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Tidal streams are complex watercourses that represent a transitional zone between riverine and marine systems; they occur where fresh and marine waters converge. Because tidal circulation processes cause substantial turbulence in these highly dynamic zones, tidal streams are the most productive of water bodies. Their rich biological diversity, combined with the convenience of land and water transports, provide sites for concentrated populations that evolve into large cities. Domestic wastewater is generally discharged directly into tidal streams in Taiwan, necessitating regular evaluation of the water quality of these streams. Given the complex flow dynamics of tidal streams, only a few models can effectively evaluate and identify pollution levels. This study evaluates the river pollution index (RPI) in tidal streams by using kriging analysis. This is a geostatistical method for interpolating random spatial variation to estimate linear grid points in two or three dimensions. A krigingbased method is developed to evaluate RPI in tidal streams, which is typically considered as 1D in hydraulic engineering. The proposed method efficiently evaluates RPI in tidal streams with the minimum amount of water quality data. Data of the Tanshui River downstream reach available from an estuarine area validate the accuracy and reliability of the proposed method. Results of this study demonstrate that this simple yet reliable method can effectively estimate RPI in tidal streams.
The complex flow of tidal streams is mainly influenced by interactions between river water and seawater. Thus, tidal streams are in constant flux as they adapt to river and climate conditions. The halfday tidal variation of the sea is the main driver of cyclic fluctuation in tidal streams [
The application of geostatistical methods or those combined with other models to water quality monitoring and estimation has been discussed extensively during the past two decades. Lo
Tidal stream water quality is difficult to simulate with the water quality model because of the many effects of runoff convergence from rainfall upstream and tide recession downstream; thus, a geostatistical method was used in this study. Four variables of water quality were obtained simultaneously from sampling stations along the tidal streams. The water quality for each station was then estimated by kriging analysis to evaluate the pollution of tidal streams. In particular, our proposed algorithm is based on 1D kriging and provides a simple and efficient solution for the complexities of boundary conditions encountered in traditional 2D hydrological models.
The geostatistical method adopted in this study for estimating RPI is based on the sampling data obtained from the Tanshui River. The use of an RPI is characterized by the fact that, although the spatial distribution of drainage area is 2D, pollution in the mainstream generally remains in a 1D variable flow transmitted from upstream to downstream. Thus, spatial estimation based on a 2D random variable domain is impossible. In this study, RPI calculations were performed at various separate sampling points in the mainstream of the Tahan River upstream from the converging point between the Hsintien and Keelung rivers to establish the testing semivariogram by geostatistics on an hourly basis. On the basis of the optimal theoretical semivariogram model, the RPI values were then estimated hourly for other areas without measuring points along the three rivers. Since the Tanshui River is a tidal stream, pollutants are likely transmitted back upstream at high tide. The overlapping estimates of RPI per river kilometer among the three rivers were averaged; data from the separate stations were estimated directly without overlapping.
Geostatistics, a scientific method that analyzes spatial structures, is based on parameters of natural phenomena in the structural characteristics of spatial distribution. In this method, regionalized variables are established in various locations, the estimation of which is based on variograms. Various research fields that apply the theory of regionalized variables include meteorology, soil physics, groundwater, mining and metallurgy, environmental monitoring, and hydrology [
Matheron [
where
where
To satisfy the optimal condition,
Equation (4) can be solved by the method of Lagrange multipliers, subsequently yielding:
where
The kriging variance (
Based on the hypothesis of secondorder stationarity, the development of kriging assumes that the mean and variogram are known. Therefore:
Variance of the increments has a finite value 2
the spherical model:
the exponential model:
and the Gaussian model:
The exponential mode is a conventionally used covariance function for modeling discontinuity at the origin of the variogram. In addition to the four basic theoretical models described above, a nested structure consisting of these models can be used to correlate with the realistic variance of a random field. Additional details of the kriging theory were reported by Journel and Huijbregts [
A diagram of the theoretical semivariogram.
A diagram of the semivariogram r(h) and covariance function C(h).
The theoretical semivariogram.
The conventionally adopted classification system in Taiwan for monitoring water quality is an RPI [
Each variable of water quality used to determine RPI is converted to one of four index scores (
where
Definition of river pollution index (RPI).
Items  Ranks  

Unpolluted  Negligibly polluted  Moderately polluted  Severely polluted  
DO (mg/L)  Above 6.5  4.6–6.5  2.0–4.5  Under 2.0 
BOD_{5} (mg/L)  Under 3.0  3.0–4.9  5.0–15  Above 15 
SS (mg/L)  Under 20  20–49  50–100  Above 100 
NH_{3}N (mg/L)  Under 0.5  0.5–0.99  1.0–3.0  Above 3.0 
Index Scores ( 
1  3  6  10 
RPI  Under 2  2.0–3.0  3.1–6.0  Above 6.0 
A length of 159 km and drainage area of 2,726 km^{2} makes the Tanshui River the third largest river in Taiwan. The Tahan, Hsintien, and Keelung rivers constitute the three main tributaries converging around Taipei from south to north. The Tahan River originates from Pintian Mountain at 3,529 m above sea level and flows through Hsinchu, Taoyuan and Taipei via the Shihmen Reservoir with a drainage area of 1,200 km^{2}. The Tanshui River begins at Jiangzicui and converges with the Tahan and Hsintien rivers; its mainstream converges with the Keelung River at Guandu, flowing through Danshui into the Taiwan Strait. The drainage area of the Hsintien River is approximately 900 km^{2}, and its upstream consists of the Beishih and Nanshih rivers. Two springs of the Beishih River originate near Kanchenkang at an elevation of approximately 620 m; the other begins near Ping Qi at an elevation of approximately 700 m. Two sources converge at the crossing and disembogue into Feitsui Reservoir. Nanshih River originates at the northern Chilan Mountain and flows northbound at an elevation of approximately 2,130 m.
The Beishih and Nanshih rivers converge near Hsintien, then discharge into the Tanshui River at Jiangzicui. The Keelung River originates at Jingtong Mountain with gorges above Badu and flows downward into a plain to converge with the Tanshui River at Guandu, which has a drainage area of 600 km^{2}.
This study analyzed water quality data from nine sampling stations along the drainage area of the Tanshui River at the Shain and Shinhai bridges along Tahan River; the Zonan and Chung Cheng bridges along the Hsintien River; the Jansho, Nanhu, and Banlin bridges along the Keelung River; and the Taipei and Guandu bridges along the Tanshui River (
Map of study area.
1D ordinary kriging analysis was performed along the river to estimate RPIs of the Tanshui River during a 13 h period. First, this study divided the drainage area of the Tanshui River into three sections. The first section included four sampling stations along the Keelung and Tanshui rivers: the Jansho, Nanhu, Banlin and Guandu bridges from upstream to downstream. The second section included four sampling stations along the Hsintien and Tanshui rivers: the Zonan, Chung Cheng, Taipei, and Guandu bridges from upstream to downstream. The third section included four sampling stations along the Tahan and Tanshui rivers: the Shain, Shinhai, Taipei, and Guandu bridges from upstream to downstream. The 1D distance between two sampling stations equaled the distance from an estuary along the river direction.
Distances in river kilometers of sampling stations in the catchment of the Tanshui River.
The 1 D ordinary kriging analysis was performed along the river to estimate RPI. The hourly testing semivariogram for the three rivers in the Tanshui River drainage area were calculated, and the data of the testing semivariograms were mixed. The results obtained were then applied to the theoretical semivariogram models, including power, sphere, index, and Gaussian models. Finally, RPIs of the Tanshui River were estimated by using the theoretical semivariogram model. Individual semivariograms were established for three estuaries.
Results of water quality sampled from nine sites and their computed RPI values.
Water quality  DO (mg/L)  BOD_{5} (mg/L)  NH_{3}N (mg/L)  SS (mg/L)  RPI  

Station  mean ± std  Min.  Max.  mean ± std  Min.  Max.  mean ± std  Min.  Max.  mean ± std  Min.  Max.  mean ± std  Min.  Max. 
Shain Bridge  4.45 ± 0.84  2.8  6.1  2.21 ± 0.33  1.7  2.6  0.02 ± 0.01  0.01  0.04  13.76 ± 3.91  10.1  23.1  1.98 ± 0.43  1.50  2.75 
Shinhai Bridge  1.22 ± 0.77  0.1  2.8  8.45 ± 2.77  4.3  12.7  5.72 ± 0.94  4.37  7.40  32.08 ± 6.55  23.2  44.2  7.04 ± 0.54  5.50  7.25 
Zonan Bridge  3.39 ± 0.43  2.7  4.0  1.89 ± 0.58  1.3  2.8  0.53 ± 0.37  0.13  1.25  17.39 ± 5.00  7.0  24.5  2.71 ± 0.42  2.25  3.50 
Chung Cheng Bridge  4.07 ± 0.75  2.9  5.1  2.48 ± 0.61  1.5  3.7  1.58 ± 0.44  0.82  2.41  23.88 ± 10.32  13.9  54.0  3.67 ± 0.59  2.75  4.50 
Jansho Bridge  3.99 ± 0.46  3.1  4.8  1.20 ± 0.15  1.0  1.4  0.01 ± 0.01  0.01  0.03  13.95 ± 9.48  3.4  28.9  2.38 ± 0.26  2.00  2.75 
Nanhu Bridge  3.32 ± 0.57  2.4  4.2  3.11 ± 0.81  1.8  4.5  0.70 ± 0.22  0.37  0.96  38.94 ± 26.07  15.8  95.8  3.52 ± 0.81  2.25  4.50 
Banlin Bridge  1.76 ± 0.22  1.4  2.1  2.98 ± 0.59  2.2  4.1  1.98 ± 0.14  1.64  2.16  13.45 ± 2.88  10.5  18.9  4.50 ± 0.46  3.50  5.00 
Taipei Bridge  1.78 ± 0.38  1.4  2.6  2.98 ± 0.96  1.6  4.4  3.73 ± 0.71  2.55  5.33  30.51 ± 14.75  14.2  61.5  5.88 ± 1.12  3.50  7.25 
Guandu Bridge  2.27 ± 0.34  1.5  2.7  1.96 ± 1.49  0.01  6.1  1.66 ± 0.44  0.53  2.21  20.81 ± 6.09  12.5  33.2  3.96 ± 0.67  2.75  5.25 
The RPI obtained at 5 a.m. on 29 September 2010, from the Tanshui River was chosen as the standard value. In this study, subsequent testing semivariograms were applied to the theoretical semivariogram models.
Parameters of the four fitted theoretical semivariograms.
Parameter  Power  Exponential  Gaussian  Spherical 


−0.005  −135.409  0.001  −0.006 

2.318  136.996  2.312  2.320 

0.417  0.002  1.000  0.480 
Least Error Sum of Squares (RSS)  3.968  4.558  3.968  3.968 
Coefficient of Determination (R^{2})  0.5289  0.4589  0.5289  0.5289 
Fitted diagram of experimental and theoretical semivariograms of data obtained at 5 a.m. on 29 September 2010.
Fitted parameters of exponential model.
Time 29 September 2010 



RSS  R^{2} 

5 a.m.  −0.005  2.528  0.420  9.949  0.3476 
6 a.m.  −0.003  2.300  0.528  6.114  0.4181 
7 a.m.  −0.006  2.773  0.424  16.512  0.2787 
8 a.m.  −0.005  2.318  0.417  3.968  0.5289 
9 a.m.  −0.007  3.549  0.437  11.234  0.4818 
10 a.m.  −0.002  2.215  0.617  6.567  0.3829 
11 a.m.  −0.004  3.355  0.631  18.835  0.3317 
noon  −0.002  2.003  0.607  5.725  0.3678 
1 p.m.  −0.015  3.201  5.662  6.497  0.5555 
2 p.m.  −0.002  2.073  0.623  11.562  0.2174 
3 p.m.  −0.002  1.906  0.692  4.571  0.3727 
4 p.m.  −0.002  1.609  0.605  4.969  0.3020 
5 p.m.  −0.002  2.221  0.609  9.146  0.3095 
This study used 1D ordinary kriging analysis to examine the RPI of the Tanshui River. During the study, four values of water quality obtained from the sampling stations were assigned to the corresponding points; the RPIs at various time intervals were calculated for each sampling station. Finally, 1D ordinary kriging analysis was performed again to estimate the RPIs along the Tanshui River. As shown in
Based on
Spatial distribution of RPIs in the Tanshui River estimated by the (
Temporal variation in RPI of the Tanshui River at approximately 23 river kilometers.
The water quality of tidal streams was calculated by using conventionally adopted hydrological and water quality models, which are timeconsuming and costly. In this study, the pollutant transfer from the upstream to downstream was first estimated by a 1D concept and later used to determine the value of pollutants in a 2D space. The spatial distribution of RPIs of the Tanshui River and its branches was simulated successfully by combining the 1D ordinary kriging method with water quality data collected in the field. This approach is simpler than simulation through conventional 2D variable hydrological models. Moreover, this approach solves the problem of determining complex initial conditions necessary for boundary building in models; instead, only the sampled data are used to represent the average water quality of the studied river section. In this method water quality along a tidal stream can be obtained efficiently. This study also analyzed the spatial distribution of RPIs obtained from various sections at given times in addition to the time distribution for each sampling station. The water quality estimation model in this study was constructed on the basis of the water quality of the tidal stream from the Tanshui River, subsequently allowing for determination of the water quality of various river sections. The results of this study demonstrate the feasibility of using the geostatistics method to estimate the complex water quality of tidal streams.
The authors would like to thank the National Science Council of Taiwan for financially supporting this research under Contract No. NSC 982625M027002.
The authors declare no conflict of interest.