LatticeVision: Image to Image Networks for Modeling Non-Stationary Spatial Data

TL;DR

LatticeVision tackles the challenge of estimating non-stationary spatial processes by treating both the spatial fields and their SAR parameter fields as images and applying image-to-image networks to predict all parameter channels in a single forward pass. By training on synthetic data that encode geophysically relevant priors and coupling the global estimators with the LatticeKrig SAR emulator, the method achieves accurate, robust parameter inference even with few replicates and enables rapid generation of thousands of synthetic fields. The approach addresses non-stationarity and long-range dependencies that hinder local methods, delivering substantial gains in accuracy and computational efficiency, with practical impact for climate ensemble emulation and uncertainty quantification. The combination of global I2I estimators and SPDE-based SAR sampling yields fast, scalable ensembles while maintaining physically interpretable parameter fields $\kappa^2(\mathbf{s})$, $\rho(\mathbf{s})$, and $\theta(\mathbf{s})$, and demonstrates superior performance over local CNN baselines on simulated and climate model data.

Abstract

In many scientific and industrial applications, we are given a handful of instances (a 'small ensemble') of a spatially distributed quantity (a 'field') but would like to acquire many more. For example, a large ensemble of global temperature sensitivity fields from a climate model can help farmers, insurers, and governments plan appropriately. When acquiring more data is prohibitively expensive -- as is the case with climate models -- statistical emulation offers an efficient alternative for simulating synthetic yet realistic fields. However, parameter inference using maximum likelihood estimation (MLE) is computationally prohibitive, especially for large, non-stationary fields. Thus, many recent works train neural networks to estimate parameters given spatial fields as input, sidestepping MLE completely. In this work we focus on a popular class of parametric, spatially autoregressive (SAR) models. We make a simple yet impactful observation; because the SAR parameters can be arranged on a regular grid, both inputs (spatial fields) and outputs (model parameters) can be viewed as images. Using this insight, we demonstrate that image-to-image (I2I) networks enable faster and more accurate parameter estimation for a class of non-stationary SAR models with unprecedented complexity.

Figures (8)

• Figure 1: An illustration of the main workflow of LatticeVision. Spatial fields are fed into an I2I network, which in turn produces estimates of the non-stationary parameter fields. These are encoded into a SAR model from which synthetic replicates are efficiently simulated.
• Figure 2: Illustration of the effects of $\kappa^2, \rho$, and $\theta$. The ellipses represent contours of constant correlation, e.g. all locations with correlation $0.5$ with the origin. $\kappa^2$ controls the radii of the ellipse, $\rho$ controls the ratio of the semi-major and semi-minor radii (i.e., the 'aspect ratio' of the ellipse), and $\theta$ is the angle the semi-major ellipse makes with the positive $x$-axis.
• Figure 3: Spatial patterns (left) and their frequency (right).
• Figure 4: True parameters $\Phi$ (left) are encoded into the LatticeKrig SAR model to simulate a testing sample $Y$, of which one replicate $\mathbf{y}^{(0)}$ is displayed. $Y$ is used as an input to STUN and a sliding window, local estimation strategy using CNN25, resulting in $\hat{\Phi}_{\text{STUN}}$ (middle), and $\hat{\Phi}_{\text{CNN25}}$ (right).
• Figure 5: Top: Standardized temperature fields drawn from the CESM1 LENS ensemble (left), the STUN-based emulator (middle), and the CNN25-based emulator (right). Bottom: Correlations with a chosen location in the Niño 3.4 region at ($212^{\circ}$E, $1^{\circ}$N) for the same three ensembles. The STUN-based emulator better preserves spatial relationships, including the zonal correlation structure along the equator and the meridional oceanic correlation range.