Wavelet analysis of the complex precipitation series in the Northern Jiansanjiang Administration of the Heilongjiang land reclamation, China

As an important component of the hydrologic cycle, precipitation is the key factor influencing regional agricultural production, drought and flood disasters, and is an important supply source of surface water and groundwater. Precipitation has vital significance for disaster prevention and mitigation, restoration and sustainable utilisation of groundwater resources and agricultural sustainable development; understanding precipitation requires the study of the variation characteristics of multi-timescales of regional precipitation series and analysis of the changing regularity of droughts and floods. In recent years, the complexity of regional precipitation series has become increasingly apparent because of the increase in human interference. The traditional precipitation time series analysis method often ignores the complexity of the series itself and leads to a full demonstration of the difficulty and poor practicability results of the structure of the precipitation series. In this context, it is necessary to perform complex measurements for obtaining the regional multiple precipitation monitoring data series, screening the most complex series as a sample series, and revealing the overall regional precipitation characteristics.

The traditional methods for determining the variation characteristics of multi-timescale precipitation mainly include Fourier analysis (Almeida et al. 2004; Prokop & Walanus 2003), spectral analysis (such as singular spectrum, power spectrum, etc.) (Maheras et al. 1992; Moten 1993; Gajić-Čapka 1994; Piervitali & Colacino 2001; Türkeş et al. 2002) and empirical mode decomposition-maximum entropy spectral analysis (Sang et al. 2012). These methods do not have time-frequency localisation properties and lack mathematical rigour in the diagnosis of an abrupt point (Liu et al. 2009), so the variation characteristics of multi-timescales of precipitation cannot be fully reflected. However, wavelet analysis, which was developed in the early 1980s, is the ideal approach of functional analysis, Fourier analysis, spline analysis, harmonic analysis and numerical analysis (Zou & Mo 1999), as it has a time-frequency multi-resolution function and rigorous analysis on abrupt points in mathematics; in addition, wavelet analysis can show the fine structure of multi-timescales of precipitation series (Wei & Wu 2011), being currently widely applied in the field of hydrology and water resources.

Many scholars have used wavelet analysis to explore the variation characteristics of multi-timescales of regional precipitation series. Özger et al. (2010) used the continuous wavelet transform (CWT) to analyse the monthly precipitation series from 43 stations in Texas, in the United States, and the results showed the annual period played an important role in the characterisation of dry spells. Zhan et al. (2011) adopted the Poyang Lake Basin daily precipitation observations of China from 1959 to 2008 and used the Mexican hat wavelet analysis to reveal annual precipitation that had a quasi-period of 20 years and rainfall in flood season that has a quasi-period of 6 years. Jury & Melice (2000) used the CWT to analyse monthly precipitation data from 1872 to 1999 in Durban, South Africa; the results showed that the monthly precipitation had 1- and 2.3–4-year periods, which accounted for 33% and 10% of rainfall variance, respectively. Subash et al. (2011) applied Morelet wavelet analysis and the power spectrum method to analyse the Central Northeast India monthly rainfall during June–September, the total rainfall during monsoon season and the annual rainfall from 1889 to 2008; the results showed that none of the precipitation series had obvious periodicity. Kim (2004) used the Mexican hat wavelet transform to analyse the spatio-temporal variation characteristics of five precipitation series from 1905 to 2001 in northern California; the results revealed that the spatial pattern of the precipitation field may have changed since 1945, and the dominant period was approximately 16 years.

Most of the studies regarding the multi-timescale variation characteristics of precipitation usually adopted the average of the multi-station as the sample series. Meanwhile, although some researchers adopted the wavelet analysis method to extract the primary cycle of precipitation, the results lacked representativeness and accuracy because they ignored the significance test process. Therefore, the aim of this article is to screen the most complex seasonal precipitation series as the sample series combining the theory of entropy, fractal and Lempel-Ziv complexity (LZC), and reveal the multi-timescale variation characteristics of seasonal rainfall and annual precipitation series in the Northern Jiansanjiang Administration of Heilongjiang land reclamation in China using the Morelet wavelet analysis method.

The rest of the paper is organised as follows. The ‘Materials and methods’ section briefly introduces the general information about the study area, describes the basic data used in this study, and presents the methods employed in this study. Then the results obtained with real rainfall series data are presented and discussed. Conclusions are given in the final section.

Figure 2 shows that the seasonal precipitation series of each farm has an obvious characteristic seasonal period variation of decreasing in the spring and winter and increasing in the summer and autumn in the Northern Jiansanjiang Administration. At the same time, although those series have obvious seasonal period characteristics, the change in their mean and variance is not smooth, belonging to the non-stationary time series. Thus, the dynamic changes include random and non-linear complexity characteristics. From Figure 3, the seasonal and annual precipitation series of each farm are both non-stationary time series in the Northern Jiansanjiang Administration, with similar variation characteristics. The spring and winter precipitation has a gradually increasing trend, whereas the summer and autumn precipitation has a gradually decreasing trend. The summer and autumn precipitation is significantly higher than that of the two seasons of spring and winter, resulting in a superimposed effect of the annual precipitation gradually reducing.

The methods for measuring hydrological series complexity include wavelet entropy (WE) (He et al. 2011), approximate entropy (ApEn) (Pincus 1991), LZC (Zhao et al. 2011), fractal theory based on resealed range analysis (R/S) (Matos et al. 2004; Wang et al. 2005), fractal theory based on discrete wavelet transform (DWT) and CWT (Wang et al. 2005), etc. The basic principles of the above-mentioned complexity measurement methods can be found in the relevant literature.

The Morlet wavelet is obtained after a periodic function goes through Gaussian function smoothing; thus, its scale factor a has a one-to-one relationship with the period of the Fourier transform: ⁠. When the constant c is 6.2, T = 1.00057a ≈ a. Thus, the Morlet wavelet can be used for period analysis.

The variation course of wavelet variance going with scale ‘a’ is called the wavelet variance map. This map can reflect the fluctuation of various scales (namely periods) included in the hydrological time series and its change characteristics of intensity (energy magnitude) with the scales. Each peak value represents each notable period in the map. Thus, we can conveniently confirm the main timescales (namely the main periods) that exist in a time series through the wavelet variance map.

If Var(a)>P, then the relevant period of that wavelet variance is significant.

To adequately consider the mutual effect of the complicated dynamic change of each farm's precipitation, the above six different methods were adopted to measure the complexity of the seasonal precipitation series of each farm in the Jiansanjiang Administration; Table 1 shows the sorted results (only northern area sorted results are listed). The range analysis of fractal theory is relatively sensitive to the series length, belonging to the biased estimation, with a slightly poor stability (Rakhshandehroo et al. 2009); however, the stability of the Wavelet fractal theory is higher (Wang et al. 2005), with the others in the middle. To give full play to the advantages of various types of complexity measurement methods (Pan et al. 2011, 2012), based on the above analysis, we determined the weight of various complexity levels, see Table 1.

To conveniently handle the measured precipitation series, data (n = 37) of the four seasons of spring, summer, autumn, winter and the annual precipitation (March to the next February) from 1970 to 2006 in Qinglongshan Farm were analysed with anomaly (centralisation). The wavelet transform coefficients W_f (a, b) of the precipitation anomaly series f(kΔt) (k = 1, 2, … 37; Δt = 1) in Qinglongshan Farm were calculated using Equation (3).

Figure 4(a) shows that the signal energy distribution strength of different timescales of the spring precipitation anomaly series, in which the signal energy change over timescales of 3 to 8 years is the strongest, are found to mainly occur in the periods from 1980 to 1988 and 1994 to 2006, with an oscillation at the centres in 1983 and 2000. The signal energy of timescales of 13 to 24 years mainly occurs in the period from 1978 to 2006. The signal energy of timescales of 1 to 3 years mainly occur in the periods from 1970 to 1972, 1988 to 1990 and 1999 ∼ 2004. The signal energy of the other cases has a lower amount of change.

Figure 5(a) shows the variation of different timescales of the spring precipitation anomaly series, and the changing point distribution and structure of the positive and negative phases; the timescales of 3 to 8 and 12 to 25 years are the most obvious, the positive and negative phase appear alternately, and the central timescales of the periods are approximately 4 years and 17 years, respectively. In addition, the timescales of 8 to 11 years also appear and the central timescale is approximately 9 years.

From Figure 6(a), it can be observed that the main peaks of wavelet variances appear in scales a = 4, 9 and 17. For the scale of a = 4, the first peak value is the corresponding wavelet variance, which indicates that rectilinear oscillation is the strongest at approximately 4 years. This first peak value is the first main period, and the second and third main periods are 17 years and 9 years, respectively. To distinguish whether the above-mentioned main periods have statistical significance, it is necessary to perform a significance test.

After calculation, the first-order autocorrelation coefficient of the spring precipitation anomaly series in Qinglongshan Farm is r(1) = −0.0380. Next, r_c was calculated according to Equation (5), ⁠. Thus, the wavelet variances were tested using the white noise spectrum with Equations (6) and (7), and the 95% confidence level line was drawn (see Figure 6(a)). It can be observed that only the 4-year period exceeds the 95% confidence level line, i.e., the significant main period of the spring precipitation series is approximately 4 years in Qinglongshan Farm.

The significant main periods of the other precipitation anomaly series were recognised using the same method (see Table 2). The results reveal that the two seasons of summer and autumn precipitation series have similar main periods to the annual precipitation series in Qinglongshan Farm, i.e., the two seasons of summer and autumn precipitations have the same changing trend of a gradual reduction at Qinglongshan Farm from Table 2. Thus, the two seasons of summer and autumn precipitation commonly control the precipitation throughout the year. In Table 2, significant main periods of approximately 2, 3, and 4 years are consistent with the period in the range of 2 to 7 years of the ENSO (Tudhope et al. 2001; Moore 2008), and the significant main periods of 9 years and 11 years are almost the same as the 11-year period of sunspot activity (Zolotova & Ponyavin 2006), which illustrates that ENSO is the main cause of the periodic variation characteristics of spring, summer and winter precipitation in Qinglongshan Farm, and the periodic variation characteristics of autumn and the annual precipitation are due to the dual influences of ENSO and sunspot activity.

The period ingredient of the precipitation series of spring, summer, autumn, winter and the year can be recognised using the harmonic analysis method based on the Fourier transform (Bu et al. 2012); the results are shown in Table 3. From Tables 2 and 3, we can see that the significant main period of the winter precipitation series is 4.1 years according to the harmonic analysis method based on Fourier transform; this period is the same as the 4-year period determined via the wavelet analysis method. The significant main periods of the annual precipitation series are 2.1 years and 9.3 years according to the method of harmonic analysis based on the Fourier transform; this is the same as the 3-year and 9-year periods recognised by the wavelet analysis method. The significant main period of the autumn precipitation series is 2.2 years according to the method of harmonic analysis based on the Fourier transform; this is the same as the 2-year period recognised by the wavelet analysis method. However, the significant main period of 12 years cannot be recognised. In addition, the main period of the spring and summer precipitation series cannot be recognised in Qinglongshan Farm. These discrepancies may be due to the short length of the precipitation series. The above analysis shows that the wavelet analysis algorithm has some advantages in terms of analysing the multiple timescale change features of the hydrological time series. Meanwhile, the pattern of extracting the precipitation main period in this paper can remedy some of the disadvantages identified in some references (Jury & Melice 2000; Kim 2004; Özger et al. 2010), such as ignoring the significance test and the lack of strictness in extracting the main period.

The changes in the drought and waterlogging processes of the rest of the precipitation series were analysed, and their variation trends were revealed (see Table 4). It can be observed that the trend of precipitation in summer and autumn is just the opposite of that in spring and winter under the condition of the smaller scales of 4 years and 9 years, demonstrating that summer and autumn precipitation dominates the annual precipitation changing trend at smaller scales. The exception is winter precipitation; the rest of the precipitation series has a similar trend to the annual precipitation on the larger scale of 17 years, indicating that the local change trend of the annual precipitation is subject to the three seasons of spring, summer and autumn.

According to Table 4, the future key points of drought resistance and waterlogging control for the Northern Jiansanjiang Administration can be formulated (see Table 5). It can be observed that drought and flood disasters are frequent in the next 10 years in the Northern Jiansanjiang Administration. The following measures are recommended to improve the ability of local farmland to withstand natural disasters (see Table 6).

This article used the complexity measures of WE, LZC, fractal theory and Morlet wavelet transform to analyse the variation characteristics of the precipitation series over multi-timescales in the Northern Jiansanjiang Administration. The following conclusions can be drawn from the study:

1. Qinglongshan Farm has the highest integrated complexity indices of the seasonal precipitation series, demonstrating that natural factors and human activities have the greatest influence on the local precipitation. Thus, taking the precipitation series of Qinglongshan Farm as the sample series of the Northern Jiansanjiang Administration can reflect the regional precipitation period change characteristics accurately and comprehensively, and the selected sample series and period analysis are highly representative.

2. The Qinglongshan Farm seasonal precipitation and annual precipitation series have significant main periods of 2 to 4 years, 9 years and 12 years, which indicate that the ENSO and sunspot activity are the main causes for the period variation characteristics. The result reveals the two seasons of summer and autumn precipitation series have a similar main period and change trend to those of the annual precipitation series in Qinglongshan Farm; thus, the two seasons of summer and autumn precipitation commonly control the precipitation throughout the year.

3. The harmonic analysis method based on Fourier transform can only recognise the 2.2-year period of precipitation in autumn, the 4.1-year period of precipitation in winter, and the 2.1-year and 9.3-year periods of annual precipitation, which confirms that the wavelet analysis is an excellent tool that can reveal the fine structure of the hydrological time series and extract the period accurately.

4. Agricultural production will continue under the threat of drought and flood disasters in the Northern Jiansanjiang Administration in the next 10 years. It is recommended that the local government should make a reasonable plan of disaster prevention and mitigation, adopting the appropriate measures to improve the ability of local farmland to withstand natural disasters.

In fact, the lack of precipitation monitoring data in Nongjiang Farm inevitably affects the complexity sorting results of each farm. Therefore, selecting a valid spatial interpolation method, such as Ordinary Kriging, Co-Kriging, Improved Inverse Distance Weighted, etc., to interpolate the precipitation series of Nongjiang Farm can amend the research results of the precipitation complexity in the Jiansanjiang Administration. This topic is worthy of further study in the future.

This study is supported by the National Natural Science Foundation of China (No. 51579044, No. 41071053, and No. 51479032), the Sub-Task of National Science and Technology Support Program for Rural Development in The 12th Five-Year Plan of China (No. 2013BAD20B04-S3), the Specialised Research Fund for the Public Welfare Industry of the Ministry of Water Resources (No. 201301096), the Specialised Research Fund for Innovative Talents of Harbin (Excellent Academic Leader) (No. 2013RFXXJ001), the Science and Technology Research Program of the Education Department of Heilongjiang Province (No. 12531012), the Science and Technology Program of Water Conservancy of Heilongjiang Province (No. 201319, No. 201501, and No. 201503), and the Northeast Agricultural University Innovation Foundation for Postgraduates (No. yjscx14069).