Large agricultural field trials may display irregular spatial trends that cannot be fully captured by a purely randomization-based analysis. For this reason, paralleling the development of analysis-of-variance procedures for randomized field trials, there is a long history of spatial modeling for field trials, starting with the early work of Papadakis on nearest neighbor analysis, which can be cast in terms of first or second differences among neighboring plot values. This kind of spatial modeling is amenable to a natural extension using splines, as has been demonstrated in recent publications in the field. Here, we consider the P-spline framework, focusing on model options that are easy to implement in linear mixed model packages. Two examples serve to illustrate and evaluate the methods. A key conclusion is that first differences are rather competitive with second differences. A further key observation is that second differences require special attention regarding the representation of the null space of the smooth terms for spatial interaction, and that an unstructured variance–covariance structure is required to ensure invariance to translation and rotation of eigenvectors associated with that null space. We develop a strategy that permits fitting this model with ease, but the approach is more demanding than that needed for fitting models using first differences. Hence, even though in other areas, second differences are very commonly used in the application of P-splines, our conclusion is that with field trials, first differences have advantages for routine use.

Environmental systems are complex, with many interacting variables (biotic and abiotic) across diverse spatial and temporal scales. Historically, modellers have used scientific knowledge of how these systems function to build tractable modelling frameworks. The resultant models are used today in multiple operational settings including natural resource management, early warning systems for hazards such as floods, or ecosystem restoration. This approach to model building is often favoured because it suggests a level of reliability when models are extrapolated to new conditions or scenarios that haven't been observed previously. Despite this, there is an explosion of interest recently in modelling approaches that capitalise on recent advances in machine learning (ML), Artificial Intelligence (AI) and related technologies. These approaches have shown promise in initial applications to environmental problems, but scepticism remains about their usefulness given their disconnect from the knowledgebase of environmental systems. This leads to a fundamental question: How can we effectively combine these innovative approaches with the established understanding of environmental processes? We demonstrate here the integration of traditional process-based environmental models with emerging modelling technologies, aiming to bolster the predictive power and trustworthiness of environmental models. We demonstrate how new hybrid approaches to ML offer a pathway to refining our understanding of natural processes, enhancing predictions, and addressing pressing environmental challenges with improved accuracy.