This article shows how information about the riverine load of pollutants at different sites in a study area can be combined with data on the spatial distribution of sources and selected river basin characteristics to carry out a statistical source apportionment of the load observed at the mouth of a river or the outlet of an arbitrary sub-basin. The suggested approach is based on the following: (i) a partitioning of the study area into disjoint subbasins partially ordered by their upstream-downstream relations; (ii) matrix formulas for the transport and retention of an arbitrary substance in such a system of sub-basins; (iii) matrix formulas for the source apportionment of the output from an arbitrary sub-basin; (iv) general principles for the parameterization of losses of substances from soil and retention of substances in lakes and watercourses. As compared to conventional source inventories, our approach offers several advantages: data from sub-basins with arbitrary upstream-downstream relationships can be combined in the same study; retention and sourcestrength parameters can be estimated simultaneously; matrix forms of all models and source apportionment formulas enable parameter estimation by the use of standard methods and software for non-linear regression analysis. Some of the formulas derived are based on the theory of Markov chains. To illustrate the different steps of the procedure, the time-averaged output of nitrogen from a Swedish river basin during a five year period was apportioned to various diffuse and point sources.