Novel Solutions to the Three-Anchor ToA-Based Three-Dimensional Positioning Problem

At least four non-coplanar anchor nodes (ANs) are required for the time-of-arrival (ToA)-based three-dimensional (3D) positioning to enable unique position estimation. Direct method (DM) and particle filter (PF) algorithms were developed to address the three-anchor ToA-based 3D positioning problem. The proposed DM reduces this problem to the solution of a quadratic equation, exploiting the knowledge about the workspace, to first estimate the x- or z-coordinate, and then the remaining two coordinates. The implemented PF uses 1000 particles to represent the posterior probability density function (PDF) of the AN’s 3D position. The prediction step generates new particles by a resampling procedure. The ToA measurements determine the importance of these particles to enable updating the posterior PDF and estimating the 3D position of the AN. Simulation results corroborate the viability of the developed DM and PF algorithms, in terms of accuracy and computational cost, in the pursuit and circumnavigation scenarios, and even with a horizontally coplanar arrangement of the three ANs. Therefore, it is possible to enable applications requiring real-time positioning, such as unmanned aerial vehicle (UAV) autonomous docking and circling a stationary (or moving) position, without the need for an excessive number of ANs.

Efficient Probabilistic Joint Inversion of Direct Current Resistivity and Small-Loop Electromagnetic Data

Often, multiple geophysical measurements are sensitive to the same subsurface parameters. In this case, joint inversions are mostly preferred over two (or more) separate inversions of the geophysical data sets due to the expected reduction of the non-uniqueness in the joint inverse solution. This reduction can be quantified using Bayesian inversions. However, standard Markov chain Monte Carlo (MCMC) approaches are computationally expensive for most geophysical inverse problems. We present the Kalman ensemble generator (KEG) method as an efficient alternative to the standard MCMC inversion approaches. As proof of concept, we provide two synthetic studies of joint inversion of frequency domain electromagnetic (FDEM) and direct current (DC) resistivity data for a parameter model with vertical variation in electrical conductivity. For both studies, joint results show a considerable improvement for the joint framework over the separate inversions. This improvement consists of (1) an uncertainty reduction in the posterior probability density function and (2) an ensemble mean that is closer to the synthetic true electrical conductivities. Finally, we apply the KEG joint inversion to FDEM and DC resistivity field data. Joint field data inversions improve in the same way seen for the synthetic studies.

Assimilation of multiple linearly dependent data vectors

Assimilation of a sequence of linearly dependent data vectors, $\{d_{l}\}^{L}_{l=1}${dl}l=1L such that ${d_{l} = B_{l}d_{L}}^{L-1}_{ l=1}$dl=BldLl=1L−1, is considered for a parameter estimation problem. Such a data sequence can occur, for example, in the context of multilevel data assimilation. Since some information is used several times when linearly dependent data vectors are assimilated, the associated data-error covariances must be modified. I develop a condition that the modified covariances must satisfy in order to sample correctly from the posterior probability density function of the uncertain parameter in the linear-Gaussian case. It is shown that this condition is a generalization of the well-known condition that must be satisfied when assimilating the same data vector multiple times. I also briefly discuss some qualitative and computational issues related to practical use of the developed condition.

Bayesian Inference Approach to Inverse Problems in a Financial Mathematical Model

This paper investigates an inverse problem of option pricing in the extended Black--Scholes model. We identify the model coefficients from the measured data and attempt to find arbitrage opportunities in financial markets using a Bayesian inference approach. The posterior probability density function of the parameters is computed from the measured data. The statistics of the unknown parameters are estimated by Markov Chain Monte Carlo (MCMC), which explores the posterior state space. The efficient sampling strategy of MCMC enables us to solve inverse problems by the Bayesian inference technique. Our numerical results indicate that the Bayesian inference approach can simultaneously estimate the unknown drift and volatility coefficients from the measured data.

A Bayesian Framework of Computing and Updating Wavelet Parameter Distributions for Identifying Early Bearing Faults

Rolling element bearings are widely used in machines to support rotation shafts. Bearing failures may result in machine breakdown. In order to prevent bearing failures, early bearing faults are required to be identified. Wavelet analysis has proven to be an effective method for extracting early bearing fault features. Proper selection of wavelet parameters is crucial to wavelet analysis. In this paper, a Bayesian framework is proposed to compute and update wavelet parameter distributions. First, a smoothness index is used as the objective function because it has specific upper and lower bounds. Second, a general sequential Monte Carlo method is introduced to analytically derive the joint posterior probability density function of wavelet parameters. Last, approximately optimal wavelet parameters are inferred from the joint posterior probability density function. Simulated and real case studies are investigated to demonstrate that the proposed framework is effective in extracting early bearing fault features.

An Improved Approach of Terrain-Aided Navigation Based on RBPF

Rao-Blackwellized Particle Filter (RBPF) is suitable for solving the linear/nonlinear mixed Terrain-Aided Navigation (TAN) problem. But the Particle Filter (PF) part of RBPF is Standard Particle Filter (SPF), causing particle diversity reduction and even filters divergence under extreme conditions. To get a better estimation of the errors of INS, this paper proposes an improved approach called Regularized Rao-Blackwellized Particle Filter (RRBPF). After updating the nonlinear state and corresponding importance weights, RRBPF resamples from the Epanechnikov kernel and then get the resampled particles through a linear transition process. Theoretically, the resampling part of RRBPF is equivalent to resampling from the approximated continuous posterior probability density function. Shuttle Radar Topography Mission (SRTM) terrain data is used in simulations to investigate the performance of RRBPF. Results show that RRBPF can provide more accurate estimation of TAN and bear larger initial position error than Sandia Inertial Terrain Aided Navigation (SITAN).

Sequential Bearings-Only-Tracking Initiation with Particle Filtering Method

The tracking initiation problem is examined in the context of autonomous bearings-only-tracking (BOT) of a single appearing/disappearing target in the presence of clutter measurements. In general, this problem suffers from a combinatorial explosion in the number of potential tracks resulted from the uncertainty in the linkage between the target and the measurement (a.k.a the data association problem). In addition, the nonlinear measurements lead to a non-Gaussian posterior probability density function (pdf) in the optimal Bayesian sequential estimation framework. The consequence of this nonlinear/non-Gaussian context is the absence of a closed-form solution. This paper models the linkage uncertainty and the nonlinear/non-Gaussian estimation problem jointly with solid Bayesian formalism. A particle filtering (PF) algorithm is derived for estimating the model’s parameters in a sequential manner. Numerical results show that the proposed solution provides a significant benefit over the most commonly used methods, IPDA and IMMPDA. The posterior Cramér-Rao bounds are also involved for performance evaluation.

Joint estimation of porosity and saturation using stochastic rock-physics modeling

Sediment porosity and saturation affect bulk modulus, shear modulus, and density. Consequently, estimating hydrocarbon saturation and reservoir porosity from seismic data is a joint estimation problem: Uncertainty in porosity will lead to errors in saturation prediction, and vice versa. Porosity and saturation can be jointly estimated using stochastic rock-physics modeling and formal Bayesian estimation methodology. Knowledge of shear impedance reduces the uncertainty in porosity and thus also reduces uncertainty in saturation estimation. This study investigates joint estimation of porosity and saturation by using rock-physics, stochastic modeling, and Bayesian estimation theory to derive saturation and porosity maps of expected pay sands. In the field example, the uncertainty in porosity, quantified by the standard deviation (STD) associated with the posterior probability density function (pdf), derived from inversion of seismic data is much less than the uncertainty in the derived saturation. For a typical case, the STD associated with saturation is [Formula: see text] while porosity STD is about 1.34 porosity units given seismic-derived inversion attributes with reasonable accuracy. Comparison of these numbers with prior estimates showed that inversion of seismic data decreased the uncertainty in porosity to 15% of the prior uncertainty while saturation uncertainty was only reduced to 92% of the prior uncertainty. Although these results may vary from one location to another, the methodology is general and can be applied to other locations.