Stochastic Realization Research Articles

Exploring the Adequacy of Steady-State-Only Calibration

This paper explores the adequacy of steady-state-only calibration as a precursor to use of a groundwater model for decision-support. First, it reviews metrics by which a decision-support model should be judged. On the basis of these metrics, it establishes the shortcomings that a decision-support model may incur through foregoing transient calibration. These are 1) failure to reduce the uncertainties of management-salient model predictions to the extent that available data allows, and 2) creation of unquantifiable bias in management-salient predictions. Two methodologies for quantification of these deficiencies are proposed. The first of these addresses uncertainty reduction. This is relatively easy to implement, as it requires only that sensitivities of pertinent model outputs to a model’s parameters be calculated. The second methodology addresses predictive bias. Implementation of this second methodology is more expensive as it requires repeated calibration of a steady state model against stochastic realizations of a transient model.These methods are demonstrated using a synthetic case which explores the viability of steady-state-only calibration of models deployed to examine the impacts of pumping on stream flows and groundwater levels. It is demonstrated that, for some predictions of management interest, steady-state-only calibration is more than sufficient for this kind of decision-support modelling.

Clearness index forecasting: A comparative study between a stochastic realization method and a machine learning algorithm

With the increase in solar energy penetration in electric power systems, it is necessary to predict future solar radiation. With this, the solar power generation capacity can be predicted, allowing accurate scheduling for other sources. The solar radiation or the clearness index (CI) prediction was already presented using classical time series methods, artificial intelligence algorithms, among other approaches. In this paper, two methods are proposed to predict the CI. The first one solves a stochastic realization problem to find a mathematical model for the stochastic process that describes the CI. Then, a state observer is implemented to predict its future values. The second one performs the k-Nearest Neighbors (k-NN) regression algorithm for time series forecasting using two different approaches: the traditional k-NN and a recursive version proposed by the authors. As far as the authors know, the statistical and the recursive k-NN methods are different from the ones found in the literature to predict the CI. The methods were implemented and compared for short and long-term predictions. The results illustrate that one approach is more accurate than the other, depending on the prediction horizon. The comparison between the methods for different horizons is another contribution of this paper.

Probability elicitation using geostatistics in hydrocarbon exploration

The exploratory phase of a hydrocarbon field is a period when decision-supporting information is scarce while the drilling stakes are high. Each new prospect drilled brings more knowledge about the area and might reveal reserves, hence choosing such prospect is essential for value creation. Drilling decisions must be made under uncertainty as the available geological information is limited and probability elicitation from geoscience experts is key in this process. This work proposes a novel use of geostatistics to help experts elicit geological probabilities more objectively, especially useful during the exploratory phase. The approach is simpler, more consistent with geologic knowledge, more comfortable for geoscientists to use and, more comprehensive for decision-makers to follow when compared to traditional methods. It is also flexible by working with any amount and type of information available. The workflow takes as input conceptual models describing the geology and uses geostatistics to generate spatial variability of geological properties in the vicinity of potential drilling prospects. The output is stochastic realizations which are processed into a joint probability distribution (JPD) containing all conditional probabilities of the process. Input models are interactively changed until the JPD satisfactory represents the expert’s beliefs. A 2D, yet realistic, implementation of the workflow is used as a proof of concept, demonstrating that even simple modeling might suffice for decision-making support. Derivative versions of the JPD are created and their effect on the decision process of selecting the drilling sequence is assessed. The findings from the method application suggest ways to define the input parameters by observing how they affect the JPD and the decision process.

Measuring trajectories of environmental noise

In classical mechanics, a natural way to simplify a many body problem is to ``replace'' some of the elements of the composite system with surrogate force fields. In the realm of quantum mechanics, however, such a description is rarely compatible with the formalism of the theory. Nevertheless, the quantum version of external field models---the so-called noise representations---can be employed in certain circumstances. The mathematics behind these models indicate that the appearing fields typically exhibit random fluctuations, hence, the name noise field is more apt. In principle, measuring a classical force field is a trivial task; all that is needed is a probe equipped with an accelerometer that can be moved around while taking the measure of forces that affect it. Unfortunately, an analogous method cannot work in the quantum case. As indicated by the theory, the result of any measurement performed on the quantum system affected by a noise field always appear as an average over fluctuations, therefore, it is impossible to observe the noise fluctuations by measuring the state of the probe. Here, we will demonstrate that this limitation can be circumvented and that it is possible to expose the noise field fluctuations to direct observation. This allows one to sample the noise trajectories (stochastic realizations), store them, and later use them to simulate the dynamics of open quantum systems affected by the noise.

Quantitative uncertainty analysis of gravity disturbance. The case of the Geneva Basin (Switzerland)

Gravity data from the International Gravimetric Bureau and the Gravimetric Atlas of Switzerland have been used to evaluate their application and limitations as a subsurface investigation tool to constrain key geological structures in support of the georesources exploration in the Geneva Basin (GB). In this context, the application of an effective processing workflow able to produce a topography-free gravity disturbance and quantify its uncertainty is a crucial first step to delineate gravitational effects correlated to geologic sources of geothermal interest. This study focuses on an innovative processing workflow applied to publicly available gravity data in order to produce stochastic realizations of residuals of the topography-free gravity disturbance. The resulting realizations are compared to a standard interpolation technique, demonstrating the potential of the stochastic approach for georesources exploration in sedimentary basins.

Noise suppression in stochastic genetic circuits using PID controllers.

Inside individual cells, protein population counts are subject to molecular noise due to low copy numbers and the inherent probabilistic nature of biochemical processes. We investigate the effectiveness of proportional, integral and derivative (PID) based feedback controllers to suppress protein count fluctuations originating from two noise sources: bursty expression of the protein, and external disturbance in protein synthesis. Designs of biochemical reactions that function as PID controllers are discussed, with particular focus on individual controllers separately, and the corresponding closed-loop system is analyzed for stochastic controller realizations. Our results show that proportional controllers are effective in buffering protein copy number fluctuations from both noise sources, but this noise suppression comes at the cost of reduced static sensitivity of the output to the input signal. In contrast, integral feedback has no effect on the protein noise level from stochastic expression, but significantly minimizes the impact of external disturbances, particularly when the disturbance comes at low frequencies. Counter-intuitively, integral feedback is found to amplify external disturbances at intermediate frequencies. Next, we discuss the design of a coupled feedforward-feedback biochemical circuit that approximately functions as a derivate controller. Analysis using both analytical methods and Monte Carlo simulations reveals that this derivative controller effectively buffers output fluctuations from bursty stochastic expression, while maintaining the static input-output sensitivity of the open-loop system. In summary, this study provides a systematic stochastic analysis of biochemical controllers, and paves the way for their synthetic design and implementation to minimize deleterious fluctuations in gene product levels.

Conditional stochastic inversion of common-offset ground-penetrating radar reflection data

We have developed a stochastic inversion procedure for common-offset ground-penetrating radar (GPR) reflection measurements. Stochastic realizations of subsurface properties that offer an acceptable fit to GPR data are generated via simulated annealing optimization. The realizations are conditioned to borehole porosity measurements available along the GPR profile or equivalent measurements of another petrophysical property that can be related to the dielectric permittivity, as well as to geostatistical parameters derived from the borehole logs and the processed GPR image. Validation of our inversion procedure is performed on a pertinent synthetic data set and indicates that our method is capable of reliably recovering strongly heterogeneous porosity structures associated with surficial alluvial aquifers. This finding is largely corroborated through application of the methodology to field measurements from the Boise Hydrogeophysical Research Site near Boise, Idaho, USA.

Time-delayed and stochastic effects in a predator-prey model with ratio dependence and Holling type III functional response.

In this article, we derive and analyze a novel predator-prey model with account for maturation delay in predators, ratio dependence, and Holling type III functional response. The analysis of the system's steady states reveals conditions on predation rate, predator growth rate, and maturation time that can result in a prey-only equilibrium or facilitate simultaneous survival of prey and predators in the form of a stable coexistence steady state, or sustain periodic oscillations around this state. Demographic stochasticity in the model is explored by means of deriving a delayed chemical master equation. Using system size expansion, we study the structure of stochastic oscillations around the deterministically stable coexistence state by analyzing the dependence of variance and coherence of stochastic oscillations on system parameters. Numerical simulations of the stochastic model are performed to illustrate stochastic amplification, where individual stochastic realizations can exhibit sustained oscillations in the case, where deterministically the system approaches a stable steady state. These results provide a framework for studying realistic predator-prey systems with Holling type III functional response in the presence of stochasticity, where an important role is played by non-negligible predator maturation delay.

Sequential Gaussian simulation for geosystems modeling: A machine learning approach

Sequential Gaussian Simulation (SGSIM) as a stochastic method has been developed to avoid the smoothing effect produced in deterministic methods by generating various stochastic realizations. One of the main issues of this technique is, however, an intensive computation related to the inverse operation in solving the Kriging system, which significantly limits its application when several realizations need to be produced for uncertainty quantification. In this paper, a physics-informed machine learning (PIML) model is proposed to improve the computational efficiency of the SGSIM. To this end, only a small amount of data produced by SGSIM are used as the training dataset based on which the model can discover the spatial correlations between available data and unsampled points. To achieve this, the governing equations of the SGSIM algorithm are incorporated into our proposed network. The quality of realizations produced by the PIML model is compared for both 2D and 3D cases, visually and quantitatively. Furthermore, computational performance is evaluated on different grid sizes. Our results demonstrate that the proposed PIML model can reduce the computational time of SGSIM by several orders of magnitude while similar results can be produced in a matter of seconds.

Semiflexible Polymer Enclosed in a 3D Compact Domain

The conformational states of a semiflexible polymer enclosed in a volume V:=ℓ3 are studied as stochastic realizations of paths using the stochastic curvature approach developed in [Rev. E 100, 012503 (2019)], in the regime whenever 3ℓ/ℓp>1, where ℓp is the persistence length. The cases of a semiflexible polymer enclosed in a cube and sphere are considered. In these cases, we explore the Spakowitz–Wang–type polymer shape transition, where the critical persistence length distinguishes between an oscillating and a monotonic phase at the level of the mean-square end-to-end distance. This shape transition provides evidence of a universal signature of the behavior of a semiflexible polymer confined in a compact domain.

Estimating Caprock Impact on CO2 Migration in the Gassum Formation Using 2D Seismic Line Data

Realizable CO2 storage potential for saline formations without closed lateral boundaries depends on the combined effects of physical and chemical trapping mechanisms to prevent long-term migration out of the defined storage area. One such mechanism is the topography of the caprock surface, which may retain CO2 in structural pockets along the migration path. Past theoretical and modeling studies suggest that even traps too small to be accurately described by seismic data may play a significant role. In this study, we use real but scarce seismic data from the Gassum Formation of the Norwegian Continental shelf to estimate the impact of topographical features of the top seal in limiting CO2 migration. We seek to estimate the amount of macro- and sub-scale trapping potential of the formation based on a few dozen interpreted 2D seismic lines and identified faults. We generate multiple high-resolution realizations of the top surface, constructed to be faithful to both large-scale topography and small-scale statistical properties. The structural trapping and plume retardation potential of these top surfaces is subsequently estimated using spill-point (static) analysis and dynamical flow simulation. By applying these techniques on a large ensemble of top surface realizations generated using a combination of stochastic realizations and systematic variation of key model parameters, we explore the range of possible impacts on plume advancement, physical trapping and migration direction. The stochastic analysis of trapping capacity and retardation efficiency in statistically generated, sub-seismic resolution features may also be applied for surfaces generated from 3D data.

A methodology for phenomenological analysis of cumulative damage processes. Application to fatigue and fracture phenomena

Sample functions, i.e., stochastic process realizations, are used to define cumulative damage phenomena which end into an observable terminal state or failure. The complexity inherent to such phenomena justifies the use of phenomenological models associated with the evolution of a physical magnitude feasible to be monitored during the test. Sample functions representing the damage evolution may be identified, once normalized to the interval [0,1], with cumulative distribution functions (cdfs), generally, of the generalized extreme value (GEV) family. Though usually only a fraction of the whole damage evolution, according to the specific problem handled, is available from the test record, the phenomenological models proposed allow the whole damage process to be recovered. In this way, down- and upwards extrapolations of the whole damage process beyond the scope of the experimental program are provided as a fundamental tool for failure prediction in the practical design. The proposed methodology is detailed and its utility and generality confirmed by its successive application to representative well-known problems in fatigue and fracture characterization. The excellent fittings, the physical interpretation of the model parameters and the good expectations to achieve a complete probabilistic analysis of these phenomena justify the interest of the proposed phenomenological approach with possible applications to other cumulative damage processes.

Evaluating probability of containment effectiveness at a GCS site using integrated assessment modeling approach with Bayesian decision network

AbstractImproved scientific and engineering understanding of the behavior of geologic CO2 storage together with established regulatory framework and incentive structures raise the prospects for accelerated, large‐scale deployment of this greenhouse gas emissions reduction approach. Incentive structures call for the establishment of appropriate verification and accounting approaches to support claims of the integrity of a geologic storage complex and to justify taking credit for long‐term storage. In this study, we present a framework for assessing the probability of containment effectiveness over the lifetime of a geologic carbon storage site (e.g., after 70 years of injection and postinjection site performance) using forward stochastic model realizations based on site characterization data and using a monitoring‐informed Bayesian network based on hypothetical detectability from surface seismic surveys over the site injection and post‐injection phases. The National Risk Assessment Partnership's open‐source Integrated Assessment Model (NRAP‐Open‐IAM) was utilized to develop an ensemble of 10,000 a priori stochastic forecasts of CO2 containment. Those simulations were used to train the Bayesian network model to estimate the prior probabilities of the CO2 leakage mass into overlying, monitorable aquifers considering the uncertainties in the reservoir properties, permeability of potentially leaky wells and the overlying aquifers. The conditional probabilities in the Bayesian network were either learned from the NRAP‐Open‐IAM simulations or derived from the predefined detection thresholds for the monitoring method. Observations obtained from monitoring, over time during the site operation phases were then used to generate updated posterior probabilities of containment (and any loss from containment) in the Bayesian network by propagating the prior probabilities through the conditional probabilities. We demonstrate how to construct and use the Bayesian network for verifying the long‐term storage complex effectiveness informed by monitoring based on the NRAP‐Open‐IAM simulations previously developed for the FutureGen 2.0 site. This approach may have relevance for stake holders to demonstrate secure geologic storage, provide a defensible, probabilistic approach to claim credit for geologic storage, and to estimate the likelihood that any fraction of the claimed credit may need to be refunded to the creditor based on available monitoring information. © 2021 Society of Chemical Industry and John Wiley & Sons, Ltd.

Faults detection and classification in a centrifugal pump from vibration data using markov parameters

One of the strategies to detect and classify faults in mechanical systems is to use a time domain family of techniques known as output-only methods. Those methods are based on the analysis of sample covariance matrices, which are estimated from vibration data extracted from mechanical systems under unmeasured natural excitation. Using the stochastic realization theory, it is possible to derive Markov parameters from sample covariance matrices. Those parameters contain only the significant spectral components from data. In this paper, a novel output-only method based on the Markov parameters is proposed to diagnose faults. The idea is to use the Markov parameters estimated from vibration data as features in classification algorithms based on convex optimization. The method was applied to diagnose incipient cavitation failures in a water supply network centrifugal pump. A low-cost triaxial vibration sensor developed by one of the authors was used to register the vibration data. The proposed method was compared to the analysis based on sample covariance matrices demonstrating the advantages related to the use of the Markov parameters.

Independently Controlling Stochastic Field Realization Magnitude and Phase Statistics for the Construction of Novel Partially Coherent Sources

In this paper, we present a method to independently control the field and irradiance statistics of a partially coherent beam. Prior techniques focus on generating optical field realizations whose ensemble-averaged autocorrelation matches a specified second-order field moment known as the cross-spectral density (CSD) function. Since optical field realizations are assumed to obey Gaussian statistics, these methods do not consider the irradiance moments, as they, by the Gaussian moment theorem, are completely determined by the field’s first and second moments. Our work, by including control over the irradiance statistics (in addition to the CSD function), expands existing synthesis approaches and allows for the design, modeling, and simulation of new partially coherent beams, whose underlying field realizations are not Gaussian distributed. We start with our model for a random optical field realization and then derive expressions relating the ensemble moments of our fields to those of the desired partially coherent beam. We describe in detail how to generate random optical field realizations with the proper statistics. We lastly generate two example partially coherent beams using our method and compare the simulated field and irradiance moments theory to validate our technique.

Propagation of uncertainty of soil hydraulic parameterization in the prediction of water balance components: A stochastic analysis in kaolinitic clay soils

Hydrological models are powerful tools to understand soil–vegetation–atmosphere transport (SVAT) processes. These processes are partially controlled by soil water retention and hydraulic conductivity, in process-based models described by analytical functions (K-θ-h) that define the dependence between pressure head (h), soil water content (θ) and hydraulic conductivity (K). Given the nonlinearity of these process-based SVAT models, stochastic analysis is an interesting tool to get insight in the pattern of model outputs given the uncertainty in the K–θ–h functions. We developed a stochastic framework to evaluate outputs of the SWAP hydrological model according to the uncertainty in the van Genuchten-Mualem K–θ–h analytical functions (VGM). A multivariate Gaussian joint distribution analysis stochastically sampled VGM parameters using two statistical methods, one considering and one disregarding parameter correlations. We evaluated the relations of VGM parameters and simulation outputs (i) for an internal drainage scenario to predict pressure head at flux-based field capacity and (ii) for a cropped scenario to predict water balance components. We also assessed the relative importance of each parameter using linear regression and random forest methods. The hydrological stochastic simulations showed that the developed algorithm allowed to perform multiple SWAP model runs in a quick and unassisted manner on a common personal computer. Although the degree of VGM uncertainty of the studied soils did not affect the SWAP simulation results considerably, model output uncertainty increased when uncorrelated stochastic realizations of VGM parameters were used. The relative propagation of parameter uncertainty showed to be dependent on the simulated scenario (boundary conditions) and on the soil itself. For the correlated VGM stochastic realizations, both criteria used to analyze the parameter importance detected the VGM parameter l as one of the most relevant in determining water balance components variation. This is an interesting outcome, as this parameter is hardly ever determined, and commonly assumed to a predefined value.

Comfort improvement in railway vehicles via optimal control of adaptive pneumatic suspensions

ABSTRACT This paper presents an adaptive optimal control strategy focused on improving ride comfort by minimising the root mean square of accelerations measured at seven locations on the carbody floor. It requires preview knowledge of both the train speed and the standard track quality (not the deterministic track disturbances) and differs from other works in considering pointwise constraints (upper and lower secondary suspension travel limits). Adaptive control is achieved by varying the diameter of the restriction orifice which connects the pneumatic spring to the auxiliary air reservoir, which confers different dynamic stiffness and damping to the front and rear secondary suspensions. Decision maps with the front and rear optimal diameters for five track irregularity levels and six different speeds were obtained using a Monte Carlo strategy based on solving the optimal control problem for 240 stochastic realisations of time-varying track irregularities of 300 s. Results show that the adaptive optimal control provides a successful compromise between comfort and safety which overshadows the conventional passive approach: it improves comfort up to 50% with respect to stiff passive configurations and it guarantees constraint fulfilment unlike compliant passive configurations.

Integration of Scheduling and Control Under Stochastic Parametric Uncertainty with Varying Unit Operation Times for Chemical Batch Plants: A Back-Off Approach

Abstract A new back-off methodology is presented in this work as an approach for solving MIDO formulations arising for the optimal scheduling and control of flow-shop batch plants under stochastic parametric uncertainty. The proposed algorithm decomposes the MIDO problem into a scheduling problem, a dynamic optimization problem and a unit time operation minimization problem. These problems are solved iteratively using back-off terms. Parametric uncertainty is modeled using statistical distribution functions and are embedded in the algorithm to ensure dynamic feasibility of the optimal control actions under stochastic realizations in those parameters. The proposed algorithm identifies scheduling and control decisions offline. To exemplify this methodology, the integration of scheduling and control of a flow-shop batch plant is considered. The results show that unit operation times chosen from optimization are better suited to accommodate stochastic parametric uncertainty while the control actions enforce process operational and product quality constraints at reasonable costs.

Simulation of Surface Pressure Fluctuations on an Airbus-A320 Fuselage at Cruise Conditions

The fuselage surface pressure fluctuations on an Airbus-A320 aircraft at cruise conditions are simulated by solving a Poisson equation. The right-hand-side source terms of the Poisson equation, including both the mean-shear term and the turbulence–turbulence term, are realized with synthetic turbulence generated by the fast random particle-mesh method. The stochastic realization is based on time-averaged turbulence statistics derived from a Reynolds averaged Navier–Stokes simulation under the same condition as in the flight tests conducted with DLR, German Aerospace Center’s Airbus-A320 research aircraft. The fuselage surface pressure fluctuations are calculated at three streamwise positions from front to rear, corresponding to the measurement positions in the flight tests. Features of the pressure fluctuations relevant to the fuselage surface excitation are obtained and analyzed. A good agreement for one- and two-point spectral characteristics of the surface pressure fluctuations is found between the simulation and flight-test results.

Reliability assessment of the performance of granular column in the nonuniform liquefiable ground to mitigate the liquefaction-induced ground deformation

ABSTRACT Granular columns have been widely used to mitigate the liquefaction-induced ground deformation. The increment in lateral stress due to densification, shear reinforcement, and drainage capacity of granular columns are believed to increase the liquefaction resistance of the ground. However, several case histories and recent research development exhibited the limitations of the effectiveness of granular columns under strong earthquakes. Besides, the mechanism of shear reinforcement governed by granular columns is poorly understood. Moreover, the spatial nonuniformity of the ground should be considered for a reliable engineering assessment of the performance of granular columns. A series of three-dimensional nonlinear stochastic analyses are carried out using the OpenSees framework with PDMY02 elasto-plastic soil constitutive model to map the reliability of the performance of equally-spaced granular columns. Soil variability is implemented with stochastic realizations of overburden and energy-corrected, equivalent clean sand, (N1)60cs values using spatially correlated Gaussian random field. The reliability of the performance of granular column is assessed based on the stochastic distributions of average surface settlement and horizontal ground displacement associated with the degree of confidence. The implications of cumulative absolute velocity, Arias Intensity and peak acceleration of different ground motions on the efficacy of the granular column are also discussed.