• An application of Bayesian variable selection to international economic data

      Tian, Xiang; Goddard, Scott; Barry, Ron; Short, Margaret; McIntyre, Julie (2017-06)
      GDP plays an important role in people's lives. For example, when GDP increases, the unemployment rate will frequently decrease. In this project, we will use four different Bayesian variable selection methods to verify economic theory regarding important predictors to GDP. The four methods are: g-prior variable selection with credible intervals, local empirical Bayes with credible intervals, variable selection by indicator function, and hyper-g prior variable selection. Also, we will use four measures to compare the results of the various Bayesian variable selection methods: AIC, BIC, Adjusted-R squared and cross-validation.
    • Bayesian methods in glaciology

      Brinkerhoff, Douglas; Truffer, Martin; Aschwanden, Andy; Tape, Carl; Bueler, Ed (2017-12)
      The problem of inferring the value of unobservable model parameters given a set of observations is ubiquitous in glaciology, as are large measurement errors. Bayes' theorem provides a unified framework for addressing such problems in a rigorous and robust way through Monte Carlo sampling of posterior distributions, which provides not only the optimal solution for a given inverse problem, but also the uncertainty. We apply these methods to three glaciological problems. First, we use Markov Chain Monte Carlo sampling to infer the importance of different glacier hydrological processes from observations of terminus water flux and surface speed. We find that the opening of sub-glacial cavities due to sliding over asperities at the glacier bed is of a similar magnitude to the opening of channels due to turbulent melt during periods of large input flux, but also that the processes of turbulent melting is the greatest source of uncertainty in hydrological modelling. Storage of water in both englacial void spaces and exchange of water between the englacial and subglacial systems are both necessary to explain observations. We next use Markov Chain Monte Carlo sampling to determine distributed glacier thickness from dense observations of surface velocity and mass balance coupled with sparse direct observations of thickness. These three variables are related through the principle of mass conservation. We develop a new framework for modelling observational uncertainty, then apply the method to three test cases. We find a strong relationship between measurement uncertainty, measurement spacing, and the resulting uncertainty in thickness estimates. We also find that in order to minimize uncertainty, measurement spacing should be 1-2 times the characteristic length scale of variations in subglacial topography. Finally, we apply the method of particle filtering to compute robust estimates of ice surface velocity and uncertainty from oblique time-lapse photos for the rapidly retreating Columbia Glacier. The resulting velocity fields, when averaged over suitable time scales, agree well with velocity measurements derived from satellites. At higher temporal resolution, our results suggest that seasonal evolution of the subglacial drainage system is responsible for observed changes in ice velocity at seasonal scales, and that this changing configuration produces varying degrees of glacier flow sensitivity to changes in external water input.
    • Bayesian predictive process models for historical precipitation data of Alaska and southwestern Canada

      Vanney, Peter; Short, Margaret; Goddard, Scott; Barry, Ronald (2016-05)
      In this paper we apply hierarchical Bayesian predictive process models to historical precipitation data using the spBayes R package. Classical and hierarchical Bayesian techniques for spatial analysis and modeling require large matrix inversions and decompositions, which can take prohibitive amounts of time to run (n observations take time on the order of n3). Bayesian predictive process models have the same spatial framework as hierarchical Bayesian models but fit a subset of points (called knots) to the sample which allows for large scale dimension reduction and results in much smaller matrix inversions and faster computing times. These computationally less expensive models allow average desktop computers to analyze spatially related datasets in excess of 20,000 observations in an acceptable amount of time.
    • Edge detection using Bayesian process convolutions

      Lang, Yanda; Short, Margaret; Barry, Ron; Goddard, Scott; McIntyre, Julie (2017-05)
      This project describes a method for edge detection in images. We develop a Bayesian approach for edge detection, using a process convolution model. Our method has some advantages over the classical edge detector, Sobel operator. In particular, our Bayesian spatial detector works well for rich, but noisy, photos. We first demonstrate our approach with a small simulation study, then with a richer photograph. Finally, we show that the Bayesian edge detector performance gives considerable improvement over the Sobel operator performance for rich photos.
    • Probabilistic decline curve analysis in unconventional reservoirs using Bayesian and approximate Bayesian inference

      Korde, Anand A.; Awoleke, Obadare; Goddard, Scott; Dandekar, Abhijit (2019-08)
      In this work, a probabilistic methodology for Decline Curve Analysis (DCA) in unconventional reservoirs is presented using a combination of Bayesian statistical methods and deterministic models. Accurate reserve estimation and uncertainty quantification are the primary objectives of this study. The Bayesian inferencing techniques described in this work utilizes three sampling mechanisms, namely the Gibbs Sampling (implemented in OpenBUGS), the Metropolis Algorithm, and Approximate Bayesian Computation (ABC) to sample parameter values from their posterior distributions. These different sampling mechanisms are applied in conjunction with DCA models like Arps, Power Law Exponential (PLE), Stretched Exponential Production Decline (SEPD), Duong and Logistic Growth Analysis (LGA) to estimate prediction intervals. Production is forecasted, and uncertainty bounds are established using these prediction intervals. A complete workflow and the summary steps for each of the sampling techniques are provided to permit readers to replicate results. To examine the reliability, the methodology was tested over 74 oil and gas wells located in the three main sub plays of the Permian Basin, namely, the Delaware play, the Central Basin Platform, and the Midland play. Results show that the examined DCA-Bayesian models are successful in providing a high coverage rate, low production prediction errors and narrow uncertainty bounds for the production history data sets. The methodology was also successfully applied to unconventional reservoirs with as low as 6 months of available production history. Depending on the amount of production history available, the combined deterministic-stochastic model that provides the best fit can vary. It is therefore recommended that all possible combinations of the deterministic and stochastic models be applied to the available production history data. This is in order to obtain more confidence in the conclusions related to the reserve estimates and uncertainty bounds. The novelty of this methodology relies in using multiple combinations of DCA-Bayesian models to achieve accurate reserve estimates and narrow uncertainty bounds. The paper can help assess shale plays as most of the shale plays are in the early stages of production when the reserve estimations are carried out.
    • Using rate transient analysis and bayesian algorithms for reservoir characterization in hydraulically fractured horizontal gas wells during linear flow

      Yuhun, Pirayu; Awoleke, Obadare; Ahmadi, Mohabbat; Hanks, Catherine (2019-05)
      Multi-stage hydraulically fractured horizontal wells (MFHWs) are currently a popular method of developing shale gas and oil reservoirs. The performance of MFHWs can be analyzed by an approach called Rate transient analysis (RTA). However, the predicted outcomes are often inaccurate and provide non-unique results. Therefore, the main objective of this thesis is to couple Bayesian Algorithms with a current production analysis method, that is, rate transient analysis, to generate probabilistic credible interval ranges for key reservoir and completion variables. To show the legitimacy of the RTA-Bayesian method, synthetic production data from a multistage hydraulically fractured horizontal completion in a reservoir modeled after Marcellus shale reservoir was generated using a reservoir (CMG) model. The synthetic production data was analyzed using a combination of rate transient analysis with Bayesian techniques. Firstly, the traditional log-log plot was produced to identify the linear flow production regime, which is usually the dominant regime in shale reservoirs. Using the linear flow production data and traditional rate transient analysis equations, Bayesian inversion was carried out using likelihood-based and likelihood-free Bayesian methods. The rjags and EasyABC packages in statistical software R were used for the likelihood-based and likelihood-free inversion respectively. Model priors were based (1) on information available about the Marcellus shale from technical literature and (2) hydraulic fracture design parameters. Posterior distributions and prediction intervals were developed for the fracture length, matrix permeability, and skin factor. These predicted credible intervals were then compared with actual synthetic reservoir and hydraulic fracture data. The methodology was also repeated for an actual case in the Barnett shale for a validation. The most substantial finding was that for all the investigated cases, including complicated scenarios (such as finite fracture conductivity, fracturing fluid flowback, heterogeneity of fracture length, and pressure-dependent reservoir), the combined RTA-Bayesian model provided a reasonable prediction interval that encompassed the actual/observed values of the reservoir/hydraulic fracture variables. The R-squared value of predicted values over true values was more than 0.5 in all cases. For the base case in this study, the choice of the prior distribution did not affect the posterior distribution/prediction interval in a significant manner in as much as the prior distribution was partially informative. However, the use of noninformative priors resulted in a loss of precision. Also, a comparison of the Approximate Bayesian Computation (ABC) and the traditional Bayesian algorithms showed that the ABC algorithm reduced computational time with minimal loss of accuracy by at least an order of magnitude by bypassing the complicated step of having to compute the likelihood function. In addition, the production time, number of iterations and tolerance of fitting had a minimal impact on the posterior distribution after an optimum point--which was at least one-year production, 10,000 iterations and 0.001 respectively. In summary, the RTA-Bayesian production analysis method implemented in relatively easy computational platforms, like R and Excel, provided good characterization of all key variables such as matrix permeability, fracture length and skin when compared to results obtained from analytical methods. This probabilistic characterization has the potential to enable better understanding of well performance, improved identification of optimization opportunities and ultimately improved ultimate recovery from shale gas resources.