Modelling of inspection data for large scale physical systems is critical to assessment of their integrity. We present a general method for inference about system state and associated model variance structure from spatially distributed time series that are typically short, irregular, incomplete and not directly observable. Bayes linear analysis simplifies parameter estimation and avoids often-unrealistic distributional assumptions. Second-order exchangeability judgements facilitate variance learning for sparse inspection time-series. The model is applied to inspection data for minimum wall thickness from corroding pipe-work networks on a full-scale offshore platform, and shown to give materially different forecasts of remnant life compared with an equivalent model neglecting variance learning. © 2013 IMechE.