This study analyzes involved trends in stream flow and precipitation data at monthly, seasonal and annual timescales observed at six precipitation and four stream flow stations of Tampa Bay using non-parametric Mann–Kendall (MK) and discrete wavelet transform (DWT) methods. The MK test and sequential MK analysis were applied to different combinations of DWT after removing the effect of significant lag-1 serial correlation to calculate components responsible for trend of the time series. Also, the sequential MK test was used to find the starting point of changes in annual time series. The results showed that negative trend is prevalent in the case study; generally, short-term periods were important in the involved trend at original time series. Thus, the precipitation data at three scales showed short-term periods of 2 months, 6 months and 2 years in monthly, seasonal and annual scales, respectively. In the greatest stream-flow time series at three timescales, wavelet-based detail at level 2 plus the approximations time series was conceded as the dominant periodic component. Finally, the results of Sen's trend analysis, applied to the original annual time series, also confirmed the results of the proposed wavelet-based MK test in most cases.

INTRODUCTION

The climate change is controlled by many factors, and there are many different scientific opinions regarding which of these factors is the most significant. There are also established links between atmospheric circulation, climate and stream flow. Trend identification of hydro-climatologic time series is one of the commonly used methods to detect climate change. Changes in hydrological processes may in turn affect the overall availability and quality of water resources and alter the spatiotemporal characteristics of hydrologic occurrences, such as the timing of flow events, and the frequency and severity of floods and droughts. Investigations into detection of trend in hydrological time series have been focused on changes in stream flow (Zhang et al. 2001; Kahya & Kalayci 2004; Cigizoglu et al. 2005; Kumar et al. 2009; Nalley et al. 2012), precipitation (Lettenmaier et al. 1994; Türkes 1996; Zhang et al. 2000; Penalba & Vargas 2004; Paul et al. 2011; Nalley et al. 2012) and water quality (Yu et al. 1993; Van Belle & Hughes 1984; Kalayci & Kahya 1998; Daneshvar et al. 2013).

Nowadays, there are different statistical methods for trend analyzing hydrological time series such as t-test, regression analysis, Pearson correlation coefficient, the Spearman's Rho, Sen's slope, Wald–Wolfowitz and most commonly used method of Mann–Kendall (MK). The advantage of the MK method is that it does not follow any specific statistical distribution. The MK test is simple to compute, and resilient to non-stationary data and missing values (Partal & Küçük 2006; Adamowski et al. 2009). Gautam et al. (2010) investigated precipitation and stream flow trends at no-snow fall Nepal region using MK and modified MK tests for data during 1974–1994. Türkeş (1996) analyzed the spatial and temporal characteristics of Turkish annual mean rainfall data for long-term trends, fluctuations and changes in the runs of dry and wet years. Partal & Kahya (2006) used MK test and Sen's t-test to verify regional hydroclimatic conditions comprising 96 precipitation stations around Turkey and observed significant trends (in annual scale), especially during January, February and September. Xu et al. (2008) investigated the spatial and temporal changes of stream flow in the Yangtze River basin using the data of 154 stations and MK method. The results showed a significant increasing trend at down-stream for precipitation and stream flow time series in summer; but at up-stream areas a significant decreasing trend was observed for precipitation and stream flow data. Yonghui & Tian (2009) studied the severe changes of stream flow and factors influencing such changes. They reported the human activities (mainly agricultural) as the main factors for the decreasing trend of stream flow. Xu et al. (2010) estimated the precipitation-stream flow trend for major rivers in China using MK test in order to evaluate the role of human activities in stream flow changes. Danneberg (2012) analyzed the stream flow data of different sub-basins of Thuringia watershed in Germany. He concluded that the stream flow has a significant increasing trend at mountainous areas during winter.

The classic statistical methods usually cannot be used to analyze multiscale trends and their assumptions may cause the results to have limited use and low precision (Xie et al. 2010; Zou et al. 2011). The wavelet transform as a multi-scale pre-processing method can be utilized to extract a variety of features from the data, such as short-term and long-term fluctuations, by decomposing the time series into different sub-components. The wavelet decomposition of a non-stationary time series into various scales provides an interpretation of the time series structure and extracts significant information about its history. As a result of these features, the wavelet transform has been applied to time series analysis of non-stationary hydrological signals (e.g. Adamowski 2008a, b; Nourani et al. 2009). Also, wavelet transform analysis has the ability to elucidate simultaneously both spectral and temporal information within the signal. This overcomes the basic shortcoming of Fourier analysis, which is that the Fourier spectrum contains only globally averaged information.

In recent years, various investigations have been performed in the field of change and trend analysis of hydrological time series using wavelet transform. Jung et al. (2002) explored simple linear trends in temperature and precipitation over South Korea, and statistical significance of the calculated linear trend was tested using the t-test. Temperature time series were decomposed by wavelet transform to reveal decadal variability. They showed a positive linear trend on the wavelet coefficient of winter-time temperatures at decadal–inter-decadal variability. Pisoft et al. (2004) analyzed the cycles and trends in temperature time series of Czech by wavelet transform. Non-parametric methods such as the wavelet analysis and MK test were used to determine the possible trends in annual precipitation time series of Marmara, Turkey (Partal & Küçük 2006). Almasri et al. (2008) used discrete wavelet transform (DWT) to assess trends in the temperature time series. Xu et al. (2009) studied the impact of climatic changes in the Tarim River basin in China for the period of 1959–2006, by approximating nonlinear trends in annual temperature, precipitation and relative humidity time series using a wavelet-based decomposition and reconstruction technique. They found that all variables show nonlinear trends and/or fluctuating patterns, especially at the 4- and 8-year scales. Paul et al. (2011) applied the wavelet transform to the annual precipitation data of four different locations in northern Maharashtra region for determining the trend of the time series data. The analysis revealed an increasing trend in the total annual precipitation, and evidence of 4 and 8-year periodicity was found in the local climatic behavior. Wang et al. (2012) examined the hydroclimatic trends for the lower reach of the Shiyang River Basin in NW China using the wavelet analysis and MK test. In order to identify the optimal combination of the hydroclimatic time series obtained via DWT, the results from DWT were tested by the MK test. The results revealed that the discharge has been significantly decreased during the last five decades, while there was no noticeable change in precipitation. Temperature increased significantly in all periods, and evaporation decreased slightly in spring, summer and autumn. Nalley et al. (2012) applied the same approach to mean flow and total precipitation over southern parts of Quebec and Ontario, Canada. They found that most of the trends are positive and started during the mid-1960s to early 1970s and that intra- and inter-annual events (up to 4 years) are more influential factors, affecting the observed trends.

In the current paper, a new hybrid wavelet transform-MK method is proposed to increase the confidence level of trend analysis. The purpose of this study is to analyze the trends in stream flow and precipitation time series at different timescales in order to determine the dominant periodic components and to detect the change points of trends. Also, Sen's innovative method is used to compare and investigate the type of trend change of annual time series in six precipitation and four stream flow stations of the Tampa Bay watershed.

MATERIALS AND METHODS

Original Mann–Kendall test (MK1)

MK test was introduced and developed by Mann (1945) and Kendall (1975), respectively. The advantage of this method is that it does not follow any specific statistical distribution. It has been widely used to analyze the monotonic trends in hydrological time series. The MK test for trend detection assumes that the sample data are serially independent, even though a few hydrological series show significant serial correlation. Consider the time series of the stream flow and precipitation data for the given scale (seasonal, monthly) in each of the n years and at t stations. Each observation is denoted by which represents the observation collected in year i (i = 1,2,…n), the given scale g (g = 1,2, …,m) and station k (k = 1,2,…,t). The series for the given scale g at station k may be expressed as {X1gk, X2gk, X3gk,…,Xngk}. The MK test statistic for the series, Sgk, is the sum of all signs of consecutive observation differences defined as (Panda et al. 2007) 
formula
1
where 
formula
2
Under the null hypothesis of no trend, Sgk is asymptotically normally distributed with zero mean, and the variance is given as (Panda et al. 2007) 
formula
3
where d is the extent of any tie (i.e. length of tie), and the summation is over all the ties. For these series, having no repeated observations, d becomes 0. For a time series equal to more than 10 years, i.e. , the MK1 statistics are nearly normal distributed. Applying continuity correction, the test statistic becomes, which follows normal distribution. For testing the null hypothesis, the Z-value associated with the test statistic can be calculated 
formula
4
The null hypothesis is accepted when where are the standard normal distribution quantities. The MK test is applied to the yi series to determine the significance of the trend.
Another very useful index to quantify the monotone trend is Kendall slope , initially proposed by Sen (1968) and later extended by Hirsch et al. (1982). It is defined as 
formula
5

The estimator is the median over all combination of record pairs for the whole data set and is resistant to the extreme observations. A positive value of indicates an upward trend, and the negative value indicates a downward trend with time.

MK1 with trend-free pre-whitening (MK2)

An alternative MK test is to remove first the serial correlation such as lag-1 autoregression or higher-order process from the time series prior to application of the test. This process is called pre-whitening (Zhang et al. 2001; Burn & Hag Elnur 2002). The presence of positive serial correlation increases the probability that MK test detects trend even though no such trend exists. One approach to handle serial correlation is to consider a subset of data that ensures the data independence (Gan 1998). The MK2 test described by Yue et al. (2002) involves the following steps (Kumar et al. 2009):

  • (a) Compute the lag-1 autocorrelation coefficient (r1) using 
    formula
    6
  • (b) If then the data are assumed to be serially independent at 5% significance level (CL = 95%) and no pre-whitening is required. If, on the other hand, the calculated r1 is found to be outside the range, the corresponding data set is assumed to exhibit a significant autocorrelation at the 5% significance level and pre-whitening is required before applying the MK1.

  • (c) Compute slope in the sample data set using Equation (5) and remove the trend from the series to get a de-trended series using the following equation: 
    formula
    7
  • (d) Compute the lag-1 autocorrelation of the de-trended series, i.e. r1 using Equation (6).

  • (e) Remove the lag-1 autoregressive component (AR (1)) from the de-trended series to get a residual series as given below 
    formula
    8
  • (f) The trend is added back to the residual series to get a new series as follows: 
    formula
    9

Sequential Mann–Kendall test

The sequential MK test is used to test an assumption about the beginning of the trend development within a sample and see the change point of trend with time. The test is computed using ranked values, yi, of the original analyzed values, xi. The magnitudes of yi, (i =1,..., N) are compared with yj (j = 1,..., i −1). For each comparison, the cases where yi > yj are counted and denoted by ni.

A statistic ti can therefore be defined as ti = Σni (Mohsin & Gough 2010). For a large N, ti is normally distributed with mean E(ti) and variance Var(ti) given by 
formula
10
 
formula
11
The sequential values of the statistic U(ti) can then be computed as (Sneyers 1990) 
formula
12

U(t) is similar to the z value.

Innovation trend analysis technique of Sen

As an innovative and simple trend analysis method, Sen (2012) used subsection time series plots derived from a given time series on Cartesian coordinate system which is independent of the trend in time series, pre-assigned significance level, magnitude of trend, sample size and the amount of variation within time series. This plot shows whether any increasing or decreasing trend exists by finding the upper and lower triangle with respect to the 45° trend-free line. If all the data points lie on 1:1 line, it means that there is no trend present in the time series. In innovative trend analysis technique, scatter points above or below the 1:1 line indicate increasing or decreasing monotonic trends, respectively. Trend in the high, medium and low magnitude can be detected using this simple but still powerful innovative trend analysis methodology. This method was validated through extensive Monte Carlo simulation and then applied for two annual stream flow data sets and one precipitation time series from Turkey in addition to annual flows of the Danube River (Sen 2012).

Wavelet transform

Additional information about the properties of a time signal at different scales (resolutions) can be obtained by representing the time signal by a series of wavelet coefficients.

Wavelets can perform a local analysis, revealing aspects of data that other signal analysis techniques miss, such as trends, abrupt changes, breakdown points and discontinuities. In the field of earth sciences, Grossmann & Morlet (1984), who worked on geophysical seismic signals, introduced the wavelet transform application. A comprehensive literature survey of wavelets in geosciences can be found in Foufoula-Georgiou & Kumar (1995) and the most recent contributions have been cited by Labat (2005).

The observed hydrologic series are often discrete. As a result, the DWT is usually employed to decompose a hydrologic time series into a set of coefficients and sub-signals under different scales. According to Mallat (1998) DWT has the form of 
formula
13
where g(t) is called wavelet function or mother wavelet, m and n are integers that control the wavelet dilation and translation, respectively; a0 is a specified fined dilation step greater than 1; and b0 is the location parameter and must be greater than zero. The most common and simplest choices for parameters are a0 = 2 and b0 = 1.
This power-of-two logarithmic scaling of the translation and dilation is known as the dyadic grid arrangement. The dyadic wavelet can be written in more compact notation as (Mallat 1998) 
formula
14
Discrete dyadic wavelets of this form are commonly chosen to be orthonormal; i.e. (Mallat 1998) 
formula
15
where is Kronecker delta. This allows for an expansion of a linear combination of translates and dilates orthonormal wavelets for the complete regeneration of the original signals. For a discrete time series xi, the dyadic wavelet transform becomes (Mallat 1998) 
formula
16
where Tm,n is wavelet coefficient for the discrete wavelet of scale a = 2m and location b = 2mn. Equation (16) considers a finite time series, xi, i = 0, 1, 2,…, N − 1; and N is an integer power of 2: N = 2M. This gives the ranges of m and n as 0< n <2M m−1 and 1< m < M, respectively. At the largest wavelet scale (i.e. 2m where m = M) only one wavelet is required to cover the time interval, and only one coefficient is produced. At the next scale (2m−1), two wavelets cover the time interval, hence two coefficients are produced, and comes down to m = 1. At m = 1, then a scale is 21, i.e. 2M−1 or N/2 coefficients are required to describe the signal at this scale. The total number of wavelet coefficients for a discrete time series of length N =2M is then 1 + 2 + 4 + 8+ …+2M−1 = N − 1. In addition to this, a signal smoothed component, is left, which is the signal mean. Thus, a time series of length N is broken into N components, i.e. with zero redundancy. The inverse discrete transform is given by (Mallat 1998) 
formula
17
or in a simple format as (Mallat 1998) 
formula
18
where is called approximation sub-signal at level M and Wm(t) are details sub-signals at levels m =1, 2,…, M. The wavelet coefficients, Wm(t) (m =1, 2,…, M), provide the detail signals, which can capture small features of interpretational value in the data; the residual term, , represents the background information of data. Because of simplicity of W1(t), W2(t),…, WM (t), , some interesting characteristics, such as period, hidden period, dependence and jump can be easily diagnosed through wavelet components.

Used criteria

The correlation coefficient (Co) is usually used in statistical analysis to show linear relationship between two variables or time series. The values of Co are between 0 for a non-existent and 1 for a perfect linear correlation (Partal & Cigizoglu 2008; Rajaee et al. 2011). The Co is defined as 
formula
19
where is the average of variable X and is the average of variable Y.

Two criteria were used to select the most efficient component in producing a trend in this study. The first criterion was Co, the component with the greatest value of Co in the original time series was selected. The second criterion was the Z-MK value (Equation (4)), the component with the closest Z-MK value to the Z-MK value of original time series was selected from among the components with the highest Co in the original time series.

Case study and data

Tampa Bay, located in Gulf of Mexico on the coast of central Florida, was selected as the case study of this research. It is Florida's largest open-water estuary spanning almost 1,036 km2 and drains 5,957 km2 of land. The Tampa Bay watershed extends north of the Bay to the upper reaches of the Hills Borough River, east to the headwaters of the Alafia River, and south to the headwaters of the Manatee River. The Bay receives freshwater inflow from the Lake Tarpon Canal and the Hillsborough, Palm, Alafia, Little Manatee, and Manatee rivers. Tampa Bay empties into the Intracoastal Waterway via Boca Ciega Bay and into the Gulf of Mexico via the Southwest Channel and Passage Key Inlet. Tampa Bay is an important nursery for young fish, shrimp and crabs and provides habitat for many other types of wildlife, including wading birds, dolphins, sea turtles and manatees (Figure 1).

Figure 1

The location of stream flow and precipitation stations in Tampa Bay basin.

Figure 1

The location of stream flow and precipitation stations in Tampa Bay basin.

The data used in this study encompasses monthly stream flow and precipitation time series observed in four and six gauging stations, respectively, at Tampa Bay during 1982–2011.

The information of stations and statistics of the observed monthly data have been presented in Tables 1 and 2 for precipitation and stream flow time series, respectively. Monthly data were aggregated into seasonal and annual time series to be used in trend analysis of processes at seasonal and annual scales.

Table 1

The precipitation stations

No. stationStation nameLatitude (°)Longitude (°)Maximum monthly (mm)Standard deviation (mm)
P-1 East Lake 28.06 −82.28 714 102 
P-2 Zphyrhills 28.14 −82.10 491 91 
P-3 Richloam 28.30 −82.06 562 92 
P-4 Sarsota 27.20 −82.31 483 83 
P-5 Largo 27.54 −82.47 576 102 
P-6 Myakka 26.54 −82.9 587 96 
No. stationStation nameLatitude (°)Longitude (°)Maximum monthly (mm)Standard deviation (mm)
P-1 East Lake 28.06 −82.28 714 102 
P-2 Zphyrhills 28.14 −82.10 491 91 
P-3 Richloam 28.30 −82.06 562 92 
P-4 Sarsota 27.20 −82.31 483 83 
P-5 Largo 27.54 −82.47 576 102 
P-6 Myakka 26.54 −82.9 587 96 
Table 2

The stream flow stations

No. stationStation nameLatitude (°)Longitude (°)Drainage area (km2)Maximum monthly (m3/s)Minimum monthly (m3/s)Standard deviation (m3/s)
S-1 South Prong Alafia River at Lithia FL 27.47 −82.07 277 658.79 0.27 84.9 
S-2 Manatte River Near P-6 Head FL 27.28 −82.12 169 506.84 1.69 88.7 
S-3 Little Manatte River Near Wimauma FL 27.40 −82.21 385 1,017.61 9.88 163.08 
S-4 Alafia River at Lithia FL 27.52 −82.15 867 334.702 0.75 42.87 
No. stationStation nameLatitude (°)Longitude (°)Drainage area (km2)Maximum monthly (m3/s)Minimum monthly (m3/s)Standard deviation (m3/s)
S-1 South Prong Alafia River at Lithia FL 27.47 −82.07 277 658.79 0.27 84.9 
S-2 Manatte River Near P-6 Head FL 27.28 −82.12 169 506.84 1.69 88.7 
S-3 Little Manatte River Near Wimauma FL 27.40 −82.21 385 1,017.61 9.88 163.08 
S-4 Alafia River at Lithia FL 27.52 −82.15 867 334.702 0.75 42.87 

RESULTS AND DISCUSSION

As MK test is sensitive to the data autocorrelation, firstly the correlation of data was removed and then the time series were analyzed by wavelet and MK test at three monthly, seasonal and yearly timescales.

Results of autocorrelation analysis

Monthly, seasonal and annual stream flow and precipitation data for Tampa Bay obtained from four stream flow and six precipitation stations (from the beginning of 1982 to the end of 2011) were analyzed in this study. Firstly, the autocorrelation analysis was performed in order to investigate the significance of lag-1 autocorrelation. Lag-1 autocorrelation values on original data are presented in Tables 3 and 4. It was expected that the monthly and seasonally based data would have more autocorrelation issues compared to the annual data.

Table 3

Lag-1 autocorrelation functions of original stream flow data

No. stationMonthly dataSeasonal dataAnnual data
S-1 0.685* 0.271* 0.372* 
S-2 0.426* −0.017 0.212* 
S-3 0.446* 0.07 0.197 
S-4 0.251* −0.031 0.212* 
No. stationMonthly dataSeasonal dataAnnual data
S-1 0.685* 0.271* 0.372* 
S-2 0.426* −0.017 0.212* 
S-3 0.446* 0.07 0.197 
S-4 0.251* −0.031 0.212* 

*Indicates significant lag-1 autocorrelations at confidence level α = 5%.

Table 4

Lag-1 autocorrelation functions of original precipitation data

No. stationMonthly dataSeasonal dataAnnual data
P-1 0.199* −0.178* 0.2 
P-2 0.182* −0.076 0.27* 
P-3 0.21* −0.027 0.34* 
P-4 0.181* −0.107 0.3* 
P-5 0.229* −0.149* −0.141 
P-6 0.24* −0.73* 0.184 
No. stationMonthly dataSeasonal dataAnnual data
P-1 0.199* −0.178* 0.2 
P-2 0.182* −0.076 0.27* 
P-3 0.21* −0.027 0.34* 
P-4 0.181* −0.107 0.3* 
P-5 0.229* −0.149* −0.141 
P-6 0.24* −0.73* 0.184 

*Indicates significant lag-1 autocorrelations at confidence level α = 5%.

As seen in Tables 3 and 4, lag-1 autocorrelation values are significant in some cases, in which the original MK test (MK1) with trend-free pre-whitening (MK2) should be used for such cases.

Decomposition by DWT

In trend analysis, decomposition level of the wavelet plays a significant role as well as the type of mother wavelet used. There are many jumps in the stream flow and precipitation time series because of their sudden start and cessation over the related watershed. The structure of the Daubechies-4 (db4) wavelets is similar to the signal and they could be used to capture the signal's features (Nourani et al. 2013).

The original time series at monthly, seasonal and annual scales were decomposed into 4, 3, 3 resolution levels, respectively, via the DWT since the minimum decomposition level (K) has been suggested as (Aussem et al. 1998; Wang & Ding 2003) 
formula
20
where L is the length of the time series. In monthly scale for the case study, L = 360, that gives lower level of K = 3. To be more cautious, level 4 was considered for the monthly data and in a similar manner for seasonal and annual time series level 3 was considered.

Dyadic wavelet transform at ith level decomposition produces two types of coefficients: approximation coefficients (Ai) and detail coefficients (Dj). The Ai denotes the smoothing trend of the series. The Dj components represent a specific periodicity: in monthly data, the 2-month (D1), 4-month (D2), 8-month (D3) and 16-month periodicity (D4); in seasonal data, the 6-month (D1), 12-month (D2) and 24-month periodicity (D3); and in annual data, the 2-year (D1), 4-year (D2) and 8-year periodicity (D3). For example, Figure 2 shows the sub-series obtained via DWT applied on monthly time series of stream flow.

Figure 2

Application of DWT on monthly stream flow time series for station S-4 via db4 wavelet, at decomposition level 4.

Figure 2

Application of DWT on monthly stream flow time series for station S-4 via db4 wavelet, at decomposition level 4.

As it can be seen in Figure 2, the lower detail levels have higher frequencies, which represent the rapidly changing component of the data set, whereas the higher detail levels have lower frequencies, which represent the slowly changing component of the time series.

Trend analysis of wavelet components

After decomposing the original time series using DWT, different sub-series were created by a combination of detail and approximation components. These combinations made it possible to reveal signal features via the multi-resolution analysis. The results of trend analysis and the Co between different sub-series combinations and original time series in different scales were computed (for both precipitation and stream flow). It is important to note that generally, among all Ai + Dj compounds, the compound with the largest Co with original time series and Z-value close to the Z-value of original time series is identified as the most effective component to produce the trend of time series (such correlation coefficients and Z-MK values are presented in bold in the tables).

Trend analysis of monthly time series

The application of the MK test on the six original monthly precipitation time series (P-1 to P-6) over the study period showed an overall negative trend with negative values for Z-MK parameter. Table 5 shows the MK values for the original time series, the detail components (Dj), approximations (A4), and the combination of the Dj with the approximation.

Table 5

Mann–Kendall test results of the monthly precipitation time series

P-1
P-2
P-3
P-4
P-5
P-6
Series No.Sub-seriesZ-MK**CoZ-MKCoZ-MKCoZ-MKCoZ-MKCoZ-MKCo
Original time series −1.33 – −1.01 – −1.03 – −1.2 – −1.01 – −1.5 – 
A4 −22.6 0.68 −18.83 0.76 −15.6 0.72 −23.8 0.81 −18.8 0.82 −19.52 0.77 
D1 −0.26 0.08 0.015 −0.17 0.2 −0.18 −0.23 −0.48 −0.4 −0.19 −0.1 0.04 
D2 0.05 −0.08 0.4 −0.43 −0.2 −0.27 −0.02 −0.24 −0.4 0.07 0.04 0.2 
D3 −0.7 0.6 −0.4 0.37 −1.4 0.36 −0.36 0.32 0.02 0.37 −0.2 0.24 
D4 0.1 0.25 2.44* 0.21 −0.46 0.36 1.84 0.12 0.56 0.62 0.82 −0.003 
A4 + D1 2.920.78 1.4 0.89 1.11 0.76 2.70.87 1.5 0.87 1.6 0.86 
A4 + D2 −3.5* 0.7 −1.9 0.82 −1.5 0.7 −3.5* 0.88 −1.7 0.88 −3.2* 0.91 
A4 + D3 −3.6* 0.83 −3.2* 0.91 −3.5* 0.91 −4.7* 0.89 −1.8 0.85 −3.5* 0.92 
10 A4 + D4 −12.5* 0.65 −9.7* 0.91 −5.9* 0.77 −15.6* 0.84 −5.9* 0.86 −10* 0.86 
11 A4 + D1 + D2 −2.46 0.77 1.22 0.87 −1.1 0.71 −1.53 0.84 −0.75 0.84 −1.5 0.89 
12 A4 + D1 + D3 −1.5 0.91 −0.8 0.9 −1.24 0.95 −1.7 0.89 0.73 0.91 −1.5 0.95 
13 A4 + D1 + D4 −2.3* 0.76 −1.32 0.88 −1.3 0.82 −2.2* 0.89 −0.77 0.89 −1.6 0.87 
14 A4 + D1 + D3 + D4 −1.4 0.93 −0.74 0.97 −1.29 0.96 −1.6 0.94 −0.76 0.96 −1.4 0.96 
15 A4 + D1 + D2 + D4 −2.2* 0.77 −1.1 0.89 −0.94 0.76 −1.4 0.88 −0.59 0.91 −1.3 0.89 
16 A4 + D1 + D2 + D3 −1.4 0.92 −1.05 0.91 −1.06 0.96 −1.24 0.91 −0.8 0.87 −1.5 0.96 
17 A4 + D2 + D3 2.20.83 −1.5 0.93 −1.96 0.93 −1.98* 0.89 −0.4 0.88 −2.9* 0.95 
18 A4 + D2 + D4 −3.2* 0.7 −1.6 0.88 1.5 0.75 3.20.9 −1.8 0.92 −3.2* 0.89 
19 A4 + D3 + D4 −3.2* 0.9 −2.7* 0.93 −3.2* 0.96 −4.7* 0.92 −2.3* 0.96 −3.2* 0.92 
20 A4 + D2 + D3 + D4 −1.9 0.93 −1.44 0.97 −2.1* 0.98 −2.1* 0.97 −0.7 0.98 −2.6* 0.95 
P-1
P-2
P-3
P-4
P-5
P-6
Series No.Sub-seriesZ-MK**CoZ-MKCoZ-MKCoZ-MKCoZ-MKCoZ-MKCo
Original time series −1.33 – −1.01 – −1.03 – −1.2 – −1.01 – −1.5 – 
A4 −22.6 0.68 −18.83 0.76 −15.6 0.72 −23.8 0.81 −18.8 0.82 −19.52 0.77 
D1 −0.26 0.08 0.015 −0.17 0.2 −0.18 −0.23 −0.48 −0.4 −0.19 −0.1 0.04 
D2 0.05 −0.08 0.4 −0.43 −0.2 −0.27 −0.02 −0.24 −0.4 0.07 0.04 0.2 
D3 −0.7 0.6 −0.4 0.37 −1.4 0.36 −0.36 0.32 0.02 0.37 −0.2 0.24 
D4 0.1 0.25 2.44* 0.21 −0.46 0.36 1.84 0.12 0.56 0.62 0.82 −0.003 
A4 + D1 2.920.78 1.4 0.89 1.11 0.76 2.70.87 1.5 0.87 1.6 0.86 
A4 + D2 −3.5* 0.7 −1.9 0.82 −1.5 0.7 −3.5* 0.88 −1.7 0.88 −3.2* 0.91 
A4 + D3 −3.6* 0.83 −3.2* 0.91 −3.5* 0.91 −4.7* 0.89 −1.8 0.85 −3.5* 0.92 
10 A4 + D4 −12.5* 0.65 −9.7* 0.91 −5.9* 0.77 −15.6* 0.84 −5.9* 0.86 −10* 0.86 
11 A4 + D1 + D2 −2.46 0.77 1.22 0.87 −1.1 0.71 −1.53 0.84 −0.75 0.84 −1.5 0.89 
12 A4 + D1 + D3 −1.5 0.91 −0.8 0.9 −1.24 0.95 −1.7 0.89 0.73 0.91 −1.5 0.95 
13 A4 + D1 + D4 −2.3* 0.76 −1.32 0.88 −1.3 0.82 −2.2* 0.89 −0.77 0.89 −1.6 0.87 
14 A4 + D1 + D3 + D4 −1.4 0.93 −0.74 0.97 −1.29 0.96 −1.6 0.94 −0.76 0.96 −1.4 0.96 
15 A4 + D1 + D2 + D4 −2.2* 0.77 −1.1 0.89 −0.94 0.76 −1.4 0.88 −0.59 0.91 −1.3 0.89 
16 A4 + D1 + D2 + D3 −1.4 0.92 −1.05 0.91 −1.06 0.96 −1.24 0.91 −0.8 0.87 −1.5 0.96 
17 A4 + D2 + D3 2.20.83 −1.5 0.93 −1.96 0.93 −1.98* 0.89 −0.4 0.88 −2.9* 0.95 
18 A4 + D2 + D4 −3.2* 0.7 −1.6 0.88 1.5 0.75 3.20.9 −1.8 0.92 −3.2* 0.89 
19 A4 + D3 + D4 −3.2* 0.9 −2.7* 0.93 −3.2* 0.96 −4.7* 0.92 −2.3* 0.96 −3.2* 0.92 
20 A4 + D2 + D3 + D4 −1.9 0.93 −1.44 0.97 −2.1* 0.98 −2.1* 0.97 −0.7 0.98 −2.6* 0.95 

*Significant trend at % 95 significance level (critical value = 1.96).

**Underlined Z-MKs obtained via MK1.

Similar to previous works of Kumar et al. (2009) and Panda et al. (2007), the confidence level was set at 95% in this study. Considering this confidence level, none of the individual time series showed significant MK values. Only the combination of approximation with A4 component did not show statistically significant Z-MK values (i.e. time series Nos. 7, 8). The results of the monthly precipitation data analysis show how the approximation components affect the detail components by increasing their trend values (in most cases) as reflected by the higher MK values (Table 5). The trend directions, after adding A4 to the sub-series, were also always in agreement with those of the corresponding original data. It is clear that the approximation sub-series impact on the original data because they carry the greatest portion of the process's trend.

An example of how sequential MK graphs of the different periodic components portray their trend lines with respect to those of the original time series is given in Figure 3 (for station P-2). The upper and lower dashed lines represent the confidence limits (α = 5%). The result of trends analysis and Co between results of sequential MK test for different wavelet combinations and original time series at different scales are shown in Table 5. It should be noted that, generally, the greater correlation and closer the MK Z-value of a component to the original time series denotes its importance and effectiveness in trend analysis of the process. Z-value of compound 7 (A4 + D1) shows better agreement with the original time series.

Figure 3

Sequential MK graphs of precipitation in station P-2 at monthly scale: (a) A4 + D1, (b) A4 + D2, (c) A4 + D3, (d) A4 + D4.

Figure 3

Sequential MK graphs of precipitation in station P-2 at monthly scale: (a) A4 + D1, (b) A4 + D2, (c) A4 + D3, (d) A4 + D4.

Obviously, in a case (component) with more than one detail sub-series added to the approximation, the Co with the original time series would be greater. For instance, in Table 5 related to station P-4, it is seen that all four combinations (A4 + D1, A4 + D2, A4 + D3 and A4 + D4) show higher Co. Despite this, because its Z value is closest to the original time series, A4 + D1 is selected as the best choice. If no noticeable difference is observed among the Z values of these components, a Z with the closest value to the original time series should be selected among components 11–20 and the participating details in these components are known as the most effective details in producing the trend.

As well as precipitation, the proposed wavelet-based trend analysis was applied to the stream flow data. In this way, by applying the MK test over different combinations of DWT, it could be deduced that stream flow time series of all stations, except S-2, have approximately similar behaviors in monthly scale with negative trends. The results of MK test on different combinations of DWT are presented in Table 6. Two stations (i.e. S-3 and S-4) show significant trends at 5% level (with Z values of −1.96 and −2.72, respectively). The observed positive trend of S-2 station was low (Z = 0.29).

Table 6

Mann–Kendall test results of monthly stream flow time series

S-1
S-2
S-3
S-4
Series No.Sub-seriesZ-MK**CoZ-MKCoZ-MKCoZ-MKCo
Original time series −1.82 – 0.29 – −1.96* – −2.72* – 
2 A4 −11* 0.88 4.67* 0.89 −14.31* 0.79 −20.36* 0.56 
D1 −0.42 0.16 −0.15 −0.35 −0.39 −0.45 −0.05 −0.13 
D2 0.09 −0.08 0.16 0.19 −0.23 −0.18 −0.77 0.34 
D3 0.08 0.29 −1.54 0.07 −1.12 0.44 −1.03 0.44 
D4 −1.71 0.42 −1.74 0.13 −1.89 0.30 0.55 0.31 
A4 + D1 1.56 0.88 1.07 0.86 1.32 0.78 −1.93 0.60 
A4 + D2 −2.05* 0.82 0.37 0.89 1.75 0.79 3.150.72 
A4 + D3 −3.70* 0.85 −1.36 0.91 −4.39* 0.83 −5.49* 0.64 
10 A4 + D4 −7.63* 0.90 1.06 0.88 −5.90* 0.83 −9.88* 0.79 
11 A4 + D1 + D2 −1.36 0.83 0.30 0.86 −1.34 0.80 1.77 0.75 
12 A4 + D1 + D3 −1.49 0.87 −0.24 0.93 −1.81 0.86 1.86 0.66 
13 A4 + D1 + D4 −1.75 0.93 0.76 0.89 −1.27 0.82 1.75 0.69 
14 A4 + D1 + D3 + D4 −1.97* 0.95 −0.26 0.94 −1.91 0.92 −1.83 0.85 
15 A4 + D1 + D2 + D4 −1.85 0.94 0.33 0.91 −1.48 0.90 −1.74 0.88 
16 A4 + D1 + D2 + D3 −1.20 0.84 −0.27 0.85 −1.87 0.87 −2.05* 0.77 
17 A4 + D2 + D3 −1.65 0.87 −0.31 0.89 −2.18* 0.87 −2.73* 0.78 
18 A4 + D2 + D4 −2.78* 0.95 0.50 0.93 −1.69 0.90 −2.96* 0.87 
19 A4 + D3 + D4 −3.75* 0.93 −1.03 0.93 −4.03* 0.90 −4.72* 0.79 
20 A4 + D2 + D3 + D4 −2.09* 0.99 0.18 0.98 −1.89 0.99 −2.77* 0.96 
S-1
S-2
S-3
S-4
Series No.Sub-seriesZ-MK**CoZ-MKCoZ-MKCoZ-MKCo
Original time series −1.82 – 0.29 – −1.96* – −2.72* – 
2 A4 −11* 0.88 4.67* 0.89 −14.31* 0.79 −20.36* 0.56 
D1 −0.42 0.16 −0.15 −0.35 −0.39 −0.45 −0.05 −0.13 
D2 0.09 −0.08 0.16 0.19 −0.23 −0.18 −0.77 0.34 
D3 0.08 0.29 −1.54 0.07 −1.12 0.44 −1.03 0.44 
D4 −1.71 0.42 −1.74 0.13 −1.89 0.30 0.55 0.31 
A4 + D1 1.56 0.88 1.07 0.86 1.32 0.78 −1.93 0.60 
A4 + D2 −2.05* 0.82 0.37 0.89 1.75 0.79 3.150.72 
A4 + D3 −3.70* 0.85 −1.36 0.91 −4.39* 0.83 −5.49* 0.64 
10 A4 + D4 −7.63* 0.90 1.06 0.88 −5.90* 0.83 −9.88* 0.79 
11 A4 + D1 + D2 −1.36 0.83 0.30 0.86 −1.34 0.80 1.77 0.75 
12 A4 + D1 + D3 −1.49 0.87 −0.24 0.93 −1.81 0.86 1.86 0.66 
13 A4 + D1 + D4 −1.75 0.93 0.76 0.89 −1.27 0.82 1.75 0.69 
14 A4 + D1 + D3 + D4 −1.97* 0.95 −0.26 0.94 −1.91 0.92 −1.83 0.85 
15 A4 + D1 + D2 + D4 −1.85 0.94 0.33 0.91 −1.48 0.90 −1.74 0.88 
16 A4 + D1 + D2 + D3 −1.20 0.84 −0.27 0.85 −1.87 0.87 −2.05* 0.77 
17 A4 + D2 + D3 −1.65 0.87 −0.31 0.89 −2.18* 0.87 −2.73* 0.78 
18 A4 + D2 + D4 −2.78* 0.95 0.50 0.93 −1.69 0.90 −2.96* 0.87 
19 A4 + D3 + D4 −3.75* 0.93 −1.03 0.93 −4.03* 0.90 −4.72* 0.79 
20 A4 + D2 + D3 + D4 −2.09* 0.99 0.18 0.98 −1.89 0.99 −2.77* 0.96 

*Significant trend at % 95 significance level (critical value = 1.96).

**Underlined Z-MKs obtained via MK1.

None of the individual details showed a significant trend. However, in most cases, by adding the approximate sub-signal to the details, the trends became more significant in the same trend direction as the original time series. This means that most trend components are represented by approximates. To determine the most dominant periodicities of trends, the sequential MK graph of each detail component (added to the approximation sub-signal) was examined with respect to the original data (see Figure 4). The Co between results of sequential MK test (for different DWT combinations) and original time series are shown in Table 6. Based on the MK values and the sequential MK graphs, D2 (added to the approximation sub-signal) was determined to be the most dominant periodic component affecting the trend in three stations but in S-1, D1 was known as the most influential periodicity.

Figure 4

Sequential MK graphs of stream flow in station S-1 at monthly scale. The upper and lower dashed lines represent the confidence limits (α = 5%): (a) A4 + D1, (b) A4 + D2, (c) A4 + D3, (d) A4 + D4.

Figure 4

Sequential MK graphs of stream flow in station S-1 at monthly scale. The upper and lower dashed lines represent the confidence limits (α = 5%): (a) A4 + D1, (b) A4 + D2, (c) A4 + D3, (d) A4 + D4.

Trend analysis of seasonal time series

The seasonally based data confirm the presence of annual cycles in some data sets as the ACFs show high values at every fourth lag (Tables 3 and 4). Each seasonally based time series was decomposed into three detail components (D1–D3) and one approximation (A3). D1, D2 and D3 represent the 6, 12 and 24-month fluctuations, respectively. The A3 component corresponds to the approximation of the third decomposition level.

All MK Z-values for the original seasonally based precipitation data showed negative trends (Table 7), none of the stations experienced statistically significant trends, which is in agreement with the results of the monthly precipitation data.

Table 7

Mann–Kendall test results of the seasonal precipitation time series

P-1
P-2
P-3
P-4
P-5
P-6
Series No.Sub-seriesZ-MK**CoZ-MKCoZ-MKCoZ-MKCoZ-MKCoZ-MKCo
Original time series −0.72 – 0.8 – 0.67 – −1.3 – −0.52 – −0.9 – 
A3 −8.05 0.61 −9.64 0.55 −6.3 0.56 −10.7 0.75 −4.7 0.8 8.7 0.88 
D1 0.16 0.35 0 0.26 −0.12 0.16 0.06 0.21 0.33 0.36 0.14 
D2 0.34 0.33 0.24 −0.2 0.25 0.25 0.24 0.3 0.21 0.04 0.08 0.16 
D3 −1.13 0.57 −1.01 0.72 −1.24 0.39 0.02 0.41 −0.02 0.53 −1.5 0.61 
A3 + D1 0.9 0.67 0.77 0.88 −0.06 0.84 1.5 0.89 −0.3 0.84 0.8 0.92 
A3 + D2 −1.04 0.56 −1.03 0.85 0.69 0.73 −1.7 0.86 0.6 0.72 −1.22 0.91 
A3 + D3 −5.7* 0.72 −3.6* 0.9 −1.97* 0.76 −7.1* 0.85 −1.3 0.75 −5.1* 0.83 
A3 + D1 + D2 −0.8 0.88 −0.8 0.94 −0.43 0.9 −1.45 0.88 −0.46 0.9 0.62 0.97 
10 A3 + D1 + D3 −1.2 0.86 0.69 0.91 −0.27 0.9 −1.5 0.95 −0.48 0.9 −0.85 0.94 
11 A3 + D2 + D3 −1.3 0.8 −1.07 0.91 −1.07 0.94 −1.9 0.93 −1.01 0.79 −1.14 0.95 
P-1
P-2
P-3
P-4
P-5
P-6
Series No.Sub-seriesZ-MK**CoZ-MKCoZ-MKCoZ-MKCoZ-MKCoZ-MKCo
Original time series −0.72 – 0.8 – 0.67 – −1.3 – −0.52 – −0.9 – 
A3 −8.05 0.61 −9.64 0.55 −6.3 0.56 −10.7 0.75 −4.7 0.8 8.7 0.88 
D1 0.16 0.35 0 0.26 −0.12 0.16 0.06 0.21 0.33 0.36 0.14 
D2 0.34 0.33 0.24 −0.2 0.25 0.25 0.24 0.3 0.21 0.04 0.08 0.16 
D3 −1.13 0.57 −1.01 0.72 −1.24 0.39 0.02 0.41 −0.02 0.53 −1.5 0.61 
A3 + D1 0.9 0.67 0.77 0.88 −0.06 0.84 1.5 0.89 −0.3 0.84 0.8 0.92 
A3 + D2 −1.04 0.56 −1.03 0.85 0.69 0.73 −1.7 0.86 0.6 0.72 −1.22 0.91 
A3 + D3 −5.7* 0.72 −3.6* 0.9 −1.97* 0.76 −7.1* 0.85 −1.3 0.75 −5.1* 0.83 
A3 + D1 + D2 −0.8 0.88 −0.8 0.94 −0.43 0.9 −1.45 0.88 −0.46 0.9 0.62 0.97 
10 A3 + D1 + D3 −1.2 0.86 0.69 0.91 −0.27 0.9 −1.5 0.95 −0.48 0.9 −0.85 0.94 
11 A3 + D2 + D3 −1.3 0.8 −1.07 0.91 −1.07 0.94 −1.9 0.93 −1.01 0.79 −1.14 0.95 

*Significant trend at % 95 significance level (critical value = 1.96).

**Underlined Z-MKs obtained via MK1.

Also, Table 7 shows the periodic components that are the most influential in the trends of the seasonally based precipitation data. Most of the precipitation trends are affected by the A3 + D1 component, except for station P-3, where A3 + D2 is considered as the most dominant periodic component. It could be suggested that the periodic components that mainly affect trends in the seasonally based total precipitation data are 6- and 12-month modes.

An example of the sequential MK graph for the different periodic components portraying their trend lines with respect to the original time series is given in Figure 5 (for station P-5).

Figure 5

Sequential MK graphs of precipitation in station P-5 at seasonal scale. The upper and lower dashed lines represent the confidence limits (α = 5%): (a) A3 + D1, (b) A3 + D2, (c) A3 + D3.

Figure 5

Sequential MK graphs of precipitation in station P-5 at seasonal scale. The upper and lower dashed lines represent the confidence limits (α = 5%): (a) A3 + D1, (b) A3 + D2, (c) A3 + D3.

In the seasonal scale, none of the details showed significant trend. The Z-value of A3 + D1 compound showed better agreement with the Z-value of original time series. Most of the trends in the monthly and seasonal data are influenced by higher periodic components (i.e. lower-frequency events).

For the stream flow data, all of the stations showed negative trends in seasonal scale, among the observed trends in original time series, only S-4 station with Z = −2.12 showed a significant trend at 5% significance level. As seen in Table 8, despite the low trend of individual details, the Z-MK value was increased after adding the approximation component which led to displaying a trend, and even became significant in some cases. For example, Z-value of −0.12 for D2 at South Prong (S-4) increased to −2.15 after adding the approximation component. The analysis of sequential seasonal MK diagrams (Figure 6) and the observed Z-values (Table 8) for different components show that D1 and D2 compounds, which represent the 6- and 12-month time modes are most influential in producing seasonal trend.

Table 8

Mann–Kendall test results of the seasonal stream flow time series

S-1
S-2
S-3
S-4
Series No.Sub-seriesZ-MK**CoZ-MKCoZ-MKCoZ-MKCo
Original time series −1.10 – 0.05 – 1.08 – 2.12– 
A3 −4.24* 0.9 2.180.72 −3.19* 0.74 −6.2* 0.45 
D1 0.13 0.02 0.25 0.32 0.39 −0.06 0.40 −0.04 
D2 −0.12 0.19 −0.08 0.38 0.03 0.44 −0.20 0.33 
D3 0.03 0.49 −1.81 0.35 −2.15* 0.61 −1.87 0.41 
A3 + D1 0.91 0.81 0.38 0.86 0.80 0.79 −0.67 0.58 
A3 + D2 −2.15* 0.93 0.15 0.88 −1.37 0.87 2.070.72 
A3 + D3 −2.62* 0.93 −0.80 0.91 −3.12* 0.86 −5.10* 0.70 
A3 + D1 + D2 −0.89 0.96 0.00 0.86 −0.87 0.88 1.37 0.81 
10 A3 + D1 + D3 −0.82 0.95 0.05 0.87 0.86 0.85 1.31 0.71 
11 A3 + D2 + D3 −1.82 0.96 −0.42 0.91 −1.26 0.90 −2.26* 0.83 
S-1
S-2
S-3
S-4
Series No.Sub-seriesZ-MK**CoZ-MKCoZ-MKCoZ-MKCo
Original time series −1.10 – 0.05 – 1.08 – 2.12– 
A3 −4.24* 0.9 2.180.72 −3.19* 0.74 −6.2* 0.45 
D1 0.13 0.02 0.25 0.32 0.39 −0.06 0.40 −0.04 
D2 −0.12 0.19 −0.08 0.38 0.03 0.44 −0.20 0.33 
D3 0.03 0.49 −1.81 0.35 −2.15* 0.61 −1.87 0.41 
A3 + D1 0.91 0.81 0.38 0.86 0.80 0.79 −0.67 0.58 
A3 + D2 −2.15* 0.93 0.15 0.88 −1.37 0.87 2.070.72 
A3 + D3 −2.62* 0.93 −0.80 0.91 −3.12* 0.86 −5.10* 0.70 
A3 + D1 + D2 −0.89 0.96 0.00 0.86 −0.87 0.88 1.37 0.81 
10 A3 + D1 + D3 −0.82 0.95 0.05 0.87 0.86 0.85 1.31 0.71 
11 A3 + D2 + D3 −1.82 0.96 −0.42 0.91 −1.26 0.90 −2.26* 0.83 

*Significant trend at % 95 significance level (critical value = 1.96).

**Underlined Z-MKs obtained via MK1.

Figure 6

Sequential MK graphs of stream flow in station S-3 at seasonal scale. The upper and lower dashed lines represent the confidence limits (α = 5%): (a) A3 + D1, (b) A3 + D2, (c) A3 + D3.

Figure 6

Sequential MK graphs of stream flow in station S-3 at seasonal scale. The upper and lower dashed lines represent the confidence limits (α = 5%): (a) A3 + D1, (b) A3 + D2, (c) A3 + D3.

Trend analysis of annual time series

Each annual time series was decomposed into three levels corresponding to the 2-, 4- and 8-year variations. All MK Z-values for the original annual precipitation data showed negative trends (Table 9); none of stations experienced statistically significant trends, which is in agreement with the results of the monthly and seasonal precipitation time series. The dominant periodic components of trends in precipitation data are indicated in Table 9. Examples of dominant periodic components that affect the development of trends in annual precipitation time series are given in Figure 7. None of individual detail component (without adding approximation) showed significant trend values (Table 9). The Z-value of A3 + D1 compound at all stations and scales shows good agreement with Z-value of original time series with greatest Co compared to other compounds and its sequential graphs showed greater harmony. As a result, 2-month, 6-month and 2-year periods were taken as the dominant periods for monthly, seasonal and annual scales, respectively, in all stations. The lowest periodic mode (D1) was the most prominent affecting the trend structure involved in the precipitation time series in monthly, seasonal and annual timescales.

Table 9

Mann–Kendall test results of the annual precipitation time series

P-1
P-2
P-3
P-4
P-5
P-6
Series No.Sub-seriesZ-MK**CoZ-MKCoZ-MKCoZ-MKCoZ-MKCoZ-MKCo
Original time series 1.4 – 0.6 – −0.2 – −1.02 – 0.5 – 1.03 – 
A3 −3.8 0.5 6.88 0.54 0.54 0.8 −6.27 0.67 1.8 0.6 −5.15 0.75 
D1 0.05 0.34 0.3 −0.05 −0.5 0.4 −0.25 0.41 0.06 0.48 −0.3 0.52 
D2 0.035 0.54 0.35 0.43 1.33 0.54 0.07 0.46 −0.25 0.38 −0.03 0.7 
D3 0.7 0.74 −1.4 0.6 0.8 0.66 1.11 0.44 −0.77 0.26 0.8 0.79 
A3 + D1 1.85 0.28 0 0.63 0 0.85 0.75 0.79 0.28 0.78 0.96 0.77 
A3 + D2 0.02 0.56 −0.24 0.62 0.95 0.75 −1.18 0.86 0.54 0.62 −1.44 0.83 
A3 + D3 −3.2* 0.18 0.6 0.72 −1.3 0.82 −0.2 0.79 −1.35 0.64 −1.78 0.82 
A3 + D1 + D2 −1.1 0.72 0 0.64 0.14 0.93 0.64 0.85 0.39 0.82 1.03 0.93 
10 A3 + D1 + D3 1.9 0.5 0.46 0.88 0.2 0.92 0.85 0.95 0.35 0.88 0.9 0.88 
11 A3 + D2 + D3 −0.3 0.85 −1.2 0.79 0.6 0.76 −0.89 0.82 −0.24 0.89 1.2 0.94 
P-1
P-2
P-3
P-4
P-5
P-6
Series No.Sub-seriesZ-MK**CoZ-MKCoZ-MKCoZ-MKCoZ-MKCoZ-MKCo
Original time series 1.4 – 0.6 – −0.2 – −1.02 – 0.5 – 1.03 – 
A3 −3.8 0.5 6.88 0.54 0.54 0.8 −6.27 0.67 1.8 0.6 −5.15 0.75 
D1 0.05 0.34 0.3 −0.05 −0.5 0.4 −0.25 0.41 0.06 0.48 −0.3 0.52 
D2 0.035 0.54 0.35 0.43 1.33 0.54 0.07 0.46 −0.25 0.38 −0.03 0.7 
D3 0.7 0.74 −1.4 0.6 0.8 0.66 1.11 0.44 −0.77 0.26 0.8 0.79 
A3 + D1 1.85 0.28 0 0.63 0 0.85 0.75 0.79 0.28 0.78 0.96 0.77 
A3 + D2 0.02 0.56 −0.24 0.62 0.95 0.75 −1.18 0.86 0.54 0.62 −1.44 0.83 
A3 + D3 −3.2* 0.18 0.6 0.72 −1.3 0.82 −0.2 0.79 −1.35 0.64 −1.78 0.82 
A3 + D1 + D2 −1.1 0.72 0 0.64 0.14 0.93 0.64 0.85 0.39 0.82 1.03 0.93 
10 A3 + D1 + D3 1.9 0.5 0.46 0.88 0.2 0.92 0.85 0.95 0.35 0.88 0.9 0.88 
11 A3 + D2 + D3 −0.3 0.85 −1.2 0.79 0.6 0.76 −0.89 0.82 −0.24 0.89 1.2 0.94 

*Significant trend at % 95 significance level (critical value = 1.96).

**Underlined Z-MKs obtained via MK1.

Figure 7

Sequential MK graphs of precipitation in station P-6 at annual scale. The upper and lower dashed lines represent the confidence limits (α = 5%): (a) A3 + D1, (b) A3 + D2, (c) A3 + D3.

Figure 7

Sequential MK graphs of precipitation in station P-6 at annual scale. The upper and lower dashed lines represent the confidence limits (α = 5%): (a) A3 + D1, (b) A3 + D2, (c) A3 + D3.

For the annual stream flow data, only time series of the station S-2 experienced positive trend value (Z = +0.21) (Table 10). There was not any component with significant trend values (Table 10). The lower values of Z-MK in annual scale in comparison to the other timescales denote to the lower trend in annual time series of stream flow.

Table 10

Mann–Kendall test results of the annual stream flow series

Series No.Sub-seriesS-1S-2S-3S-4
Original time series Z-MK** Co Z-MK Co Z-MK Co Z-MK Co 
 −0.36 – 0.21 – 0.64 – 0.75 –  
2 A3 2.79* 0.53 4* 0.84 −4.71* 0.76 −6.17* 0.58 
3 D1 0.04 0.04 −0.24 −0.06 −0.09 0.46 0.28 0.36 
D2 0.25 0.42 0.25 0.72 0.28 0.53 0.32 0.47 
D3 −0.21 0.53 0.02 0.73 1.03 0.58 0.32 0.63 
A3 + D1 0.39 0.35 0.71 0.79 0.79 0.72 1.43 0.75 
A3 + D2 0.36 0.74 1.11 0.84 0.62 0.76 −1.48 0.79 
A3 + D3 −1.33 0.80 1.74 0.84 −1.48 0.76 1.11 0.71 
A3 + D1 + D2 0.18 0.81 0.39 0.88 0.82 0.88 0.89 0.88 
10 A3 + D1 + D3 −1.22 0.77 0.11 0.91 0.71 0.89 0.93 0.94 
11 A3 + D2 + D3 −0.54 0.95 0.54 0.90 −0.39 0.86 −0.88 0.79 
Series No.Sub-seriesS-1S-2S-3S-4
Original time series Z-MK** Co Z-MK Co Z-MK Co Z-MK Co 
 −0.36 – 0.21 – 0.64 – 0.75 –  
2 A3 2.79* 0.53 4* 0.84 −4.71* 0.76 −6.17* 0.58 
3 D1 0.04 0.04 −0.24 −0.06 −0.09 0.46 0.28 0.36 
D2 0.25 0.42 0.25 0.72 0.28 0.53 0.32 0.47 
D3 −0.21 0.53 0.02 0.73 1.03 0.58 0.32 0.63 
A3 + D1 0.39 0.35 0.71 0.79 0.79 0.72 1.43 0.75 
A3 + D2 0.36 0.74 1.11 0.84 0.62 0.76 −1.48 0.79 
A3 + D3 −1.33 0.80 1.74 0.84 −1.48 0.76 1.11 0.71 
A3 + D1 + D2 0.18 0.81 0.39 0.88 0.82 0.88 0.89 0.88 
10 A3 + D1 + D3 −1.22 0.77 0.11 0.91 0.71 0.89 0.93 0.94 
11 A3 + D2 + D3 −0.54 0.95 0.54 0.90 −0.39 0.86 −0.88 0.79 

*Significant trend at % 95 significance level (critical value = 1.96).

**Underlined Z-MKs obtained via MK1.

The D1 and D2 components, which represent the 2- and 4-year time modes, were also seen to be important in affecting the trends (Table 10 and Figure 8), but in S-4, D3 was detected as the most dominant periodic component of trend.

Figure 8

Sequential MK graphs of stream flow in station S-3 at annual scale. The upper and lower dashed lines represent the confidence limits (α = 5%): (a) A3 + D1, (b) A3 + D2, (c) A3 + D3.

Figure 8

Sequential MK graphs of stream flow in station S-3 at annual scale. The upper and lower dashed lines represent the confidence limits (α = 5%): (a) A3 + D1, (b) A3 + D2, (c) A3 + D3.

Figure 9 presents scatter plots between dominant details (added to approximate) and original time series at annual and seasonal scales for stations S-4 and P-6. Figure 9 indicates almost a reasonable agreement between dominant details (added to approximate) and the original time series.

Figure 9

Scatter plots between dominant detail (added to approximate) and original time series: (a) annual time series in S-4 station; (b) seasonal time series in S-4 station; (c) annual time series in P-6 station; (d) seasonal time series in P-6 station.

Figure 9

Scatter plots between dominant detail (added to approximate) and original time series: (a) annual time series in S-4 station; (b) seasonal time series in S-4 station; (c) annual time series in P-6 station; (d) seasonal time series in P-6 station.

Result of the Sen method

Despite the advantages of the compound methods of DWT and MK to identify trend in hydrological processes including precipitation and stream flow, a simple qualitative but useful method was introduced by Sen in 2012 which identifies the quantity and quality of trends in total time series. This method was used for annual time series of precipitation and stream flow in all stations of the present research in which the findings confirm the compound method applied in this study.

For this purpose, annual stream flow and precipitation time series were divided into two portions with respect to time and arranged in the ascending order. Based on the scatter diagrams prepared by annual time series from 1982 to 2011, the trend analysis via Sen's method showed good agreement with the obtained results of MK in the most cases. For example, for data of P-4 station, the original annual time series got Z-MK = −1.02 that the result of Sen method in scatter diagram (Figure 10) justifies the same result. As seen in Figure 10, the low and medium precipitation clusters showed slight decreasing trend and high values show increasing in 1997–2011 duration compared to 1982–1996.

Figure 10

Results of innovative trend detection technique: (a) P-4 station, (b) S-3 station (1982–2011).

Figure 10

Results of innovative trend detection technique: (a) P-4 station, (b) S-3 station (1982–2011).

In order to study the detailed trend, the combinations of all details with approximates were also analyzed. The scatter diagrams of these components clearly indicate a negative trend for all details for data of P-4 station (Figure 11).

Figure 11

The scatter plots of details (with approximate) in P-4 station: (a) A3 + d1, (b) A3 + d2, (c) A3 + d3 sub-series.

Figure 11

The scatter plots of details (with approximate) in P-4 station: (a) A3 + d1, (b) A3 + d2, (c) A3 + d3 sub-series.

Result of sequential MK test

The sequential MK test identifies the start-end time of a trend in an annual time series. In general, for annual stream flow data, an increasing trend was observed from 1990 to 1998, and a decreasing trend from 1999 onward (Figure 12).

Figure 12

The sequential MK test graphs for annual time series of stream flow.

Figure 12

The sequential MK test graphs for annual time series of stream flow.

The decreasing trend in most precipitation time series of all stations started in 1989 and continued to 2001, a following decreasing trend started in 2003 (Figure 13).

Figure 13

The sequential MK test graphs for annual time series of precipitation.

Figure 13

The sequential MK test graphs for annual time series of precipitation.

SUMMARY AND CONCLUSIONS

In this study, the involved trend in stream flow and precipitation time series was analyzed using the data from four hydrometery stations and six rain gauges, respectively, at three scales (i.e. monthly, seasonal and annual). DWT and MK tests were used to analyze the trend in hydrological time series and identify the detail, with the most effective role in creating trends in time series. The periodic components that affect the trends were not the same in stream flow and precipitation time series, but short-term periods influenced both phenomena. Both stream flow and precipitation time series showed similar behavior so that most observed trends were negative. Only at Manattee River, low positive trend was observed for monthly and annual scales.

The results indicate that different precipitation stations had similar observations, indicating that for all scales Ai + D1 compound had the closest Z-value to the original time series and the diagram of sequential MK test for this compound showed a rather greater harmony and correlation with the original time series; therefore, all time series had a short-term period as 2-month, 6-month and 2-year for monthly, seasonal and annual scales, respectively. For most stream flow time series at three scales, D2 (by adding approximate sub-signal) was conceded as the dominant periodic component, but at Alafia River it was observed that in annual scale, 8-year period (D3) was responsible in producing trend. Only Little Manattee River and Alifia River at Lithia FL experienced significant trends at monthly scale. As seen in Tables 5,678910, a large number of Z values obtained via MK2 test indicate that most time series and their components have significant lag-1 Co.

The overall results of the study indicate that during the period of 1982–2011, for all three scales there was a negative trend for precipitation and stream flow. Stream flow trends are related to precipitation trends, the MK analysis that was performed for the three timescales revealed this fact. Mostly, stream flow and precipitation time series showed negative trends but not significant. Climate change or abatement of precipitation plays an important role in reduction of stream flow rate. Other activities, such as slope of the stations or humanistic factors such as dams or the changes in the types of plants covering the regions, might interfere; yet precipitation and climate change remain as the most dominant factors.

Sen's innovative method for analyzing trend in hydrological time series was also applied in this study in which the findings were in accordance with MK test results and confirmed the existence of a negative trend in annual precipitation and stream flow time series.

Finally, the result of this study showed the changes in stream flow and precipitation processes during 1985–2011 in the Tampa Bay region, the changes of stream flow trend since 1999 showed a more negative slope and the negative trend was more noticeable from 2003. The findings may help designers and managers of water resources develop robust models.

Overall, we believe that this study provides an elaborate view of past precipitation and stream flow trends in Tampa Bay which should be useful for further research. By incorporating stream flow and precipitation trends, this study provides an idea regarding the effect of at least one of the natural forces (precipitation) on stream flow trends. More importantly, this study provides a base for additional research that can address the effect of regional activities such as wetland drainage on the hydroclimatology of the region.

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