Spatial fields in the Earth and environmental sciences are often available at multiple scales or resolutions. While coarse-scale data (e.g., from global circulation models) are often abundant, they lack the local detail provided by fine-scale data (e.g., from regional climate models), which are typically computationally expensive to generate. Statistical downscaling and multi-scale data fusion address this challenge by predicting high-resolution fields from low-resolution or related inputs. We propose a highly scalable Bayesian approach that can learn the joint non-Gaussian distribution and nonlinear dependence structure of nonstationary spatial fields across multiple scales from a small number of training samples. Our method employs scale-aware autoregressive Gaussian processes with suitably chosen regularization-inducing priors to model the conditional distribution of fine-scale fields given coarse-scale data. Exploiting conjugacy, the integrated likelihood is available in closed form, enabling efficient parameter optimization via stochastic gradient descent. Once trained, the method provides a closed-form characterization of the posterior distribution of fine-scale fields given coarse-scale inputs. In numerical comparisons, we demonstrate that our approach substantially outperforms existing methods and effectively characterizes and simulates fine-scale climate behavior based on output from coarse global circulation models.

翻译：地球与环境科学中的空间场通常以多种尺度或分辨率呈现。尽管粗尺度数据（如来自全球环流模型的数据）通常较为丰富，但其缺乏细尺度数据（如来自区域气候模型的数据）所提供的局部细节，而后者通常因计算成本高昂而难以大量获取。统计降尺度与多尺度数据融合通过从低分辨率输入或相关输入预测高分辨率场，旨在应对这一挑战。我们提出一种高度可扩展的贝叶斯方法，该方法能够仅凭少量训练样本，学习跨多个尺度的非平稳空间场的联合非高斯分布及其非线性依赖结构。我们的方法采用具有恰当选取的正则化诱导先验的尺度感知自回归高斯过程，以在给定粗尺度数据条件下对细尺度场的条件分布进行建模。利用共轭性，积分似然函数具有闭合形式，从而能够通过随机梯度下降实现高效的参数优化。训练完成后，该方法可提供给定粗尺度输入条件下细尺度场后验分布的闭合形式表征。在数值对比中，我们证明了该方法显著优于现有方法，并能基于粗尺度全球环流模型的输出有效表征与模拟细尺度气候行为。