2. Methodology

significant differences in the storage of water resources, determining a method by which to

Recently, Taiwan has been affected by climate change, which is resulting in distinct wet and dry seasons with stronger rainfall intensity in the wet season and a lack of rain in the dry season [7]. Variations in Taiwan's topography and uneven distribution of rainfall, with more in the mountains than in the plains and more in the northern than in the southern areas, are causing uneven distribution of water resources in time and space. Taiwan's annual average rainfall is up to 2500 mm, which is approximately 2.5 times more than the annual global average rainfall. Nevertheless, Taiwan is still facing a surface water shortage problem in the dry season. In southern Taiwan, which is the most serious area, the ratio of wet season rainfall and dry season rainfall is as high as 9:1 [8]. Therefore, in this study, the southern area is set as the study area to discuss what groundwater resource problems this area will be facing in the future. The water balance concept is used to explore the storage-discharge relationship, and a low flow analysis is used to assess the lowest groundwater storage in southern Taiwan. The assessment results can be provided to water resource agencies to assist with water resource management plans.

Low flow analysis, which is a hydrological method widely used to estimate hydraulic parameters, is used to characterize basin characteristics and assess groundwater storage trends. Brutsaert and Nieber [9] analyzed six basins in the Finger Lakes region of upstate New York using the lower envelope fitting method to characterize recession curves in order to estimate specific yield. Szilagyi et al. [10] pointed out that complexity of basin shape, hydraulic conductivity heterogeneity, and gently sloping impermeable layers do not affect the estimation of hydraulic conductivity and mean aquifer depth. Field-based estimates of these hydraulic parameters using a low flow analysis are a beneficial method for evaluating discharge data

Watershed properties such as hydrology, geology, and topography affect streamflow regimes [12]. Thus, through knowing the main factors dominating streamflow recession characteristics, it is possible to understand drainage systems. Zecharias and Brutsaert [13] selected 19 basins in the Allegheny Mountains of the Appalachian Plateaus to investigate streamflow recession curve characteristics, and their results showed that recession time is affected by mean basin slope, drainage density, and the ratio of the hydraulic conductivity and the specific yield. Vogel and Kroll [14] evaluated Massachusetts basin characteristics, and their result showed that recession time is highly correlated with basin area and basin slope. Brutsaert [15] suggested that streamflow recession time is highly correlated with basin area, basin elevation, and stream length. Kirchner [16] used a single-equation rainfall-runoff model to select streamflow data in order to estimate recession time from the headwaters of the Severn and Wye rivers at Plynlimon in mid-Wales. Sánchez-Murillo et al. [17] suggested that geology affects the length of recession time. Metamorphic and sedimentary rocks result in longer recession time. Low flow in flat basalt landscapes recesses rapidly. Knowing the dominating factors related to recession characteristics makes it easier to understand their effects on subsurface properties. Recently, several studies analyzed annual groundwater storage trends in order to assess water supply availability in the future. Sugita and Brutsaert [18] researched perennial groundwater storage and low flow trends in the Kanto region. If the water demand remained constant, there

that are not easily obtained, mainly in low population density area [11].

assess groundwater storage is a very important issue [6].

56 Aquifers - Matrix and Fluids

#### 2.1. Water balance concept

In a catchment, a hydrological system can be represented with the mass conservation equation:

$$\frac{dS}{dt} = P - E - Q \tag{1}$$

where S [LT�<sup>1</sup> ] is the unit area water stored in the catchment; P [LT�<sup>1</sup> ] is the rate of precipitation; E [LT�<sup>1</sup> ] is the evapotranspiration rate; and Q [LT�<sup>1</sup> ] is the unit area discharge.

Assuming that discharge is only related to storage, f(S) can be expressed as the storagedischarge relationship

$$Q = f(S) \tag{2}$$

Assuming that Q is an increasing single-valued function of S since storage-discharge function is invertible

$$S = f^{-1}(Q) \tag{3}$$

The discharge change rate through time is yielded by combining Eqs. (1) and (2)

$$\frac{dQ}{dt} = \frac{dQ}{dS} \left. \frac{dS}{dt} = \frac{dQ}{dS} \left( \left. P - E - Q \right. \right) \right| \tag{4}$$

Q is assumed to be an increasing single-valued function of S. Additionally, S cannot be directly measured, and thus dQ dS can be defined as a function of Q

$$\frac{d\mathbf{Q}}{dS} = f'(\mathbf{S}) = f'\left(f^{-1}(\mathbf{Q})\right) = \mathbf{g}(\mathbf{Q})\tag{5}$$

g(Q) refers to the streamflow sensitivity function because it expresses the sensitivity of discharge to changes in storage [16]. Because S cannot be directly measured in the catchment, combining Eqs. (4) and (5) to yield Eq. (6) solves the measurement problem of S, where discharge only has to be expressed as changes in storage

$$g(Q) = \frac{dQ}{dS} = \frac{\frac{dQ}{dt}}{\frac{dS}{dt}} = \frac{\frac{dQ}{dt}}{P - E - Q} \tag{6}$$

Precipitation, evapotranspiration, and artificial input events are excluded in the low flow conditions when groundwater storage is only related to the water flow in the river. Furthermore, mass

The Discharge-Storage Relationship and the Long-Term Storage Changes of Southern Taiwan

conservation law of hydrologic system can display as the following integral function:

relationship between groundwater storage and drainage from the catchment

from the trend of discharge in the river as the following function:

where QL7 is the annual lowest 7-day daily mean flows [LT�<sup>1</sup>

points after major events should be eliminated.

replace daily flow to reduce uncertainty

follows [21]:

eliminated.

be eliminated.

known base flow equation

2.2.2. The Vogel and Kroll (1992) model

<sup>S</sup> ¼ � <sup>ð</sup><sup>∞</sup> t

where S and Q are defined as Eq. (1). Substitution of Eq. (9) in Eq. (11) derives following

According to Eq. (12), the temporal trend of catchment groundwater storage can be obtained

Because daily flow varies, in this study, the annual lowest 7-day daily mean flow is used to

dt <sup>¼</sup> <sup>K</sup> dQL<sup>7</sup>

The long-term temporal trend of catchment groundwater storage is significant because it

Obtaining the trend in the catchment groundwater storage requires the characteristics of time scale K. In this study, a lower envelope was adopted, which is defined as roughly 5% data points below the logarithmic plot of change of flow rate data against the flow rate. In order to delete the effectiveness of precipitation, the criteria for the selection of streamflow data are as

a. Changes of flow rate are zero and positive, and sudden anomalous data should be

b. Three data points after the changes in the flow rate are zero and positive, and four data

c. Two data points before the changes in the flow rate are zero and positive should be eliminated. d. When there is suddenly a higher value of change in the flow rate in a dry period, it should

Vogel and Kroll [14] assumed discharge to be linearly related to storage. Eq. (15) is a well-

dS dt <sup>¼</sup> <sup>K</sup> dQ

dS

reveals whether the catchment will face a water shortage problem in the future.

Q dt (11)

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59

S ¼ KQ (12)

dt (13)

dt (14)

].

In a catchment, when the precipitation and evapotranspiration are much smaller than discharge, the relationship between discharge and groundwater storage can be most accurately expressed

$$\log(Q) = \frac{dQ}{dS} \approx \frac{-\frac{dQ}{dt}}{Q}\Big|\_{P \ll Q, E \ll Q} \tag{7}$$

Finally, the relationship between discharge and storage is derived from Eq. (7)

$$\int dS = \int \frac{dQ}{g(Q)}\tag{8}$$

#### 2.2. Low flow analysis

In the water balance conceptual method, Eq. (8) represents the relationship between discharge and storage. Brutsaert [21], Vogel and Kroll [14], and Kirchner [16] used low flow analysis models to select recession curves and the same method has been used in this study to characterize southern Taiwan's hydrological behaviors in order to quantify the lowest groundwater storage and to assess southern Taiwan whether facing a groundwater shortage problem in the future.

#### 2.2.1. The Brutsaert (2008) model

Low flow is a period without precipitation. In these periods, the hydrologic water flow system in the river is only related to the groundwater drainage from the aquifer in the catchment. As exponential decay type function is more commonly used to describe low flows

$$
\dot{Q} = \dot{Q}\_0 \, e^{-\frac{\ell}{\hbar}} \tag{9}
$$

where Q\_ is defined as volume of flow rate in the river [L<sup>3</sup> T�<sup>1</sup> ]; Q\_ <sup>0</sup> is the value of flow rate at arbitrarily chosen time such as t = 0 [L3 T�<sup>1</sup> ]; and K, which is defined as the characteristic of time scale in the catchment, is also referred as the storage coefficient [T].

A power law function is used to describe the relation between the change in the flow rate and the discharge characterized hydrologic behavior in this research (e.g., Brutsaert and Nieber [9])

$$\frac{d\dot{\mathcal{Q}}}{dt} = -a\,\dot{\mathcal{Q}}^b\tag{10}$$

in which a and b are constants [�]. These constants can be characterized to display catchment hydrologic behavior [22].

Precipitation, evapotranspiration, and artificial input events are excluded in the low flow conditions when groundwater storage is only related to the water flow in the river. Furthermore, mass conservation law of hydrologic system can display as the following integral function:

$$S = -\int\_{t}^{\infty} Q \, dt \tag{11}$$

where S and Q are defined as Eq. (1). Substitution of Eq. (9) in Eq. (11) derives following relationship between groundwater storage and drainage from the catchment

$$S = KQ \tag{12}$$

According to Eq. (12), the temporal trend of catchment groundwater storage can be obtained from the trend of discharge in the river as the following function:

$$\frac{dS}{dt} = K \frac{dQ}{dt} \tag{13}$$

Because daily flow varies, in this study, the annual lowest 7-day daily mean flow is used to replace daily flow to reduce uncertainty

$$\frac{dS}{dt} = K \frac{dQ\_{L7}}{dt} \tag{14}$$

where QL7 is the annual lowest 7-day daily mean flows [LT�<sup>1</sup> ].

The long-term temporal trend of catchment groundwater storage is significant because it reveals whether the catchment will face a water shortage problem in the future.

Obtaining the trend in the catchment groundwater storage requires the characteristics of time scale K. In this study, a lower envelope was adopted, which is defined as roughly 5% data points below the logarithmic plot of change of flow rate data against the flow rate. In order to delete the effectiveness of precipitation, the criteria for the selection of streamflow data are as follows [21]:


#### 2.2.2. The Vogel and Kroll (1992) model

g(Q) refers to the streamflow sensitivity function because it expresses the sensitivity of discharge to changes in storage [16]. Because S cannot be directly measured in the catchment, combining Eqs. (4) and (5) to yield Eq. (6) solves the measurement problem of S, where

> dQ dt dS dt ¼

In a catchment, when the precipitation and evapotranspiration are much smaller than discharge, the relationship between discharge and groundwater storage can be most accurately expressed

> dS <sup>≈</sup> � dQ dt

> > ð dQ

In the water balance conceptual method, Eq. (8) represents the relationship between discharge and storage. Brutsaert [21], Vogel and Kroll [14], and Kirchner [16] used low flow analysis models to select recession curves and the same method has been used in this study to characterize southern Taiwan's hydrological behaviors in order to quantify the lowest groundwater storage and to assess southern Taiwan whether facing a groundwater shortage problem in the future.

Low flow is a period without precipitation. In these periods, the hydrologic water flow system in the river is only related to the groundwater drainage from the aquifer in the catchment. As

<sup>Q</sup>\_ <sup>¼</sup> <sup>Q</sup>\_ <sup>0</sup> <sup>e</sup>

A power law function is used to describe the relation between the change in the flow rate and the discharge characterized hydrologic behavior in this research (e.g., Brutsaert and Nieber [9])

in which a and b are constants [�]. These constants can be characterized to display catchment

dQ\_

�t

exponential decay type function is more commonly used to describe low flows

where Q\_ is defined as volume of flow rate in the river [L<sup>3</sup> T�<sup>1</sup>

time scale in the catchment, is also referred as the storage coefficient [T].

arbitrarily chosen time such as t = 0 [L3 T�<sup>1</sup>

dQ dt

<sup>P</sup> � <sup>E</sup> � <sup>Q</sup> (6)

<sup>Q</sup> <sup>j</sup> <sup>P</sup> <sup>≪</sup> Q,E <sup>≪</sup> <sup>Q</sup> (7)

g Qð Þ (8)

<sup>K</sup> (9)

]; and K, which is defined as the characteristic of

dt ¼ �<sup>a</sup> <sup>Q</sup>\_ <sup>b</sup> (10)

]; Q\_ <sup>0</sup> is the value of flow rate at

discharge only has to be expressed as changes in storage

2.2. Low flow analysis

58 Aquifers - Matrix and Fluids

2.2.1. The Brutsaert (2008) model

hydrologic behavior [22].

g Qð Þ¼ dQ

g Qð Þ¼ dQ

Finally, the relationship between discharge and storage is derived from Eq. (7)

ð dS ¼

dS ¼

Vogel and Kroll [14] assumed discharge to be linearly related to storage. Eq. (15) is a wellknown base flow equation

$$
\dot{Q} = \dot{Q}\_0 K\_b^t \tag{15}
$$

where {X1, X2, X3,…, Xn} is streamflow data, which are 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:

The Discharge-Storage Relationship and the Long-Term Storage Changes of Southern Taiwan

<sup>σ</sup><sup>2</sup> <sup>¼</sup> n nð Þ � <sup>1</sup> ð Þ <sup>2</sup><sup>n</sup> <sup>þ</sup> <sup>5</sup>

In this study, long-term streamflow data are likely to be repeated in the data series; thus,

S � 1 σ

S þ 1 σ

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 significant. Assuming, a significance level of α, if |Z|≧Zα, the null hypothesis is rejected, which represents that the series has a significant trend; whereas, 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

The Theil-Sen slope [25] 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

> Xj � Xi j � i

X tðÞ¼ X<sup>1</sup> � Xn , t ¼ 1 � n

� �, for all i <sup>&</sup>lt; <sup>j</sup> (22)

values. Thus, it is often used with the MK test. Slope β is defined as follows:

X tðÞ¼ β t þ C

β ¼ Median

u¼1

, S > 0

, S < 0

0 , S ¼ 0

" #

<sup>18</sup> n nð Þ � <sup>1</sup> ð Þ� <sup>2</sup><sup>n</sup> <sup>þ</sup> <sup>5</sup> <sup>X</sup>

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

8 >>>>><

>>>>>:

Z ¼

Kendall modified the approximated solution Eq. (19) to Eq. (20)

<sup>σ</sup><sup>2</sup> <sup>¼</sup> <sup>1</sup>

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

trend. When it is below this level, there is no significant trend.

2.4. Slope test

2.4.1. Theil-Sen slope

<sup>18</sup> (19)

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(20)

61

(21)

(23)

u uð Þ � 1 ð Þ 2u þ 5

where Kb is base flow recession constant [�]. Eq. (16) is derived by Eq. (9) by comparing the Brutsaert (2008) model

$$K = -\frac{1}{\ln \mathcal{K}\_b} \tag{16}$$

Vogel and Kroll's model used a linear regression to define the characteristics of time scale K to represent average basin hydrological behavior. The Vogel and Kroll model has criteria for selection of the streamflow data as follows:


Only accept Q\_ <sup>t</sup> ≥ 0:7Q\_ <sup>t</sup>�<sup>1</sup>:

#### 2.2.3. The Kirchner (2009) model

Kirchner [16] suggested that when evapotranspiration, precipitation, aquifer recharge, and leakage are much smaller than discharge, they can be neglected. At this time, discharge is directly related to changes in storage as shown in Eq. (17) below:

$$\frac{d\mathcal{S}}{dt} = -\dot{\mathcal{Q}}\tag{17}$$

The power law equation proposed by Brutsaert and Nieber [9] is used to represent hydrological low flow analysis. Kirchner's model for streamflow criteria is simpler than the other available alternatives. It only requires the streamflow recession segment.

#### 2.3. Trend test

#### 2.3.1. Mann-Kendall test

Mann-Kendall (MK) [23, 24] test is 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 are that the 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 hypothesis given that if there is not significant trend in the series, test statistic S is defined as:

$$\text{Sign}\left(\mathbf{X}\_{j} - \mathbf{X}\_{i}\right) = \begin{cases} +1 & \text{ $\mathbf{X}\_{j} - \mathbf{X}\_{i} > 0$ } \\ 0 & \text{ $\mathbf{X}\_{j} - \mathbf{X}\_{i} = 0$ } \\ -1 & \text{ $\mathbf{X}\_{j} - \mathbf{X}\_{i} < 0$ } \end{cases}, \quad \mathbf{S} = \sum\_{i=1}^{n-1} \sum\_{j=i+1}^{n} \text{Sign}(\mathbf{X}\_{j} - \mathbf{X}\_{i}) \tag{18}$$

where {X1, X2, X3,…, Xn} is streamflow data, which are 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:

$$
\sigma^2 = \frac{n(n-1)(2n+5)}{18} \tag{19}
$$

In this study, long-term streamflow data are likely to be repeated in the data series; thus, Kendall modified the approximated solution Eq. (19) to Eq. (20)

$$
\sigma^2 = \frac{1}{18} \left[ n(n-1)(2n+5) - \sum\_{u=1}^{} u(u-1)(2u+5) \right] \tag{20}
$$

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

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

$$Z = \begin{cases} \frac{S - 1}{\sigma} & , S > 0 \\ 0 & , S = 0 \\ \frac{S + 1}{\sigma} & , S < 0 \end{cases} \tag{21}$$

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 significant. Assuming, a significance level of α, if |Z|≧Zα, the null hypothesis is rejected, which represents that the series has a significant trend; whereas, 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.

#### 2.4. Slope test

<sup>Q</sup>\_ <sup>¼</sup> <sup>Q</sup>\_

Brutsaert (2008) model

60 Aquifers - Matrix and Fluids

Only accept Q\_

2.3. Trend test

2.3.1. Mann-Kendall test

Sign Xj � Xi � � <sup>¼</sup>

selection of the streamflow data as follows:

<sup>t</sup> ≥ 0:7Q\_

2.2.3. The Kirchner (2009) model

moving average beginning to increase. b. Delete 30% of the front total recession length.

<sup>t</sup>�<sup>1</sup>:

directly related to changes in storage as shown in Eq. (17) below:

available alternatives. It only requires the streamflow recession segment.

where Kb is base flow recession constant [�]. Eq. (16) is derived by Eq. (9) by comparing the

<sup>K</sup> ¼ � <sup>1</sup> lnKb

Vogel and Kroll's model used a linear regression to define the characteristics of time scale K to represent average basin hydrological behavior. The Vogel and Kroll model has criteria for

a. A recession range is between a 3-day moving average beginning to decrease and a 3-day

Kirchner [16] suggested that when evapotranspiration, precipitation, aquifer recharge, and leakage are much smaller than discharge, they can be neglected. At this time, discharge is

The power law equation proposed by Brutsaert and Nieber [9] is used to represent hydrological low flow analysis. Kirchner's model for streamflow criteria is simpler than the other

Mann-Kendall (MK) [23, 24] test is 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 are that the 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-

, S <sup>¼</sup> <sup>X</sup><sup>n</sup>�<sup>1</sup>

i¼1

Xn j¼iþ1

Sign Xj � Xi � �

(18)

esis given that if there is not significant trend in the series, test statistic S is defined as:

þ1 , Xj � Xi > 0 0 , Xj � Xi ¼ 0 �1 , Xj � Xi < 0

8 ><

>:

dS

0Kt

<sup>b</sup> (15)

dt ¼ �Q\_ (17)

(16)

#### 2.4.1. Theil-Sen slope

The Theil-Sen slope [25] 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:

$$\beta = \text{Median}\left(\frac{\mathbf{X}\_j - \mathbf{X}\_i}{j - \mathbf{i}}\right), \text{ for all } \mathbf{i} < j \tag{22}$$

$$\begin{aligned} X(t) &= \beta \; t + \mathcal{C} \\ X(t) &= X\_1 \sim X\_n \; \; t = 1 \sim n \end{aligned} \tag{23}$$

#### 2.5. Change point test

#### 2.5.1. Mann-Whitney-Pettit test

The Mann-Whitney-Pettit (MWP) test [26] 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) 6¼ F2(X). The definition of Ut,n is as shown in Eq. (24). If there is not a change point in the data series, j j Ut,n 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, j j Ut,n 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

$$\text{Sign}\left(\mathbf{X}\_{i} - \mathbf{X}\_{j}\right) = \begin{cases} +1 & , \quad \mathbf{X}\_{i} - \mathbf{X}\_{j} > \mathbf{0} \\ \mathbf{0} & , \quad \mathbf{X}\_{i} - \mathbf{X}\_{j} = \mathbf{0} \\ -1 & , \quad \mathbf{X}\_{i} - \mathbf{X}\_{j} < \mathbf{0} \end{cases}, \qquad \mathbf{U}\_{t,n} = \sum\_{i=1}^{t} \sum\_{j=t+1}^{n} \text{Sign}\left(\mathbf{X}\_{i} - \mathbf{X}\_{j}\right) \tag{24}$$

$$K\_n = \text{Max} \quad |\mathcal{U}\_{t,n}| \quad , \quad 1 \le t < n \tag{25}$$

standards are used for the selection of the streamflow station. First, a streamflow station must provide long-term continuous measurement data; thus, selected stations have at least 40 years of daily flow data. Second, streamflow stations are unaffected by artificial structures such as dams. After screening, Taiwan's southern basin was determined to have six stations complying with the above two criteria. These stations are listed in Table 1 and are located in Figure 1.

The Discharge-Storage Relationship and the Long-Term Storage Changes of Southern Taiwan

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63

Figure 1. The location of streamflow gaging station.

order

Table 1. The characteristics of the study basins of streamflow gaging station [8].

Average elevation (m)

Bazhang River Changpan Bridge 4 309 22,547 0.31 0.05 98.34 Yanshui River Xinshi 4 49 29,402 0.14 0.06 143.4 Erren River Chungde Bridge 4 89 38,553 0.36 0.25 138.3 Gaoping River Laonong 5 1833 99,211 0.70 0.08 803.0

Linbian River Xinbei 5 692 36,815 0.49 0.10 311.4

Length of main stream (m)

Chaozhou 5 233 28,848 0.25 0.06 178.5

Average slope (m/m) Slope of main stream (m/m)

Area (km<sup>2</sup> )

Basin Station Stream

Donggang River

To confirm that change points exist, Eq. (25) is used to calculate the extreme value of j j Ut,n that is turning point as Kn. Eq. (26) is used to calculate the probability of a change point. In this study, P = 0.95 is set as the confidence level, where P > 0.95 judges that the time is a significant changing point

$$P = 1 - \exp\left(\frac{-6K\_n}{n^2 + n^3}\right) \tag{26}$$

However, in some data series, a change point may not exist by itself; thus, Eq. (27) is used to calculate each year's P(t) value. The P(t) value is identified when it is greater than the confidence level

$$P(t) = 1 - \exp\left(\frac{-6|U\_{t,n}|^2}{n^2 + n^3}\right) \tag{27}$$

#### 3. Study area

The study area is set in southern Taiwan, including Chiayi, Tainan, Kaohsiung, and Pingtung area. The main basin includes the Bazhang River, the Jishuei River, the Cengwen River, the Yanshui River, the Erren River, the Gaoping River, the Donggang River, and the Linbian River (see Figure 1). The annual average temperature is between 22 and 26�C, and the annual average precipitation is about 2000 mm, with distinct wet and dry seasons. The precipitation is concentrated in the May to October high flow periods, and on the contrary, November to April is the low flow period, which causes the rivers to dry up in southern Taiwan [8]. Two standards are used for the selection of the streamflow station. First, a streamflow station must provide long-term continuous measurement data; thus, selected stations have at least 40 years of daily flow data. Second, streamflow stations are unaffected by artificial structures such as dams. After screening, Taiwan's southern basin was determined to have six stations complying with the above two criteria. These stations are listed in Table 1 and are located in Figure 1.

Figure 1. The location of streamflow gaging station.

2.5. Change point test

62 Aquifers - Matrix and Fluids

Sign Xi � Xj � � <sup>¼</sup>

changing point

dence level

3. Study area

2.5.1. Mann-Whitney-Pettit test

The Mann-Whitney-Pettit (MWP) test [26] 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) 6¼ F2(X). The definition of Ut,n is as shown in Eq. (24). If there is not a change point in the data series, j j Ut,n 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, j j Ut,n 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

To confirm that change points exist, Eq. (25) is used to calculate the extreme value of j j Ut,n that is turning point as Kn. Eq. (26) is used to calculate the probability of a change point. In this study, P = 0.95 is set as the confidence level, where P > 0.95 judges that the time is a significant

<sup>P</sup> <sup>¼</sup> <sup>1</sup> � exp �6Kn

However, in some data series, a change point may not exist by itself; thus, Eq. (27) is used to calculate each year's P(t) value. The P(t) value is identified when it is greater than the confi-

P tðÞ¼ <sup>1</sup> � exp �6j j Ut,n <sup>2</sup>

The study area is set in southern Taiwan, including Chiayi, Tainan, Kaohsiung, and Pingtung area. The main basin includes the Bazhang River, the Jishuei River, the Cengwen River, the Yanshui River, the Erren River, the Gaoping River, the Donggang River, and the Linbian River (see Figure 1). The annual average temperature is between 22 and 26�C, and the annual average precipitation is about 2000 mm, with distinct wet and dry seasons. The precipitation is concentrated in the May to October high flow periods, and on the contrary, November to April is the low flow period, which causes the rivers to dry up in southern Taiwan [8]. Two

, Ut,n <sup>¼</sup> <sup>X</sup><sup>t</sup>

2

n<sup>2</sup> þ n<sup>3</sup> � �

n<sup>2</sup> þ n<sup>3</sup> !

i¼1

Kn ¼ Max Uj j t,n , 1 ≤ t < n (25)

Xn j¼tþ1

Sign Xi � Xj � �

(24)

(26)

(27)

behalf of this data series, and there may be more than one change point

þ1 , Xi � Xj > 0 0 , Xi � Xj ¼ 0 �1 , Xi � Xj < 0

8 ><

>:


Table 1. The characteristics of the study basins of streamflow gaging station [8].
