## 1.Introduction

Groundwater level fluctuation is governed by seasonal recharge and discharge rates into aquifers. The recharge and discharge rates are critically related to hydraulic parameters, aquifer boundaries, and geometry. When an aquifer is located adjacent to a river (or recharge boundary), the aquifer groundwater level varies linearly with river discharge rate. Riverbank Filtration (RBF) intake from riverside alluvium is a type of indirect artificial recharge technique which uses more surface water than groundwater. Critical issues in such a case are determining optimal pumping quantity and relating groundwater level with optimal discharge and seasonal river discharge. RBF has been used because it produces higher quality potable water compared to surface-origin public water supply, due to its natural purification during transmission through alluvium (Hiscock and Grischek, 2002). RBF was first used in 1810 in the United Kingdom by the Glasgow Waterworks Company to supply water (Ray et al., 2002). Since the 19th century, European countries especially Germany, the Netherlands, France, Austria, and Sweden have produced drinking water using the RBF method. In particular, Germany produces 15 - 16% of its total drinking water using RBF (Achten et al., 2002). The United States produces drinking water using RBF in the Columbia, Missouri, Mississippi, Ohio, Colorado, Rio Grande, Russian, and Connecticut River basins (Ray, 2001). Lee et al. (2009) characterized riverbank-filtered water and river water qualities at the Daesan-Myeon site in the lower Nakdong River basin, Korea, using chemical analyses and time series methods.

Many studies have focused on interaction between river and alluvium, relationship between hyporheic flow and RBF, and characterization of heterogenic hydraulic properties of alluvial aquifers adjacent to rivers in relation to RBF (Castro and Hornberger, 1991; Hendricks and White, 1991; Sophocleous, 1991; Rossi et al., 1994; Woessner et al., 1998; Woessner, 2000; Mayer et al., 2002; Oh et al., 2016). Sophocleous (1991) demonstrated long-distance and fast propagation of stream floodwaves with large hydraulic diffusivity in the Great bend Prairie aquifer of Kansas, using stream-aquifer numerical simulation. Mayer et al. (2002) investigated the effect of the storage coefficient controlling stream-groundwater interaction during rapid stream-stage fluctuations, and the impact of the storage coefficient on groundwater contamination scenarios, using numerical simulation. Oh et al. (2016) evaluated river-aquifer interaction at a river barrage site in the Nakdong River basin, Korea, using analytical and statistical methods considering the effect of river barrage construction.

Time series analyses are used for evaluating temporal variations in groundwater level and groundwater quality data (Duffy and Gelhar, 1986; Lacocque et al., 1998; Worrall and Burt, 1999; Lee and Lee, 2000). Larocque et al. (1998) characterized a karst aquifer in the Charente region in France using correlation and spectral analyses. Lee and Lee (2000) interpreted temporal variation patterns of groundwater levels using autocorrelation and cross-correlation analyses. Kim et al. (2005) identified the relationship between tidal level change and groundwater quality using time series analyses. Worrall and Burt (1999) applied autoregressive (AR) and autoregressive moving average (ARMA) models to nitrate concentration in river water.

In Korea, except in Jeju Island. surface water is mostly used as drinking water source. Now, Korea is confronted with gradual degradation of drinking water quality due to surface water quality loss, especially in downstream areas compared to upstream areas. In the early 2000s, Changwon City, for the first time in Korea, constructed municipal water plants of 10,000 m^{3}/d capacity which use RBF in the Daesan-Myeon and Book-Myeon areas in the vicinity of the Nakdong River (Fig. 1). Through the first stage construction lasting from 2000 to 2005, Changwon City obtained 70,000 m^{3}/d of drinking water from 36 vertical wells and one radial collector well in the Daesan-Myeon area. By the end of the second stage in 2010, drinking water production in the Daesan -Myeon area reached 130,000 m^{3}/d.

This study aimed to estimate and predict groundwater level fluctuation associated with pumping quantity and streamflow rate at the river-bank filtration site of Daesan-Myeon area, Changwon City, Korea, using autocorrelation, cross-correlation, and autoregressive analyses.

## 2.Geological and hydrogeological setting

The geology of the study area was largely composed of Cretaceous andesite, andesitic tuff, andesitic lapilli tuff and andesitic tuff breccia, Cretaceous sedimentary rock (arkosic sandstone, greenish shale, and cherty shale) and Cretaceous biotite granite, granodiorite, and acidic dykes intruding volcanic and sedimentary rocks (Kim and Lee, 1964; Cheong et al., 2008). The Cretaceous rocks were also overlain with Quaternary alluvium which covers the area around the Nakdong River. The most dominant joint groups were elongated along the N-S and NNW-SSE directions. The joints were better developed in the Cretaceous sedimentary rock than in the Cretaceous volcanic and granitic rocks.

Based on pumping and monitoring wells drill data, the alluvium was composed of upper fine sand, medium sand, lower fine sand, and highly conductive sand/gravel layers from the ground surface below to a 60-meter depth. A moderately conductive weathered andesitic tuff underlay the alluvium (Cheong et al., 2008). The main aquifer, a sand/gravel layer with an average thickness of 12.5 m, showed greater thickness near wells PW1, PW2, PW3, PW4, DS1, DS2, DS4, and DS8 (13.2 17.5 m), and lesser thickness near wells PW5, PW6, PW7, DS3, DS5, DS6, and DS7 (6.3-10.7 m), being thicker near the Nakdong River and the eastern part of the study area.

Precipitation and atmospheric temperature data from January 1977 until December 2006 (30 years) were collected from the Miryang rain gauging station, the station closest to the study area. A hydrograph of the Nakdong River measured every 30 minutes from March 1998 until September 2006 on the Susan Bridge (located about 2.7 km south of the RBF site) was obtained from the Flood Control Office of the Nakdong River. The Nakdong River flowed with a mean discharge of 843 m^{3}/s and a mean precipitation of 653 mm in the high-flow summer season (June-August), and flowed with a mean discharge of 321 m^{3}/s and a mean precipitation of 458 mm in the low-flow non-summer season (the other months) during the 30-year period. During the 30-year period, the mean atmospheric temperature was 24.2℃ in the summer season, and 9.90℃ in the non-summer season. The river water level rose during the wet season and fell during the dry season. Using river discharge data from the years 2002 and 2005-2007, extremely high water, high water, normal water, low water, and extremely low water periods were determined based on the probability of exceeding, equaling, or trailing stream discharge (Park, 2003). The wet season was defined as comprising the extremely high water, the high water, the normal water, and the low water periods which corresponded to 6 Jul 22 Sep 2002, 2 Jul 27 Aug 2005, 10 May-2 Aug 2006, and 28 May-21 Aug 2007, respectively. The dry season was defined as the period of extremely low water discharge; i.e., the months other than those of the wet season.

The groundwater level data from 1 January 2003 until 31 December 2004 were obtained from the pumping and monitoring wells in the study area (Fig. 1). Seven pumping wells (PW1 PW7) for the production of riverbank filtrate were located 45 110 meters from the Nakdong River, and eleven monitoring wells (DS1-DS8, OW11, OW13, and OW16) were located at distances of 10 500 meters from the river. Total discharge from the pumping wells ranged from 6,000 to 8,000 m^{3}/d and will be increased in the future. This RBF site was used for greenhouse agriculture until the end of 2007 and was then developed into a drinking water plant.

Groundwater elevation was apparently higher around PW1, reaching 9 meters amsl compared to 7.5 meters amsl in the other parts of the study area, although it changed seasonally and yearly due to sedimentation during the flood season, and erosion by the Nakdong River. Groundwater levels fluctuated seasonally with an rise in the wet season (May to September) and a drop in the dry season (the other months) within a range of -0.41 - 10.81 m, showing similar fluctuation patterns at all monitoring wells. Mean groundwater levels at different piezometers ranged from 0.80 to 1.25 m. Groundwater level fluctuation at DS1 indicated that water levels were higher (approximately 3 - 10 m amsl) in the wet season, and lower (approximately 0 2 m amsl) in the dry season (Fig. 2). DS7 displayed the largest fluctuation from -0.18 to 10.81 m, while OW16 showed the narrowest variation from 0.41 to 3.99 m. Relatively low water levels at DS4 (lowest level of -0.49 m) and DS5 (lowest level of -0.29 m) were due to pumping from PW2 PW5 and groundwater use for nearby greenhouse agriculture.

## 3.Methods

Autocorrelation analyses effectively draw out a specific property and repetition, and estimate self-similarity of temporal or spatial data. The autocorrelation function estimates linear tendency of successive values over a time period and memory effects (Padilla and Pulido-Bosch, 1995; Angelini, 1997; Larocque et al., 1998; Lee and Lee, 2000). The autocorrelation function, *r*(*k*), is expressed as(1)(2)(3)

where *n* is the total number of observations, *x _{t}* is an observation of the time series,

*x*is the mean of the observations, k that has the number from 0 to m (truncation point) is the lag number of intervals between observations

*x*and

_{t}*x*. For autocorrelation analysis, orderly data with constant interval is required.

_{t-k}Cross-correlation analysis is used to establish a link between the input time series *x _{t}*, (

*x*

_{1},

*x*

_{2}, …,

*x*) and the output time series

_{n}*y*(

_{t}*y*

_{1},

*y*

_{1}, …,

*y*). The cross-correlation function with

_{n}*m*, for

*k*= 0, 1, 2, …,

*m*, is defined as (Padilla and Pulido-Bosch, 1995)(4)

where *s _{x}* and

*s*are the corresponding standard deviations of the input time series

_{y}*x*and the output time series

_{t}*y*.

_{t}*C*(

_{xy}*xy*), covariance between the series

*x*and

_{t}*y*, is expressed as

_{t}

where *x* and *y* are the means of the time series *x _{t}* and

*y*, respectively. The cross-correlation function,

_{t}*r*(

_{xy}*k*), is not symmetrical [

*r*(

_{xy}*k*) ≠

*r*(

_{yx}*k*)] and if

*k*< 0,

*x*replaces

*y*and vice versa in eqs. (5) and (6). If

*r*(

_{xy}*k*) > 0 with

*k*> 0, the input time series influences the output time series; If

*r*(

_{xy}*k*) > 0 with

*k*< 0, the output influences the input (Larocque et al., 1998).

Autoregressive (AR) model analysis can estimate and forecast temporal tendency of time series data, using linear regression. When the AR model is not temporally independent of the error term i.e., when the error term is auto-correlated, the efficiency of ordinary least-squares parameter estimates is adversely affected and standard error estimates become biased. In such a case, AR analysis uses the AR random error technique for considering the autocorrelated error term. The AR model is expressed as follows.(7)

where *a _{i}* represents autoregression coefficients,

*x*is the

_{t}*t*th dependent variable, and

*x*is the (

_{t-i}*t*-

*i*)th independent variable. The value of

*i*is the order, and

*n*is the highest order considered. The noise term or residue (Gaussian white noise) ε

_{t}, indicates that the AR model is normally and independently distributed with zero mean and unit variance

*s*

^{2}.

## 4.Results and Discussion

### 4.1.Seasonal trends in stream discharge, groundwater level, and pumping quantity

Differences between river water level and groundwater level are a main factor in aquifer-stream interaction. River water level is influenced by artificial activity (discharge from upstream dams or pumping quantity in the vicinity of the river) as well as natural phenomena (rainfall and groundwater level in riverside aquifers). Groundwater levels were measured at 11 monitoring wells (DS1 DS8, OW11, OW13, and OW16) from January 2003 to August 2006 (Fig. 3). The water level was automatically measured every minute using pressure sensors (Solinst Level logger with a measuring range of 30 m and Eijkelkamp Diver with a measuring range of 10 m). In the study area, the water levels ranged from -0.45 m to 10.33 m, with an average level of 1.21 m. Stream discharge of the Nakdong River varied from 61 to 4,244 m^{3}/s. It was the highest in 1998 and the least in 2001, and was dependent on precipitation. River water level directly and positively influences groundwater level. In general, river water level rises with an increase in precipitation. However, despite no noticeable increase in precipitation, significant rises in river water level occur due to the opening of dam gates meant to prevent storm water floods. Conversely, despite increase in precipitation, significant rises in river water level are prevented owing to dam water storage.

Groundwater levels at most of the monitoring wells fluctuated seasonally with rises in the wet season (June, July, August, and September) and drops in the dry season (the other months) within a range of -0.41 10.81 m, showing similar fluctuation patterns at all monitoring wells. For instance, groundwater level fluctuation at DS1 indicated that water levels were higher (about 3 - 10 m amsl) in the wet season from June to August, and lower (about 0 - 2 m amsl) in all other dry months (Fig. 3). Groundwater levels had a linear tendency in the low-water season (January 2003 - April 2003) and the high-water season (May 2003 September 2003). The relationship between groundwater level and river water level was represented as a correlation coefficient of 0.8353 in the low-water season and a correlation coefficient of 0.9466 in the high-water season (Table 1). The groundwater level at DS3, closest to the river, showed the highest correlation coefficient of 0.98; that of OW13, farthest from the river, showed the lowest correlation coefficient of 0.35. River water levels in the high-water season showed a wider fluctuation and higher influence compared to those in the low-water season. Groundwater levels in the low-water season showed short-term fluctuation due to the influence of pumping quantity, with a rising trend towards the high water season (Fig. 3). On the other hand, groundwater levels in the high-water season showed long-term fluctuation due to the effect of river water level, with a falling trend towards the low water season.

Daily pumping quantity at the seven pumping wells (PW1 PW7) was related to groundwater level at the eleven monitoring wells. The pumping quantity varied from 0 to 8,639 m^{3}/d and the difference between the maximum and minimum rate was 2,000 m^{3}/d in most cases. The pumping quantity increased with the potable water supply and seasonally increased in the summer (June, July, and August), while drawdown decreased despite increase in the pumping quantity. On the contrary, in the winter, even if pumping quantity decreased, drawdown increased due to the decrease of groundwater recharge with the river level dropping. Increases in stream discharge were found to reduce drawdown ratio (the ratio of drawdown to pumping quantity) in the pumping area in the wet season. The seasonal trend in drawdown may also be related to water viscosity changes depending on water temperature changes, which can alter hydraulic conductivity of the aquifer. This trend was different from commonly observed increase in drawdown with increase in pumping quantity. A more quantitative relationship between drawdown at the monitoring wells and stream discharge will be examined in the following statistical analyses.

### 4.2.Auto- and cross-correlation analyses

The autocorrelation method was used to estimate temporal properties of groundwater level fluctuation and river water level. Autocorrelation analysis was conducted on groundwater levels in 2003 at eight monitoring wells (DS1 DS8) (Fig. 4). The autocorrelation results for the wet and dry seasons were distinctly different. Except DS7 and DS8, autocorrelation of groundwater levels in the wet season exhibited strong linearity and long memory, with a lag time of 31 hours (Fig. 4a), compared to weak linearity and short memory with 1 5 hours of lag time in the dry season (Fig. 4b).

The autocorrelation function for the wet season could be grouped into three types (types 1, 2, and 3). The boreholes DS1-DS4, belonging to type 1 showed behavior similar to river fluctuation, with lag time ranging from 16.5 to 17.2. Type 2 comprising DS5 DS7 were influenced by the river and groundwater abstraction. Neither the effect of the river nor that of well pumpage was seen in type 3 (DS8). In contrast to that in the wet season, the autocorrelation function for the dry season exhibited largely two types. Type 1 comprising DS2, DS3, DS4, and DS6 showed the effect of groundwater pumping (DS4 in particular), while type 2 comprising DS1, DS5, DS7, and DS8 showed little effect due to pumpage and little variation in water level. The river level showed autocorrelation results similar to type 2.

Cross-correlation analysis was performed on the Nakdong river level (input time series) and groundwater levels at the eight monitoring wells (output time series) in wet and dry seasons, using data from 2003 (Fig. 5). During the wet season, high cross-correlation functions (0.42 0.70) were seen at boreholes DS7 and DS8 with a 0-hour lag time. In contrast, low cross-correlation functions from 0.21 to 0.41 were obtained at boreholes DS1 DS6 with a 0-hour lag time. On the other hand, during the dry season, high cross-correlation functions (0.42 0.57) were seen at boreholes DS4 and DS8 with a 0-hour lag time. In contrast, low cross-correlation functions (0.21-0.35) were obtained at boreholes DS1-DS3, and DS5-DS7, with a 0-hour lag time. The cross-correlation was higher during the wet season than during the dry season because the river level rise greatly influenced groundwater levels despite the effect of discharge from the pumping wells. In addition, the general pattern of short lag time at most boreholes indicated a pumping effect.

### 4.3.Autoregressive analysis

AR analysis was carried out in order to estimate temporal properties of groundwater level fluctuation in relation to river water level and the RBF intake quantity in the study area. The AR analysis adopted multivariate approaches using hourly groundwater level as the dependent variable, and river water level and RBF intake quantity as independent variables. The analysis was carried out for the low-water season (January to April, 2003) and the high-water season (May to September, 2003), using hourly groundwater level data at the monitoring wells (DS1 DS8) neutralizing auto-correlation effects of the error term (Fig. 4).

The AR model analysis showed very good agreement between measured groundwater levels and predicted groundwater levels at the monitoring wells (Fig. 6), primarily due to neutralization of autocorrelation effects of the error term. Residuals determined using the AR model were small in the dry season, with the smallest value of -0.1 0.08 m for the well DS3; in contrast, residuals in the wet season were large, with the smallest value of -0.21-0.22 m for the well DS6 (Table 2). The regression model equation showed that groundwater level in the high-water season rised more rapidly than the river water level variation, compared to that in the low-water season. Ground -water level in the low-water season was more affected by RBF intake quantity than river discharge; whereas groundwater level in the high-water season was more influenced by river discharge than RBF intake quantity. Therefore, the water level decline effect induced by RBF intake was not significant in the high-water season compared to that in the low-water season. Groundwater level varied at different monitoring wells depending on their distance from the pumping wells and the Nakdong River; i.e., greater water level variation took place at the monitoring wells located in the vicinity of the pumping wells but far from the river; in the opposite case, smaller water level variation occurred.

Groundwater levels estimated using the autoregressive model for the dry and wet seasons resulted in very good agreement between measured groundwater levels and forecasted levels at the monitoring wells. Therefore, the AR model that only uses simple inputs of river discharge and pumping quantity data, can be predict groundwater levels in a specific RBF area, in the unnecessity of using numerical models which require physical property data and hydraulic parameters of the alluvial aquifers.

## 5.Conclusion

In the wet season, drawdown ratio increased due to river level rise and river discharge even when pumping quantity increased; the drawdown ratio decreased in the dry season. The relationship between drawdown ratio and river discharge was very strong with a correlation coefficient of 0.96. Therefore, the relationship between river discharge and drawdown ratio can be used for predicting optimal yield of riverbank filtrate and for effective joint utilization of surface-origin water and groundwater.

Based on the autocorrelation function, groundwater levels in the wet season could be grouped into three types (types 1, 2, and 3). Type 1 represented behavior similar to river fluctuation with a lag time ranging from 16.5 to 17.2, type 2 showed the influence of both river fluctuation and groundwater pumping, and type 3 showed behavior irrelevant to river fluctuation and groundwater pumping. In the dry season, groundwater levels could be classified into two types. Type 1 was affected by groundwater pumping, and type 2 displayed little effect due to pumping effect and little groundwater level fluctuation. Cross -correlation was higher in the wet season than in the dry season because river level rise greatly influenced groundwater level rise, even if groundwater pumping caused groundwater level decline.

Autoregressive analysis was conducted on groundwater level data at the monitoring well DS1 in low- and high-water seasons in order to neutralize auto-correlation effects of the error term. Results of autoregressive model analysis for DS1 show excellent agreement between measured groundwater levels and predicted groundwater levels. From the model equations, it was found that groundwater level in the wet season rised more rapidly due to rapid river water level rise compared to that in the in dry season. Groundwater levels in the wet season were closely related to pumping quantities at pumping wells PW3 and PW4; while groundwater levels in the dry season were closely related to pumping quantities at pumping wells PW3, PW4, PW5, and PW6. Therefore, drawdown effect due to pumping r quantity was smaller in the wet season than in the dry season.

Therefore, when RBF intake and streamflow discharge do not vary greatly in a specific area, the AR approach can be a simple, efficient tool for estimating and predicting groundwater level change, without using numerical models which require physical data and hydrologic properties of the alluvium.