无线电干涉仪测量方程的再探讨。IV。一种广义张量形式
Revisiting the radio interferometer measurement equation. IV. A generalized tensor formalism
摘要：
射电干涉仪测量方程（RIME），特别是它的2x2形式，提供了一个全面的基于矩阵的形式主义来描述经典的射电干涉测量和偏振测量，如本系列的前三篇论文所示。然而，最近的实践和理论的发展，如相控阵饲料（PAF），孔径阵列（AA）和宽场偏振，暴露出的形式主义的局限性。本文的目的是开发一个更一般的形式主义，可以用来明确定义矩阵RIME的限制，并描述这些限制之外的观测场景。对矩阵RIME的一些基本假设进行了详细的解释和分析。为此，阵列相关矩阵（ACM）的形式主义进行了探讨。这证明了有限的使用，它表明，矩阵代数根本不是一个足够灵活的工具的工作。为了克服这些局限性，一个更一般的形式主义张量和爱因斯坦符号的基础上提出和探索理论上，并着眼于实际的实现。张量形式主义优雅地产生广义RIME描述波束形成，互耦合，和宽场偏振在一个方程。结果表明，在明确的假设下，张量方程约化为2 × 2 RIME。从实用的角度出发，提出并分析了张量方程的优化实现方法。张量RIME是描述不服从矩阵RIME的观测场景的有力手段。即使在后者仍然适用的情况下，张量形式主义也可以成为理解这种适用性限制的有价值的工具。
英文摘要：
The radio interferometer measurement equation (RIME), especially in its 2x2 form, has provided a comprehensive matrix-based formalism for describing classical radio interferometry and polarimetry, as shown in the previous three papers of this series. However, recent practical and theoretical developments, such as phased array feeds (PAFs), aperture arrays (AAs) and wide-field polarimetry, are exposing limitations of the formalism. This paper aims to develop a more general formalism that can be used to both clearly define the limitations of the matrix RIME, and to describe observational scenarios that lie outside these limitations. Some assumptions underlying the matrix RIME are explicated and analysed in detail. To this purpose, an array correlation matrix (ACM) formalism is explored. This proves of limited use; it is shown that matrix algebra is simply not a sufficiently flexible tool for the job. To overcome these limitations, a more general formalism based on tensors and the Einstein notation is proposed and explored both theoretically, and with a view to practical implementations. The tensor formalism elegantly yields generalized RIMEs describing beamforming, mutual coupling, and wide-field polarimetry in one equation. It is shown that under the explicated assumptions, tensor equations reduce to the 2x2 RIME. From a practical point of view, some methods for implementing tensor equations in an optimal way are proposed and analysed. The tensor RIME is a powerful means of describing observational scenarios not amenable to the matrix RIME. Even in cases where the latter remains applicable, the tensor formalism can be a valuable tool for understanding the limits of such applicability.
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