Hydrologic connectivity assessment among natural hydrosystem components using emerging entropy and evolutionary computing methods

Hydrosystems, also known as hydrological systems, encompass various natural components that interact to regulate the movement, distribution, and quality of water. The key natural components of hydrosystems include (but are not limited to) surface and groundwater bodies, precipitation, soil moisture, vegetation, temperature, etc. The groundwater level (GWL) is an essential aspect of hydrological systems and plays a vital role in water supply, agriculture, and ecosystems. It refers to the depth at which water is found beneath the Earth's surface in underground aquifers or water-bearing formations. The GWL can vary depending on several factors, including precipitation, evaporation, surface water interactions, geological conditions, and human activities such as groundwater pumping and recharging. When precipitation exceeds evaporation and water infiltrates into the ground, the GWL tends to rise. Conversely, it can decline during periods of low rainfall or excessive pumping.

Soft computing techniques (SCTs) have been increasingly used in the field of civil engineering, especially hydrology, to detect underlying patterns and accurately predict environmental variables (Creaco et al. 2022; Ghasemi et al. 2023; Yin et al. 2023). Regarding the changing trends of GWL, several studies have been conducted on regional and catchment scales (Roushangar et al. 2020; Nourani et al. 2023). Among different SCTs, artificial neural networks (ANNs) and support vector regression (SVR) have been frequently used as an efficient way to solve problems in the groundwater-related field, such as GWL prediction (Daliakopoulos et al. 2005; Sreekanth et al. 2009; Nourani et al. 2011; Gholami & Sahour 2022; Roshani & Hamidi 2022; Dąbrowska et al. 2023). For example, Nourani et al. (2016) used the Self-Organizing Map (SOM)-based clustering method and the hybrid wavelet-Adaptive Neuro-Fuzzy Inference System (ANFIS) approach for GWL prediction and showed the strength of ANNs to provide estimation accuracy. Adamowski & Chan (2011) proposed a new method based on the wavelet and ANN predictions for GWL prediction. The exact prediction results for both St-Remi and Mercier sites in the Chateauguay basin indicate that the Wavelet Analysis based ANN (WA-ANN) method is a suitable method for predicting GWLs. Hosseini et al. introduced a program of evolved ANNs using a simple genetic algorithm to simulate monthly GWLs in the coastal aquifer in the Shabestar Plain, Iran. This research shows that employing a genetic algorithm to initialize neural connection weights in complex space avoids the risk of becoming stuck in local minima, which can prevent high costs and time-wasting in water resource management projects for drilling more piezometers. Gholami (2022) combined modular ANNs with adaptive fuzzy inputs to increase the accuracy of the annual GWL modeling process on the southern coasts of the Caspian Sea (Iran). More recently, Roshani & Hamidi (2022) used ANNs to predict GWL fluctuations in the Sari-Neka aquifer, the Caspian Sea basin. The authors showed that the groundwater depth of the Sari-Neka basin is expected to decrease with a gentle slope during the period 2021–2040.

Genetic programming (GP) is rather a new class of SCTs that is applied to a wide range of problem domains, including symbolic regression, classification, control systems, and optimization. It allows for the automatic generation of programs without the need for explicit programming by humans (Danandeh Mehr et al. 2023). GP has been used successfully in various fields, such as data mining, robotics, bioinformatics, and finance, to tackle complex problems and discover novel solutions. Various versions of GP such as monotonic GP, linear GP, multigene GP, and gene expression GP have been satisfactorily used in hydrological studies (Uyumaz et al. 2014; Rahmani-Rezaeieh et al. 2020; Samadi et al. 2020, 2021; Mehr & Gandomi 2021). A recent review study on the application of SCTs for GWL prediction conducted by Tao et al. (2022) indicated that GP was used only in 5.0% of the artificial intelligence models applied for GWL modeling, so that there is room for more research on the application of GP for GWL prediction. Some of the recent implications of GP variants for GWL estimation include (not limited to) the study of Fallah-Mehdipour et al. (2013), Kasiviswanathan et al. (2016), Aryafar et al. (2019), and Abdolahzadeh & Schmalz (2022). The existing studies mostly trained a standalone GP engine or hybridized it to find the best short-term (less than a season) GWL estimating model. Differing from the earlier studies, we applied GP to identify how different hydrometeorological variables interconnect with GWL fluctuations from a temporal perspective. Thus, standalone monotonic GP was used in this study without any attempt to enhance its performance via common hybridization techniques.

In recent years, there has been a burgeoning interest in the interconnectedness of ecosystems and how various components respond to both natural variations and human-induced activities. This approach has gained traction as a valuable tool for comprehending the fundamental principles that inform policy-making and ecosystem management (Rinderer et al. 2018; Zhang et al. 2023; Zhou & Yang 2023). For instance, Rinderer et al. (2018) explored hydrologic connectivity among groundwater and streamflow time series from a Swiss catchment and demonstrated that employing ground-based high-frequency sensors as well as emerging remote sensing techniques can provide new opportunities for a better understanding of connectivity across ecosystems. Another noteworthy study by Zhou et al. (2022) introduced a novel algorithm for mining prevalent negative co-location patterns in spatial data. Meanwhile, Jia & Zhou (2023) proposed a response surface methodology to estimate the global karst carbon sink, demonstrating that functional connectivity analysis could significantly reduce numerical calculation time, making it a viable tool for global weathering models. These investigations collectively highlight the versatility of the concept of connectivity in both ecology and hydrology. In ecology, it is defined as the extent to which the landscape facilitates or hinders movement among resource patches. In hydrology, on the other hand, it pertains to the water-mediated transfer of matter, energy, and/or organisms within or between elements of the hydrologic cycle (Rinderer et al. 2018). The integration of these perspectives underscores the interdisciplinary nature of connectivity studies, offering valuable insights for informed decision-making in environmental science and management.

Monitoring GWL is important for hydrosystem management and assessing the overall health of aquifers. This information helps determine the availability of water for various uses, including drinking water, irrigation, and industrial purposes. Additionally, GWL estimation helps to prevent issues such as land subsidence, saltwater intrusion, and ecological disruptions caused by significant changes in water table depths. Therefore, the purpose of this study is to use emerging statistical methods and soft computing tools (i) to assess the hydrologic connectivity between various hydrosystem components and then (ii) to identify the underlying pattern of GWL based on the hydrologic connectivity and evolutionary GP technique. The study focuses on the southeastern Caspian Sea region that covers three coastal catchments and an unconfined aquifer. This article is expected to contribute to the development of a more effective and sustainable water supply for the domestic and agricultural water consumption in the study area, that is, facing water scarcity and negative climate change impacts these years.

To assess hydrologic connectivity between the hydrometeorological variables (i.e., H, E, T_max, T_min, R) and the GWL in the study area, first, a representative GWL time series (hereafter GWL_ave) was obtained for the whole area by calculating the arithmetic mean of GWL at all piezometers. Then, a set of linear and nonlinear connectivity measures including the pairwise correlation coefficient (PCC) and mutual information (MI) was calculated. In hydrological time series, the value of a variable at a particular time can be influenced by previous values, so-called lags, of the same variable or other indicator variables. Therefore, we also looked for leading indicators when conducting conductivity analysis. In other words, in addition to evaluating the linear and nonlinear correlations between two concurrent time series, we investigate the correlation between the GWL and the independent variables with a few time lags (here 36 months). Thus, lagged PCC and lagged MI (LMI) were also calculated to find the dominant lags. Finally, a monotonic jittered genetic programming (JGP) method was employed to link the GWL in the study area with the most dominant independent variables and their lags. To cope with the existing trends and dimensional inconsistency of both dependent (i.e., GWL_ave) and independent variables, they were standardized first, and then made stationary via a differencing method before the modeling stage. Due to the rather short period of historical observations, jittered-based augmented time series were used for training and testing the GP model.

A value of 0.0 implies no linear correlation between the signals. A value of 1 indicates a perfect positive linear relationship, where an increase in one variable is associated with a proportional increase in the other variable. A value of −1 indicates a perfect negative linear relationship, where an increase in one variable is associated with a proportional decrease in the other variable.

MI measures the statistical dependence or information shared between two random variables. It quantifies how much information is obtained about one variable by knowing the value of another variable. In other words, it measures the amount of information that two variables share. MI is often used in the fields where a nonlinear relationship between variables is aimed to be detected (Batina et al. 2011).

To cope with the comment overfitting problem of GP models, particularly when they encounter a small number of samples, we introduced the JGP model in which the jittering technique was used to create additional samples for training and testing GP. This technique can be valuable for improving the robustness and generalization ability of all machine learning models trained on time series data. It involves adding small amounts of random noise to the data to create variations while maintaining the overall structure of the time series. This technique can be valuable for improving the robustness and generalization ability of machine learning models trained on time series data. To this end, first, the level of jitter was specified as one-third of the standard deviation of each indicator. Then, random noise values that follow a Gaussian distribution were created. Finally, the random noise value was added to each data point in the original time series. The result is a new time series that was added to the end of existing ones and used for training and testing GP.

The graphical analysis of leading indicators (Figure 6) illustrates the temporal offset between the signals at a time delay 12 that reflects the common annual periodicity existing in hydrological time series. It is observed from Table 3 and Figure 6 that almost all PCC values of P remain in the insignificant connectivity range and therefore, their influence on average groundwater depth in the study area could be considered negligible. Despite being insignificant, an increasing trend in the PCC values of P is seen, indicating that P may influence GWL_ave at longer lags. The associated lagged PCC values had predominantly positive values after a 30-month delay, signaling that the mean GWL at the site has a direct relation with P after two and half years. All the PCC values of H and Q fell in the blue cells and hence gave an impression of widespread adverse connectivity across the groundwater monitoring wells and the catchment humidity and runoff. Despite the increasing trend, almost all PCC values of H and Q are in the range of negative connectivity. Therefore, it is concluded that increasing H and Q in the basin may decrease GWL_ave. The associated PCC values of E, T_min, and T_max show almost the same periodicity pattern including both positive and negative connectivities in the range of −0.43 to +0.43. The peaks and troughs appear with 3–4- and 9–10-month lags, respectively.

According to the figure, The MI values are insignificant for P and Q in all lags. This reveals the insignificant influence of concurrent and past rainfall and runoff on the current behavior of GWL_ave. On the other hand, H, E, T_min and T_max are highly correlated to GWL_ave after 18 lags (Figure 8(b)), and the maximum association is seen between GWL_ave and its lagged values back to the 27-month lag (Figure 8(a)). This long significant correlation might be a consequence of our decision to limit our threshold to 0.25. The LMI series (Figure 8(a)) shows that antecedent GWL_ave back to 3 months together with its past year value (i.e., GWL_ave (t − 12)) can provide valuable information for modeling and understanding the groundwater dynamic system in the Jafakende aquifer. The magnitudes of LMI of T_min were greater than T_max in the first 6 months lag and contrasting results were obtained after the 18-month lag, thus suggesting that the interaction between groundwater dynamics and temperature should be assessed at different seasons separately.

The figure shows that the evolved model can well detect the fluctuating dynamics of the GWL. However, the extreme peaks and troughs are not well predicted. This is mainly due to the intentional implementation of the ad hoc GP technique to discover the simplest nonlinear relation (Figure 9(b)) between GWL_ave and the indicators. The use of advanced optimization techniques for fine-tuning of GP may increase the estimation accuracy of extreme values as seen in Sepahvand et al. (2019). Compared to the literature, our study showed some agreements and conflicting results with those of similar studies. For example, like the works of Kasiviswanathan et al. (2016), our study showed that GP effectively captures the non-linearities in GWL fluctuations, demonstrating a reasonable performance using observed hydrometeorological and antecedent GWL data, all without necessitating explicit knowledge of the underlying physics of the system. In contrast to Kasiviswanathan et al. (2016), we found E as a dominant parameter in GWL. Such conflicting results could be due to the higher perspectives of lag times (up to 36 months) that were considered in our study. Thus, the authors believe that our findings could better represent GWL dynamics from a long-term perspective. In addition, our findings showed that precipitation is not a dominant factor in GWL fluctuations in our study area.

Our study revealed apprehensions regarding the feasibility of developing a forecasting model for GWL estimation. Like previous research, our study was constrained to datasets obtained from in situ hydrometeorological measurements. Subsequent investigations could incorporate additional factors, including demographic and geological data such as topographic wetness index, pumping rate, groundwater recharge, lithology, land use, population growth, and more.

Hydrologic connectivity analysis was conducted across the Jafakende River catchment to identify potential correlation and causality between hydrometeorological parameters and the GWL at the Jafakende aquifer. First, linear and nonlinear methods were utilized to determine the dominant features, and then, their temporal variation was linked to the target GWL using an innovative GP technique. Our study generally showed that the relationship between GWL and hydrometeorological factors is complex and challenging to be modeled. The detailed conclusions are itemized below.

• Both linear and nonlinear statistics suggested the absence of significant connectivity between average groundwater depth and P and Q at the catchment.

• There is a strong correlation between H, E, T_min, T_max, and GWL_ave after 18 lags.

• Incorporating antecedent GWL_ave up to 3 months, along with its value from the previous year (GWL_{ave (t−12)}), can offer valuable insights for modeling and comprehending the dynamics of the groundwater system in the Jafakende aquifer.

• Jittering is useful for preventing the overfitting of GP. However, it's essential to strike a balance in the level of jitter introduced, as excessive noise can hinder model performance.

All relevant data are included in the paper or its Supplementary Information.

The authors declare there is no conflict.