Wavelet and statistical analyses of river water quality parameters: a case study in the Lower Minnesota River

Surface water systems can be influenced by many factors – i.e., they are multi-scale, non-linear and non-stationary. Surface water pollution induced by physical, chemical and biological contaminants has become an epidemic and comprehensive problem worldwide (Pavlidis & Tsihrintzis 2018). River water quality at any location reflects several major influences, including the basin's lithology, atmospheric and anthropogenic inputs, and climatic conditions. Rivers also play a significant role in the assimilation or transport of municipal and industrial wastewater and runoff from agricultural lands (Noori et al. 2010).

Water quality analysis is the basis of assessment, and can be used to analyze and predict water quality variation trends according to the monitoring data (Najah et al. 2012). In recent decades many mathematical methods have been used to analyze water quality monitoring data. Parmar & Bhardwaj (2013) used fractal dimensional analysis to evaluate the water quality index at Harike Lake, India, and showed that the water was severely contaminated and unsuitable for domestic or industrial use. Kisi & Ay (2014) applied a new trend method (Sen trend test) to analyze water parameters in the Kizilirmak River in Turkey, and showed that the Sen trend test could be used successfully for water parameter trend analysis.

Wavelet analysis was first introduced in 1974 by Grossman & Morlet (Nourani et al. 2014). It can be used to obtain the detailed features of different sub-series, which facilitates interpreting global trends and local detail, especially those restricted by strong global trends in the original series. Thus, it is a powerful tool for studying non-stationary and multi-scaled process details. Sang (2013) reviewed wavelet analysis applications in hydrologic time series and highlighted its advantages in multi-faceted information analysis. He et al. (2008) studied a water quality management system using wavelet analysis, applying the approach to a river water quality system to demonstrate its practicability in parameter estimation. Almasri et al. (2008) adopted discrete wavelet transform (DWT) to evaluate temperature time-series trends in Sweden for the period 1850 to 1999. Wavelet analysis has also been used to investigate water quality in other drainage basins (Yu et al. 2010; Parmar & Bhardwaj 2012; Zhou et al. 2012).

In this study, both wavelet and statistical analysis tools were applied to the Lower Minnesota River (LMR), USA, to analyze river water quality parameters to assist in clarifying study of their statistical characteristics, inter-relationships and time-series evolution features. Comprehensive analytical results provide rewarding insights to the Minnesota Pollution Control Agency (MPCA), and the method can also be a reference for other regions’ environmental and water resource management.

The Minnesota River is a main tributary of the Mississippi River. It is approximately 534 km long and drains a watershed of nearly 44,000 km², with annual average flow of 125 m³/s. The upper watershed is used primarily for agriculture but the extent of development increases as the river flows toward its confluence with the Mississippi River. Figure 1 is a map of the study area. The upstream-downstream boundary of the study reach is between river miles (RM) 39.4 and 3.5, including major tributaries, wastewater treatment plants, power plants and an airport. An analytical tool to study the dynamic features of water quality parameters is needed for regional planning and state regulatory agencies, to evaluate the increased wastewater impact on water quality in LMR.

Water quality parameters, including dissolved oxygen (DO), nitrate (NO₃), ammonium (NH₄), phosphate (PO₄), total phosphorus (TP), total Kjeldahl nitrogen (TKN), chlorophyll a (CHLA), total suspended solids (TSS) and water temperature (TEMP) were monitored (at an average sampling interval of 12 days) for six years (from 8 September 2001 to 2 September 2006) at five stations along the study reach: Minnesota River at Jordan, Shakopee, Savage, Black Dog and Ft. Snelling (Figure 1).

Statistical analysis includes various coefficients – e.g., means, medians, minima, maxima, standard deviations, kurtosis and skewness (Bendat & Piersol 1986). In particular, kurtosis refers to the degree of flatness or peakedness in a region about the mode of a frequency curve. Normally, the kurtosis of a normal distribution is 3, so, for convenience, 3 is subtracted from the kurtosis, changing the kurtosis of a normal distribution to 0 in this study. Positive kurtosis indicates a narrow peak and heavy tails, with more variance due to infrequent extreme deviations. Skewness concerns data symmetry and the skewness of a normal distribution is 0. Positive skewness indicates a longer distribution tail for relatively infrequent and more extreme values exceeding the mean.

The correlation coefficient (R) is usually used in statistical analysis to show the correlation relationship between two variables. The absolute value of R is between 0 for a non-existent and 1 for a perfect linear correlation. R > 0 means that two variables are positively correlated, while R < 0 means that they are negatively correlated (Aitchison 1986).

The most important concerns in wavelet decomposition are appropriate selection of the mother wavelet and the decomposition level. Such selection has been studied by others (Nourani et al. 2011; Sang 2012). Nourani & Parhizkar (2013) stated that the guideline for choosing the appropriate mother wavelet should be based on similarity in shape between it and the time-series. The Daubechies-5 (Db5) wavelet has been used extensively, since its wavelet coefficients can capture the maximum amount of signal energy (He et al. 2008; Parmar & Bhardwaj 2012; Soni et al. 2017). In addition, the shape of the Db5 wavelet is similar to the water quality parameter time-series, so it was used in this study.

Decomposition level is often chosen based on the data length, and to yield a desired low-pass cutoff frequency (Soni et al. 2017). Given a set of daily data, decomposition leads to 2ⁿ-day mode resolutions (e.g., 2¹-day mode, 2²-day mode, 2³-day mode, which is roughly weekly, 2⁴-day mode, 2⁵-day mode, which is roughly monthly, and…, 2⁸-day mode, which is roughly annually, etc.), approximating the periodicity of the water quality time-series (Nourani et al. 2014). In this study, given a set of 12-day data, decomposition leads to 12 × 2ⁿ-day mode resolutions. To detect water quality characteristics between years (approximately 12 × 2⁵-day mode), five-layer multi-scale was selected.

Wavelet decomposition of some key parameters was presented in seven parts – s/ss, a5, d5(1Y), d4(27 W), d3(14 W), d2(7 W) and d1(3 W) (Figure 3), where “s/ss” represents the original/synthesized signal (dashed line); the low frequency part “a5” gives an approximation of the signal at level 5; and the relatively high frequency parts d5 (approximately one year), d4 (27 weeks), d3 (14 weeks), d2 (7 weeks) and d1 (three weeks) contain the details of “s” at different scales, respectively.

Ten water quality parameters were evaluated at the five monitoring stations using statistical methods. The results show that the values for the parameters generally remain at the same level between the five stations. The basic water quality statistical parameters for the LMR at Jordan are listed in Table 1 and their probability density distributions (P) are presented in Figure 2.

DO: the mean and median values are approximately equal. The standard deviation suggests that the DO data are slightly spread. Skewness and kurtosis are both approximately 0, showing that the DO data are approximately symmetrical and mesokurtic. The probability density distribution has two peaks with values of 8.5 and 13.5 respectively.

BOD: the mean and median values are approximately equal. The standard deviation suggests that the BOD data are spread. The data are approximately symmetrical and mesokurtic, with skewness and kurtosis both close to 0.

NO₃: the mean and median values are approximately equal. The standard deviation suggests that the NO₃ data are spread out. NO₃ skewness and kurtosis approximate to 0, indicating that the data are approximately symmetrical and mesokurtic. The probability density distribution has two peaks with values of 3 and 8 respectively.

NH₄: the mean and median values are different. The standard deviation suggests that the NH₄ data are spread out. The probability density distribution is leptokurtic and has a long distribution tail.

PO₄: the mean and median values are approximately equal. The standard deviation suggests that the PO₄ data are spread out. The probability density distribution is leptokurtic and approximately symmetrical (skewness approximates to 1).

TP: the median and mean values differ. The standard deviation suggests that the TP data are spread out. The probability density distribution is leptokurtic and has a long distribution tail.

TKN: the mean and median values are approximately equal. The standard deviation suggests that the data are slightly spread. The probability density distribution is leptokurtic and has a long distribution tail.

CHLA: the mean and median values differ. The standard deviation suggests that the CHLA data are spread out. The skewness and kurtosis both approximate to 1, indicating that CHLA is approximately symmetrical and leptokurtic.

TSS: the mean and median values differ. The standard deviation suggests that the TP data are spread out. The probability density distribution is leptokurtic and has a long distribution tail.

TEMP: the mean and median values are approximately equal. The standard deviation suggests that the TEMP data are spread out. The probability density distribution is platykurtic and approximately symmetrical (skewness approximates to 0). The probability density distribution has three peaks at 5, 15 and 25 respectively.

The calculated water quality correlation coefficients at all five monitoring stations are approximately equal, so comprehensive correlation coefficients were computed for the study reach – see Table 2. As can be seen, the water quality parameters from the full set of stations present strongly positive correlation between CHLA-BOD (R = 0.677), TP-TKN (R = 0.78), TP-TSS (R = 0.709) and TKN-TSS (R = 0.601), and a strongly negative DO-TEMP correlation (R = −0.762).

DO, NH₄ and PO₄ are important quality parameters for eutrophication in LMR (Cole & Buchak 2002). DO is a critical biogenic parameter and an important water health indicator, reflecting the status of biological growth and water health. NH₄ is an important oxygen consumer, and is toxic for aquatic organisms. PO₄ is a nutrient for primary producers and plays an extremely important role in phytoplankton growth. In order to inspect the selection rationality of mother wavelet and decomposition level in this study, TEMP is also selected and analyzed for its inherent inter-annual cycle characteristics.

The wavelet decomposition results for TEMP and the three other important parameters – DO, NH₄ and PO₄ – are presented in Figure 3. The original signals “s” are unstable as the time-series are chaotic, non-linear and multi time-scaled, while the detailed evolution processes of signals at different scales present quasi-periodicity after DWT. The DWT results show that the evolution processes at different scales for the five monitoring stations are similar, because of which only the DWT results from Ft. Snelling are discussed here. The minima, maxima, and value ranges for the parameters at different scales are shown in Table 3.

The Db5 analyses for the water quality parameters show that:

The TEMP data present obvious quasi-periodicity over one year at d5 level, which reflects the seasonal cycle. At d1 level, the data between 40 and 44 (roughly a 48-day period) show obvious shake, reflecting slight fluctuations in the original time-series at s/ss level. The values of d5, d4, d3, d2 and d1 vary from −12.33 to 12.27, −13.59 to 12.39, −7.19 to 6.03, −2.96 to 3.7, and −3.88 to 5.64, respectively.

The DO data also present relative quasi-periodicity of about one year at d5 level. Taking the TEMP time-series as reference, the peak time-series at d5 level corresponds well with the TEMP time-series trough at d5, reflecting the strongly negative correlation between DO and TEMP (Figure 2). The values of d5, d4, d3, d2 and d1 vary from −3.41 to 3.6, −2.62 to 2.69, −3.05 to 3.25, −2.21 to 2.42 and −2.28 to 2.86, respectively.

The quasi-periodicity of NH₄ is not as obvious as it is for TEMP or DO. The values of d5, d4, d3, d2 and d1 vary from −0.07 to 0.06, −0.18 to 0.21, −0.17 to 0.18, −0.21 to 0.22, and −0.13 to 0.22, respectively. The minimum fluctuation range of NH₄ is at the d5 sub-scale, indicating that its inter-annual variation is relatively weak.

The quasi-periodicity of PO₄ is also less clear than for TEMP or DO. Comparing the decomposition results for PO₄ and NH₄, the PO₄ time-series at d5, d4 and d3 are almost synchronous with those of NH₄ at d5, d4 and d3, correspondingly, reflecting the relative strongly positive correlation between these parameters (Figure 2). The values of d5, d4, d3, d2 and d1 vary from −0.03 to 0.03, −0.06 to 0.07, −0.08 to 0.09, −0.09 to 0.12, and −0.14 to 0.13, respectively. The minimum fluctuation range of PO₄ is at d5 level, indicating that the inter-annual variation of PO₄ is relatively weak.

Statistical, correlation and DWT analyses were used to study water quality parameters covering six years on LMR. Water quality parameter basic conditions – e.g., means, medians, minima, maxima, standard deviations, kurtosis and skewness – were inspected using statistical methods, statistical results are a useful aid to judging river water quality. Water quality parameters are not independent, but have interactive effects on the river system, so correlation analysis was adopted to analyze the relationships between water quality parameters. On the basis of the correlation coefficient of water quality parameter pairs and the condition of one water quality parameter, another parameter condition could be inferred qualitatively – e.g., on the basis of relatively high river water TEMP, relatively low DO concentration could be inferred because of the strongly negative correlation between TEMP and DO (R = −0.762). DWT analysis, a powerful tool for studying time-series processes, was introduced to study non-stationary and multi-scaled water quality dynamic processes – e.g., the low frequency part “a5” of TEMP rising gradually over the research period, reflecting the inter-annual trend, while TEMP also presents obvious quasi-periodicity over one year at the d5 level (the seasonal cycle). The DWT analytical results help managers to analyze and predict the dynamic tendency of parameters at different decomposition levels. In addition, the wavelet transform for temporal pre-processing is a useful tool in de-noising and reconstructing water quality parameter time-series, which are used as inputs for hydrodynamic and water-quality forecasting models of LMR–e.g., the CE-QUAL-W2 model developed by the U.S. Army Corps of Engineers (Smith et al. 2012).

In general, the comprehensive analytical results provide useful insights for MPCA, and the method can also be a reference for environmental and water resources management in other regions.

Water quality parameters covering six years on LMR were studied using statistical, correlation and DWT analyses. The statistical results show that DO, BOD, NO₃ and CHLA are all approximately symmetrical and mesokurtic, while NH₄, TP, TKN and TSS are leptokurtic and have long distribution tails. PO₄ is leptokurtic and approximately symmetrical, and TEMP platykurtic and approximately symmetrical. The probability density distributions of DO, NO₃ and TEMP have multiple peaks.

The correlation analysis results show that there are strongly positive correlations between CHLA-BOD (R = 0.677), TP-TKN (R = 0.78), TP-TSS (R = 0.709) and TKN-TSS (R = 0.601), with strongly negative correlation between DO-TEMP (R = −0.762) for the data from all stations monitored.

The DWT results show that TEMP and DO have the relative quasi-periodicity at about one year; while the quasi-periodicity of NH₄ and PO₄ is less obvious. The fluctuation ranges of different parameters and correlations between some pairs were also analyzed on the basis of wavelet decomposition results.

Statistical, correlation and DWT analysis all proved feasible and effective in this study. Their results provide valuable references for understanding water quality.

This work was jointly funded by the National Key R&D Program of China (2016YFC0401506), the Projects of National Natural Science Foundation of China (51679146; 51479120), and Projects of Nanjing Hydraulic Research Institute (Y117009; Y118009; Y118012).

The authors declare no conflict of interest.