Residual maximum likelihood (REML) methods for analysing hydrological data series

Much hydrological data can be displayed as two-way tables with observations classified (for example) by years (rows) and sites (columns), commonly with many missing entries; data classified by three factors or more (e.g. gauge sites within drainage basins; drainage basins; years) can also be put in this form. On an appropriate scale, the observations in such tables can frequently be represented by linear, additive models of components, some of which can be considered as random variables. Residual maximum likelihood (REML) is a technique for fitting models in which each observation is expressed additively in terms of fixed and random effects. When the model contains only one such random effect, the linear model reduces to a restricted form of multiple regression; REML can be regarded as an extension of multiple regression to the case where there are several error terms with different statistical characteristics. Models of this kind are appropriate in the hydrological context where the effects of the-years (or other periods) of observation can be regarded as a sample from a hypothetical population of years (periods), or where sites can be regarded as random. The paper discusses two examples where REML was used: one in estimating mean areal monthly rainfall in Amazonia, using incomplete records from 48 raingauge sites, and the other using incomplete records of annual floods from 19 gauging stations on the Rio Itajai-Acu, in southern Brazil. In both cases, the assumptions of the REML model were satisfied and the objectives of the analysis achieved. Given the prevalence of incomplete hydrological records, the REML method may well have wider application.