Stochastic Space Research Articles

Experimental investigation of stochastic space charge effects on pattern resolution in ion projection lithography systems

Coulomb interactions in charged particle beams are known to blur beam profiles as the total current is increased. While this has been studied in single ion beam columns, similar studies have not been published, so far, for masked ion beam projection systems. Experiments were performed using highly transparent stencil grid masks of 40 mm×40 mm area with resolution test patterns in central and peripheral positions. Such grid test masks have been inserted into the ion projection lithography machine IPLM-02 and into the ‘‘Alpha ion projector’’ operated at 5× ion–optical reduction (Alpha-5×). A standard duoplasmatron ion source was used for both ion projectors. Operating the IPLM-02 at 8.7× demagnification with He+ ions of 73 keV ion energy at the wafer plane (about 50 keV at the crossover), no significant degradation in resolution was found changing the ion beam current through the column from 300 nA to 2 μA. Using the Alpha-5×, the total current through the column was changed either by varying the ion source emission or by blanking parts of the mask, using appropriate apertures. The special design of the Alpha ion projector with an ion energy of 5–7 keV at the crossover region between the imaging lenses is a sort of worst case scenario for space charge effects. Thus, the influence of stochastic space charge effects on resolution could be clearly identified. Using helium ions with 7 keV at the crossover the exposure latitude (change of linewidth in nm with 10% change of exposure dose) increased by 60 nm when changing the current through the column from 3 to 120 nA. Based on this experimental result with the Alpha ion projector and on well-known scaling laws, it can be stated that a newly designed ion projector (ALG-1000) with 200 keV ion energy at the crossover will allow sub-0.15-μm resolution for total ion beam currents through the column of 3 μA, as needed for future high volume production of dense patterns within ≥20 mm×20 mm exposure fields. Furthermore, improved designs of ion optical columns will allow a reduction of system length, thus reducing the influence of stochastic space charge effects. Consequently, ion projection lithography can be extended to the high volume fabrication of sub-0.10-μm feature patterns in large exposure fields.

Stochastic space transforms in subsurface hydrology — Part 2: Generalized spectral decompositions and plancherel representations

In earlier publications, certain applications of space transformation operators in subsurface hydrology were considered. These operators reduce the original multi-dimensional problem to the one-dimensional space, and can be used to study stochastic partial differential equations governing groundwater flow and solute transport processes. In the present work we discuss developments in the theoretical formulation of flow models with space-dependent coefficients in terms of space transformations. The formulation is based on stochastic Radon operator representations of generalized functions. A generalized spectral decomposition of the flow parameters is introduced, which leads to analytically tractable expressions of the space transformed flow equation. A Plancherel representation of the space transformation product of the head potential and the log-conductivity is also obtained. A test problem is first considered in detail and the solutions obtained by means of the proposed approach are compared with the exact solutions obtained by standard partial differential equation methods. Then, solutions of three-dimensional groundwater flow are derived starting from solutions of a one-dimensional model along various directions in space. A step-by-step numerical formulation of the approach to the flow problem is also discussed, which is useful for practical applications. Finally, the space transformation solutions are compared with local solutions obtained by means of series expansions of the log-conductivity gradient.

State space model for river temperature prediction

A stochastic state space model for the estimation of river temperature is presented, setting the stage for its implementation in optimal control algorithms. Physical processes modeled include advection, diffusion, and environmental heat exchange; the mathematical formulation is one dimensional. The deterministic formulation uses a hybrid characteristics‐finite differences numerical scheme for the solution of the governing equations. The stochastic formulation accounts for uncertainty due to model assumptions, errors in the model inputs and parameters, and river temperature measurements. Model use is demonstrated by the estimation of temperatures in a 5‐mile (8 km) reach of the Des Plaines River, Illinois, below the Joliet power station. River temperature measurements and hydrological and meteorological data are used to estimate the model parameters; model formulation and limitations are assessed based on the model application results. The stochastic formulation improves river temperature estimation, as measured by the mean, variance‐covariance, correlation, and range of the residuals.

Subspace algorithms for the stochastic identification problem

In this paper, we derive a new subspace algorithm to consistently identify stochastic state space models from given output data without forming the covariance matrix and using only semi-infinite block Hankel matrices. The algorithm is based on the concept of principal angles and directions. We describe how they can be calculated with QR and Quotient Singular Value Decomposition. We also provide an interpretation of the principal directions as states of a non-steady state Kalman filter bank.

A kalman filter‐based technique for combined digital estimation of voltage flicker and phasor in power distribution systems

AbstractElectric power quality is of prime importance when sensitive electronic equipment is involved. One of the aspects of power quality is known as flicker level. The present paper proposes an algorithm for real‐time monitoring of voltage flicker based on the Kalman filtering technique. First a stochastic state space model of the system voltage with a linear time‐invariant transition matrix has been established, yet non‐linear in the scalar observation model. Using noisy measurements of a single‐phase voltage, an extended Kalman filter leads to instantaneous value of rms voltage and phase. Then the flicker level has been derived by appropriate filtering and statistical processing on short records of the rms voltage measurements. Simulated and actual data demonstrating the effectiveness of the overall scheme in estimating voltage phasor and flicker for a wide range of swing frequencies are presented and discussed. Among the several power system applications of the new stochastic observer considered, a remote load control concept based on the carrier amplitude modulation is shown being possible and included are test results confirming the filter capability to retrieve sequences of amplitude‐modulated bit patterns travelling through the power line to customer sites.

The development of stochastic space transformation and diagrammatic perturbation techniques in subsurface hydrology

This paper develops concepts and methods to study stochastic hydrologic models. Problems regarding the application of the existing stochastic approaches in the study of groundwater flow are acknowledged, and an attempt is made to develop efficient means for their solution. These problems include: the spatial multi-dimensionality of the differential equation models governing transport-type phenomena; physically unrealistic assumptions and approximations and the inadequacy of the ordinary perturbation techniques. Multi-dimensionality creates serious mathematical and technical difficulties in the stochastic analysis of groundwater flow, due to the need for large mesh sizes and the poorly conditioned matrices arising from numerical approximations. An alternative to the purely computational approach is to simplify the complex partial differential equations analytically. This can be achieved efficiently by means of a space transformation approach, which transforms the original multi-dimensional problem to a much simpler unidimensional space. The space transformation method is applied to stochastic partial differential equations whose coefficients are random functions of space and/or time. Such equations constitute an integral part of groundwater flow and solute transport. Ordinary perturbation methods for studying stochastic flow equations are in many cases physically inadequate and may lead to questionable approximations of the actual flow. To address these problems, a perturbation analysis based on Feynman-diagram expansions is proposed in this paper. This approach incorporates important information on spatial variability and fulfills essential physical requirements, both important advantages over ordinary hydrologic perturbation techniques. Moreover, the diagram-expansion approach reduces the original stochastic flow problem to a closed set of equations for the mean and the covariance function.

Physically Based Simulation of Radar Rainfall Data Using a Space–Time Rainfall Model

A scheme for simulating radar-estimated rainfall fields is described. The scheme uses a two-dimensional stochastic space–time model of rainfall events and a parameterization of drop-size distribution. Based on the statistically generated drop-size distribution, radar observables, namely, radar reflectivity and differential reflectivity, are calculated. The simulated measurable variables are corrupted with random measurement error to account for radar measurement process. Subsequently, radar observables are used in rainfall estimation. Generated fields of the simulated rainfall and the corresponding radar observables are presented. Rainfall estimates from radar simulations are also presented. Use of the described radar-data simulator is envisioned in those applications where the effects of radar rainfall errors are of interest.

Identification of Nonlinear Stochastic Grey Box Models: Theory, Implementation, and Experiences

Abstract Grey box identification refers to the practice of identifying dynamical systems in model structures exploiting partial prior information. If this leads to nonlinear stochastic state space models, there is a question as to whether the complexity of analysis and software implementation is justifiable in terms of improved control and/or prediction. The paper covers the following three issues. Firstly, the theoretical foundations for solving the problem are reviewed. Secondly, the implementation of a general purpose grey box identification software is reported. And thirdly, with help of two industrial applications, it is demonstrated that the complexity of the method can indeed pay off in terms of simplifying control (case study 1) or in terms of improved prediction and parameter accuracy (case study 2).

Identification of State Space Models Using a Stochastic Realization Algorithm with Application to Modal Identification of Flexible Aerospace Structures

Abstract Accurate Control of Large Space Structures (LSS) for pointing and tracking applications requires accurate knowledge of the vibrational characteristics of the LSS. Since it is not practical to duplicate the space environment during ground vibration tests, accurate models of LSS can only be obtained via System Identification algorithms applied to real data collected from the LSS after deployment in orbit. A similar problem arises in Aircraft Flutter Testing, which is done to verify that the flight envelop does not contain any aeroelastic or aeroservoelastic instabilities. In this paper, we show that System Identification car be done with high accuracy using State Space Models and a Stochastic Realization Algorithm (SRA) It is shown that the Stochastic Realization Algorithm (SRA) outperforms other available techniques and produces excellent results for the AFAL Flexible GRID test data and the X-29 Aircraft Flutter data under natural turbulence conditions. SRA also produces robust estimates of frequencies, dampings and mode shapes for repeated frequency modes of an elastic membrane even under conditions of 300% multiplicative noise. SRA identifies a multi-input multi-output stochastic state space model for the data using a noniterative technique based on Singular Value Decomposition of a Hankel matrix.

Positivity conditions for stochastic state space modelling of time series

This short paper clarifies some aspects of the balancing method for state space modelling of observed time series. This method may fail to satisfy the so-called positive real condition for stochastic processes. We illustrate this by theoretical spectral analysis and also by simulating univariate ARMA (1,1) models.

節理分布性状の確率統計学的モデル化に関する研究 （その２） 節理分布特性要素と確率モデル

Joint distribution is regulated by 7 joint characteristic elements ; 1) orientation, 2) persistence, 3) density, 4) aperture, 5) filling, 6) roughness and 7) conductivity. For the purpose of making a stochastic space model of each joint characteristic element is defined as the results of the field investigations, Each joint set has its own distributional character. Therefore, each probabilistic space model must be probability density function conditional upon joint set.

A High Precision Track Controller for Ships

The general features of a ship guidance system is discussed and the structure of a high precision track controller is presented. The controller includes the planning of a full manoeuvre trajectory for track changes. The ship is kept to this reference path using feedforward control from the trajectory construction in combination with a stochastic state space (LQG) controller. The dynamic ship model used for manoeuvre computation and controller design has to be identifiable from on board measurements. This favours a reduced version of the usual second order core model of ship motion. The track controller has proved high performance in several full scale sea trials.

THE DEVELOPMENT OF A PRACTICAL REAL-TIME STOCHASTIC FLOOD PREDICTION MODEL FOR DAM OPERATION

Using stochastic state space techniques, the authors developed a practical real-time stochastic flood prediction method for dam operation. Here we consider the series tanks model and the storage function model as the candidate for run-off model. In order to select a suitable stochastic model, predicted results are evaluated by a measure which is taken notice of accuracy in the rising part of a flood. The procedure of model set-up adopted here is applied for two dam basins: Tonoyama dam basin (294km2) and Tsunokawa dam basin (724km2). It is shown, by using the historical data, that the proposed method is practical and useful. The developed models are going to be utilized for real dam operation in near future.

Kalman Filtering and Control of Water Level in an Interconnected Multi-Reservoir Power System

This paper deals with combined state estimation and optimal control of water level in an interconnected multi-reservoir power system. A mathematical model of the system is described as a dscrete-time stochastic state space model taking into account the mass balance of water, uncertainties which cannot be explicitly included in the model, and measurement noise. Under this circumstance Kaiman filtering technique is applied to estimate states which are the water storage volume in the reservoir and the total inflow of water into the reservoir. An optimal closed-loop control system is designed to track the water level in the reservoir as close as the desired level where the control variable is the ammount of water released from the upper stream reservoir through the hydroelectric power generation. Throughout the simulation study of combined real-time estimation and optimal control, we point out the importance of predicting so-called residual inflows into the reservoir, and we evaluate the effectiveness of Kaiman filtering for stochastic control in relation to the amount of measurement noise.

Stochastic analysis of rigid foundation filtering

AbstractThe effect of the space–time variation of earthquake ground motion on the translational response of structures supported on large rigid mat foundations is considered. A stochastic space–time ground motion model, based on the analysis of recordings from the SMART‐1 seismograph array in Lotung, Taiwan, is used. Random vibration theory is utilized to obtain an expression for the reduction in the maximum structural response, for a specified probability level and strong motion duration. Numerical computations are performed to examine the sensitivity of the reduction in structural response to various ground motion and structural parameters. The results indicate that spatial correlation and travelling wave effects give rise to significant reductions in structural responses when the seismic waves have low apparent propagation velocities. Utilization of this result in the design of structures on large foundations should yield substantial cost savings.

A poincaré gauge-invariant formulation of general-relativistic quantum geometry

We formulate a general-relativistic concept of quantum geometry capable of describing the behaviour of quantum Lorentz frames in free fall in arbitrary external gravitational fields. The mathematical framework suited to this task turns out to be that of Hilbert bundles over a space-time manifold of mean stochastic locations of quantum test particles, with fibres soldered to the base manifold at each given mean space-time location. Furthermore, these fibre bundles incorporate connections compatible with the Hermitian structure. Quantum propagation results from the parallel transport governed by these connections and applied to generalized coherent states, whereas the Hermitian structure supplies the transition probabilities required for the construction of quantum frame and particle propagators. All resulting fibre bundles are shown to possess nonzero curvature forms, which in the generic case incorporate the symplectic 2-form of the cotangent bundle of the base manifold, the pseudo-Riemannian curvature of that manifold and additional components resulting from Maxwell and Yang-Mills connections. In addition, their structure groups incorporate stochastic phase space representations of the Poincare group in a manner that supplies manifest Poincare gauge invariance of the resulting quantum space-time framework.

Extended harmonic analysis of phase space representations for the Galilei group

The spectral resolution of phase space representations of the Galilei group is achieved by deriving all possible decompositions into irreducible representations corresponding to reproducing kernel Hilbert spaces. Spectral syntheses in terms of eigenfunction expansions, as well as in terms of continuous resolutions of the identity, are achieved. For the latter, the existence, uniqueness and other basic properties of resolution generators are established. This is shown to lead to systems of covariance related to measurements of stochastic phase space values performed with extended quantum test particles, whose proper wavefunctions are the aforementioned resolution generators.

Unsharp reality and joint measurements for spin observables.

The Einstein-Podolsky-Rosen (EPR) reality criterion is generalized to fit with the notion of positive-operator-valued observables occurring in quantum optics, stochastic quantum mechanics, and other fields of quantum physics. The resulting concept of unsharp reality for quantum systems is illustrated within stochastic spin space where it leads to a notion of unsharp spin property. Finally we investigate the possibility of joint spin measurements and give a brief discussion of the EPR-Bell argument for unsharp spin properties.

Scattering theory for extended elementary particles in nonrelativistic stochastic quantum mechanics

Scattering theory for extended elementary particles in stochastic phase space is studied. It is shown that the interacting Hamiltonian is equivalent to an effective potential in configuration representation. Asymptotic completeness can be studied by investigating the behavior of the effective potential. The sharp-point limit of the extension of these particles is studied. It is also shown that scattering theory can also be studied directly in stochastic phase space in the optimal case.

Stochastic phase spaces, fuzzy sets, and statistical metric spaces

This paper is devoted to the study of the notion of the phase-space representation of quantum theory in both the nonrelativisitic and the relativisitic cases. Then, as a derived concept, the stochastic phase space is introduced and its connections with fuzzy set theory and probabilistic topological (in particular, metric) spaces are discussed.