Spatiotemporal processes, such as floods, rainfall-runoff, and land-use changes, continuously evolve over space and time with high dynamism and complex nonlinearity. Accurate and efficient spatiotemporal process prediction is crucial for understanding their underlying patterns. Recently, deep learning has effectively addressed spatiotemporal prediction issues in Earth science. However, most existing studies address either short-term or long-term dependencies, but ignore the multiscale characteristics and spatial heterogeneity inherent to spatiotemporal processes and critical for practical applicability. This study develops a Spatiotemporal Adaptive Multiscale Transformer (SAMT) model for spatiotemporal process prediction. First, we design an enhanced multiscale spatial heterogeneity module to extract multiscale spatial heterogeneity. Then, we introduce the adaptive scale selection that assigns weights to features at different scales based on their contributions. In addition, we incorporate a spatiotemporal transformer block to simultaneously capture short-term and long-term dependencies. We conduct extensive experiments on three representative spatiotemporal datasets of rainfall, temperature, and flood. Compared to state-of-the-art models, the SAMT model achieves significant improvements across all evaluation metrics. The developed SAMT model critically improves the performance of spatiotemporal process prediction for more accurate and effective modelling of spatiotemporal evolution patterns in the field of Earth sciences.

Landmarks serve as critical reference points for determining spatial orientations. Owing to the complexity and diversity of the shapes of landmark buildings, numerous fine visual details can hinder the clear identification of three-dimensional (3D) landmark models, posing a challenge for their automatic generation. To address this issue, we propose a method based on topological perception to simplify 3D landmark models, focusing on enhancing global perception features by exaggerating topology-related features. This method involves three key steps: voxelization, hole exaggeration and model generation. We evaluated the effectiveness of exaggeration and conducted a quantitative analysis of its degree of application in landmark buildings. The results demonstrate that topology-based exaggeration significantly improves the perception of 3D landmark models, and the degree of exaggeration is inversely correlated with the proportion of topology-related visual features in the models. Furthermore, a comparative analysis of four commonly used simplification algorithms shows that our method outperforms the other methods across five key evaluation metrics.

Model validation ensures that mathematical models accurately represent real-world phenomena and meet scientific requirements in the Earth sciences. Current validation approaches for spatial modelling mainly use accuracy metrics such as goodness of fit and prediction error to validate regression, machine learning, spatial models and geospatial intelligence models. However, existing spatial validation still faces challenges in assessing models’ capacity for spatial interpretability. This study develops a degree of spatial interpretability (DSI) indicator to evaluate the effectiveness and ability of spatial models to capture spatial characteristics such as spatial autocorrelation and heterogeneity and to support model selection. DSI is implemented to evaluate ten common models for spatial prediction of Australian vascular plant species diversity using nineteen abiotic explanatory variables. The results reveal that strong goodness of fit and low prediction error do not necessarily correspond to high spatial interpretability and demonstrate the need to employ DSI in model evaluation. The experiments confirm the importance of assessing a model’s ability to explain spatial characteristics. The DSI indicator enhances model evaluation by providing a spatially informed assessment perspective and has strong potential to advance model evaluation systems.

Understanding the relationships between geographical variables is a fundamental task in spatial analysis. However, existing spatial methods often underperform in scenarios involving nonlinear relationships and complex interactions among geographical variables. Identifying the relationships between individual variables (i.e. univariate effect) within multiple interacting variables remains challenging and long-lasting. In this study, we propose a novel model—Geographical Pattern Interaction (GPI)—based on the premise that the spatial pattern of a response variable emerges from the interaction of spatial patterns in explanatory variables. GPI leverages decision trees and Shapley value explanations to quantify both global and local univariate effects by measuring the alignment between the spatial distribution of the target variable and those of the predictors. Through simulation experiments, we demonstrate GPI’s superior performance compared to traditional regression-based spatial explanation methods. Notably, the GPI framework is stable across varying spatial scales and sample sizes, making it particularly suitable for spatial explanation tasks under small data and multi-scale conditions. A case study on homelessness risk in Australia demonstrates GPI’s ability to reveal nonlinear spatial associations and interaction effects. By capturing overlooked pattern similarities and interactions, GPI offers an interpretable and transferable tool for analyzing complex spatial relationships.

Spatial prediction aims to accurately estimate attributes at unsampled locations based on spatial dependencies, patterns, variability, and covariates, providing knowledge of complex spatial systems and supporting diverse applications. However, existing methods for spatial prediction ignore geographic and environmental characteristics outside sample locations, particularly spatial outliers, significantly impacting prediction accuracy. This study introduces the concept of second-dimension outliers (SDO) and SDO models that incorporate local outlier information at unsampled locations to enhance prediction accuracy. SDO models generate SDO variables that capture samples’ external geographic and environmental characteristics in the spatial prediction. This study develops SDO-based machine learning to predict wheat production in Australia, using cross-validation to evaluate prediction accuracy. Results demonstrate that SDO-based support vector machines (SVM) improve spatial prediction accuracy, with the R2 increasing from 0.555 to 0.671 compared to aspatial SVM, particularly for extreme values. The developed local outlier strength index that examines the strength of SDO ensures more accurate and smooth spatial predictions. The SDO concept provides more in-depth explanatory information from an innovative spatial perspective and a detailed understanding of local outliers for spatial prediction, making it a robust and effective tool for spatial statistical inference and geographic computation across various fields.

Spatial stratified heterogeneity measures spatial association by power of determinant (PD) that compares variations within strata and across space. Models based on stratified heterogeneity have been extensively used across various fields to analyze PD from a spatial perspective. However, stratified heterogeneity in the local regions has not been investigated, although it can significantly influence the overall PD measurements in large-scale studies. This study proposes a local indicator of stratified power (LISP) to analyze local spatial stratified association and demonstrate how spatial stratified association changes spatially and in local regions. The LISP model was implemented to examine the local potential determinants of the thickness variations in lake-terminating glaciers of the Greater Himalayas. The results indicate that LISP can reveal spatial association at various local positions, effectively mitigating the underestimation or overestimation of PD values in local regions that are probably missed in global spatial stratified association models. The developed LISP model provides a reliable methodological framework for exploring local spatial associations and identifying local determinants in broad fields.

Spatial autoregressive (SAR) models are often used to explicitly account for the spatial dependence underlying geographic phenomena. However, traditional SAR models are specified using a single SAR coefficient, assuming constant spatial dependence over space. This assumption oversimplifies the situation where the true spatial autoregressive process varies in strength; the consequences of ignoring heterogeneous autocorrelation remain to be discussed. This study proposes a heterogeneous spatial autocorrelation model by extending the spatial lag model (SLM). The new model includes change point detection for identifying patterns of spatially varying autocorrelation strengths, a SAR coefficient matrix for representing heterogeneous spatial autocorrelation, and maximum likelihood estimation for determining multiple SAR coefficients. Monte Carlo simulations demonstrate that the proposed method is effective in modeling SAR processes with heterogeneous autocorrelation patterns, while traditional SLM inflates uncertainties in the regression coefficients when a heterogeneous autocorrelation structure is not accounted for. We further applied the new method to an empirical analysis of traffic crashes in the Greater Perth Area, Australia. The heterogeneous spatial autocorrelation model reduces model RMSE by 42% (compared with traditional SLM). Results from both simulation and empirical studies indicate that spatially varying autocorrelation strengths should be considered for SAR processes and relevant applications.

Spatial heterogeneity refers to uneven distributions of geographical variables. Spatial interpolation methods that utilize spatial heterogeneity are sensitive to the way in which spatial heterogeneity is characterized. This study developed a Generalized Heterogeneity Model (GHM) for characterizing local and stratified heterogeneity within variables and to improve interpolation accuracy. GHM first divides a study area into multiple spatial strata according to the sample values and locations of a variable. Then, GHM estimates simultaneously the spatial variations of the variable within and between the spatial strata. Finally, GHM interpolates unbiased estimates and uncertainty at unsampled locations. We demonstrated the GHM by predicting the spatial distributions of marine chlorophyll in Townsville, Queensland, Australia. Results show that GHM improved both the overall interpolation accuracy across the study area and along strata boundaries compared with previous interpolation models. GHM also avoided bull’s eye patterns and abrupt changes along strata boundaries. In future studies, GHM has the potential to be integrated with machine learning and advanced algorithms to improve spatial prediction accuracy for studies in broader fields.

The explanation of spatial errors in geospatial modelling has long been a challenge. This study introduces an index that captures the complexity of local spatial distribution, which can partially provide insight into spatial errors. While previous studies have explored the complexity of geographical data from various perspectives, there is limited knowledge on assessing the complexity while taking spatial dependence into account. This study proposes a measure of geocomplexity, i.e. the spatial local complexity indicator, which characterizes the complexity of local spatial patterns while considering spatial neighbor dependence. We used both aspatial and spatial models to estimate the economic inequality in Australia, and applied the spatial local complexity indicator to explain spatial errors in these models. Results show that the developed geocomplexity indicator, using a binary spatial matrix, can effectively explain spatial errors arising from models, including 17%-47% of errors in aspatial models and 14% in a spatial model. The experiments in this study support our hypothesis that geocomplexity is an essential component in explaining spatial errors. The proposed geocomplexity indicator, along with our hypothesis, has the potential for advancing the understanding complex geospatial systems and enabling applications in various fields related to spatial data analysis.

Geographical variables are usually not independent of each other. Hence, it is necessary to investigate the effect of interactions among explanatory variables on a response variable to characterize spatially enhanced or weakened relationships among all variables. The geographical detector (GD) model identifies zones for each explanatory variable, divides the study area into spatial units by overlapping these zones, and quantifies spatial associations as the power of interactive determinant (PID) between a response variable and explanatory variables. Consequently, the PID values depend upon the distributions of explanatory variables (i.e. spatial characteristics) and the subsequent division of spatial units out of these explanatory variables. This study has therefore proposed an Interactive Detector for Spatial Associations (IDSA) to optimize spatial division and improve PID. IDSA utilizes spatial autocorrelation of each explanatory variable and optimizes spatial units based on spatial fuzzy overlay to compute PID. We test the IDSA on both a simulation study and practical case that analyzes road deterioration in Australia. Results showed that the IDSA model could effectively assess the PID while existing GD overestimated PID. Hence, the IDSA improves the GD with refined spatial units based on explanatory variables to enhance their local spatial associations with a response variable.

Urban forms reflect spatial structures of cities, which have been consciously and dramatically changing in China. Fast urbanisation may lead to similar urban forms due to similar habits and strategies of city planning. However, whether urban forms in China are identical or significantly different has not been empirically investigated. In this paper, urban forms are investigated based on two spatial units: city and block. The boundaries of natural cities in terms of the density of human settlements and activities are delineated with the concept of ‘redefined city’ using points of interests (POIs), and blocks are determined by road networks. Urban forms are characterised by city-block two-level spatial morphologies. Further, redefined cities are classified into four hierarchies to examine the effects of different city development stages on urban forms. The spatial morphology is explained by urbanisation variables to understand the effects. Results show that the urban forms are spatially clustered from the perspective of city-block two-level morphologies. Urban forms tend to be similar within the same hierarchies, but significantly varied among different hierarchies, which is closely related to the development stages. Additionally, the spatial dimensional indicators of urbanisation could explain 41% of the spatial morphology of redefined cities.