Function Space Optimization: A Symbolic Regression Method for Estimating Parameter Transfer Functions for Hydrological Models

Estimating parameters for distributed hydrological models is a challenging and long studied task. Parameter transfer functions, which define model parameters as functions of geophysical properties of a catchment, might improve the calibration procedure, increase process realism, and can enable prediction in ungauged areas. We present the function space optimization (FSO), a symbolic regression method for estimating parameter transfer functions for distributed hydrological models. FSO is based on the idea of transferring the search for mathematical expressions into a continuous vector space that can be used for optimization. This is accomplished by using a text generating neural network with a variational autoencoder architecture that can learn to compress the information of mathematical functions. To evaluate the performance of FSO, we conducted a case study using a parsimonious hydrological model and synthetic discharge data. The case study consisted of two FSO applications: single-criteria FSO, where only discharge was used for optimization, and multicriteria FSO, where additional spatiotemporal observations of model states were used for transfer function estimation. The results show that FSO is able to estimate transfer functions correctly or approximate them sufficiently. We observed a reduced fit of the parameter density functions resulting from the inferred transfer functions for less sensitive model parameters. For those it was sufficient to estimate functions resulting in parameter distributions with approximately the same mean parameter values as the real transfer functions. The results of the multicriteria FSO showed that using multiple spatiotemporal observations for optimization increased the quality of estimation considerably.