Data Cubes and Matrix Formulae for Convenient Handling of Physical Flow Data

If the technosphere and the biosphere are divided into cells, the presence and turnover of a substance in a study area can be summarized in a vector of stocks and a matrix of flows between different pairs of cells. Likewise the stocks and flows of several substances or materials in one or more time periods can be summarized in multidimensional data cubes. In this article, we provide a theoretical framework for handling physical flow data, and we demonstrate how a set of matrix operations can facilitate exploratory analysis and quality assessment of such data regardless of the number of substances, materials, and time periods considered. In particular, we show how matrices and cubes of flow data can be recalculated when the collection of cells is modified by joining cells, and also what information is required to recalculate flows when cells are split. Furthermore, we illustrate how and under what circumstances substance‐flow data originating from different studies with different collections of cells can be compared or merged. The generic character of the given formulae facilitates the development of software for physical flow data.