Introduction

Effective management of freshwater reservoirs is very important for the sustainability of natural resources. Freshwater stored in coastal aquifers is particularly vulnerable in terms of water quality degradation due to its proximity to seawater, in combination with the intensive water demands that accompany the higher population densities of coastal zones. The water level in an aquifer is an important parameter in groundwater hydrology, and a careful and detailed analysis of its variation in time and space reveals useful information on the aquifer system behavior (Aflatooni and Mardaneh 2011). Various factors affecting the groundwater level are groundwater pumping, recharge, precipitation, water pressure, evapo-transpiration, interaction with surface-water bodies, and the tidal oscillation of the sea level in coastal aquifers (Ataie-Ashtiani et al. 1999; Maréchal et al. 2002). The pressure of tidal waves causes periodic fluctuation of the groundwater level in the coastal aquifer. This damping effect progressively diminishes the amplitude of groundwater head in the inland direction (Maréchal et al. 2002). The monitoring of primary hydrological parameters (water pressure, precipitation, electrical conductivity and tide) takes into account the principal sources of information on the seawater intrusion against the coastal aquifer groundwater system. (Winter et al. 2000; Moon et al. 2004; Ahmadi and Sedghamiz 2007; Christophe et al. 2016).

Hydrological processes are usually nonlinear, complex, dynamic, and widely scattered due to the influence of physical processes involving variability in space and time. Further complexity is added when such processes are observed at small spatiotemporal scales, but estimates are needed for very large ones; therefore, it is vital to evaluate whether these processes have different behaviors at different scales. These processes and properties vary at different spatiotemporal scales, and their variations essentially affect groundwater level and quality fluctuations in coastal aquifer systems. Time series of groundwater parameters usually exhibit complex fluctuations owing to the interactions of many factors, and time series analysis has been widely used in the evaluation and forecast of groundwater monitoring data, because it elucidates the characteristics of these complicated features. Many research projects decompose groundwater-level variations into trend, periodicity and random components, and combine the components together as a model to forecast the future variations (Hu et al. 2001; Kim et al. 2005, 2007; Zhao et al. 2007; Zhou et al. 2007; Yang et al. 2009; Chung et al. 2015).

The permeation of seawater through coastal shores and eventual intrusion into coastal aquifers affects the salinity levels of the groundwater. The degree of seawater intrusion varies with the water pressure, precipitation and tidal fluctuations. Time series analysis can gain insight into the relationship between seawater intrusion and groundwater parameters. In this study, the freshwater–saltwater interface was considered as a sharp interface, and a TLC meter was installed at 25-m depth in a monitoring well to observe the interface (Fig. 1). The TLC device measured temperature (T), groundwater level (L) and electrical conductivity (C) of the groundwater. Spectral analysis methods outlined by Shumway and Stoffer (2011) were used to understand the fluctuation behavior of selected hydrological parameters.

Fig. 1
figure 1

Schematic diagram of the interface between freshwater and saline water and the monitoring design in the study area

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The purpose of this study is to understand the variation of the hydrological parameters EC, water pressure, precipitation and tide, and their correlations at a coastal area in Busan, South Korea, using monitoring techniques and time series analysis of these hydrological parameters. Prediction of the extent and influence of seawater intrusion is possible for the long period covered through this study.

Materials and methods

Study area and field measurements

The study area (Fig. 2) is in the Nam-gu District in Busan City, and includes Pukyong National University Campus and the surrounding coastal area. The geology of the study area is mainly composed of tuffaceous sedimentary rocks and andesitic volcanic breccia. The geological time of the rocks is the Cretaceous Period of the Mesozoic Era. Groundwater in the study area is relatively vulnerable to seawater intrusion, because the coast is composed of a sandy beach and reclaimed land.

Fig. 2
figure 2

Location map of the study area

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The study area has four climatological seasons, and received 1,222 and 1,773 mm of precipitation in 2000 and 2009, respectively. The precipitation of 2000 was much smaller than that of 2009 because of drought. Most of the precipitation falls from May to September, with averages 963.5 and 1,352.2 mm in 2000 and 2009, respectively (Fig. 3a,b). During summer, a typhoon usually comes to the Korean Peninsula, accompanied by heavy rain.

Fig. 3
figure 3

Precipitation data of the study area in 2009

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Water level and EC were measured at a monitoring well at Pukyong National University from August 2009 to October 2009, using an automatic TLC sensor with a time interval of 5 min. The monitoring period covered a short dry season (see Fig. 3 precipitation data). The reason for gathering hydrogeological data in the dry season is to reduce the rainfall effect on the groundwater level and quality data. The hydrological data in the dry season was monitored because groundwater quality was not much influenced by rainfall. This well is developed in an unconfined aquifer, and is located 650 m from the coastline. It is 120 m in depth, and its topographic elevation is 4.1 m above mean sea level. The early water level was about 3.9 m below the ground surface in 2009. In this area, the aquifer is almost homogeneous and consists of tuffaceous sedimentary rocks.

The freshwater–saline water interface was measured at the depth of about 25 m below ground surface (−GL). The interface between the freshwater and saline water was considered as a sharp interface. The TLC meter (model 107) made by Solinst Canada Ltd. was used to collect data and was installed at 25 m (−GL). Tide was monitored by the Korea Hydrographic and Oceanographic Administration, and precipitation was measured by the Korea Metrological Administration.

Time series analyses

Time series models have been developed for various hydrologic and environmental applications for effectively forecasting risk (Li et al. 2009; Xu et al. 2009; Erdogan and Gulal 2009). A time series can be composed of a case observed at discrete times, averaged over a time interval, or recorded continuously with time. A sequence of observations made in a time series can be influenced by three components: (1) a trend or long-term component, (2) a cyclical or oscillating component and (3) a random or irregular component. Several techniques are available for separating the trend component from the oscillating fluctuations and random variations in a time series. An important aspect of hydrogeological studies is the determination of long- and short-term trends, which can show the effect of all processes that influence the aquifer over a long and short period (Box and Jenkins 1976).

A short overview of periodogram, spectral density function, auto and partial auto correlation analysis (ACF and PACF) and cross correlation functions (CCF) is presented. The mathematical expressions of the functions are shown in detail by Jenkins and Watts (1968), Mangin (1984), Box et al. (1994), Padilla et al. (1994) and Laroque et al. (1998). The simple auto-correlation analysis quantifies the linear dependency of successive values over a time period. The auto-correlation function, ρ(k) is expressed as

$$ \rho (k)=\raisebox{1ex}{$C(k)$}\!\left/ \!\raisebox{-1ex}{$C(0)\ $}\right. $$
(1)
$$ C(k)=\frac{1}{n-k-1}\sum \limits_{t=1}^{n-k}\left({x}_t-\overline{X}\right)\left({x}_{t+k}-\overline{X}\right)\kern0.5em $$
(2)

where k is the time lag (k = 0 to m) for which m is the cut point, n is the number of events, xt is a single event at time t, and \( \overline{X} \) is the mean of events. C(k) is an auto-covariance of time series data, x(t). The cutting point is usually determined based on the interval of the analysis and the given circumstance. If the time series has strong interdependency and a long memory effect, the auto-correlation function shows a gently decreasing slope and nonzero values over a long time lag. However, if the time series is uncorrelated such as for rainfall, the auto-correlation function decreases very quickly and reaches a zero value in a short time (Lee and Lee 2000; Liang 2011; Chung et al. 2015).

The partial autocorrelation function (PACF) of a stationary process, xt, denoted Øhh, for h = 1, 2, …, (where h is the separation between xt + h and xt) is

$$ {\mathrm{\O}}_{11}=\mathrm{corr}\left({x}_{t+1},{x}_t\right)=\rho (1) $$
(3)

and

$$ {\mathrm{\O}}_{hh}=\mathrm{corr}\left({x}_{t+h}-{\widehat{x}}_{t+h},{x}_t-{\widehat{x}}_t\right),\kern0.5em h\ge 2 $$
(4)

Both \( \left({x}_{t+h}-{\widehat{x}}_{t+h}\right) \) and \( \left({x}_t-{\widehat{x}}_t\right) \) are uncorrelated with {xt + 1, …, xt + h − 1}. The PACF, Øhh, is the correlation between xt + h and xt with the linear dependence of {xt + 1, …, xt + h − 1} on each, removed (Shumway and Stoffer 2011). If the process xt is Gaussian, then Øhh = corr(xt + h, xt ⃓ xt + 1, …, xt + h − 1); that is, Øhh is the correlation coefficient between xt + h and xt in the bivariate distribution of (xt + h, xt) conditional on {xt + 1, …, xt + h − 1}.

ACF is not as useful in the identification of the order of an auto-regressive process (AR) for which it will most likely have a mixture of exponential decay and damped sinusoidal expressions. PACF is useful for the analysis of the time series with the AR structure (Montgomery et al. 2008). The simple spectral density analysis is complementary to the auto-correlation analysis. The spectral density function corresponds to change from a time mode to a frequency mode through a Fourier transformation of the auto-correlation function. The Blackman and Turkey (1958) method was used for the power spectral density (PSD) in this study. The method corresponds to change from a time mode to a frequency mode through a complex Fourier transformation of the auto-correlation function. The actual computation of the power spectrum can only be performed at a finite number of different frequencies by employing the Fast Fourier Transformation (FFT; Trauth 2015). The FFT is a method of computing a discrete Fourier transform with reduced execution time.

$$ {X}_{\mathrm{x}}(f)=\sum \limits_{k=0}^M{\gamma}_{\mathrm{k}}(k){e}^{-i2\pi fk/{f}_{\mathrm{s}}} $$
(5)
$$ \mathrm{PS}{\mathrm{D}}_{\mathrm{x}}(f)=\frac{{\left|\ {X}_{\mathrm{x}}(f)\right|}^2}{f_{\mathrm{s}}} $$
(6)

where fsis sampling frequency, M is maximum lag, Xx(f) is a complex Fourier transformation of the auto-correlation function, and PSDx(f) is power spectral density.

The cross-correlation analysis is used to establish a link between the input time series and the output time series. If the input time series is random, the cross-correlation function, rxy(k), corresponds to the impulse response of the system. In other cases, the cross-correlation function provides information on the interrelationship between the input and the output time series data as well as the importance of these relationships. The definition of a cross-correlation coefficient is as follows:

$$ {\rho}_{xy(k)}=\frac{C_{xy}\ (k)\kern0.5em }{\sigma_x{\sigma}_y} $$
(7)
$$ {C}_{xy}\ (k)=\frac{1}{n-k-1}\sum \limits_{t=1}^{n-k}\left({x}_t-\overline{X}\right)\left({y}_{t+k}-\overline{Y}\right) $$
(8)

where Cxy(k) is the cross-covariance of x(t) and y(t) time series, and σx, σy are the standard deviations of two time-series, respectively. The time series analyses were performed by SPSS (Ver. 21).

Results and discussion

Variations of water pressure and EC at the interface between freshwater and saline water

Figure 4a,b illustrate the EC variations with depth in the monitoring well in June 2000 and August 2009, respectively. At the beginning of the well development in June 2000, EC increased with depth at three intervals, i.e., from 22 to 79 m, from 79 to 101 m, and from 117 to 120 m (Fig. 4a). Thus, three transition zones were formed by seawater intrusion at that time. However, the transition zones changed into a saline water zone due to the complete mixing of groundwater with seawater in August 2009 (Fig. 4b). The EC value was 47,000 μS/cm below the interface in 2009, and it was nearly seawater quality. Then, the interface between freshwater and sea water was located only at 25 m depth (−GL).

Fig. 4
figure 4

Electrical conductivity (EC) logs of groundwater with depth in a June 2000 and b August 2009

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Figure 5a,b show the variation of water pressure and EC with precipitation and tide at the interface at 25 m (−GL). The minimum, maximum and average values of measured parameters are presented in Table 1. The water pressure varied from 20.96 to 21.28 m at the fresh–saline water interface, and formed a quite complex shape due to the several kinds of influences such as tide, air pressure and precipitation. EC varied from 1,258 to 3,149 μS/cm, and showed a very smooth variation. EC was largely affected by precipitation rather than by tide. This was examined by the relation of precipitation with EC from 22 August 2009 to 17 October 2009 (Fig. 5a,b). EC began to decrease due to the increase of precipitation at the beginning of October 2009.

Fig. 5
figure 5

Relation of water pressure and EC with a precipitation and b tide

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Table 1 Minimum, maximum and average values of measured parameters in the study area
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If the water pressure at the fresh-saline water interface was increased, EC decreased due to the increase of freshwater, and vice versa. When water pressure was increased at the end of August 2009, EC decreased due to the increase of freshwater. When water pressure decreased during September 2009, EC increased due to the decrease of freshwater. Figure 5b shows the relation of water pressure and EC with tide. Water pressure was influenced by tidal variation. Water pressure generally increased according to the rise of tide, and in case of the fall of tide, water pressure decreased. EC is decreased by the increase of water pressure, and vice versa; thus, the level and quality of groundwater were affected by seawater intrusion in the study area.

Periodograms and spectral density functions (SDF)

The periodograms of hydrological parameters according to frequency and period are given in Fig. 6. All periodograms of the frequency domain are in inverse proportion to those of the period domain, because frequency is inversely proportional to period. The periodogram of tide is similar to that of precipitation on the basis of frequency, but the periodogram of tide is different from that of precipitation on the basis of period. Tide has a big periodogram value on the very low frequency, and many small periodogram values on the higher frequencies. Tide has a big periodogram value at the period of less than 5 × 103. Precipitation also has a big periodogram value on the very low frequency; however, its periodogram value on the period domain continuously increases and ends as a big value. Water pressure and EC also have high periodogram values at the very low frequency, but their periodogram values are different on the period domain. Water pressure has very low periodogram values until the period of 5 × 103, and it increases after that period value. The periodogram value of EC increases continuously.

Fig. 6
figure 6

Periodograms of hydrological parameters with respect to a frequency and b period

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Spectral density functions (SDF) of selected hydrological parameters such as EC, water pressure, precipitation and tide are shown in Fig. 7. The SDF was calculated using the Blackman-Turkey method. This method used the complex Fourier transformation for the autocorrelation functions of time series data. The spectral density functions of hydrologic parameters are quite similar to the periodograms of hydrologic parameters in this study. The behaviors of spectral density functions of hydrologic parameters are also similar to each other, and their highest peaks have a single point on the very small frequency. The spectral density function of EC shows the similar trend as those of tide, precipitation and water pressure on both the frequency and period domains. This suggests that EC fluctuations were affected by tide, water pressure and precipitation (Diggle 1992; Lu et al. 2014). All hydrological parameters show quite similar spectra with an initial break in slope and gradual fall-off until the increased asymptotic frequency. To explain the variation in frequency and period, additional factors must play controlling roles. This behavior suggests the preponderance of long memory in the interaction between groundwater and seawater. The results of these analyses show that hydrological parameters are useful for predicting the seawater intrusion in the monitoring well.

Fig. 7
figure 7

Power spectral density of hydrological parameters with respect to a frequency and b period

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Autocorrelation and partial autocorrelation functions (ACF and PACF)

Figure 8 shows the ACF and PACF of the hydrological parameters EC, water pressure, precipitation and tide. The ACFs and PACFs of precipitation and EC present nearly the same pattern. The ACFs almost approach 1.0 regardless of the number of the time lag; however, the PACFs are 1.0 only at the time lag 1, and almost 0 at more than the time lag 1. The ACF of tide gradually goes down according to the increase of the time lag, and approaches 0.75. The PACF of tide is 1.0 only at the time lag 1, and its absolute value is decreased to 0 with the increasing time lag number. The ACF of water pressure is 1.0 regardless of the number of time lag, but the PACF of water pressure is gradually decreased to 0 with the increasing time lag number. The ACFs of all the hydrological parameters show very good correlations over the entire time lag; however, the PACFs of precipitation and EC reveal good correlations only at time lag 1, and they are almost 0 over the rest of the time lags. The PACFs of tide and water pressure show good correlations only at time lag 1, and their correlations decrease with the increasing time lag number; thus, the PACF cuts off after the first time lag, and the only significant sample PACF value is at time lag 1, suggesting that for the first-order autoregressive process, the AR(1) model is indeed appropriate to the hydrologic data (Montgomery et al. 2008). The ACF and PACF of hydrological data are effective analyses for forecasting seawater intrusion against groundwater in the coastal area (Tularam and Keeler 2006).

Fig. 8
figure 8

a Auto correlation functions (ACF) and b partial auto correlation functions (PACF) of hydrological parameters

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Cross correlation functions (CCF)

The CCF was used to evaluate the correlations between the hydrological parameters EC, water pressure, precipitation and tide, and the characteristics of seawater intrusion in a coastal aquifer system. In this analysis, EC and water pressure were the objects of cross correlation using tide, precipitation and water pressure. Figure 9 presents the graphs of CCFs for these hydrological parameters.

Fig. 9
figure 9

Cross correlation functions (CCF) of hydrological parameters: a EC and tide, b EC and natural log-transformed Tide, c EC and precipitation, d EC and water pressure, e water pressure and tide, and f water pressure and precipitation

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The cross correlation between EC and tide was −0.024 to –0.026 (Fig. 9a), and the correlation was largely increased to −0.063 to −0.065 after the natural log transformation of tide data (Fig. 9b). The cross correlation between EC and precipitation was 0.316 to 0.317 (Fig. 9c) and it suggests that EC was largely affected by precipitation rather than tide. The cross correlation between EC and water pressure was −0.190 to −0.195 (Fig. 9d), and their relation was in inverse proportion. This means that the increase of water pressure decreased the EC, because the increase of water pressure resulted from the increase of freshwater. EC generally has a close relationship with water pressure or precipitation; however, their cross correlation coefficients were not large in this study, because the amount of data was not enough for the definite identification of their relations. The cross correlation between water pressure and tide was −0.049 to −0.089 (Fig. 9e), and that between water pressure and precipitation was −0.234 to −0.241 (Fig. 9f); thus, water pressure was much more influenced by precipitation than tide. The variation of water pressure was partially caused by precipitation events acting as a recharge source (Namdar Ghanbari and Bravo 2011). From the cross correlations between the hydrologic parameters, it is understood that EC was much more influenced by precipitation, and water pressure was also much affected by precipitation in the study area.

Conclusions

The transition zone between fresh and saline water was changed into a saline water zone due to the complete mixing of groundwater with seawater in the monitoring well after 9 years of well development. The interface existed only at 25 m depth in August 2009, although it was located at 22, 79, 101 and 117 depths at the time of well development in 2000. Moreover, EC greatly increased from 30,000 to 47,000 μS/cm over the 9-year period; thus, seawater intrusion expanded widely during the 9 years of study and greatly affected the groundwater quality in this coastal aquifer of tuffaceous sedimentary rock.

EC and water pressure were compared to the variations of precipitation and tide. EC was largely affected by water pressure and precipitation, rather than tide. Water pressure was influenced by precipitation as well as tide. The periodograms of hydrological parameters were produced on the basis of frequency and period. All periodograms on the frequency domain were in inverse proportion to those on the period domain. The periodograms of all hydrological parameters were quite similar to each other, and had a big periodogram value on the very low frequency. The power spectral density (PSD) of selected hydrological parameters such as EC, water pressure, precipitation and tide were produced by the Blackman-Turkey method, and were nearly the same as the periodograms of the hydrologic parameters used in this study. This behavior highlights the significance of long-term behavior in the interaction between groundwater and seawater.

The ACF and PACF of the hydrological parameters were produced to understand their self-correlations. The ACFs of all hydrologic parameters showed very good correlation over the entire time lag, but the PACFs revealed that their correlations were good only at time lag 1. The CCF was used to evaluate the correlations between the hydrological parameters and the characteristics of seawater intrusion in the coastal aquifer system. The large cross correlation between EC and precipitation suggests that EC was largely affected by precipitation rather than tide. The inverse cross correlation between EC and water pressure means that the increase of water pressure decreased the EC, because the increase of water pressure resulted from the increase of freshwater. From the CCFs between hydrologic parameters, it is understood that EC was much influenced by precipitation, and water pressure was also much affected by precipitation in the study area. The monitoring and time series analysis of hydrological parameters such as water pressure, EC, precipitation and tide have been found to be very useful to understand the characteristics of the parameter variations and their correlations, and the extent and influence of seawater intrusion. The continuous monitoring of hydrological parameters will be very useful for preventing the further deterioration of groundwater quality in the coastal regions.