Spatial Multi-Sensor Fusion with Heterogeneous Error Characteristics

Fusing spatial observations from sensors with heterogeneous error characteristics is a persistent challenge in geostatistics. Classical kriging assumes a Gaussian likelihood for all observations, an assumption that fails when sensors exhibit one-sided or asymmetric noise. We present a Variable Rank Kriging (VRK) formulation that supports per-observation heterogeneous likelihoods where each observation may define its own likelihood function, thus enabling principled fusion of sensors whose noise structures are significantly different in terms of distribution family and magnitude. Within this framework, we use the exponential (one-sided) likelihood as a case study to demonstrate the method and compare it with sampling-based numerical alternatives for general likelihoods without closed forms. A non-collocated RTK calibration workflow uses kriging predictions from a sparse high-accuracy reference to characterise sensor-specific likelihood parameters without requiring co-located paired observations. Synthetic 1-D and 2-D experiments show that correct per-point likelihood specification reduces RMSE by up to 92% (1-D) and 57% (2-D) relative to a misspecified Gaussian model while also eliminating systematic positive bias. A demonstration using NEON Airborne Observation Platform lidar data at Harvard Forest confirms these findings in a practical, real-world scenario. Across multiple subsamples of the lidar dataset, the exponential likelihood reduces vegetated-zone RMSE by 20.6% (open zone: 18.6%) and mean absolute bias by 26.5% relative to a heteroscedastic Gaussian baseline. The open-source vrk Python (>=3.10) package provides a reproducible implementation that can be applied to any spatial domain that requires multi-sensor spatial fusion with heterogeneous error structures.