It is no secret that statistical modelling often involves making simplifying assumptions when attempting to study complex stochastic phenomena. Spatial modelling of extreme values is no exception, with one of the most common such assumptions being stationarity in the marginal and/or dependence features. If non-stationarity has been detected in the marginal distributions, it is tempting to try to model this while assuming stationarity in the dependence, without necessarily putting this latter assumption through thorough testing. However, margins and dependence are often intricately connected and the detection of non-stationarity in one feature might affect the detection of non-stationarity in the other. This work is an in-depth case study of this interrelationship, with a particular focus on a spatio-temporal environmental application exhibiting well-documented marginal non-stationarity. Specifically, we compare and contrast four different marginal detrending approaches in terms of our post-detrending ability to detect temporal non-stationarity in the spatial extremal dependence structure of a sea surface temperature dataset from the Red Sea.

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