A Forecast Heuristic of Back Propagation Neural Network and Particle Swarm Optimization for Annual Runoff Based on Sunspot Number

Abstract

Runoff prediction is of great importance to water utilization and water-project regulation. Although sun activity has been considered an important factor in runoff, little modeling has been constructed. This study put forward a forecast heuristic combining back propagation neural network (BPNN) and particle swarm optimization (PSO) for annual runoff based on sunspot number and applied it to the Yellow River of China for the period 1956–2016 and assessed the contribution of the sunspot number by placing sole BPNN modeling on the time series as a contrast. First, the heuristic is made up of BPNN calibration and PSO optimization: (1) we use historical data to calibrate BPNN models and obtain a prediction of the sunspot number for training and testing stages; (2) we use the PSO to minimize the difference between the predicted runoff of both BPNN and a linear equation for forecasting stage. Second, the application offers interesting findings: (1) while BPNN calibration obtains first-class forecasting with the ratio >85% with <20% absolute error in training and testing stages, the PSO can achieve similar performance in the forecasting stage; (2) the heuristic can achieve better prediction in years with a lower sunspot number; (3) besides the influence of the sun activity, atmospheric circulation, water usage, and water-project regulation do play important roles on the measured or natural runoff to some extent. This study could provide useful insights into further forecasting of measured and natural runoff under this forecast heuristic in the world.

1. Introduction

Runoff forecast is key to the optimal usage of water resources, sediment regulation and water-related disaster mitigation [1,2,3,4]. However, annual runoff time series are currently being reshaped by natural and human factors, like climate change [5,6], solar activity [7], dam constructions [8], and water intake [9]. Currently, although complicated [10], there is ample observational evidence that shows that annual runoff has a close relationship with sunspot number [5,7,10,11], an arbitrary numerical value that is used to describe the sun’s spottiness.

On one hand, much important research [5,7,10,11,12,13,14,15,16,17,18,19,20] shows the direct influence of solar variability on runoff. First, a statistical relationship between solar activities (the sunspot number) and the precipitation for some time was confirmed in the Beijing area of China [16]. This shows that annual precipitation is closely related to the variation of sunspot numbers, and that solar activity probably plays an important influence on the precipitation.

Second, a clear relationship between solar activities and natural runoff was found [10,15,20]. Based on the 1901–2020 discharges, precipitation, and air temperature of three Central European rivers in Poland, wavelet transform analysis highlighted the frequency–time distribution of the coherences between solar and discharge variability and led to the nonlinear relationship for most links between solar activity and discharges [20].

Third, after disentangling the contributions made by system feedback, natural variability and other forcings on a wide range of time scales, the potential solar forcing mechanism interaction with system feedback or variability may be stronger than the solar forcing itself [12]. However, the recent rise in global mean surface air temperature mostly can be explained by greenhouse forcing rather than solar forcing [14]. Moreover, although global average temperature changes were small, regional temperatures like the colder temperatures over the Northern Hemisphere continents could be quite large due to a forced shift toward the low index state of the Arctic Oscillation/North Atlantic Oscillation as solar irradiance decreases [13]. And, the time shift of wet and dry cycles in different regions is influenced by general oceanic and atmospheric circulation [21].

Fourth, the periods of sunspot indicate a strong relationship between solar activity and flood/drought disasters [7,22]. Solar activity behaves as a factor of astronomy [23] beyond the hydrological cycle system and plays a cyclical role on climate. Constantly affected by natural geographic characters, atmospheric circulation generates seasonal climate change and provides basic random foundations for different scales of weather system activities [23]. A study on annual discharge time series of selected large rivers in the world for wet and dry periods [19] found that extreme occurrence has 28–29 year and 20–22 year cycles. The temporal shift for discharge extremes relies on the longitude and latitude. A total of 18 major European rivers during the period 1850–1997 were analyzed and it was found that dry cycles are about 13.5 and 28–29 years long [21]. Similar patterns were found in the Amazon [18] and Congo [17].

Fifth, observations during the past 130 years show a close association between long-term variations in Earth’s temperature and variations in the solar cycle length [5]. Further, this fact can lead to a real physical mechanism to predict long-term climate changes (possibly the runoff inside a watershed) by appropriate modeling of the sun’s dynamics. The prerequisite for such prediction is to make a firm conclusion regarding anthropogenic changes.

In a word, runoff is related to solar activity, general oceanic and atmospheric circulation, and anthropogenic changes. The relationship between runoff and solar activity exists and is nonlinear. In other words, solar activity exerts a general influence on runoff, while the spatiotemporal variability is influenced by general oceanic and atmospheric circulation and anthropogenic changes, indicated by temperature increase and evapotranspiration.

On the other hand, as summarized [24], there is a wide range of techniques used for annual or monthly discharge forecasting, including time series modeling methods, statistical approaches, artificial intelligence (AI) methods, and hybrid models. Time series modeling methods include the AR model, MA model and ARIMA model with the assumption of stationarity and specific distribution [8]. In order to better consider the nonlinear, non-stationary and non-normal information, artificial neural networks (ANN), support vector machines (SVM), chaotic theory models and fuzzy predictive models have emerged [8]. After comparison among the Back Propagation Neural Network (BPNN) model, group method of data handling (GMDH) model, and autoregressive integrated moving average (ARIMA) model, it is found that BPNN can achieve better prediction results [9]. The comparison between multiple linear regression (MLR) and BPNN [25] shows that BPNN can use its self-learning and self-adaptive ability to realize complex non-linear mapping to better solve nonlinear problems [9].

Moreover, PSO has become widely applied due to its simplicity and its effectiveness with low computational cost [26,27]. PSO is a swarm-based intelligence algorithm that imitates the social behavior of a flock of birds finding a food source or a school of fish protecting themselves from a predator. The movement of each particle is coordinated by a velocity that has both magnitude and direction. Each particle’s position at any instance of time is influenced by its best position and the position of the best particle in a problem space. The performance of a particle is measured by a fitness value, which is problem-specific.

The combination of BPNN and PSO has been used in other fields [25,28]. The most common way of the combination is to use PSO to optimize the initial weights and thresholds of BP network neurons [25,28]. Here, in contrast, we just use the BPNN model as a constraint condition of the PSO.

When it comes to the relationship between runoff and sunspot number in the Yellow River, China, as the sun activity gets stronger abruptly in the exact or following year, runoff will become larger than ones in other years [29]. Moreover, according to historical data, flood years mostly occur in three phases in the Yellow River of China: solar maximum year, years after solar maximum year and solar minimum year. In addition, there is an alternate roughly 10-year cycle of flood/drought in the Yellow River of China [29]. A similar pattern was found in the Yangtze River [30], and it is possible to improve the accuracy of the monthly hydrology forecasting by using the sunspot number.

Based on the 1700–2003 records of annual natural runoff in the Yellow River of China and the sunspot relative number, one wavelet analysis method [10] shows the correlation between sunspot relative number and runoff may be lower due to other factors like precipitation, land cover and land use, soil, vegetation, and others. On an 11-year scale, there exists an obvious correlation in some periods, like 1711–1767 (0.86). As stated [10], since the effects of solar activities on runoff are related to a number of complex physical processes, a more robust method is desired in future studies. Moreover, based on previous studies on the Yellow River, it seems that solar activity plays a key influence on its runoff generation [29].

However, there has been little research on modeling the relationship between the sunspot number and runoff. Our main contributions in this paper are as follows:

• Formulating a BPNN model for annual runoff in training and testing stages;
• Designing a heuristic that uses PSO to solve the forecasting problem based on the proposed BPNN model in the forecasting stage.

This paper is organized as follows: Section 2 describes the design of the forecasting heuristic based on the BPNN and PSO. Section 3 describes the study region and datasets involved. Section 4 presents results and discussion based on the performance of our heuristic. Lastly, Section 5 concludes the paper, summarizing the results and providing suggestions for further work.

2. Forecast Heuristic of BPNN and PSO

2.1. General Framework

The runoff forecast problem can be formularized as a PSO-based problem with the BPNN model as one constraint condition. First, for one site in $S=\{1,\dots ,\mathrm{e}\}$ with e as the quantity of the forecast sites, we use the historical data of time series for training and testing a BPNN model, where $T_{j,n}=\{1,\dots ,j,1,\dots ,n\}$ represents the data with j as the quantity of time series and n as the time. Second, using PSO to calculate the targeted index, where $I_m=\{1,\dots ,m\}$ with m as the time.

We put forward a forecasting heuristic for dynamically carrying out the annual runoff forecasting based on the sunspot number (Figure 1 and Algorithm 1). First, we either directly adopt the predicted sunspot number or carry out its forecasting by artificial intelligence models. Here, for simplicity, we adopt the officially predicted values of the sunspot number. Second, we obtain a fully trained BPNN model for the relationship between the annual sunspot number and the annual runoff based on historical data. One unique trained BPNN model input is the difference between one annual sunspot number and its corresponding runoff value. Third, by particle swarm optimization technique, the heuristic minimizes the difference between one predicted runoff by the trained BPNN model and the other predicted runoff by the linear relationship among the unique input of the trained BPNN model, the sunspot number, and the runoff.

The problem can be stated as: “Find a difference such that the predicted runoff by the trained BPNN model equals the predicted runoff by the linear equation between the historical runoff and sunspot number.”

We use the R², maximum error, minimum error, average absolute error, the ratio with <20% error, and Kling Gupta efficiency for evaluating the performance of BPNN models in the training and test stages and forecasting heuristic through PSO in the forecasting stage. As R² is larger than 0.8, the modeling performs well. The closer the Kling Gupta efficiency goes to 1, the better the modeling is. We used the SPSS modeler and MATLAB to carry out the modeling and analysis.

Here, we will provide a brief description of the forecast heuristic of BPNN and PSO.

$$Minimize{Ob}_t=\left|R_{B,t}-R_{L,t}\right|$$

(1)

Subject to:

$$R_{L,t}=S_{P,t}-D_t$$

(2)

$$R_{B,t}=f_B(S_{P,t},D_t)$$

(3)

where:

${Ob}_t$: the objective, the difference between the runoff predicted by BPNN and the runoff by the linear relationship;

$R_{B,t}$: the runoff predicted by BPNN at time t;

$R_{L,t}$: the runoff by the linear relationship at time t;

$D_t$: the difference between the runoff and the corresponding sunspot number at time t;

$f_B\left(\right)$: the trained BPNN model based on training data;

$S_{P,t}$: the predicted sunspot number at time t on our own., or by NASA and NOAA [31];

Algorithm 1: Forecast heuristic

1: Calculate the difference ${Ob}_t$ between the runoff predicted by BPNN and the runoff by the linear relationship

2: Train and test the BPNN model with historical data, including the sunspot number and runoff and so on

3: Set an initial value $D_t$ as one input of the trained BPNN model, other inputs could be predicted either on your own, or by other official organizations

4: Compute $R_{B,t}$ based on $D_t$ with the trained BPNN model

5: Minimize ${Ob}_t$ through the PSO algorithm

6: repeat

7:  for all the targeted index $I_m=\{1,\dots ,m\}$ do

8:   Calculate the $R_{L,t}$ step by step

9:   Calculate the absolute value of ${Ob}_t$ step by step

10:  Update the $D_t$ step by step

11:  end for

12:  Compute PSO for $I_m=\{1,\dots ,m\}$

13: until there are no forecasting tasks

2.2. Back Propagation Neural Network

Machine learning is a subfield of artificial intelligence. Under machine learning, there exists unsupervised learning, supervised learning, deep learning, and so on. Artificial neural networks are representative of deep learning. Moreover, BPNN is one kind of ANN.

BPNN is a typical multilayer ANN on the basis of error backpropagation [32]. It applies the slope reduction algorithm to minimize error. It is made up of three layers, namely the input layer, hidden layer, and output layer (Figure 2). While multiple inputs are included in the input layer, one output is included in the output layer. In the hidden layer, multiple neurons bear no direct contact with the outside world, but express the relationship between the input layer and output layer.

A conventional three-layer BPNN is used to establish the prediction model of the annual runoff series in this paper. Tan-sigmoid is the transfer function between output and hidden layers, and the nonlinear Levenberg–Marquardt (LM) algorithm is the training function of BPNN. The maximum number of iterations is 100. The number of input layer nodes is the same as the number of input variables. The optimal value is determined by continuously adjusting the number of hidden layer neurons in the range of 2 to 13. The original datasets fall into training samples (70%) and testing samples (30%).

After trial and error, two BPNN models were trained and tested with inputs, including the difference of measured or actual runoff and sunspot number at the years ($D_t$, $D_{t-1}$, $D_{t-2}$), sunspot number at the years ($S_{P,t-1}$, $S_{P,t-2}$). Both have 4 neuros in the hidden layer.

The mathematical principle of the BPNN model is as follows [32]:

$$y_i=\sum _{j=0}^m{\omega }_{ij}{x}_j+{\beta }_j$$

(4)

where $x_j$ is input neuron and $j\in (0,m)$, $m$ is the number of input neurons, ${\omega }_{ij}$ is weight of the jth neuron in the input layer corresponding to the jth neuron in the hidden layer, ${\beta }_j$ is bias-related weight of hidden neurons, $y_i$ is input of the hidden layer node (i = 0, 1, …, n), and n the number of neurons in the hidden layer. Tan-sigmoid is the transfer function between the layer output and the hidden layer, and its form is as follows [32]:

$$l_i={\displaystyle \frac{1}{1+e^{-y_i}}}$$

(5)

The output layer is estimated by the following equation [32]:

$$g_k=\sum _{i=0}^n{\omega }_{ik}{l}_i+{\beta }_k$$

(6)

$$O=\mathrm{m}\mathrm{a}\mathrm{x}(0,g_k)$$

(7)

Among them, $g_k$ and O represent input and output values of the output layer, respectively.

The formulas above are the principles of the feedforward propagation mode of the BPNN model. In the process of cyclic simulation, errors generated by the system are collected and returned to the output value (Algorithm 2). By adjusting the weights and thresholds of neurons, network parameters corresponding to the minimum error are determined to generate an ANN system that can simulate the original problem.

Algorithm 2: Back Propagation Neural Network

Input: Training and Testing Set $T_{j,n}$

Learning rate η

1: Data normalization (here, data preprocessing needs to be carried out according to the actual situation of the data and algorithm requirements)

2: Create a network

3: Training Network

Repeat for $T_{j,n}$

3.1: Positive propagation

3.2: Back propagation

Until the end condition is met

4: Using the network

5: Data denormalization

Output: Trained BP neural network

6: Test the trained network

2.3. Particle Swarm Optimization

A brief description of the PSO algorithm is as follows [26,27]:

$$v_a^{k+1}=\omega v_a^k+c_{1}ran{d}_1\times \left(pbes{t}_a-x_a^k\right)+c_{2}ran{d}_2\times \left(gbest-x_a^k\right)$$

(8)

$$x_a^{k+1}=x_a^k+v_a^{k+1}$$

(9)

where:

$v_a^k$: velocity of particle a at iteration k;

$v_a^{k+1}$: velocity of particle a at iteration k + 1;

$\omega$: inertia weight;

$c_b$: acceleration coefficients; b = 1, 2;

$ran{d}_b$: random number between 0 and 1;

$x_a^k$: current position of particle a at iteration k;

$pbes{t}_a$: best position of particle a;

$gbest$: position of best particle in a population;

$x_a^{k+1}$: position of the particle a at iteration k + 1;

In PSO, the population is the number of particles in a problem space (Algorithm 3). Particles are initialized randomly. Each particle’s fitness value will be evaluated by a fitness function to be optimized in each generation. Each particle knows its best position pbest and the best position so far among the entire group of particles gbest. The pbest of a particle is the best result (fitness values) at a given time reached by the particle, whereas gbest is the best particle in terms of fitness in an entire population. In each generation, the velocity and the position of particles will be updated as in Equations (8) and (9), respectively.

Algorithm 3: PSO algorithm

Set particle dimension as equal to the size of Learning rate η

1: Data normalization (here, data preprocessing needs to be carried out according to the actual situation of the data and algorithm requirements)

2: Create a network

3: Training Network

Repeat for $T_{j,n}$

3.1: Positive propagation

3.2: Back propagation

Until the end condition is met

4: Using the network

5: Data denormalization

Output: Trained BP neural network

6: Test the trained network

3. Application in the Yellow River, China

3.1. Study Region

Wanjiazhai Reservoir [23] is located on the main stream of the Yellow River (see Figure 3). Its watershed covers 39.48 × 10⁴ km², with 8.96 × 10⁸ m³ storage capacity at normal water level, 108 × 10⁴ kW total installed capacity of the power station, and 27.5 × 10⁸ kWh annual design power generation. Its average annual runoff is 249 × 10⁸ m³, and its capacity of annual design water supply is 14 × 10⁸ m³. Its maximum discharge capacity is 21,100 m³/s.

The Liujiaxia Reservoir [29] in the upper reaches of the Yellow River was put into operation in 1968, with a total storage capacity of 6.15 billion cubic meters and a regulating storage capacity of 4.2 billion cubic meters. The Longyangxia Reservoir [29] was put into operation in 1986, with a total storage capacity of 26.6 billion cubic meters and a regulating storage capacity of 19.36 billion cubic meters.

The most important function of Wanjiazhai Reservoir is to provide an annual water supply of 1.4 billion cubic meters, including 200 million cubic meters to Jungar Banner, Inner Mongolia and 1.2 billion cubic meters to Shanxi Province. The installed capacity of the hydropower station is 1.08 million kW, with a designed annual power generation of 2.75 billion kW·h.

3.2. Datasets

Two datasets are involved in this study. One is the actual measured annual runoff from 1956 to 2016, and the corresponding natural runoff has been officially determined by the Ministry of Water Resources, China. Each year, the hydrological section of the Ministry of Water Resources of China uses all available data on hydrological, precipitation, water project regulation records, and others to generate the natural runoff in important crosssections, which are official products.

The other data are the actual measured annual sunspot number from 1950 to 2016. The predicted sunspot number is obtained either based on their time series on our own by AI techniques or obtained by The Solar Cycle 25 Prediction Panel, an international group of experts co-sponsored by NASA and NOAA [31].

3.3. Study Design under the Heuristic

First, we use the historical 1956–2006 datasets of sunspot numbers to train and test one BPNN model (see Algorithm 2) for measured annual runoff and the corresponding natural runoff. One unique input of the BPNN model is the difference between the annual runoff and the sunspot number.

Second, under the heuristic (see Algorithm 1), we solve the optimization to make the forecasting of annual runoff during the period 2007–2016 based on the predicted sunspot number by using the PSO (see Algorithm 3). Sunspot future prediction can be carried out using AI of Solar cycle 25 [31], but in this study, we did not use it.

Third, we carried out time series modeling by BPNN without considering the sunspot number for measured and natural runoff as a contrast. We assume that the runoff at time t is related to the runoff data at time t − 1, t − 2, t − 3, and t − 4.

4. Results and Discussion

4.1. Training, Testing, and Forecasting

As can be seen from

Table 1. Performance of only BPNN models without sunspot number.

Item	Measured Runoff			Natural Runoff
	Training	Testing	Forecasting	Training	Testing	Forecasting
The ratio with <20% error	72.4%	50.0%	78.3%	56.3%	83.7%	78.1%
Average absolute error	27.6%	22.0%	21.7%	16.3%	16.3%	21.9%
Minimum error	0.9%	1.4%	1.4%	0.7%	0.3%	0.9%
Maximum error	110.6%	63.7%	57.6%	42.9%	38.7%	51.7%
R2	0.046	0.227	0.003	0.033	0.072	0.157
Kling Gupta efficiency	0.055	0.364	−0.068	-0.044	0.204	−0.466

, the only BPNN models do not perform well with R² quite small as a contrast. As can be seen from

Table 2. Performance of forecasting heuristic through BPNN-PSO with the sunspot number.

Item	Measured Runoff			Natural Runoff
	Training	Testing	Forecasting	Training	Testing	Forecasting
The ratio with <20% error	90.5%	90.7%	89.3%	93.3%	93.5%	92.3%
Average absolute error	9.6%	9.3%	9.7%	6.7%	6.5%	6.4%
Minimum error	0.1%	2.4%	3.6%	0.5%	1.3%	1.6%
Maximum error	34.5%	26.2%	26.8%	26.9%	21.1%	21.7%
R2	0.887	0.837	0.831	0.887	0.875	0.864
Kling Gupta efficiency	0.912	0.735	0.802	0.881	0.764	0.885

, both BPNN models perform well with R² larger than 0.80, implying that the sun activity bears a major influence on the runoff generation. Generally, the BPNN model for natural runoff works a little bit better than the one for measured runoff. For example, the average absolute error increases from 6.7% in the BPNN model of natural runoff to 9.6% in the BPNN model of measured runoff in the training stage. This fact can be explained by only considering the influence of the sun activity, and not considering human factors to carry out the forecasting.

According Standard for hydrological information and hydrological forecasting (GB/T 22482-2008) [33], as the ratio with <20% absolute error is larger than 85%, the forecasting is considered first-class. In a word, our forecasting (see Table 2) meets the first-class criterion.

This study (see Figure 4) shows that, in general, the absolute error is larger in years with high sunspot numbers than in years with low sunspot numbers for both models. This may be related to the fact that as the sun’s activity gets stronger, the runoff gets larger in nearby years [29]. As the runoff gets larger, the difficulty of forecasting gets larger to some extent. This may imply that although the sun’s activity bears a strong influence on the generation of runoff, other activities like atmospheric circulation [29] can play an important role in the runoff.

Another important finding (see Figure 4) is that while the absolute error for natural runoff in each sunspot cycle is steady, the absolute error for measured runoff gets larger step by step. This probably is related to the fact that the human factor was becoming increasingly more important since water abstraction in the Yellow River became larger and larger during that period [2].

Here, for simplicity, we use prediction of sunspot numbers from NASA and NOAA and only make one-year-forward forecasting of measured and natural runoff. Based on the trained BPNN models and predictions of sunspot numbers during the period 2007–2016, Algorithm 3 was used to solve the optimization problem. The results show that the heuristic performs well for both measured and natural runoff (see Table 2).

The absolute error ratio during the period 2006–2016 reached the peak right one year after the peak of the sunspot number in 2014 (see Figure 4). In the sunspot cycle, as the sun activity gets on a high platform, the absolute error for measured runoff experienced a fluctuating-increase period. This confirms that as the sun activity gets strong abruptly, in the exact or following year runoff will become larger than the one in other years [29].

Runoff in most places of the whole world is influenced by not just solar activity, but also other factors, such as general oceanic and atmospheric circulation, and anthropogenic changes [23]. This is because the historical data of the Yellow River show that the sun activity can likely exert a strong effect on the runoff generation [10,29]. This is likely the reason why this approach, only considering the sunspot number, works well for the Yellow River of China. Meanwhile, this may be the largest limitation of this approach, only considering the sunspot number to make prediction of runoff, which is quite insufficient in most cases.

4.2. Forecasting Performs Best during the Period 2004–2014 and Extreme Analysis

The absolute error ratio during the period 1956–2016 had the lowest values during the period 2004–2014 (see Figure 4), one sunspot cycle. Meanwhile, the sunspot number also experienced the lowest values in this sunspot cycle. This implies that this forecasting heuristic works well for low values of sunspot numbers. Moreover, as the sunspot number is less than 50, this forecasting heuristic has great performance: 67% of years have less than 10% absolute error for measured runoff and 90% of years have less than 10% absolute error for natural runoff.

Figure 5 shows that during the period 2004–2014, water usage represented by the difference between natural and measured runoff experienced the fiercest fluctuation with the peak in 2005 and reduced sharply to 2006. Meanwhile, the absolute error for measured runoff reached the lowest number in 2005 between 1956 and 2016. This may imply that as the sunspot number gets small and water usage varies sharply, this forecasting heuristic works well for both measured and natural runoff.

Obviously, water usage has two significant stages in the operation of Longyangxia Reservoir in 1986 (see Figure 5) [29]. Before 1986, although Liujiaxia Reservoir’s operation in 1968 [29] generated a notable water usage increase in 1969, the water usage maintained a steady increase with small fluctuations. After 1986, the water usage fluctuated quite fiercely. This suggests that the regulation of Longyangxia Reservoir played a significant role in the fluctuation of water usage.

The two extremes are interesting. On the one hand, as the measured annual runoff reached the smallest value in 1997, its absolute error reached its peak. This implies that as the Yellow River became dry, the forecasting heuristic performed badly to some extent. On the other hand, the smallest water usage goes to 2002 [10], while its absolute error ratio for natural runoff reaches the peak. The larger the anthropogenic interference, the larger the difficulty of modeling by sunspot number. It is easy to understand this since our modeling only uses the sunspot number (one natural factor). As the anthropogenic interference becomes stronger, this modeling performs badly, which is normal. This is because runoff is influenced by solar activity, general oceanic and atmospheric circulation, and anthropogenic changes [23]. This implies that as water usage became small, the forecasting heuristic performed badly to some extent. The above two facts bear further potential for modeling improvement of the forecasting heuristic.

5. Conclusions

It is important to model the relationship between runoff and the sunspot number. Inspired by related researchers, we put forward a forecast heuristic by combining the BPNN and PSO for measured and natural runoff forecasting. First, it uses historical data to calibrate BPNN models and obtain predictions of sunspot numbers. It uses the PSO to minimize the difference between the predicted runoff by BPNN and a linear equation.

This forecasting heuristic was applied in the Wanjiazhai reservoir’s 1956–2016 datasets. The application offers interesting findings: (1) While BPNN calibration obtains first-class forecasting with the ratio >85% with <20% absolute error in the training and testing stages, the PSO can achieve similar performance in the forecasting stage with the calibrated BPNN model as a constrain condition; (2) the heuristic can achieve better prediction in years with lower sunspot number; (3) besides sun activity, other factors, like atmospheric circulation, water usage, and water-project regulation, exert important influences on the measured or natural runoff to some extent.

Admittedly, the forecast heuristic is new, and requires further improvement, like considering more factors such as atmospheric circulation, water usage, and water-project regulation. For example, admittedly, land-use and land-cover (LULC) changes influence the runoff generation. In our opinion, the LULC changes could possibly be integrated into an indicator, which is involved in this modeling approach. In the future, we will study how to quantify the influence of other factors.