**2. Methodology**

Taiwan being 1740 m3

28 Groundwater - Contaminant and Resource Management

management in Taiwan.

system and its recharge volume.

are ignored) [22–26].

resources.

/year [2]. Climate changes and increasing temperature experienced in

recent decades caused the precipitation in Taiwan to increase during the rainy season, while decreasing in the dry season, resulting in a drier dry season and a wetter wet season [3]. Consequently, stream flow in both dry and wet seasons varied significantly. Thus, it is urgent to assess and discuss groundwater recharge characteristics, toward effective water resource

When analyzing the hydrogeological model and groundwater system of a region, the study on recharging groundwater through precipitation is a very important but complex issue. Meteorological factors (e.g., intensity and delay of precipitation, temperature, humidity, and wind speed), thickness of the soil layer, prevailing groundwater level, topographical condi‐ tions on the surface, vegetation cover, and land uses, all have direct relationships with the system and must be considered [4–10]. It is very easy to measure precipitation and run‐off amounts when analyzing water balance, but it is very difficult to quantify the recharging process. Its evaluation requires not only precipitation data, but also other factors, such as prevailing climatic conditions, soil type, soil moisture status, vegetation cover, and evapo‐ transpiration conditions [11, 12]. The infiltration of precipitation into the soil leads to the recharging process and is an important factor determining circulation of the groundwater

Groundwater recharge can be quantitatively estimated using two methods. The first is the water balance model [13–16], which is applicable for humid regions; the second is for arid regions and involves the use of tensiometers, tracers, infiltrometers, and other instruments on site to observe the movement of water in the unsaturated zone, before estimating the groundwater recharge for that area [17–19]. However, it is generally more difficult to implement the second method because of high costs and the need for long‐term monitoring on site. The existing methods for estimating groundwater recharge at the regional level using the water balance model are further divided into two types: (i) precipitation, infiltration, run‐ off, evapotranspiration, and groundwater recharge are treated as components of an interrelated system, with the soil moisture status being that of an ever‐changing soil water balance model [7, 20, 21]; and (ii) the hydrograph of a stream flow is used to estimate its base flow, with the latter being treated as the groundwater recharge (based on the assumption that the heterogeneous hydrogeological conditions within the catchment area

In this study, the base flow estimation method was used to determine groundwater recharge and the evapotranspiration from the unsaturated zone was not considered. The calculation method to determine the effective recharge is simple, and it neither requires any complicated hydrogeological models, nor factors such as weather conditions and soil types. In this study, the calculation method was used to assess long‐term changes of groundwater recharge in Northern Taiwan, and the findings can serve as reference for the management of water

## **2.1. Base flow model**

Based on the concept of water balance and the data from stream flow gauging stations in the main river basins of the catchment area, this study employed the stream flow PARTitioning (PART) program (a base flow analysis method) developed by the US Geological Survey [27]. This method utilized the stream flow data to separate the base flow, which is regarded as the groundwater recharge. However, for rainy and humid regions and steep mountainous areas, if the estimation of groundwater recharge is based solely on base flow separation, it will often lead to overestimation of the base flow during the wet season [7, 28]. Therefore, the calculation of a steady base flow separation is required. In this study, a method for steady base flow analysis was adopted for this calculation to subsequently estimate a reasonable groundwater recharge.

Using Grey theory, the steady base flow analysis obtains data trends with reference to the rearrangement and accumulation of data. The steady base flow analysis uses trends for a low‐ flow period, a steady base flow period, and an overestimated base flow period, as shown in **Figure 1**. These are obtained after the rearrangement and accumulation of the separated base flows, and then, the steady base flow period is linearly extrapolated to achieve the steady base flow.

**Figure 1.** The diagram of the stable base flow analysis.

The steps in the analytical process are as follows:


#### **2.2. Mann‐Kendall test**

Mann‐Kendall (MK) [29, 30] test is a nonparametric method developed from Kendall's tau (τ). It can be used to test the relationship between two sets of data. The advantages of this method is that extreme values and missing data problems will not seriously affect the certification value. The MK test assesses the trend in a series via comparing the value of the series before and after to determine whether the series exhibits a specific degree of trend. The null hypoth‐ esis given that if there is not significant trend in the series, test statistic *S* is defined as follows:

$$Sign\left(\mathbf{X}\_{j} - \mathbf{X}\_{i}\right) = \begin{cases} +1, \ \mathbf{X}\_{j} - \mathbf{X}\_{i} > 0 \\ 0, \ \mathbf{X}\_{j} - \mathbf{X}\_{i} = 0 \\ -1, \ \mathbf{X}\_{j} - \mathbf{X}\_{i} < 0 \end{cases}, \quad \mathbf{S} = \sum\_{i=1}^{n-1} \sum\_{j=i+1}^{n} \text{Sign}\left(\mathbf{X}\_{j} - \mathbf{X}\_{i}\right) \tag{1}$$

where {*X1*, *X2,X3*, …, *Xn*} is stream flow data which is arranged in accordance with time {*T1, T2,T3,…, Tn*}. *n* is the number of data. When *n* is close to infinity, the probability of the *S* distribution curve will present as a normal distribution with a mean of 0. In addition, when *n* is more than 10, the variance of *S* can be substituted into the following approximate solution:

$$
\sigma^2 = \frac{n(n-1)(2n+5)}{18} \tag{2}
$$

In this study, long‐term stream flow data are likely to be repeated in the data series; thus, Kendall modified the approximated solution Eq. (2) to Eq. (3).

Spatiotemporal Analysis of Groundwater Recharge Trends and Variability in Northern Taiwan http://dx.doi.org/10.5772/63508 31

$$
\sigma^2 = \frac{1}{18} \begin{bmatrix} n(n-1)(2n+5) - \sum \ u(n-1)(2u+5) \\ u = 1 \end{bmatrix} \tag{3}
$$

where *u* is the duplicate value number of the data series.

Finally, the normalized statistical test *S* values becomes the Z value, as follows:

$$Z = \begin{cases} \frac{S - 1}{\sigma} & , S > 0 \\ 0 & , S = 0 \\ \frac{S + 1}{\sigma} & , S < 0 \end{cases} \tag{4}$$

When *Z* is a positive value, this indicates that the series is exhibiting an increasing trend; in contrast, when the value is negative, it indicates that the series has decreased. At this time, the obtained *Z* value should be tested by a significance test to assess whether the series is signifi‐ cant. Assuming a significance level of α, if |*Z*|≧*Z*α, the null hypothesis is rejected, which represents that the series has a significant trend, otherwise the series has no significant trend. In the study, the significance level is set as α=0.05. When |*Z*|≧1.96, the series has a significant trend. When it is below this level, there is no significant trend.

#### **2.3. Theil‐Sen slope**

The steps in the analytical process are as follows:

30 Groundwater - Contaminant and Resource Management

and obtain the steady base flow period.

lation of the linear regression equation.

ì ï ï í ï ï î

**2.2. Mann‐Kendall test**

**1.** Obtain the base flow of each month by base flow separation.

long‐term mean base flow on a month‐by‐month basis.

*j i*

*X X*

*j i*

s

Kendall modified the approximated solution Eq. (2) to Eq. (3).

**2.** Sum the base flow per month over several years and then average the sum to achieve the

**3.** Sort the long‐term mean base flow of each month in descending order and accumulate them to obtain the base flow accumulated per month and the trend of such base flows.

**4.** Determine the rising point of the base flow by the trend line of the accumulated base flow

**5.** Obtain the annual base flow, namely the annual groundwater recharge, by the extrapo‐

Mann‐Kendall (MK) [29, 30] test is a nonparametric method developed from Kendall's tau (τ). It can be used to test the relationship between two sets of data. The advantages of this method is that extreme values and missing data problems will not seriously affect the certification value. The MK test assesses the trend in a series via comparing the value of the series before and after to determine whether the series exhibits a specific degree of trend. The null hypoth‐ esis given that if there is not significant trend in the series, test statistic *S* is defined as follows:

*ji ji j i*

( ) 0, 0 , ( ) 1 1 1, 0

+ -> - - = -= = å å -

*i ji X X*

where {*X1*, *X2,X3*, …, *Xn*} is stream flow data which is arranged in accordance with time {*T1, T2,T3,…, Tn*}. *n* is the number of data. When *n* is close to infinity, the probability of the *S* distribution curve will present as a normal distribution with a mean of 0. In addition, when *n* is more than 10, the variance of *S* can be substituted into the following approximate solution:

= = + - -<

<sup>2</sup> ( 1)(2 5)


18

In this study, long‐term stream flow data are likely to be repeated in the data series; thus,

*Sign X X X X S Sign X X*

1, 0 <sup>1</sup>

*n n*

(2)

(1)

The Theil‐Sen slope [31] is used to estimate the magnitude of the trend slope. The Theil‐Sen slope estimation method is different from the slope values calculated using a linear regression, because it selects the median value, and therefore, the properties are less affected by extreme values. Thus, it is often used with the MK test. Slope *β* is defined as follows:

$$\beta = \text{Median}\left(\frac{X\_j - X\_i}{j - i}\right), \text{ for all } i < j \tag{5}$$

$$\begin{aligned} X(t) &= \beta \ t + \mathcal{C} \\ X(t) &= X\_1 \sim X\_n, t = 1 \sim n \end{aligned} \tag{6}$$

#### **2.4. Mann‐Whitney‐Pettit test**

The Mann‐Whitney‐Pettit (MWP) test [32] can be used to search for significant change points in a data series. The definition of a change point is when a data series {*X1, X2,…, Xn*} has a change point *Xt* , Order {*X1, X2,…, Xt* } is *F1*(*X*) and {*Xt+1, Xt+2, …, Xn*} is *F2*(*X*), then *F1*(*X*)*≠F2*(*X*). The definition of is as shown in Eq. (7) shown. If there is not a change point in the data series, |*Ut*,*<sup>n</sup>* | on the function of time, *t* will continue to rise, and there will be no turning point. On the contrary, if there is a change point, |*Ut*,*<sup>n</sup>* | on the function of time *t*, there will be a decreasing turning point. In the same data series, the turning point may occur several times on behalf of this data series, and there may be more than one change point.

$$\text{Sign}\,(X\_l - X\_j) = \begin{cases} +1, X\_l - X\_j > 0 \\ 0, X\_l - X\_j = 0 \\ -1, X\_l - X\_j < 0 \end{cases}, \quad U\_{t,n} = \sum\_{i=1}^{t} \sum\_{j=t+1}^{n} \text{Sign}\,(X\_l - X\_j) \tag{7}$$

$$K\_n = \text{Max}\left|U\_{\iota,n}\right|, \ 1 \le t < n \tag{8}$$

To confirm that change points exist, Eq. (8) is used to calculate the extreme value of |*Ut*,*<sup>n</sup>* | that is turning point as *Kn*>. Equation (9) is used to calculate the probability of a change point. In this study, *P* = 0.95 is set the as confidence level, where *P* > 0.95 judges that the time is a significant changing point.

$$P = 1 - \exp\left(\frac{-6K\_n^{-2}}{n^2 + n^3}\right) \tag{9}$$

However, in some data series, a change point may not exist by itself; thus, Eq. (10) is used to calculate each year's *P*(*t*) value. The *P*(*t*) value is identified when it is greater than the confidence level.

$$P(t) = 1 - \exp\left(\frac{-6\left|U\_{t,n}\right|^2}{n^2 + n^3}\right) \tag{10}$$
