Multi-scale flood prediction based on GM (1,2)-fuzzy weighted Markov and wavelet analysis

Flood caused by rainstorm is one of the severe and abrupt natural disasters, which poses a great threat to the national economy and people's security. With the continuous development of urbanization in China, cultivated land and grassland around the city is gradually being replaced by construction land, resulting in the change of relationship between runoff generation and confluence (Milly et al. 2002). The underlying surface variation has caused significant hydrological changes such as increased frequency of floods, increased peaks of floods, and increased flood volume. Therefore, flood prediction model is ever more important to be established (Lee et al. 2018).

The grey model is mainly suitable for forecasting problems with less sequence data and small volatility (Huang & Shen 2013), while the Markov chain is suitable for forecasting problems with large sequence volatility (Bonakdari et al. 2019). Combining the grey model with the weighted Markov chain can determine the relationship between variables and make preliminary predictions, and then it predicts the range of random fluctuation by the Markov chain method to optimize and adjust the grey predicted result, which can enhance the preciseness of the prediction (Kumar & Jain 2010). Therefore, some scholars combined the grey theory with the weighted Markov chain to predict runoff in recent years (Li et al. 2007). However, these studies were mainly based on the autocorrelation of the runoff sequence, which is GM (1,1). Nevertheless, flood is closely related to rainfall (Qin et al. 2013), so these studies cannot reflect the correlation between flood and rainstorm, they also cannot accurately predict the flood caused by extra heavy rainstorms. In addition, there is no uniform weighted Markov method, so different weighted Markov methods have a certain influence on forecast accuracy.

The basic purpose of wavelet analysis is to determine the frequency (or scale) content of the signal and to evaluate and determine the time variation of the frequency content (Labat 2005). As for discrete wavelet analysis, it is generally used to decompose a series into sub-signals given proper wavelet and decomposition levels, and then to guide various time series analysis, such as wavelet decomposition, wavelet denoising, and wavelet-assisted hydrological forecasting (Sang 2013; Ling et al. 2017). Wavelet analysis has been widely used in catastrophe detection and periodic change analysis of hydrological variables such as precipitation and runoff (Xu et al. 2009; Zhang et al. 2019). Moreover, wavelet analysis can also be combined with neural networks and other methods to predict runoff (Kalteh 2013; Sahay & Srivastava 2014).

It is more reasonable to construct a GM (1,2) model of rainstorm-flood to predict flood than a GM (1,1) model because of the good correlation between rainfall and flood volume. Besides, a fuzzy weighted Markov method which can theoretically improve forecast accuracy is introduced to predict the residual because it has large volatility. However, whether it is the weighted Markov model or the grey model, their prediction of variables is only reflected on the macro scale. The predicted results do not timely reflect the variation trend of variables on the micro scale, nor can they reveal the uncertainty relationship between variables at the micro level. Combining wavelet analysis with GM (1,2)-fuzzy weighted Markov can effectively solve this problem.

Based on the relationship between rainfall and flood volume, it is important to reflect the change of flood volume at the macro level, but the physical mechanism between them can be better reflected by analyzing the change of flood volume at the micro level, which needs to combine grey model and wavelet analysis. Therefore, the innovation of this paper is to combine the two methods and reveal the relationship between rainfall and flood volume at the micro level, then forecasted results are reflected not only at the macro level, but also at the micro level.

In this paper, wavelet transform is used to stratify rainfall-flood volume and analyze their changes at the micro level. The GM (1,2) model is constructed to link the rainfall and flood volume on different components, and the fuzzy weighted Markov method is used to predict residual and correct the forecast value of the grey model. The flood forecasting value is obtained by superimposing the forecasted component values. In order to verify the reliability of the model, the maximum daily, 3-, 5- and 7-day flood volume of the next five floods in Zhongmu and Jiangang hydrological stations are forecasted in this paper.

From Equation (4), we can determine the basic principle of wavelet transform, that is, obtaining low-frequency or high-frequency information of signals by increasing or decreasing a, and then analyzing the signal details to realize different time scale and spatial local characteristics of the signals (Seo et al. 2017).

Deng (1989) proposed the grey system theory to study the uncertainty of the system. Grey models, including GM (1,1) and GM (1,2), were built based on grey system theory and are very effective to forecast linear and irregular data series. The meaning of the GM (1,2) model is to use a first-order differential equation to model two variables. The GM (1,2) characteristics were established based on the indicators that define the model, given the fact that these indicators are:

• (a)
  The sequence of dependent variable:

  
• (b)
  The sequence of independent variable:

  
• (c)
  The sequence obtained by first-order accumulation of data series:

  
• (d)
  The sequence consisting of the average of consecutive neighbors in the data series:

  

The Markov chain occupies an important position in modern prediction methods. The state at the next moment can be predicted by the state transition probability matrix and the current state, and the prediction has high accuracy, scientificity and adaptability (Wang et al. 2007).

The traditional Markov chain can only make qualitative predictions for variables, but there is a large error in quantitative prediction. With the development of the Markov chain, many weighted Markov methods have been developed to make quantitative predictions (Alyousifi et al. 2020). In this paper, a method combining fuzzy mathematics with weighted Markov is introduced. According to the characteristics of the data at different times, the level eigenvalue method and multi-step state transition matrix are combined to quantitatively calculate the numerical value of the predicted target.

The sequences of rainfall and flood volume are decomposed into several component sequences by wavelet analysis. Based on the periodicity analysis and correlation analysis, the corresponding components of rainfall and flood volume are correlated by establishing GM (1,2) models at the micro level. The residual sequence is obtained by comparing the simulated sequence and the actual component sequence of each GM (1,2) model. Taking the rainfall component as an independent variable, the component value of flood volume is calculated. Based on residual sequence, fuzzy weighted Markov models are constructed to predict the residual error, thus, the predicted component value is corrected. Finally, flood volume is obtained by superposition. The specific process of flood prediction by the model is shown in Figure 1.

The Jialu River originates in Xinmi City (affiliated to Zhengzhou City in China), and eventually flows into the Shaying River, which is the main tributary of the Huaihe River Basin. The Jialu River in Zhengzhou City has many tributaries, including the Jinshui River, Suoxu River, Xiong'er River, Qili River, and the Dongfeng Canal (Yang et al. 2012; Wang et al. 2018). Additionally, both Jiangang and Zhongmu hydrological stations are located in the Zhengzhou section of the Jialu River, where the former is located upstream and the latter is located downstream, as shown in Figure 2.

Daily rainfall exceeding 50 mm is defined as rainstorm (Li et al. 2016). According to the daily rainfall data of four precipitation stations including Jiangang, Changzhuang, Sizhao and Sanli, the rainstorm process corresponding to the flood process of Zhongmu hydrological station was selected and compared. It can be concluded that the rainfall of Changzhuang rainfall station has a good correlation with the flood of Zhongmu hydrological station, and 51 rainstorm floods were screened out from 1980 to 2012. Similarly, 56 rainstorm floods were screened out according to the rainfall data of Jiangang rainfall station and flow data of Jiangang hydrological station (1980–2012). On this basis, the maximum daily, 3-, 5- and 7-day rainfall and flood volume of each storm flood can be calculated.

Taking the maximum daily rainfall and maximum daily flood volume of 51 rainstorm floods in Zhongmu hydrological station as an example, the change curves of rainfall and flood volume are shown in Figure 3.

Daubechies wavelet is generally abbreviated to dbN, where N is the order of the wavelet. In the wavelet analysis of hydrological series, db6 or db4 are usually used (Roushangar et al. 2018; Guo et al. 2019). In this paper, db6 wavelet function is selected to decompose the original sequence. Moreover, the appropriate number of layers should be selected to decompose the variable in order to analyze its periodic variation. By the time the wavelet decomposition reaches the 5th layer, its high-frequency signal still fluctuates periodically and its low-frequency signal has no obvious periodic fluctuation, thus, the number of decomposition layers of wavelet transform is determined to be five layers. Five detail components and one trend component were obtained, that is d1, d2, d3, d4, d5 and a5, as shown in Figure 4. The original sequence consists of these six component sequences, and a5 reflects the overall change trend of the original sequence (Adewusi & Albedoor 2001).

It can be seen from Figure 4(a) that both P_d1 and R_d1 have fluctuation periods of quasi-2–4; (b) shows that both P_d2 and R_d2 have fluctuation periods of quasi-4–6; (c) shows that P_d3 has fluctuation periods of quasi-8–15 while R_d3 has fluctuation periods of quasi-9–14; (d) shows that P_d4 has fluctuation periods of quasi-23 while R_d4 has fluctuation periods of quasi-17; (e) shows that P_d5 has fluctuation periods of quasi-28. In addition, the fluctuation trend of the maximum daily rainfall and flood volume is basically the same in d1, d2 and d3 while the fluctuation trend of two variables is the opposite in d5; (f) shows that the sequence of the maximum daily rainfall decreased slightly and then increased slowly on the whole, while the sequence of the maximum daily flood volume shows a more obvious trend of decreasing first and then increasing on the whole. Combined with the actual situation, this may be due to the increasing urbanization of Zhengzhou City, the decrease of the permeable area and the acceleration of the confluence process in recent years (Wang et al. 2017).

In order to analyze the correlation between rainfall and flood volume in different components, the related coefficients between the components of two variables are calculated in Table 1. Moreover, the correlation coefficient represents the similarity of the changes of two variables, it also means that the correlation coefficient can measure the stability of the relationship between variables, so the magnitude of correlation coefficient can be used as a reference for reasonable construction of the GM (1,2) model.

It can be seen from Table 1 that the d1, d2, d3 and a5 of maximum daily rainfall-flood volume are closely and positively related, the d5 of two variables are closely and negatively correlated, while the correlation between them is the worst in the d4. The GM (1,2) model built by the two variables with poor correlation has a poor simulation effect. Moreover, the amounts of rainfall and flood on the d5 component are so small that it has little effect on the forecast of the flood. In order to improve the forecasting accuracy, d4, d5 and a5 are superimposed. Therefore, four GM (1,2) forecasting models are constructed for P_d1-R_d1, P_d2-R_d2, Pd₃-R_d3, P_d4+d5+a5–R_d4+d5+a5 respectively.

Since the decomposed sequences such as d1, d2, and d3 obtained by wavelet transform have multiple negative values, the six data sequences are equivalently replaced to satisfy the non-negative requirements of the GM (1,2) model for data. That is, each item of the original sequence subtracts the minimum value M of the sequence to obtain new sequences including P_b1 ∼ P_b3, R_b1 ∼ R_b3. The minimum values of the sequences are shown in Table 2.

Combining Equation (9), the simulated values of each component of the maximum daily flood volume are obtained by inverse transformation of the equivalent substitution, and four groups of residual sequence are obtained by comparing with R_d1, R_d2, R_d3 and R_(d4+d5+a5). In addition, the component simulation values of maximum daily flood volume are obtained from the rainfall of the next five floods. Taking the 52nd flood as an example, the calculation results are shown in Table 3.

According to Equation (19), when the upper and lower limit values appear negative, they cannot satisfy the requirements. Therefore, in order to avoid unnecessary calculation problems, Y_d1, Y_d2, Y_d3 and Y_d4+d5+a5 are obtained from equivalent substitution of four residual sequences. The method is the same as above under ‘Determination of equivalent substitution value’.

According to the mean value μ and standard deviation σ of the new sequence after the equivalent substitution, the state values of the variables are divided into five levels with μ−σ, μ−0.5σ, μ+0.5σ, μ+σ as the boundaries. The specific criteria for division are shown in Table 4.

The coefficients and weights of each order obtained by Equations (12) and (13) are shown in Table 5. In addition, the prediction probability matrix is constructed to predict the sequence value of the 52nd flood, as shown in Table 6.

From Table 6, the transition probability of state three is the largest. Therefore, the sequence value of the 52nd flood is predicted to be in state three. Combining Equation (19), the predicted value of new series is 33.528. After restoring, the residual correction value in d1 of the maximum daily flood volume is 3.594. Similarly, the residual correction values of other components are obtained.

Residual correction is carried out on the simulation results of each layer of grey model and the corrected results are superimposed to obtain the maximum daily flood forecasting value of the 52nd to 56th floods in Zhongmu hydrological station, which is compared with the actual measured value. The results are shown in Table 7.

The maximum daily flood components of the original sequence and the predicted components are combined (Figure 5), and it can be concluded that during the forecasting period, the d1 of maximum daily flood volume has a quasi-2 fluctuation period with slightly reduced fluctuation amplitude when the d2 presents a quasi-6 fluctuation period with increased fluctuation amplitude compared with the previous period; the fluctuation amplitude of d3 component has obvious change compared with the previous period with a short sudden change in fluctuation period; the fluctuation amplitude of d4 + d5 + a5 increases slightly, but generally tends to be flat. In a word, the influence of rainfall on the flood volume is the most significant in the d3 component during the process of using the model to predict the maximum daily flood volume of Zhongmu hydrological station.

The flood volume for each period of the next five floods in two hydrological stations is forecast. In the process, we can conclude that both rainfall and flood volume of storm floods in two areas decreased first and then increased, and the change trend of flood volume was more obvious. In addition, the flood volume corresponding to the same level of rainfall gradually increases. In all periods rainfall and flood volume have high consistency in these components including d1, d2 and d3. The relative errors of predicted flood are shown in Figures 6 and 7.

From Figures 6 and 7 it can be concluded that the multi-scale flood forecasting model based on GM (1,2)-fuzzy weighted Markov and wavelet analysis has great prediction accuracy. The relative error meets the requirement of less than 20%.

Comparing the forecasted flood volume in each period of the next five floods with the actual measurement, the average relative errors in Zhongmu hydrological station are 7.177, 7.554, 5.240 and 9.937%, while the average relative errors in Jiangang hydrological station are 5.239, 7.967, 4.544 and 5.803%. In addition, the forecast accuracy of the maximum 5-day flood volume is higher than that of other periods in both hydrological stations.

Using wavelet analysis, a GM (1,2) model and the fuzzy weighted Markov method, the model is constructed which not only reveals the relationship between rainfall and flood at the micro level, but also forecasts flood volume at the micro and macro level.

The multi-scale flood forecasting model based on GM (1,2)-fuzzy weighted Markov and wavelet analysis has high forecasting precision. Whether it is Zhongmu hydrological station or Jiangang hydrological station, the average relative errors of the maximum daily, 3-, 5- and 7-day flood volume are all less than 10% when compared with the actual value. In addition, the forecasting accuracy of maximum 5-day flood volume is higher than other periods, with average relative errors of 5.240 and 4.544%. Because of the time lag between rainstorm and flood, the model can forecast the downstream flood according to the rainfall at the upstream rain station, and then relevant departments can take appropriate measures to avoid unnecessary losses.

In the process of forecasting flood, it can be concluded that the flood volume of two hydrological stations decreased first and then increased, while the corresponding flood volume of the same rainfall level increased gradually. At the micro level, the fluctuation trend and period of rainfall-flood volume in d1, d2 and d3 are basically the same. Among the components of forecasted flood, the fluctuation periods and amplitudes of d1 and d2 are basically the same as those of the actual measuring period, while d3 has an obvious abrupt change, that is, the impact of rainstorm on flood is more significant in the d3 component.

In summary, the model is effective in forecasting flood and researching the uncertainty relationship between rainfall and flood volume, so it provides a new idea and method for flood prediction. Moreover, many factors, such as the stationary of the original data series, the fineness of the state partition in the Markov chain and the fuzzy mathematics method, may affect the accuracy of the model, thus, how to coordinate these factors may be a research direction.

This research was supported by the National Key R&D Program of China (Grant No. 2018YFC0406501), Program for Innovative Talents (in Science and Technology) at University of Henan Province (Grant No. 18HASTIT014), and Foundation for University Youth Key Teacher of Henan Province (Grant No. 2017GGJS006).

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