Data Assimilation for a Geological Process Model Using the Ensemble Kalman Filter

We consider the problem of conditioning a geological process-based computer simulation, which produces basin models by simulating transport and deposition of sediments, to data. Emphasising uncertainty quantification, we frame this as a Bayesian inverse problem, and propose to characterize the posterior probability distribution of the geological quantities of interest by using a variant of the ensemble Kalman filter, an estimation method which linearly and sequentially conditions realisations of the system state to data. A test case involving synthetic data is used to assess the performance of the proposed estimation method, and to compare it with similar approaches. We further apply the method to a more realistic test case, involving real well data from the Colville foreland basin, North Slope, Alaska.Comment: 34 pages, 10 figures, 4 table

The value of imperfect borehole information in mineral resource evaluation

Abstract In mineral resource evaluation a careful analysis and assessments of the geology, assay data and structural data is performed. One critical question is where to position the exploration boreholes that render it possible to classify as much of the deposit as possible as a measured or indicated resource. Another important question is what method to use when analyzing the grade in the collected material. For the deposit we consider, a challenge is to assess whether one should analyze the collected core samples with accurate and expensive XRF equipment or the less accurate and less expensive XMET equipment. A dataset of 1,871 XMET and 103 XRF observations is available, along with relevant explanatory variables. At the 103 sites where XRF data is acquired, 103 XMET measurements are also available. We first derive estimates of the regression and covariance parameters of a Gaussian random field model for the log XMET and log XRF data. Next, the model is used to predict the decisive grade parameter on block support. To improve the predictions, the mining company has planned to drill and collect 265 core samples along new boreholes. The associated reduction in prediction variance, with XRF or XMET data collection, is studied. Moreover, we compute the value of the XRF or XMET information using the statistical model, the expected development costs and revenues. The value of information is a useful diagnostic here, comparing the actual price of the XRF or XMET data with its added value

The value of information for correlated GLMs

We examine the situation where a decision maker is considering investing in a number of projects with uncertain revenues. Before making a decision, the investor has the option to purchase data which carry information about the outcomes from pertinent projects. When these projects are correlated, the data are informative about all the projects. The value of information is the maximum amount the investor would pay to acquire these data. The problem can be seen from a sampling design perspective where the sampling criterion is the maximisation of the value of information minus the sampling cost. The examples we have in mind are in the spatial setting where the sampling is performed at spatial coordinates or spatial regions. In this paper we discuss the case where the outcome of each project is modelled by a generalised linear mixed model. When the distribution is non-Gaussian, the value of information does not have a closed form expression. We use the Laplace approximation and matrix approximations to derive an analytical expression to the value of information, and examine its sensitivity under different parameter settings and distributions. In the Gaussian case the proposed technique is exact. Our analytical method is compared against the alternative Monte-Carlo method, and we show similarity of results for various sample sizes of the data. The closed form results are much faster to compute. Model weighting and bootstrap are used to measure the sensitivity of our analysis to model and parameter uncertainty. A general guidance on making decisions using our results is offered. Application of the method is presented in a spatial decision problem for treating the Bovine Tuberculosis in the United Kingdom, and for rock fall avoidance decisions in a Norwegian mine

Comparison of Ensemble-Based Data Assimilation Methods for Sparse Oceanographic Data

For oceanographic applications, probabilistic forecasts typically have to deal with i) high-dimensional complex models, and ii) very sparse spatial observations. In search-and-rescue operations at sea, for instance, the short-term predictions of drift trajectories are essential to efficiently define search areas, but in-situ buoy observations provide only very sparse point measurements, while the mission is ongoing. Statistically optimal forecasts, including consistent uncertainty statements, rely on Bayesian methods for data assimilation to make the best out of both the complex mathematical modeling and the sparse spatial data. To identify suitable approaches for data assimilation in this context, we discuss localisation strategies and compare two state-of-the-art ensemble-based methods for applications with spatially sparse observations. The first method is a version of the ensemble-transform Kalman filter, where we tailor a localisation scheme for sparse point data. The second method is the implicit equal-weights particle filter which has recently been tested for related oceanographic applications. First, we study a linear spatio-temporal model for contaminant advection and diffusion, where the analytical Kalman filter provides a reference. Next, we consider a simplified ocean model for sea currents, where we conduct state estimation and predict drift. Insight is gained by comparing ensemble-based methods on a number of skill scores including prediction bias and accuracy, distribution coverage, rank histograms, spatial connectivity and drift trajectory forecasts

VNIbCReg: VICReg with Neighboring-Invariance and better-Covariance Evaluated on Non-stationary Seismic Signal Time Series

One of the latest self-supervised learning (SSL) methods, VICReg, showed a great performance both in the linear evaluation and the fine-tuning evaluation. However, VICReg is proposed in computer vision and it learns by pulling representations of random crops of an image while maintaining the representation space by the variance and covariance loss. However, VICReg would be ineffective on non-stationary time series where different parts/crops of input should be differently encoded to consider the non-stationarity. Another recent SSL proposal, Temporal Neighborhood Coding (TNC) is effective for encoding non-stationary time series. This study shows that a combination of a VICReg-style method and TNC is very effective for SSL on non-stationary time series, where a non-stationary seismic signal time series is used as an evaluation dataset

Block composite likelihood models for analysis of large spatial datasets

Abstract Large spatial datasets become more common as a result of automatic sensors, remote sensing and the increase in data storage capacity. But large spatial datasets are hard to analyse. Even in the simplest Gaussian situation, parameter estimation and prediction are troublesome because one requires matrix factorization of a large covariance matrix. We consider a composite likelihood construction built on the joint densities of subsets of variables. This composite model thus splits a datasets in many smaller datasets, each of which can be evaluated separately. These subsets of data are combined through a summation giving the final composite likelihood. Massive datasets can be handled with this approach. In particular, we consider a block composite likelihood model, constructed over pairs of spatial blocks. The blocks can be disjoint, overlapping or at various resolution. The main idea is that the spatial blocking should capture the important correlation effects in the data. Estimates for unknown parameters as well as optimal spatial predictions under the block composite model are obtained. Asymptotic variances for both parameter estimates and predictions are computed using Godambe sandwich matrices. The procedure is demonstrated on 2D and 3D datasets with regular and irregular sampling of data. For smaller data size we compare with optimal predictors, for larger data size we discuss and compare various blocking schemes

A revised implicit equal-weights particle filter

Particle filters are fully non-linear data assimilation methods and as such are highly relevant. While the standard particle filter degenerates for high-dimensional systems, recent developments have opened the way for new particle filters that can be used in such systems. The implicit equal-weights particle filter (IEWPF) is an efficient approach which avoids filter degeneracy because it gives equal particle weights by construction. The method uses implicit sampling whereby auxiliary vectors drawn from a proposal distribution undergo a transformation before they are added to each particle. In the original formulation of the IEWPF, the proposal distribution has a gap causing all but one particle to have an inaccessible region in state space. We show that this leads to a systematic bias in the predictions and we modify the proposal distribution to eliminate the gap. We achieved this by using a two-stage proposal method, where a single variance parameter is tuned to obtain adequate statistical coverage properties of the predictive distribution. We discuss properties of the implicit mapping from an auxiliary random vector to the state vector, keeping in mind the aim of avoiding particle resampling. The revised filter is tested on linear and weakly nonlinear dynamical models in low-dimensional and moderately high-dimensional settings, demonstrating the suiccess of the new methodology in removing the bias

Learning excursion sets of vector-valued Gaussian random fields for autonomous ocean sampling

Improving and optimizing oceanographic sampling is a crucial task for marine science and maritime resource management. Faced with limited resources in understanding processes in the water-column, the combination of statistics and autonomous systems provide new opportunities for experimental design. In this work we develop efficient spatial sampling methods for characterizing regions defined by simultaneous exceedances above prescribed thresholds of several responses, with an application focus on mapping coastal ocean phenomena based on temperature and salinity measurements. Specifically, we define a design criterion based on uncertainty in the excursions of vector-valued Gaussian random fields, and derive tractable expressions for the expected integrated Bernoulli variance reduction in such a framework. We demonstrate how this criterion can be used to prioritize sampling efforts at locations that are ambiguous, making exploration more effective. We use simulations to study and compare properties of the considered approaches, followed by results from field deployments with an autonomous underwater vehicle as part of a study mapping the boundary of a river plume. The results demonstrate the potential of combining statistical methods and robotic platforms to effectively inform and execute data-driven environmental sampling

Dynamic stochasticmodeling for adaptive sampling of environmental variables using an AUV

Discharge of mine tailings significantly impacts the ecological status of the sea. Methods to efficiently monitor the extent of dispersion is essential to protect sensitive areas. By combining underwater robotic sampling with ocean models, we can choose informative sampling sites and adaptively change the robot’s path based on in situ measurements to optimally map the tailings distribution near a seafill. This paper creates a stochastic spatio-temporal proxy model of dispersal dynamics using training data from complex numerical models. The proxy model consists of a spatio-temporal Gaussian process model based on an advection–diffusion stochastic partial differential equation. Informative sampling sites are chosen based on predictions from the proxy model using an objective function favoring areas with high uncertainty and high expected tailings concentrations. A simulation study and data from real-life experiments are presented.publishedVersio