Stochastic partial differential equations have been used in a variety of contexts to model the evolution of uncertain dynamical systems. In recent years, their applications to geophysical fluid dynamics has increased massively. For a judicious usage in modelling fluid evolution, one needs to calibrate the amplitude of the noise to data. In this paper we address this requirement for the stochastic rotating shallow water (SRSW) model. This work is a continuation of [LvLCP23], where a data assimilation methodology has been introduced for the SRSW model. The noise used in [LvLCP23] was introduced as an arbitrary random phase shift in the Fourier space. This is not necessarily consistent with the uncertainty induced by a model reduction procedure. In this paper, we introduce a new method of noise calibration of the SRSW model which is compatible with the model reduction technique. The method is generic and can be applied to arbitrary stochastic parametrizations. It is also agnostic as to the source of data (real or synthetic). It is based on a principal component analysis technique to generate the eigenvectors and the eigenvalues of the covariance matrix of the stochastic parametrization. For SRSW model covered in this paper, we calibrate the noise by using the elevation variable of the model, as this is an observable easily obtainable in practical application, and use synthetic data as input for the calibration procedure.

翻译：随机偏微分方程已被广泛应用于多种情景中，用于模拟不确定动力系统的演化。近年来，其在地球物理流体动力学中的应用大幅增加。为了在流体演化的建模中合理使用这些方程，需要根据数据标定噪声的振幅。本文针对随机旋转浅水（SRSW）模型解决了这一需求。本工作是对文献[LvLCP23]的延续，其中提出了一种适用于SRSW模型的数据同化方法。在[LvLCP23]中使用的噪声被引入为傅里叶空间中的任意随机相位偏移，这未必与模型降阶过程所引起的不确定性一致。本文提出了一种新的SRSW模型噪声标定方法，该方法与模型降阶技术兼容。该方法是通用的，可应用于任意随机参数化方案，且与数据来源（真实或合成）无关。该方法基于主成分分析技术，用于生成随机参数化协方差矩阵的特征向量和特征值。对于本文涵盖的SRSW模型，我们利用模型的高程变量（该变量在实际应用中易于观测）来标定噪声，并使用合成数据作为标定过程的输入。