Simulated annealing coupled with a Naïve Bayes model and base flow separation for streamflow simulation in a snow dominated basin

Abstract

Streamflow simulation in a snow dominated basin is complex due to the presence of a high number of interrelated hydrological processes. This complexity is affected by the delayed responses of the catchment to snow accumulation and snow melting processes. In this study, long short-term memory (LSTM) and artificial neural network (ANN) models were utilized for rainfall–runoff simulation in a snow dominated basin, the Carson River basin in the United States (US). The input structure of the models was determined using the simulated annealing algorithm with a naïve Bayes model from a high dimensional feature space to represent the long-term impacts of historical events (i.e. the hysteresis effect) on current observations. Further, to represent the different responses of the catchment in the model structure, a base flow separation method was included in the simulation framework. The obtained performance indices, root mean square error, percentage bias, Nash–Sutcliffe and Kling–Gupta efficiencies are 0.331 m³ s⁻¹, 13.00%, 0.848, and 0.852 for the ANN model and 0.235 m³ s⁻¹, − 0.80%, 0.923, and 0.934 for the LSTM model, respectively. The proposed methodology was found to be promising for improving the streamflow simulation capability of LSTM and ANN models by only considering precipitation, temperature, and potential evapotranspiration as input variables. Analysing the flow duration curves indicated that the LSTM model is more efficient in representing different flow dynamics within the basin due to embedded cell states. Further, the uncertainty and reliability analyses were conducted by using expanded uncertainty ($U_{95}$), reliability, and resilience indices. The obtained $U_{95}$, reliability and resilience indices are 1.78–1.72 m³ s⁻¹, 31.28–66.67% and 11.58–38.27% for the ANN and LSTM models, respectively, showed that the LSTM model produced less uncertainty and is more reliable. However, while lacking a memory component, the proposed methodology significantly contributes to the simulation capability of the ANN model in rainfall–runoff modelling. The results of this study indicated that the proposed methodology could enhance the learning capabilities of machine learning models in rainfall–runoff simulation.

Appendices

Appendix 1

The simulated annealing algorithm starts with a randomly selected subset of features in addition to the selected number of iterations. Then, a base learner model is built, and its predictive performance or initial error cost (${\text{Cost}}_{1}$) is evaluated.

The current feature set is slightly updated (e.g. 1–5%) by randomly including or excluding the features and associated predictive error is calculated (${\text{Cost}}_{2}$). If the performance of the new model is better than the previous model, the new feature set is accepted. However, if the performance of the new model is worse than the previous model, the new feature set is still not discarded. The acceptance probability of ith iteration, $p_{i}^{accept}$, is calculated. If this probability is larger than the randomly generated number from a uniform distribution between zero and one, the new feature set is accepted. Otherwise, the new feature set is rejected. This randomness enables the algorithm to continue to search in solution space to avoid local minima (Aarts et al. 2005).

Appendix 2

Expanded uncertainty can be defined as:

$$ U_{95} = 1.96\left( {SD^{2} + RMSE^{2} } \right)^{0.5} $$

(10)

where SD indicates the standard deviation of the observed values and RMSE shows the root mean square error between the observed and simulated values.

Reliability represents the overall consistency of a model and can be expressed as:

$$ {\text{Rel = }}\left( \frac{100}{N} \right)\sum\limits_{i = 1}^{N} {k_{i} } $$

(11)

where $N$ indicates the length of the data series and $k$ shows whether the simulated value is within the allowed threshold value of a relative absolute error. $k_{i}$ is equal to one if the calculated relative absolute error (RAE) is lower than the threshold (in this study, it is fixed as 0.20 as done in Saberi-Movahed et al. (2020)) otherwise it is equal to zero. The relative absolute error can be calculated as:

$$ RAE_{i} = \left| {\frac{{Q_{obs} - Q_{sim} }}{{Q_{obs} }}} \right| $$

(12)

Resilience indicates the ability of a model to return to its acceptable simulation performance whenever its simulations are unacceptable. It can be defined as:

$$ {\text{Res }}=\left( {\frac{{\sum\nolimits_{i = 1}^{N - 1} {r_{i} } }}{{N - \sum\nolimits_{i = 1}^{N} {k_{i} } }}} \right) \times 100 $$

(13)

where $r_{i}$ is the number of times that it is possible for the model to recover from an unacceptable estimate to an acceptable one in the ith data value. Here, if a simulation value’s relative absolute error is higher than 0.20, it is assumed as unacceptable.