We study the multifractal properties of water level with a high-frequency and massive time series using wavelet methods (estimation of Hurst exponents, multiscale diagram, and wavelet leaders for multifractal analysis (WLMF)) and multifractal detrended fluctuation analysis (MF-DFA). The dataset contains more than two million records from 10 observation sites at a northern China river. The multiscale behaviour is observed in this time series, which indicates the multifractality. This multifractality is detected via multiscale diagram. Then we focus on the multifractal analysis using MF-DFA and WLMF. The two methods give the same conclusion that at most sites the records satisfy the generalized binomial multifractal model, which is robust for different times (morning, afternoon, and evening). The variation in the detailed characteristic parameters of the multifractal model indicates that both human activities and tributaries influence the multifractality. Our work is useful for building simulation models of the water level of local rivers with many observation sites.
Long-range dependence (LRD) and multifractality are the inherent characteristics of many natural phenomena. Long-range dependence (LRD) was found in the flood process by Hurst [
The concept of multifractality was originally introduced by Mandelbrot [
Previous studies on the LRD and multifractality of hydrology (see, e.g., [
It is known that the hydrologic data are often affected by trends or other nonstationarities, for example, due to the seasonal cycle or a change in climate, which may lead to an overestimation of the Hurst exponent. Furthermore, under the influence of the trends, the uncorrelated data may behave as long-term correlated data. Thus, the methods applied in this paper are based on wavelet analysis (estimation of Hurst exponents, multiscale diagram, and wavelet leaders for multifractal analysis (WLMF)) and multifractal detrended fluctuation analysis (MF-DFA). The wavelet method can eliminate some trends as a result of the vanishing moment property, and MF-DFA can avoid spurious detection of correlations that are artifacts of nonstationarity in the records [
In this paper, we first apply the wavelet-based method to estimate the Hurst exponent with different sampling and wavelet scales and observe both the LRD phenomenon
The logical structure and organization of the paper are described in Figure
Data processing flow chart.
The remaining parts of the paper are organized as follows. We introduce the data source and data extraction method for analysis in Section
The records that are collected in real time from one river (a branch of the Haihe River in China) represent the water level. The river is located in northern China, which has a significant continental monsoon climate. It is cold and dry in the winter and warm and humid in the summer. From upstream to downstream, 10 artificial canals have been built to control the water level. Next to these canals, 10 water level observation sites (Site 1–Site 10) have been set up along the river (see Figure
Sampling information.
Site 1 | Site 2 | Site 3 | Site 4 | Site 5 | Site 6 | Site 7 | Site 8 | Site 9 | Site 10 | |
---|---|---|---|---|---|---|---|---|---|---|
Number of records | 182855 | 152307 | 189967 | 190959 | 188074 | 185557 | 190840 | 190445 | 185822 | 111600 |
Sampling period (days) | 904 | 837 | 904 | 904 | 904 | 904 | 904 | 904 | 903 | 698 |
Sampling frequency | 6 min | 6 min | 6 min | 6 min | 6 min | 6 min | 6 min | 6 min | 6 min | 6 min |
Site 1–Site 10 locations.
The water level records
The water level of the river is always influenced by the weather conditions and human activities. Every year, during the traditional rainy season, the upstream canals are always filled with much rainwater. During the dry season, the upstream canals discharge the stored water. Both the storage and discharge of water should be adjusted at any time according to the growing condition of crops along the bank of the river and the actual demand of human activities. According to our survey, water storage and discharge at Site 2 and Site 7 are conducted frequently for crop growth and human activities.
The following three preprocessing steps were conducted before the multifractal analyses: (
Veitch et al. [
Let
Let
Because of the stationary property of
As a result,
Put
Because of the different variances of
Abry et al. [
Wavelet procedures provide us with a way to estimate the Hurst parameter. Sometimes, the Hurst index depends on time
Wavelet processes are an effective tool for obtaining
If
The wavelet-based multiscale diagram can detect whether the process
In MF-DFA, one investigates the series:
Let
In FA, the fluctuation
For a multifractal description of records, one must consider all moments
For very large
It is known that [
One can also get the Renyi scaling exponent
If
The three parameters
For comparison with MF-DFA, we introduce the method of wavelet leaders for multifractal analysis (WLMF) proposed by Wendt et al. (see [
Assume that
The weights
To estimate the multifractal spectrum, Wendt et al. [
We emphasize that the wavelet transform with
This section describes the wavelet-based estimation of Hurst parameters for the water level records of the 10 sites. We estimate the Hurst exponents at different scales and different sampling interval times. The results indicate that there is an interesting connection between the wavelet scale and the sampling period. For comparison, we repeat the procedure for a synthetic exponential long-range dependent process (ELRD) with length of 180000, which is close to the actual sampling data length. The vanishing moment of the used Daubechies wavelet is
Simulation data analysis for
Interval |
| ||||
---|---|---|---|---|---|
1 |
2 |
3 |
4 |
5 |
|
1 | 0.61 (0.009) | 0.61 (0.013) | 0.60 (0.018) | 0.59 (0.025) | 0.59 (0.036) |
2 | 0.61 (0.013) | 0.61 (0.018) | 0.60 (0.025) | 0.59 (0.036) | 0.60 (0.051) |
4 | 0.61 (0.018) | 0.61 (0.025) | 0.60 (0.036) | 0.59 (0.051) | 0.61 (0.071) |
8 | 0.60 (0.025) | 0.60 (0.036) | 0.60 (0.051) | 0.61 (0.071) | 0.62 (0.106) |
Simulation data analysis for
Interval |
| ||||
---|---|---|---|---|---|
1 |
2 |
3 |
4 |
5 |
|
1 | 0.83 (0.009) | 0.82 (0.013) | 0.81 (0.018) | 0.80 (0.025) | 0.80 (0.034) |
2 | 0.83 (0.013) | 0.82 (0.018) | 0.81 (0.025) | 0.80 (0.034) | 0.82 (0.049) |
4 | 0.83 (0.018) | 0.82 (0.025) | 0.81 (0.034) | 0.81 (0.049) | 0.80 (0.072) |
8 | 0.82 (0.025) | 0.82 (0.034) | 0.82 (0.049) | 0.80 (0.072) | 0.79 (0.100) |
The simulation results show that the simulations for both
Tables
Multiscales and multisampling for
Interval |
| |||
---|---|---|---|---|
1 |
2 |
3 |
4 |
|
6 min | 0.84 (0.062) |
|
|
|
12 min |
|
|
|
— |
24 min |
|
|
— | — |
48 min |
|
— | — | — |
Multiscales and multisampling for
Interval |
| |||
---|---|---|---|---|
1 |
2 |
3 |
4 |
|
6 min | 0.80 (0.101) |
|
|
|
12 min |
|
|
|
|
24 min |
|
|
|
— |
48 min |
|
|
— | — |
Multiscales and multisampling for
Interval |
| |||
---|---|---|---|---|
1 |
2 |
3 |
4 |
|
6 min | 0.79 (0.047) |
|
|
|
12 min |
|
|
|
|
24 min |
|
|
|
— |
48 min |
|
|
— | — |
We can also see that the estimations for the Hurst exponent are not stable under changes in the interval length and estimation scale, while the estimations along the secondary diagonal direction of the tables are still stable. We conclude that the massive hydrologic data have a multiscale consistent property, while the simulation data do not. This indicates that there may exist multifractality in the records.
In this section, we use the multiscale diagram to test the multifractality that may exist in the records. The multiscale diagram is the graph of
We use a linear multiscale diagram to examine the alignment in the multiscale diagram, in which
To determine the alignment of hydrological data, we impose the synthesized FBM (
|
0.5 | 1 | 1.5 | 2 | 2.5 | 3 | 4 | 5 | 6 | 8 |
---|---|---|---|---|---|---|---|---|---|---|
Site 1 | 0.52 | 0.86 | 1.13 | 1.33 | 1.49 | 1.61 | 1.80 | 1.97 | 2.15 | 2.52 |
Errors bars | 0.008 | 0.015 | 0.025 | 0.041 | 0.064 | 0.095 | 0.170 | 0.250 | 0.319 | 0.423 |
Site 2 | 0.51 | 0.97 | 1.34 | 1.56 | 1.68 | 1.76 | 1.87 | 1.97 | 2.09 | 2.39 |
Errors bars | 0.010 | 0.023 | 0.046 | 0.083 | 0.123 | 0.160 | 0.233 | 0.308 | 0.383 | 0.526 |
Site 3 | 0.52 | 1.00 | 1.39 | 1.68 | 1.85 | 1.96 | 2.07 | 2.12 | 2.16 | 2.26 |
Errors bars | 0.007 | 0.014 | 0.024 | 0.045 | 0.077 | 0.117 | 0.201 | 0.282 | 0.357 | 0.471 |
Site 4 | 0.59 | 1.16 | 1.61 | 1.70 | 1.59 | 1.47 | 1.27 | 1.15 | 1.07 | 1.00 |
Errors bars | 0.007 | 0.017 | 0.064 | 0.156 | 0.222 | 0.272 | 0.362 | 0.441 | 0.511 | 0.618 |
Site 5 | 0.55 | 0.94 | 1.10 | 0.98 | 0.76 | 0.49 | −0.06 | −0.61 | −1.13 | −2.10 |
Errors bars | 0.013 | 0.031 | 0.086 | 0.161 | 0.228 | 0.288 | 0.394 | 0.481 | 0.552 | 0.655 |
Site 6 | 0.42 | 0.80 | 1.12 | 1.37 | 1.58 | 1.73 | 1.91 | 2.02 | 2.10 | 2.26 |
Errors bars | 0.008 | 0.015 | 0.023 | 0.037 | 0.060 | 0.090 | 0.161 | 0.227 | 0.285 | 0.386 |
Site 7 | 0.48 | 0.77 | 1.02 | 1.23 | 1.42 | 1.58 | 1.86 | 2.08 | 2.25 | 2.56 |
Errors bars | 0.016 | 0.031 | 0.050 | 0.075 | 0.110 | 0.157 | 0.282 | 0.420 | 0.537 | 0.684 |
Site 8 | 0.58 | 1.07 | 1.52 | 1.77 | 1.82 | 1.80 | 1.74 | 1.71 | 1.71 | 1.76 |
Errors bars | 0.015 | 0.027 | 0.067 | 0.188 | 0.334 | 0.437 | 0.542 | 0.589 | 0.615 | 0.653 |
Site 9 | 0.46 | 0.88 | 1.25 | 1.51 | 1.66 | 1.76 | 1.98 | 2.22 | 2.46 | 2.94 |
Errors bars | 0.007 | 0.015 | 0.035 | 0.079 | 0.140 | 0.202 | 0.304 | 0.373 | 0.423 | 0.495 |
Site 10 | 0.59 | 1.12 | 1.59 | 1.86 | 1.95 | 1.98 | 2.03 | 2.11 | 2.22 | 2.50 |
Errors bars | 0.012 | 0.024 | 0.057 | 0.135 | 0.220 | 0.286 | 0.383 | 0.455 | 0.513 | 0.599 |
Testing for multiscaling. (a) Multiscale diagram. Compared with the alignment for FBM, hydrological data with no alignment suggests multiscale behaviour. (b) Linear multiscale diagram. The FBM series exhibits horizontal alignment. The errors bars are based on the weighted linear least square fit which is used to estimate
In Figure
In this section, we analyze the daily water level records at the 10 observation sites using MF-DFA analysis for the multifractal behaviour. In order to ensure that there are enough fluctuations in each segment so that MF-DFA can be applied, the high-frequency collected daily records are used. For sufficient samples, we get 100 sets of daily records randomly at different times of one day. The number of records of each sets is about six hundred. We also apply wavelet leaders for multifractal analysis (WLMF) based on the daily water level records for a comparison with MF-DFA.
Following the test method reported by Khalil et al. [
Statistical tests at Site 4. (a) and (e) show the log-log plots of MF-DFA and WLMF; different colors indicate different orders of polynomial (or different vanishing moments of wavelet). (a) 2-order (red), 3-order (blue), and 4-order (green). (e) 2-moment (red), 3-moment (blue), and 4-moment (green). (b) shows the pdf of
Statistical tests at Site 7. (a) and (e) show the log-log plots of MF-DFA and WLMF; different colors indicate different orders of polynomial (or different vanishing moments of wavelet). (a) 2-order (red), 3-order (blue), and 4-order (green). (e) 2-moment (red), 3-moment (blue), and 4-moment (green). (b) shows the pdf of
Figures
The log-scaling plots of the water level dynamics of all of the sites. The different symbols and colors indicate different moments:
The results of MF-DFA and WLMF for 10 sites.
MF-DFA | WLBMF | |||||||
---|---|---|---|---|---|---|---|---|
|
|
|
|
|
|
|
|
|
Site 1 | 0.88 | 0.069 | 0.40 | 0.70 | 0.81 | 0.48 | 0.97 | 1.00 |
Site 2 | 1.10 | 0.043 | 0.39 | 0.56 | 0.52 | 0.39 | 0.66 | 0.76 |
Site 3 | 0.90 | 0.040 | 0.34 | 0.66 | 0.95 | 0.26 | 0.83 | 1.66 |
Site 4 | 0.89 | 0.028 | 0.30 | 0.67 | 1.17 | 0.36 | 0.87 | 1.27 |
Site 5 | 0.92 | 0.014 | 0.26 | 0.70 | 1.44 | 0.26 | 0.83 | 1.66 |
Site 6 | 0.95 | 0.029 | 0.26 | 0.63 | 1.27 | 0.32 | 0.77 | 1.26 |
Site 7 | 0.88 | 0.039 | 0.37 | 0.66 | 0.82 | 0.52 | 0.86 | 0.73 |
Site 8 | 0.99 | 0.018 | 0.16 | 0.63 | 1.99 | 0.19 | 0.95 | 2.28 |
Site 9 | 1.20 | 0.027 | 0.29 | 0.54 | 0.88 | 0.35 | 0.64 | 0.87 |
Site 10 | 1.37 | 0.017 | 0.25 | 0.46 | 0.85 | 0.30 | 0.53 | 0.81 |
Figure
Figure
Comparison of the shape of
To further confirm the results of MF-DFA, we apply MF-DFA to the daily records at different times (morning, afternoon, and night) of a day. As seen in Figure
The generalized Hurst exponents
From Figure
From Table
The long-range dependence property and multifractality we detected in the water level records are identical to the previous results of hydrologic data (see, e.g., [
In the previous results (see, e.g., [
In light of the results obtained by MF-DFA, the values of
As previously mentioned, the spurious multifractality may be induced by intricate nonstationarities, which often exist in the hydrologic data, for example, due to the seasonal cycle or a change in climate. We consider that the obtained multifractality is reliable. First the method we used are based on wavelet analysis and MF-DFA which can eliminate some trends and avoid spurious detection of correlations. Second the multifractality is confirmed consistently via multiple methods (multiscale diagram, MF-DFA, and WLMF). The MF-DFA and WLMF also give the same multifractal model, and this multifractal model is robust in different location and different time. Last the results obtained from the analyzed river are consistent with the previous study of actual example.
In this paper, we proceed in an orderly manner to study the LRD and multifractal properties of high-frequency water level records of a river in northern China by using wavelet analysis and MF-DFA.
We observe the long-range dependence property and multiscale behaviour in the time series, which indicates the multifractality. This multifractality is detected via multiscale diagram. Then we focus on the multifractality of records using MF-DFA and WLMF. At most sites, the generalized Hurst exponent
We obtain and list the values of
As a technical improvement in the estimation of the Hurst parameter, we show that the sampling intervals and choice of scales of the discrete wavelet transformation should be consistent (Tables
In contrast to the previous studies, our studies are based on a short sampling time, which we have not seen before. This study contributes a new insight into previous research applications, and it may also be the extension of the study of multifractality. Further investigations are needed, especially for the massive data cases, for example, comparing the fitting of
As we know that the process
Assuming
The multiscale diagram is implemented in Matlab by Veitch and available from his homepage at
The multifractal detrended fluctuation analysis (MF-DFA) is implemented in Matlab by Ihlen and available at
The wavelet leaders for multifractal analysis (WLMF) is implemented in Matlab by Wendt and available from his homepage at
The authors declare that there are no conflicts of interest regarding the publication of this paper.
This work is supported in part by the National Basic Research Program of China (973 Program, Grant no. 2013CB910200), National Science Foundation of China (no. 61103136), the open project of the Hubei Province Key Laboratory of Intelligent Robot (no. 201007), and the Fundamental Research Funds for the Central Universities (no. JBK170164, no. JBK120509, and no. JBK140507).