A Statistical Method for Generating Temporally Downscaled Geochemical Tracers in Precipitation

Sampling intervals of precipitation geochemistry measurements are often coarser than those required by fine-scale hydrometeorological models. This study presents a statistical method to temporally downscale geochemical tracer signals in precipitation so that they can be used in high-resolution, tracer-enabled applications. In this method, we separated the deterministic component of the time series and the remaining daily stochastic component, which was approximated by a conditional multivariate Gaussian distribution. Specifically, statistics of the stochastic component could be explained from coarser data using a newly identified power-law decay function, which relates data aggregation intervals to changes in tracer concentration variance and correlations with precipitation amounts. These statistics were used within a copula framework to generate synthetic tracer values from the deterministic and stochastic time series components based on daily precipitation amounts. The method was evaluated at 27 sites located worldwide using daily precipitation isotope ratios, which were aggregated in time to provide low resolution testing datasets with known daily values. At each site, the downscaling method was applied on weekly, biweekly and monthly aggregated series to yield an ensemble of daily tracer realizations. Daily tracer concentrations downscaled from a biweekly series had average (+/- standard deviation) absolute errors of 1.69‰ (1.61‰) for δ2H and 0.23‰ (0.24‰) for δ18O relative to observations. The results suggest coarsely sampled precipitation tracers can be accurately downscaled to daily values. This method may be extended to other geochemical tracers in order to generate downscaled datasets needed to drive complex, fine-scale models of hydrometeorological processes.

Stochastic Response of Hysteresis System Under Combined Periodic and Stochastic Excitation Via the Statistical Linearization Method

A statistical linearization approach is proposed for determining the response of the single-degree-of-freedom of the classical Bouc–Wen hysteretic system subjected to excitation both with harmonic and stochastic components. The method is based on representing the system response as a combination of a harmonic and of a zero-mean stochastic component. Specifically, first, the equation of motion is decomposed into a set of two coupled non-linear differential equations in terms of the unknown deterministic and stochastic response components. Next, the harmonic balance method and the statistical linearization method are used for the determination of the Fourier coefficients of the deterministic component, and the variance of the stochastic component, respectively. This yields a set of coupled algebraic equations which can be solved by any of the standard apropos algorithms. Pertinent numerical examples demonstrate the applicability, and reliability of the proposed method.

ACCOUNTING FOR UNCERTAINTY AND RANDOM RISKS OF ACTIVATION OF COMPLEX NATURAL PROCESSES USING THE BAYES TRUST NETWORK

In the research, modeling, and forecasting of complex systems associated with various natural processes, scientists have encountered uncertainty and unpredictability. In order to eliminate or reduce the impact of uncertainty on the prediction of complex natural processes and to determine possible catastrophic consequences, traditional methods and models are not applicable and do not show reliable results, since they do not take into account the stochastic component. The article deals with the use of Bayesian theory in various spheres of life, and will also consider and highlight the factors that have a significant impact on the activation and flow of the processes and phenomena under consideration. The Bayes trust network is constructed and trained by filling in tables of conditional probabilities by experts and introducing numerical values of factors to determine the unconditional (a priori) probabilities of the factors under consideration. The constructed network can be supplemented, if necessary, with vertices and connections between them.

A stochastic-statistical residential burglary model with independent Poisson clocks

Residential burglary is a social problem in every major urban area. As such, progress has been to develop quantitative, informative and applicable models for this type of crime: (1) the Deterministic-time-step (DTS) model [Short, D’Orsogna, Pasour, Tita, Brantingham, Bertozzi & Chayes (2008) Math. Models Methods Appl. Sci.18, 1249–1267], a pioneering agent-based statistical model of residential burglary criminal behaviour, with deterministic time steps assumed for arrivals of events in which the residential burglary aggregate pattern formation is quantitatively studied for the first time; (2) the SSRB model (agent-based stochastic-statistical model of residential burglary crime) [Wang, Zhang, Bertozzi & Short (2019) Active Particles, Vol. 2, Springer Nature Switzerland AG, in press], in which the stochastic component of the model is theoretically analysed by introduction of a Poisson clock with time steps turned into exponentially distributed random variables. To incorporate independence of agents, in this work, five types of Poisson clocks are taken into consideration. Poisson clocks (I), (II) and (III) govern independent agent actions of burglary behaviour, and Poisson clocks (IV) and (V) govern interactions of agents with the environment. All the Poisson clocks are independent. The time increments are independently exponentially distributed, which are more suitable to model individual actions of agents. Applying the method of merging and splitting of Poisson processes, the independent Poisson clocks can be treated as one, making the analysis and simulation similar to the SSRB model. A Martingale formula is derived, which consists of a deterministic and a stochastic component. A scaling property of the Martingale formulation with varying burglar population is found, which provides a theory to the finite size effects. The theory is supported by quantitative numerical simulations using the pattern-formation quantifying statistics. Results presented here will be transformative for both elements of application and analysis of agent-based models for residential burglary or in other domains.

Formulation of Statistical Linearization for M-D-O-F Systems Subject to Combined Periodic and Stochastic Excitations

A formulation of statistical linearization for multi-degree-of-freedom (M-D-O-F) systems subject to combined mono-frequency periodic and stochastic excitations is presented. The proposed technique is based on coupling the statistical linearization and the harmonic balance concepts. The steady-state system response is expressed as the sum of a periodic (deterministic) component and of a zero-mean stochastic component. Next, the equation of motion leads to a nonlinear vector stochastic ordinary differential equation (ODE) for the zero-mean component of the response. The nonlinear term contains both the zero-mean component and the periodic component, and they are further equivalent to linear elements. Furthermore, due to the presence of the periodic component, these linear elements are approximated by averaging over one period of the excitation. This procedure leads to an equivalent system whose elements depend both on the statistical moments of the zero-mean stochastic component and on the amplitudes of the periodic component of the response. Next, input–output random vibration analysis leads to a set of nonlinear equations involving the preceded amplitudes and statistical moments. This set of equations is supplemented by another set of equations derived by ensuring, in a harmonic balance sense, that the equation of motion of the M-D-O-F system is satisfied after ensemble averaging. Numerical examples of a 2-D-O-F nonlinear system are considered to demonstrate the reliability of the proposed technique by juxtaposing the semi-analytical results with pertinent Monte Carlo simulation data.

Reconstruction of Dynamics of SO2 Concentration in Troposphere Based on Results of Direct Measurements

A method for the reconstruction of the dynamics of processes with discrete time, developed in our previous papers, has been applied for study the dynamics of concentration of sulfur dioxide in lower troposphere. For the analysis, recordings of sulfur dioxide concentration from four measurement stations located in Poland (two of them has been located in huge cities and two in rarely inhabited regions) were used. We managed to obtain the deterministic and stochastic component of this dynamics. In result, we estimate the lifetime of sulfur dioxide in troposphere and the increase of sulfur dioxide concentration influenced by anthropogenic sources.

Evaluating a stochastic parametrization for a fast–slow system using the Wasserstein distance

Constructing accurate, flexible, and efficient parametrizations is one of the great challenges in the numerical modeling of geophysical fluids. We consider here the simple yet paradigmatic case of a Lorenz 84 model forced by a Lorenz 63 model and derive a parametrization using a recently developed statistical mechanical methodology based on the Ruelle response theory. We derive an expression for the deterministic and the stochastic component of the parametrization and we show that the approach allows for dealing seamlessly with the case of the Lorenz 63 being a fast as well as a slow forcing compared to the characteristic timescales of the Lorenz 84 model. We test our results using both standard metrics based on the moments of the variables of interest as well as Wasserstein distance between the projected measure of the original system on the Lorenz 84 model variables and the measure of the parametrized one. By testing our methods on reduced-phase spaces obtained by projection, we find support for the idea that comparisons based on the Wasserstein distance might be of relevance in many applications despite the curse of dimensionality.