  Robotic Information Gathering (RIG) is a foundational research topic that
answers how a robot (team) collects informative data to efficiently build an
accurate model of an unknown target function under robot embodiment
constraints. RIG has many applications, including but not limited to autonomous
exploration and mapping, 3D reconstruction or inspection, search and rescue,
and environmental monitoring. A RIG system relies on a probabilistic model's
prediction uncertainty to identify critical areas for informative data
collection. Gaussian Processes (GPs) with stationary kernels have been widely
adopted for spatial modeling. However, real-world spatial data is typically
non-stationary -- different locations do not have the same degree of
variability. As a result, the prediction uncertainty does not accurately reveal
prediction error, limiting the success of RIG algorithms. We propose a family
of non-stationary kernels named Attentive Kernel (AK), which is simple, robust,
and can extend any existing kernel to a non-stationary one. We evaluate the new
kernel in elevation mapping tasks, where AK provides better accuracy and
uncertainty quantification over the commonly used stationary kernels and the
leading non-stationary kernels. The improved uncertainty quantification guides
the downstream informative planner to collect more valuable data around the
high-error area, further increasing prediction accuracy. A field experiment
demonstrates that the proposed method can guide an Autonomous Surface Vehicle
(ASV) to prioritize data collection in locations with significant spatial
variations, enabling the model to characterize salient environmental features.
