Developing novel hybrid models for estimation of daily soil temperature at various depths

•Estimating daily soil temperature at different depths.•Applying the FARIMA, GEP, and FFBPNN models.•Combining the GEP and FFBPNN models with the FARIMA.•A better accuracy of the GEP and FFBPNN compared with the FARIMA.•The best performance of proposed hybrid models at different soil depths. Estimation of soil temperature (ST) as one of the vital parameters of soil, which has an impact on many chemical and physical characteristics of soil, is of great importance in soil science. This study applies a time series-based model, namely fractionally autoregressive integrated moving average (FARIMA), as well as two machine learning-based models consisting of feed-forward back propagation neural networks (FFBPNN) and gene expression programming (GEP) for daily ST estimation. In doing so, the daily ST data of three stations at four depths (5, 10, 50, and 100 cm) in Iran were used for the time period from 1998 to 2017. Studied stations were selected from different climates including arid (Isfahan station), semi-arid (Urmia station), and very humid (Rasht station) to evaluate the performance of models and generalize the outcomes in different climate classes. The performances of the developed models are evaluated via three statistical metrics including the root mean square error (RMSE), mean absolute error (MAE), and relative RMSE (RRMSE). Results obtained demonstrated that the machine learning-based FFBPNN and GEP models performed better than the time series-based FARIMA approach at all depths. As a result, negligible differences were observed between the accuracies of FFBPNN and GEP. In addition, this study developed novel hybrid models through combining the FFBPNN and GEP techniques with the FARIMA to enhance the accuracy of traditional FARIMA, FFBPNN, and GEP. The developed hybrid models named GEP-FARIMA and FFBPNN-FARIMA were found to achieve better estimates of daily ST data at different depths in comparison with the classical models. The daily ST estimates with the highest accuracy were observed at a depth of 50 cm via the GEP-FARIMA at Isfahan station (RMSE = 0.05 °C, MAE = 0.03 °C, RRMSE = 0.25% for the testing phase), the GEP-FARIMA at Urmia station (RMSE = 0.04 °C, MAE = 0.03 °C, RRMSE = 0.26% for the testing phase), and the FFBPNN-FARIMA at Rasht station (RMSE = 0.07 °C, MAE = 0.05 °C, RRMSE = 0.35% for the testing phase).