Short–long-term streamflow forecasting using a coupled wavelet transform–artificial neural network (WT–ANN) model at the Gilgit River Basin, Pakistan

Accurate streamflow forecasting is of instrumental importance in the domain of water resources, especially in the sub-disciplines of flood control, water supply, hydropower, reservoir operation, etc. (Senthil Kumar et al. 2013; Nacar et al. 2017). The streamflow generation is dictated by a number of parameters that simultaneously interact at varying spatio-temporal scales. These include the climate variables (precipitation, temperature, evapotranspiration, etc.), the ground conditions (land use, roughness, etc.), radiation, anthropogenic activities, etc. (Nacar et al. 2017). All these factors make flow processes highly non-linear and complex. The researchers, faced with this bewildering complexity of nature, have resorted to the development of streamflow forecasting models that can adequately simulate the flow dynamics suitably and with sufficient accuracy (Kalteh 2013).

Typically, modeling methodologies are broadly divided into two categories: process-based models and data-driven models (Patel & Joshi 2017; Zhang & Yang 2018). Process-based modeling assumes the river basin processes as interlinked mathematical functions (Islam 2011; Hassan et al. 2014). The process-based models are subdivided into conceptual and physically based models. Conceptual models are mostly based on simplified assumptions and hence their utility is limited to simple uses, while physically based models can incorporate complex processes (Islam 2011). These types of models, generally, require a higher number of data inputs, such as climate, physiographic inputs, etc., and their processing is based on the holistic consideration of these inputs for modeling. Physiographic inputs such as land use and vegetation are generally derived from the satellite via indirect measurements and could contain uncertainty (Hosseini et al. 2022). Land use/cover is also incorporated as a single layer and therefore, considered time-stationary, which means that it is assumed that during modeling the land use dynamics (monthly and annual variations) will not affect the flow generation. These limitations along with the paucity of data are adjusted by extensive alterations in other model parameters during calibration, which results in an improper model parameterization and hence degrades the quality of the simulation (Nesterova et al. 2021). Secondly, physical models necessarily require future weather data for forecasting, which is an issue in itself. Until now, generally, weather forecasting accuracy has been 80% for a 7-day forecast and 50% for a 10-day forecast (Scijinks n.d). Nevertheless, predicting quantity is even more challenging (precipitation, etc.) (Šaur 2015). Therefore, in view of the aforementioned limitations (in the model and weather forecast), forecasting using physical modeling is an onerous task that is prone to high uncertainty. Alternatively, ensemble-based modeling could be a potent option as compared to a standalone model, as it could define probabilistic ranges against a forecasting sequence. However, the development and calibration of ensembles are computationally intensive and rigorous tasks.

Besides physically based models, Artificial Neural Network (ANN) black box models are another popular choice for hydrological modeling. ANNs are parallel-distributed processing systems analogous to the biological neural networks of the human brain (ASCE 2000; Govindaraju 2000). Due to this, these models are robust in approximating intricate issues such as approximating non-linear relations (Patel & Joshi 2017), pattern recognition (Jianjin et al. 2017), classification, prediction, and sequential analysis (Akhtar et al. 2009). Several types of ANN models are in use for water resources applications, such as Hayder et al. (2022) implementation of Non-Linear Autoregressive with Exogenous Inputs (NARX) and Long Short-Term Memory (LSTM) for forecasting river flows. Wunsch et al. (2020) use LSTM, NARX, and a Convolutional Neural Network (CNN) for forecasting groundwater levels. Further, Apaydin et al. (2020) investigate three types of ANNs: NARX, LSTM, and the popular Multi-Layer Perceptron (MLP) for reservoir outflow forecasting. The specialty of ANN models is that they can directly relate inputs to the required target without considering the underlying hydrological processes (Noori & Kalin 2016; Pitta et al. 2016). The ANN model benefits from the affiliated correlation present between the input and target variable and, therefore, for optimum convergence, unlike the physical model, it does not enforce the fulfillment of specific input requirements. Any exogenous or non-exogenous variables with necessary affiliation/correlation could be used for result derivation. This unique attribute of ANN can compensate for physically based forecasting's shortcomings. However, the inclusion of these variables is subject to data availability and access. That's why parsimonious approaches are often more preferred and commonly used worldwide (Kalteh 2013).

The wavelet transforms are physically based signal processing techniques that are recently introduced and are being used in conjunction with ANN and other machine learning approaches for hydrological applications. They deliver the information of the primary signal or data in time and frequency domains, thus providing valuable information regarding their structure and also giving a time-frequency depiction of the primary signal at multiple temporal resolutions (Daubechies 1990). Several studies have used ANN-Wavelet Transform (WT) hybrids for obtaining daily forecasts. For instance, Khan et al. (2021) and Partal & Cigizoglu (2008) used a WT–ANN hybrid for forecasting the daily suspended sediment load. Adamowski & Sun (2010) used ANN and WT for forecasting the flow of non-perennial rivers and reported that hybrid modeling provides better results than the exclusive ANN model. Adamowski & Chan (2011) used the ANN-wavelet model for forecasting groundwater levels. Kalteh (2013) used wavelet and ANN along with other deep learning approaches for forecasting streamflow at Kharjegil and Ponel stations in Northern Iran. Freire et al. (2019) reported that coupling WT–ANN provides better results than standalone ANN for short-term streamflow forecasting. Recently, Wang et al. (2022) coupled several machine learning algorithms with wavelet theory for determining monthly scale river discharge. Qasem et al. (2019) modeled monthly pan evaporation by hybridizing deep learning Support Vector Regression (SVR) and ANN schemes with wavelet transformations. Most of these studies concluded that WT–ANN hybrids are more robust in forecasting than exclusively ANN, which informs that WT–ANN hybridization possesses a strong harnessing ability that can leverage model performance.

The Gilgit Basin, being part of the Upper Indus Basin (UIB), is characterized by high-altitude and cold terrain. As one of the largest tributaries of the UIB, it provides crucial input into the Indus water system that tends to regulate the downstream water fluctuations (flooding and droughts). Physiographically, a large portion of the basin remains covered with snow all year long (Adnan et al. 2016a), which signifies that the snow-melting process regulates the annual perennial flow cycles of the Gilgit River. However, the melting process is convolutionally uneven and heavily relies upon numerous intricate parameters (precipitation distribution and amount, temperature distribution, rain on snow, snow density, atmospheric pressures, solar radiation, surface canopy, hill shades, etc.). Previously, most studies on the Gilgit Region were based on simplified physical assumptions (e.g., Degree Day) that only used a few generic parameters (precipitation, temperature, etc.) for imitating the melting process, which made either the results or the model setting ambiguous (Bashir & Rasul 2010; Adnan et al. 2016b; Latif et al. 2019). Nevertheless, these studies were relevant to streamflow prediction/hydro-climatic modeling rather than forecasting. No works of literature were found that directly addressed streamflow forecasting in our region of interest. Unsatisfactory meteorological information in the region is another constraint that creates a complication for streamflow forecasting. Various studies have reported a paucity of hydrometeorological stations and other inherent deficiencies at UIB (Syed et al. 2022). Further, stochastic variation in precipitation gradient at a limited spatial scale also adds up to uncertainty in flow simulation that obscures the streamflow generation process (Schreiner-Mcgraw & Ajami 2020). Faced with limitations, we presume that ANN would be a better option for flow forecasts, as it directly relates an input to an output, skipping the underlying intricate physical processes and solving based on a given relation between these variables. Further, lags due to snowmelt and other unknown reasons can also be adjusted by simply applying time-lag adjustments to inputs. Further, the wavelet transforms can be used in conjunction with ANN, as it allows to decomposition of the signals to multiple degrees, which could remove unwanted noise and clarify the most associative hidden signals. In this way, unlike standalone ANN, coupled WT–ANN could be better at tracking the exposed signals and calculating the flow magnitudes in an appropriate way.

The objective of this study is to forecast the Gilgit River's flow using WT and ANN hybrids at several forecasting intervals (short, intermediate, and long-terms). It will comprehensively discuss each phase of the model's implementation and highlight internal and external strengths and limitations.

Transformations are mathematical functions that are used to gather signal information, represented by the frequency with a certain magnitude. Among several transformation functions, Fourier transforms are the most popular. The Fourier transform can process stationary signals (constant frequency for a long time) but cannot provide frequency information of non-stationary or complex signals. The Window Fourier Transform (WFT) addresses this frequency localization (what frequency at what time instants) problem by processing the frequency using a localized, time-based window. The window slides over time, assuming the portion of the non-stationary signal inside the window as a stationary signal and calculates the Fast Fourier Transform (FFT) at each instant. However, despite the improvement, its accuracy highly depends upon its window function, which affects its frequency resolution. For instance, if we consider fixing the window size when sliding over a non-stationary signal, frequency localization could be lost if the frequency inside the window is either too low or too high (Santos & Silva 2014).

The back-propagation algorithm is a supervised learning technique that is used for error reduction in MLP. It works in forward and backward passes. In the forward pass, inputs are passed through the intervening layers to the output layer, where losses are calculated by means of the loss function (model residual of target and output). Errors are then back-propagated or back passed for adjustment. In this way, the process repeatedly adjusts the weights (which were initially randomly assigned) at every iteration (Jimeno-Sáez et al. 2018; Ali & Shahbaz 2020; Hosseini et al. 2022). This process keeps iterating until the following rules are met: epochs reach the set limit, or the gradient becomes 0, or the model fulfills early stopping conditions. Previously, Aqil et al. (2007) attested several training algorithms (e.g., Levenberg–Marquardt (LM) back-propagation, Bayesian regularization, and gradient descent with momentum) and found the LM back-propagation to be more effective than other training algorithms. Further, studies also reported LM's better performance and a faster convergence rate as compared to other popular back-propagation algorithms such as conjugate gradient and resilient back-propagation (Kisi & Uncuoğlu 2005; Tabbussum & Qayoom 2020).

This study considered four commonly used statistical evaluators or metrics, namely, Pearson Correlation (R), Coefficient of Determination (R²), Nash–Sutcliffe Efficiency (NSE), and Root Mean Square Error (RMSE), also shown in Table 1. Pearson Correlation (R) explains the strength and direction of relationships. Its values range from −1 to 1. The value −1 indicates a perfectly inverse relation, while 1 represents a perfectly proportional relation; nearly zero or zero shows poor or no relation at all. The R² is the square of the Pearson Correlation that highlights the collinearity or strength of the relationship in terms of a positive number. It is a common metric for measuring the goodness of fit of the models. The least value 0, represents no relation, while the maximum value 1, represents a perfect fit. Similar to R², NSE is nearly identical to R² except it quantifies model simulation. Its value ranges from -∞ to 1. The NSE value can also be written in terms of RMSE, as given in Table 1. According to the relationship, if RMSE is equal to the standard deviation of observation, then NSE will be 0, while if the RMSE value is half of the standard deviation of observation, then NSE will be 0.75. As such, lower RMSE values are highly desirable.

Table 1

Statistical evaluator and details

Obj. functions .	Equations .	Range .
Coefficient of Determination (R2)		(0, 1)
R		(1, 1)
NSE		(-⁠, 1)
RMSE		(0, ⁠)

Obj. functions .	Equations .	Range .
Coefficient of Determination (R2)		(0, 1)
R		(1, 1)
NSE		(-⁠, 1)
RMSE		(0, ⁠)

Note: N, the number of flow values; Q_obs and Q_sim, the observed and simulated flows for ith observations; Q*_obs and Q*_sim, the means of the observed and simulated flow values; S_obs and S_sim, the standard deviation of the corresponding observed and simulated flows.

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For this study, 10-daily temporal resolution stream flow from 1999 to 2015 was used as the main input and output, comprising of in total of 612 data points. The 10-daily is a moderate temporal resolution commonly used by engineering practitioners, where daily records are either unavailable or ambiguous. It is more suitable than commonly used daily and monthly resolutions since it smooths out the daily noise and at the same time preserves sub-monthly trends. The first 10-daily represents the mean of the first 10 days of the month; the second 10-daily represents the mean from 11th to 20th days of the month and the third 10-daily represents the mean of the rest of the remaining days in a month. Flow was selected as an exclusive predictor. Exogenous climate variables (precipitation, temperature, etc.) were not taken into account in this study as they are dependent upon several factors such as data availability, public accessibility, and reliability (errors in data; wind-induced, random, etc.). As explained earlier in the introduction, the Gilgit River Basin has deficient and biased observation climatology, which could degrade the model simulation; therefore, we opted to omit the inclusion of exogenous climate variables. Further, we have found no explicit guidelines regarding the selection of forecasting intervals in the literature; however, we opted to adopt an interval range as a multiple of two rather than taking constant intervals. In this way, inferences about larger ranges can be computed by only considering a few intervals. Additionally, it is also allowed to estimate inferences on sub-intervals via interpolation with reasonable accuracy. Following, monthly intervals were selected for forecasting based on 10-daily data (T_0.33; 0.33-month, T_0.66; 0.66-month, T₁; 1-month, T₂; 2-month, T₄; 4-month, and T₈; 8-month). Further, we categorize intervals as short-term (T_0.33 and T_0.66), intermediate-term (T₁), and long-term (T₂, T₄ , and T₈) forecasting intervals.

In total, two types of input variables were considered for the ANN modeling, namely, flow and wavelet-based. The flow-based included the stream flow (Q), while the wavelet-based comprised Approximation (A) and Details (D). Monthly interval flow-based input variables (QT) are as follows: QT_0.33, QT_0.66, QT₁, QT₂, QT₄, and lastly QT₈. For the wavelet-based input variables, stream flow (Q) was decomposed using the Daubechies wavelet setting. This resulted in first-level Approximation (A₁) and Details (D₁). The A₁ was then further decomposed to give A₂ and D₂, and this procedure was then repeated until obtaining A₁₀ and D₁₀ (subscript represents the level of decomposition). At the end, the series were adjusted for the given monthly intervals. The input, monthly interval Approximations (T) and Details (T) are as follows: Approximations, A_n = [T_0.33, T_0.66, T₁, T₂, T₄, and T₈]; Details, D_n = [T_0.33, T_0.66, T₁, T₂, T₄, and T₈], where n = (1,…, 10) represented the number of level of decomposition. Further, cross-correlation analysis using Pearson Correlation was performed between the input variable and stream flow to find the highly correlated variables. This analysis was used for good effect as highly correlated variables were selected for developing the input configurations for the ANN modeling. The configurations were selected based on two criteria; firstly, the best variables were selected considering the same interval. Secondly, the best variables were selected from the other intervals, provided that the interval was less than or equal to the reference interval. For instance, for 4 months streamflow forecasting, only (AT₄, AT₈, QT₄, QT₈, etc.) can be used as input variables and no other variables with T < 4 will be considered eligible. By doing so, the model was able to make use of good relationships existing at large intervals.

The development of an ANN model requires determining the optimum configuration of neurons and layers in the model. The number of neurons in the input and output layers of the model is equal to the number of input and output variables. However, generally, the number of neurons in the intervening layers is adjusted by trial-and-error. Our study opted for the following settings, preferring parsimonious conditions: input layer with four neurons denoting a maximum of 4 inputs from streamflow, for the hidden layer and 1 single layer with 10 neurons was selected after trial-and-error and 1 neuron for the output layer denoting stream flow output. For better performance and rapid processing, the ReLU transfer function was selected. The ANN models were randomly trained using a typical 70% of the data and then tested on 30% of the remaining data. Early stopping condition was considered for avoiding overfitting. LM was used as a back-propagation algorithm for better results and faster convergence. The models were statistically evaluated using NSE, RMSE, and CC (Correlation Coefficient). Details are provided in Section 3.3.

A comprehensive cross-correlation analysis was carried out between the flow and the input variables, shown in Table 2. Synoptically, correlation reduces as forecasting intervals increase. However, interestingly, it was also found that for T₄ and T₈ intervals, correlation slightly improved in terms of negative correlation, which might be indicating an inverse transformation of signals. Nevertheless, this type of behavior also helped model learning and assimilation of data.

Table 2

Cross-correlation analysis of flow and input variables

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Analysis showed that in the case of flow, Q, a strong correlation was achieved corresponding to the short-term intervals (T_0.33 and T_0.66) followed by the intermediate interval T₁, which also fell in a fair range. However, signal correlation drastically reduced to a fairly low value for the T₂ interval, though minor improvement was observed for the long-term (T₄ and T₈) intervals. In the case of approximations, 10 decomposed approximation series were compared to the flow at different intervals, and a correlation was obtained. The results showed that the A₁ and A₂ series were among the best performers, as they were able to preserve most of the information and yield the maximum correlation with the flow. The Details series depicted a relatively lower correlation with the flow, nevertheless, the pattern remained the same as for the approximation series.

There are no explicit guidelines regarding the number of inputs for an ANN model, however, it should be carefully selected as the input variables are instrumental in mapping the underlying non-linear processes. More than necessary inputs can reduce the learning rate of ANNs while a deficient number of inputs could result in the formation of a weak model (Jimeno-Sáez et al. 2018). In our case, four highly correlated variables from the cross-correlation analysis were used to prepare each of the nine input configurations. Additional details and criteria for developing input configuration are provided in Section 3.4 of this article.

Individual inspection shows that two configurations of the short-term (T_0.33) forecasting interval stand out in terms of performance, both in training and testing epochs. Both configurations adequately fit into observation data, as can be seen in Figure 2(a1-2). The given configuration against short-term (T_0.66) also yielded good and consistent performance, as depicted in Figure 2(b1). However, a significant drop in metric values was observed compared to the latter case. For intermediate intervals, the model produced comparatively low but moderate inferences. In Figure 2(c1), discrepancies under high flow conditions could be observed. Interestingly, a relatively fair performance was recorded in the case of the long-term T₂ interval, considering the poor correlation of its associated variables. This might be either due to the unique ability of ANN to track hidden links of poorly related variables or else the weak variables, when in combination, were able to show a higher correlation value. The later hypothesis can be further explained by conjecturing that the multiple weak variables in an ensemble formation might have raised the signal strength, which allowed the generation of a better signal and supported ANN to assimilate appropriately. Nevertheless, the model poorly produced high flow conditions Figure 2(d1-2). In the case of long-term (T₄) prediction, the model showed improved results as compared to the T₂ interval. This was most probably because of improved correlation due to the inverse transformation of stream signals, as indicated by an improved negative correlation in Table 2. This might be an indication of the influence of annual recurrent pattern of the Gilgit River on long-term forecasts. Figure 2(e1-2) displays the T₄ flow series and it can be seen that its performance is below par in predicting extreme flow conditions. Likewise, at 8-month time interval, T₈ the model performance was suboptimal as shown in Figure 2(f1). Overall, given the fact that the training and testing scores are comparable (in some cases slightly better), it is clear that the model's lack of ability to predict extreme values represents a major hurdle in achieving a perfect score.

Forecasting performance was evaluated at different instants of time using a Flow Duration Curve (FDC). FDC was discretized into three sets of exceedance probability based on flow conditions as follows: high flow, ⁠, intermediate flow, and low flow, ⁠.

In this study, a hybrid model of Wavelet Transforms and ANN (WT–ANN) was tested for forecasting flows at short (T_0.33, T_0.66), intermediate (T₁), and long (T₂, T₄ , and T₈) time intervals. The model was built using 10-daily flows of the Gilgit River.

The Gilgit River Basin is influenced by two major factors, i.e., the snow-melting process and local orography. While modeling its streamflow signals, the stochastic disturbances caused by these factors could add up to uncertainty in the simulation results. Our study proposed to leverage the utility of the ANN model by resorting to a different level of streamflow signal decomposition (approximation and details) using DWT. We decomposed the signal up to 10 levels using Daubechies wavelets. Further, in order to enhance the ANN processing, we first performed a cross-correlation analysis and selected highly correlated inputs. Then, from the selected inputs, nine input configurations were formed based upon two criteria: firstly, the best variables were selected considering the same interval. Secondly, the best variables were selected from other intervals, provided the interval was less than or equal to the reference interval. The simulations were evaluated using statistical metrics, i.e., R², NSE, and RMSE. The results showed that the model performed best for short-term forecasting. At intermediate intervals, its performance declined and, interestingly, recovered back for long-term forecasting. This performance recovery might be due to the inverse transformation of the signal pattern (especially after T₂), as indicated by the incrementing negative correlation in Table 2. This can be attributed to the snow-melting and accumulation process that plays a pivotal role in manipulating the streamflow signal in the Gilgit Basin and hence signifies the role of the annual recurrent pattern. It is also to be noted that research has exclusively considered flow as a primary input for modeling, which itself is the product of climate variables (precipitation and temperature fluxes). Therefore, the ingested role of these variables also played a part in the signal transformation. Further FDC analysis showed that short-term forecasting performed well at low and intermediate flow time scales, while beyond that the signal starts to deteriorate. This might be owing to fewer training samples for the extremes (low and high flows).

Although short-term forecasting provided better results, it did not necessarily mean that long-term forecasting will yield poor results, as it depends on several factors, e.g., the time resolution of forecasting interval (daily, 10-daily, monthly, etc.) and the behavior of the stream in that time resolution (perennial, non-perennial). The typical behavior of streamflow at short time intervals follows a predictable extrapolated trend (continuation of previous series) and hence yields good results. Depending upon behaviors and type of stream, the longer intervals could also benefit from the larger recurrent pattern of the stream (in our case perennial annual cycle regulated by snow-melting) and can improve its inference (in the case of a non-stochastic, perennial stream). However, forecasting at intermediate intervals could be challenging as they lack in capacity to capture both predictable patterns and larger recurrent patterns due to interval differences. Furthermore, the analysis also revealed that the lesser number of samples related to extreme values (low and high flow) in the training set limits the predictability of extreme events and could induce biases in results, thus adversely affecting the forecasting model.

While this study, adopted a suitable methodology, justified by peer-reviewed literature, limitations persist that must be addressed. It is also important to emphasize that although ANNs provide ease in utility, they still require an overabundance of data to train appropriately, while physically based models have no such restriction. Thereby, a physically based model may be considered a more suitable choice in low data constraints. Another challenge is non-stationarity in hydroclimatology, which has become more pronounced after the emergence of climate change. ANN is weak in predicting values that are out of bound to their training samples; therefore, the modeler must be vigilant to train a model on sufficiently longer records and use variables that coincide with extreme values of flow. Moreover, as described earlier in the conclusion that streamflow ingests the effects of climate variables, which can be reflected in predicted values, thereby validity of forecast models maybe also be subject to the intensity of climate change.

S.H., S.A, and K.Z.J. conceptualized the study; Z.S., RF., and K.A. did data curation; Z.S and P.M. did formal analysis; Z.S. and P.M. investigated the study; S.A., K.Z.J., and Z.S. performed the methodology; S.H., S.A., and S.S. collected resources; S.H. and Z.S wrote the original draft.

This research received no external funding.

The authors have no relevant financial or non-financial interests to disclose.

Data cannot be made publicly available; readers should contact the corresponding author for details.

The authors declare there is no conflict.