Geostatistical models for multivariate applications such as heavy metal soil contamination work under Gaussian assumptions and may result in underestimated extreme values and misleading risk assessments (Marchant et al, 2011). A more suitable framework to analyse extreme values is extreme value theory (EVT). However, EVT relies on replications in time, which are generally not available in geochemical datasets. Therefore, using EVT to map soil contamination requires adaptation to be used in the usual single-replicate data framework of soil surveys. We propose a bivariate spatial extreme mixture model to model the body and tail of contaminant pairs, where the tails are described using a stationary generalised Pareto distribution. We demonstrate the performance of our model using a simulation study and through modelling bivariate soil contamination in the Glasgow conurbation. Model results are given as maps of predicted marginal concentrations and probabilities of joint exceedance of soil guideline values. Marginal concentration maps show areas of elevated lead levels along the Clyde River and elevated levels of chromium around the south and southeast villages such as East Kilbride and Wishaw. The joint probability maps show higher probabilities of joint exceedance to the south and southeast of the city centre, following known legacy contamination regions in the Clyde River basin.

翻译：针对重金属土壤污染等多变量应用的地统计模型通常基于高斯假设，这可能导致低估极端值并产生误导性风险评估（Marchant等，2011）。更适用于分析极端值的框架是极值理论（EVT）。然而，EVT依赖于时间序列中的重复观测数据，而地球化学数据集通常不具备这一条件。因此，将EVT用于土壤污染制图时，需对其调整以适应土壤调查中常见的单重复数据框架。我们提出一种双变量空间极端混合模型，用于描述污染物对的分布主体与尾部特征，其中尾部采用平稳广义帕累托分布进行建模。通过模拟研究及对格拉斯哥城市群双变量土壤污染的分析，验证了模型性能。模型结果以预测边际浓度图和土壤指导值联合超标概率图的形式呈现。边际浓度图显示克莱德河沿岸铅浓度较高，而南郊与东南部村镇（如东基尔布赖德和维肖）铬浓度较高。联合概率图显示，市中心南部与东南部区域（沿克莱德河流域已知历史污染区）的联合超标概率更高。