwater flow model where (1) only the recharge rate is spatiotemporally variable, (2) only transmissivity is spatially variable with σlnT2 = 1.0 or (3) with σlnT2 = 2.7, and (4) both recharge rate and transmissivity are uncertain (a combination of (1) and (3)). The performance of EnKF for simultaneous state and parameter estimation in saturated groundwater flow problems is investigated in dependence of the number of stochastic realizations, the updating frequency and updating intensity of log‐transmissivity, the amount of measurements in space and time, and the method (iterative versus noniterative EnKF), among others. Satisfactory results were also obtained if both transmissivity and recharge rate were uncertain. However, it was found that filter inbreeding is much more severe if hydraulic heads and transmissivities are jointly updated than if only hydraulic heads are updated. The filter inbreeding problem was investigated in more detail and could be strongly reduced with help of a damping parameter, which limits the intensity of the perturbation of the log‐transmissivity field. An additional reduction of filter inbreeding could be achieved by combining two measures: (1) inflating the elements of the predicted state covariance matrix on the basis of a comparison between the model uncertainty and the observed errors at the measurement points and (2) starting the flow simulations with a very large number of realizations and then sampling the desired number of realizations after one simulation time step by minimizing the differences between the local cpdfs (and bivariate cpdfs) of hydraulic head for the large ensemble and the corresponding cpdfs for the reduced ensemble. The two measures, which cause very limited CPU costs, allowed making 100 stochastic realizations for the reproduction of the states as efficient as 200–500 untreated stochastic realizations.

Kolmogorov–Smirnov test for spatially correlated data

The Kolmogorov–Smirnov test is a convenient method for investigating whether two underlying univariate probability distributions can be regarded as undistinguishable from each other or whether an underlying probability distribution differs from a hypothesized distribution. Application of the test requires that the sample be unbiased and the outcomes be independent and identically distributed, conditions that are violated in several degrees by spatially continuous attributes, such as topographical elevation. A generalized form of the bootstrap method is used here for the purpose of modeling the distribution of the statistic D of the Kolmogorov–Smirnov test. The innovation is in the resampling, which in the traditional formulation of bootstrap is done by drawing from the empirical sample with replacement presuming independence. The generalization consists of preparing resamplings with the same spatial correlation as the empirical sample. This is accomplished by reading the value of unconditional stochastic realizations at the sampling locations, realizations that are generated by simulated annealing. The new approach was tested by two empirical samples taken from an exhaustive sample closely following a lognormal distribution. One sample was a regular, unbiased sample while the other one was a clustered, preferential sample that had to be preprocessed. Our results show that the p-value for the spatially correlated case is always larger that the p-value of the statistic in the absence of spatial correlation, which is in agreement with the fact that the information content of an uncorrelated sample is larger than the one for a spatially correlated sample of the same size.

Reservoir characterization with iterative direct sequential co-simulation: Integrating fluid dynamic data into stochastic model

This paper presents a novel approach of inverse modelling to integrate fluid dynamic data into static model of reservoirs. The inverse procedure for history matching is achieved with an iterative process of successive co-simulations of reservoir characteristics in order to minimize an objective function of dynamic responses. Two main versions of inverse algorithm with a global and regional field perturbation are proposed. Direct Sequential Simulation and Co-simulation ( Co-DSS) are used to obtain convergent stochastic realizations to the dynamic objectives. Either of the algorithms, global and regional perturbation, proposed and evaluated led to satisfactory results with an acceptable iterations number.

Investigating Error Metrics for Satellite Rainfall Data at Hydrologically Relevant Scales

Abstract This paper addresses the following open question: What set of error metrics for satellite rainfall data can advance the hydrologic application of new-generation, high-resolution rainfall products over land? The authors’ primary aim is to initiate a framework for building metrics that are mutually interpretable by hydrologists (users) and algorithm developers (data producers) and to provide more insightful information on the quality of the satellite estimates. In addition, hydrologists can use the framework to develop a space–time error model for simulating stochastic realizations of satellite estimates for quantification of the implication on hydrologic simulation uncertainty. First, the authors conceptualize the error metrics in three general dimensions: 1) spatial (how does the error vary in space?); 2) retrieval (how “off” is each rainfall estimate from the true value over rainy areas?); and 3) temporal (how does the error vary in time?). They suggest formulations for error metrics specific to each dimension, in addition to ones that are already widely used by the community. They then investigate the behavior of these metrics as a function of spatial scale ranging from 0.04° to 1.0° for the Precipitation Estimation from Remotely Sensed Information Using Artificial Neural Networks (PERSIANN) geostationary infrared-based algorithm. It is observed that moving to finer space–time scales for satellite rainfall estimation requires explicitly probabilistic measures that are mathematically amenable to space–time stochastic simulation of satellite rainfall data. The probability of detection of rain as a function of ground validation rainfall magnitude is found to be most sensitive to scale followed by the correlation length for detection of rain. Conventional metrics such as the correlation coefficient, frequency bias, false alarm ratio, and equitable threat score are found to be modestly sensitive to scales smaller than 0.24° latitude/longitude. Error metrics that account for an algorithm’s ability to capture rainfall intermittency as a function of space appear useful in identifying the useful spatial scales of application for the hydrologist. It is shown that metrics evolving from the proposed conceptual framework can identify seasonal and regional differences in reliability of four global satellite rainfall products over the United States more clearly than conventional metrics. The proposed framework for building such error metrics can lay a foundation for better interaction between the data-producing community and hydrologists in shaping the new generation of satellite-based, high-resolution rainfall products, including those being developed for the planned Global Precipitation Measurement (GPM) mission.

Passage Time Statistics in a Stochastic Verhulst Model

Using the stochastic paths perturbation approach analytic individual realizations of a stochastic Verhulst model are introduced. The escape of the unstable state is studied for any kind of noise from these individual realizations. We infer from these paths the statistics of the first passage time distribution invoking the solution of an explicit equation with a random coefficient. A stochastic population Verhulst's dynamics with small perturbations of the Wiener class is explicitly worked out. The method can also be implemented for other type of stochastic perturbations like Poisson-noise (shot white pulses), etc.

Stochastic analysis of dynamic interaction between train and railway turnout

The performance of a railway turnout (switch and crossing) is influenced by a large number of input parameters of the complex train–turnout system. To reach a robust design that performs well for different traffic situations, random distributions (scatter) of these inputs need to be accounted for in the design process. Stochastic analysis methods are integrated with a simulation model of the dynamic interaction between train and turnout. For a given nominal layout of the turnout, using design of experiments methodology and a two-level fractional factorial screening design, four parameters (axle load, wheel–rail friction coefficient, and wheel and rail profiles) are identified to be the most significant. These parameters are further investigated using a three-level full factorial design and stochastic analysis. The random distributions of transverse wheel profile and set of transverse rail profiles along the switch panel are accounted for by the Karhunen–Loève expansion technique. The influence of the random distributions of the input parameters on the statistical outputs of wheel–rail contact forces, wear and rolling contact fatigue is assessed using Latin hypercube sampling to generate a number of stochastic load realizations.

Sub-grid parameterisation of fluvio-deltaic processes and architecture in a basin-scale stratigraphic model

We present a parameterisation of fluvio-deltaic drainage network evolution and alluvial architecture in a basin-scale 2-DH model. The model setup is capable of producing convergent and divergent channel networks. Major elements are the alluvial-ridge aggradation and the coupled overbank deposition, the dimension and style of the channel belt and the sub-grid stratigraphic expression. Avulsions are allowed to develop out of randomly instigated crevasses. Channel stability is modelled one dimensionally by calculating the flow and sediment transport at prospective avulsion nodes. The ultimate fate of crevasses (failed avulsion, successful avulsion, stable bifurcation) depends on the ratio of cross-valley and in-channel gradients in the local neighbourhood of the grid cell under consideration and on the amount and distribution of the suspended sediment load in the water column. The sub-grid parameterisation yields implicit knowledge of the alluvial architecture, which may be analysed stochastically. Stochastic realisations of the alluvial architecture allow us to investigate the relationship between basin-fill architecture and small-scale alluvial architecture, which is likely to improve geological reservoir modelling of these notoriously complex deposits. Modelling results under conditions of time-invariant forcing indicate significant quasi-cyclic autogenic behaviour of the fluvio-deltaic system. Changes in the avulsion frequency are correlated with the number and length of distributary channels, which are in turn related to alternating phases of progradational and aggradational delta development. The resulting parasequences may be difficult to distinguish from their allogenically induced counterparts.

A hydrogeophysical synthetic model generator

HGmod is a computer program that builds on stochastic realizations of porosity fields to derive electrical conductivity, dielectric permittivity and hydraulic permeability models. The presence of clay, the influence of salinity as well as temperature of the fluid of imbibition are taken into account in the underlying formulations. The saturated and unsaturated zones are also considered through the application of a saturation profile on the porosity field. A micro-geometrical model is used to relate the porosity to the clay fraction. This model also used to derive an expression for the pore specific surface of the sand–clay mixture. The specific surface is subsequently used to compute the conduction at the pore surface when building electrical conductivity models. Dielectric permittivity fields are built by successive applications of either the Hanai–Bruggeman or Maxwell–Garnett mixing models, depending on the relative proportions of sand, clay, water or air. In addition, the dielectric permittivity of water and clay follow a Cole–Cole behavior. HGmod is therefore a versatile tool useful to generate synthetic datasets needed to anticipate the geophysical response under specific conditions and to study hydrogeophysical sensitivity or resolution analysis.

Stochastic realization approach to the efficient simulation of phase screens

The phase screen method is a well-established approach to take into account the effects of atmospheric turbulence in astronomical seeing. This is of key importance in designing adaptive optics for new-generation telescopes, in particular in view of applications such as exoplanet detection or long-exposure spectroscopy. We present an innovative approach to simulate turbulent phase that is based on stochastic realization theory. The method shows appealing properties in terms of both accuracy in reconstructing the structure function and compactness of the representation.

Modeling thermal effects on nonlinear wave motion in biopolymers by a stochastic discrete nonlinear Schrödinger equation with phase damping

A mathematical model is introduced for weakly nonlinear wave phenomena in molecular systems like DNA and protein molecules that includes thermal effects: exchange of heat energy with the surrounding aqueous medium. The resulting equation is a stochastic discrete nonlinear Schrödinger equation with focusing cubic nonlinearity and "Thermal'' terms modeling heat input and loss: PDSDNLS. <br> New numerical methods are introduced to handle the unusual combination of a conservative equation, stochastic, and fully nonlinear terms. Some analysis is given of accuracy needs, and the special issues of time step adjustment in stochastic realizations. Numerical studies are presented of the effects of thermalization on solitons, including damping induced <em>self-trapping</em> of wave energy, a discrete counterpart of single-point blowup.

Scenario-based reservoir modelling: the need for more determinism and less anchoring

Abstract The scenario-based reservoir modelling method places a strong emphasis on the deterministic control of the model design, contrasting with strongly probabilistic approaches in which effort is focused on the ‘richness’ of a geostatistical algorithm to derive multiple stochastic realizations. Scenario-based approaches also differ from traditional ‘rationalist’ modelling, which often involves the construction of only a single, best-guess or base-case model. The advantage of scenario modelling is that there is no requirement to anchor on a preferred, base-case model, and it is argued here that selection of a base case is detrimental to achieving appropriately wide uncertainty ranges. Multiple-deterministic scenario modelling also carries the advantage of maintaining explicit dependency between model parameters and the ultimate model outcome, such as a development plan. The approach has been applied widely to new fields, where multiple deterministic reservoir simulations of a suite of static models can be easily handled. The approach has also been extended to mature fields, in which practical approaches to multiple-history matching are required. Mature field scenario modelling, in particular, illustrates the weaknesses of base-case modelling, and delivers a strong statement on the non-uniqueness of modelling in general. Current issues are the need to develop better methodologies for multiple-history matching, and for linking discrete, deterministic, scenario-based outcomes to probabilistic reporting. Experimental design methods offer a solution to the latter issue, and a simple, practical workflow for its application is described.

Important Moments in Systems and Control

The moment problem matured from its various special forms in the late 19th and early 20th centuries to a general class of problems that continues to exert profound influence on the development of analysis and its applications to a wide variety of fields. In particular, the theory of systems and control is no exception, where the applications have historically been to circuit theory, optimal control, robust control, signal processing, spectral estimation, stochastic realization theory, and the use of the moments of a probability density. Many of these applications are also still works in progress. In this paper, we consider the generalized moment problem, expressed in terms of a basis of a finite-dimensional subspace P of the Banach space $C[a,b]$ and a “positive” sequence c, but with a new wrinkle inspired by the applications to systems and control. We seek to parameterize solutions which are positive “rational” measures in a suitably generalized sense. Our parameterization is given in terms of smooth objects. In particular, the desired solution space arises naturally as a manifold which can be shown to be diffeomorphic to a Euclidean space and which is the domain of some canonically defined functions. The analysis of these functions, and related maps, yields interesting corollaries for the moment problem and its applications, which we compare to those in the recent literature and which play a crucial role in part of our proof.