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Register a new userUncertainty quantification in imaging and automatic horizon tracking: a Bayesian deep-prior based approach
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In inverse problems, uncertainty quantification (UQ) deals with a probabilistic description of the solution nonuniqueness and data noise sensitivity. Setting seismic imaging into a Bayesian framework allows for a principled way of studying uncertainty by solving for the model posterior distribution. Imaging, however, typically constitutes only the first stage of a sequential workflow, and UQ becomes even more relevant when applied to subsequent tasks that are highly sensitive to the inversion outcome. In this paper, we focus on how UQ trickles down to horizon tracking for the determination of stratigraphic models and investigate its sensitivity with respect to the imaging result. As such, the main contribution of this work consists in a data-guided approach to horizon tracking uncertainty analysis. This work is fundamentally based on a special reparameterization of reflectivity, known as deep prior. Feasible models are restricted to the output of a convolutional neural network with a fixed input, while weights and biases are Gaussian random variables. Given a deep prior model, the network parameters are sampled from the posterior distribution via a Markov chain Monte Carlo method, from which the conditional mean and point-wise standard deviation of the inferred reflectivities are approximated. For each sample of the posterior distribution, a reflectivity is generated, and the horizons are tracked automatically. In this way, uncertainty on model parameters naturally translates to horizon tracking. As part of the validation for the proposed approach, we verified that the estimated confidence intervals for the horizon tracking coincide with geologically complex regions, such as faults.
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Uncertainty quantification is essential when dealing with ill-conditioned inverse problems due to the inherent nonuniqueness of the solution. Bayesian approaches allow us to determine how likely an estimation of the unknown parameters is via formulating the posterior distribution. Unfortunately, it is often not possible to formulate a prior distribution that precisely encodes our prior knowledge about the unknown. Furthermore, adherence to handcrafted priors may greatly bias the outcome of the Bayesian analysis. To address this issue, we propose to use the functional form of a randomly initialized convolutional neural network as an implicit structured prior, which is shown to promote natural images and excludes images with unnatural noise. In order to incorporate the model uncertainty into the final estimate, we sample the posterior distribution using stochastic gradient Langevin dynamics and perform Bayesian model averaging on the obtained samples. Our synthetic numerical experiment verifies that deep priors combined with Bayesian model averaging are able to partially circumvent imaging artifacts and reduce the risk of overfitting in the presence of extreme noise. Finally, we present pointwise variance of the estimates as a measure of uncertainty, which coincides with regions that are more difficult to image.
Deep Learning methods are known to suffer from calibration issues: they typically produce over-confident estimates. These problems are exacerbated in the low data regime. Although the calibration of probabilistic models is well studied, calibrating extremely over-parametrized models in the low-data regime presents unique challenges. We show that deep-ensembles do not necessarily lead to improved calibration properties. In fact, we show that standard ensembling methods, when used in conjunction with modern techniques such as mixup regularization, can lead to less calibrated models. In this text, we examine the interplay between three of the most simple and commonly used approaches to leverage deep learning when data is scarce: data-augmentation, ensembling, and post-processing calibration methods. We demonstrate that, although standard ensembling techniques certainly help to boost accuracy, the calibration of deep-ensembles relies on subtle trade-offs. Our main finding is that calibration methods such as temperature scaling need to be slightly tweaked when used with deep-ensembles and, crucially, need to be executed after the averaging process. Our simulations indicate that, in the low data regime, this simple strategy can halve the Expected Calibration Error (ECE) on a range of benchmark classification problems when compared to standard deep-ensembles.
The idea to distinguish and quantify two important types of uncertainty, often referred to as aleatoric and epistemic, has received increasing attention in machine learning research in the last couple of years. In this paper, we consider ensemble-based approaches to uncertainty quantification. Distinguishing between different types of uncertainty-aware learning algorithms, we specifically focus on Bayesian methods and approaches based on so-called credal sets, which naturally suggest themselves from an ensemble learning point of view. For both approaches, we address the question of how to quantify aleatoric and epistemic uncertainty. The effectiveness of corresponding measures is evaluated and compared in an empirical study on classification with a reject option.
Hyperspectral pansharpening aims to synthesize a low-resolution hyperspectral image (LR-HSI) with a registered panchromatic image (PAN) to generate an enhanced HSI with high spectral and spatial resolution. Recently proposed HS pansharpening methods have obtained remarkable results using deep convolutional networks (ConvNets), which typically consist of three steps: (1) up-sampling the LR-HSI, (2) predicting the residual image via a ConvNet, and (3) obtaining the final fused HSI by adding the outputs from first and second steps. Recent methods have leveraged Deep Image Prior (DIP) to up-sample the LR-HSI due to its excellent ability to preserve both spatial and spectral information, without learning from large data sets. However, we observed that the quality of up-sampled HSIs can be further improved by introducing an additional spatial-domain constraint to the conventional spectral-domain energy function. We define our spatial-domain constraint as the $L_1$ distance between the predicted PAN image and the actual PAN image. To estimate the PAN image of the up-sampled HSI, we also propose a learnable spectral response function (SRF). Moreover, we noticed that the residual image between the up-sampled HSI and the reference HSI mainly consists of edge information and very fine structures. In order to accurately estimate fine information, we propose a novel over-complete network, called HyperKite, which focuses on learning high-level features by constraining the receptive from increasing in the deep layers. We perform experiments on three HSI datasets to demonstrate the superiority of our DIP-HyperKite over the state-of-the-art pansharpening methods. The deployment codes, pre-trained models, and final fusion outputs of our DIP-HyperKite and the methods used for the comparisons will be publicly made available at https://github.com/wgcban/DIP-HyperKite.git.
In the presence of background noise, arrival times picked from a surface microseismic data set usually include a number of false picks that can lead to uncertainty in location estimation. To eliminate false picks and improve the accuracy of location estimates, we develop an association algorithm termed RANSAC-based Arrival Time Event Clustering (RATEC) that clusters picked arrival times into event groups based on random sampling and fitting moveout curves that approximate hyperbolas. Arrival times far from the fitted hyperbolas are classified as false picks and removed from the data set prior to location estimation. Simulations of synthetic data for a 1-D linear array show that RATEC is robust under different noise conditions and generally applicable to various types of subsurface structures. By generalizing the underlying moveout model, RATEC is extended to the case of a 2-D surface monitoring array. The effectiveness of event location for the 2-D case is demonstrated using a data set collected by the 5200-element dense Long Beach array. The obtained results suggest that RATEC is effective in removing false picks and hence can be used for phase association before location estimates.
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