Independent Probability Density Functions Research Articles

A Bayesian Hierarchical Framework for Postprocessing Daily Streamflow Simulations across a River Network

Abstract Despite the potential and increasing interest in physically based hydrological models for streamflow forecasting applications, they are constrained in terms of agility to generate ensembles. Hence, we develop and test a Bayesian hierarchical model (BHM) to postprocess physically based hydrologic model simulations at multiple sites on a river network, with the aim to generate probabilistic information (i.e., ensembles) and improve raw model skill. We apply our BHM framework to daily summer (July–August) streamflow simulations at five stations located in the Narmada River basin in central India, forcing the Variable Infiltration Capacity (VIC) model with observed rainfall. In this approach, daily observed streamflow at each station is modeled with a conditionally independent probability density function with time varying distribution parameters, which are modeled as a linear function of potential covariates that include VIC outputs and meteorological variables. Using suitable priors on the parameters, posterior parameters and predictive posterior distributions—and thus ensembles—of daily streamflow are obtained. The best BHM model considers a gamma distribution and uses VIC streamflow and a nonlinear covariate formulated as the product of VIC streamflow and 2-day precipitation spatially averaged across the area between the current and upstream station. The second covariate enables correcting the time delay in flow peaks and nonsystematic biases in VIC streamflow. The results show that the BHM postprocessor increases probabilistic skill in 60% compared to raw VIC simulations, providing reliable ensembles for most sites. This modeling approach can be extended to combine forecasts from multiple sources and provide skillful multimodel ensemble forecasts.

Wet gas metering by cone throttle device with machine learning

Abstract The online measurement of gas and liquid flow rates in wet gas is of great significance. This paper presents a new method to measure gas and liquid flow rates of wet gas by combining the cone throttle device and machine learning techniques. The equivalent diameter ratio of the cone device is 0.45. Experiments are carried out in a horizontal pipe of diameter 50 mm and the operating pressure ranges from 100 kPa to 250 kPa. The working fluids are the mixture of air and water with the Lockhart-Martinelli parameter (XLM) less than 0.3. The multilayer feedforward neural network is used for developing the measurement model. The model requires representative features as inputs and uses the gas and liquid flow rates as outputs. In addition to the mean values of the permanent pressure loss and the upstream-throat differential pressure, the probability density function (PDF) and power spectral density (PSD) of the upstream-throat differential pressure fluctuation are also extracted as representative features. With the principal component analysis method, the independent PDF and PSD features are obtained. The untrained dataset is used to evaluate the performance of the neural network model. Predictions of the flow rates are in good agreement with the experiments. The mean relative errors of the gas and liquid flow rates are 0.05% and −3.66%, respectively. The results show that the proposed method is capable of establishing the implicit correlations between the characteristic parameters of wet gas and the corresponding flow rates.

Likely striping in stochastic nematic elastomers

For monodomain nematic elastomers, we construct generalised elastic–nematic constitutive models combining purely elastic and neoclassical-type strain energy densities. Inspired by recent developments in stochastic elasticity, we extend these models to stochastic–elastic–nematic forms, where the model parameters are defined by spatially independent probability density functions at a continuum level. To investigate the behaviour of these systems and demonstrate the effects of the probabilistic parameters, we focus on the classical problem of shear striping in a stretched nematic elastomer for which the solution is given explicitly. We find that, unlike the neoclassical case, where the inhomogeneous deformation occurs within a universal interval that is independent of the elastic modulus, for the elastic–nematic models, the critical interval depends on the material parameters. For the stochastic extension, the bounds of this interval are probabilistic, and the homogeneous and inhomogeneous states compete, in the sense that both have a a given probability to occur. We refer to the inhomogeneous pattern within this interval as ‘likely striping’.

Uncertainty quantification of elastic material responses: testing, stochastic calibration and Bayesian model selection

Motivated by the need to quantify uncertainties in the mechanical behaviour of solid materials, we perform simple uniaxial tensile tests on a manufactured rubber-like material that provide critical information regarding the variability in the constitutive responses between different specimens. Based on the experimental data, we construct stochastic homogeneous hyperelastic models where the parameters are described by spatially independent probability density functions at a macroscopic level. As more than one parametrised model is capable of capturing the observed material behaviour, we apply Baye theorem to select the model that is most likely to reproduce the data. Our analysis is fully tractable mathematically and builds directly on knowledge from deterministic finite elasticity. The proposed stochastic calibration and Bayesian model selection are generally applicable to more complex tests and materials.

A comparison of Monte Carlo fission models for safeguards neutron coincidence counters

Monte Carlo simulation is a powerful tool used to model neutron coincidence detectors for international safeguards. The simulation has typically sampled properties such as the fission neutron multiplicity, energy, and direction, from independent probability density functions. However, multiplicity counters detect event-based neutron correlations and thus more accurate fission event modeling is needed. To respond to this need, the Fission Reaction Event Yield Algorithm (FREYA) and the Cascading Gamma-ray Multiplicity with Fission (CGMF) models were added in the newest version of MCNP, MCNP6.2. The models simulate individual fission events conserving momentum, energy, and angular momentum such that correlated particles are emitted.The effects of the new models on simulations of safeguards neutron coincidence counters were studied and compared to standard MCNP simulations. The MCNPX-PoliMi model was also included in the comparison. The properties of fission neutrons from safeguards relevant isotopes were compared to literature references. Then a hypothetical simplified detector was modeled to isolate the effects of specific differences between models. Experimental measurements from previous work were modeled and agreements were compared. Finally, the probabilities of correlated events occurring in the experimental measurements were calculated with the different models. For example, the probability was calculated of detecting neutrons from both induced fission in uranium and spontaneous fission of Cf-252 in the same fission chain.

Direct dialling of Haar random unitary matrices

Random unitary matrices find a number of applications in quantum information science, and are central to the recently defined boson sampling algorithm for photons in linear optics. We describe an operationally simple method to directly implement Haar random unitary matrices in optical circuits, with no requirement for prior or explicit matrix calculations. Our physically motivated and compact representation directly maps independent probability density functions for parameters in Haar random unitary matrices, to optical circuit components. We go on to extend the results to the case of random unitaries for qubits.

A Robust Maximum‐Likelihood Earthquake Location Method for Early Warning

This study proposes a robust method that estimates a reliable earthquake location from only a small number of P ‐wave arrival times. The method is based on the maximum‐likelihood estimation from differential P ‐wave arrivals. We formulate the problem using a probability density function (PDF) of the residual between observed and predicted differential P ‐wave travel times between two seismic stations and construct the likelihood function from the sum of the products of the independent PDFs. The hypocenter is determined by an iterative grid‐search algorithm that finds the point with the largest probability on successively finer grids. To reduce the effect from outliers possibly concealed within a small number of observations, the Student’s t ‐distribution is used for the PDF of the location likelihood. The jackknife resampling technique is also used to discriminate outliers from the observations. The robustness of the method is tested using the Monte Carlo experiments that locate 10,000 events from small numbers of P ‐wave arrivals observed within an epicentral distance of 100 km, including both arrival‐time error and velocity‐model error. The earthquakes are located within an epicentral distance of 8.5±10.8 km and 20.6±33.1 km for events inside the seismic network and outside the network, respectively, using only five P ‐wave arrivals, including a large arrival‐time error between ±1 and 5 s. This shows that this method can estimate the location of the event reliably with only a few P ‐wave arrivals, even when contaminated by an outlier. Therefore, it is believed that this location method could significantly improve the robustness of an earthquake early warning system.

Robust optimization of base isolation devices under uncertain parameters

Base isolator devices are widely used for mitigation of vibrations induced in structures by seismic actions. In order to achieve high performances in the mitigation of seismic effects, base isolator mechanical properties should be designed by an optimum criterion. In common approaches, the nature of dynamic loads is assumed as the only source of uncertainty. In the present paper a robust optimization criterion for base isolator devices design is proposed, considering the unavoidable effects of uncertainty in structural properties and seismic action. Uncertain parameters are modeled as random variables and are represented by bounded independent probability density function, with uniform law. The structure is described by a single-degree-of-freedom model and is protected by a linear base isolator in order to reduce vibration levels induced by base acceleration, here modeled by the stationary Kanai-Tajimi stochastic process. The optimal design is formulated as a constrained minimization problem, assuming as an objective function a suitable measure of the isolator efficiency and imposing a constraint on the maximum isolator displacement. A sensitivity analysis is carried out on the robust solution in order to assess characteristics and differences with respect to the conventional deterministic solution.

A Dual Mass Flux Framework for Boundary Layer Convection. Part II: Clouds

Abstract This paper presents the extension of the eddy diffusivity mass flux (EDMF) framework for turbulent transport into the statistical modeling of boundary layer clouds. The advection–diffusion decomposition that defines EDMF is projected onto the turbulent distribution as used in the statistical cloud model. Each EDMF component is thus assigned its own independent probability density function (PDF), resulting in an updraft PDF and a diffusive PDF. This double PDF system is configured and integrated in conserved variable space, with the position and orientation of each PDF determined by its unique nature. The parameterization of the associated updraft/diffusion decomposition of variance introduces close ties to the transport scheme; whereas the grid box mean variance is reconstructed using a prognostic variance budget, the variance of the updraft component is parameterized as a function of the spread among various resolved model updrafts. Individual model components and the scheme as a whole are evaluated in detail against large-eddy simulations of a number of prototype subtropical trade wind cases. The results show that various structures in cloud fraction, condensate, and variance are reproduced. The diffusive PDF acts to represent stratiform clouds; the advective PDF represents cumuliform clouds in conditionally unstable layers. This allows representation of complex scenarios in which both cloud forms occur, such as the transitional trade wind regime featuring cumulus rising into stratocumulus.

Two-State Model of Acto-Myosin Attachment-Detachment Predicts C-Process of Sinusoidal Analysis

The force response of activated striated muscle to length perturbations includes the so-called C-process, which has been considered the frequency domain representation of the fast single-exponential force decay after a length step (phases 1 and 2). The underlying molecular mechanisms of this phenomenon, however, are still the subject of various hypotheses. In this study, we derived analytical expressions and created a corresponding computer model to describe the consequences of independent acto-myosin cross-bridges characterized solely by 1), intermittent periods of attachment ( t att) and detachment ( t det), whose values are stochastically governed by independent probability density functions; and 2), a finite Hookian stiffness ( k stiff) effective only during periods of attachment. The computer-simulated force response of 20,000 ( N) cross-bridges making up a half-sarcomere ( F hs( t)) to sinusoidal length perturbations ( L hs( t)) was predicted by the analytical expression in the frequency domain, ( F ˜ hs ( ω ) / L ˜ hs ( ω ) ) = ( t ¯ att / t ¯ cycle ) N k ¯ stiff ( i ω / ( t ¯ att − 1 + i ω ) ) , where t ¯ att = mean value of t att, t ¯ cycle = mean value of t att + t det, k ¯ stiff = mean stiffness, and ω = 2 π × frequency of perturbation. The simulated force response due to a length step ( L hs) was furthermore predicted by the analytical expression in the time domain, F hs ( t ) = ( t ¯ att / t ¯ cycle ) N k ¯ stiff L hs e − t / t ¯ att . The forms of these analytically derived expressions are consistent with expressions historically used to describe these specific characteristics of a force response and suggest that the cycling of acto-myosin cross-bridges and their associated stiffnesses are responsible for the C-process and for phases 1 and 2. The rate constant 2 πc, i.e., the frequency parameter of the historically defined C-process, is shown here to be equal to t ¯ att − 1 . Experimental results from activated cardiac muscle examined at different temperatures and containing predominately α- or β-myosin heavy chain isoforms were found to be consistent with the above interpretation.

Probability density functions for radial anisotropy from fundamental mode surface wave data and the Neighbourhood Algorithm

SUMMARY We applied Sambridge’s Neighbourhood Algorithm (NA) to degree-8 fundamental mode Love and Rayleigh wave phase velocity maps between 40 and 150 s to find models of radial anisotropy in the upper 220 km of the mantle. The NA is a powerful tool to explore a multidimensional model space and retrieve an ensemble of models from which statistical inferences (posterior probability density functions (PPDFs) and trade-offs) can be made. We sought solutions for density anomalies and perturbations in the five elastic coefficients that describe transverse isotropy and obtained independent probability density functions for S-wave anisotropy, P-wave anisotropy, intermediate parameter η, Vp, Vs and density anomalies. We find robust departures from PREM in S-wave anisotropy (ξ ) under cratons and oceans alike, with a clear change of sign in the anomalies with respect to the reference model at approximately 100 km depth. No significant difference is observed between cratons and oceans, both in the amplitude and depth variation of ξ . The signal within continents is clearly age related, with platforms and tectonically active regions characterized by a rapid decrease in ξ with depth, while cratons display a more constant signal. A similar age dependence in S-wave anisotropy is also observed beneath oceans: a strong and rapidly decreasing anisotropy for young oceans and a more constant anisotropy for older oceans. Perturbations in P-wave anisotropy (φ) are small and limited to the shallowest part of the continents. A small age-dependent signal for φ is observed beneath oceans. Anomalies in intermediate parameter η are similar to those in φ (but not globally correlated), but the deviation from PREM is stronger for η than for φ. Cratons appear to be devoid of any η anisotropy in the top 100 km. There is no obvious global correlation between deviations in φ and deviations in ξ , and the ratio between dη and dξ is clearly regionally variable, which cautions against the use of commonly used proportionality factors between these variables in inversions. In all regions, we found a good correlation between the equivalent isotropic P- and S-wave velocity anomalies, with a ratio d ln Vs/d ln Vp close to 1. Density anomalies were not sufficiently well resolved with fundamental mode data alone, but do not influence the results for anisotropy.

State space sensitivity to a prescribed probability density function shape in coal combustion systems: Joint β-PDF versus clipped Gaussian PDF

The turbulent transport of three coal off-gas mixture fractions is coupled to a prescribed joint β -probability density function ( β -PDF) mixing model. This physical transport and subgrid joint β -PDF mixing model is used to explore the incorporation of coal off-gas compositional disparities between the devolatilization and the char oxidation regime in detailed pulverized-coal combustion simulations. A simulation study of the University of Utah pulverized-coal research furnace is presented to evaluate the sensitivity of different mixing model assumptions. These simulation studies indicate that using a variable composition to characterize the process of coal combustion does not appreciably change the predicted gas-phase temperature field. Moreover, neglecting fluctuations in the char off-gas stream was found to change gas-phase temperature predictions by approximately 15%. State space variable sensitivity to the assumed shape of the PDF (clipped Gaussian vs. joint β ) is presented. Simulation results indicate differences in temperature profiles of as much as 20% depending on the chosen shape of the PDF. Integration accuracy issues for the joint β -PDF are presented and are found to be acceptable. A robust β -PDF function evaluation procedure is presented that accommodates arbitrarily high β -PDF distribution factors. This robust algorithm simply transforms the joint β -PDF function evaluation into a logarithmic form. The assumption that a joint PDF, as rigorously required within a prescribed subgrid mixing model, can be written as the product of N − 1 statistically independent probability density functions is quantified and shown to be less accurate.