A non-stationary Gaussian random field model is developed based on a combination of the stochastic partial differential equation (SPDE) approach and the classical deformation method. With the deformation method, a stationary field is defined on a domain which is deformed so that the field becomes non-stationary. We show that if the stationary field is a Matérn field defined as a solution to a fractional SPDE, the resulting non-stationary model can be represented as the solution to another fractional SPDE on the deformed domain. By defining the model in this way, the computational advantages of the SPDE approach can be combined with the deformation method’s more intuitive parameterisation of non-stationarity. In particular it allows for independent control over the non-stationary practical correlation range and the variance, which has not been possible with previously proposed non-stationary SPDE models.

The model is tested on spatial data of significant wave height, a characteristic of ocean surface conditions which is important when estimating the wear and risks associated with a planned journey of a ship. The model parameters are estimated to data from the north Atlantic using a maximum likelihood approach. The fitted model is used to compute wave height exceedance probabilities and the distribution of accumulated fatigue damage for ships traveling a popular shipping route. The model results agree well with the data, indicating that the model could be used for route optimization in naval logistics.

基于随机偏微分方程（SPDE）方法和经典变形方法的组合，开发了一个非平稳高斯随机场模型。使用变形方法，在变形的区域上定义了一个固定的场，以使该场变得不平稳。我们证明，如果固定场是定义为分数SPDE的解的Matérn场，则所得的非平稳模型可以表示为变形域上另一个分数SPDE的解。通过以这种方式定义模型，可以将SPDE方法的计算优势与变形方法的非平稳性更直观的参数化结合起来。特别是，它允许对非平稳的实际相关范围和方差进行独立控制，

该模型在显着波高的空间数据上进行了测试，这是海洋表面状况的特征，当估算船舶的计划航行所伴随的磨损和风险时，这一点很重要。使用最大似然法对来自北大西洋的数据估计模型参数。拟合模型用于计算波高超出概率以及在流行航线上行驶的船舶的累积疲劳损伤分布。模型结果与数据吻合良好，表明该模型可用于海军物流中的路线优化。