Interpolation of monthly runoff along rivers applying empirical orthogonal functions: Application to the Upper Magdalena River, Colombia

- We present an approach for interpolation of discharge along river networks.
- We use the theory of Karhunen-Loève expansion.
- We adapt this theory to apply to discharge data.
- We tested theoretical findings on time series from the Magdalena River, Colombia.
- The results are good and confirm the plausibility of the approach.

SummaryAn approach for interpolation of discharge along rivers is presented applying empirical orthogonal functions. From a theoretical point of view this approach for hydrological applications should be looked upon as a generalisation of a Karhunen-Loève expansion. The eigenvalue problem is then represented by a Fredholm integral equation of the second kind over the spatial domain, which in a hydrological application is defined by the principal drainage area. Most methods for numerical solution of this equation entail reduction to an approximately equivalent algebraic problem. The problem is to correctly account for drainage areas and drainage patterns, as the irregularity in sizes of drainage areas might violate the orthogonality if not accounted for correctly. A basic solution to the problem is developed and demonstrated on discharge observations from the Upper Magdalena drainage basin, Colombia. Seven discharge time series are used to determine the empirical orthogonal functions and principal components and four series are used for validation of the application for gap filling and estimation at ungauged sites. The results show a high accuracy for larger basins while those for mountain headwater stations are moderately good. For these latter stations the gain of a short period of observations compared to no observations at all is 5% increase in the coefficient of determination. The results confirm the plausibility of the theoretical approach.