# Investigating Trends in Streamflow and Precipitation in Huangfuchuan Basin with Wavelet Analysis and the Mann-Kendall Test

^{*}

## Abstract

**:**

## 1. Introduction

^{2}, which is mainly comprised of arid and semi-arid environments [12]. The Huangfuchuan basin, an important semiarid watershed in the middle reaches of the Yellow River, was selected as a meso-scale catchment representative of the semiarid climates that predominate across the Yellow River watershed, in order to detect the effects of climate variability and change. A better understanding of climate variability and change on both a basin and regional scale is obviously critical to water management and sustainable ecological conservation of arid and semiarid regions. Many studies which consider both climate variability and change have centered on the assessments in hydro-climate parameters such as temperature, precipitation and streamflow [13,14,15,16,17,18]. Hydrological variables have been considered as useful indicators of how the climate has changed and varied over time, therefore, it is needful to research trends associated with hydrological events [19,20].

## 2. Study Area and Data

^{2}that is characterized by a semi-arid continental climate. The basin’s average precipitation and mean temperature from 1961 to 2000 were 388 mm and 7.5 °C, respectively [12]. The Huangfu gauging station started in 1954 with 3175 km

^{2}of control area, which accounts for 98% of the area of the whole watershed. This area has complex geomorphological types including a feldspathic sandstone hilly-gully region, the loess hilly-gully region and the sanded loess hilly-gully region [51]. The Huangfuchuan basin is considered to be fairly vulnerable to climate change due to vegetation deterioration, soil erosion and land desertification [12,52].

## 3. Methodology

#### 3.1. Wavelet Transforms (WTs)

#### 3.1.1. Continuous Wavelet Transform (CWT)

_{Ψ}of CWT for the time series x(t) are computed by using the convolution of x(t) with the scaled and translated versions of the wavelet, Ψ(η) [20,22]:

_{0}is non-dimensional frequency and ω

_{0}= 6 is used here to satisfy the admissibility condition. The advantage of the Morlet wavelet function provides a conductive definition of the signal in the spectral-space [54].

#### 3.1.2. Global Wavelet Spectrum

#### 3.1.3. Discrete Wavelet Transform (DWT)

_{0}is a fixed dilation step whose value is greater than 1; and γ

_{0}is the location parameter whose value is greater than zero. In general, for practical reasons, the parameters s

_{0}and γ

_{0}are 2 and 1, respectively [56]. This is the DWT dyadic grid arrangement. Supposing a discrete time series x

_{t}, where x

_{t}occurs at a discrete time t, the wavelet coefficient for the DWT becomes

_{Ψ}(a, b) are computed at scale s = 2

^{a}and location γ = 2

^{a}b which reveal the variation of signals at different scales and locations.

#### 3.1.4. Time Series Decomposition via DWT

_{i}represents the original data value of an signal with a record length of n, and a

_{i}is the approximation value of x

_{i}. However, this criteria cannot be used in the series that contains zero value, and there are zero values in the datasets that are used in this study. Therefore, the smallest mean absolute error (MAE) was applied in this study as the first criteria to give a similar evaluation, which was computed as

_{a}and Z

_{o}represent the MK Z-value of the last approximation for the decomposition level used and the original data, respectively.

_{i}, n is the length of data record, the total energy of x

_{i}can be computed as below [42]

_{o}is the total energy of the original series and E

_{a}is the total energy of the last approximation for the decomposition level used.

#### 3.2. Trend Analysis

#### 3.2.1. The Mann-Kendall (MK) Trend Test

_{0}assumes that the deseasonalized data (x

_{1},..., x

_{n}) denotes a sample of n independent and randomly ordered variables. The alternative hypothesis H

_{1}of a two-sided test states that the distribution of x

_{i}and x

_{j}is not identical for all i, j≤n with I ≠ j [20].

_{0}should be accepted if $\left|Z\right|\le {Z}_{\raisebox{1ex}{$\alpha $}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$ at the α level of significance in a two-sided test for trend. If Z > 0, the time series has an upward trend and if Z < 0, then there is a downward trend. Critical value Z

_{α/2}at α = 5% significance level of trend test equals ±1.96.

#### 3.2.2. Mann-Kendall Test with Trend-free Pre-whitening (TFPW)

- Calculate the lag-1 (k = 1) autocorrelation coefficient (r
_{1}) by:$${r}_{k}=\frac{{\displaystyle \sum _{t=1}^{n-k}({x}_{t}-\overline{x})({x}_{t+k}-\overline{x})}}{{\displaystyle \sum _{t=1}^{n}{({x}_{t}-\overline{x})}^{2}}}$$If $\frac{\{-1-1.96\sqrt{n-2}\}}{n-1}\le {r}_{1}\le \frac{\{-1+1.96\sqrt{n-2}\}}{n-1}$, then the data are considered to be serial-independent at the 5% significant level and it is not necessary to conduct TFPW. Elsewhere data are assumed to be serial-correlated and TFPW is required. - The magnitude of trend in sample data is estimated by the Theil-Sen approach (TSA) [59,60], the TSA slope β is computed as:$$\beta =median\left[\frac{{x}_{j}-{x}_{i}}{j-i}\right]\text{for all}ij$$Then the series are detrened by using the following equation:$${x}_{i}^{d}={x}_{i}-\beta \times i$$
- Calculate the r
_{1}of the detrended series by using the Equation (18) and the AR(1) is removed from the detrended series to get a residual series by:$${x}_{i}^{r}={x}_{i}^{d}-{r}_{1}\times {x}_{i}^{d}$$ - The identified trend (β × i) is added back to the residual series to get a blended series by using the following$${x}_{i}^{b}={x}_{i}^{r}+\beta \times i$$

#### 3.2.3. Modified Mann-Kendall Test by Variance Correction

_{1}is variance correction factor according to Hamed and Rao (1998) [24], cf

_{2}according to Yue and Wang (2004) [26], ρ

_{s}(i) is the lag-i significant autocorrelation coefficient of time series ranks; ρ(i) is the lag-i significant autocorrelation coefficient of time series. The values of ρ

_{s}(i) and ρ(i) must be estimated from the detrended sample data and only significant values are used in Equations (24) and (25) since the insignificant values will have an adverse effect on the accuracy of the estimated variance of S [24,26]. The modified MK test calculated by using the cf

_{1}and cf

_{2}are referred as to MK1998 and MKDD, respectively.

_{1}= ρ

_{1}(Yue and Wang, 2004).

_{1}that is estimated form the detrended sample data and the modified MK test calculated by using the cf

_{3}is referred as to MKDD1.

#### 3.2.4. Sequential Mann-Kendall Test

#### 3.2.5. Determining the Dominant Periodic Components for the Observed Trends

## 4. Results and Discussion

#### 4.1. Mann-Kendall Analysis

#### 4.2. Decomposition via DWT

#### 4.3. Decomposition and Analysis of Monthly Data

#### 4.3.1. Monthly Mean Streamflow Series

_{0}was basically the same with the change of energy in detail component combinations. D3 with A5, which has the highest energy, also has the highest C

_{0}(0.645). In addition, two sequential MK (the original MK and MKDD) graphs of the different periodic components with approximations corresponding to the original series of the monthly mean streamflow are shown in Figure 10, in which the trend line of D3 with A7 is most similar to the trend line of the original series. This evidence proves that D3 is the dominant periodic component for the observed trend in the monthly mean streamflow.

#### 4.3.2. Monthly Total Precipitation

_{0}, which corresponds with the results of the monthly mean streamflow. Three sequential MK graphs of the different periodic components with respect to the original series of the monthly total precipitation are shown in Figure 11, in which the trend line of D3 with A6 is most similar to the trend line of the original series. It makes sense that D3 is the most effective periodic component for the real trend seen in the monthly total precipitation. It is clear that the dominant periodic components in the monthly precipitation and streamflow are consistent. In addition, MK1998 and MKDD tests were not applicable in the trend examination of D5 due to the negative values in the calculations of the correction factor cf which can result in incorrect results. However, detailed are not explored in this study due to the relatively lesser impact on our results.

#### 4.4. Decomposition and Analysis of Seasonal Data

#### 4.4.1. Seasonal Mean Streamflow Series

_{0}in the periodic components. Three sequential MK graphs of the seasonal mean streamflow are presented in Figure 12, and the trend line of D1 with A4 is most harmonious with the trend line of the original series. Obviously, D1 is the most effective periodic component for the real trend observed in the seasonal streamflow series.

#### 4.4.2. Seasonal Total Precipitation Series

_{0}and energy indicate that D1 and D2 are probably the dominant periodicities. Further analysis was conducted and results showed that the energy of D1 + D2 + A4 accounted for 96% of the energy of the original series while the correlation coefficient C

_{0}of D1 + D2 + A4 and the corresponding original series was up to 0.961 (Table 7). The results from all three MK tests show that the MK values of the D1 + D2 + A4 and the MK values of the original series are very close to each other. Apparently, D1 and D2 are the dominant periodic component for the trend of the seasonal total precipitation. It is important to say that the energy of the detail component combinations is a very important index to indicate the most effective components. In addition, D2, which represented the annual (12-month) periodicity in the seasonal time and was considered as the most dominant periodic component, t indicating that annual cycles can explain the trends found in streamflow.

#### 4.5. Decomposition and Analysis of Annual Aata

#### 4.5.1. Periodicities of Annual Streamflow and Precipitation Data

#### 4.5.2. Annual Mean Streamflow Series

_{0}as well as the highest energy which implied that D1 might be the most influential periodic component for the trend. The sequential MK analysis of the annual mean streamflow is exhibited in Figure 16, the trend line of D1 with A4 approaches to the trend line of the original series, suggested by three MK tests. The sequential MK graphs tests also indicate that D3 (MK1998) and D4 (MK1998 and MKDD) could be the most effective periodic component for the trend found in the annual mean streamflow series. The reason why the original sequential MK test does not suggest that D4 is the dominant periodicity is that D4 with A4 has a very high autocorrelation coefficient (see Table 8) (the presence of the positive autocorrelation will overestimate the significance of trends). In addition, it is vital to note that decadal events are the major periodicities found in the CWT and GWS, which makes sense that D3 and D4 represent the decadal events found in the CWT figure and are the most effective periodic components. Therefore, D1, D3 and D4 are the most dominant periodic components for the trend in the annual mean streamflow.

#### 4.5.3. Annual Total Precipitation Series

_{0}, which also indicates that the difference of D4 with A5 and the original series is high. Therefore, D4 is not considered to be the most effective component in influencing the trend. This also illustrates that when analyzing the dominant periodic component for the trend, both MK values and their sequential MK graph should be taken into consideration. In addition, MK1998 and MKDD also indicate that D5 is the most effective periodic component for the trend found in the annual total precipitation series. It is worth mentioning that the intense 28-year scale oscillation which is defined as a decadal event is found in the CWT and GWS, which figures out that D5 represents the decadal events found in the CWT figure and is the most influential periodic component. According to previous analysis, the combination of the periodic component and respective approximation with higher energy has more significant impacts on the trend of the observed series. From the results of trend assessment, D5 is the most effective periodic component for the observed trend, but it has the lowest energy. This might be due to the fact that the energy of A5 accounts for 92% of the energy of the annual total precipitation series while the energy of the detail components only account for about 8% of the total energy. There is no big energy difference among each detail component with A5. In this case, energy is no longer an important index to evaluate the most effective periodic component for the observed trends. To sum up, D5 is the most dominant periodic component for the trend in the total precipitation streamflow.

#### 4.6. Factors Related to Precipitation and Streamflow Variations

## 5. Conclusions and Recommendations

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Stojković, M.; Ilić, A.; Prohaska, S.; Plavšić, J. Multi-temporal analysis of mean annual and seasonal stream flow trends, including periodicity and multiple non-linear regression. Water Resour. Manag.
**2014**, 28, 4319–4335. [Google Scholar] [CrossRef] - Yue, S.; Pilon, P.; Cavadias, G. Power of the Mann-Kendall and spearman’s rho tests for detecting monotonic trends in hydrological series. J. Hydrol.
**2002**, 259, 254–271. [Google Scholar] [CrossRef] - Jones, P.D.; Lister, D.H.; Osborn, T.J.; Harpham, C.; Salmon, M.; Morice, C.P. Hemispheric and large-scale land-surface air temperature variations: An extensive revision and an update to 2010. J. Geophys. Res. Atmos.
**2012**, 117, 214–221. [Google Scholar] [CrossRef] - Rohde, R.; Curry, J.; Groom, D. Berkeley earth temperature averaging process. Geoinform. Geostat. Overv.
**2013**, 1. [Google Scholar] [CrossRef] - Huntington, T.G. Evidence of intensification of the global water cycle: Review and synthesis. J. Hydrol.
**2006**, 319, 83–95. [Google Scholar] [CrossRef] - Renner, M.; Bernhofer, C. Long term variability of the annual hydrological regime and sensitivity to temperature phase shifts in Saxony/Germany. Hydrol. Earth Syst. Sci.
**2011**, 15, 1819–1833. [Google Scholar] [CrossRef] - Zhou, Y.; Shi, C.; Fan, X.; Shao, W. The influence of climate change and anthropogenic activities on annual runoff of Huangfuchuan basin in northwest China. Theor. Appl. Clim.
**2015**, 120, 137–146. [Google Scholar] [CrossRef] - Mishra, A.K.; Singh, V.P. Changes in extreme precipitation in Texas. J. Geophys. Res. Atmos.
**2010**, 115, 1307–1314. [Google Scholar] [CrossRef] - Burn, D.H.; Mohammed, S.; Kan, Z. Detection of trends in hydrological extremes for Canadian watersheds. Hydrol. Process.
**2010**, 24, 1781–1790. [Google Scholar] [CrossRef] - Cong, Z.; Yang, D.; Gao, B.; Yang, H.; Hu, H. Hydrological trend analysis in the Yellow River basin using a distributed hydrological model. Water Resour. Res.
**2009**, 45, 335–345. [Google Scholar] [CrossRef] - Hu, Z.; Wang, L.; Wang, Z.; Hong, Y.; Zheng, H. Quantitative assessment of climate and human impacts on surface water resources in a typical semi-arid watershed in the middle reaches of the Yellow River from 1985 to 2006. Int. J. Climatol.
**2015**, 35, 97–113. [Google Scholar] [CrossRef] - Xu, H.; Taylor, R.G.; Xu, Y. Quantifying uncertainty in the impacts of climate change on river discharge in sub-catchments of the Yangtze and Yellow River basins, China. Hydrol. Earth Syst. Sci.
**2011**, 15, 333–344. [Google Scholar] [CrossRef] - Lettenmaier, D.P.; Wood, E.F.; Wallis, J.R. Hydro-climatological trends in the continental United States, 1948–88. J. Clim.
**1994**, 7, 586–607. [Google Scholar] [CrossRef] - Zhang, X.; Vincent, L.A.; Hogg, W.D.; Niitsoo, A. Temperature and precipitation trends in canada during the 20th century. Atmos. Ocean
**2000**, 38, 395–429. [Google Scholar] [CrossRef] - Zhang, X.; Harvey, K.D.; Hogg, W.D.; Yuzyk, T.R. Trends in canadian streamflow. Water Resour. Res.
**2001**, 37, 987–998. [Google Scholar] [CrossRef] - Birsan, M.V.; Molnar, P.; Burlando, P.; Pfaundler, M. Streamflow trends in Switzerland. J. Hydrol.
**2005**, 314, 312–329. [Google Scholar] [CrossRef] - Hannaford, J.; Buys, G. Trends in seasonal river flow regimes in the UK. J. Hydrol.
**2012**, 475, 158–174. [Google Scholar] [CrossRef] [Green Version] - Hannaford, J.; Buys, G.; Stahl, K.; Tallaksen, L.M. The influence of decadal-scale variability on trends in long European streamflow records. Hydrol. Earth Syst. Sci.
**2013**, 17, 2717–2733. [Google Scholar] - Burn, D.H.; Elnur, M.A.H. Detection of hydrologic trends and variability. J. Hydrol.
**2002**, 255, 107–122. [Google Scholar] [CrossRef] - Nalley, D.; Adamowski, J.; Khalil, B. Using discrete wavelet transforms to analyze trends in streamflow and precipitation in Quebec and Ontario (1954–2008). J. Hydrol.
**2012**, 475, 204–228. [Google Scholar] [CrossRef] - Bihrat, O.O.; Bayazit, M.C. The power of statistical tests for trend detection. Turk. J. Eng. Environ. Sci.
**2003**, 27, 247–251. [Google Scholar] - Partal, T.; Kucuk, M. Long-term trend analysis using discrete wavelet components of annual precipitations measurements in marmara region (Turkey). Phys. Chem. Earth
**2006**, 31, 1189–1200. [Google Scholar] [CrossRef] - Adamowski, K.; Prokoph, A.; Adamowski, J. Development of a new method of wavelet aided trend detection and estimation. Hydrol. Process.
**2009**, 23, 2686–2696. [Google Scholar] [CrossRef] - Hamed, K.H.; Rao, A.R. A modified Mann-Kendall trend test for autocorrelated data. J. Hydrol.
**1998**, 204, 182–196. [Google Scholar] [CrossRef] - Yue, S.; Pilon, P.; Phinney, B.; Cavadias, G. The influence of autocorrelation on the ability to detect trend in hydrological series. Hydrol. Process.
**2002**, 16, 1807–1829. [Google Scholar] [CrossRef] - Yue, S.; Wang, C.Y. The Mann-Kendall test modified by effective sample size to detect trend in serially correlated hydrological series. Water Resour. Manag.
**2004**, 18, 201–218. [Google Scholar] [CrossRef] - Ishak, E.H.; Rahman, A.; Westra, S.; Sharma, A.; Kuczera, G. Evaluating the non-stationarity of Australian annual maximum flood. Hydrology
**2013**, 494, 134–145. [Google Scholar] [CrossRef] - Gautam, M.R.; Acharya, K. Streamflow trends in Nepal. Hydrol. Sci. J.
**2012**, 57, 344–357. [Google Scholar] [CrossRef] - Kumar, S.; Merwade, V.; Kam, J.; Thurner, K. Streamflow trends in indiana: Effects of long term persistence, precipitation and subsurface drains. J. Hydrol.
**2009**, 374, 171–183. [Google Scholar] [CrossRef] - Khaliq, M.N.; Ouarda, T.B.M.J.; Gachon, P.; Sushama, L.; St-Hilaire, A. Identification of hydrological trends in the presence of serial and cross correlations: A review of selected methods and their application to annual flow regimes of Canadian rivers. J. Hydrol.
**2009**, 368, 117–130. [Google Scholar] [CrossRef] - Daubechies, I. The wavelet transform, time-frequency localization and signal analysis. IEEE Trans. Inf. Theory
**1990**, 36, 961–1005. [Google Scholar] [CrossRef] - Torrence, C.; Compo, G.P. A practical guide to wavelet analysis. Bull. Am. Met. Soc.
**1998**, 79, 61–78. [Google Scholar] [CrossRef] - Wang, W.; Hu, S.; Li, Y. Wavelet transform method for synthetic generation of daily streamflow. Water Resour. Manag.
**2011**, 25, 41–57. [Google Scholar] [CrossRef] - Kulkarni, J.R. Wavelet analysis of the association between the southern oscillation and the Indian summer monsoon. Int. J. Climatol.
**2000**, 20, 89–104. [Google Scholar] [CrossRef] - Dai, X.; Wang, P.; Jifan, C. Multiscale characteristics of the rainy season rainfall and interdecadal decaying of summer monsoon in north China. Sci. Bull.
**2003**, 24, 2730–2734. [Google Scholar] [CrossRef] - Pišoft, P.; Kalvová, J.; Brázdil, R. Cycles and trends in czech temperature series using wavelet transforms. Int. J. Climatol.
**2004**, 24, 1661–1670. [Google Scholar] [CrossRef] - Zume, J.T.; Tarhule, A. Precipitation and streamflow variability in northwestern Oklahoma, 1894–2003. Phys. Geogr.
**2006**, 27, 189–205. [Google Scholar] [CrossRef] - Xu, J.; Chen, Y.; Li, W.; Ji, M.; Shan, D.; Hong, Y. Wavelet analysis and nonparametric test for climate change in Tarim River basin of Xinjiang during 1959–2006. Chin. Geogr. Sci.
**2009**, 4, 306–313. [Google Scholar] [CrossRef] - Partal, T. Wavelet transform-based analysis of periodicities and trends of Sakarya basin (Turkey) streamflow data. River Res. Appl.
**2010**, 26, 695–711. [Google Scholar] [CrossRef] - Adarsh, S.; Reddy, M.J. Trend analysis of rainfall in four meteorological subdivisions of southern India using nonparametric methods and discrete wavelet transforms. Int. J. Climatol.
**2015**, 35, 1107–1124. [Google Scholar] [CrossRef] - Hirsch, R.M.; Slack, J.R. A nonparametric trend test for seasonal data with serial dependence. Water Resour. Res.
**1984**, 20, 727–732. [Google Scholar] [CrossRef] - Sang, Y.F. A practical guide to discrete wavelet decomposition of hydrologic time series. Water Resour. Manag.
**2012**, 26, 3345–3365. [Google Scholar] [CrossRef] - Percival, D.B.; Walden, A.T. Wavelet Methods for Time Series Analysis; Cambridge University Press: Cambridge, UK, 2002; p. 359. [Google Scholar]
- Popivanov, I.; Miller, R.J. Similarity search over time-series data using wavelets. In Proceedings of the 18th International Conference on Data Engineering, San Jose, CA, USA, 26 February–1 March 2002; Agrawal, R., Dittrich, K., Ngu, A.H.H., Eds.; IEEE: San Jose, CA, USA, 2002; pp. 212–221. [Google Scholar]
- Artigas, M.Z.D.; Elias, A.G.; Campra, P.F.D. Discrete wavelet analysis to assess long-term trends in geomagnetic activity. Phys. Chem. Earth
**2006**, 31, 77–80. [Google Scholar] [CrossRef] - Vonesch, C.; Blu, T.; Unser, M. Generalized daubechies wavelet families. IEEE Trans. Signal Process.
**2007**, 55, 4415–4429. [Google Scholar] [CrossRef] - Su, H.; Liu, Q.; Li, J. Alleviating border effects in wavelet transforms for nonlinear time-varying signal analysis. Adv. Electr. Comput. Eng.
**2011**, 11, 55–60. [Google Scholar] [CrossRef] - Baddoo, T.D.; Guan, Y.; Zhang, D.; Andam-Akorful, S.A. Rainfall variability in the Huangfuchuang watershed and its relationship with ENSO. Water
**2015**, 7, 3243–3262. [Google Scholar] [CrossRef] - Zhao, G.; Tian, P.; Mu, X.; Jiao, J.; Wang, F.; Gao, P. Quantifying the impact of climate variability and human activities on streamflow in the middle reaches of the Yellow River basin, China. J. Hydrol.
**2014**, 519, 387–398. [Google Scholar] [CrossRef] - Tian, P.; Zhao, G.; Mu, X.; Wang, F.; Gao, P.; Mi, Z. Check dam identification using multisource data and their effects on streamflow and sediment load in a Chinese Loess Plateau catchment. J. Appl. Remote Sens.
**2013**, 7, 073697. [Google Scholar] [CrossRef] - Li, E.; Mu, X.; Zhao, G.; Gao, P.; Shao, H. Variation of runoff and precipitation in the Hekou-Longmen region of the Yellow River based on elasticity analysis. Sci. World J.
**2014**, 2014. [Google Scholar] [CrossRef] [PubMed] - Wen, Z.M.; Jiao, F.; Jiao, J.Y. Prediction and mapping of potential vegetation distribution in Yanhe River catchment in hilly area of Loess Plateau. J. Appl. Ecol.
**2008**, 19, 1897–1904. [Google Scholar] - Fu, G.; Chen, S.; Liu, C.; Shepard, D. Hydro-climatic trends of the Yellow River basin for the last 50 years. Clim. Chang.
**2004**, 65, 149–178. [Google Scholar] [CrossRef] - Coulibaly, P.; Burn, D.H. Wavelet analysis of variability in annual Canadian streamflows. Water Resour. Res.
**2004**, 40, 225–236. [Google Scholar] [CrossRef] - Chou, C.M. Applying multi-resolution analysis to differential hydrological grey models with dual series. J. Hydrol.
**2007**, 332, 174–186. [Google Scholar] [CrossRef] - Mallat, G. A theory for multiresolution signal decomposition: The wavelet representation. IEEE Trans. Pattern Anal. Mach. Intell.
**1989**, 11, 674–693. [Google Scholar] [CrossRef] - Douglas, E.M.; Vogel, R.M.; Kroll, C.N. Trends in floods and low flows in the United States: Impact of spatial correlation. J. Hydrol.
**2000**, 240, 90–105. [Google Scholar] [CrossRef] - Yue, S.; Wang, C.Y. Applicability of prewhitening to eliminate the influence of serial correlation on the Mann-Kendall test. Water Resour. Res.
**2002**, 38. [Google Scholar] [CrossRef] - Theil, H. A Rank-Invariant Method of Linear and Polynomial Regression Analysis; Springer: Amsterdam, The Netherlands, 1992; Volume 23, pp. 345–381. [Google Scholar]
- Sen, P.K. Estimates of the regression coefficient based on kendall’s tau. J. Am. Stat. Assoc.
**1968**, 63, 1379–1389. [Google Scholar] [CrossRef] - Matalas, N.C.; Langbein, W.B. Information content of the mean. J. Geophys. Res.
**1962**, 67, 3441–3448. [Google Scholar] [CrossRef] - Kharitonenko, I.; Zhang, X.; Twelves, S. A wavelet transform with point-symmetric extension at tile boundaries. IEEE Trans. Image Process.
**2002**, 11, 1357–1364. [Google Scholar] [CrossRef] [PubMed] - Wang, H.; Yang, Z.; Saito, Y.; Liu, J.P.; Sun, X. Interannual and seasonal variation of the Huanghe (Yellow River) water discharge over the past 50 years: Connections to impacts from enso events and dams. Glob. Planet. Chang.
**2006**, 50, 212–225. [Google Scholar] [CrossRef] - Zhang, Q.; Xu, C.-Y.; Jiang, T.; Wu, Y. Possible influence of enso on annual maximum streamflow of the Yangtze River, China. J. Hydrol.
**2007**, 333, 265–274. [Google Scholar] [CrossRef] - Prokoph, A.; Adamowski, J.; Adamowski, K. Influence of the 11 year solar cycle on annual streamflow maxima in southern Canada. J. Hydrol.
**2012**, 442–443, 55–62. [Google Scholar] [CrossRef] - Chunhui, L.; Zhifeng, Y. Relationship between solar activities and precipitation in the Yellow River basin. Meteorol. Mon.
**2006**, 31, 42–44. [Google Scholar] - Fu, C.; James, A.L.; Wachowiak, M.P. Analyzing the combined influence of solar activity and el niño on streamflow across southern Canada. Water Resour. Res.
**2012**, 48. [Google Scholar] [CrossRef] - Zhao, G.; Mu, X.; Tian, P.; Wang, F.; Gao, P. The variation of stream flow and sediment flux in the middle reaches of Yellow River over the past 60 years and the influencing factors. Resour. Sci.
**2012**, 34, 1070–1078. [Google Scholar]

**Figure 2.**Monthly, seasonal and annual streamflow and precipitation plots of the study used. (

**a**) annual streamflow; (

**b**) annual precipitation; (

**c**) seasonal streamflow; (

**d**) seasonal precipitation; (

**e**) monthly streamflow; (

**f**) monthly precipitation.

**Figure 3.**The correlograms of monthly, seasonal and annual series. The

**upper**and

**lower**solid lines represents the confidence intervals (95% confidence level). (

**a**) monthly streamflow; (

**b**) monthly precipitation; (

**c**) seasonal streamflow; (

**d**) seasonal precipitation; (

**e**) annual streamflow; (

**f**) annual precipitation.

**Figure 4.**Er values of monthly mean streamflow series with different extension mode in six and seven levels (sym: symmetrization extension; per: periodic extension; zpa: zero-padding extension). (

**a**) L = 6; (

**b**) L = 7.

**Figure 5.**Linear fit of original series and approximate components of annual total precipitation series in three extension modes (sym: symmetrization extension; per: periodic extension; zpa: zero-padding extension) with the decomposition level L = 4. The dash lines are the linear fits of the approximate components decomposing from different db types. (

**a**) sym; (

**b**) per; (

**c**) zpa.

**Figure 6.**Linear fit of original series and approximate components of monthly, seasonal and annual streamflow and precipitation series in the three criteria. (

**a**) monthly streamflow; (

**b**) monthly precipitation; (

**c**) seasonal streamflow; (

**d**) seasonal precipitation; (

**e**) annual streamflow; (

**f**) annual precipitation.

**Figure 7.**Original monthly mean streamflow series and its approximation (A7) and detail components (D1–D7) decomposed via DWT. (

**a**) original data; (

**b**) A7; (

**c**) D1; (

**d**) D2; (

**e**) D3; (

**f**) D4; (

**g**) D5; (

**h**) D6; (

**i**) D7.

**Figure 8.**Original monthly precipitation series and its approximation (A6) and detail components (D1–D6) decomposed via DWT. (

**a**) original data; (

**b**) A6; (

**c**) D1; (

**d**) D2; (

**e**) D3; (

**f**) D4; (

**g**) D5; (

**h**) D6.

**Figure 9.**The correlograms of A7 and each detail component with A7 decomposing from the monthly streamflow series via DWT. The upper and lower solid lines represent the confidence intervals (95% confidence level). (

**a**) A7; (

**b**) A7 + D1; (

**c**) A7 + D2; (

**d**) A7 + D3; (

**e**) A7 + D4; (

**f**) A7 + D5; (

**g**) A7 + D6; (

**h**) A7 + D7.

**Figure 10.**Two sequential Mann-Kendall (MK and MKDD) graphs of monthly streamflow series exhibiting the progressive trend lines of each detail component (with the addition of the approximation) with respect to the original series. The upper and lower dashed lines represent the confidence limits (α = 5%). (

**a**) MK; (

**b**) MKDD.

**Figure 11.**Three sequential Mann-Kendall graphs of monthly total precipitation series exhibiting the progressive trend lines of each detail component (with the addition of the approximation) with respect to the original series. The upper and lower dashed lines represent the confidence limits (α = 5%). (

**a**) MK; (

**b**) MK1998; (

**c**) MKDD.

**Figure 12.**Three sequential Mann-Kendall graphs of seasonal mean streamflow series exhibiting the progressive trend lines of each detail component (with the addition of the approximation) with respect to the original series. The upper and lower dashed lines represent the confidence limits (α = 5%). (

**a**) MK; (

**b**) MK1998; (

**c**) MKDD.

**Figure 13.**Three sequential Mann-Kendall graphs of seasonal total precipitation series exhibiting the progressive trend lines of each detail component (with the addition of the approximation) with respect to the original series. The upper and lower dashed lines represent the confidence limits (α = 5%). (

**a**) MK; (

**b**) MK1998; (

**c**) MKDD.

**Figure 14.**Continuous wavelet spectrum (

**a**) and global wavelet spectrum (

**b**) of the annual mean streamflow series.

**Figure 15.**Continuous wavelet spectrum (

**a**) and global wavelet spectrum (

**b**) of the annual total precipitation series.

**Figure 16.**Three sequential Mann-Kendall graphs of annual mean streamflow series exhibiting the progressive trend lines of each detail component (with the addition of the approximation) with respect to the original series. The upper and lower dashed lines represent the confidence limits (α = 5%). (

**a**) MK; (

**b**) MK1998; (

**c**) MKDD.

**Figure 17.**Three sequential Mann-Kendall graphs of annual total precipitation series exhibiting the progressive trend lines of each detail component (with the addition of the approximation) with respect to the original series. The upper and lower dashed lines represent the confidence limits (α = 5%). (

**a**) MK; (

**b**) MK1998; (

**c**) MKDD.

**Table 1.**Lag-1 autocorrelation functions (ACFs) and MK values (using different MK methods) of the original monthly, seasonal, and annual streamflow and precipitation series (MS: monthly streamflow; SS: seasonal streamflow; AS: annual streamflow; MP: monthly precipitation; SP: seasonal precipitation; AP: annual precipitation).

Type | ACF | MK | TFPW | MK1998 | MKDD | MKDD1 |
---|---|---|---|---|---|---|

MS | 0.399 * | −13.179 | −12.410 | −5.503 | −4.025 | −8.779 |

SS | 0.065 | −7.624 | −7.764 | −4.055 | −3.188 | −7.578 |

AS | 0.166 | −5.239 | −5.376 | −5.239 | −5.239 | −6.312 |

MP | 0.482 * | 0.202 | 0.380 | 0.228 | 0.193 | 0.120 |

SP | −0.051 | 0.168 | 0.107 | 0.086 | 0.067 | 0.176 |

AP | −0.266 * | −0.902 | −0.984 | −1.286 | −1.344 | −1.196 |

**Table 2.**MAE, er, and Er of monthly mean flow series (sym: symmetrization extension; per: periodic extension; zpa: zero-padding extension).

Extension Mode | Decomposition Levels | Criterion | db5 | db6 | db7 | db8 | db9 | db10 |
---|---|---|---|---|---|---|---|---|

sym | L = 6 | MAE | 4.637 | 4.658 | 4.627 | 4.632 | 4.643 | 4.638 |

er | 1.772 | 1.613 | 1.723 | 2.005 | 1.643 | 1.783 | ||

Er | 0.798 | 0.803 | 0.805 | 0.799 | 0.8 | 0.805 | ||

L = 7 | MAE | 4.667 | 4.68 | 4.642 | 4.634 | 4.66 | 4.666 | |

er | 1.425 | 0.863 | 1.295 | 1.687 | 0.990 | 0.616 | ||

Er | 0.792 | 0.798 | 0.812 | 0.805 | 0.795 | 0.802 | ||

per | L = 6 | MAE | 4.672 | 4.679 | 4.697 | 4.699 | 4.677 | 4.684 |

er | 2.287 | 2.732 | 3.252 | 2.628 | 2.297 | 1.759 | ||

Er | 0.811 | 0.817 | 0.813 | 0.808 | 0.812 | 0.816 | ||

L = 7 | MAE | 4.799 | 4.814 | 4.736 | 4.736 | 4.792 | 4.806 | |

er | 1.352 | 1.793 | 1.142 | 1.142 | 1.787 | 1.505 | ||

Er | 0.82 | 0.811 | 0.829 | 0.829 | 0.82 | 0.813 | ||

zpa | L = 6 | MAE | 4.577 | 4.6 | 4.606 | 4.591 | 4.587 | 4.604 |

er | 1.907 | 1.131 | 1.469 | 1.656 | 1.715 | 1.341 | ||

Er | 0.816 | 0.819 | 0.817 | 0.813 | 0.816 | 0.819 | ||

L = 7 | MAE | 4.577 | 4.606 | 4.616 | 4.595 | 4.581 | 4.604 | |

er | 0.922 | 0.747 | 0.516 | 0.583 | 1.066 | 0.624 | ||

Er | 0.828 | 0.823 | 0.829 | 0.834 | 0.829 | 0.824 |

**Table 3.**The results of extension mode, decomposition levels and db type that were used in DWT of monthly, seasonal and annual flow and precipitation series in the three criteria (MS: monthly streamflow; SS: seasonal streamflow; AS: annual streamflow; MP: monthly precipitation; SP: seasonal precipitation; AP: annual precipitation; sym: symmetrization extension; per: periodic extension; zpa: zero-padding extension).

Criterion | Data Type | Extension Mode | Decomposition Levels | db Type |
---|---|---|---|---|

MAE | MS | zpa | 6 | db5 |

MP | zpa | 7 | db5 | |

SS | zpa | 4 | db7 | |

SP | zpa | 5 | db7 | |

AS | zpa | 4 | db6 | |

AP | sym | 4 | db5 | |

er | MS | zpa | 7 | db7 |

MP | per | 6 | db6 | |

SS | sym | 5 | db10 | |

SP | per | 4 | db10 | |

AS | zpa | 4 | db7 | |

AP | per | 4 | db10 | |

Er | MS | sym | 7 | db5 |

MP | per | 6 | db8 | |

SS | sym | 4 | db5 | |

SP | sym | 4 | db9 | |

AS | sym | 4 | db9 | |

AP | per | 5 | db8 |

**Table 4.**Slope β (computed by TSA), Lag-1 ACFs, Mann-Kendall values (three MK tests) and energy of monthly mean streamflow series: original data, detail components (D1–D7), approximations (A7) and a set of combinations of the details and their respective approximations. C

_{0}is the correlation coefficients between the decomposition combinations and the original series.

Series | Slope (β) | ACF | MK | MK1998 | MKDD | C_{0} | Energy |
---|---|---|---|---|---|---|---|

Original | −0.0027 | 0.399 | −13.179 * | −5.503 * | −4.025 * | – | 69,705 |

A7 | −0.0103 | 0.995 | −34.542 * | −5.855 * | −9.761 * | 0.212 | 14,476 |

D1 | −0.0003 | −0.437 | −0.647 | −0.849 | −0.942 | 0.556 | 18,159 |

D2 | 0 | 0.338 | −0.066 | −0.171 | −0.122 | 0.385 | 8783 |

D3 | 0.0003 | 0.83 | 0.366 | 0.444 | 0.337 | 0.611 | 22,011 |

D4 | 0.0003 | 0.932 | 0.614 | 1.485 | 3.356 * | 0.252 | 4569 |

D5 | −0.0003 | 0.98 | −1.287 | −1.016 | −0.806 | 0.171 | 1895 |

D6 | −0.0002 | 0.995 | −0.797 | −0.579 | −0.911 | 0.126 | 985 |

D7 | 0.0006 | 0.999 | 3.682 * | 1.307 | 2.145 * | 0.023 | 348 |

D1 + A7 | −0.0102 | −0.227 | −16.555 * | −15.260 * | −20.989 * | 0.595 | 32,641 |

D2 + A7 | −0.0103 | 0.512 | −15.830 * | −12.121 * | −15.987 * | 0.438 | 23,381 |

D3 + A7 | −0.0098 | 0.852 | −10.262 * | −9.374 * | −7.963 * | 0.645 | 36,696 |

D4 + A7 | −0.0098 | 0.964 | −18.084 * | −12.563 * | −16.646 * | 0.333 | 18,692 |

D5 + A7 | −0.0101 | 0.984 | −23.363 * | −10.236 * | −15.233 * | 0.275 | 16,204 |

D6 + A7 | −0.0098 | 0.993 | −25.500 * | −8.628 * | −11.843 * | 0.244 | 15,522 |

**Table 5.**Lag-one ACFs, Mann-Kendall values (three MK tests) and energy of monthly total precipitation series: original data, details components (D1–D6), approximations (A6) and a set of combinations of the details and their respective approximations. C

_{0}is the correlation coefficients between the decomposition combinations and the original series.

Series | ACF | MK | MK1998 | MKDD | C_{0} | Energy |
---|---|---|---|---|---|---|

Original | 0.482 | 0.202 | 0.228 | 0.193 | - | 2,438,503 |

A6 | 0.999 | −9.241 * | −1.510 | −2.187 * | 0.071 | 828,972 |

D1 | −0.626 | 0.317 | 0.692 | 0.526 | 0.434 | 305,735 |

D2 | 0.369 | −0.168 | −0.270 | −0.190 | 0.455 | 336,637 |

D3 | 0.852 | 0.325 | 0.385 | 0.313 | 0.716 | 840,561 |

D4 | 0.954 | −0.047 | −0.084 | −0.269 | 0.25 | 98,838 |

D5 | 0.987 | −1.789 | enable | enable | 0.13 | 25,058 |

D6 | 0.997 | 0.682 | 0.465 | 0.696 | 0.08 | 12,201 |

D1 + A6 | −0.581 | −1.748 | −1.841 | −2.013 * | 0.44 | 1,134,729 |

D2 + A6 | 0.385 | −1.042 | −0.938 | −0.986 | 0.461 | 1,168,091 |

D3 + A6 | 0.853 | −0.336 | −0.337 | −0.309 | 0.72 | 1,663,805 |

D4 + A6 | 0.958 | −1.512 | −0.899 | −1.581 | 0.26 | 928,184 |

D5 + A6 | 0.989 | −5.125 * | −1.716 | −2.319 * | 0.148 | 854,880 |

D6 + A6 | 0.997 | −4.703 * | −1.178 | −1.892 | 0.108 | 839,865 |

**Table 6.**Lag-1 ACFs, Mann-Kendall values and energy of seasonal mean streamflow series: original data, details components (D1–D4), approximations (A4) and a set of combinations of the details and their respective approximations. C

_{0}is the correlation coefficients between the decomposition combinations and the original series.

Data | ACF | MK | MK1998 | MKDD | C_{0} | Energy |
---|---|---|---|---|---|---|

Original | 0.065 | −7.624 * | −4.055 * | −3.188 * | - | 14,100 |

A4 | 0.969 | −17.946 * | −8.343 * | −16.597 * | 0.313 | 4878 |

D1 | −0.359 | 1.5 | 0.772 | 0.571 | 0.765 | 6040 |

D2 | 0.205 | 0.56 | 0.993 | 0.677 | 0.449 | 2497 |

D3 | 0.745 | 0.044 | 0.1 | 0.059 | 0.246 | 491 |

D4 | 0.952 | 1.557 | 1.57 | 1.886 | 0.191 | 477 |

D1 + A4 | −0.131 | −6.562 * | −3.633 * | −2.651 * | 0.822 | 11,032 |

D2 + A4 | 0.464 | −8.928 * | −8.132 * | −7.344 * | 0.552 | 7293 |

D3 + A4 | 0.869 | −13.844 * | −7.722 * | −8.074 * | 0.39 | 5462 |

D4 + A4 | 0.947 | −13.443 * | −9.512 * | −12.182 * | 0.393 | 5036 |

**Table 7.**Lag-one ACFs, Mann-Kendall values and energy of seasonal total precipitation series: original data, details components (D1–D4), approximations (A4) and a set of combinations of the details and their respective approximations. C

_{0}is the correlation coefficients between the decomposition combinations and the original series.

Data | ACF | MK | MK1998 | MKDD | C_{0} | Energy |
---|---|---|---|---|---|---|

Original | −0.051 | 0.168 | 0.086 | 0.067 | - | 600,926 |

A4 | 0.988 | −5.513 * | −1.902 | −3.115 * | 0.11 | 277,514 |

D1 | −0.507 | −0.072 | −0.050 | −0.033 | 0.556 | 100,382 |

D2 | 0.055 | −0.143 | −1.107 | −0.955 | 0.783 | 196,786 |

D3 | 0.791 | −0.129 | −0.139 | −0.116 | 0.243 | 20,315 |

D4 | 0.948 | −0.446 | −0.479 | −0.542 | 0.116 | 4836 |

D1 + A4 | −0.445 | −1.106 | −0.727 | −0.501 | 0.567 | 376,755 |

D2 + A4 | 0.075 | −0.920 | −2.745 * | −4.076 * | 0.79 | 475,719 |

D3 + A4 | 0.819 | −1.865 | −1.131 | −1.171 | 0.268 | 297,078 |

D4 + A4 | 0.959 | −3.744 * | −1.779 | −2.304 * | 0.161 | 281,563 |

D1 + D2 + A4 | −0.126 | 0.05 | 0.025 | 0.019 | 0.961 | 578,056 |

**Table 8.**Lag-one ACFs, Mann-Kendall values and energy of annual mean streamflow series: original data, detail components (D1–D4), approximations (A4) and a set of combinations of the details and their respective approximations. C

_{0}is the correlation coefficients between the decomposition combinations and the original series.

Data | ACF | MK | MK1998 | MKDD | C_{0} | Energy |
---|---|---|---|---|---|---|

Original | 0.166 | −5.239 * | −5.239 * | −5.239* | - | 1528 |

A4 | 0.948 | −9.823 * | −5.058 * | −8.497* | 0.6 | 1184 |

D1 | −0.618 | 0.145 | 0.208 | 0.151 | 0.562 | 220 |

D2 | 0.322 | 0.048 | 0.101 | 0.088 | 0.45 | 149 |

D3 | 0.818 | −0.805 | −0.882 | −1.295 | 0.252 | 18 |

D4 | 0.903 | 0.296 | 0.212 | 0.215 | 0.049 | 18 |

D1 + A4 | 0.18 | −5.831 * | −5.609 * | −4.645 * | 0.827 | 1390 |

D2 + A4 | 0.653 | −6.877 * | −8.021 * | −7.182 * | 0.753 | 1328 |

D3 + A4 | 0.937 | −8.791 * | −5.595 * | −10.864 * | 0.629 | 1229 |

D4 + A4 | 0.915 | −10.925 * | −5.980 * | −5.717 * | 0.631 | 1156 |

**Table 9.**Lag-one ACFs, Mann-Kendall values and energy of annual total precipitation: original data, detail components (D1–D5), approximations (A5) and a set of combinations of the details and their respective approximations. C

_{0}is the correlation coefficients between the decomposition combinations and the original series.

Data | ACF | MK | MK1998 | MKDD | MKDD1 | TFPW | C_{0} | Energy |
---|---|---|---|---|---|---|---|---|

Original | −0.266 | −0.902 | −1.286 | −1.344 | −1.196 | −0.984 | - | 10,820,675 |

A5 | 0.921 | −2.265 | −0.817 | −0.890 | −0.542 | −10.994 * | −0.182 | 9,977,842 |

D1 | −0.691 | −0.186 | enable | enable | −0.426 | −0.434 | 0.807 | 616,764 |

D2 | 0.27 | −0.103 | enable | enable | −0.079 | −0.379 | 0.493 | 208,741 |

D3 | 0.796 | 0.062 | 0.092 | 0.105 | 0.022 | −0.213 | 0.292 | 87,441 |

D4 | 0.938 | −0.778 | −0.529 | −0.836 | −0.162 | −2.072 | 0.201 | 17,038 |

D5 | 0.93 | −4.619 | −1.636 | −1.530 | −1.163 | −10.994 * | 0.228 | 38,918 |

D1+A5 | −0.688 | −0.227 | enable | enable | −0.518 | 0.213 | 0.8 | 10,628,434 |

D2+A5 | 0.276 | −0.172 | enable | enable | −0.130 | −0.516 | 0.479 | 10,148,639 |

D3+A5 | 0.793 | −0.062 | −0.089 | −0.101 | −0.022 | −1.095 | 0.272 | 10,078,044 |

D4+A5 | 0.944 | −0.750 | −0.632 | −1.450 | −0.152 | −2.141 * | 0.174 | 10,095,310 |

D5+A5 | 0.938 | −5.032 * | −1.834 | −1.723 | −1.299 | −10.994 * | 0.222 | 9,718,913 |

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons by Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Chen, Y.; Guan, Y.; Shao, G.; Zhang, D.
Investigating Trends in Streamflow and Precipitation in Huangfuchuan Basin with Wavelet Analysis and the Mann-Kendall Test. *Water* **2016**, *8*, 77.
https://doi.org/10.3390/w8030077

**AMA Style**

Chen Y, Guan Y, Shao G, Zhang D.
Investigating Trends in Streamflow and Precipitation in Huangfuchuan Basin with Wavelet Analysis and the Mann-Kendall Test. *Water*. 2016; 8(3):77.
https://doi.org/10.3390/w8030077

**Chicago/Turabian Style**

Chen, Yuzhuang, Yiqing Guan, Guangwen Shao, and Danrong Zhang.
2016. "Investigating Trends in Streamflow and Precipitation in Huangfuchuan Basin with Wavelet Analysis and the Mann-Kendall Test" *Water* 8, no. 3: 77.
https://doi.org/10.3390/w8030077