Reza Madankan

Polynomial-chaos-based Bayesian approach for state and parameter estimations

Two new recursive approaches have been developed to provide accurate estimates for posterior moments of both parameters and system states while making use of the generalized polynomial-chaos framework for uncertainty propagation. The main idea of the generalized polynomial-chaos method is to expand random state and input parameter variables involved in a stochastic differential/difference equation in a polynomial expansion. These polynomials are associated with the prior probability density function for the input parameters. Later, Galerkin projection is used to obtain a deterministic system of equations for the expansion coefficients. The first proposed approach provides means to update prior expansion coefficients by constraining the polynomial-chaos expansion to satisfy a specified number of posterior moment constraints derived from Bayes’s rule. The second proposed approach makes use of the minimum variance formulation to update generalized polynomial-chaos coefficients. The main advantage of the prop...

Computation of probabilistic hazard maps and source parameter estimation for volcanic ash transport and dispersion

Volcanic ash advisory centers are charged with forecasting the movement of volcanic ash plumes, for aviation, health and safety preparation. Deterministic mathematical equations model the advection and dispersion of these plumes. However initial plume conditions - height, profile of particle location, volcanic vent parameters - are known only approximately at best, and other features of the governing system such as the windfield are stochastic. These uncertainties make forecasting plume motion difficult. As a result of these uncertainties, ash advisories based on a deterministic approach tend to be conservative, and many times over/under estimate the extent of a plume. This paper presents an end-to-end framework for generating a probabilistic approach to ash plume forecasting. This framework uses an ensemble of solutions, guided by Conjugate Unscented Transform (CUT) method for evaluating expectation integrals. This ensemble is used to construct a polynomial chaos expansion that can be sampled cheaply, to provide a probabilistic model forecast. The CUT method is then combined with a minimum variance condition, to provide a full posterior pdf of the uncertain source parameters, based on observed satellite imagery.The April 2010 eruption of the Eyjafjallajokull volcano in Iceland is employed as a test example. The puff advection/dispersion model is used to hindcast the motion of the ash plume through time, concentrating on the period 14-16 April 2010. Variability in the height and particle loading of that eruption is introduced through a volcano column model called bent. Output uncertainty due to the assumed uncertain input parameter probability distributions, and a probabilistic spatial-temporal estimate of ash presence are computed.

Polynomial Chaos Quadrature-based Minimum Variance Approach for Source Parameters Estimation

Abstract We present a polynomial chaos based minimum variance formulation to solve inverse problems. The utility of the proposed approach is evaluated by considering the ash transport problem arising due to volcanic eruption. Volcanic ash advisory centers generally makes use of mathematical models for column eruption and advection and diffusion of ash cloud in atmosphere. These models require input data on source conditions such as vent radius, vent velocity and distribution of ash-particle size. The inputs are usually not well constrained, and estimates of the uncertainty in the inputs is needed to make accurate predictions of cloud motion. The recent eruption of Eyjafjallajokull, Iceland in April 2010 is considered as test example. For validation, the puff advection and diffusion model is used to hindcast the motion of the ash cloud through time concentrating on the period 14-16 April 2010. Variability in the height and loading of the eruption is introduced through the volcano column model bent. Output uncertainty due to uncertain input parameters is determined with a polynomial chaos quadrature (PCQ)-based sampling of the multidimensional puff input vector space. Furthermore, the posterior distribution for input parameters is obtained by assimilating satellite imagery data with PCQ predictions using a minimum variance approach.

Challenges in developing DDDAS based methodology for volcanic ash hazard analysis - Effect of numerical weather prediction variability and parameter estimation

In this paper, we will present ongoing work on using a dynamic data driven application system (DDDAS) based approach to the forecast of volcanic ash transport and dispersal. Our primary modeling tool will be a new code puffin formed by the combination of a plume eruption model Bent and the ash transport model Puff. Data from satellite imagery, observation of vent parameters and windfields will drive our simulations. We will use ensemble based uncertainty quantification and parameter estimation methodology – polynomial chaos quadrature in combination with data integration to complete the DDDAS loop.

Temporal, probabilistic mapping of ash clouds using wind field stochastic variability and uncertain eruption source parameters: Example of the 14 April 2010 Eyjafjallajökull eruption

Uncertainty in predictions from a model of volcanic ash transport in the atmosphere arises from uncertainty in both eruption source parameters and the model wind field. In a previous contribution, we analyzed the probability of ash cloud presence using weighted samples of volcanic ash transport and dispersal model runs and a reanalysis wind field to propagate uncertainty in eruption source parameters alone. In this contribution, the probabilistic modeling is extended by using ensemble forecast wind fields as well as uncertain source parameters. The impact on ash transport of variability in wind fields due to unresolved scales of motion as well as model physics uncertainty is also explored. We have therefore generated a weighted, probabilistic forecast of volcanic ash transport with only a priori information, exploring uncertainty in both the wind field and the volcanic source.

Fast Construction of Surrogates for UQ Central to DDDAS– Application to Volcanic Ash Transport

In this paper, we present new ideas to greatly enhance the quality of uncertainty quantification in the DDDAS framework. We build on ongoing work in large scale transport of geophysical mass of volcanic origin – a danger to both land based installations and airborne vehicles. The principal new idea introduced is the concept of a localized Bayes linear model as a surrogate for the expensive simulator. Probability of ash presence is compared to earlier work.

Optimal information collection for source parameter estimation of atmospheric release phenomenon

In this research, the effect of dynamic data measurement on source parameters estimation is studied. The concept of mutual information is exploited to identify the optimal location for each sensor, while performing the dynamic data measurement to improve accuracy of estimation. For validation purposes, an advection - diffusion simulation code, SCIPUFF (Second-order Closure Integrated PUFF) is being used as a modeling testbed to study the effect of using dynamic data measurement. A Bayesian estimation framework is being used to characterize the source parameters, while data measurement is performed by mobile sensors, which are located based on the concept of maximizing the information content. As our numerical simulations show, using dynamic data measurement, based on maximum information collection, leads to considerably better estimates of source parameters.

Application of Conjugate Unscented Transform in source parameters estimation

A polynomial chaos based minimum variance method is introduced to solve inverse problems. Two different set of quadrature points, Conjugate Unscented Transform and Gauss Legendre quadrature points, are being used to perform the estimation process. The main concentration of this paper is to show the efficiency of Conjugate Unscented Transform points versus Gauss Legendre quadrature scheme. For validation purposes, an advection - diffusion simulation code, SCIPUFF (Second-order Closure Integrated PUFF) is being used as a modeling testbed to study the effect of both these quadrature schemes on solution of an inverse problem. Simulation results show efficiency of Conjugate Unscented Transform versus Gauss-Legendre quadratures.

A robust data assimilation approach in the absence of sensor statistical properties

A convex optimization based approach is presented to perform model-data assimilation of spatial temporal dynamical systems where sensor error characteristics are not available. The key idea of the proposed technique is that one should not make any assumption regarding the statistical properties of sensor data when they are not available. Recently developed quadrature scheme, Conjugate Unscented Transformation in conjunction with convex optimization tools is used to obtain an approximation of posterior density function. The proposed approach is validated by considering the problem of source parameter estimation for toxic material release in the atmosphere. The numerical experiments provides a basis for optimism for the robustness of the proposed methodology.

Parameter Estimation of Atmospheric Release Incidents Using Maximal Information Collection

The effects of data measurement on source parameter estimation are studied. The concept of mutual information is applied to find the optimal location for each sensor to improve accuracy of the overall estimation process. For validation purposes, an advection - diffusion simulation code, called SCIPUFF, is used as a modeling testbed to study the effects of using dynamic data measurement. Bayesian inference framework is utilized for model-data fusion using stationary and mobile sensor networks, where in mobile sensors, the proposed approach is used to locate data observation sensors. As our numerical simulations show, using the proposed approach leads to a considerably better estimate of parameters comparing with stationary sensors.