UCLA

In the age of global warming, there is a crucial need to accurately assess uncertainty levels when analyzing observed changes in the climate. For many climate problems, the development of statistical methods that appropriately account for uncertainty is challenging due to the complexity of the underlying climate processes and the various sources of uncertainty involved. This thesis addresses methodological challenges in modeling uncertainty for two climate problems with important real-world applications. The first problem is concerned with quantifying the heat content of the global ocean and its change over time. Understanding the trend in ocean heat content is particularly important as it informs estimates of transient climate sensitivity, a physical parameter that largely determines the amount of warming that will be expected in the years to come. This problem is nevertheless made difficult by the challenge of representing the complex covariance structure of the ocean heat content field, as well as the challenge of quantifying the uncertainty in the estimation of this structure. The second problem is concerned with separating the influence of warming caused by human activities from natural variability in the observed climate, a problem that is often referred to as climate change ``detection and attribution''. While various sources of uncertainty in this problem have been addressed in the literature, recent results have suggested that commonly-used methods under-estimate uncertainty in their conclusions. Producing reliable detection and attribution confidence intervals is difficult in part due to the challenge of modeling the uncertainty in the estimation of the natural variability covariance structure from limited climate model simulations.

This thesis proposes methods for addressing statistical challenges in these two problems with respect to three overarching themes. The first theme is the use of spatially-coherent statistical models to represent the covariance structures of the underlying physical processes. For the ocean heat content problem, a novel cylindrical kernel-convolution Gaussian process model is developed to flexibly represent the complex spatial correlation patterns of the global ocean heat content field. For the detection and attribution problem, a Laplacian basis vector parameterization of the covariance matrix is proposed to enforce spatially-coherent correlation patterns. This parameterization is also able to avoid the uncertainty in the traditional approach of estimating principal component vectors from limited numbers of climate model runs. The second theme is the use of hierarchical Bayesian models to propagate the uncertainty in estimating the covariance structure to the final results. In the ocean heat content problem, the spatially-varying parameter fields describing the kernel-convolution Gaussian process are themselves modeled as Gaussian processes in a hierarchical framework. This allows for the uncertainty in estimating these parameters to be propagated to the final posterior distribution for the ocean heat content trend. In the detection and attribution problem, the parameters of the Laplacian parameterization of the covariance matrix, as well as the number of Laplacians to use, are both represented in a Bayesian hierarchical framework that prioritizes the accurate modeling of uncertainty. Finally, the third theme concerns the evaluation of the statistical properties of the Bayesian posterior distributions. For the ocean heat content problem, this is done using cross-validation on the observations with respect to a metric for evaluating both the mean and uncertainty implied by the posterior predictive distributions. For the detection and attribution problem, climate model simulations are used to evaluate the accuracy of the posterior means and credible intervals produced by the proposed methods in the context where the true value can be assumed to be known.

Chapter 1 begins by introducing the broader context and implications of the two climate problems and proceeds to give a brief overview of the three statistical themes. Chapter 2 develops the proposed methodology for the ocean heat content problem in a restricted context focusing on spatial variability. A cross-validation study is presented showing that the proposed framework achieves higher accuracy in the predictive posterior distributions than a commonly-used previous method as well as simpler Bayesian approaches. This framework is then extended to the full spatio-temporal context in Chapter 3 and is applied to the quantification of the trend in ocean heat content from 2007 to 2021. The detection and attribution problem is addressed in Chapter 5, where a climate model validation study shows that the proposed approach achieves higher accuracy in the posterior mean and more accurate credible intervals than a traditional approach. While the validation results for each of these proposed methods show quantitative improvements over previous approaches, the results suggest several promising opportunities for additional improvements and extensions. Several of these potential avenues for future research are discussed in Chapter 6.