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\begin{document}

\title{Function Space Optimization: A symbolic regression method for estimating
parameter transfer functions for hydrological models}



\author[1]{Moritz Feigl}%
\author[1]{Mathew Herrnegger}%
\author[2]{Daniel Klotz}%
\author[1]{Karsten Schulz}%
\affil[1]{University of Natural Resources and Life Sciences, Vienna}%
\affil[2]{Johannes Kepler University Linz}%


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  \date{May 26, 2024}


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\begin{abstract}
Estimating parameters for distributed hydrological models is a
challenging and long studied task. Parameter transfer functions, which
define model parameters as functions of geo-physical 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 multi-criteria 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 multi-criteria
FSO showed that using multiple spatiotemporal observations for
optimization increased the quality of estimation considerably.%
\end{abstract}%



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\textbf{Hosted file}

\verb`supporting-information_wrr_feigl_etal.docx`  available at \url{https://authorea.com/users/435642/articles/604750-function-space-optimization-a-symbolic-regression-method-for-estimating-parameter-transfer-functions-for-hydrological-models}

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