Comparative study on stream flow prediction using the GMNN and wavelet-based GMNN

discharge at an upstream section at time t+Δt in a river system

discharge at downstream section at time t+Δt in a river system

flow depth at an upstream section at time t+Δt in a river system

flow depth at downstream section at time t+Δt in a river system

upstream and downstream flow depth at time t

equivalent inflow at a point r in the basin for N upstream flows

muskingum routing coefficients

upstream hydrograph evolution parameters

discharge through a river section

parameters for computing discharge from the depth

depth of flow at a section

⁠,

upstream and downstream flow depth at time t+Δt

parameters for computing discharge from depth at upstream

kernel function

memory resolution

p

memory order

A hydrological forecast represents the preliminary assessments of future hydrological events and their characteristics. Hydrological forecasts are necessary for effective water management and to mitigate the effects of natural hazards, such as floods and droughts. The prediction of flow in a water system is required to minimize the impact of uncertainties of the climate on the management of water resources, considered one of the main challenges related to integrated knowledge of climatology and hydrology. The use of expected flows for the optimization of river flow operation is suggested by many authors such as Owuor & Mwiturubani (2022) and Beker & Kansal (2022).

Currently, many hydrological models are being used in flood forecasting studies, distributed models or concentrated, conceptual or empirical, discrete or continuous. The objective of studying most of these models is the basis of the simplicity they offer in applying the water balance and mass balance principle. One of the models of the artificial neural network (ANN), the time-delay neural network (TDNN), has been chosen from various models for being an empirical model which has been widely accepted as potentially useful for modeling complex nonlinear systems with a large amount of data. A TDNN can well approximate a number of functional relationships describing outputs that are some functions of the same inputs. These models are especially helpful when it is unclear how the functional link between the input and output data works. The multiple gamma memory neural network (MGMNN) models can also be used in place of and in conjunction with mathematical models.

With this in context, channel inundation, being a multiplex process, is distinguished by spatial and time-related alteration in flow constraints. Applications can be seen in the literature regarding flood flow modeling having the use of the ANN. In most of the ANN models, a multilayer perceptron (MLP) is employed to predict flow variations and storage change except for models deploying the use of time lag feed-forward delays such as the TDNN. Memory in the ANN is assimilating mostly in delays having both feed-forward and feedback. The MLP cannot identify time-related modifications in input series, which is why MLP with a sliding window approach, equivalent to a TDNN, is being implicated in the modeling of temporal variations. Self-recurrent network by Lang et al. (1990) can also be an instance of feedback delays in which the input holds the state of past and neural states. Memory depth remains static in the case of the TDNN while training since it cannot scale the time axis. An adaptive ANN can be best suited to find the robust memory depth for which the MGMNN can be best employed.

In an unsteady flow channel, storage and flow rates change concurrently with respect to time. Thus, it may be said that during unsteady flow along with flow rate, flow depths, storage and storage characteristics in reach evolve in time. Hence, a river flow model must also be applicable to river systems with a number of upstream flows and a common downstream flow. Flood flow modeling, being spatially and temporally dependent, requires effective investigation in quantifying the relationship between the use of flood susceptibility maps and the applied machine learning model, since the use of different ANN models on different watershed basins produces results that are dependent on the topographical and geomorphological structures of the river basins (Avand et al. 2022).

In 2015, Choudhury and Roy predicted multiple rate flow in river systems using the ANN. The model is based on a time-based application useful in real-time forecasting as flood predictions are time constraints. It is known that during unsteady flow in river reaches, flow rate as well as storage in the channel change continuously with time and are governed by the same causes. Such changes are dependent on common causes such as river reach, watershed basin characteristics and properties, and hydrological and geomorphological properties. Storage change also indicates unsteady flow; therefore, any model utilized for predicting unsteady flow must incorporate storage characteristics. Considering this as a novelty, this study has considered flow rates implicitly as well as explicitly incorporating storage properties. Since storage is as important as flow, therefore, the model meant for predicting unsteady flow must incorporate storage characteristics also. Different storage indicators have been taken to develop a model based on flows and storage characteristics, which is the novelty of this work. In this work, extension and modification of multiple inputs–multiple outputs (MIMO)-based and multiple inputs–single output (MISO)-based model (Aboutalebi et al. 2016) have been utilized for using different storage characteristics of which one may be the storage based on the instantaneous storage rate change and the other is the storage based on the gauge height when calculated from an arbitrary datum of the Tar River in North Carolina.

The principle of continuity plays a vital role and has tremendous significance when it comes to river flow modeling. Routing ANN does not incorporate the latter while modeling and training the network architecture. In this study, the basic fundamental continuity principle has been incorporated in order to predict the storage, rate of change of storage, and flow depth variation, which implies the average rate of change in storage properties in the case of MIMO–ANN models. The models in this study obey the continuity equation in the case of MIMO-1 and MIMO-2. But the MISO model does not follow the continuity equation since the weights in the output node are set to zero for all the river sections, excluding the forecasting section. The result shows that all ANNs very accurately matched the pre-decided zero value at the sections.

River flow forecasting is an essential aspect of flood damage mitigation and plays a vital role in water resource management. The characteristics of river flow are governed by various factors and results of different factors, such as precipitation, evapotranspiration, and groundwater discharge (Ardiclioglu et al. 2022). Recently, various artificial intelligence techniques, such as MLP, TDNN, gamma memory, and adaptive neuro-fuzzy inference system (ANFIS), have been employed for forecasting hydrological phenomena, which are highly nonlinear in nature and are continuously changing with time (Choudhury & Roy 2015). The MLP has the potential to match flow parameters almost with full precision but does not comply with continuity equation norms in the mass balance of river flow.

Since the last decade, artificial intelligence has promised many advantages in data feeding in computers directly without any simplification, and it has been utilized in various applications in hydrology (Kerh & Lee 2006). Kerh & Lee (2006) projected and forecasted the river flow for flood using the information at upstream stations, and Karunanithi et al. (1994) predicted the streamflow using an ANN model. The authors also found that ANNs have better performance in comparison with an analytic nonlinear power model. Furthermore, Tayfur & Guldal (2006) employed ANN models for monthly streamflow forecasting and observed that neural network methods perform better than statistical methods. Tingsanchali & Keokhumcheng (2006) also discussed the ANN-based literature to forecast river flows ranging from 1 day to 1 year in the River of Thailand using only past flow observations.

Mostly, ANN-based flow models in river flow studies purposely comply with matching input–output sequences to forecast the flow, as in the case of the study of Kerh & Lee (2006), where they used a routing-type ANN in river outreaches. Routing-type ANN models should observe continuity norms to satisfy the mass balance in a river reach/river system. Nonetheless, the routing-type ANN models that are available in the literature do not consider storage variation and, hence, may not fully satisfy the law of conservation of mass in river reaches while issuing a forecast. ANN models for river flow studies commonly employ static MLPs to predict the flow variables. Maier & Dandy (2000) reviewed 43 studies that dealt with the prediction and forecasting of water resource-related variables and reported that all but only two studies used static MLP networks. Moreover, flood flow in river reaches is highly nonlinear and time-varying and is characterized by changes in channel storage and flow rates at bounding sections over time (Thirumalaiah & Deo 1998). An MLP is a feed-forward and static network with no recursion or memory elements; mapping in the MLP being instantaneous, it cannot recognize and integrate temporal variations in input sequences (Giles et al. 1997). Thus, the applications of the static MLP in forecasting flood flows, which is a time-varying process, may not be preferable if the accuracy and timeliness of forecasts must be catered to. Shukla et al. (2022) studied that depth–discharge forecasting requires actual field data and has a significant effect on model performances. They also emphasized the use of the actual length of training data by using multi-artificial intelligence models. They found that an ANFIS gives better results than simple ANN models. They also formulated depth–discharge relations by using various ANN techniques coupled with wavelet.

Given that the state of a time-dependent process is a function of its previous states, ANN models that can store and utilize past information are found to be more efficient in analyzing these processes. Broadly, memory by feed-forward delays and memory by feedback delays classify ways to incorporate short-term memory in the neural network system. Lang et al. (1990) classified memory by feed-forward delays as a TDNN, which involves the explicit inclusion of delays in the neural network system. In the case of memory by feedback delays, recurrent units hold a trace of the past input and self-recurrent network of Elman & Zipser (1988) and Jordan (1986). Static MLP-type ANN models with a sliding window approach, which are equivalent to a TDNN, have been utilized in modeling temporal variations (Jain et al. 1999; Tokar & Johnson 1999; Coulibaly et al. 2000a, 2000b). In this case, a fixed number of past information selected by the user is presented as inputs to the MLP network.

As fixed numbers of past samples are used as inputs, the network possesses a fixed or static memory. Most of the ANN-based flood forecasting models available in the literature are capable of providing forecasts at a single location and do not possess forecast updating capability (Agarwal et al. 2021b). This restricts the applicability of these models in real-time situations (Coulibaly et al. 2001). A TDNN creates memory by delaying the input sequences, and its application in river flow studies is available in the work of Coulibaly et al. (2000b). A limitation of TDNN and MLP with memory is that with memory depth being fixed and pre-decided, the selected memory depth does not match the temporal characteristics of river flow studies, thereby giving poor results. Hence, transforming a river reach on a large scale and specifying calculating characteristics for river flow even with a small catchment now depend on the geomorphological river reach characteristics (Varentsova et al. 2020). Singh et al. (2021) categorized the flood plain into critical zones and developed it by considering topographical aspects. Therefore, topography also plays a critical role in flood flow modeling. On the basis of the literature study, the following broad objectives are attempted in the present study: to formulate an ANN-based flow forecasting model for a river system incorporating storage characteristics, and to evaluate the performances of different memory-based ANN models along with the use of wavelet technique in forecasting flows incorporating storage variations in river reaches.

A MIMO-1 ANN has (n+1) output nodes of which the rth nodes provide the real flow forecast for the rth bounding section while the remaining nodes are fixed to zero-rate flow as forecasts for other sections.

The MISO models rely on matching flow usually at one downstream section only using a number of upstream flows. Equations (1) and (2) describe the fractional-storage change in the MIMO model form having different functional values on each input. Fractional-storage change in MIMO-1 form combined together may depict the actual storage variation since MIMO-1 models are complementary to each other and sum together, depicting actual storage change.

where c₁ and c₃ stand for the Muskingum model parameters, whereas α and β are upstream hydrograph evolution parameters. Equivalent flow is split into two complementary parts for upstream and downstream flows having storage, and in Equation (6) and ⁠, in Equation (7).

It may be mentioned that the MIMO-1 models, when trained using flow variables, only implicitly learn storage change characteristics for the reach during a period. To make the networks learn the storage variation characteristics explicitly from the data set rate of storage change data computed using instantaneous inflow and outflow rates and instantaneous average flow depth data computed using known concurrent flow depths at the bounding sections in the reach may be used along with the flow rate/flow depth data in training the network.

Based on the literature survey, Agarwal et al. (2021a) computed storage rate change implicitly. With this in context, storage rate change has been computed explicitly in this work, satisfying the continuity norm. MIMO-1 is mapping all outputs to zero except for the forecasting sections. MIMO-1 ANN, learns characteristic fractional-storage changes providing a forecast for flow variables at a single section only.

As the MIMO-1 is mapping all outputs to zero except for the forecasting sections, two MIMO-1 networks mapping upstream and downstream future flow, respectively, for a single reach may be summed, obtaining a single MIMO-2 network that has the same (two) inputs as the MIMO-1 network and two outputs representing real flow forecast for the upstream and downstream sections, respectively.

It is evident that a MIMO-2 network explicitly learns flow variations in the upstream and downstream sections, and as indicated by the continuity principle, the networks also implicitly learn the actual storage variation in the reach during the period. It may be pointed out that a MIMO-2 network learns actual storage variation, whereas a MIMO-1 learns a fractional-storage variation such that (N+1) fractional-storage variations learned by the MIMO-1 ANNs always sum to the actual variation learned by the MIMO-2 network.

In the case of a river system having (N+1) flow series, all the series can be forecasted by training a single MIMO-2 network with concurrent (N+1) inputs and concurrent (N+1) outputs separated by a time interval Δt.

Here, μ is a memory parameter representing memory resolution. The generating kernels (t) are the integrands of the gamma function (Γ(x)≡dt), and thus the memory is called the gamma memory.

The gamma memory has an adjustable memory parameter μ that can be optimized during the training period selecting the best possible memory depth for a time-varying pattern. In the case of a focused MGMNN model, a number of gamma memories, one at each node in the input layer, are used, and, thus, the model can process multiple time-varying inputs using different memory depths as in Shukla et al. (2022). In this study, the applicability of gamma neural networks having adaptive memory has been investigated since gamma memory incorporates both feedback and feed-forward memory parameters, and while training, it can adjust its memory resolution for a time-varying pattern. The MGMNN possesses an adaptive memory to select the best duration in the past that the network should remember for giving the best result.

Most of the flood forecasting models were developed using discharge and/or depth variables only. Although during the flood flow, storage in river reaches continuously changes along with the bounding section flow rates, these storage variables are required to be included in the model to accurately compute desired flow rates and the damages caused. In the present study, different categories of the model using the artificial intelligence technique and incorporating storage change-related variables have been developed and applied for real-life river system flood flow analysis (Equations (13a)–(13d)).

Studies have been conducted to capture variation in one variable at a time and also to capture variation in all the variables simultaneously (Equations (12a) and (12c)). Both the models agree with the continuity equation. The application of the widely used MISO model that does not explicitly agree with the continuity equation has been covered in the study.

A wavelet neural network (WNN) transform is a new signal analysis and processing technology that has emerged in the past one or two decades. WNN is a new type of feed-forward network combining wavelet theory and ANN. In forward transmission, the input signal is processed layer by layer from the input layer through the hidden layer to the output layer, and the error is backpropagated. Unlike Back Propagation (BP) neural networks, in WNNs, the wavelet basis function is used instead of the Sigmoid function as the excitation function.

The reduction and translation variables make WNNs have both the characteristics of the time–frequency localization of the wavelet transform and the self-learning characteristics of the neural network, and the network has a strong approximation ability and also improves generalization (Choudhury & Roy 2015).

When modeling natural systems such as floods, the problem of noisiness and nonlinearity of real data often occurs. This problem is partly solved with the help of such filtration. Data filtering always arises when it is necessary to separate the signal from the noise that distorts it. Here, the signal is ordered as a set of numerical information about the process, i.e., time-series; the signal source can be any change ratio system. The purpose of data filtering is to restore change of the original signal against the background of interference or the definition of a proper signal and, therefore, the reduction of data unreliability. There are many filtering methods, and many of them can be applied in the creation of information systems of analysis and forecasting, but the one of wavelet is considered the most promising transformation. One of the urgent tasks of digital processing signals is cleaning the signal from noise.

The task of filtering is to maximize the possible reduction of the second term in (15). Filtering a signal using wavelet transformations is performed in the following four stages:

• The decomposition of the signal in terms of the wavelet basis.

• The choice of the noise threshold value for each decomposition level.

• The threshold filtering of detail coefficients.

• Signal recovery.

For the wavelet transform, two indicators are critical: the order of the wavelet and the depth of decomposition. Wavelet depth—an indicator that determines the smoothed sharpness—the higher this value, the smoother the reconstructed signal. It should be noted that smoother wavelets create a smoother signal approximation and vice versa (wavelets with a small depth of the peaks of the approximated functions). The depth of decomposition determines the scale of shifted parts: the more significant this value, the larger signal changes will be discarded, i.e., the depth of decomposition can be called the degree of signal chopping.

Wavelet analysis employs standard wavelets. The fact that wavelets are spatially localized is their most significant advantage. Data compression, data processing, and the solution of differential equations all use these algorithms.

Past recorded concurrent flow information for five gauging sites—Rocky Mount, Hilliardston, Enfield, Tarboro, and Greenville—is used in the present study to forecast flow at each of the gauging sites. Concurrent flow records for the gauging sites of the basin starting from 29 July 2004 to 20 October 2004 with an interval of 15 min were collected from the United States Geological Survey (USGS) instantaneous streamflow Data Archive (http://ida.water.usgs.gov/ida/available_records.cfm). Instantaneous streamflow values are continuous streamflow values recorded at intervals of 15 min and are converted into 2-h intervals.

In the case of delta S, calculated on the basis of the average storage rate, assuming that initially the storage is zero at the beginning and multiplying the delta S with the time interval gap, the total amount of water can be obtained.

These values imply that there is a moderate-to-very low chance of occurrence of flood during the stipulated time interval. Since the storage values calculated do not show much variation for the given time in Peak Flow Criteria (PFC) calculation. Tables 3 and 4 exhibit the value of RMSE, coefficient of efficiency (CE), and coefficient of correlation (R) before and after wavelet decomposition. The results verify that there is little improvement in these values updated due to the pre-processing of the input data set. Wavelet has removed noise from the input data set, thereby making it more precise to obtain feed-forward from ANNs. Tables 3 and 4 forecast and validate various statistical criteria in the MIMO-1 and MIMO-2 forms, whereas Tables 5 and 6 show that the values obtained from statistical criteria showcase the differences in value in inflow and outflow stations, respectively. Enfield, Hilliardston, and Rocky Mount are considered various upstream flows, whereas Tarboro is regarded as an outflow station.

Weights and parameters are assigned to the optimized WNN, and from July to October 2004, forecast samples were applied. In addition, the daily hourly flow from July to October 2004 was predicted. The prediction results are compared with the WNN and BP network prediction values. The forecast test uses three methods, namely MISO, MIMO-1, and MIMO-2. Tables 5 and 6 show the comparative results, and it can be observed that the prediction result of the WNN is better than that of the simple MLP network and that after the gamma algorithm optimization (Liu et al. 2021), the prediction accuracy has greatly been enhanced. In the wavelet model optimized by the gamma algorithm, in July, August, and September 2004, the reported value was almost consistent with the actual measured value, and the relative error was less than the results obtained in the simple MLP which achieved a very accurate forecast accuracy. The dotted lines show forecasted flow discharge, and the regular lines in the graph predict the observed or desired flow discharge in feet cube per second.

All relevant data are available from https://waterdata.usgs.gov/nc/nwis/current/?type=flow.

The authors declare there is no conflict.